[theory and applications of transport in porous media] computational methods for flow and transport...

COMPUTATIONAL METHODS FOR FLOW AND TRANSPORT IN POROUS MEDIA

Theory and Applications of Transport in Porous Media

Series Editor: Jacob Bear, Technion -Israel Institute o/Technology, Haifa, Israel

Volume 17

The titles published in this series are listed at the end of this volume.

Computational Methods for Flow and Transport in Porous Media

Edited by

1.M. Crolet University of Franche-Comte, Besan~on, France

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A c.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-5440-1 ISBN 978-94-017-1114-2 (eBook) DOI 10.1007/978-94-017-1114-2

Printed an acid-free paper

All Rights Reserved © 2000 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, inciuding photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

Table of Contents

Foreword IX

Numerical methods

The simulation of the transport of contaminants in groundwater flow: error estimates for a finite volume scheme 3

Eymard, R., Herbin, R., Hi/horst, D. and Ramarosy, N

Inertial-flow anisotropy in oblique flow through porous media 29 Firdaouss, M and Tran, P.

An adaptive method for characteristics-finite element method for solute transport equation in unsaturated porous media 39

Gabbouhy, M and Mghazli, Z.

Estimation of parameter geometry 53 Heredia, J., Medina Sierra, A. and Carrera, J.

Fast real space renormalization for two-phase porous media flow 83 Hoffmann, MR.

Solution of convection-diffusion problems with the memory terms 93 Kaeur, J.

Optimal control approach for a flow in unsaturated porous media 107 Murea, CM and Crolet, J.M

Splitting the saturation and heterogeneity for time dependent effective phase 115 permeabilities

Panjilov, M and Tchijov, A.

Fuzzy simulation of waterflooding: a new approach to handling uncertainties in multiple realizations 141

Zolotukhin, A.B.

Mass transport and heat transfer

A two-domain model for infiltration into unsaturated fine-textured soils Abdallah, A. and Masrouri, F.

163

vi

Numerical study of heat and mass transfer in a cubical porous medium heated by solar energy "Boubnov-Galerkin method" 175

Al Mers, A., Mimet, A. and Boussouis, M

Cylindrical reactor performance evaluation for a solar adsorption cooling machine 193

Aroudam, El H. and Mimet, A.

RETRASO, a parallel code to model REactive TRAnsport of SOlutes 203 Benet Llobera, I., Ayora, C. and Carrera, J.

A numerical study of the coupled evolutions of micro-geometry and transport properties of simple 3D porous media 217

Bernard, D. and Vignoles, G.

Pore-scale modelling to minimize empirical uncertainties in transport equations 231

Du Plessis, J.P.

Modelling contaminant transport and biodegradation in a saturated porous media 237

Kammouri, S.A., EI Hatri, M and Croiet, J.M

Water phase change and vapour transport in low permeability unsaturated soils with capillary effects 245

Olivella, s., Gens, A. and Carrera, J.

Behaviour of infiltration plume in porous media. Adequacy between numerical results and a simplified theory 273

Oitean, C. and Bues, MA.

A comparison of two alternatives to simulate reactive transport in groundwater 287

Saaltink, M w., Carrera, J. and Ayora, C.

Case studies

Modeling of organic liquid entrapment and surfactant enhanced recovery in heterogeneous media 303

Abriola, L.M

Application of the back-tracking method to the definition of sanitary zones of catchwork protection for drinking water supply 319

Bonnet, M and Bertone, F.

Experimental and numerical investigation of porosity variations in saline media induced by temperature gradients

Castagna, S., Olivella, s., Lloret, A. and Alonso, E.E.

CLOG: A code to address the clogging of artificial recharge systems Perez-Paricio, A., Benet, I., Saaltink, M w., Ayora, C. and Carrera, J.

Vll

327

339

Groundwater flow modelling of a landslide 353 Rius, J., Mora, J. and Ledesma, A.

Trace gas absorption by soil. Simulation study on diffusion processes of trace gases, CO, H2 and CH4 in soil 371

Yonemura, S., Yokozawa, M., Kawashima, S. and Tsuruta, H.

List of Contributors 383

FOREWORD

The first Symposium on Recent Advances in Problems of Flow and Transport in Porous Media was held in Marrakech in June '96 and has provided a focus for the utilization of computer methods for solving the many complex problems encountered in the field of solute transport in porous media. This symposium has been successful in bringing together scientists, physicists, hydrogeologists, researchers in soil and fluid mechanics and engineers involved in this multidisciplinary subject.

It is clear that the utilization of computer-based models in this domain is still rapidly expanding and that new and novel solutions are being developed. The contributed papers which form this book reflect the recent advances, in particular with respect to new methods, inverse problems, reactive transport, unsaturated media and upscaling. These have been subdivided into the following sections:

I. Numerical methods II. Mass transport and heat transfer III. Comparison with experimentation and simulation of real cases

This book contains reviewed articles of the top presentations held during the International Symposium on Computer Methods in Porous Media Engineering which took place in Giens (France) in October 1998. All of the presentations and the optimism shown during the meeting provided further evidence that computer modeling is making remarkable progress and is indeed becoming an essential toolkit in the field of porous media and solute transport. I believe that the content of this book provides evidence of this and furthermore gives a comprehensive review of the theoretical developments and applications. I thank the invited speakers, authors, delegates and session chairmen for their many stimulating presentations and lively discussions that contributed to the success of the meeting. Finally, I thank the members of the Technical Advisory Panel for their support.

October 1999

lM. Crolet

ix

Part I

Numerical methods

The simulation of the transport of contaminants in groundwater

flow: error estimates for a finite volume scheme

R. Eymard* , R. Herbin t , D.Hilhorstt , and N. Ramarosy t

Abstract. We present some error estimates for a finite volume scheme applied to a nonlinear advection-reaction-diffusion equation, where the velocity is deduced from Darcy's law together with a uniformly parabolic equation for the liquid pressure. This system describes the transport of a contaminant in groundwater flow.

Keywords: Finite volume scheme, error estimates, transport equation.

1. Introduction

We consider a model advection-reaction-diffusion equation which can for instance model the transport of contaminants in the ground. More precisely we study the nonlinear parabolic problem

(P)

(()U + PbW (u))t = div (k(x, t)V'u) - div (v(x, t)u)

ou = 0 on on

u(x, 0) = Uo (x)

-,,\ (Ou + PbW (u)) - (qs)- u + (qs)+ Us In n X (0, T)

on x (0, T)

x E n,

where u denotes the concentration of a contaminant in groundwater flow and n is a smooth bounded domain of Rd ) d 2:: 1, and a+ = max (a, 0) and a- = - min (a, 0).

We suppose that the velocity is given by Darcy's law

v = -I< (h) V'h,

where I< = I< (h) is the permeability and h satisfies the uniformly parabolic problem

• Ecole Nationale des Ponts et Chaussees, 6 et 8 Avenue Blaise Pascal - Cite Descartes - Champs-sur-Marne, 77455 MARNE-LA-VALLEE Cedex 2, France.

t CMI, Universite de Provence, 13453 MARSEILLE, France. t Analyse Numerique et EDP, CNRS et Universite de Paris-Sud (bat.425), 91405

ORSAY Cedex, France.

3 J.M. Crolet (ed.), Computational Methods for Flow and Transport in Porous Media, 3-27. © 2000 Kluwer Academic Publishers.

4 R. EYMARD, R.HERBIN, D.HILHORST and N.RAMAROSY

(PR) { :t (0 (h)) + div (v) = qs on n x (0, T) v.n = ° on an x (0, T) ,

(1)

where the smooth strictly increasing positive function 0 = 0 (h) is the water content.

We suppose that the following hypotheses hold:

(i) k is a strictly positive smooth function in n such that k ~ K, > 0; (ii) v is a smooth vector function in Rd; (iii) q is a smooth function on QT = n x [0, T] ;

(Ho) (iv) Uo is a smooth function such that 0::; uo::; M for some positive constant M which satisfies suitable compatibility conditions on an;

the constant Pb is the bulk density of the porous media;

the function w ( u) denotes the concentration of contaminants sorbed on the porous medium: the most common sorption isotherms are for s ~ 0,

(i) W (s) = K sP where K and p are positive constants, with p ~ 1 in the case of the Freundlich isotherm;

( ';';) ( ) VI S h d . t' t t H W S = were VI an V2 are POSI lve cons an s

1 + V2S in the case of the Langmuir isotherm;

k(x, t) represents the hydrodynamic dispersion coefficient;

A is the rate constant of the first-order rate reactions;

q{x, t) = qtus is the volumetric water flux representing sources and sinks;

Us = Us (x, t) is the concentration of the sources. In what follows we suppose that 0 ::; Us ::; M;

We suppose that they are such that Problem (P) has a unique solution u E C3+a , 3t" (n x [0, T)).

We discretize the partial differential equation in Problem (P) by means of a finite volume method for the space discretization and an implicit time discretization. The purpose of this paper is to present error estimates on the approximate solution.

THE SIMULATION OF THE TRANSPORT OF CONTAMINANTS 5

Problem (P) models the transport of contaminants in the ground. It describes the variation of concentration of single species miscible contaminants in groundwater flow systems where advection, dispersion and chemical reactions are involved. The finite volume method is very popular when solving problems arising in the modelling of the extraction of oil; nowadays this method is becoming increasingly popular in environmental sciences. Therefore it is essential to be able to give convergence proofs and error estimates.

In Section 2 we introduce the finite volume scheme and define the approximate Problem (Ph,At).

In Section 3 we prove the existence and uniqueness of the solution Uh,At of Problem (Ph,At) together with a discrete maximum principle; the proof is based on a contraction fixed point argument.

We present in Section 4 error estimates for approximate solutions of Problem (P) in Loo (0, T; L2 (Q)) and in a discrete space corresponding to the L2 (0, T; HI (Q)) .

Our analysis would also carry through in the case of an inhomogeneous Dirichlet boundary condition.

We refer to [RH], [FS], [LM] and [LMV] for error estimates in the elliptic case, and to a forthcoming article [EH H R] for error estimates for the approximation of boundary value problems associated to the single equation

(iJ( u))t = div (k(x, t)Vu) - div (v(x )u) - F( u) + q,

where div v 2: 0.

2. The finite volume scheme

Let Th be a mesh of Q. The elements of Th are the control volumes. For each (p, q) E T~ with p "# q, we denote by e pq = P n q their common interface, which is supposed to be included in a hyperplane of Rn, which does not intersect either p or q.

m (epq ) denotes the measure of epq , and npq the unit vector normal to epq oriented from p to q.


We denote by £ the set of pairs of adjacent control volumes, defined by £ = {(p, q) E TK, p i- q, m (epq ) i- O} .

We also use the notation N (p) = {q E Th, (p, q) E £}.

Finally, we assume that there exist positive constants a}, a2, C1 and C2 such that for all h > 0 and p E Th there holds

(i) &(p) ~ h for all p E Th;

(ii) a1hd :::; m (p) :::; a2hd for all p E Th ;

(iii) C1hd- 1 ~ m (e pq ) ~ C2hd- 1 ;

(iv) We assume furthermore that for all (p, q) E £, . Xq - xp

there eXIst Xp E p, Xq E q, such that I I = npq , Xq - xp

where & (p) denotes the diameter of the control volume p and m (p) its measure in Rd.

We denote by dpq = IX q - xpl the Euclidian distance between xp and m (epq )

Xq and we set Tpq = d . pq

The implicit finite volume scheme is then defined by the following equations:

Let 0 = to :::; t1 ~ ... ~ tN max = T denote the time step, b.ti = ti - ti-1, and b.t = max b.ti.

1SiSNmax

(i) the initial condition for the scheme is

~ = Uo (Xp) , for all p E Th;

(ii) the source term is taken into account by defining values

( ±) n 1 1 Jtn J ± qs p = 6.tn m (p) qs (x, t) dxdt, tn-l P

for all p E Th,

tn

(us); = 6.~n m ~p) J J Us (x, t) dxdt, tn-l p

for all p E Th,

(2)

(iii) the implicit finite volume scheme is defined by


where

and where we also set

0; = m ~p) J 0 (x, tn) dx, p

(3; (s) = O;S + PbW (8) ,

F; (S) = A (0; S + Pb W (s)) + (q 5 ) : s,

and where {u};q denotes the upwind approximation of u at the point (xpq, tn)

{U} n _ { U; if V;q ~ 0 pq - U; otherwise.

Integrating equation (1) on p X (tn-I, tn) , we deduce the relation

on _ on-I p !1.t p m (p) + L [v;qm (epq )] = (qS); m (p), (4)

n qEN(p)

which will be very useful in the sequel.

The scheme defined above allows to build an approximate solution Uh,L::it of Problem (P), namely

Uh,L::it (x, t) = u; for all x E p and for all t E [tn-I, tn).


3. Existence and uniqueness of the solution of the discrete scheme

We first give an alternative expression for the convection term.

Lemma 3.1

We have that

for all p E Th and n E [1, Nmaxl.

Proof

We substitute V;q = (v;q) + - (v;q) - and equality (4) into the left

hand-side of (5) to obtain the result.

The numerical scheme is given by

onun + PbW (un) - on-1un-1 - PbW (un-I) p p p p p P m() ~tn p

2: [k;qTpq(~-~)]- 2: [(V;q) {u};qm(epq )] (6) qEN(p) qEN(p)

- A (o;~ + Pb W (~ )) m (p) - (q"S): ~ m (p) + (qt): (us); m (p) .

Corollary 3.2

The numerical scheme for Problem (P) can be rewritten as:

on-1un + PbW (un) - on-1un-1 - PbW (un-I) p p p p p p () mp=

~tn


Proof

Corollary 3.2 is a direct consequence of Lemma 3.1 and equality (4) .

Next we introduce the iteration scheme

un,o = un-I p p

and

on-lun,v+1 + p W (un,V+I) - on-I un-I - p W (un-I) P P b p p p b p 6.tn m (p) =

I: [k;qTpq (u~'V - U;,V+1) ] + I: [( V;q) - (u~'V - U;,V+1) m (e pq )]

qEN(p) qEN(p)

(8)

and define the function <1>; for each p E Th and for each time step n by

(9)


which is strictly increasing for s ~ O. The equality (8) can be rewritten in the form

on-l ::-:-n-l + (::-:-n-l) = p up PbW up ( ) + '" [kn T -n,lI] !::1t m p 0 pq pqUq

n qEN(p) (10)

+ L [(V;q)-u~'Vm(epq)] + (qt);(us);m(p). qEN(p)

Lemma 3.3

Let U;,v+I be the solution of Problem (8) and suppose that 0 ::; U;,i ::; M for all i = 0, .. " v and 0 ::; (us); ::; M for all p E Thj

then 0 ::; U;,II+I ::; M for all p E Th.

Proof

We estimate the right-hand-side of (10) to obtain

on-1un- 1 + P W (un-I) o < P P b p m (p) + '" [kn T 'it",v] -!::1t 0 pq pq q

n qEN(p)

+ L [(V;q)-~'lIm(epq)] + (qt);(us);m(p)::; ~;(M)j (11) qEN(p)

therefore it follows from (10) that

Since both the functions 8 -+ ~; (s) and s -+ ). (0; 8 + PbW (8)) m (p) are strictly increasing and invertible functions, and since furthermore

~; (0) = 0,


we deduce from the first inequality in (12) that

o < un,v+l - p ,

which implies that

A (e;u~,v+1 + PbW (u~,v+1)) m (p) ~ 0,

and thus in view of (12) that

~~ (u~'V+l) :S ~~ (M).

Since ~; (x) is strictly increasing, it follows that

o < un,v+1 < M. - p -

This completes the proof of Lemma 3.3.

Next we prove the existence and uniqueness of the discrete solution.

One now rewrites the numerical scheme (8) in the form:

where

F;(s) = A (e;S+PbW(S))

for all p E Th and for n E [1, Nmax].

We define

and the operator

T:

(13)


such that w,v+1 = T W'V is defined by (13) . We also define a vector norm, namely Ilqlloo =sup Iqpl·

pETh

Using arguments quite similar to those of [EH H R] one can prove the following result.

Theorem 3.4

The operator T is a strict contraction from RP to RP.

Theorem 3.4 directly implies the following existence and uniqueness result.

Theorem 3.5

Let (U;-I) be given. There exists a unique U; for each p E Th pETh

such that {U;,V} converges to {U;} as v --+ 00. It satisfies

Finally we state a discrete maximum principle

Theorem 3.6 (Discrete maximum principle)

There exists a positive constant Cmax such that 0:::; U; :::; Cmax for all p E Th and n E [1, Nmax] •

Proof

This is a direct consequence of Lemma 3.3 and Theorem 3.5.

4. Error estimates in L OO (0, Tj L2 ([2)) and in L2 (0, Tj HI ([2))

The idea is to compare the solution of the continuous and the approximate problems. We define u; = u (xp, t n ) , where u is the unique


solution of Problem (P), and integrate the reaction-diffusion equation on a time volume element to obtain,

tn tn

!l~n j j f3(x, t, u)t dxdt - !l~n j j k(x, t)Vu.Ttdxdt tn-l p tn-lOp

tn tn

+ !l~n j j UV(X, t).Ttdxdt + !l~n j j F (x, t, u) dxdt (15)

where

and

tn-lOp tn-l P

tn

1 j jqdxdt = O. !ltn

tn-l P

13 (x, t, u) = 0 (x, t, u) u (x, t) + PbW (u (x, t)) ,

F (x, t, u) = )..13 (x, t, u) + (qS)- u (x, t).

We introduce the following consistency error terms:

tn

j j k(x, t)Vu.Ttdxdt (17)

tn -=n n {}n 1 1 j j -4 Rpq = Vpq u pq - ~ () uv.ndxdt L.l.tn m epq

tn-l epq

(18)


tn tn

R; = F;(u;)+q;- il~n m~p) J J F(x, t, u)dxdt- il1tn m~p) J J q(X, t)dxdt. tn-l p tn-l p

We also define

Then

N ext we define

L D ( u;) - L ( u; ) R~m(p)

+ I: [R~q + R;q] m(epq ) + R;m(p), qEN(p)

e; = u; - U; for all p E Th and n E [1, Nmaxl and state the main result of this paper.

Theorem 4.1

(19)

(20)

(21)

There exists a positive constant C = C (1IuIIC3+a'~(QT)) such that

1

sup (f (u (x, t) - Uh,At(X, t))2) 2" ~ C (h + ilt) (22) tE[O,T] In

and

1

(t I: [Tpq(e; - e;)2] iltn) 2" ~ C (h + ilt) (23) n=l (p,q)Et:

The proof of Theorem 4.1 is divided into several steps. We first estimate the error terms on the right-hand-side of (21) which amounts


to proving the consistency of the numerical scheme. We then prove the error estimates stated in Theorem 4.1.

Lemma 4.2

(i) IR;I ~ C(h+~t); (ii) /R;q/ ~ C (h + ~t);

(iii) lR;q/ ~ C (h + ~t) ;

(iv) /R;/ ~ C (h + ~t).

Proof

One uses formulas based on Taylor expansions to show these inequalities.

In view of the fact that

B Be Bu Bw (u) Bt (eu + PbW (u)) = u Bt + eat + Pb Bt

we rearrange the discrete derivative term in order to obtain the equivalent form

enun + P W (un) - en- I un- I - P w (un-I) pp b p p p b P

~tn

en _ ()n-I Un _ Un- I W (un) - W (un-I) = Un- I P P + en p p + p p. p ~t p ~t Pb At n n ti n

We split R; into three parts and set


where each part is the flux difference between the continuous form and the corresponding discrete quantity, i.e.

en _ en-Ill tn ae (R )n = un- 1 P P f f u dxdt

1 p P ~tn - ~tn m(p) at' tn-l P

1 tn

Un - un - 1 1 f f a (R ) n = en P P e u d dt 2 P P ~tn - ~tn m(p) at x ,

tn-l P

and

(R )n _ W (u;) - W (u;-l) __ 1 __ 1_ ftn f aw(u)d d 3 P - Pb" "( ) Pb a x t. utn utn m p t

tn-l P

Next we give an estimate for each of these differences.

By the definition of e;, we have that

tn

( ) n 1 1 f f (n-l ) ae Rl P = ~tn m(p) up - u at dxdt, tn-l P

Since for x E p and t E [tn-I, tn]

lu; - ul ~ C (1IuIIC2+a'~(QT)) (h + ~t) (24)

d f)() . . . Q d d h an f)t IS contmuous m T, one can e uce t at

For the second term (R2); , we have that


tn

1 1 J J au +--(-) (0 (x, tn) - 0 (x, t)) -a (x, t) dxdt iltn m p t tn-l P

Since 0 is continuously differentiable in QT, and since

10 (x, tn) - 0 (x, t)1 ~ ilt 1IOIIcl(QT) ,

one can show that

In order to estimate the term (R3 ); , one proceeds as in the proof of point (i) of Lemma 4.2 in [H EH R] to obtain that

Finally, we can deduce by (25) , (26) and (27) that

We refer to [EH H R] for the proofs of points (ii) and (iii). Next we present the proof of point (iv).

We have that

R; = >. (O;u; + PbW (u;)) + (qs); u; + (qt); (us); tn

- il~nm~p) J J (>'(OU+PbW(U))+qsu+qtus)dxdt. tn-l P

In order to simplify the error estimates, we split R; into four parts as well; we set


where

and

Next we successively consider each term defined above. We have that

tn

(Rl): = Il~n m~p) J J () (x, tn) (u; - u (x, t)) dxdt tn -1 p

tn

+ Il~n m~p) J J (B (x, tn) - B (x, t)) udxdt t n - 1 p

Since for x E p and t E [tn-I, tn] we can use the estimate in (24) and

to show that

1 (R1):1 ~ (h + Ilt) C (1IuIIC2+a.~ (QT) , IIBllc1 (QT)) .

Applying point (iv) of Lemma 4.2 in [EH H R] to w instead of F, we have that

Since


one can conclude using (24) that

We also have that

Finally we obtain

IR;I ~ (h + ~t) C (lluIlC3+Q'~(QT)' IIqSIICl(QT) ' Ilw I1c1 (QT) ' Ilusllcl+O.l¥-(QT)) .

We now state a technical lemma due to [MW] which will be useful for the proof of Theorem 4.1.

Lemma 4.3

Suppose that un, un-I, vn, vn - 1 are real numbers, and let the backward time difference operator be given by

un _ un- 1 6~u= ---

~tn

Suppose also that "'( : R -+ R satisfies 0 ~ "'(' ~ C < 00 and 1",("1 ~ C. Then

where

E ~ C' {(un - vn)2 + (un- 1 _ vn- 1)2 + (~tn)2} for some constant C' only depending on 16~un I and C.


Remark 4.4

Lemma 4.3 cannot be directly applied to Problem (P2) since f3 also depends on x and t. However, one can apply it to the function w to deduce that

8f( w( u) - w(v)) (un - vn) = 8f (lU [w(s) - w( v)] dS) - E (29)

where

E S; C' {(un - vn)2 + (un- 1 - vn- 1 )2 + (~tn)2}.

We also define the following sequences (}p = (();) and f3p = n=l,Nrnax

(f3;) . n=l,Nmax

Since

we have that

+(};-l 8f (up - vp) (u; - v;) .

Substituting (29) into (30) and using the relation

(a - b) a = ~ (a2 + (a - b)2 - b2) we deduce that

8r(f3p(up) - f3p(vp))(u; - v;)

= Pb8f (1:' [w(s) - w( vp)] dS) Hf (Op) (u; - v;) 2 +~0;-18f( (Up - vp)')

(31)


1 ((un - vn) - (un- 1 _ vn- 1))2 +_on-l p p p p - E

2 p ~tn

We are now in position to prove Theorem 4.1.

Proof of Theorem 4.1

Since LD (~) = 0 , we deduce from (20) and (21) that

LD (u;) - LD (~) =

f3n (un) - f3 n- 1 (un-I) f3n (un) - f3 n- 1 (un-I) p p p p () p p p p () A mp - A mp ~tn ~tn

- L [k;qTpq (€; -€;)] + L [v; {O;qm (epq )] (32) qEN(p) qEN(p)

+ F; (u;) m (p) - F; (u;!) m (p)

= R;m(p) + L [R;q + R;q] m(epq ) + R;m(p). qEN(p)

We multiply (32) by €; ~tn and sum on p E Th to obtain

L [8~ (f3p (up) - f3p (up))€;~tnm(p)]- L [k;qTpq(€; - €;)€;] ~tn pE'Th (p,q)Ee

+ L [v;q {O;q€;m(e pq )] ~tn+ L (F;(u;) - F;(u;!)) €;~tnm(p) (p,q)Ee pE'Th

(33)

= L (R;q + R;q)€;m(epq)~tn + L: (R; + R;)€;m (p) ~tn· (p,q)Ee pE'Th

In view of the symmetry of E and since k;qTpq = k;pTqp, we have the following relationships

- L [k;qTpq(€; - €;).€;] = ~ L [k;qTpq(€; - €;)2] . (34) (p,q)Ee (p,q)Ee


Also if we use the notation

£ = {(p, q) E £ such that Vpq > O}, (35)

we have that

L [v;q {e};qe;m(epq )] = L [v;q(e; - e;)e;m(epq )] . (36) (p,q)Et' (p,q)Ee

Substituting (34) and (36) into (33) gives after summation on n = 1 to N with 1::; N ::; Nmax

N

L L <Sf (;3p (up) - ;3p (up)) e;m(p)6.tn n=l pETh

N

+~ L L [k;qTpq(e; - e;)2] 6.tn n=l (p,q)Et'

N

=-L L [v;qe;(e;-e;)]m(epq)6.tn (37) n=l (p,q)Ee

N

- L L (F;(u;) - F;(tt;)) e;m(p)6.tn n=lpETh

N N + L L (R;q + R;q)e;m(epq )6.tn + L L (R; + R;)e;m(p)6.tn'

n=l (p,q)Et' n=l pETh

Since ~ = -nqt the consistency error term for the diffusion gives:

R;q = -R~p

This leads us to the relationship :

L R;qe;m(epq ) = ~ L R;q (e; - e;) m(epq ) (38) (p,q)Et' (p,q)Et'

For the convection term, we also have that

Ir;q = -R;p

so that


I: R;q~;m(epq) = ~ I: R;q (~; -~;) m(epq ). (39) (p,q)EE (p,q)EE

Substituting (38) and (39) in (37), we finally have,

N N I: I: c5~ ({3p (up) - {3p (up)) ~;m(p)~tn+~ I: I: [k;qTpq(~; - ~;)2] ~tn n=l pETh n=l (p,q)EE

N

= - I: I: [v;q~;(~; - ~;)] m(epq)~tn n=l (p,q)E[

N

- I: I: (F;(u;) - F;CV:';)) ~;m(p)~tn (40) n=l PETh

N N +~ I: I: R;q (~; - ~;) m(epq)~tn+~ I: I: R;q (~; - ~;) m(epq)~tn

n=l (p,q)EE n=l (p,q)EE

N N + I: I: R;~;m(p)~tn + I: I: R;~;m(p)~tn'

The first term on the left hand side of (40) can be bounded by using Lemma 4.3 and Remark 4.4 as well as the Young inequality for all a, b E Rand f > 0, ab :::; fa2 + lE b2 • The following equality will also be useful to the next step of the proof

t 0;-1 ((un - Vn)2 - (un- 1 - vn- 1f) n=l

N-l = 0~-1 (uN - vN) 2 _ I: c5~ (Op) (Un - Vn)2 ~tn. (41)

n=l In order to bound the right-hand-side of (40) we use the inequalities

tn

v;qm (e pq ) = ~~n J lpq v (x, t) .npqdxdt:::; Ilvll(c(QT))d m (epq ) , tn-l


where L is such that IFul :::; L, and a consequence of the hypotheses on the triangularization, namely

There exists a constant v > 0 such that m(epq ) d(xp, Xq) :::; vm (p). (42)

Next we substitute (31) , (41) into (40) and use the fact that ISf (Or I :::; 1I 01l e i (Zh) to obtain

Pb p~, [1: (( w (s) - w (u~)) dS) 1 m(p)

+~ L 0:-1 (u: - u:) 2 m(p) + ~ ~ L [k;qTpq(~; - ~;)2] D.tn pETh n=1 (p,q)E£

N v ( )2 N v (-=11)2 + L L -;;;:;; R;q m(p)D.tn + L L ~ Rpq m(p)D.tn n=1 (p,q)E£ pq n=1 (p,q)E£ pq

(43)

N (2 2 ) +G'L L (~;) + (~;-1) + (D.tn)2 m(p)D.tn n=1 pETh

+Pb p~, [1~: (( W (09) - W (~)) dS) 1 m(p)

In what follows, we write the first term on the left-hand-side of (43) as a function of ~ and also remark that the hypotheses on w implies that 0 :::; w' :::; f.1" to obtain


lv J.L o ~ u (w (s) - w (u)) ds ~ 2" (v - U) 2

Therefore (43) becomes also in view of Lemma 4.2:

We have that for any t E [0, TJ

and thus

N ( tn ) S 2 ~ ,.f. P~h (u(xp, tn ) - u(xp, i))2 m(p)dt

N ( tn ) +2 ~ ,I '~h (,,(xp , i) - "h,""(X" tn))' m(p)di

Since

lu(xp, tn) - u(xp, t) I ~ (tn - t) Il u II C2+Q '¥ (OT) ,

and since Uh,ilt is constant on (tn-I, tn) (45) becomes

~ (l p~. (~;)' m(p)dt)

S C (II"II~'+"''¥(QJ ("'t)'

( 45)

(46)


Next we substitute (46) into (44), and deduce from the Gronwall lemma that

where Q3 depends in particular on T. Finally,

Therefore we have that

Furthermore we deduce from (44) the inequality

N

l2:: 2:: [k;qTpq(~; - ~;)2] i3.tn

n=l (p,q)E£

::; Ql {~P~" (~;)' m(p)Lltn+ Q, (h Ht)' } ,

which in view of the lower bound on k and of (47) implies that

1

(f 2:: [Tpq(~; - ~;)2] i3.tn ) 2" ~ C (h + i3.t) . n=l (p,q)E£

This completes the proof of Theorem 4.1.


References

[BRZ] H. Brezis, Analyse Fonctionnelle, Theorie et Applications, Masson, Paris (1983).

[EHHR] R Eymard, R Herbin, D. Hilhorst and N. Ramarosy, A finite volume scheme for a nonlinear advection-reaction-diffusion equation: error estimates, to appear.

[EGH] R Eymard, T. Gallouet and R Herbin, Finite Volume Methods, to appear in the Handbook of Numerical Analysis, Ph. Ciarlet and J.1. Lions eds.

[FS] P.A. Forsyth and P.H. Sammon, Quadratic convergence for cell-centered Grids, Appl. Num. Math. 4, 377-394 (1988).

[LM] RD. Lazarov and I.D. Mishev, Finite volume methods for reaction-diffusion problems, in F. Benkhaldoun and R Vilsmeier eds, Finite Volumes for Complex Applications, Problems and Perspectives, Hermes, Paris (1996).

[LMV] RD. Lazarov, I.D. Mishev and P.S. Vassilevski, Finite volume methods for convection-diffusion problems, SIAM J. Numer. Anal. 33, 31-55 (1996).

[LSV] O.A. Ladyzenskaja, V.A. Solonnikov, and N.N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. of Math. Monographs 23 (1968).

[MW] T. Arbogast, M. Obeyesekere and M.F. Wheeler, Numerical methods for the simulation of flow in root-soil systems, SIAM J. Numer. Anal. 30, 1677-1702 (1993).

[RH] R Herbin, An error estimate for a finite volume scheme for a diffusion convection problem on a triangular mesh, Num. Meth. P.D.E. 11, 165-173 (1995).

INERTIAL-FLOW ANISOTROPY IN OBLIQUE FLOW THROUGH POROUS MEDIA

M. FIRDAOUSS, P. TRAN LIMSI-CNRS and UFR 923, UPMC Paris VI, B.P. 133, 91403 Orsay, France

Abstract. The extension of the Darcean momentum equation to the inertialflow is considered using the results of direct numerical simulation of flow through two-dimensional ordered porous media. Using oblique flows, the inertial-flow regime is examined for Reynolds numbers (based on the unitcell length) up to 300. The results show that the inertial-flow regime is marked at the very beginning by a low-Reynolds number subregime (0 ::; Re ::; 10), where the deviation from the Darcean-flow pressure drop is quadratic in Re (cubic in the Darcean velocity). After a relatively extended intermediate-Reynolds number subregime (10 ::; Re ::; 50), a highReynolds subregime is observed (for Re > 50) which seems to be linear in Re (quadratic in the velocity). It is shown that for ordered arrangements the Darcean-isotropic structures become inertially anisotropic, i.e., the pressure gradient and the Darcean velocity vectors are not parallel in the inertial-flow regime, even though they are in the Darcean regime. The Darcean-anisotropic structures remain anisotropic in the inertial-flow regime. For strongly inertial flows, we study the transition to unsteady periodic solutions. The values of the critical Reynolds number depend on the angle e that the incoming flow makes with the x-axis. The critical reynolds number increases with the angle of the flow. The Re dependence of the angle a that the drag force makes with the x-axis don't confirm the new results of [5] The authors claim that for high Reynolds numbers, for low angle oblique flows « 12°), the angle a reaches a maximum and the drag force becomes horizontal, and that for high angle oblique flows (> 12°, < 45°), the drag force makes a 45° angle with the x-axis. Concerning the Darcean-anisotropic structures, we present results which contradict their conclusions.

29 J.M. Crolet (ed.), Computational Methods/or Flow and Transport in Porous Media, 29-37. © 2000 Kluwer Academic Publishers.

30 M. FIRDAOUSS and P. TRAN

1. Introduction

The knowledge of the permeability tensor of porous media is important for many applications: petroleum extraction, hydrology. When the permeability is measured, the medium is generally assumed to be isotropic. The purpose of this work is to study how the anisotropy due to nonlinear effects can be added to the anisotropy already possibly present in Darcean flows, especially for oblique flows. We also study the transition to unsteadiness as a function of the Reynolds number (Re). The extension of the Darcean momentum equation to the inertial-flow is considered using the results of direct numerical simulation of flow through two-dimensional ordered porous media. Using oblique flows, the inertial-flow regime is examined for Reynolds numbers (based on the unit-cell length) up to 300.

2. Numerical method

2.1. EQUATIONS

The Navier-Stokes equations are solved simultaneously in the fluid and solid parts of the porous medium. The penalty term in solids replaces the no slip condition at the solids boundary.

av + V. VV = -Vp+ ~V2V - _1_V in n (1) at Re ReDa

V.V = 0 in n (2)

v, p, periodic in n (3)

The Darcy numbers is taken to be Da = 10-6 in solids, and Da = 1030 in fluide.

2.2. ALGORITHM

The algorithm of march in time is based on a projection method, known as a modified fractional time step method and introduced by [7]. In the first intermediate time step a velocity field V* = (u*, v*) is computed by making implicit the viscous and penalty terms, whereas the convective term and the pressure gradient are made explicit. In order to obtain a good prediction in time, a discretization which is formally precise at the second order was used for viscous and convective terms. The provisional value of the horizontal component of the velocity u* is computed by :

INERTIAL FLOW ANISOTROPY IN OBLIQUE FLOW... 31

This penalization technique has been introduced by [4]; it is more or less similar to the fictitious domain methods. Then we have to solve two problems of Helmholtz type. The velocity field u* is not divergence free. The second intermediate step consists in projecting V* onto the vector field of free divergence and zero normal trace. This can be done by defining a new scalar variable ¢ by :

Vn+l - V* = IltV¢ (5)

The solution ¢ can be computed by taking the divergence of the previous equation. Taking into account V . vn+l = 0, we have to solve:

(6)

This equation being elliptic is associated with the boundary conditions 8¢/8n = 0 onf. The final velocity is obtained by :

Vn+l = V* + IltV¢ (7)

The variable ¢ can be interpreted as a pressure. The pressure field is calculated from: pn+l = pn + 1.5 ¢. The problem one may encounter with this type of fractional step method is that the field V* is not divergence free but verify the momentum equation. The field Vn+l is divergence free but does not verify the momentum equation. The presence of the term pn in the prediction step, facilitates the convergence to the station nary solution, which may verify the equality vn = V* = V n+1 and ¢ = o.

2.3. CONVERGENCE

The spatial discretization is based on a staggered mesh where the discrete equations are based on finite volume formulation. The convective terms are treated with centred differences. The resulting heptadiagonal systems corresponding to the implicit part of the Helmoltz equations (and periodicity) are solved by an ADI factorization. For solving the Poisson equation, a multigrid method is used, it allows convergence to the solution ¢. Most of the results presented in this paper were obtained on a regular cartesian grid corresponding to 128*128 points.

3. Deviation from Darcy law

3.l. ANISOTROPY

If we assume that a porous medium is periodic, we can use homogenization theory, either in the two-scale form developed by [1] or by using the energy


technique pioneered by [2]. The flow through the porous medium is driven by a macroscopic pressure gradient. Numerical calculations can be done for one period, which may be a square or a rectangle. The anisotropic media will be defined by the distance between the axes of the grey cylinders and the center of the lattice period. In figure 1, the configuration (left) which is referred to (00) is isotropic, the configuration (right) refers to (20). Now we try to give a definition of isotropy: If the macroscopic pressure gradient and the mean velocity vector are parallel, this direction is said to be principal. The angle between Vp and (j is zero. In case of 2D flow, if there exist two principal directions which are not orthogonal, then the medium is said to be isotropic. If the medium is anisotropic, one can calculate the two orthogonal principal axes X and Y. If the medium has a central symmetry, the principal axes X, Y, are confounded with the x and y axes. If not, the principal axes can be determined from the eigenvectors associated with the eigenvalues of the permeability tensor.

o • 0 Figure 1. Array of cylinders for different arrangement

3.2. FORCHHEIMER'S ANALYSIS

In two articles on the non-linear-deviations to Darcy's law, [3] proposes three ad hoc formulae

-'\lp = G'v+f3v2 ,

where G' is the ratio of the pressure drop (measured in height of water) to the thickness of the sand bed and v is the seepage velocity (measured in meters per day). Here, we emphasize that although Forchheimer's analysis may be of some engineering value, no physical conclusion can reasonably be drawn from it by comparing the relative value of coefficients G', (3, and I, for it uses the dimensionalized quantity v.

INERTIAL FLOW ANISOTROPY IN OBLIQUE FLOW.. 33

3.3. NORMALIZATION

However, in interpreting numerical experiments that have a limited accuracy, the ambiguity concerning the definition of the Reynolds number is important. The risks of misinterpretations are increased if instead of using dimensionless quantities one uses physical data. For instance, assume that we want to test the following one-dimensional non-linear filtration law

1 a a2

--Ko.\lp = u + a-lulu + j321u12u + ... ~ v v

(9)

and in dimensionless form :

(10)

In order to avoid this ambiguity, we propose the following normalization proced ure [6]. Since we are interested in measuring the relative effects of some phenomenon on some range of Reynolds number, say 0 :::; Re :::; Re,max, we choose Re,max as a new reference by introducing the new variable

Re x=

Re,max (11)

Furthermore, since we are interested in the deviation to Darcy's law, we should consider the magnitude of (u + Re Ko.\lp)/lul. In order to compare only dimensionless numbers, we introduce the following variable y :

(-KoRe \7p) - (-KoRe \7p)o y = (-KoRe \lP)max - (-KoRe \7p)o

(12)

where the subscript "max" refers to the value measured at Re,max' With this new definition we have

~ 2 3 Y = ax + px + ')'x + ... , (13)

Of course, this rescaling makes sense only if the deviation is experimentally significant. If the experimental data are on the line y = x, it means that the deviation is linear, whereas if the data collapse on the parabola y = x2 , the deviation is quadratic and the contribution of the linear term is zero. The non-linear deviation will be zero within some range of Reynolds numbers if the data are on the line y = O.

4. Numerical results

4.1. LOW INERTIAL-FLOW

In figure 2 we plot the drag force normalized by its maximum as a function of the relative Reynolds number Rei Re,max for two values of Re,max


: 5 and 10. The results show that the inertial-flow regIme is marked at the very beginning by a low-Reynolds number subregime, where the deviation from the. Darcean-flow pressure drop is quadratic in Re for Re,max ~ 10. The left figure corresponds to Darcean-isotropic (dev = 00) structure (K = 4.308910-3). The right one corresponds to Darcean-anisotropic structure (dev = 20). If one defines the importance of the Darcean-anisotropy as a ratio of the vertical to the horizontal permeability, then we obtain Kx = 2.095310-3 , and Ky = 6.303410-3 . The ratio r = K y/ Kx = 3.0084 indicates that this structure is truely anisotropic.

dev=OO, Remax = 5, 10 dev=20, Remax = 5, 10

0.8 0.8

c: a: 1l! 0.6 Q)

$!I 0.6 a;

"0 "0

~ 0 ~

Q) 0.4 Q) 0.4 a: a: , ,

0.2 0.2

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

Figure 2. Deviation from Darcy law up to Re = 10

4.2. HIGH INERTIAL-FLOW

dev=oo, tetha= O. 5,10,15.30.45 500 r---r---.--,....---r---.--,....---T"-,.

450

0: 400 .\l! .g 350 Gl a:

..!.. 300

250

200 '----'--"""---'-----'---'---'-----'--' o 20 40 60 80 100 120 140

800

0:700 .\l! .g 600

r! ..!.. 500

dev=20. letha= 0,15.30.45

.-.-, ..•.

~O'----'---'--~'--~--'--~'--~-'

o 20 40 60 80 100 120 140

Figure 3. Drag force versus Reynolds number up to Re = 150

After a relatively extended intermediate-Reynolds number subregime, a high-Reynolds subregime is observed (for Reynolds numbers larger than 50) which seems to be linear in Re (quadratic in the velocity). In figure 3 we plot the drag force versus Re. Every line corresponds to the angle

INERTIAL FLOW ANISOTROPY IN OBLIQUE FLOW.. 35

that the flow forms with the x-axis. One can note that the curves are notably distinguishible one from another when the angle () that the flow makes with the x-axis is respectively 0°, 5°, 10°, 15°, 30°,45° on the figure 3-a (left), and 0° , 15°, 30°,45° on the figure 3-b (right). This confirms clearly the dependence of the drag force on the oblicity of the flow.

dev=OO, Remax = 50, 100,150 dev=20, Remax = 50, 100,150

0.8 0.8

a: c. as .s == 0.6 0.6 Q) Q) 'C 'C

~ 0 ~

Q) 0.4 Q) 0.4 II: II:

0.2 0.2

Figure 4. Deviation from Darcy law up to Re = 150

The results presented in the figure 4 are normalized in the same manner as those of figure 2, and correspond to 3 values of Re,max : 50, 100, 150. In this case we observe that the points are located between y = x and y = x 2 •

The coefficients Q and f3 seem to be of the same order. If one fits the data using the relation y = x"'t, then 'Y ~ 1.5.

It is shown that for ordered arrangements the Darcean-isotropic structures become inertially anisotropic, i.e., the pressure gradient and the Darcean velocity vectors are not parallel in the inertial-flow regime, even though they are in the Darcean regime. The Darcean-anisotropic structures remain anisotropic in the inertial-flow regime. The value of the angle Q that the drag force makes with the x-axis depends on the Reynolds number in a way that is in disagreement with recent results of [5]. These authors claim that, at sufficiently high Reynolds numbers, the angle of the drag passes through a maximum and the drag alignes t self with the x-axis for () ~ 12°, whereas the drag tends to become parallel to the oblique direction of 45° for 12° ~ () ~ 45°. The results plotted on figure (5-a) disagree with these conclusions. This disagreement is even more severe for Darcean-anisotropic structures, as shown in figure (5-b). These comparisons are not satisfactory and demand for further investigations to reach a coherent interpretation

36 M. FIRDAOUSS and P. IRAN

of the hydrodynamics of oblique inertial flows in porous media-an effort that will be pursued in our future work.

dev=OO, tetha= 0, 5, 10, 15, 30, 45 45~~--~--+-~45~--~-.··~·~·~~;

40 ~""""30"" 35 .~.

jg 30 ~...-~. 0. ro 25

O~~ __ ~ __ +-~~O~~~ __ -+~ o 20 40 60 80 100 120 140

dev=20, tetha= 0, 15,30,45 35~~--~--~~~~--~--~

30

.e." '·"·""·'30'·"·""·"·"·'·' ....... '.'." .

is -"._' .. -._. ..._..... . ... .1

o O~----~--~~--~--~--~ -5~~--~--~~--~--~--~

o 20 40 60 80 100 120 140

Figure 5. Reynolds number (Re) dependence of the angle a

4.3. STRONG INERTIAL-FLOW

150 30

15

0 ~tN -15

.3() 0 IS 18 20 22 2- 0 3 12

Figure 6. Time dependence of the angle a

For strongly inertial flows, we study the transition to unsteady periodic solutions. The values of the critical Reynolds number depend on the angle () that the incoming flow makes with the x-axis. The critical Reynolds number increases with the angle of the flow. Figure 6 deals with the transition

INERTIAL FLOW ANISOTROPY IN OBLIQUE FLOW .. 37

to unsteadiness for Darcean-isotropic structures and shows the time dependence of the angle Q' that the macroscopic gradient of pressure makes with the x-axis. The three plots correspond to horizontal macroscopic velocity (e = 0°), and to flow with angles e = 15° and 30°. In the first case (figure 6-a), the average angle of the pressure gradient is constant and is equal to zero, the amplitude of its oscillations being L\Q' = ±25°. In the second case (figure 6-b), we have obtained Q' = 29° ± 21°, and in the third case (figure 6-c) Q' = 39° ± 1°. The critical Reynolds number is found to increase with Q' and the respective values have been computed to be aproximately 150, 250 and 250.

5. Conclusion

The Reynolds number dependence of the angle Q' that the drag force makes with the x-axis don't confirm the new results of [5] The authors claim that for high Reynolds numbers, for low angle oblique flows « 12°), the angle Q' reaches a maximum and the drag force becomes horizontal, and that for high angle oblique flows (> 12°, < 45°), the drag force makes a 45° angle with the x-axis. Concerning the Darcean-anisotropic structures, we present results which contradict the conclusions of [5].

References

1. E. Sanchez-Palencia, Non homogeneous media and vibration theory, Lecture Notes in Physics, Springer-Verlag, 1980.

2. L. Tartar, Convergence of the homogenization process, Appendix of SanchezPalencia, 1980.

3. P. Forchheimer, Wasserbewegung durch Boden, Zeitschrift des Vereines deutscher Ingenieure, XXXXV, 49, pp. 1736-1741, and 50, pp. 1781-1788, 1901.

4. E. Arquis & J. P. Caltagirone, Sur les conditions hydrodynamiques au voisinage d'un interface milieu fluide - milieu poreux: application it la convection naturelle C. R. Acad. Sci., Serie II, 299, pp. 1-4, 1984.

5. D. L. Koch and A. J. C. Ladd, Moderate Reynolds number flows through periodic and random array of aligned cylinders, J. Fluid Mech., 349, pp. 31-66 ,1997.

6. M. Firdaouss, J.L. Guermond, P. Le Quere, Non linear corrections to Darcy's law at low Reynolds numbers, J. Fluid Mech., 343, pp. 331-350, 1997.

7. R. Temam, Navier-Stokes equations, North-Holland, Amsterdam, 1979.

AN ADAPTIVE METHOD FOR CHARACTERISTICS-FINITE ELEMENT METHOD FOR SOLUTE TRANSPORT EQUATION IN UNSATURATED POROUS MEDIA

M.GABBOUHY,Z.MGHAZU Laboratoire SIANO, Dep. de Mathematiques et d'Informatique FaculU des sciences, UniversiU Ibn Tofail, B.P. 133, 14000 Kenitra, Morocco

Abstract An adaptive method for the solution of equation modeling the transport of solute

by dispersion and advection in unsaturated porous media is presented. In many ap

plications, when the peclet number is quite large, advection dominates diffusion and

the concentration often develops sharp fronts. So finite elements are combined with

the method of characteristics to treat this problem. Because a good approximation

of velocities is necessary to calculate the advective term of the equation, the flow

equation is approximated by parabolic mixed finite element method.

An a posteriori error estimator is presented for adaptivity. This estimator yield upper

and lower bounds on the error measured in the energy norm with constants which do

not depend neither on meshsize nor on time step. Numerical examples presented here indicate that this method gives nearly exact ap

proximations of sharp fronts.

1. Introduction

We are interested to the basic problem in subsurface hydrology that is the contaminant transport in unsaturated porous media. The model takes the form:

O~~ + q\lc '- div(OD(O,q)\lc) = f (1)

where c denote the solute concentration, 0 the water content, q the Darcy's velocity, f a source/sink term, and D is the hydrodynamic dispersion co-


40 M. GABBOUHY, Z. MGHAZLI

efficient which is given in one-dimensional case by,

D(O, q) = Do + A~ (2)

here Do is the molecular diffusion and A is the dispersivity of the medium. These partial differential equations are, usually, strongly advection

dominated. Generally, standard methods introduce nonphysical oscillations into the numerical solution and exhibit excessive numerical dispersion near the front (cf. [7]). Due to the almost hyperbolic type of these problems, characteristic methods have been successfully applied to solve them (See e.g. [4], [9], [3], [1] and the bibliography therein). However, some amount of numerical dispersion will still exist near the front when applying these methods.

Generally, in the presence of certain local phenomena (e.g. wells, sharp fronts, ... ), fine meshes are required to achieve a proper resolution of the local physical behavior. In order to solve this problem (especially in the case of two and three dimensions) in a efficient manner, local grid refinement is advantageous, and can be done automatically.

In this paper, we propose a Self-Adaptive technique for CharacteristicFinite Element Method (SA-C-FEM), which is based on a robust a posteriori error estimator, to improve the accuracy of the concentration near the sharp front.

Because a good approximation of velocities is necessary to compute the advective term and the mechanical dispersion coefficient of the transport equation, the flow equation is approximated by parabolic mixed finite element method. The flow of water is described by the well-known O-based Richad's equation (cf. [2]):

8~~h) _ div(K(O(h))\7(h - z)) = f (3)

where h is the pressure head, K(h) is the hydraulic conducivity, and z the vertical direction. K (h) is assumed to be a strictly positive bounded function.

An outline of the paper is as follows. In sections 2 and 3, the discretization is presented. The flow equations will be approximated by a mixed finite element method in section 2. The transport equation is discretized by the Characteristic-Finite Element Method (C-FEM) in section 3. An a posteriori error estimator for the C-FEM is given in section 5. In the last section we show numerical results for the introduced types of discretization and for the self-adaptive technique.

AN ADAPTIVE METHOD 41

2. Approximation of Flow Equations

First of all we give some definitions and notations. Let 0 C Rd , d = 1, 2 or 3, be a bounded domain with sufficiently smooth boundary 00, and let T be a strictly positif number such that 0 < T < 00. For any bounded subset w of 0, we denote by Hk(w), k E IN, and L2(w) the usual Sobolev and Lebesgue spaces equipped with the standard norms II . Ilk,w=11 . IIHk(w) and II . Ilo,w=11 . 1I£2(w)' Similarly, (., .)w denote the scalar product of L2(w). If w = 0 we will omit the index O. For k E IN we denote by Pk the set of all polynomials of degree at most k.

In this paragraph we develop a fully-discrete mixed finite element procedure for solving following initial boundary-value problem corresponding to equation (3):

{ 8~;} _ div(K(8(h))\7(h - z)) = f in OxlO, T[ h = hD on oOx]O, T[ h = ho in 0 x {O}

(4)

To obtain the below mixed formulation, we take Darcy's velocity as auxiliary variable

q = -K(8(h))\7(h - z) (5)

Let a(h) = (K(8(h)))-1. For almost every time t E10, T[, a mixed variational form of (4) is

(a(h)q,v) - (h,divv) - < hD, V.l! > +(ez , v), \Iv E V

08(h) . (----at,W) + (dlvq,w) = (j,w), \lwEW

where V = H(div,O) = {v E (L2(0))d : divv E L2(0)}, W = L2(0), and (., .), < ., . > denote respectively the inner product in L2(0) and the duality pairing between Hl/2(00) and H-1/ 2(00).

Let Th be a quasi-uniform partitions of 0 into elements T. Let Vh x Wh C V X W be the Raviart-Thomas finite element spaces of order k (cf. [10], [8]). The semidiscrete mixed finite element method is given as follows. For almost every t E]O, T[, denote by (Q(., t), H(., t)) E Vh X Wh the approximation of (q(.,t),h(.,t)) such that:


where v is the unit outward vector to n. Let tn = nl:1t, n = 0, ... , N, where N E IN*, be a partition of [0, T]

with to = ° and tN = T. We define a fully discrete scheme by using the backward Euler difference. So, for each n ~ 0, let (Qn+1, Hn+1) E Vh X Wh be the approximation of (qn+1, hn+1) such that:

( () (Hn+ 1) - () (Hn) w ) + (di VQll + 1 W ) = (fll + 1 W ) l:1t ,h , h , h,

In [6] we have been obtained the optimal L2-error estimates for the pressure head h and for the Darcy's velocity q

N

(L: II Qn - qn 112 l:1t)1/2 n=l

O(hk+1 + l:1t)

o(hk+1 + l:1t)

3. Approximation of the Transport Equation

In this section we give a Characteristic-Finite Element Method for solving the convection-dispersion equation (1) completed by the initial condition c(x,O) = co(x) and by a boundary condition of Dirichlet type c(x, t) = CD.

In the following we rewrite (1) in the total derivative form. For this we define the characteristic curve at point x in time t as,

{ dX(r) _ q(X(r),r) for T EjO, t[

dr - 6(X(r),r) X(t) = x

So we have the total derivative

dc 8c 'I/J dT = () dt + qV'c (6)

with'I/J = (II q 112 + I () 12)1/2. Then the equation (1) can be written as

'I/J :~ - div(()D((), q)V'c) = f (7)

A backward difference scheme for (7) is

()(x, tn+1) c(x, tn+1~~ c(x, tn) _ div(()ll+lD(()ll+l, qll+1 )V'cll+1) = fll+1 (8)


where x is such that

Set (32 = e~~\ U = Cn+1, Da = en+1 D(en+!, qn+l), 9 = in+! + (32C(X, tn). Then we obtain at each time step the elliptic boundary-value problem

{ -div(Da V'u) + (32u = g in n U = 0 on an (9)

The homogeneous boundary condition is taken for the sake of simplicity. In the sequel we suppose that e is independent of space variable x and

that e and Da are a bounded coefficients that is:

0< d* :::; Da :::; d*

0< e* :::; e :::; e*

Let M = HJ(n). A variational form of (9) is: Find u E M such that

a(3(u,v) = (g,v), \Iv E M

where a(3(u, v) = In Da V'uV'v dx + (32 In uv dx.

(10)

(11)

(12)

Let Mh C M be the set of CD-piecewise linear functions on Th. The discrete variational problem of (12) is: Find Uh E Mh such that

(13)

4. An a Posteriori Error Estimator for C-FEM

In this section we give an a posteriori error estimator. The arguments given here are analogous to those developed in [11] for the reaction-diffusion equation. We restrict ourself to one dimensional case. Let n = I =]a, b[.

We define the norm II . 11(3 as

II u 11(3= (II D~/2 ~~ 116,1 +(32 II u 116,1 )1/2 (14)

The restriction of the above norm on Hl(w), where w is any bounded subset of [2, will be denoted by II . 1I(3,w' We have

(15)


From (12) and (13) we obtain

(16)

By integration by parts, we have

where £( is the set of all interior nodes of n and Uk denote the jump of f on Xi.

In virtue of (16) and (17), we have the equation (18):

Let v E !vI. Let T E Th, with T = [Xj, Xj+1], and let Xi be an interior node to O. We have

a au 2 a au 2 (g+ ax(Daax )-(3 Uk,V-Vhh S;II g+ ax(Daax )-(3 Uk IIo,TII V-Vk Ilo,T

(19) Let h be the interpolation operator which associate for each v E HJ (0)

the CO-piecewise linear function hv E Mk such that

(20)

The following estimates hold for h

II v - hv lIo,T < C hT II \Iv Ilo,wT

II v - hv Ilo,T < C II V Ilo,wT

This imply that

II v - hv Ilo,TS; c min{hT, (3-1} II V 11!3,WT (21)

where WT = [Xj-I, Xj+l].

By (20), the jump term, in (18), vanishes for Vh = hv.


From (15), (18), (19), and (21) we get the upper bound,

'"""" 2 a au 2 2 1/2 II U - uh II.B~ c{ ~ aT II 9 + ax (Da ax) - j3 Uh Ilo,T} TETh

(22)

where aT = min{hT,j3-I}. In order to obtain a lower bound, we take an arbitrary function 9h

approximation of g. Equation (17) imply that,

'"""" a au 2 '"""" au '"""" 6 (gh+ax(Daax)-j3 Uh,V)T- 6 [Daax]XiV(Xi) =aj3(U-Uh,V)+ 6 (gh-g,vh TETh xiEt:( TETh

(23) Let T an arbitrary element of Th. We denote by FT an affine transforma

tion which maps Tonto T where T is the reference segment T = [-1, +1] = [el, e2J. Set

'l/JT = { -J; 0 Fi 1 sur T o sur n \ T

where -J; = 4~1~2 with ~i is the hat function that takes 1 at vertex ei and vanishes at ej, for j f i. We have the following inequalities:

Lemma: For all v E Pk, and all k Em, we have

II A II < (A n!.A)I/2 v O,t c V, 'f'V t

II "V(-J;V) lIo,t < c II-J;v Ilo,t < II v Ilo,t

where c depend only on k.

The proof of this lemma is given in [11].

Set WT = 'l/JT(gh + tx (Da al::) - j32Uh)· By taking v = WT in (23) we obtain

a aUh 2 (gh+ ax (Da ax )-j3 Uh, wTh ~II U-Uh II.B,TII WT II.B,T + II 9h-9 lIo,TII WT Ilo,T

(24) Let v = (gh + tx(Da aau:) - j32Uh) oFT E Pl. The above lemma imply that

I a aUh 2 2 I gh + ax (Da ax ) - j3 Uh Ilo,T < ChT II v 11~,t


In other hand

a aUh 2 II WT Ilo,T < II 9h + ax (Da ax ) - (3 Uh Ilo,T

II WT ILB,T < c(h:r1 + (3) II WT Ilo,T < c2a:r1 II WT Ilo,T

Now we recapitulate

2 2 a aUh 2 - (3 Uh IloT ~ c(9h + -a (Da-a ) - (3 Uh,WT)r , x x < II U - Uh 1It3,TII WT 11t3,T + II 9h - 9 Ilo,TII WT Ilo,T < 2a:r1 II U - Uh 1It3,TII WT Ilo,T + II gh - 9 Ilo,TII WT Ilo,T

1 a aUh 2 < 2a:r II U - Uh 11t3,TII 9h + ax (Da ax ) - (3 Uh Ilo,T +

a aUh 2 + \I 9h - 9 Ilo,TII 9h + ax (Da ax ) - (3 Uh Ilo,T

Finally, we obtain

a aUh 2 aT II 9h + -a (Da-a ) - (3 Uh 110 T~II U - Uh 11(3 T +aT II 9 - 9h 110 T (25) x x ' , ,

In conclusion, by replacing U by en +1 , we have the following estimates for the concentration equation:

Theorem: Let

a acn+ 1 ()n+ 1 '11 = a 119 + _(()n+lDn+1_h_) - __ en+1 II 'IT T h ax ax ll.t h O,T (26)

where CiT = min{ hT' (3-1 }. The following inequalities hold for the a posteriori error estimator "1T,

II cn+1 - c~+1 1If3~ { L ["1~ + a~ II 9 - 9h 116,T]}1/2 TE'Th


where 9 = In+! + 8~~1 c~(x)

5. N umericaI results

Our adaptive technique for mesh control is based on the error estimator given in section 4. Here, the basic strategy is to get an "equidistribution" of the local estimators 1JT given by (26). The criterion to decide which element have to be refined is: Suppose that one has computed, for each element T of the partition Th, the estimator rrr of the error in T. Put "1 := maXTE1'h rrr, then an element T is subdivided if 1JT ~ ~"1.

We present here computations for two examples. The first one consists of advection· dominated model problem with constant coefficients fJ = 1, q = 1 mid, D(fJ, q) = 5.10-2 m2/d, L = 200 m. Initial condition is c(x,O) = 0 and boundary conditions are as follows c(O, t) = 10, c(L, t) = O.

In figure 1 the solution, for ~t = 1 and T = 50, by adaptive technique is compared with a result on a uniform mesh of 40 elements. The qualitative behavior of SA-C·FEM solution is better than C-FEM solution with approximately the same computational costs. The solution obtained through the C·FEM exhibited more numerical dispersion than that ob. tained through the SA·C-FEM. To obtain the same accuracy, a uniform mesh of 250 elements is needed for the C-FEM. The comparison is shown in figure 2. In figure 3 the concentration profiles and the adaptive meshes for the time 50, 75 and 100 are shown. We see that the evolution of the grid closely follows the evolution of the concentration profile.

--·:·F~&" ql ... e!~merJ." }

-":.~ ·(·-FE~ . .f ~ 35 ~!em~!_[s)

. ~ A I.!.lpC\Oe mt;h

2

o ~;~«> ~: .....::~-----<r:'---." G )U 1')0 150

dtpth z

Figure 1. Concentration profiles for uniform and adaptive meshes with approximately the same number of elements.

48 M.GABBOUHY,Z.MGHAZU

\

-:~A·C~FE1'.o{ {3~ ~itmer.ts"

• (' -FE~ '2)!) e:e-rr.enti

:'.A 1.':

depth l

Figure 2. Concentration profiles for uniform and adaptive meshes.

The second example treat the solute transport in unsaturated zone, we investigate here the Gharb (Morocco) soil properties (cf. [5])

O(h) Os[1 + (: trm 9

}((h) }(s(~)~ A 5.10-2

where Os = 0.3995, }(s = 30.2735 cm/h, m = 0.168, n = 2.4038, TJ = 6.8015, and hg = -27.543 cm. Initial and boundary conditions for flow equation are h(x,O) = -100, h(O, t) = -115, h(L, t) = -100 and for transport equation are c(x,O) = 0, c(O, t) = 1, c(L, t) = O.

AN ADAPTIVE METHOD

:0 T= 50 d

.. S 'j

6

i 4 " a " 2

;...::-.-:----<(:----~

tDI----_~ T= !.~ d

iC~----_~ T ~ 100 d

--.- -.~- - ...... :J'::·'.::::-.t.""'A·:;:..:.~~,', •. "" -.-~-,-'--. SI) l:.)r'I i:!) '::')1

depth z

Figure 3. Concentration profiles and adaptive meshes for times 50 d, 75 d, and lOOd.

49

We show in figures 4 and 5 the pressure and concentration profiles for different times ( T = 720 s, T = 1440 s, and T = 2880 s ). Like in first example we notice, in figure 5, that the grid follows the concentration profile. In figure 6 we present the solution at T = 14408 for different time steps. Like classical characterisics methods, the SA-C-FEM is able to use large time steps with no loss of accuracy.


depth z

• T-720s

• T = 14.tO s --T=lSSOs

Figure 4. Pressure profiles at different times for Gharb data.


1.0

\ T = no s

0,2 'J

5 0,15 .... • ~ II .. OA § u

IJ,2

'1.0 \

T= l~~O s

, .-'J,e..

U

g (i,6 '.::1

'" Ji :c ~ I)"';

8

-)~ I "),',1

1,'.1 T, ;8S0 s

'_.',.j

" 5 :j,t i -3 t;

\ '-' ;,-l :::

" ...

"',t) w--~~.' ::!:.:......2-~ J, , (t :11 : (t(i ~ :' :' _(,I_I ~:_1)

dl'pth z

Figure 5. Concentration profiles at different times for Gharb dat::l

52

'" c:

,',' v. ~

.9 0/ .

~ ti o 1).4 '-'

M. GABBOUHY, Z. MGHAZLI

--~I - IO

jl '" 18

~I = J6 ,

)''',"-) - --'-.....lIII-..-----I ..... ·'t)---'--I..L:,C-.I -~----:-'::'LO

depth z

Figure 6. Concentration profiles at T = 1440 s, for different time steps.

References 1. Arbogast, T., and Wheeler, M.F.,: A Characteristics-Mixed Finite Element Method

for Advection-Dominated Transport Problems, SIAM J. Numer. Anal. 32(1995), 404-424.

2. Bear, J.,: Dynamics of Fluids in Porous Media, Dover, New York, 1972. 3. Dawson, C.N., Russell, T.F., and Wheeler, M.F.,: Some Improved Error Estimates

for the Modified Method of Characteristics, SIAM J. Numer. Anal. 26(1989), 1487-1512.

4. Douglas, J., and Russell, T.F.,: Numerical Methods for Convection-Dominated Diffusion Problems Based on Combining the Method of Characteristics with Finite Element or Finite Difference Problems, SIAM J. NtJ.mer. Anal. 19(1982), 871-885.

5. Gabbouhy, M., Maslouhi, M., Mghazli, Z., and Saadi, Z.,: Modelisation numerique du transport de solute dans la zone non sature d'un sol tres sableux, 5emu Journees d'Analyse Numerique et Optimisation, Universite Ibn Tofai' I, Kenitra, 28-30 Avril, 1998.

6. Gabbouhy, M., and Mghazli, Z.,: Analyse d'une methode d'elements finis pour la resolution du probleme d'ecoulement et de transport de solute dans un milieu poreux non sature, 5eme $ Journees d' Analyse Numerique et Optimisation, Universite Ibn Tofail, Kenitra, 28-30 Avril, 1998.

7. Hughes, T.J.R., ed.,: Finite Element Methods for Convection Dominated Flows, American Society of Mechanical Engineers, New York, 1979.

8. Nedelec, J.,: Mixed Finite Elemnts in R 3 , Numer. Math. 35(1980), 315-341. 9. Pironneau, 0.,: On the Transport-Diffusion Algorithm and its Application to Navier

Stokes Equations, Numer. Math., 38(1982), 309-332. 10. Raviart, P., and Thomas, J.-M.,: A Mixed Finite Element Method for Second Order

Elliptic Problems, Lecture Notes in Math. 606, Springer, Berlin, pp. 292-315, 1977.

11. Verfiirth, R.,: Robust A Posteriori Error Estimators for a Singularly Perturbed Reaction-Diffusion Equation, Numer. Math. 78(1998), 479-493.

Estimation of parameter geometry

J. Heredia 1, A. Medina Sierra 2, J. Carrera 2

1 INGEMISA. Madrid. Spain

2 ETSECCPB, UPC. Barcelona. Spain

Abstract

It is well known that to use a groundwater model as a predictive tool, model

parameters have to be calibrated against measurements (heads, concentrations or

other state variables). For this reason, groundwater literature is plenty of results

and theory on inverse problems. Usually, the physical parameters are discretized

using a parameterization defined through some variables (the so called model

parameters) that can be estimated (calibrated). Most of the effort on inverse

modeling has been done in estimating the values of the model parameters, but not

their spatial variability, that customarily is considered as fixed except in some few

works.

Among different alternatives, we have chosen zonation as the way of

discretizing spatial variability of parameters, that is one of the most employed

ways of parameterization. In this paper a methodology for the estimation of both

parameter values at the zones and their shape (geometry) is presented. A model

structure identification criterion (developed on the framework of the bayesian

theory) has been defined to account for the consistency between model and real

system. This criterion leads to a optimization problem with a objective function

that is minimized using several integer algorithms. The differences and similarities

between our proposed methodology and other approaches are highlighted, as well

as their respective limitations. Examples are included to show the applicability

and restrictions of the methodology.

Introduction

The use of numerical models in hydrology is widespread nowadays. This is

done for economy and enlargement of analysis capacity. However, frequently arises

53 J.M. Crolet (ed.), Computational Methods jor Flow and Transport in Porous Media, 53-81, © 2000 Kluwer Academic Publishers.

54 J. HEREDIA et al.

relevant differences between model results and real system measurements. Most

times these are due to an incorrect conceptualization of the system, rather than

numerical difficulties or parameter estimation. As it is not possible to include all

the complexity of any real system, any model is made of important simplifications.

The make up of a numerical model involves the following tasks.

- Identification of the relevant processes. This requires to know the physico

chemical processes involved in the system and to extract which ones are

necessary (and possible, depending on the limitations of our modeling tools) to

include in the model. Usually, at the end of this step one or several equations

including all the processes are defined.

- Identification of model structure. In this step, the domain is discretized in

both space and time. At this moment, the parameters are also discretized.

- Calibration or inverse problem. At this point, the values of the parameters

are quantified. Usually, the values of the parameters have a large uncertainty

at the beginning of the conceptual model due to several reasons: difficult

measurements, different scales between data and model, different physical

conditions, etc. For this reason, in thos step, the parameters are modified in

such a way that model results (in terms of heads or concentrations) match

their respective measurements.

This separation is somewhat arbitrary, but it is useful to set the main ideas.

The domain has to be discretized both in space and time, attending

the requirements of the numerical model. In addition, physical parameters

(transmissivity, etc.) are discretized using a predefined parameterization.

Parameterization is the way to convert the physical parameters (that may display

a large unknown variation) into a relatively small number of parameters, the so

ESTIMATION OF PARAMETER GEOMETRY

called model parameters. Most of parameterizations can be described as

p(x, t) = ~ Fi(X, t)Pi i

55

(1)

where P is the physical parameter, Fi is an interpolation function and Pi are the

model parameters. In this work, we will use zonation. In this case, the domain

is partitioned into subdomains. Inside subdomain i, function Fi is constant or

varies in a predefined manner. Outside this subdomain, Fi is zero.

The model structure should represent the heterogeneity of the medium as

well as possible. To do so, both the number of zones and their shape should

be optimized. It is worth noting that optimum dimension is strongly related to

quantity and quality of data.

The proposed methodology to find the optimal transmissivity zonation is

based on the minimization of an objective criterium obtained from Bayes' theorem.

With this formulation, it is possible to include the "a priori" probabilities of a

model to be correct. The usefulness of these probabilities stand on the fact that

they are defined using the geological (soft) information. The identification of

the optimum transmissivity zonation is made joint and automatically with the

parameter estimation of all model parameters. This requires to solve the direct

and inverse problems.

State of the art

There are several approaches to find the optimum parameterization. Some of

them are automatic and some of them contain some trial and error steps. The

different methods can be grouped into four approaches.

- Least squares criterion (LSC) on the residuals and the covariance

matrix, Cov. In this line, we can find the works of Yoon and Yeh (1976),

Shah et al. (1978), Yeh and Yo on (1981), and Sun and Yeh (1985). The

parameterization method is based on the FEM. To define the dimension

of array p and optimum parameterization implies to find the number and

56 1. HEREDIA et al.

location of interpolation points. The criterion is based on the trace of the

covariance matrix. The model's fit is based on LSC. The last of those works

were the first that developed an automatic optimum parameterization. Other

authors (Carrera, 1984, Carrera and Neuman, 1986) argue that the trace of

Cov could not be a good indicator of the stability of the model when model

parameters are highly correlated.

- Linear discrimination based on statistical tests. Cooley et al. (1986)

discriminate the models using a generalization of standard linear test of

hypothesis (Graybill, 1976). It is based on assuming that any model

may posess features that describe the real system correctly. The selected

model is the simplest one that including the main features of the system

is coherent with the data. Nevertheless, many researchers (Norton, 1986,

Yeh, 1986, Carrera and Neuman, 1986) pointed that the simplest model do

not necessarily posess the optimum structure as suggested by the parsimony

principle. The model may need some features supported by the qualitative

information, but not reflected in the quantitative data as Cooley also suggest.

- The use of model identification criteria (MIC). Those models were

first developed for the analysis of time series. We will talk only about works

related to hydrogeology. The maximum likelihood criterion (MLC) allows

the comparison between different hypothesis of both hydrogeological and

statistical parameters given a fixed model structure. For this reason, it is

not really useful for the discrimination between different models and if used,

it leads to the model with the largest number of parameters (Schwarz, 1978,

Carrera, 1984). The most used criteria are AIC (Akaike, 1974), BIC (Akaike,

1977, Rissanen, 1978, Schwarz, 1978), <p (Hannan, 1980) and dk (Kashyap,

1982). Those criteria are based on information theory, maximum entropy and

bayesian theory. The expresion of dk is

d = S + Mln(N/27r) + lniF/NI (2)

Those criteria were first introduced by Carrera (1984) and they have been used

to discriminate between different models with encouraging results (Carrera,

ESTIMATION OF PARAMETER GEOMETRY 57

1984, Hoeksema and Kitanidis, 1985, Carrera and Heredia, 1988, Carrera

et al., 1990, Medina, 1993 among others). Our methodology attacks the

automatic transmissivity zonation identification by using a new identification

criterion based on the bayesian theory that can incorporate soft (geological)

information.

- Cluster analysis. This approach was first proposed by Eppstein and

Dougherty (1994) using an automatic method to identify the transmissivity

zonation and its parameter values. The method is a combination of the

extended Kalman filter (EKF) and a cluster algorithm. In a more recent

work, an extension is presented (Eppstein and Dougherty, 1996). It consists

of an approximation to EKF that increases the computational speed, a more

complex cluster algorithm and a better way of combining transmissivity zones.

Both works are applied to steady or quasi steady state synthetic examples.

The authors suggest that it could be possible to include soft information.

Works based on geostatistical methods require to obtain an optimum structure

of the parameters that define the statistical distribution of the hydrogeological

parameters (usually transmissivity). In this sense, those works find in some way the

optimum parameter structure. In this line can be included the works of Kitanidis

and Vomvoris (1983), Hoeksema and Kitanidis (1985). They usually divide the

process into four steps: 1) define a structure for the statistical distribution of

hydrogeological parameters, 2) estimation of statistical parameters, computation

of the covariance matrix and the MLC, 3) parameters' coherence is evaluated using

statistical tests and 4) model structure is accepted or rejected (in this case return

to step 1). In some of these works they use AIC to discriminate between different

structures of ststistical parameters. They conclude that the AIC method is fast

and easy to use to select between different models, even though the process used to

identify model structure is more complex than AIC. However, the models selected

by AlC were the ones that other statistical tests selected as the more likely in

general.

The paper is divided into six sections. Next section is devoted to the

58 J. HEREDIA et aI.

description of both direct and inverse problems. Third section contains the

theoretical framework. Fourth section is related to numerical implementation

details. Fifth section presents the application example. The paper ends with

the conclusions and references.

2. Direct and inverse solutions.

The direct problem is the one that solves one or more state equations. In our

case, the state equation is groundwater flow equation, which can be expressed as:

oh V.(T Vh) + q + qL = S ot in n (3)

where h is head level, T is transmissivity tensor, S is storage coefficient, n is problem domain, q is sink/sources term and qL stands for input/output from

aquitards below and/or above the aquifer. Equation (3) is solved under appropriate

initial and boundary conditions. Numerical solution is done by applying the finite

element method in space and implicit finite differences in time.

Inverse problem consists of estimating model parameters (transmissivity,

storativity, etc.) using measurements made in the system (both heads and prior

information of parameters). Inverse problem solution has been made by using

many different approaches (see Carrera, 1987 and Yeh, 1986 for reviews in this

topic). Those approaches differ in the adopted statistical framework and in

the degree of resolution, but they are quite similar in practice. Our approach

is based on Maximum Likelihood theory (ML, see Carrera, 1984, Carrera and

Neuman, 1986, Medina and Carrera (1996) among others). ML estimation leads

to maximizing (Edwards, 1972):

where N = nh + L:i ni, nh and ni are the total number of head data, z* is

an array made by h* and p*, that are head measurements and prior information

of parameters. N is the dimension of Z, whose first nh components are computed

heads h with parameters p and the rest up to N components are parameters with


given prior information, C is the covariance matrix of errors

(5)

where Ch and Cp are the covariance matrices of heads and type i parameters

and are assumed to be as (Neuman and Yakowitz, 1979):

(6)

(7)

where V h and Vi are known positive definite matrices and a~, a; are

unknown scalars. The array of parameters, 0, is composed by model parameters

(transmissivity, etc), p, and statistical parameters (J = (a~, a;) (and some that

may define the structure of matrices V h and Vi). In practice one minimizes

5(a, z*) = -2ln(L(a/ z*)) (8)

If the structure of parameter errors is known (ie. parameters (J are known), then

minimization of (7) is equivalent to minimize

i (9)

where hand h* are the computed and measured heads respectively, Pi and pi are

type i computed and prior information parameters respectively, and Ai = a~/ at are scalars that depend on the error structure of the model.

Parameters Ai can be obtained by succesive approximations (Carrera and

Neuman, 1986). It can be shown that if Ai is optimum, then a~ = J / N (Carrera,

1984). Otherwise, a~ = J Inh and a; = J Ini. Function J (Eq. 9) is taken

as the objective function to minimize, ie., parameters that minimize this function

are the calibrated parameters (the solution of the inverse problem).

Assymptotically, the maximum likelihood estimator follows a gaussian

distribution. As a result, it is fully defined by its expectation and its covariance


matrix. The expectation is the estimator itself and a lower bound of its covariance

matrix may be computed as the inverse of Fisher's information matrix, F (Bury,

1975):

p.. - ~E [ a2s 1 ZJ - 2 aQiQj

(10)

where S is given by eq. (8) and a E JRnp is the parameter array, in which both

model parameters (transmissivity, porosity, etc) and statistical parameters (J"~, (J"r) are included.

3. Model structure identification

Identifying the model structure is related to the conceptualization of the

hydrogeological system through model parameters. This includes the definition

of model geometry, boundary conditions, parameterization schemes, model

parameters, etc. It is assumed that the geological information and, sometimes,

the hydrochemical one are enough to construct the appropriate model structure.

In addition, the size and spatial and temporal correlation of residuals (differences

between computed and measured heads) suggest which model areas should be

modified, leading to a new structure. This process is repeated until a satisfactory

model is achieved.

The main characteristic of this process is its inherent subjectivity. All the

decision making in the modeling process lacks some kind of quantification. In

addition, none of the model structure identification techniques evaluates explicitly

the coherence of the model with respect to the geological information.

The main objective of the proposed methodology is to obtain the best

transmissivity structure (the shape of transmissivity zones) assuming the rest of

elements that define the model structure. Under these assumptions, the best

transmissivity structure is related to the best model structure. To achieve these

objectives, the methodology should:

1.- Include the quantifiable geological information to the modeling process, in a

rigorous and objective way


2.- Define an objective function that takes into account this information. This

would allow to compare the coherence of the models with respect to the real

system.

3.- Redefine automatically the transmissivity structure by changing the shape of

its zones. This redefinition should be based on (1) the fit between computed

and measured heads, (2) the computed parameters and (3) the quantification

of the available geological information in the different parts of the domain.

These steps are explained in the next sections.

3.1 The objective function dk.

The Bayesian Theory provides a rigorous and intuitively clear framework to

incorporate the geological information by defining an objective function. This

is done by defining the "a priori" probability that model Mi be the right one.

Let P(Mi) denote this probability. The "a priori" probability that a model be

the "correct" (no matter what is the true one), evaluated from the geological

information, could be a good indicator of the coherence degree of the model with

respect to the real system. Our objective function, dk is then derived from the

bayesian theory and accepts as the "best model" (optimum model) the one that

minimizes the expected value of selecting the wrong model.

Given r models MI, ... ,Mr mutually exclusives (none of them can be

expressed as a function of the rest), the right model (in our terminology the

"best"), MZ satisfies the following property:

P(Mdz*) ~ P(Mj/z*), j = 1, ... , r (11)

where P(MzI z*) is the "a posteriori" probability that model Mj be the correct,

given the observed data z* = (h*, a*), where h* are the measured heads and a*

is the prior information of parameters. Probability P(MzI z*) can be expressed

as (Edwards, 1972)

P(Mdz*) = klP(Z* /MI)P(MI) (12)


where P(z* / Mz) is the "a posteriori" probability of observing z* if model MZ is

correct. P(MZ) is the "a priori" probability that model MZ be the right one and

k1 is a proportionality constant. It is supposed that errors in h* and a* are both

normally and independently distributed. Approximating S (Eq. 8) by its Taylor

development up to second order around iiz (iiz are the parameters that minimize

S), and using the fact that density function is proportional to likelihood, equation

(12) can be approximated by

(13)

where na = L:i ni is a dimension and F is the Fisher information matrix.

Neglecting constant terms in the previous expression and taking logarithms, we

obtain our model comparison criterion, dkM : I

Careful examination of equation (14) allows us to conclude the following: (1)

The first term accounts for the fit between computed and measured heads and

computed and prior parameters. This part weighs the information considering the

adopted error structure by means of the covariance matrices Ch and Cpo (2) The

second term refers to the number of parameters, na, and the number of data,

N = nh + Li ni, favoring the parsimony principle (under same conditions,

select the simplest model). (3) The third part is related to the uncertainty of

the estimated parameters by means of the Fisher information matrix (under same

conditions, the model whose parameters have a weaker dependency on the data

is selected), and (4) the last term points to the model coherence with respect to

the geological information. Under same conditions, model that best fits the prior

knowledge is selected.

3.2 Definition of prior probability

A basic step in this formulation is the definition of model prior probability

P(M). To define it, a first step is considered. Using the knowledge of the system

(basically geological information) one defines the probability that one element of


the mesh belongs to a given transmissivity zone (it can be also generated by means

of geostatistical methods, eg., indicator kriging, block kriging, etc). In this way,

matrix Pezt is defined as:

Pezt = (15)

where P(ei E Zt) is the probability that element ei belongs to transmissivity

zone Zt, NUMEL is the number of elements and NZTR is the total number of

transmissivity zones.

In general, all this events will not be independent. That is, it can be expected

that the probability of one element belonging to a given zone is a function of

adjacent elements belonging to the same zone. However, for simplicity, we will

assume that all those events are independent. Under this assumption, the "a

priori" probability of model M[ being correct is:

NZTR (NELzt ) P(MZ) =]] ]] P(eiEzt ) (16)

where N ELzt is the total number of elements that compose transmissivity zone

number zt in model MZ. To assign the probabilities that set up matrix P (eq. 15)

one may choose between subjective values obtained from site knowledge or values

acquired by geostatistical methods.

3.3 Description of the methodology

The first step consists of compiling all the information needed for the model.

This includes head data and prior parameters, domain discretization, zonation of

all parameters


INPUT

Calibration of current model

Generation of alternative zonations

Selection of the next model

No Recover previous model status

Relax selection condo of elements

Figure 1. Simplified flow diagram, showing the two embeeded processes:

calibration and zonation

except transmissivity, boundary and initial conditions and definition of model prior

probability.

Second step includes the definition of the initial zonation of transmissivity.

This initial zonation can be arbitrary (given by the modeler) or automatically

generated from model prior probability matrix (eq. 15).

It should be noticed that two convergence processes are embeeded (see Figure


1). The inner minimization loop is the minimization of 8(0:, z*) (eq. 8) for a

given zonation. The outer minimization loop is the minimization offunction dkM ' l

ie. the minimum against different zonation structures.

N ext step consists of repeating the following sequence until convergence (of

zonation):

• Selection of the elements of model M{ that may change zone (ie., definition

of NseZ' the set of those elements) and computation of parameter NELC (maximum number of elements that simultaneously change their zone -these

elements belong to Nsez-)

• It should be noticed that gradient of J (Eq. 9) with respect to transmissivities

at each element and the corresponding prior probabilities are needed to select

the elements of set NseZ'

• Generation of the alternative zonations Mi and selection of model MZ+ 1

zonation. Those processes are done simultaneously. The selection of model

MZ+1 among all the alternatives should be based on the minimum of dki . However, as it may represent an un affordable computational cost (each dki represents one calibration), we compute approximately the value tl.dki, which

is the increase in the objective function (12) between model M Z and model

Mi·

• Calibration of model M Z+1'

• Estimation of objective function dkM ,and comparison with dkMl 1+1 (eventually check the convergence conditions of the zonation process)

• Depending on the result of the previous step, one of the following decisions is

taken: 1) end of the zonation process, 2) definition of a new model MZ+2 or

3) relax the selection conditions of the elements that form Nsel and make a

new trial of model M{ + 1 .


We understand as number of iterations the number of times that the above

described process is repeated. Good iteration is the one that this process ends

defining a new model MZ+1 such that dkM < dkM . One should be careful 1+1 1

not confussing the iterations of the zonation process with the ones of the calibration

problem, because the latter are included in the first (Figure 1).

4. Implementation of the methodology

As it has been stated previously, the computations have to be done carefully

to obtain a practical algorithm. In this sections we decribe the basic features of

the algorithm, as well as the main assumptions underlying it.

4.1 Generation of alternative transmissivity zonations.

programming algorithms

Integer

Zonation of model MZ+1 is obtained by choosing between a set of alternative

zonations to model M l . Each alternative zonation corresponds to the simultaneous

change of zone of N E LC elements belonging to set Nsel. The number of

alternative zonations may be huge, due to two reasons: (1) The number of different

subsets that can be defined on NseZ' Nelc can be large and, (2) many of the

chosen elements can belong to more than two transmissivity zones. To attack this

problem, we employ integer programming methods.

The Integer programming algorithms can be classified into the following

types:

- Algorithms of optimum type. These algorithms go through all alternative

zonations and select the best. In this type of algorithms we can find "branch

and bound".

- Algorithms of suboptimum type. These algorithms do not look at all the

possibilities, so they cannot ensure that the real minimum is achieved. In this

group we find "succesive exchanges" and "succesive inclusions" .

Difficulties with the first type of methods arise when the number of alterative


zonations is large, because of its huge computational cost (eg. if the size of Nsel

is40 and NELC = 20 elements, the number of combinations is above 1011).

We consider the following integer programming methods: "branch and bound" ,

a variant of it that we call "branch and bound T" (developed in this work),

"succesive exchanges", "succesive inclusions" and combinations of these last with

the rest. The implementations of "branch and bound" and "branch and bound T"

consider alternative zonations generated by subsets Nelc of dimension smaller or

equal than NELC.

4.2 Definition of Lldk' ~

Scalars Aj = CT~I CTJ can be thought as weighting parameters between the

different terms of J (Equation 9). These parameters depend upon the assumed

error structure for each model. This implies different expressions for Lldki depending on the optimality of Aj. If parameters Aj are optimum (CT~ = JIN) , then

(17)

where LlJi = Ji - J 1, Ji and Jl are the minimum values of the objective

functions of models Mi and Ml respectively. Hi and HZ are the hessian matrices

(at the optimum jj). MZ is the model to be updated and Mi is a given alternative

model.

If parameters Aj are not optimum (CT~ = J Inh), then

_ ~ t>.Jj ( t>.Ji - "L:j )..jt>.Jj) (IWI) (P(Mi )) t>.dk· - 6 -2-nh In 1 + I +In --I -In , j O"j Jh IH I P(MI)

(18)

where LlJ] = J] - J;, J] and J; are the part of Ji and Jl related to parameter

j respectively. Jh is the term of head residuals in Jl (eq. 9).


Functions Ji and J l, their corresponding head residuals Jh and Jh, their

respective parameter counterparts J~ and J~, and their hessian matrices, Hi and

Hl should be computed at pi and pl respectively. Parameters pi and pi are the

optimum values of objective functions Ji and Jl respectively. It should be noticed

that parameters pi are obtained only after calibration of model Ml'

4.3 Approximated computation of !.l.dk" t

A large number of evaluation of function !.l.dk. have to be done to obtain the t

zonation of model Ml+ 1, actually as many as the number of alternative zonations

to model MZ are considered. This makes the computation of !.l.dki to be a critical

point that has to be carefully analyzed. The elements that compose the evaluation

of !.l.dki related to model Ml are known:

i (hl (pl),pl) ,Hllp=pl' Jh (hl (pl)) , J~ (pl) ,P(Ml) (19)

where pl are the parameters that minimize Jl (hl (pl) , pl) (Eq. 9) and hi

are the computed heads of model M l . However, the corresponding values of

(19) for any alternative model Mi are unknown, because pZ (parameters that

minimize JZ) are unknown. To obtain these parameters, model Mi would have

to be calibrated, which results in an unaffordable work. To solve this problem,

function !.l.dki is approximated to avoid calibration of alternative models. Using

this approximation, computation of !.l.dk' is fast and easy, allowing us to analyze t

an important number of alternatives. To simplify the algorithm notation, only

log-transmissivities (Y = log T) are taken into account.

4.3.1 Basic assumptions

Computation of J i , Hi and J~ is based on the relathionship between g~ (jacobian matrix of heads with respect to log-transmissivity of any zone) and

aBfJe (jacobian matrix of heads with respect to log-transmissivity of any element

belonging to any zone) and some simplification assumptions.

It is assumed that the jacobian matrix of heads has low sensitivity to the


variation in log-transmissivity on one element due to change in model. That is,

8hi 8hZ

8yJy=yl ~ 8yJy=yl (20)

where g~: iy =yl is the jacobian matrix of heads of model Mi with respect to

y at element e and at point yl; Mi is an alternative model generated by changes

in transmissivity zonation; and yl is the calibrated value of log-transmissivity in

model M l. Under this hypothesis, the jacobian matrix of heads of model Mi with

respect to a transmissivity zone and at the point yl can be expressed as:

where ei and ej are the elements that are included on or removed from

transmissivity zone T when changing from model Ml to alternative model Mi respectively.

As hi can be thought as hl evaluated in a different point (value of log-T),

hi(yZ) can be approximated as

where en are the elements that change its zone in model Mi with respect to model

Ml (ie. change from zone j in model Mj to zone i in model Mi), and Y~ - y1 is

the difference in the values of log-transmissivity at element en between zonations

of models Mi and MZ-

For the computation of Hi it is also assumed that the change of first and

second order derivatives of heads with respect to log-T caused by a variation of log

T, b.yi = yi - yl, is negligible (yi are the values obtained after minimization

of objective function JZ, that are not computed to avoid calibration of model Mi). This assumption allows us to presume that

(23)


where IP is the hessian matrix of Ji .

At this point we are already prepared to briefly describe the approximated

computation of expressions Ji (hi(pi),pi) ,Hi \ .,Jt(pi) ,P(Mi), that p=pz

are needed for the computation of Ildki . Objective function of model Mi is

approximated by its Taylor development up to second order:

Ji (hi (yl), yl) is the objective function of alternative model Mi at yl and it

is computed using the approximation of heads of model Mi at point yl given by

Equation (12) and expression (9). The second term of gradient, is computed by

deriving Equation (9) with respect to Y and using approximations of equations

(21) and (22). The hessian matrix at Y = yl is computed by deriving (9) twice

with respect to y, neglecting second order derivatives and using approximation

(21). Finally, for the computation of (24) it is necessary to know Ilyi, which is

computed as

(25)

Details as well as the full expressions of all the terms needed in the computation

of Ildki can be found in Heredia (1994).

4.3.2 Computation of P(Mi)

The "a priori" probability that alternative model Mi be the true one can be

computed as: NELC

P(Mi) = P(M1) IT P(en E Zi) n=1 P(enEZj)

(26)

where P(Mi) and P(Ml ) are the "a priori" probabilities of model Mi and

Mz being the "correct" respectively, en is one element that changes zone in the

alternative model Mi, P(en E zs) is the probability that element en belongs to

zone zs; zi and zn are the zones to which element en belongs in models Mi and

MZ respectively.


Approximations of the objective function, J i , its gradient, gi, and its hessian

matrix, Ht are applicable to every alternative model Mi ,derived from the current

model Mz by modifying the transmissivity zonation of the latter. The saving

in computing time stands on avoiding the calibration of alternative models Mi and the exact computation of its own objective function, its own gradient and its

own hessian matrix. Comparing with a typical case of calibration, as in Carrera

and Neuman (1986) or Medina and Carrera (1996), additional elements have to

be computed. Apart from objective function, gradient and hessian matrix at

y = YZ, that are derivatives with respect to zonal transmissivities, we need the

computation of derivatives with respect to transmissivities at every grid element

to define set NseZ.

All these computations make the process quite high expensive from the

computational point of view, notably the computation of hessian matrices (second

order derivatives)

4.3.3 Computational Highlights

The highlights of the numerical requirements of the methodology are:

• We avoid calibration of alternative models, Mi

• Computation of J i , gi and Hi is done by simple algebraic operations with

two vectors, three matrices and scalar J l .

• We need Jl (hl (pl), pl), gl lp=pl' Hl lp=pl' which are obtained as a result

of the calibration of model Ml. In addition we have to compute g~e Ip=pl

(gradient of JZ with respect to log-Tat element e, evaluated at parameter

pl), H~e Ye Ip=pl (hessian matrix of Jl with respect to log-Tat element

e, evaluated at parameter pl) and H~y)p=pl (hessian matrix of Jl with

respect to log-T at element e and one zonal parameter, evaluated at parameter

pl).


• Computation of P(Mi ) using Eq. (26) is immediate, exact and fast.

• It should be noticed that the approximation of Ji, gi and IP is appliable to

any alternative model Mi obtained by modifying the transmissivity zonation

of a given one, Mi'

Table 1. Number of linear systems of order k (number of nodes) needed to

compute g~ and ~0' NUMEL is the number of elements and m is the number

of transmissivity zones.

Direct derivation Adjoint state

oj NUMEL 1 aYe o2J m2+m m+1 oyl 2

. oj o2J The most expensive computatIOns are aYe' oyl' These derivatives are

usually obtained by using direct derivation or the adjoint state method (Carrera

et al., 1990). We use the latter because of the saving of CPU time. These savings

can be observed in Table 1. It should be stressed the big difference between the

two methods from the point of view of the computational time. This is strongly

increased in transient problems, because all these expressions have to be computed

at every time step.

5 Exarnpl~

In this example we want to test the sensitivity of the proposed methodology

to the quantifiable geological information. For this reason, we will present two

cases, with and without including this information. These examples belong to an

extensive test set (more than 100 synthetic examples) to study the sensitivity of

this methodology to several features of both hard and soft information.

The example is defined in Figure 2. One simulation of flow equation was

done with the parameter values (and the shape of transmissivity zones) displayed

ESTIMATION OF PARAMETER GEOMETRY

True zonation

2

T 1= 500 m2/day T2=1000 m2/day T3=5000 m2/day S=0.3 H1=Om

E H2=5 m

•• •• •• I

u=O.Ol m3/day m • = Obs. point

I ~_., _. __ to. __ '- _I _-'L_--..,,-- ~

\I -" f'

1 m o o Leakage condition

PI =0.00 PI =0.00 PI =0.00 PI =0.00 PI =0.20 P2 =0.20 P2 =0.50 P2 =0.50 P2 =0.60 P2 =0.80 P3 =0.80 P3 =0.50 P3 =0.50 P3 =0.40 P3 =0.00

PI =0.00 PI =0.20 PI =0.20 PI =0.20 PI =0.20 P2 =0.20 P3 =0.80

P2 =0.40 P3 =0.40

P2 =0.40 P3 =0.40

P2 =0.50 P3 =0.30

P2 =0.80 P3 =0.00


PI =0.50 PI =0.33 PI =0.33 PI =0.33 PI =0.50 P2 =0.25 P3 =0.25

P2 =0.33 P3 =0.33

P2 =0.33 P3 =0.33

P2 =0.33 P3 =0.33

P2 =0.50 P3 =0.00


Figure 2 Description of the problem

Table 2. Flow rates

Time (days) 0 2 4 8 12 16

Q1 (m3 jday) 0 0 2.5 2.5 2.5 2.5

Q2 (m3 jday) 0 25 25 25 25 25

20 24 28

2.5 2.5 2.5

25 25 25

73


in this Figure. Flow values Q1 and Q2 are shown in Table 2. A white noise

of 0"=5 cm was added to simulated heads to obtain the "measurements". The

true zonation is also presented on Figure 2. It is also shown the quantification

of the geological information on a grid of 5 X 5 elements. The optimal zonation

was searched departing from two different initial geometries as displayed in this

Figure. The number of data points is 10 and they were "measured" at 11 times

(Table 2).

As it is show in Figure 2, the probability of belonging to a given transmissivity

zone is defined through 25 subregions. This information is translated to a more

discretized domain directly. In the case where the geological information was not

considered, a constant value of P(e E ZTJ=O.333, i=1,2,3 was taken.

5.1 Results

We have tested two different discretizations: 5 X 5 elements (same size as

the geological information, Figure 2) and 15 X 15 elements. Sensitivity results

are presented on Table 3. Table 4 shows the results of the last zonation of the

example. The cases of 5 X 5 discretization led to the true zonation in all cases

but the one depicted in Figure 3a, without considering the geological information

and departing from zonation B. In Figure 3b are displayed the final results with

the 25 X 25 grids. The cases including geological information performed better

than the rest. In tables 3 and 4 we can observe the values of some of the

criteria employed in the minimization (objective function for both the zonation

and calibration, Kashyap's criterion, etc.). In Table 3 we present the results using

the true transmissivity field. The two simulation rows are obtained by using the

"true" transmissivity field (those runs are done to obtain the "measurements").

It is important to notice that even with the "true" transmissivity field, the

calibration do not leads to the "true" transmissivity values at every zone due to the

"measurement error" (a white noise of 0"=5 cm was added to the simulated values).

The contents of Table 4 are similar to those of Table 3, but departing from initial

zonations A and B. In the case of the 25 X 25 grid, it is worth noting that even


the final shapes of the transmissivity field look quite different, the fits between

computed and measured heads (column J) are very similar, as the computed

transmissivity values also are.

Table 3. Simulation and calibration using the "true" zonation

# of Initial Process d kM DEK S J InlFI In(P{M» Tl T2 T3 elem. Zonation [m2/day] [m2 /day] [m2/dayJ

Simulation ·211 ·244 ·286 0.251 33.9 ·16.7 SOO 1000 5000

5x5 True

(25) Calibration ·212 ·245 ·287 0.248 33.9 ·16.8 496.083 1004.024 4975.117

Simulation 56.6 ·245 ·286 0.251 33.7 ·151 SOO 1000 5000

15x15 True

(225) Calibration 55.6 ·246 ·287 0.248 33.7 ·151 495.775 1004.068 4972.285

6. Discussion and conclusions

A methodology to obtain the optimum geometry of transmissivity has been

presented. The main features of this methodology are (some of the conclusions

outlined here were obtained from other examples, Heredia, 1994):

• The proposed criterion, dkM allows us to treat simultaneously hard (head

and prior information) as well as soft data (geological information)

• The minimization of dk leads to both, the geometry and the values of

model parameters still maintaining the coherence between prior and geological

information.

• The inclusion of soft information to dk is done by the estimation of the

"a priori" probability of a model to be correct. The actualization of this

probability to any model is easily updated, provided that the latter comes


••••• ••••• ..... ' . ••••• •••••• ••• • ••••••• •••••• • •••••• ••••• ••• ••••••••••• ••••••• ••••••• •••••• • •••• ••••••••••••••• ••••••••••••••• •••••••••• ••••

• ••• •••• • •••• •••• ••••

_II _11-_II~:.JLJ['][] t.Jr~jUU

••• • •• •••••• •••• • ••••••• ••• ••••••• ••••• ........ ........ ••••••••••••••• •••••••• • • ••••••••••••••• ••••••••••••••• •••••••••••••••

Figure 3a) Final geometry with 5 X 5 grid, departing from initial zonation

B and without including geological information. b) Final

geometry with 25 X 25 grid in all the cases.

from the modification of the zonation of a given model.

• Our methodology allows the inclusion of soft information. However, it has two

drawbacks: 1) it do not reflects reality (although any model does) , because

our probabilistic model treats the probabilities of elements belonging to

transmissivity zones as independent events, and 2) the "a priori" probability

depends on the number of elements in the grid.

• As it has been seen in the example, this type of problem is ill-conditioned.

This is probably due to non-uniqueness and instability. We think that those

problems are inherent to the automatic optimum zonation problem.


Table 4. Sensitivity analysis to initial zonation, soft information and

# of Soft. info Inicial dkM DEK S J InlFI In(P(M)) Tl T2 T3 elem. included Zonation [m2 /day] [m2 /dayJ [m2 /dayJ

A -212 -245 -287 0.248 -33.9 -16.7 496.083 1004.024 4975.117

YES

B -212 -245 -287 0.248 -33.9 -16.7 496.083 1004.024 4975.117

5x5

(25) A -190 -245 -287 0.248 -33.9 -27.5 496.085 1004.022 4975.109

NO

B 38.6 -16.3 -50.4 3.17 -26.0 -27.5 440.576 524.499 3840.876

A 46.7 -255 -297 0.223 -33.7 -151 465.551 1045.507 4945.981

YES B 48.6 -254 -296 0.227 -33-4 -151 452.934 1055.403 5184.495

15x15

(225) A 237 ·258 -299 0.218 -33.4 -247 452.172 1014.617 5557.921

NO

B 244 -251 -292 0.236 -32.8 -247 430-414 884.295 5096.257

dkM = zonation objective function (9); DEK= Kashyap criterion (1); S = -2In(L), Eq. (5); J= calibration objective function (6); In IFI= natural logarithm of the determinant of Fisher's information matrix; In(P(M)) = natural logarithm of the "a priori" probability that the model be correct; Ti = computed transmissivity at the zone i

• The increase of quantity and quality of data allow a better definition of dkm

reducing the ill-possedness of the problem.

• Many times, the initial zonation conditions the final geometry due to the

non-linearity of the problem of automatic estimation zones. Sun and Yeh

(1985) also detected this problem trying to solve the automatic optimum

parameterization problem.

• The inclusion of P( e E Zi) helps the process of finding the optimum zonation.


• The influence of initial zonation is reduced.

• A smaller number of alternative zonations is needed, wich reduces the

CPU cost.

• The influence of head data noise is reduced. In addition, some features

that otherwise will be missed, can be introduced in the model. This

can be very helpful in places were heads have low sensitivities to some

parameters.

• The most effective integer algorithms were the ones mixing succesive inclusions

and branch and bound or branch and bound T (this method is a modification

of branch and bound developed in this work). However, the dimension of the

problem cannot be too large. For larger dimensions, the method of succesive

exchanges should be used.

• The use of optimum Ai performed quite similar to the other case. This is

because the character of being optimum is closely related to a given model.

If the model is changed, the optimality is lost.

• One of the contributions of this work is the computation of J i , gi and Hi of any model Mi (provided that it is derived from another by modifying

transmissivity zones) in an easy and efficient way (simple matrix and vector

operations) .

• The exact computation of Ht using the adjoint state method has been another

contribution of the present work. Nevertheless, working with the first order

approximation of the hessian matrix is more stable and economic.

The obtained results invite to make a reflexion about the correctness of talking

about the optimum or right zonation. Our answer is that we should talk about

the coherence of a given zonation and the whole knowledge of the system. Such

a knowledge has a given associated uncertainty, wich will be partially reflected on the ill-possedness of the problem.


Acknowledgements This work was done under the funding of ENRESA (spanish

radioactive waste company).

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FAST REAL SPACE RENORMALIZATION FOR TWO-PHASE POROUS MEDIA FLOW

MARC R. HOFFMANN [email protected] Department of Environmental Sciences Wageningen Agricultural University Nieuwe K anaal 11 6709 PA Wageningen The Netherlands

Abstract. Upscaling of hydraulic conductivity data in space is one of the important problems in hydrological modelling.

In this article the basics of Real Space Renormalization (RSR) for upscaling hydraulic conductivity are introduced. The RSR procedure is described on a 2 - D hydraulic conductivity grid. RSR is chosen because it can cope with correlated and anisotropic media. An up to now unanswered problem is the non-unique relationship between water fluxes and water contents or heads.

In special cases analytical solutions for two phase flow systems allow RSR to be as fast as single phase RSR. In the general case, numerical solution of the small scale flow equations are required. A new solution procedure which reduces the amount of computations is described.

1. Introduction

Upscaling of hydraulic conductivity data in space is one of the important problems in hydrological modelling. For the modelling of ground water flow and the input of climate models soil hydraulic conductivity data have to be provided which match the scale of the model. Several different techniques are proposed in the literature to solve this problem. Up to know no upscaling technique gives satisfactory results in all cases and is used widely. Current state of the art are lumped parameter models. The procedures to derive these lumped parameters vary according to the goal of the modeller


84 MARC R. HOFFMANN

and the data available, e.g. an effective conductivity which reproduces the mean flow in a grid block, will normally differ from a conductivity which reproduces the mean head.

The main question is: how to estimate with reasonable precision the conductivity on larger scales from small scale data? When going from the Darcy scale to the larger model grid block scale or to a scale suitable for use in weather or climate models, a closure problem arises. In addition to the spatial scaling problem, also the time step of the model changes. This article is focused only on the spatial scaling problem.

In this presentation we introduce the basics of Real Space Renormalization (RSR) [4] and apply this to a 2 - D conductivity field. The RSR is chosen because it can cope with spatially correlated and anisotropic media.

1.1. DEFINITION OF THE PROBLEM

An important issue in modelling subsurface flow is the problem of scales. Scales refer to the spatial extent or base at which porous media properties are defined. The general two-phase flow problem is highly nonlinear due to the dependency of the hydraulic conductivity on the fluid pressure. If the porous medium is heterogeneous, additional nonlinearities arise in solving the flow equations due to the spatial variable parameters of the hydraulic conductivity-pressure relation.

Upscaling of hydraulic conductivity is a computational process, that transforms small scale information obtained by measurements and geostatistical methods to larger scale information suitable for numerical flow simulators. The small scale hydraulic conductivity information is usually given as sampled or interpolated point values or given as grid block values. In this article the small scale information is assumed given. The process of upscaling is necessary, because numerical flow simulators can't handle all small scale information. Quite often the output of the flow simulator is interesting only on larger scale. Because of uncertainty in the small scale data, there is a need for fast simulations. With different small scale input distributions Monte Carlo simulations can be done, which improve prediction certainty.

Given the distribution of the hydraulic conductivity k(h) in space we want to calculate the flux q across the boundaries of the flow domain. With the aid of numerical techniques we can calculate this fluxes. If the flow domain is large or the information on k(h) is detailed, the calculation procedure takes a lot of time. To simplify this calculation, we want to replace k(h) by an effective up scaled K, take the gradient over the entire flow domain and expect the same fluxes.

The techniques for upscaling vary from simple averaging of the heterogeneous conductivities to more sophisticated inversion and probabilistic

FAST REAL SPACE RENORMALIZATION ... 85

methods. During the last years Real Space Renormalization (RSR) has been increasingly used in upscaling conductivity from the measurement scale to the grid block scale required for numerical simulators.

RSR is a computational procedure to estimate characteristics of disordered systems. It is used in statistical physics and magneto-hydrodynamics. In the field of percolation theory it is used to describe the flow characteristics in network models of porous media. This use is binary (wet vs. dry) only, but the technique is powerful enough to derive critical scaling exponents of the networks considered, like the percolation threshold [7].

Recently RSR was also applied in the field of reservoir engineering. King e.a. [4] used it to scale up the hydraulic conductivity from the measurement scale to the flow simulator scale. Strengths and weaknesses are discussed in his publications. Also Hinrichsen e.a. [3] present RSR in the case of a fractal hydraulic conductivity distribution.

The main strength of RSR compared to effective medium approaches or other perturbation methods is that it does not rely on on the "small perturbation" expansion, basically a linear operation.

1.2. SOLUTIONS UP TO NOW

Up to now several RSR schemes exist for one-phase fluid flow [5, 3], which work quite fast and give in most cases satisfactory approximations of the actual flow. These schemes work either by replacing the finite difference grid by a modified grid, which is obtained by locally decoupling grid blocks and analytically solving the decoupled equations [4] or by removing bonds in a bond oriented network and redistributing the "lost" pore space by an averaging procedure [2].

For two-phase flow also schemes exist, but they still require a large amount of computing resources [4, 2]. As the one-phase RSR these schemes decouple the finite difference equations, but they solve the decoupled equations numerically. In general, no analytical solutions for the decoupled equations can be obtained.

1.3. OVERVIEW OF THE ANALYTICAL AND NUMERICAL APPROACH

Both the analytical and numerical approach to RSR require small scale information on the hydraulic conductivity. Both deliver an upscaled K function. In the case of the analytic approach, K is given by an explicit function. The numerical solution only gives a table, mapping the boundary conditions to the flow. This table is an implicit description of K and can be parameterised by a function.

Two-phase flow is described with Darcy's law and and a conductivity function dependent on the fluid pressure. By directly integrating the flow

86 MARC R. HOFFMANN

equation over a grid block explicit solutions are obtained. These algebraic equations are coupled to form the RSR grid and then solved analytically. By making use of these solutions two-phase RSR is nearly as fast as one-phase RSR. In contrast to earlier two-phase RSR schemes [4, 2], in which the computing costs grow with a power function with volume, in the presented scheme computing costs grow super-linear with volume. With this method steady-state upscaled conductivity-pressure relationships are found. The method handles anisotropy of the porous medium.

Further a new, simplified numerical method is described which handles arbitrary k(h) relations. It is similar to the method described by [4], but simplifies the numerical procedure by using the special boundary conditions in the RSR procedure explicitly.

2. Description of flow equations and RSR

In this article the formulation of the equations is used as in standard vadose zone hydrology. This implies two-phase flow (air and water). For the gaseous phase Richards approximation is used (high mobility of the gaseous phase). In the equations air never enters explicitly.

2.1. THE BASIC FLOW EQUATIONS

The basic equation which relates fluid flow through porous media to a pressure gradient is Darcy's law:

q = -k(h)\1h (1)

with q the flow rate, k(h) the hydraulic conductivity depending on the pressure hand \1 the gradient operator. For the hydraulic conductivity different functional forms are in widespread use, e.g. van Genuchten [8], Gardner [1] and Pullan [6]. When solving (1) numerically, the exact parameterisation of the k(h) relation doesn't matter. Analytical exact solutions are only possible in heterogeneous media when using the form given in [6]:

k(h) = koe-ah (2)

with ko the saturated hydraulic conductivity and a a spatial constant parameter.

In heterogeneous media the above equations are assumed valid on the small scale. The equations on the large scale are assumed to have the same form, just the functional form of the k(h) relation will change. The gradient operator is replaced by an average gradient.

FAST REAL SPACE RENORMALIZATION ...

2.2. REAL SPACE RENORMALIZATION

87

As an example of RSR, the (adapted) algorithm by [5] is described. The description is for a 2 by 2 block (figure 1). The main point of the renormalization procedure is that small scale blocks may be replaced by a network of resistors, using the analogy between Darcy's and Ohm's law (figure 2). Special assumptions on the boundary conditions for the small scale blocks have to be made. In our case we assume no flow boundary conditions, but periodic boundary conditions should also be possible. For the four small scale blocks the mass conservation equations are solved and the small scale k(h) functions are replaced by an "upscaled" K function. The functional form of K is different from the k(h) and has to be parameterized. Because a non-homogeneous k(h) on the local scale implies anisotropy on a larger scale, the renormalised conductivity for the larger scale will be a second order tensor. This procedure is repeatedly applied to the conductivity field up to the required scale.

Figure 1. Basic RSR geometry

Thus the following steps have to be taken for the RSR:

- Take four small scale blocks and solve for the flow - Use the flow and the boundary values hI and h2 to find the upscaled

K - Rotate the four blocks by 90 degree and repeat the above two steps to

find K in the other principal direction - Assemble four "new" small scale blocks made of the upscaled K's - Repeat the above steps up to the required scale

88 MARC R. HOFFMANN

ql q4 q7

q3

q6 q2 qs qs

Figure 2. Flows in the small scale blocks

3. Analytical approach

In order to solve the RSR on four small scale blocks, the flow equations are discretized using a standard block-centered finite difference formulation. By combining equations (1) and (2) one obtains:

(3)

Integrating the above equation over the following domain and assuming q constant:

Figure 3. 1 - D discretized domain for analytical integration

Because of continuity of the fluxes and assuming equidistance between Xl, X12, X2 we get:

2kol k02 (-ah 2 -ah1 ) q= * e -e a(X2 - Xr)(k02 + kot}

(4)

This equation is now an algebraic equation in terms of q and e-ah and can be solved for these variables.


For each flow in figure 2 equation (4) is established and a system of eight algebraic equations is obtained. The system has the form:

ql

q2 q3 q4 qs

q6 q7 qs

with:

(~x/2)-1(ksda)(HI - HL) (~x/2)-1(ks3/a)(H3 - HL) (~x)-1((2kslks3)/(a(ks3 + ks1 )))(H3 - HI) (~x)-1((2kslks2)/(a(ksl + ks2 )))(H2 - HI) (~x)-1((2ks3ks4)/(a(ks3 + ks4 )))(H4 - H3) (~x)-1((2ks2ks4)/(a(ks2 + ks4)))(H4 - H2) (~x/2)-1(ks2/a)(HR - H2) (~x/2)-1(ks4/a)(HR - H4)

HI e-ah1

H2 _ e-ah2

H3 e-ahg

H4 HL HR -

Additionally we can use the conservation of mass in the nodes:

ql - q3 + q4 q4 q6 + q7

q2 qs - q3

(5)

(6)

(7)

The above set set of equations can be solved analytically. With this solution the large scale K is determined.

4. Numerical approach

When using a different functional form for k(h} than above, numerical solutions to the RSR have to be found. In principle this is easy. What counts is the amount of computations. A straight forward implementation can be formulated by using a standard finite difference scheme. This leads to a matrix equation of the following form:

Ah=b (8)

90 MARC R. HOFFMANN

with A a four by four matrix, h a vector with the four small scale h's and b another four element vector.

By numerically solving the set of FD equations on the small scale grid, one obtains upscaled k(h) functions. Even ifthe small scale k's are isotropic, the up scaled ones will be anisotropic and non-symmetric in V h. Previous studies [4] used a straight forward FD implementation to solve the small scale equations. This approach still requires a numerical intensive iterative procedure to solve the equations.

By carefully analyzing the geometry and FD discretization, a simplification of the mass balance equations occurs. The given head boundary conditions and the geometry inside the four small scale blocks constrain the values the small scale h's can take. These h's are not independent, but are related to each other by:

(9)

Each occurrence of e.g. h4 in the mass conservation equations can directly be replaced by:

(10)

The new equation (10) reduces the four above mass conservation equations (7) to only three. The new equations written in matrix form and suitable for a numerical implementation have the form:

(11)

This system can be solved in fewer steps than the straight forward FD implementation. Further reduction in the computations is possible by rewriting the iterations to implicitly include previous iteration steps.

5. Discussion

In this article the basics of RSR are described and a new analytical and numerical technique is developed. Especially the new numerical technique can help in practical up scaling problems. This is because of the flexible parameterisation of the k (h) relationship. A further step in the development will be the testing of this new technique with a standard flow simulator.

Still open questions are:

- It is still open which equation or functional form to use for the up scaled k(h) relation.


- Up to now I looked only at the first renormalization step. In the next renormalization step four already upscaled k(h) functions are combined. Because of their different form from the original small scale description, this requires a different solution strategy.

- Still open is also the problem of time varying fluxes. Up to now the upscaling only deals with steady state fluxes.

Despite these open questions RSR holds promise for practical upscaling, especially in two-phase flow. In two phase flow there is nearly always a trend in the hydraulic conductivity because of a gradient in the fluid pressure. This also causes spatial correlation in the conductivity field. These two problems exclude simple stochastic techniques and defy up to now the solution by advanced stochastic techniques. In these cases RSR is a strong candidate for the upscaling.

Acknowledgements

Thanks to R. M. Stallman, D. E. Knuth and L. Torvalds for providing this project with the necessary software.

References

1. W. R. Gardner. Some Steady-State Solutions of the Unsaturated Moisture Flow Equation with Application to Evaporation from a Water Table. Soil Science, 85( 4):228 - 232, Apr. 1958.

2. A. Hansen, S. Roux, A. Aharony, J. Feder, T. J0ssang, and H. H. Hardy. RealSpace Renormalization Estimates for Two-Phase Flow in Porous Media. Transport in Porous Media, 29(3):247 - 279, Dec. 1997.

3. E. L. Hinrichsen, A. Aharony, J. Feder, A. Hansen, T. J0ssang, and H. H. Hardy. A Fast Algorithm for Estimating Large-Scale Permeabilities of Correlated Anisotropic Media. Transport in Porous Media, 12(1):55 - 72, July 1993.

4. P. King, A. Muggeridge, and W. Price. Renormalization Calculations ofImmiscible Flow. Transport in Porous Media, 12(3):237 - 260, Sept. 1993.

5. P. R. King. The Use of Renormalization for Calculating Effective Permeability. Transport in Porous Media, 4(1):37 - 58, Feb. 1989.

6. A. J. Pullan. The Quasilinear Approximation for Unsaturated Porous Media Flow. Water Resources Research, 26(6):1219 - 1234, June 1990.

7. M. Sahimi. Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches. VCH, Weinheim, New York, Basel, Cambridge, Tokyo, 1995.

8. M. T. van Genuchten. A Closed-form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils. Soil Sci. Soc. Amer. 1., 44:892 - 898, 1980.

SOLUTION OF CONVECTION-DIFFUSION PROBLEMS WITH THE MEMORY TERMS

J. KACUR, Department of Numerical Analysis, Faculty of Mathematics and Physics, Comenius University, 842 15 Bratislava, Slovakia

1. Introd uction

In this paper 1 an approximation solution of the following convection diffusion problem is discussed

8t b(u) + div(F(t, x, u) - k(t, x, u)\7u) = f(t, x, u, s), s(t, x) = J~ K(t, z)7jJ(u(z, x))dz

in (0, T) x 0, (1)

where 0 C ~N is a bounded domain with a Lipschitz continuous boundary 80, T < 00.

We consider the mixed boundary conditions

u = 0 on I x r l ,

-k(t, x, u)\7u . v = g(t, x, u) on I x r 2 (2)

where I == (O,T), r 1,r2 C 8n, r 1 n r 2 = 0 and mesN-1r1 + mesN-1r 2 = mesN_1aO. Together with (1), (2) we consider the initial condition

b(u(O, x)) = b(uo(x)) in O. (3)

We assume that b(s) is strictly increasing in s, F(t, x, s) is Lipschitz continuous in x, sand f, g, 7jJ are sub linear in u. Problem (1)-(3) has been studied in [9] for a special case when f == f(t, x, u), i.e., without memory term.

As an example we present a model of contaminant transport in porous media intensively studied in the last years, see [1, 3, 4, 5, 10, 11, 12]

8t (8C + pS) + div(vC - D\7C) = 0 p8t S = d(7jJ(C) - S) (E)

where C is the concentration of the contaminant, v is (Darcy) velocity field of water, D is diffusion tensor, p is bulk density, 7jJ is sorption isotherm of the porous

lThis research was partially supported by scientific grant VEGA and by grant 201/97/0153 of Grant Agency of Czech Republic.

93 J.M. Crolet (ed.), Computational Methodsfor Flow and Transport in Porous Media, 93-lO6. © 2000 Kluwer Academic Publishers.

94 J. KACUR

media with porosity O. Here 8 is the mass of contaminant adsorbed by unit mass of porous medium. Coefficient d describes the rate of adsorption. If d -+ 00 then equilibrium sorption process occurs and consequently 8 = 'ljJ( C). Then b(s) = Os + p'ljJ(s) generates the parabolic term in (1) with f == O. Moreover, when 'ljJ(s) = esP, 0 < P (so called Freundlich isotherm) then b'(O) = 00 for p < 1 which occurs in most practical situations. In that case (1) is of porous media type with convective term and the support of contaminant develops with the finite speed. In the non equilibrium case (d < 00) we can eliminate 8 from ODE and we obtain

b(z) == z, f(t, x, u, s) == s - d'ljJ(u)

and

s = da lot e-a(t-z) 'ljJ(u(z, x))dz . h d WIt a = -

p (8(0) = 0) .

In the case of Freundlich isotherm with p < 1 the function 'ljJ is not Lipschitz continuous.

Numerical analysis of the model with the equilibrium sorption process (i.e. s == 0) is included in our previous paper [9]. The contribution of the present paper is the numerical analysis of the mathematical model (1) which includes both equilibrium and nonequilibrium sorption process in (E). The degeneracy b' = 0,00 in some points is included and thus convective term can be strongly dominant. Numerical solution of (1)-(3) thus represents a delicate problem. We extend our concept of approximation introduced in [9] for the case s == O. We prove the convergence of the approximate solution. The existence and uniqueness of the variational solution is discussed in [12] where F(t, x, u) == v(x)u.

Our concept of approximation is based on the relaxation schemes developed by W. Jager and J. Kacur in [6, 7] and on the method of characteristics initiated by O. Pironneau [14] and J. Douglas and T. Russel [2].

2. A pproximation scheme

The transport part of (1) is of the form

8t b(u) + divF(t,x,u) = 0

which formally we can rewrite into the form

8 F~(t, x, u) . V' __ divxF(t, x, u) t U + b'(u) u - b'(u)

and hence the corresponding velocity field v = F~b~t('~t) is depending on the unknown u and cannot be expected to be smooth. Thus we regularize b by bn

SOLUTION OF CONVECTION-DIFFUSION PROBLEMS 95

with the properties listed below. The transport part with the velocity field v can be realized by means of characteristics X (s; t, x) governed by ODE

dX ds = veX, s); X(t; t, x) = x.

Their Euler backwards type approximation between time levels t = ti, t = ti+1 is given by <pHI (x) = x- (ti+! -ti)V(X, ti+!). Then concentration profile Ui(X) at t = ti after the transport along approximated characteristics prolonging ti+! - ti became Ui 0 <pi+!o The transport can be realized if characteristics X or their approximations <p do not intersect each other which requires the boundedness of IV'vioo :S c and small time step T = tHI -ti. In our case we cannot guarantee it. As it was proven in [9] a smoothing (or averaging) of velocity field can guarantee that the corresponding characteristic will not intersect provided the time step T = tH I - ti is small. Applying the method of characteristics the points <pH I (x) can cross the boundary 80 and in such case we understand by Ui 0 <pi := Ui 0 <pi where Ui is an extension ofui E Wi(O) to Ui E Wi(o*) with 0* ::J n so that Iluillw~(n.) :S Iluillw~(n)·

We realize smoothing of v by convolution Wh *v where Wh is standard mollifier

with Wh(X) = ,!wWI(~) where WI(X) = Kexp(l~pl~I)' fRNwIdx = 1.

We consider nonstandard time discretization of (1) with time step T == Tn = Tin (n E N) and Ui is an approximation of u(x, ti) at time level ti = T.i, i = 1, ... , n. We have to determine Ui from linear elliptic equation coupled with a relaxation parameter 0 < Ai E Loo (0)

Ai( ui - Ui-I 0 <P~J - div(ki V'ud = T Hi + T f(ti, x, ui-I Si) (4)

Ui=O onrl , -kiV'Ui·V=gi==g(ti,ui-d onr2 ,

where

Si = L~:~ Ctij1f;(Uj)T, Ctij = ~ ftt:_l K(ti,s)ds, Hi := divxF(x,ti,ui-d,

<p~(x) := x - TWh * (F~(t;,:,Ui_l ) with 0 < v E Loo(O), h = TW , wE (0,1)

and the following" convergence conditions" (5), (6) have to be satisfied

where

(II ·110 is L2 norm and bn is a regularization of b) and

bn(Ui) - bn(Ui-1 0 <P~J Ai - .

Ui - ui-I 0 <P~; < T i3 ,

o f3 E (0,1).

(5)

(6)

96 J. KACUR

The scheme is implicit and to guarantee (5), (6) we propose the iterations

Ai,k-l(Ui,k - Ui-l 0 <P~J - Tdiv(kiV'ui,k) = THi + Tfi (7k)

These iterations are not coupled with (5). If

IIAi,ko - Ai,ko-ilia < T{3

then we put Ui := Ui,ko' Ai := Ai,ko-l'

To obtain <p~;11 we propose fixed point type iterations

/Li,l = Gi+l(/Li,l-d, l = 1, ...

and when II/Li,lo - /Li,lo-ilia < TO. then we put /Li+l := /Li,lo-l and obtain (5) (with i + 1 in the place of i). Then we continue (4) on the next time level t = ti+l.

3. Assumptions and convergence of (8z)

By c we denote generic positive constants. We shall assume

(Hd b is increasing, absolutely continuous function satisfying b(O) = O. We assume that there exist bn E C1(IR), bn(O) = 0 (T = Tn = Tin) with b~(s) locally Lipschitz continuous such that:

(i) bn(s) ---+ b(s) locally uniformly;

(ii) cn-d S b~(s) S cn'Y Vs E 1R; d, 'Y E (0,1);

(iii) sUPlzl<K Ibn(z)1 S c(K) < 00 0 < K < 00;

(iv) min{b'(s),c:} S cb~(s) for some c > 0;

(v) Ib~(s)1 S cnP, p E (0,1);

(H2 ) k(t,x,s) : I x 0 x IR ---+ IRNxN is continuous and

(H3) F(t, x, s), F~ == BsF(t, x, s) : I x 0 X 1R1 ---+ IRN are continuous and

IBsFI S c, IB;FI S c, IBxF(t, x, 8)1 + IBxF~(t, x, s)1 S c(L(t, x) + lsI) for a.e. (t, x) E QT == I x 0, s E IR .

We also assume that F(t,x,s) can be extended to 0* :J (2 so that the estimates hold true for x E 0* and L E L 2 (0* x 1);


(H4) f(t,x,s,'f}), g(t,x,s) are continuous in their variables and

If(t, x, s, 'f})1 ~ c(1 + lsi + l'f}l), Ig(t, x, s)1 ~ c(1 + lsi)

'l/J(s) : R -+ JR is continuous and I'l/J(s) I ~ c(1 + lsi); K(t, s) E Loo(I x 1) .

(H5) Uo E Wi (0) n Loo(O) .

We denote the standard functional spaces by L2 = L2 (0), Loo(O), V = {v E Wi(O);v = 0 on rd, L 2 (1, V) - see [13]. By 11·110,11·1100,11·11, 1I·lIr 2 we denote the norms in L2, Loo(O), Wi(O), L2(r2), respectively. We denote by (u, v) = In uvdx, (u, v)r2 = Ir2 uvdx and V* the dual space to V. In the sequel we drop the variable x in the terms k, f, g, F, A, J1-.

We use the concept of variational solution. Let < u,v > represents the duality between V* and V.

Definition g. u E L 2 (1, V) is a variational solution of (1)-(3) iff

(i) b(u) E Loo (I,L1(0), Otb(u) E L2(I, V*);

(ii) I[ < Otb(u),v > + I[(divF(t,u),v > + I[(k(t,u)\1u, \1v)+ + I[(g(t,u),v)r2 = I[(f(t,u,s),v) 'Iv E L2 (I, V) , s(x, t) = I~ K(t, s)'l/J(u(x, s))ds a.e. (t, s) E 1 x 0;

(iii) I[ < Otb(u), v >= I[(b(u) - b(uo), Otv) 'Iv E L2(1, V) n Loo(I x 0), OtV E Loo(QT), v(T) = 0

Also Ui E V in (4) is a variational solution of (4)

(Ai(Ui - Ui-1 0 <p~.),v) + T(ki\1ui, \1vd + T(gi,V)r2 = = T(Hi' v) + T(f(ti' Ui-1, Si) "Iv E V

(10)

If Ui-1, Ui-1 0 <P~i E L2 then the existence of Ui E V in (10) is guaranteed by Lax-Milgram lemma. To obtain the a priori estimates for Ui and to prove that Ui-1 0 <pi E L2 the crucial role place the estimate (see [9J Lemma 11)

1 . . 2"lx - yl ~ l<p~i (x) - <P~i (y)1 ~ 21x - yl (11)

uniformly for i = 1, ... , n, provided J1-i = Gi(l/) for any 0 < 1/ E Loo(O) and w+d < 1, T ~ TO. Then Ui-1 O<P~i E L2 provided Ui-1 E L2. Hence {udi=l E V satisfying (10) is guaranteed.

In the following we shall assume that

lIuilioo ~ c uniformly for n, i = 1, ... , n (12)

without any structural restrictions on b, F, g, f. If f == f(t, x, u) is not dependent on s (Le. the memory term is not considered in (1)) then (12) has been proved in [9], Lemma 15 under the following structural restrictions:

98 J. KACUR

(i) f,g,divxF(t,x,s) == 0; or

(ii) 9 == 0, Ai ~ q > 0, L E Loo(I x 0) in H3 ); or

(iii) f(t,x,s)s ~ 0, g(t,x,c)s ~ 0, H(t,x,s)s ~ 0

(in the case (iii) li == f(ti, x, Ui), gi = g(ti' x, Ui), Hi = H(ti, x, Ui) in (10)).

In our situation the result (12) hold true in the case {ii}. Indeed as in [9] we obtain estimate

where we estimate Hi, fi in terms of Ilujlloo using (H3), (H4)' Then we obtain

Iluili oo S; Il ui-lll oo + CT (1Iui-11100 + L~:'~ Iluj 1100 T + 1) Iluili oo S; (1 + CT) Il u i-lll oo + CT L~:'~ Iluj 1100 T + CT S;

S; (1 + CT)i (Cl + C2 L~:'~ Ilujlloo T)

since (1 + CT)i S; ecT , Gronwall argument implies the estimate (12).

The convergence of iterations (8l) has been proved in [9] under the assumptions:

lI\7uillo:S c, 'lin, i = 1, ... ,n and p+d+ ~ w < 1, T:S TO

where N =dim O.

Remark 13 The convergence of iterations in {7k} has been analysed in {8}.

Remark 14 The regularization bn(s) of b(s) in (Hd is not so much restrictive with respect to asymptotic behaviour of b~, b~. For example, let us consider b(s) = IslPsgn s, p E (0,1). We can take for bn(s)

b (s) _ { (s + n-O)P - n-PO for SI ~ S ~ 0 n - -Is - n-olP + n-PO for - SI < s :S 0

and bn(s) = bn(sd + b~(SI)(S - sI) for s ~ SI and similarly for s S; -SI' Then we have Cln-(I-p)O S; b~(s) S; C2n(l-p)o {i.e. d = 'Y = (1 - p)8 for any 8 > O) and b~(s) S; cnl with p = (2 - p)8. We can verify easily {i}- -{v} in HI)'

4. Convergence of the method

To obtain a priori estimates for {ui}i=1 we follow [9] and only sketch the additional terms on R.H.S. in (10) concerning memory. We obtain


Lemma 15 Under the assumptions (H1 )-(H5 ), {12} and w+d < 1, ,),+2d+p < W, a > d the following a priori estimates hold true

uniformly for u, where Bn(s) := bn(s)s - J; b(z)dz.

SKETCH OF THE PROOF. We put v = Ui into (10) and sum it up for i = 1, ... ,j. Similarly as in [9] we obtain

j 2 maXl::;j::;i In Bn(Uj)dx + 2:i=l Iluill T:::;

:::; C (E + ;T2w-2oy-4d-2p) 2:1=1 Il u il1 2 T + c€ + C 2:1=1 2:~=1 Ilukll~ T,

where the last term arises in estimation of memory terms. Since Iluilio < C Iluilioo :::; c we estimate the last term by a constant. Then we take E sufficiently small and consequently T :::; TO(E) we obtain the required a priori estimate.

By means of {ui}i=l we construct Rothe's functions

un(t) := Ui

un(t) := Ui-1 + t-~-l (Ui - ui-d fortE (ti-1,ti>, i=I, ... ,n

with un(O) = Uo.

Lemma 16 {un} is compact in Ls(Ix 0) Vs> 1, i.e. there exists U E Loo(Ix 0) and {ii} C {n} such that un -+ U in Ls(I x 0).

The proof is based on the a priori estimate

n-k L(bn(Ui+k) - bn(Ui),Ui+k - UdT:::; CkT i=l

which can be obtained in the same way as in [9] estimating the memory terms using (12). This a priori estimates can be rewritten in the form

foT-

Z (bn(un(t + z) - bn(un(t)), un(t + z) - un(t))dt :::; cz

uniformly for z E (0, zo). From this and the estimate JoT Ilun (t)11 2 dt :::; c (see Lemma 15) we deduce that un(t, x) -+ u(t, x) a.e. in I x 0 because b is strictly increasing and because of the regularization properties of bn - see [7]. Then (12) implies Ls convergence.

As a consequence we obtain U E L2(I, V) and un ->. U in L2(I, V).

Now we can prove our main result.

100 J. KACUR

Theorem 17 Let the assumptions (HI) -(H5) , (12) and 0: > d, 'Y + 2d + P < w be satisfied. Then un -t u in Ls(I x 0) Vs> 1 and un ->. U inL2(I, V) where un is from (4)-(6) and U is a variational solution of (1). If the variational solution u is unique then the original sequence {un} is convergent.

SKETCH OF THE PROOF. We rewrite (10) in the form

(bn(Ui) - bn(Ui-d,v) = -(bn(Ui-d - bn(Ui-1 0 'Pti)'V)+

+TtJ(t\;i(Ui - Ui-I 0 'PtJ,v) - T(ki'VUi, 'VV) - T(gi,v)r 2 + (13)

+T(Hi'V) + T(1i, v) Vv E V

where IIt\;illo ~ 1. Then we consider v E L2(I, V)ncl(I, COO(O)) with v(x, t) = 0 for t in a neighbourhood of T and put it into (13). Then we integrate it over I and denote the corresponding terms by JI,n - h,n. Similarly as in [9] we obtain

JI,n -t -(b(uo),v(O)) - j(b(u),8tV)dt for n -t 00

J2 ,n -t j(P~(t,u). 'Vu,v)dt for n -t 00

where we have rearranged

( ) ( i) _ bn(Ui-l)-bn(Ui-lDip~.) ( i ) bn Ui-I - bn Ui-I 0 'PI" - u. -u' Dip' • Ui-I - Ui-I 0 'PI"

I .-1 .-1 P.i I

Ui-l - Ui-l 0 'P~i = J~ 'VUi-1 (x + s(cjP - x))ds [Fi + (Wh * Fi - Fi)]

with Fi := P;(ti~~i-l). Here we use also In -t 1, Mn -t 0 in L2(I x 0) for n -t 00 where

for t E (ti-I,ti), i = 1, ... ,n. Similarly we obtain

for n -t 00 •

Easily we deduce

J4 ,n -t j (k(t, u)'Vu, 'Vv)dt for n -t 00

and

for n -t 00 .


For the last term we have to use the following facts

sn(x, t) = I~-T K(t, s)7jJ(u~(s))ds --+ I~ K(t, s)7jJ(u(s))ds

a.e. in (t, x) E 1 x nand Ilsnlloo ~ c \In.

Hence as a special case we obtain

sn --+ s == lot K(t, z)7jJ(u(z))dz

and consequently

h,n --+ jU(t,u,s),V)dt.

Then we take the limit n --+ 00 in

and obtain

- II(b(u),Otv)dt - (b(uo),v(O)) = - II(divF(t,x,u),v)dt-

- II (k(t, u)V'u, V'v)dt + II(g(t, u), v)r2dt + IIU(t, u, s), v)dt

where we have added together terms hand J6 . Hence we deduce that there exists Otb(u) E L 2 (/, V*) similarly as in [9]. Then we conclude that u is a variational solution of (1)-(3).

The uniqueness of variational solution has been studied in [12].

The more strong convergence results we obtain under the regularity assumptions on b.

Theorem 19 Let the assumptions of Theorem 17 be satisfied. Suppose that (5), (6) are satisfied with the norm 11.1100 in the place of 11.110 and let 0 < c :::; b'(s) :::; M < 00 a.e. in ~ and d = 'Y = 0 in (Hd. Then un --+ u in L 2 (/, V).

The proof of Theorem 19 is the same as that one in [9] (Theorem 48) and the presence of the memory term represents no substantial difficulties.

5. Numerical implementation

The numerical implementation of (4)-(6) is rather costly also without the presence of memory term - see [9]. The additional difficulties arise including the

102 J. KACUR

memory terms since at the time level t = ti we need for evaluation of Si the values of Uj for all j = 0, ... ,i -1. In the numerical realization of (4) we project it into finite dimensional space VA C V using FEM. We assume that VA ~ V for ..\ ~ 0 in canonical sense (..\ being the discretization parameter). Then instead of Ui E V we obtain u~ E VA as a solution of projected equation (4). Then by means of {U;}r=l we construct Rothe's function U a (a = (7,..\)) as our approximate solution. For the convergence U a ~ U for a ~ 0 we obtain the same results as in Theorems 17, 19. In the projected problem (4) we assume that u~ is a projection of Uo E W21 into VA and that u~ ~ Uo in L2(O). The evaluation of Ui-l 0 'Pi is the most costly from the numerical point of view. As an alternative to the standard back tracing we can use the following procedure. On each time level ti we construct a new basis {'ljIH .7=\i which we obtain by shifting along characteristics of the basis {'ljI ;-1 } .7~'i i -1 corresponding to t = ti-1. The new basis elements are locally completed by new elements or locally reduced with respect to the density of the grid points which is changing by means of characteristics. This process is simply realizable when, e.g., piecewise linear elements are used. Then, in the place of back tracing in Ui-l 0 'P~i we obtain immediately the values in nodal points for the new basis on time level ti. Additional treatment needs the extension of Ui-1 outside 0 (using boundary conditions).

We shall discuss now the treatment of memory terms. For realistic contaminant transport problem (E) with sorption isotherms 'ljI( C) (e.g. 'ljI( C) = kl CP for Freunlich type, 'ljI(C) = k2~faC for Langmuir type etc.) we can express

S(t) = S(O) e-at +a lot e-a(t-s) 'ljI(C(s))ds

and when we insert ats into transport equation we obtain a memory term with f(t,x,u,s) == s - 'ljI(u) and

s = ad lot e-a(t-s) 'ljI(C(s))ds, (K(t, s) == e-a(t-s») provided 5(0) = 0 .

In that case we can verify that aiH,j = e-ar ai,j since ai,j = ~ ft:j_1 e-a(ti- S ) ds and

Si+l = e-ar Si + ai+1,i'ljl(Ui) for i = 1, ... , n .

As a consequence we do not need to store the values of Uj (j = 1, ... , i-I) for evaluation of Si.

Example 1. We apply the proposed method in numerical solution for the problem (E) with equilibrium sorption isotherm i.e. d ~ 00 which implies 5 = 'IjJ(C) == ",CP. Then (E) with specific data (in 1D) reduces to

at (~U + 1.5UP) + ax (3u - O.05ax u) = 0 .


We shall consider p = 1.2, 0.8, 0.6, 004, n = (0,100), T = 30 and the Dirichlet boundary condition

(i) u(O, t) = 1

(ii) u(O, t) = ° with the corresponding initial conditions

(i) u(x,O) = u6(x)

(ii) u(x,O) = u6(x) .

Here ub(x) (i = 1,2) are piecewise linearfunctions ofthe following form: uA(x) = 1 for x E (0,0.1), u6(0.2) = 0, uA(x) = ° for x > 0.2; u6(x) = ° for x E (0,100)\(0.1,0.5), u6(0.2) = u6(OA) = 1. The solutions are drawn in three time moments for various p (p = 1.2 - dash-dash-dotted line, p = 1.0 - full line, p = 0.8 - dash-dotted line, p = 0.6 - dashed line, p = 004 - dotted line) in Fig. 1 for t = 2, in Fig. 2 for t = 4 and in Fig. 3 for t = 6 in the case (i). The case (ii) is drawn in Fig. 4 for t = 1, in Fig. 5 for t = 2 and in Fig. 6 for t = 6.

o 2 4 6 8 10

FIG. 1

IF===~======~~,,~~~~~~~~~~~~~--~~~

" 0.8

0.6

0.4

0.2

2 4 6 8 10 12 14

FIG. 2

104 J. KACUR

1~~~~,~.,~~~~-'~--~~~--~--~~~--~--~~~

0.8

0.6

0.4

0.2

6

0.5

0.4

0.3

0.2

0.1

0

0

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

0

,

"' \ \ \

"'

8

\

"' \ "' '.

2

10

1

12 14 16 18 20

FIG. 3

.... -.-

2 3 4 5

FIG. 4

\

'. \ '. \ '.

\ '.

\ \ , ,

' ... '" --0-

4 6 8 10

FIG. 5


0.2 ;\ , \

\ \ 0.15 '.

/\ \ ! i \

! i \ / ! \,

; ! ., ' /' .'. "" -' I.' !

..... "./ l// 1 ' ..............

0.1 0.05

•••••• ;,-::.:.~ - ( .-._ ..... • .... 1 - ..... _-. __ • __ • __

0c=~==~====================~======~=J o 5 10 15 20

FIG. 6 Acknowledgement. I want to express my thanks to D.Kostecky for his help

with numerical experiments.

References

[lJ C. N. Dawson, C. J. Van Duijn and R. E. Grundy: Large time asymptotics in contaminant transport in porous media. SIAM J. Appl. Math. Vol. 56, N4, (1996), pp. 965-993.

[2J J. Douglas, T. F. Russel: Numerical methods for convection dominated diffusion problems based on combining the method of the characteristics with finite elements or finite differences. SIAM J. Numer. Annal. 19 (1982), pp. 871-885.

[3] C. J. Van Duijn, P. Knabner: Solute transport in porous media with equilibrium and non-equilibrium multiple-site adsorption: Traveling waves. J. Reine Angewandte Math., 415, (1991), pp. 1-49.

[4] C. J. Van Duijn, P. Knabner: Transport in porous media 8 (1992), pp. 167-226.

[5J R.E. Grundy, C.J. Van Duijn:Asymptotic profiles with finite mass in onedimensional contaminant transport through porous media: The fast reaction case. Q. J Mech. appl. Math., Vol. 47, pp. 69-106

[6J W. Jager, J. Kacur: Solution of porous medium systems by linear approximation scheme. Num.Math. 60, pp. 407-427 (1991).

[7J W. Jager, J. Kacur: Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes. M2 AN Mathematical modelling and numerical analysis Vol. 29, N5, pp. 605-627 (1995).

106 J. KACUR

[8] J. Kacur: Solution to strongly nonlinear parabolic problems by a linear approximation scheme. Mathematics Preprint No. IV-Ml-96, Comenius University Faculty of Mathematics and Physics, (1996), pp. 1- 26, appear in IMA J. Num.

[9] J. Kacur: Solution of degenerate convection- diffusion problems by the method of characteristics, to appear

[10] P. Knabner: Meth. Verf. Math. Phys.,36 (1991)

[11] P. Knabner: Finite-Element-Approximation of Solute Transport in Porous Media with General Adsorption Processes. "Flow and Transport in Porous Media" Ed. Xiao Shutie, Summer school, Beijing, 8-26 August 1988, World Scientific (1992), pp. 223-292.

[12] P. Knabner, F. Otto: Solute transport in porous media with equilibrium and non-equilibrium multiple-site adsorption: Uniqueness of the solution, to appear

[13] A. Kufner, O. John, S. Fucik: Function spaces. Noordhoff, Leiden, 1977.

[14] O. Pironneau: On the transport-diffusion algorithm and its applications to the Navier-Stokes equations. Numer. Math., 38 (1982), pp. 309-332.

OPTIMAL CONTROL APPROACH FOR A FLOW IN UNSATURATED POROUS MEDIA

C.M. MUREA University of Bucharest, Faculty of Mathematics, 14, str. Academiei, 70109 Bucharest, Romania, e-mail: [email protected]

AND

J.-M. CROLET Universite de Franche-Comte, Equipe de Calcul Scientijique, 16, route de Gray, 25030 Besan90n Cedex, France, e-mail: [email protected]

Abstract. The aim of this paper, dealing with the management of fresh water, is to present an optimal control approach for the steady flow in a rectangular aquifer there are two wells. The classical problem is a free boundary problem. After a change of variable transformation, we obtain an optimal control problem in a fixed domain, where the control appears in a Dirichlet boundary condition and in the coefficients of the state equation. After a finite element discretization, we obtain an optimization problem where the cost function is differentiable and the gradient could be computed analytically.

1. Introd uction and notations

The management of fresh water is a very important problem of our actual society. The typical example can be described by an aquifer of large size with several wells of small size in which the piezometric level evolves in time with pumping. The wish is to know the upper boundary of this aquifer. This problem is crucial if the aquifer is locally polluted in a well known area because in this case, the pumping well has to be correctly chosen.

Such a problem has already be solved with the software EOS which is based on a finite element method ([5] or [4]). But it was observed that the

107 J.M. Crolet (ed.), Computational Methodsfor Flow and Transport in Porous Media, 107-114. © 2000 Kluwer Academic Publishers.

108 C.M. MUREA AND J.-M. CROLET

computational time was too important: in fact, in the method used by EOS, the situation of the unconfined aquifer was obtained by an iterative process on confined aquifers with remeshing at regular time steps. This study has been pursued in order to get a faster algorithm.

The real case is a tree-dimensional situation with many wells. In order to simplify the presentation of the method, only a bidimensional case with two wells is considered (cf figure 1).

D c

dry region

H E --r..:...:··c:..:··:..:..··:...:...·~······································· ......... .

water G saturated region seepage

F

water A B

Figure 1. The geometrical configuration

The right and left boundary represent the wells. A very strong assumption is made: the domain is divided in two regions (dry and saturated): there is no unsaturated area in the aquifer. The reason of this assumption will be given later. It can be noted that such a general framework can also describes the situation of a dam.

OPTIMAL CONTROL IN UNSATURATED MEDIA 109

We denote by V = [ABeD] the rectangular open domain of the porous media. The levels of the water on the left and right sides of the dam are hI and h2 respectively. We assume that hI > h2.

The coordinates of the significant points are A(O,O), B(L,O), E(O, hI), F(L, h2), G(L, h3) and H(L, hI). We suppose that hI ~ h3 ~ h2.

The boundary [AB] is impermeable and the boundary [FG] is the seepage face. The saturated region is bounded by the boundaries [EA], [AB], [BF], [FG] and [GE].

The boundary [EG], which separates the saturated region from the dry region is unknown.

The problem is to find the boundary [EG] and the pressure of the water in the saturated region.

We assume that the boundary [EG] is the graph of the function

u : [0, L] --t JR

and we denote

The saturated region is denoted by au' Let us denote by p the pressure in the saturated region and by z the

piezometric head given by the equality

z: au --t JR, z (Xl, X2) = P (Xl, X2) + X2·

This condition can be written only if there is no unsaturated area in the aquifer.

We assume that all the constants concerning the porous media are equal to one.

The classical equations for the flow in the saturated region are

~z (Xl, X2) ° on au (1)

Z(XI,X2) hI on [AE] (2)

Z(XI,X2) h2 on [BF] (3)

Z (Xl, X2) X2 on [FG] (4) oz a (XI,X2) X2

° on [AB] (5)

oZ on (Xl, X2) ° on ru (6)

Z(XI,X2) X2 on r u (7)


where n is the outward unit normal to r u.

After the Baiocchi transformation (see [1]), i.e.

for (Xl, X2) E nu ,

for (Xl, X2) E V \ nu ,

this problem can be reformulated as a variational inequality: find y E HI (V) such that

Iv V'yV'(y-v}dx+ Iv(y-v)dx<s:o, VvEK

where K = {v E HI (V); V = 9 on fJD, v ~ ° in V} .

A proof of the existence and uniqueness for the solution of this variational inequality could be found in [2, p. 55J.

The aim of this paper is to present an optimal control approach for the system of equations (1)-(7). Some numerical aspects will be discussed.

2. Optimal control setting

2.1. SHAPE DESIGN FORMULATION

Let u E Cl ([0, L)) be given. The variational formulation for the equations (1)-(6) is the following:

Find

Zu E HI (nu) (8)

Zu hI on [AE] (9)

Zu h2 on [BF] (10)

Zu X2 on [FGJ (11)

such that

{ V'zu V'¢dx = 0, V¢ E HI (nu) ,¢ = ° on [AE] U [BF] U [FG]. (12) Jnu

The system of equations (1)-(7) can be reformulated as a shape design problem.


Find ru (or equivalent, find u in C1 ([O,L])), such that zu, the solution of the variational problem (8)-(12), verifies the equality

Zu = X2 on ru· (13)

2.2. CHANGE OF VARIABLE TRANSFORMATION

Let us denote 0 0 = ]0, L[ x ]0, hI[ .

For each u E C1 ([0, L]) given, let us consider the following one-to-one continuous differentiable transformation:

Tu : 0 0 ---+ Ou,

(Xl,X2) M Tu (X1,X2) = (Xl, (1 + U~l)) X2),

which admits the continuous differentiable inverse mapping

T;;l : Ou ---+ 0 0 ,

-1 _ ( hlX2) (Xl, X2) M Tu (Xl, X2) - Xl, hI + U (xt) .

We have

and we denote G' , P'.

2.3. OPTIMAL CONTROL APPROACH

In order to pose the variational formulation in the reference configuration let us consider the form

defined by

a;(u,v,w)

Using the change of variable transformation above, we obtain from (8)(12) the following variational formulation:

112

Find

such that

C.M. MUREA AND J.-M. CROLET

Zu E HI (no)

Zu hI on [AE] ZU h2 on [BF']

Zu (1 + U ~~)) X2 on [F'G']

(14)

(15) (16)

(17)

a (u, zu, i» = 0, vi> E HI (no), i> = 0 on [AE] U [BF'] U [F'G']. (18)

The problem (8)-(13) can be rewritten as following: Find U in CI ([0, L]), such that zu, the solution of the variational problem

(14)-(18), verifies the equality

(19)

This is an exact controllability model. If we treat the equality (19) by the Least Square Method, we obtain the optimal control problem:

(20)

subject to conditions (14)-(18). We observe that the control U appears in a Dirichlet boundary condition

(17) and in the coefficients of the equation (18). The existence of this kind of problem was studied in [6] for an external

flow and in [7] for an internal flow.

3. N umedcal aspects

3.1. FINITE ELEMENT DISCRETIZATION

We use the Hermite finite element for the approximation Uh of U. If Xo = 0, ... ,Xn = L is a discretization of the interval [0, L], the degrees of freedom are:

Uh (xt), ... ,Uh (Xn-I) , Uk (Xo) , Uk (xt), ... ,uk (Xn-d, Uk (xn).

We know that Uh (xo) = 0 and Uh (xn) = h3 - hI' For the approximation Zh of Z, we can use the standard finite elements

triangle or rectangle. Remeshing is not necessary. The discretizations for Uh and respectively for Zh don't depend one on

the other one.


3.2. THE FINITE-DIMENSIONAL OPTIMIZATION PROBLEM

We denote by Vh the finite element space for Zh. After the finite element discretization, the control optimal problem becomes the finite dimensional optimization problem below.

subject to:

and

Zh(Uh) E

Zh (Uh)

Zh (Uh)

Zh (Uh)

3.3. NUMERICAL TEST

Vh

hI on [AE] h2 on [BF/]

(1 + U\~L)) X2 on [F'G']

(21) (22) (23)

(24)

We perform the computations for L = 1, hI = 1, h2 = 0.25 and h3 = 0.5. We consider Uh : [0, 1] -+ IR of the form

Uh (x) = x3 - ~X2 + (x3 - x2) uk (1)

where uk (1) is a real number to be identified. We have used a mesh of 164 triangles PI Lagrange in order to compute

Zh (Uh). The values ofthe cost function obtained for different values of uk (1) are presented in the table below.

u~ (1) cost function

-1.00 0.001555 -0.70 0.000521 -0040 0.000161

u~ (1)

-0.30 -0.20 0.00

cost function

0.000187 0.000286 0.000701

The optimal value is obtained for uk (1) = -DAD. These numerical results were produced using the software freefem+ (see

[3]).


3.4. COMPUTATION OF THE GRADIENT

In order to compute Zh (Uh), we have to solve the linear system

where A (Uh) is a matrix of entries aij (Uh)' The applications

Uh E lR,2n -+ aij (Uh) E lR,

are differentiable. From the Implicit Function Theorem, we obtain that

is differentiable and consequently the cost function of the optimization problem is differentiable. The gradient could be computed analytically using the Implicit Function Theorem. Also, the gradient could be computed using automatic differentiation.

In order to solve numerically our problem, we can use the gradient like algorithms.

4. Conclusions

It can be hoped that this algorithm is faster as the previous one of EOS because the size of the problem to solve has been minimized and remeshing is not necessary. The use of the implicit function theorem in order to compute the gradient of the cost function increases this potentiality.

References

1. Baiocchi C. (1972) Su un problema a frontiera lib era conesso a questioni di hidraulica, Ann. Mat. Pure ed Applicata 92, 107-127.

2. Barbu V. (1984) Optimal control of variational inequalities, Research Notes in Mathematics 100, Pitman.

3. Bernardi D., Hecht F., Ohtsuka K. and Pironneau O. (1998) FREEFEM+ for Macs, PCs, Linux, ftp:/ /ftp.ann.jussieu.fr/pub/soft/pironneau

4. Crolet J.M. and Jacob F. (1998) Numerical dispersivity in modelling of saltwater intrusion into a coastal aquifer, in J. Bear (ed.) Theory and applications of transport in porous media, Vol. 11, Kluwer Academic Publishers, pp. 131-142.

5. Jacob F., Crolet J.M., Lesaint P. and Mania J. (1995) A three dimensional finite element model for fluid flow and transport in confined or unconfined aquifer, in Water Pollution, Vol. 3, Computational Mechanics Publications, Ashurst Lodge, Southampton, pp. 89-96.

6. Murea C.M. and Maday Y. (1997) Existence of an optimal control for a nonlinear fluid-cable interaction problem, Rapport of research CEMRACS 96, C.I.R.M. Luminy.

7. Murea C.M. and Vazquez C. (to appear) Shape Sensitivity of the Stokes Equations. Application to the Fluid-Structure Interaction Problems.

SPLITTING THE SATURATION AND HETEROGENEITY

FOR TIME DEPENDENT EFFECTIVE PHASE

PERMEABILITIES

MIKHAIL PANFILOV OCR!, Russian Academy of Sciences and Moscow Lomonosov University [email protected]

AND

ALEXEI TCHIJOV OCR!, Russian Academy of Sciences

Abstract. The method of splitting the saturation and heterogeneity is developed as a new fast numerical tool to compute the effective relative permeabilities (e.r.p), including in case of time-dependent, dynamic, permeabilities. Description of dynamic model, explanation of dynamic capillary non-nequilibrium effects, tensor properties of e.r.p., solution of cell problems, calculation of e.r.p. and qualitative analysis are presented.

1. INTRODUCTION

One of the recognized defects of the classical equations of two-phase flow through porous media is the hypothesis of local capillary equilibrium, this theory is based on. As the result, the attempt to introduce the time dependent functions as relative permeabilities constitutes a basic way how to modify the conventional two-phase equations: [12], [8], [9], [10], [11].

It is known from laboratory experiments that the characteristic time of stabilization for two-phase flow is very large. According to experimental data, the stabilization is very slow and can proceed some days in a porous sample of a small volume less than 100 cm3 . The stabilization is caused by fluid redistribution in space in such a way, that the local capillary equilibrium would be reached in each point. Even in homogeneous medium, the

115 J.M. Crolet (ed.). Computational Methods for Flow and Transport in Porous Media. 115-140. © 2000 Kluwer Academic Publishers.

116 MIKHAIL PANFILOV AND ALEXEI TCHIJOV

process of fluid redistribution is very slow. So, in media with high degree of heterogeneity the non-equilibrium phenomena of fluid redistribution in space are expected to be very important.

Hence, we can expect that a highly heterogeneous medium is a good model which could lead to definition of the characteristic eigen times of fluid redistribution in the porous space.

Really, in papers [15], [6], [14] new macroscale equations of two-phase flow through highly heterogeneous media have been deduced by homogenization. These equations contain the non-equilibrium, time dependent, function of capillary pressure and effective permeability.

In this paper we present some numerical results showing behavior of relative permeability and capillary pressure in time, as well as some numerical estimations for characteristic relaxation times.

To compute effective relative permeability, the method of decomposition is applied, which is based on asymptotic expansion for relative permeability tensor with respect to the heterogeneity parameter. This expansion allows to solve the problem of "splitting the geometry and the saturation" . More strictly speaking, the problem is as follows. In classical cases, the effective relative permeability is determined via cell functions, which are solutions of the boundary-value problems in a heterogeneity cell. These functions parametrically depend on saturation. One solution of the cell problem yields only one point of the function effective relative permeability versus saturation. The problem is how to eliminate saturation from this algorithm, and how to obtain whole the function of effective relative permeability basing on a single solution of the cell problem.

This method may be considered as a powerful numerical algorithms of up-scaling two-phase flow.

2. DEFINITION OF THE PROBLEM ON THE MICROSCALE

2.1. FLUID-MEDIUM PROPERTIES

Consider the flow of two incompressible phases (the a-phase and the ,a-phase) in a medium nCIR3 made of a highly permeable connected "matrix", nIl, and of a system of periodically located lowly permeable "inclusions" or" blocks" , nJ , such that n=nI unIl, as shown on Fig.I. Let r x be the interface separating two media, nJ and nIl.

Each of such sub-domains ni , i = I, I I, is characterized by a rock permeability tensor J{i and a porosity mi, by the relative phase permeability functions k~ (s), k~ (s), and the capillary pressure function P~ (s). The capillary pressure is the difference between the pressures in two fluids: PC=P/3-POI '

The viscosity jJ. of each fluid phase is supposed to be constant. The original relative permeabilities are assumed to be scalar functions of the saturation.

SPLITTING THE SATURATION AND HETEROGENEITY... 117

Figure 1. Structure of the medium with highly permeable connected matrix II and lowly permeable inclusions I

We assume first the heterogeneity is fast oscillating, such that the heterogeneity scale l is more small than the macroscopic scale L: €=l/L«l.

Secondly, we assume the medium is highly heterogeneous with respect to the permeability and to the capillary pressure, such that:

1) the permeability of the inclusions I is must lower than the permeability of the matrix II;

2) the order of capillary forces in the inclusions is must higher than in the matrix.

In third, we assume the medium is periodic, such that the heterogeneity scale € is equal to the heterogeneity period.

Assume the rock permeabilities gland gIl, as well as the porosities m I and mIl to be constant.

The parameter gI

WK= gIl

will be used to characterize the heterogeneity degree. The medium is highly heterogeneous when WK-70.


2.2. PROPERTIES OF NONLINEAR FUNCTIONS

Let P be the pressure and s be the saturation of the a-phase. Let s*' s* (s*<s*) denote the residual saturations correspondent to the percolation threshold for each phase.

Assume both the relative permeabilities ki(S) and the capillary pressure Pc(s) are different in the media I and II, as shown on Fig.2.

Figure 2. Micro-scale relative permeabilities (a) and capillary pressures (b) for blocks I and matrix II

The relative ratio between s; and s;l, as well as between sh and slh is any.

Assume the following properties for the functions, where i=I, I I:

Si*

J P~(s)ds <00, p~(i*)=o s· •

(1)

kb(Si)E[O, 1], -00< ~, <0, when s~::;si::;si*j kb(si*)=O (3)

where s~, si* are the residual saturations corresponding to the percolation threshold for each phase in the medium i (i=I, 11) j

In inequalities (2) and (3), the principal fact consists in that the relative permeability has a non-zero derivative at the percolation threshold saturation, that follows from basic concepts of percolation theory. The properties (1) define singular behavior of the capillary pressure curves, but in such a


manner that these functions remain integrable. Hence, the averaged capillary pressures exist:

Si*

p?= . 1 . J P~(s)ds, Sl*-S~ .

i=I, I I (4)

s~

due to property (1).

2.3. FLOW EQUATIONS

Equations describing two-phase flow in n with respect to the a-phase saturation s and to the a-phase pressure P may be written in the following form, for any t>O:

In nl:

(5a)

(5b)

(6a)

(6b)

On fx:

[PJr = 0 , (7a) pflr = Pfllr,. , (7b) J{lk~(s) apl J{IIk~l(s) apII --=-:.-'-~nilr = -a-nilr ' (7c)

/-La UXi /-La Xi

J{lk~(s) a [pl+Pf(s)] J(IIk~l(s) a [pII+PcII(s)] /-Lf3 aXi nilr = /-L(3 aXi nilr' (7d)

The initial condition:

s(X, y, O)=so(x). (8)


The boundary condition requires that

Plan = Pan(x, t), and sian = san (x, t) (9)

where Pan(x,t) and san(x,t) are known functions. Conditions on r mean the continuity for the flow rate and the pressure

in each phase.

3. MACROSCALE MODEL OF TWO-PHASE FLOW

3.1. NONEQUILIBRIUM MODEL

The generalized macroscale model for flow has been deduced in [15], [6], [14]. Before writing it, we introduce some necessary notations.

To any point x in the domain n we associate the image y=x/c in a single cell Y={-1/2<Yi<I/2, i=1,2,3}. The image of the surface r will be denoted in the same way, i.e., as the surface r. It divides the cell into two sets yI (the dense block) and yll (the highly permeable matrix), and r is assumed to be not intersecting the external boundary of Y: fnaY =0. Then f=ayI.

The symbols ( . )1 and ( . )2 mean the integration over the subdomains yI and yll and:

( . )1 = J (-)dy, (. )2 = J (-)dy yl yll

The macroscale model corresponding to the problem (5) - (7) has the following form:

where


The block pressure is determined as:

~I ~II * I ( I ) OSI I _ k;(SI) Pex=Pex +t (S) 1-Fex(S) Tt, Fex(S)= kI(SI)+k~(SI)~; (12)

where e is the volume fraction of the blocks; p~ is the microscale capillary pressure in the medium i, (i=I,II); P~ is the macroscale (effective) capillary pressure; Kij and Ky are the effective phase permeabilities; Pex is

the macroscale pressure in the fluid a; SI and SII are the macroscale block and matrix saturations.

Effective phase permeabilities Kij and KY, and capillary relaxation time t* are defined in the next sections.

Equation (lOc) with (lla) and (llb) defines a function:

SI = SI (SII, t) (13)

which is the solution of the following ordinary differential equation:

t*(SI) 8~I p!,a _ p! (SI) = _p!I (SII)

3.2. EQUILIBRIUM MODEL

(14)

To compare results obtained using the above model, we will examine also the well known model deduced first in [18] and [17] for moderately heterogeneous media, i.e. when wK"'l. This model describe the capillary equilibrium behavior of the system:

(m) 8S _ ~ (Kf}(S) 8Pex ) = 0 8t 8Xi flex 8x j

8S 8 (KY(S)8[Pex+Pc(S))) -(m)-- -8t 8Xi fLf3 8x j

=0

Pc(S)=p! (SI)=p!f (SII) where

p! (Sf)=p! (Sf), p!I (SII)=p!f (SII)

S= (m)l Sf+(m)2SII - (m)

(15a)

(15b)

(15c)

(15d)

(15e)

(15f)

Where Sf, SII and S are the averaged saturation for the blocks, the matrix and the whole medium; Pex is the macroscale pressure in the aphase; pII (Sf), pJI (SII) and Pc(S) are the macroscale capillary pressure for the blocks, matrix and the whole cell.


Parameters K~ (S), K~ (S) and Pc(S) are determined via the cell functions 7f.;,,!,i for each fluid" ,=a, (3, i=l, 2, 3:

j{~ (S) =( K;; (y) k~( Sf) (~:'; +0;; ) ) 1 +

(K;; (y)k~f (SIl) (~~'; +0;; ) ),

where 7f.;,,!,k is defined as solution of the cell problem:

{ a~; (K;; (y) ko (y, Sf, SIl) (at;;' +Ok; ) ) = 0,

(7f.;,,!,k) = 0, 7f.;,,!,k is y-periodic

where

(16)

yEY (17)

We see that the global block pressure and the global high conductivity pressure are equivalent.

4. CAPILLARY RELAXATION PHENOMENON

4.1. CAPILLARY RELAXATION TIME

In (11), the capillary relaxation time t* determines the characteristic time, after which the saturation on blocks reaches the equilibrium state. It is defined explicitly as

(18)

where r.p is defines as the solution of the Dirichlet problem:

{ ~ (KIf ar.p) = -1; aYi aYk

r.plr = 0

(19)

1 is the dimensional period of heterogeneity, kQ (s) and kj3 (s) are the microscale relative permeabilities.


If the block is a sphere of radius E and I< II is a scalar, the problem (19) has an analytical solution: <p(r)= - (r2-E2)/6, and we obtain then an explicit expression for the value (<P)1 in (18):

2e (<P)1 = 15

Then:

In Fig.3 the dependence T.(S)=t·(S)/t. is drown, for the following relative permeabilities: kcr (S)=S2, k.a(S)=(1-S)3 and J.Lcr/J.L.a=l. The value t. is determined as t. = ( mIl J.L.a L 2) / (I< II liP), where liP=PB - P A, PB and PA being the maximum and the minimum values of the boundary function Pan in (9).

1000-H---------+--l

100+--'<--------+---1

10 +---=..-=:..-----j

s

Figure 3. Capillary relaxation time vs. block saturation

As seen, the relaxation rate is very small, when the system becomes single-phase (8-+0, or 8-+1). In these cases any relaxation is stopped. Nonlinear behavior of the relaxation time shows, that although the formal order of T. is ..j€, its real order can by sufficiently more large. Hence, the phenomena of phase redistribution in space can be very slow, that corresponds to experimental observation.


4.2. DYNAMIC CAPILLARY PRESSURE ON BLOCKS

The saturation field in the matrix II is in equilibrium state according to (llb), while the block saturation Sf defined by Eqs. (lOc) and (lla) varies in time and follows the relaxation law.

This relaxation is equivalent to a global process of saturation redistribution in space, caused by capillary forces. According to experimental data, this relaxation is very slow and can proceed some days in a porous sample of a small volume less than 100 cm3 .

Due to this non instantaneous relaxation the macroscale block capillary pressure, fit, becomes dependent not only on the block saturation, but on the time also, as seen from (lla).

So, the effective capillary pressure curve in dense blocks displays dynamic behavior, and is varying in time in such a way that it tends to the microscale curve, p{ (8), at t--tOO, as shown in Fig. 4. where the flesh shows

AI r--r--------, ~

a Sl 81

b

Figure 4. Dynamic capillary pressure in blocks when the block saturation grows (a), or decreases (b) in time

the direction of time growth. Note that the capillary pressure in high conductive matrix is in equi

librium state, and is equal to the microscale function P{l(s), according to (llb).

Contrarily to moderately heterogeneous media [18], [17] the concept of a mean effective capillary pressure for the whole medium becomes useless when the medium is highly heterogeneous. Any unique capillary pressure curve is not able to describe behavior of this system.


4.3. DIFFERENCE BETWEEN THE MATRIX AND THE BLOCK PRESSURES

Relation (12) shows that a difference between the matrix and the block pressure arises due to capillary phenomena.

It is necessary to note, that in a moderately heterogeneous medium, no difference exists in the matrix and the block pressures for each fluid phase.

The difference P~ - p~! is governed by a kinetic law (12) with the same order of relaxation time as in (14).

However, it is necessary to note, that the difference in the phase pressures is small enough, whereas two saturations S! and SII are considerably distinct.

4.4. EXPLANATION OF NON-EQUILIBRIUM BEHAVIOR

A relation of the same type as (14) was first proposed in [2], [5], [4], citeNiko68 as a general approach to describe non-equilibrium capillary processes, with a constant relaxation time. In our model, the relaxation time is not constant, it depends on the saturation in the blocks as described in (18). To explain the physical meaning of this formula, we, first, note, that the relaxation process happens in the form of an invasion of the a-phase into the blocks. During this invasion, the ,6-phase has to leave the blocks, hence a counter-flow is forming. The relaxation time is equal to the characteristic length of a block divided by the velocity of the invading phase flow during such a two-phase counter-motion. If the relative permeability for one of the fluids is very small, invasion and flowing out are complicated, therefore filling of the blocks is very slow, and the relaxation time tends to infinity, as shown in Fig. 3.

5. EFFECTIVE PHASE PERMEABILITIES

5.1. GENERAL DEFINITION

For each phase " with ,=a,,6, the effective phase permeability tensor Kif is determined in the following way:

(21a)

where three constant tensors are:

-},II \}'II (B7j;£! r: )) \.ij = \.ik BYj +Ukj 2 (21b)

}~! f},! (B7j;£ f )) \.i j=\ \.ik 8Yj +Ukj 1 (21c)

126 MIKHAILPANFILOV AND ALEXEI TCHIJOV

(21d)

and three types of cell functions '1j;£I(y), '1j;£(y) and 'Ij;~I,I(y) are solutions of the fOlJe>wing cell problems:

o ( .II (O'lj;£I £ )) yEyII OYi Rij oYj +Ukj = 0,

(22)

'lj;F is y-periodic

(23)

(24)

.,,11 I . . d' 'l-'k' IS y-peno lC

Note, when both the saturations are equal to 0 or 1 we obtain an analogous relation for the effective global permeability from (21a):

(25)

Eq. (21) presents the effective phase permeability in a splitted form, where some constant coefficients are multiplied by some functions of saturation.

The constant tensor Kg is the effective global permeability of the highly permeable matrix II only, i.e. when the blocks are assumed to be impenetrable.

The constant tensor Kl is an additive contribution to the effective global permeability of the medium caused by translation type of flow through the blocks. Note that Kl is not equal to the effective permeability of the blocks, as the averaged permeability can not be obtained by summation of the permeabilities of all parts.


The constant tensor ~I,I is an additional contribution to the effective global permeability caused by the secondary influence of the translation flow through the blocks on the flow through the matrix II.

Functions k; and k;I are the microscale relative permeabilities for the fluid ,.

5.2. TENSOR PROPERTIES OF THE EFFECTIVE PHASE PERMEABILITIES

The effective phase permeabilities Kfj and K¥ keep the same tensor prop

erties as the effective global permeability Rj (25). In particular, the following is easy to be proved.

Diagonal case 1 Let three conditions be satisfied: - the block shape is uniformly symmetrical respectively to all the axis; - the microscale permeability tensor is diagonal, i.e. J(lI (y)=O,

J(l(y)=O, Vilj; - the microscale permeabilities J(ii(y) are even with respect to each axis. Then the effective phase permeabilities are diagonal:

ViI j

Certainly, the effective global permeability Kij is also diagonal.

Diagonal case 2 Let three conditions be satisfied: - the block shape is symmetrical respectively to each axis;

(26)

- the microscale permeability is scalar, i.e. J(H(y)=J(II(y)8ij , J(l(y)= J(I(y)8ij, Vi,j;

- the microscale permeabilities J(II(y) and J(I(y) are even with respect to each axis.

Then the effective phase permeabilities are diagonal, i.e. (26) is true too.

Scalar case Let in the last case the block shape be uniformly symmetrical respec

tively to all the axis. Then the effective phase permeabilities are scalar:

Kij(SII t)=K (SII t)8·· a , a , ~J' -}"?ij(SII ) --;- (SII ) 5: \{3 ,t=R{3 ,tuij; Vi,j (27)

The "Diagonal case 1" shows that the anisotropy of the microscale permeability field and the asymmetrical form of the blocks are not sufficient to arising of non-diagonal terms in the macroscale permeability tensor.


The "Diagonal case 2" shows that the anisotropy of the macroscale permeability field may be produced only by an asymmetrical shape of the blocks, although the microscale permeability is isotropic.

The "Scalar case", i.e. the scalar macroscale field may be obtained only when the microscale permeability is isotropic, and the block shape is uniformly symmetrical.

6. EFFECTIVE RELATIVE PERMEABILITIES

On practice of reservoir engineering, the relative permeabilities are more widely used than the phase permeabilities Rj} and Ry introduced in (21) . In scalar case, the relative permeability for the phase a (or (3) is determined as kcx(s)=Kcx(s)/K, where K is the global permeability. When the effective phase permeability Rj} and the effective global permeability Rij are tensors, definition of relative permeability becomes not obvious.

6.1. DEFINITION OF THE TENSOR EFFECTIVE RELATIVE PERMEABILITY

~ ~

In general tensor case, the effective relative permeability kcx or k{3 may be introduced only via a convolution of two tensors, i.e.

Rij(SIl t)-R· kkj(SIl t) 'Y ,- lk 'Y " ,=a, (3; i,j=l, 2, 3 (28)

In Rn , for a fixed fluid phase" Eqs. (28) constitute a system of n2

equations for n 2 components {Je~j} n. of a tensor function of second rank. k,J=l

This system is closed and the relative permeabilities are completely defined. For instance, in R2 this system takes the form:

]( Jell +]( k21 = ](11 11 'Y 12 'Y 'Y

(29)

y p2 + j{ k22 = j{22 ~21 'Y 22 'Y 'Y

When the medium is highly heterogeneous, the system (28) has an analytical solution. It may be deduced using decomposition (21), (25) for the effective phase and Q'lobal permeabilities, and taking into account

-1 :>J.:..Il 1 that all the values Kij and Kij' are of order WK. Then at wK < < 1 we can obtain from (28) and (21) for any phase ,=a, (3:

k1 (SIl, SI) ~ k;1 (SIl) + ~1 (SIl, SI), i, j=l, 2, 3 (30)


where the components of the tensor ~1 are solutions of a linear algebraic system:

(31)

It follows that the macroscale relative permeabilities do not depend on the tensor ~1,I, although the phase permeabilities K1 depend on it.

6.2. DIAGONAL EFFECTIVE RELATIVE PERMEABILITIES

As it follows from the" diagonal case I" and the" diagonal case 2", if the effective phase permeabilities are diagonal, then the microscale global permeability is diagonal too. In this case it follows from (28) that the effective relative permeabilities are also diagonal:

---}"ii }----;- ~kii \Q = \ii Q'

}-{ii ---},," ~kii. {3= '-ii{3, i=1, 2, 3

In these formulas no of summation is made over i.

(32)

Using (21) we can represent the relative permeability in the explicit form for each fluid ,:

(no of summation over i). This relation may be simplified, if we take into account that K~I and

K~ are of order WK. Using asymptotic expansion of (33) at WKtoO, we obtain for each fluid T

----;-1

k~(SII, t) ~ k~I (SII) + ;;> [k~(SI) - k~1 (SII)] , i=1, 2, 3 (34) n

(no of summation over i). This relation can be deduced directly from Eqs. (30) and (31).

6.3. SCALAR EFFECTIVE RELATIVE PERMEABILITY

Scalar effective relative permeability can be obtained in two cases. The first one is rather trivial. When effective global permeability field is

scalar, then the effective relative permeability is scalar too. In other words, if

then kii(SJI SI) - k (SII SI)8·· "Y ' - "Y , 1)


where --:-I

k,(SII, SI) ~ k;I (SII) + ;II [k;(SI) - k;I (SII)] (35)

In the second case, the effective global permeability may be a tensor, but all the ratios Kl/KIf for all i,j=l, 2, 3 must be equivalent. Thus, if

-I [{ .. -~I =lI=const, Vi, j [{ ..

IJ

(no of summation over repeating indexes). Then:

Property (36) will be called further as the "uniform anisotropy".

6.4. SPLITTING THE SATURATION AND HETEROGENEITY

(36)

One of the most important problems of up-scaling two-phase flow is that the standard algorithms performing calculation of phase permeability is very time expensive. More strictly speaking, in the classical models [18], [17] the cell problem defines some intermediary functions like 'l/Jk (y; S) and 'I/J% (y; S) in (16), (17), which depend on the saturation S as the parameter. It means that the cell problem has to be computed for each value of saturation S. The cell problem being solved once gives only one point of the effective relative permeability curve. The problem is how to extract the saturation from the algorithm of phase permeability calculation.

In other words, this fundamental problem of scale-up theory is formulated as a problem of splitting saturation and heterogeneity geometry.

Eqs. (21), (30), and (37) show that the problem is automatically solved in case of highly heterogeneous medium. All the cell problems (22) - (24) do not include saturation. One solution of all the cell problems gives the whole effective phase permeability vs. saturation curve.

This is reached due to splitting the functions dependent on saturation, as microscale relative permeabilities, k;(s) and k;I(s), Cr=a,(3), and the

constant factors dependent on heterogeneity geometry, as Kif and Ki~' Splitting of phase permeability (21) remains true only for highly het

erogeneous media, i.e. when WK < < 1, because it can be considered as an asymptotic expansion of the problem (8) - (9) over WK when WK-rO.

However calculation shown validity ofEq. (21) even when wK~0.5-0.6.


6.5. NON-EQUILIBRIUM BEHAVIOR OF RELATIVE PERMEABILITY

According to (21), or (30), each phase permeability is a function of two saturations, SII and 51. Due to Eq. (14) defining the dependence between two saturations (13) as S1=S1(SIl, t), it follows that Eqs. (21) and (14) describe" dynamic" phase permeabilities, while (30) and (14) describe "dynamic" phase permeabilities:

or:

j{ij (SII 51) = Kij (SII t) a' a' , KJ (SIl, 51) = KJ (SIl, t)

k~ (SIl, 51) = k~ (SIl, t)

Kf!f,~j (SIl, 51) = K& [k~(S1) - k~1 (SIl)] , (38)

t* (51) 8~1 p[,Q _ p[ (51) = _ p[1 (SIl)

In scalar case, this system becomes more simple:

{ k~ (SIl, 51) = k!/ (SIl) + II [k~(S1) - k~1 (SIl)] ,

(39) t*(S1) 8~I p[,Q _ p[ (51) = _p[1 (SIl)

Fig.5 shows the behavior of two effective relative permeabilities in time. The dashed lines correspond to the original microscale relative permeabilities, k~, k~ 1 , k~ and k~1. The solid lines represent the effective relative permeability. the horizontal axis corresponds to the macroscale matrix saturation SIl, where the flesh shows the direction of time growth.

These figures have been computed using (lla) in the following way. The exponential grow of the saturation SII was accepted as the given information, which can be associated to some process of injection of the a-phase into the medium. The parameter II was calculated basing on solution of the cell problem as will be explained in the section 7. The initial saturation 51 is any.

It seen, that the effective relative permeability is equal to the relative permeability of the matrix at the initial time moments, but it grows in time due to the increasing contribution of the blocks.


k .--------, II '

II

S

Figure 5. Dynamic behavior of the effective relative permeabilities. Results of calculation

7. NUMERICAL SIMULATION OF EFFECTIVE RELATIVE PERMEABILITY

7.1. NUMERICAL SOLUTION OF CELL PROBLEMS

Splitting (21), (30) is a very powerful tool to compute the effective phase permeabilities. Along with "extraction" of saturation described above it leads to three types of cell problems each of them is linear and is posed in homogeneous sub-domain.

7.2. REDUCTION OF THE PERIODIC PROBLEM TO THE BOUNDARY-VAL UE

Problems (22) and (24) comprise periodic conditions, while standard numerical methods are developed for boundary-value problems. In general case, solution of a problem with periodic conditions is performed basing on penalty method, Fourier series and domain decomposition method. In the penalty method [3] the periodic problem is replaced by an equivalent sequence of boundary-value problems. Expansion of the solution into Fourier series, applied for instance in [19], is powerless for perforated structures. The method of domain decomposition was used in [7]. In particular cases, when some symmetry of heterogeneity is observed, the periodic problem may be replaced by a single equivalent boundary-value problem. Such a reduction have been described in [1].

In this section we will show the problem (22) or (24) may be transformed into a boundary-value in more general case, when the symmetry is required with respect to one coordinate axis only.

Let us consider a periodic problem in Y, where Y is a square in R2, i.e.,


y= - 1/2<Yi<1/2, i=1,2:

&~; ( K;; (~~: +01; ) ) = 0, yEY

(1P1)2 = 0 'lj;l is y-periodic

and a boundary-value problem in Y:

&~; (K;; (:~ +01;) ) = 0, yEY

17I Y1 =-1/2=17I Y1 =1/2=17*

817 I - 817 I -0 8Y2 Y2=_1/2- 8Y2 Y2=1/2-

with any constant value 17*. The problem (41) is correctly formulated, see [1]. Proposition:

(40a)

(40b)

(40c)

(41a)

(41b)

(41c)

- When the tensor J{ij is diagonal, i.e., J{ij=O, ilj, and both the functions J{ii(Y) are even with respect to Y2, then:

'lj;l (y) = 17(Y) + C, where C= - (17) (42)

The structure of the cell may be any, only the symmetry with the axis Yl must take place.

In fact, due to the symmetry of parameters with respect to Y2, the solution 17(Y) is symmetrical too, i.e., it is even with respect to Y2. Therefore periodicity condition (40c) is satisfied along the axis Y2. Due to condition (41 b) , the function 17(Y) is periodic along the axis Yl too.

The function 17(Y) satisfies Eg. (41a) and is periodic. Then the function 'lj;l=TJ-(TJ) satisfies Eg. (41a) , is y-periodic and has zero mean value in Y, i.e. satisfies the problem (40). The proposition is proved.

Certainly, the constant value C in (42) is related to the boundary value TJ* in (41). Our proposition does not give this connection, but it is not necessary, because the effective permeability is defined via derivatives of the functions 'lj;k(Y), as follows from (21) and (8).

Thus to compute the effective permeability in (8), or in (21), it is sufficient to solve the boundary-value problem (41) with any TJ* instead of periodic problem (40), when the condition of symmetry along one of axis is satisfied.


The function WI =1](Y) +Yl has meaning of a dimensionless pressure within a cell and satisfies the following problem:

yEY

(43)

8111 1 1 -0 8Y2 Y2=±1/2-

We see the pressure drop along the axis Yl is equal to 1, i.e., ~Wl= (-1/2+1]*) - (-1/2+1]*)=1 and do not depend on 1]*. In fact, the structure of flow depends only on the pressure drop and does not depends on the shift 1]* of the whole pressure field.

Note that the second function, 'lj;2(Y) can not be computed in the same way, when only the symmetry along the axis Y2 holds.

The similar theorem can be proved for the problems (22) and (24) posed in Y II only. But in this case, for the function 'Ij;[I the form of the internal boundary r must be also symmetrical with respect to the axis Y2.

7.3. SOLUTION OF CELL PROBLEMS

All the cell problems have been computed using standard solvers based on finite-difference or finite-element method.

A symmetrical cell structure was examined with a rectangular block as shown in Fig. 6.

,w. ... y'

-~

y' ~

-c. . 0:

of,

..

Figure 6. Structure of a cell used for numerical solution of cell problems

The microscale permeability has been assumed to be a scalar function, . }"lI_}(IU" }"I _}(ls: . . l.e. '-ij - Ol]' \ij- Ol)"


Solution of the problem (22) is shown in Fig. 7, where both the functions 1/7[1 (y) (a) and wF (y)= 1/7[1 (Y)+Yl (b) are presented on one quarter of the cell Y.

a b

Figure 7. Numerical solution of cell problem

(22)

Function wF is a dimensionless local pressure field within the subdomain Y II of a cell, and, therefore, has more physical sense, than the function 1/7 [ I.

a b

Figure 8. Numerical solution of cell problems

(23) and (24)

Solutions of cell problems (23) and (24) are shown in Fig. 8, such that the figure (a) presents the function W{(y)=1/7{(Y)+Yl , while the figure


(b) corresponds to the function 'r/Ji'll (y). All functions are shown on one quarter of the cell Y.

Behavior of each cell function on all parts of the cell Y may be reconstructed basing on the symmetry properties. In particular, the functions 'r/J[l(y) and 'r/J[(y) are even with respect to the variables Y2 and Y3, but they are odd with respect to the variable Yl' The function 'r/J{I,I (y) is even with respect to all the variables.

7.4. CALCULATION OF THE EFFECTIVE RELATIVE PERMEABILITIES. METHOD OF SPLITTING THE SATURATION AND HETEROGENEITY

In this section we show some exam pIes of the effective relative permeabilities computed basing on equations (21), or its sequences (30), (34), (35), (37) .

We present the results concerning the large values of time, after the macroscale capillary equilibrium state has reached. Transient behavior in time is shown in section 6.5.

The solutions of the cell problems obtained below have been used to perform integration according to (25) . The scalar case (35) is investigated. The original microscale relative permeabilities are shown in Fig. 9:

k

....

" 'I /', '\P

1.29 >: G.2IiI if." a.Oft 8.E8 S

Figure 9. Microscale relative perrneabilities used in calculations

In the figures shown below the solid lines correspond to the method of splitting the saturation and heterogeneity, while the dashed lines are obtained by solving the full, non-split ted , cell problem deduced in classical theory, [18], [17], and presented in the form of (16) and (17). Several cases have been examined with various volume fraction of blocks e and various heterogeneity degree WK.

Three figures 10 show the effective relative permeabilities for a rather small volume fraction of blocks, 8=0.3.

The next three figures 11 correspond to 8=0.5.


",---------, k '-

....

8=0.3 w=O.1

II

S

.... 8=0.3 w=0.02

9. 9. *' 9.08 9.. 11

S

Figure 10. Effective relative permeabilities for a highly heterogeneous medium; the volume fraction of blocks is 8=0.3

, , 8=0.5

.. =0.1

0.40 0.80 n S

.... 9.2111 9 . .e i.oW 9.. IT

S


The next three figures 12 correspond to 8=0.7. The last three figures 13 show behavior of the effective relative perme

ability when the volume fraction of blocks is very high, 8=0.9 .

.. '"



k" " " 8=0.9 .... ",=0.01

Iit.2IJ e.. 1.68 e.. n S


It is necessary to note that the effective relative permeability curve is not obligatory located between two original microscale curves correspondent to the matrix and to the blocks. However it is always above the microscale relative permeability for the highly permeable matrix due to the contribution of blocks. When the microscale relative permeability of the matrix is higher than that of the blocks, the effective relative permeability curve is located above both the microscale curves.

The method of splitting the saturation anf heterogeneity gives satisfactory results in various ranges of the parameter WK depending on the reciprocal location of the microscale relative permeability curves.

In case when the microscale relative permeability of the matrix is under that of the blocks, the method of splitting gives the good results for any wK<O.6 and any B. In our example, this situation has been chosen for the a-phase.

In case when the microscale relative permeability of the matrix is above that of the blocks, the method of splitting gives the good results for wK<O.08 and any B. In our example, this situation has been chosen for the ,8-phase.

CONCLU~IONS

Behavior of the macroscale relative permeability dependent on time is studied for two-phase flow in highly heterogeneous media. The macroscale model describes a non-equilibrium system with a slow transient process of stabilization for the macroscale saturation field due to capillary forces. As the result, the saturation in low permeable blocks and that in highly permeable connected matrix are independent one of other, contrarily to the equilibrium case. More strictly speaking, two saturations are related by a nonlinear ordinary differential equation with respect to the time.


The effective phase permeabilities (e.p.p) and the effective relative permeabilities (e.r.p.) are shown to be dependent on both the saturations, or on one of saturation and on the time explicitly.

It is shown that in some particular case named as uniform anisotropy, the microscale relative permeability may be a scalar function, although the permeability field is anisotropic.

The structure of e.p.p. and e.r.p. is splitted in such a way, that only two cell functions should be solved to calculate the e.r.p .. Moreover, these cell problems are independent of saturation. So the problem to select the saturation form the cell problem is solved. This result ensures development of a very fast numerical method to up-scale the e.r.p.

Strictly speaking, the method of splitting the saturation is true for highly heterogeneous media, when the block/matrix permeability ratio WK

is small. However the numerical 2D experiments have shown this method to be valid even for moderately heterogeneous media, until WK~O.l.

ACKNOWLEDGMENTS

The work was supported by the Russian Foundation in Basic Research (grant N 98-01-00460).

References

1. Bakhvalov, N.S. and Panasenko, G.P. (1989) Homogenization of processes in periodic media. Ed. Nauka, Moscow. (Version in English: Bakhvalov N. and Panasenko, G. Homogenization: Averaging Processes in Periodic Media. Kluwer Academic Publishers, Dordrecht)

2. Barenblatt, G.!. (1971) Flow of two immiscible liquids through the homogeneous porous medium, Izvestiya Academii Nauk SSSR, Mekhanika Zhidkosti i Gaza, no 5, pp. 144-151.

3. Bougemaa, A. (1998) Private communication. 4. Barenblatt, G.!. and Vinichenko, A.P. (1980) Non-equilibrium flow of immiscible

fluids in porous media, Uspekhi Mathematicheskikh Nauk, no 3, pp. 35-50. 5. Barenblatt, G.!" Entov, V.M. and Ryzhik, V.M. (1972) Theory of Non-Stationary

Flow of Liquids and Gases Through Porous Media. Nedra, Moscow. 6. Bourgeat, A. and Panfilov, M. (1998) Effective two-phase flow through highly het

erogeneous porous media: Capillary nonequilibrium effects, Computational Geosciences, 2, no. 3, pp. 191-215.

7. Charpentier, 1., and Maday, Y. (1995) Deux methodes de decomposition de domaine pour la resolution d'equations aux derivee partielles avec conditions de periodicite: application a. la theorie de l'homogeneisation, C. R. Acad. Sci. Paris, Ser.I, 321, pp. 359-366.

8. Hassanizadeh, S.M., and Gray, W.G. (1993) Thermodynamic basis of capillary pressure in porous media, Water Resour. Res., 29, pp. 3389-3405.

9. Hassanizadeh, S.M., and Gray, W.G. (1993) Toward an improved description of the physics of two-phase flow, Adv. Water Resour., 16, pp. 53-67.

10. Hassanizadeh, S.M., and Gray, W.G. (1990) Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries, Adv. Water Resources, 13, pp. 169-186.


11. Hassanizadeh S.M. (1997) Dynamic effects in the capillary pressure-saturation relationship, Proc. 4th Int. Conf. Civ. Eng., Sharif Univ. Techno!., Tehran, Iran, 4-6 May, 1997, VA: Water Resources and Environmental Engineering, pp. 141-149.

12. Kalaydjian, F. (1992) Dynamic capillary pressure curve for water/oil displacement in porous media, theory vs. experiment, SPE Conf., Washington, DC, October 4-7, Paper 24813, pp. 491-506.

13. Nikolaevskii, V.N., Bondarev, E.A., Mirkin M.l. et al, (1968) Motion of Hydrocarbon Mixtures in Porous Media. Nedra, Moscow (in Russian).

14. Panfilov, M., and Bourgeat, A. (1996) Capillary Relaxation Model for Two-Phase Flow Through Dual-Porosity Media Proc, 4th Int. Sympos. On Evaluation of Reservoir Wettability and its Effects on Oil Recovery, Sept. 11-13, 1996, Montpellier, France.

15. Panfilov, M. (1996) Homogenized model with capillary nonequilibrium for two-phase flow through highly heterogeneous porous media, C. R. Acad. Sci. Paris, Ser.II, no. 3.

16. Panfilov, M. and Panfilova, 1. (1996) Averaged models of flows with heterogeneous internal structure. Ed. Nauka, Moscow (in Russian).

17. Quintard, M. and Whitaker, S. (1988) Two Phase Flow in Heterogeneous Porous Merna: The Method of Large-Scale Averaging, Transport in Porous Media, no. 3, pp. 357-413.

18. Saez, A. E., Otero, C. J. and Rusinek, 1. (1989) The Effective Homogeneous Behaviour of Heterogeneous Porous Media, Transport in Porous Media, no. 4, pp. 212-238.

19. Suquet, P. (1990) Une metode simplifiee pour des proprietes elastiques de materiaux Mterogenes a. structure periornque, C. R. Acad. Sci. Paris, Ser.II, 311, pp. 769-774.

FUZZY SIMULATION OF WATERFLOODING

A New Approach to Handling Uncertainties in Multiple Realizations

A.B. ZOLOTUKHIN Stavanger College P.O. Box 2557, Ullandhaug,N-4004, Stavanger, Norway

1. Abstract

Evaluation of field perfonnance and a long-tenn production forecast require considerable resources spent on [numerical] reservoir simulation. Uncertainty in reservoir characterization and future prospects concerning oil price, operating cost, etc. advocates for sound sensitivity analysis which requires even more resources.

Most of the problems associated with the uncertainty of evaluation can be handled either by running a sensitivity analysis or by probabilistic methods. The latter being extensively used in the past for resovling numerous engineering problems are often limited by lack of data with statistical properties. Moreover, in many engineering applications amount of accessible infonnation is often not sufficient for its processing by statistical methods. In such cases fuzzy methods seem to be more appropriate technique to solve the problems.

From a mathematical perspective, the difference between probabilistic and fuzzy methods is based on the definition of membership function that does not necessarily rest on probability, but rather on relative preference among the members of the reference set. As a result, probability theory evaluates the likelihood of outcomes, while fuzzy mathematics models the possibility of occurence. Fuzzy methods can handle uncertainty directly, without running the sensitivity analysis. Another advantage of fuzzy technique is that it links uncertainty of input data to the reliability estimation of the final decision.

From a computational point of view, fuzzy methods, being based on rules resembling axioms of deterministic mathematics, are much faster as compared to stochastic methods. However, little effect can be gained when applying those methods to a volumetric reserve estimate, material balance equation, decline curve analysis, etc. Considerable effect can be foreseen in handling problems related to reservoir characterization. In areas of [numerical] reservoir simulation fuzzy technique outperfonns probabiliastic methods in the most effective way and seems to have no rivals.

Examples of the application of fuzzy methods to petroleum engineering problems like resources and reserves estimate, reservoir description and characterization, reservoir simulation, optimization and decision making, have been discussed earlier in the literature[ 16,

141 1.M. Crolet (ed.), Computational Methods/or Flow and Transport in Porous Media, 141-160. © 2000 Kluwer Academic Publishers.

142 A.B. ZOLOTUKHIN

11,5, 19,20]. However, little attention has been paid to numerical simulation of a multiphase flow in porous media. The emphasis in this paper is given to reservoir simulation problems illustrated by comparizon of classical deterministic and fuzzy solutions to a two-phase flow of incompressible fluids in porous media known as a fractional flow or a Buckley-Leverett problem.

2. Introduction

Fuzzy sets and methods based on them were introduced by L. Zade [15] for solving socalled "hard-to-formulate" problems. Advantage of this approach is based on a mathematical formalism that associates degree of confidence of a statement with the membership grade of a fuzzy set describing a certain class of objects, features of phenomenon etc. Extension of this approach to fuzzy numbers representing a special case of fuzzy sets, has been shown very fruitful in describing a scattered environment where data are either incomplete or uncertain and where statistics is not available and probabilistic methods could fail.

One of important merits of a fuzzy approach is that it enables to show how uncertainties of the input and output data relate to each other. Interest to fuzzy methods was constantly growing since first publications gradually extending areas of their application to classification, clustering, ranking, optimization and many others[3, 7, 13,8, 12, 1].

Although the application of fuzzy methods have been appreciated in many technical areas, and in some of them, like economics, successfully competed with conventional [deterministic] mathematics, petroleum industry was almost ignorant to this innovative technique. Only since the beginning of 90-s appeared first publications on this matter in petroleum literature [8, 6, 17, 18]. However, interest to these methods has been shown mostly by scientific community rather than by the industry. Explanation to this fact is obvious and can be stated as follows:

• Fuzzy calculus is not straightforward. Its axioms differ from those of deterministic mathematics [3], and calculations should be done with care.

• Different fuzzy solutions can be obtained for the same problem depending on the experience of a person dealing with it. It is often difficult to prove what solution is the most correct one.

• Software for solving fuzzified problems almost does not exist.

Nevertheless, intensive research on possible applications of fuzzy methods and interval analysis to petroleum related problems has been continued and resulted in a number of recent publications [16, 10, 11, 5, 14, 21, 19, 20]. Scope of problems considered in these publications was mostly related to decision making in fuzzy environment [5, 8], solution of fuzzy algebraic equations [10, II] and application of fuzzy methods to reserves estimate, fuzzy reservoir characterization, material balance calculations, decline curve

FUZZY SIMULATION OF WATERFLOODING 143

analysis and similar analytical techniques for field performance evaluation [16, 11, 5, 21, 19,20]. In particular, these publications have shown that fuzzy methods were competitive and, in some cases, even superior to deterministic and probabilistic approaches.

Still one important question was unanswered: can fuzzy technique be applied to a problem of a multiphase flow of fluids through porous media? First results of formal application of fuzzy methods to a two phase flow of incompressible fluids in porous media known as a Backley-Leverett or fractional flow problem, were very discouraging [9]. In particular, it has been shown that after several time steps the water and oil saturation profiles were fuzzified to such an extent that solution became meaningless. Another attempt to solve the same problem showed non-physical results of fuzzy numerical simulation [4]. Still this area of possible applications of fuzzy methods is one of the most interesting ones since it allows to acknowledge the data available at early stages of reservoir characterization (this data are apriory fuzzy) and substitute (if possible) the time-and-resources consuming sensitivity analysis by a single-run simulation.

3. Fuzzy Arithmetic and Interval Numbers

Fuzzy numbers (FN) were first introduced by Dubois and Prade [2]. A "non-linear" triangular FN (often called a FN of RL-type) can be represented by the following generalization [11]:

tLA(X) = max(O, 1 -iyn, (1)

where

Fig.I represents a scope of fuzzy numbers obtained by using different values of the power t. As follows from Fig.} a linear triangular FN corresponds to t = 1, and an interval number corresponds to the extreme case of a fuzzy number with t ~ 00.

Fuzzy number can be represented by an interval number and vice versa. For example, a linear triangular FN can be defined [7]:

1. By its membership function tLA(X, h):

2. By the interval of confidence in function of h:

ai(h) < x < a2

a2 :S x :S a3(h)

(2)

144 A.B. ZOLOTUKHIN

1=0 0<1<1 (=1

1.0 1.0

0.0 '--~-. 0.0 ~_-+-__ -..

1>1 1»1

1.01----,.......,. 1.0

0.0 ~_--+-_~

Figure 1: Examples of fuzzy numbers of RL-type

Here h is the level of presumption, which represents a degree of confidence (0 ~ h ~ 1), and [ai(h), a3(h)] - the corresponding interval of confidence or, simply, an interval number.

Operations on fuzzy numbers originate from classical operationas with the interval of confidence [1, 2]:

[alt a3] ED ebb b3] = Cal + bI, a3 + b3]

[al,a31 e [b1 , b3 ] = [al - b3,a3 - b11

[al,a3] 0 [bl ,b3 ) = [min( II aib;),max( II aibj)) i,;=1,2 i,;=I,2

(4)

In the following sections analysis of a two-phase flow will be based on interval numbers. However, results will be valid for fuzzy numbers since any FN can be represented by an interval number and vice versa, as follows from equations (2), (3) and (1).

4. A Non-Fuzzy Setup of a ID Waterflooding Problem

Let us consider the simpliest example of numerical simulation, namely, a ID simulation of waterflooding based on a fraction flow theory (FFf). A material balance equation for


the water phase can be written in the following form:

8Sw 8F(Sw) 7ft = -v 8x ' (5)

Initial and boundary conditions for Sw(x, t) are standard, i.e.

(6)

Here F(Sw) is a fractional flow function, F(Sw) = uw/(uw + uo ), v = u/¢, u is a superficial (Darcy) velocity ofa two-phase flow, u = U w + uo , and Sw - water saturation.

Explicit finite difference scheme for a non-fuzzy model can be written as follows:

S!'l+1 = S!'- + v6t (F!'- - F!'-) • • 6x .-1 •

(7)

The same explicit scheme will be used later on for solving fuzzified fractional flow equations.

5. Fuzzy Setup of a ID Waterflooding Problem

Let us assume now that some of the input data is not known with certainty and, thus, can be represented as interval numbers. For example, fuzzification of porosity can be expressed according to (3) as

(8)

In general, fuzzification of the input data will result in fuzzification of an output, and thus, equation of continuity ( 5) in a fuzzy environment can be written as:

88w _ 8F(Sw) -- = -v 8 --::---8t 8x '

(9)

As follows from (5) a superstitial (true) velocity ii is calculated as

_ U1 U3 V = [VI, V3] = [¢3 ' 1>1 ]

There are two new terms in equation (9), namely, 88w/8t and 8F(Sw)/8x that should be defined. As follows from Fig.2, the following four different cases can be considered:

a) An arbitrary function f(x) is an increasing function of x, and the interval of its fuzzified analog /(x) is a non-decreasing function of x, as shown in Fig.2, a). In this case

8j _ [8f 1 8h ] 8x - ax' 8x (10)

146 A.B. ZOLOTUKHIN

a) b)

x x

c) d)

x x

Figure 2: Defining a derivative for a fuzzified function

b) An arbitrary function f(x) is an increasing function of x, and the interval of its fuzzified analog J(x) is a non-increasing function of x, (Fig.2, b». In this case

aj _ [ah aft) ax - ax' ax

(11)

c) An arbitrary function f(x) is a decreasing function of x, and the interval of its fuzzified analog j(x) is a non-decreasing function of x, (Fig.2, c». In this case

(12)

d) An arbitrary function f(x) is a decreasing function of x, and the interval of its fuzzified analog j(x) is a non-increasing function of x, (Fig.2, d». In this case

aj _ [813 8II ] 8x - 8x' 8x

(13)

It can be proven that this definition of the extended derivative is consistent with the definition used in [3) for the extended derivative for fuzzy numbers of RL-type.

It follows from the problem setup (5), (6) that the water saturation Swat any time is a non-increasing function of the distance x. It means that the derivative of the fractional


flow function F[Sw(x)] with respect to x is non-positive, i.e.

_ aF(Sw) > 0 ax -

On the other hand, at any fixed distance x water saturation is a non-decreasing function of time t, i.e.

asw > 0 at -This simple analysis leads to the following transcription of the right-hand side term of eq. (9):

asw , I asw , 3 . aft al3 aft al3 [~,~] = [VI ·mm(- ax'- ax ),V3 ·max(- ax' - ax)] (14)

Here 11 (Sw) and 13 (Sw) are fuzzified analogs of F(Sw).

Initial and boundary conditions for eq. (14) can be written according to (6) as follows:

(15)

As follows from (15) lower and upper bounds of water saturation should satisfy the following constraints:

(16)

Fractional flow function F(Sw) and its fuzzified analogs ft (Sw) and 13 (Sw) are defined below:

F(S ) - krw(Sw) Swe ~ Sw ~ SJ (17) w - krw(Sw) + (J-Lw/J-Lo)kro(Sw)'

jl(Sw) = krw,l(Sw) S < S < S (18) k (S) ( / )k (S) we, 1 - w - J,3

rw, 1 w + J-Lw, 3 J-Lo, 1 ro, 3 w

I3(Sw) = () krw ,3(Sw) ) Swe, 3 ~ Sw ~ SJ,l (19) krw, 3 Sw + (J-Lw, d J-Lo, 3)kro, 1 (Sw

Here I (Sw - Swe) a",

krw = krw S J - Swe ;

( S S )aW'3

I w - we, I krw, 1 = krw, I S _ S ;

J,3 we, I

( S S) ao. 3 I J,l - w

kro, 1 = kro, 1 S _ S J,l we,3

( S S ) a", 1

I W - we 3 ' krw, 3 = krw, 3 S _ S ' ;

J,l we,3 ( S S )

ao 1 I J,3 - w .

kro, 3 = kro, 3 S _ S ' J,3 we,l

and k~i' k~i, l' k~i, 3 are end-point values for the i-th phase relative permeability curve, i = W,o.

148

k'ro3 ~ k' " ro " , ,

A.B. ZOLOTUKHIN

F. f

k· .... 3

o Swc1 swcSwc3

Figure 3: Crisp F and fuzzified b, /3 fractional flow functions

Fig.3 exemplifies relative phase permeabilities to oil and water, fractional flow function F(S,J and its fizzified analogs.

According to eq. (14), (18) and (19) a fuzzified fractional flow function F(Sw) = [11 (Sw), 13(Sw)] that will appear in eq. (9) should be defined as follows:

IdSw) : 13(Sw) :

R(Sw) = min{f~ (Sw), I~(Sw)),

l~(Sw) = max{f{ (Sw), I~(Sw)), Swc.l $ Sw $ SJ,3

Swc, 1 $ Sw $ SJ, 3

A function satisfying these conditions is depicted in Fig.4 and is defined below:

S* $ Sw $ SJ, 1

Swc.l $ Sw $ S*

S* $ Sw $ SJ,3 Swc. 3 :::; Sw :::; S*

(20)

(21)

(22)

Here S* is the water saturation at which derivatives of both II (Sw) and h (Sw) are equal, i.e.

Eqs. (21), (22) and Fig.4 clearly show that functions 11 (Sw) and 13 (Sw) are defined on the corresponding intervals defined earlier by relation (16)1.

With the defined in such way fuzzy fractional flow functions the studied problem (14)-(15) can be stated as follows2 :

I Note that defined in such a way fuzzified fractional flow function becomes a crisp function F( Sw) when the interval of uncertainty for each of the variable parameter in (17)-( 19) becomes zero.

2 Here and further on a subscript w denoting the water saturation, will be omitted.

FUZZY SIMULATION OF WATERFLOODING

-'

· · · · · · · · , , , /

,/

!

Figure 4: Defining fuzzified fractional flow function [11 (Sw), 13(Sw)]

i = 1,3

S-(O t) - SJ--, , - ,&, i = 1,3

149

(23)

(24)

Solution of this system of equations is an interval function S(x, t) = [SI (x, t), S3 (x, t)] where SI(X, t) and S3(X, t) should represent the minimum possible and maximum possible water saturations, respectively. However, it is not correct due to the non-linearity of the studied problem. For example, the mean water saturation distribution can not be obtained as a half-sum of the minimum and maximum values. Results of preliminary numerical simulations proved that the mean water saturation at some points of a reservoir might be higher than the maximum possible water saturation corresponding to the upper bound of the function S(x, t). To avoid this inconsistency, one has to define modified functions Fl (S), F3 (S) that will "appreciate" all possible fractional flow functions that can be derived within the given range of uncertainties of the input data_ Corrected in such a way functions will generate "true" minimum and maximum possible water saturations.

150 A.B. ZOLOTUKHIN

As shown in App.endix, solution of this problem belongs to a class of objects (often called as Cantor sets) with the following distinguished property: derivative of such a function does not coincide with its slope. However, the function's derivative can be described by a spline-function that should satisfy certain conditions and whose integrant we will call a auxiliary function3. Thus, instead of two functions 11 (S) and 11 (S) describing a fuzzified fractional flow function F(S) we will have four new ones: two modified fractional flow functions Fl (S) and F3(S) representing the lower and upper bounds of F(S), and two auxiliary functions Ft (S) and F; (S) representing integrants of the derivatives of the corresponding modified fractional flow functions. Using these definitions one can finally describe a fuzzy setup of a ID waterflooding problem by the following equation:

8Si 8Ft (Si) 8t = -Vi 8x' i = 1,3 (25)

that should satisfy initial and boundary conditions (24).

In this problem setup functions Ft (S) and F;(S) are used for calculating the water saturation from a finite difference analogs of eqs. (23) while modified fractional flow functions Fl (S) and F3 (S) are used for evaluation of the water cut and oil and water flowrates.

As follows from the definition of fuzzy and interval numbers (3), the most likely solution can be obtained from system (5) - (6).

6. Simulation Runs and Analysis of the Results

Several runs with different degrees of uncertainty of the input data are presented below. In all simulation runs an explicit numerical scheme (7) has been used.

Fig.5 corresponds to the case when only two parameters, i.e. injected and intitial water saturation, are fuzzified (see Table 1, Run # 1).

Fig. 6 shows simulation results for the case when all input parameters were considered as interval numbers (Table 1. run # 2). As expected, the latter case is quite different from the previous one although the most likely values of all the input parameters in both cases were the same. As follows from the problem setup the distribution of the most likely water saturation coincides with that obtained by conventional methods (bold lines in Fig. 6).

Note that results of fuzzy simulation include all possible outcomes that can be generated by running a sensitivity analysis. Simple evaluation shows that if a conventional sensitivity analysis is carried out for n variable parameters and if for each of them minimum, maximum and most likely values are taken than fuzzy simulator runs n times faster bringing the same information.

Fig. 7 shows simulation results of waterflooding in a reservoir with the uncertainty in the porosity evaluation4 increasing linearly with the distance from injection and production

3 Definintion and construction of this function is given in Appendix. 4For a I D case it is equivalent to the uncertainty in reservoir characterization.


Table I: Input Data for Fuzzy Simulation of Waterflooding

Parameter Run # I Run#2

Min. value Max. value Min. value Max. value

Number of grid nodes, M", Reservoir length, X res , m Viscosity of oil, Jto, cp Viscosity of water, Jtw, cp Reservoir porosity, I/> Connate water saturation, Swc Initial water saturation, Swin Saturation of the injected water, Swinj Fluid velocity, U, mIhr End-point reI. penneability to water, k~w End-point reI. penneability to oil, k~o Power expo for water reI. penn. curve, WE Power expo for oil reI. penn. curve, 0 E

s:: o

.,-i

oW III

0.9

0.7

WI 0.80

2.00 2.00 1.95 1.00 1.00 0.95

0.200 0.200 0.190 0.190 0.210 0.190 0.190 0.210 0.190 0.790 0.810 0.790 0.070 0.070 0.069 0.300 0.300 0.290 0.500 0.500 0.490 2.00 2.00 1.95 2.00 2.00 1.95

~ 0.5~ ................................. i····· .. ·~11····· .............. : .............................. ~ .. ~ ... i ...................................... ··1 oW III !II

t1=0.3 hrs

WI 0.80

2.05 1.05

0.210 0.210 0.210 0.810 0.071 0.310 0.510 2.05 2.05

)..f (J)

oW III S

0.3

, 'lImm' mmnnnI_M __ om _on ____ :Lnm_nm 0.1L---------~-------+--------~--------~------~

o 20 40 60 80

Nodes

Figure 5: Fuzzy simulation of waterflooding. Distribution of water saturation at different time steps. Connate water and water saturation at the injection site are interval numbers (see Table 1, run # 1)

100

sites (Run # 3). As follows from the figure, saturation profiles propagate more or less uniformly at early stages of waterflooding when the influence of the uncertainty in porosity description is not strong. Gradually they start to disperse, as far as the displacement front

152 A.B. ZOLOTUKHIN

0.9

; . o . 5--- -_ .... :._ ... _ .... _._ .... -1-...... -

• I . - --... -.. - .... ~ ... -.---... -..... L_._~_

1 I

1. ____ _ . . ,

0.3 , t1=O.3 hrs! ~=O.6 hrs : l:!=O.9 hrs ~

-----irt~J::J]IT---:In;---;-~r i

0.1L-________ ~ ________ _L ________ ~ __________ ~ ________ ~

o 20 40 60 80

Nodes

Figure 6: Fuzzy simulation of waterflooding. Distribution of water saturation at different time steps. All input parameters are interval numbers (Table 1, run # 2)

100

approaches the middle part of the reservoir, where uncertainty level is the highest. Interesting to note that saturation profiles continue to diverge even when the fuzzified front passed the area with the highest uncertainty, bringing highly dispersed water saturation front to the production point. This fact is reflected by the shape and divergence of the cumulative oil recovery and a water cut curves as shown in Fig. 8.

Note that introduction of a non-uniform distribution of uncertainty to the reservoir characterization does not complicate simulation and has no effect on the elapsed time. The latter is valid for a more general case of heterogeneous reservoir with a non-uniform distribution of uncertainty.

7. Concluding Remarks

• A new approach based on fuzzy numbers has been introduced to a problem of fluid flow through porous media. The method is based on fuzzification of parameters describing reservoir/fluid properties (input data) and results, via application of operations on fuzzy numbers, in 3 equations of continuity (instead of one conventional equation) describing the mean, minimum and maximum possible solutions to the fractional flow problem.

• Fuzzy simulation of I D waterflooding problem bring sound physical results fuzzified to the extent dominated by the input data uncertainty and not by the method

0.9

0.7

0.5

0.3

0.1


Ell ......... s ==. ~:' . . '"'w3 .....

----.. \'

\ ; I I t,=0.3 hrs ~=0.6 hrs

\ : \. t3=0.9 hrs

I I' \.

\ . \ I ··1 . I'

I \ I t \ ' ~:.. _-== .:::_ -=. -= :.. _ t

o 20 40 60 80 100

Nodes

Figure 7: Fuzzy simulation of waterflooding in a reservoir with unceratinty in porosity description increasing linearly with the distance from injection and production sites (Run # 3). Distribution of water saturation at different time steps.

itself. In case when all the input data are set as crisp numbers the simulator will generate a classical deterministic solution.

• Solutions to these equations encapsulate all possible outcomes that can be obtained by running a sensitivity analysis within a given range of uncertainty of the input data.

• Simple evaluation shows that the method is approximately three times slower than conventional deterministic simulation without running the sensitivity analysis, performing equally fast with conventional simulator when running the sensitivity analysis for a single variable parameter, and runs n times faster than the conventional simulator when the sensitivity analysis is applied to n input parameters.

• In case of a heterogeneous reservoir advantages of a fuzzy approach are even more obvious since the introduction of heterogeneity in the reservoirlfluid parameters description does not complicate the [fuzzy] calculations and does not increase the elapsed time.

154

+J ~ t)

H Q)

+J III 3:

1.0 r----I

0.8 t

0.6

0.4

0.2

o -!--_-.;...,L,

0.1 1.1

A.B. ZOLOTUKHIN

> ____ -_v.~

...... , 0.20

0.16

0.12 .~

Je_.- .. _·-"_·-_·'·_-"

.-.- .., .. _.,.._-

0.08 -------,--~------

'-... ~

0.04

o 2.1 3.1 4.1 5.1

Time, hrs

Figure 8: Run # 3. Cumulative oil production and water cut versus time.

Acknowledgement

>, H ~ Q)

> 0 t) Q) H

E ~ U

The author wishes to acknowledge Stavanger College for the opportunity to carry out this project and for financial support.

We thank Prof. Ya. Khurgin from the Gubkin State Academy of Oil and Gas whose work and insight has contributed to this paper. The author also wishes to thank Prof. A. Ulanovsky from Stavanger College and Prof. N. Eremin from the Oil and Gas Research Institute of the Russian Academy of Sciences for fruitful technical discussions.

The view and opinions presented in this article are entirely the responsibility of the author.


Nomenclature

EB e o <2> S~

t

S S* S** F(S)'f(S) p. krp(S) k~p u v

Subscripts

w water 0 oil

extended summation extended subtraction extended multiplication extended division value of variable S at the time level t = tk at the grid node i saturation saturation at which derivatives of functions It (S) and 13 (S) coincide frontal water saturation for the function h(S) fractional flow function viscosity relative phase permeability to phase p end-point relative permeability to phase p total superficial (Darcy) velocity of 2-phase flow total interstitial (true) velocity of 2-phase flow porosity distance time power exponent for water reI. permeability curve power exponent for oil reI. permeability curve interval number

we connate water I initial J injected

Appendix

Situation when solution S3(X, t) does not represent an upper bound for all possible solutions to the studied problem is shown in Fig. A-I. As follows from the figure, in some areas, due to the non-linearity ofthe problem, the most likely value ofthe water saturation Sex, t) is greater than S3(X, t). A corrected fuzzy solution is shown in Fig. A-2.

Construction of such corrected solution involves functions (modified fractional flow functions) which belong to a class of objects (often called as Cantor sets) with the following distinguished property: derivative of such a function does not coincide with its slope. Construction of the modified fractional flow function F3 (s) with such properties is shown in Fig. A-3 and A-4. Note that the function F3(S) within the path 1 - 2 is generally a

156

0.1

c: ~ 0,6

!'! ~ en 0.4

~ ~ 0.2

A.B. ZOLOTUKHIN

Interval numbers Interval numbers + MLV

~=O.3 hrs I

I

20

'..,=0.6 hrs

• I

I

40 60

Grid nodes

~ ~-s..,

~=o.g I'Irs

100

c: 0

~ ~ '" l

0.4 .. ~

0.2

~=O.3 hrs .,=0 .....

20 40 60

Grid nodes

Figure A-I: Example of a non-correct solution to the fractional flow problem: the most likely value of the water saturation S(x, t) is greater than S3 (x, t).

~,.-.. -- I ..

I

Figure A-2: Corrected fuzzy solution to the fractional flow problem.

1:,=0,9 tws

80 100

curved line. However, taking a reasonable assumption that the interval of uncertainty is not very wide one can approximate this part of the function F3 (S) by a spline aS2 + bS + c satisfying obvious boundary conditions at points 1,2 (Fig. A-4):

S = SJ,3: S = S**: S = S** :

as;, 3 + bS + c = 11 (SJ, 3)

as**2 + bS** + c = h(S**) 2aS** + b = I~(S**)

Corrected in such a way function F3(S) is depicted in Fig. A-4 (path 1 - 2 - 3).

(A-I)

Derivative of the modified fractional flow function I; (S) within the same interval can be


1 2

3 4

1=-=

Figure A-3: A corrected fuzzy solution to the fractional flow problem.

1

F. f

Figure A-4: Constructing corrected fuzzified fractional flow function.

slope

function

approximated by a straight line AS + B given the following boundary conditions at points 1,2 (Fig. A-4):

S = SJ,3 : S = SOOOO :

ASJ,3 + B = f~(SJ, 3) ASOO" + B = fHSOOOO) (A-2)

Thus, eqs. (A-2) identify a derivative of the modified fractional flow function F3(S) denoted as 13 (S). However, in the problem setup it is better to use, instead of the function

158 A.B. ZOLOTUKHIN

f; (S), its integrant F; (S) which can be called as an auxiliary function. The following procedure is used to secure the smoothness of the function Fj (S) important for numerical calculations (see Fig. A-5):

i) F;(S) coincides with the function F3(S) within the path 2 - 3.

ii) Within the path 1 - 2:

F;(S) = / J;(8)dS = ~A82 + B8 + C, S**::; 8 ::; 8J,3

iii) A constant C appeared due to integration is determined at point 2, i.e.:

!A(S**)2 + BS* + C = F;(S**) 2

Constructed in such a way auxiliary function F; (S) has continuous first derivative along the path 1 * - 2 - 3 (see Fig. A-5, a» and, therefore, secures the smoothness of its finitedifference approximation used in numerical simulation.

Modified function Fl (S) and auxiliary function Ft (5) are constructed in a similar way and are depicted in Fig. A-5, b). Note that here the path 4* - 5* is obtained by simple downward shifting of the path 4 - 5 of the function Fl (5) until it joines the rest of the curve (path 5* - 6).

1* ~ 1,---------~~~~~

1 b) 1~----------~~~~

F. f F. f

6

o Swcl Swc3 S".

Figure A-5: Modified fractional How functions Fi (S) and F3(S) and auxiliary functions Fi, F; .

Note that auxiliary functions Ft (8) and F; (S) are used for calculating the water saturation from the finite difference analogs of eqs. (23) while modified fractional flow functions Fl (S) and F3 (S) are used for evaluation of the water cut and oil and water flowrates.


8. References

G. Alepheld and Herzberger. Introduction to Interval Computations. Academic Press, New York,1983.

2 D. Dubois and H. Prade. Operations on fuzzy numbers. Int. 1. Systems SCI, 7:613-626, 1978.

3 D. Dubois and H. Prade. Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York, 1980.

4 N.A. Eremin. On solution to the problem of fluid flow through porous media by fuzzy mathematics. Oil Industry, (4):33-35, April 1990. (in Russian).

5 N.A. Eremin. Hydrocarbon Field Simulation by Fuzzy Logic Methods. Nauka, Moscow, 1994. (in Russian).

6 1.H. Fang and H.C. Chen. Uncertainties are better handled by fuzzy arithmetic. AAPG Bull., 74(8):1228-1233,1990.

7 A. Jones, A Kaufmann, and H.-J. Zimmermann. Fuzzy Set Theory and Applications. NATO ASI Series. Reidel, Dordrecht, 1985.

8 E.V. Kalinina, AG. Lapiga, V.V. Polyakoff, Ya.l. Khurgin, M. Wagenknecht, M. Peshel, K. Haintze, and K. Hartmann. Optimization o/Quality: Complex Products and Systems. Chemistry, Moscow, 1989. (in Russian).

9 Ya.l. Khurgin. Personal contacts. 1995-1997.

10 Ya.l. Khurgin. Fuzzy equations in petroleum geophysics. Techn. Cybernetics Journal, (5): 141-148, 1993.

11 Ya.l. Khurgin. Fuzzy methods in petroleum industry. State Gubkin Academy of Oil and Gas, Moscow, 1995. 131 p.

12 George J. Klir and Tina A Folger. Fuzzy Sets, Uncertainty, and In/onnation. Prentice Hall, Englewood Cliffs, New Jersey, 1988.

13 T. Terano, K. Asai, and M Sugeno. Fussy Systems Theory and Its Applications. Academic Press, New York, 1992.

14 J-R. Ursin and A.B. Zolotukhin. Optimization of a gas condensate field production performance by fault block modelling and decesion under uncertainty technique. In Proceedings of the 5th European Conference on the Mathematics of Oil Recovery, pages 365-374, Leoben, Austria, Sept. 1996.

15 L.A Zade. Fuzzy sets. In/onn. and Controll, 8(3):338-353, 1965.

16 I.P. Zhabrev and Ya.l. Khurgin. Fuzzy mathematical model for reserves estimate. In Proceedings VNIGNI, pages 37-40. VNIGNI, 1993.

17 AB. Zolotukhin. Methodology of a Multiobjective Systems Engineering 0/ Natural Hydrocarbon Fields Development. USSR Academy of Sciences, Oil & Gas Research Institute, Moscow, 1990. (in Russian).

18 A.B. Zolotukhin. A new approach to decision making in petroleum engineering. In Proceedings o/the International Meeting on Petroleum Engineering, pages 247-254, Beijing, China, March 1992. SPE.

19 AB. Zolotukhin. Managing uncertainties in resources evaluation and field development planning. In Proceedings 0/ the 9th European Symposium on Improved Oil Recovery, The Hague, The Netherlands, 20-22 Oct. 1997. EAGE. Paper No. 009.

160 A.B. ZOLOTUKHIN

20 A.B. Zolotukhin. Handling multiple realizations in a long-term production forecast. In Proceedings of the 6th European Conference on the Mathematics of Oil Recovery (ECMOR VI). Peebles. Scotland. 8-11 Sep. 1998. EAGE. Paper No. 8030.

21 A.B. Zolotukhin and Ya.l. Khurgin. Fuzzy approach and its applications in petroleum sciences. In Proceedings of the Conference on Fundamental Problems in Petrolewn Sciences. Moscow. Jan. 1997. State Gubkin Academy of Oil and Gas.

Part II

Mass transport and heat transfer

A TWO-DOMAIN MODEL FOR INFILTRATION INTO UNSA TURA TED FINE-TEXTURED SOILS

Abstract

A. ABDALLAH and F. MASROURI LAboratoire Environnement, Geomecanique et Ouvrages, Ecole Nationale Superieure de Geologie Rue du Doyen Marcel Roubault, BP 40 54501 Vandceuvre-les-Nancy, FRANCE

In this paper, a one-dimensional model handling the vertical infiltration process within a compacted clay column submitted on its top to a constant hydraulic head is presented. The model is based on the flow mechanism described by Philip (1968) and Jayawickrama (1990). The infiltration is supposed to take place simultaneously in two different porosity-domains. The fIrst is the soil intact matrix and the second represents a net of interconnected channels presenting much higher hydraulic conductivities. The flow in the two domains is described by Richards' pressure-based formulation and an exchange term reflects the interaction on the basis of Darcy's law. A time-centered Finite Difference scheme is used to accommodate the transient nature of the problem. No attempt is made here to reproduce the geometry of the two domains, thus no assumption is needed concerning macropore-distribution, only an estimation of macropore hydraulic conductivity is required. Numerical simulation of laboratory infiltration tests yielded good agreement with measured data.

1. Introduction

Fine-textured soils as clays and silts, are widely used in landfills and other facilities in order to constitute barriers to water and contaminant transport preventing water-table pollution. With the increasing environmental concern, the unsaturated flow within compacted soils has known in recent years, a growing interest among scientists from different domains (hydrology, soil physics, agronomy, etc.). One aspect of the problem which has been recognized and studied by numerous authors, concerns the presence within these porous media of preferential flow paths through voids much larger than the soil pores. These voids also named in the literature "macropores" may be formed by plant roots, fractures or simply due to non-uniform compacting energy and offer lower resistance to water infiltration than the intact soil matrix.


164 ABDALLAH A. and MASROURI F.

This paper presents a model for infiltration into unsaturated layers of compacted finetextured soils. The model is based on a two-domain concept which considers that the infiltration takes place simultaneously in two porosity-domains: the first represents the soil intact matrix; and the second is formed by connected macropores. In the next paragraph, a rapid overview of different approaches used to simulate macroporous flow is presented.

2. Literature review

The existence of preferential flow paths within soil has been mentioned since the early 1864 by Shumacher (1864), but the first attempts to integrate their effects in flow models appeared at the end of the 1970's (Scotter, 1978 ; Edwards et al., 1979 .. etc.). The available models dealing with preferential flow through macropores may be classified in two different groups according to the manner they choose to accommodate macropores.

The first approach defines the soil macropores by their structural composition i.e. by their form, diameter, spacing or by their proportion compared to the total porosity of the medium (Beven and Germann, 1981 for example).

The second one defines them by their hydraulic significance i.e. by their characteristic pressure (pressure at which they are emptied of water) or by their contribution to total infiltration rate.

As pointed out by Chen et al.(1993), the numerical values defining macropores with any of the above properties are not exact but are associated in all cases with matrix properties. There is therefore no universal manner to define macropores. Table 1 summarizes some of the most interesting attempts to model flow in macroporous soils.

Models based on macropores' structural organization

TABLE 1. Most interesting models for flow in macroporous soils.

Author (year) Description

Scotter (1978) Macropores are assimilated to cylindrical tubes which size and spacing are estimated from soil saturated hydraulic conductivity.

Beven and Germann (1981) Continuous approach based on an interconnected net of macropores presenting a constant hydraulic conductivity equal to soil permeability.

Davidson (1984) 2D homogeneous medium containing regularly spaced vertical fractures supposed to be instantaneously filled (no resistance to flow).

TWO-DOMAIN MODEL FOR INFILTRATION

Beven and Clarke (1986)

Anderson et ai. (1989)

Benson (1989)

Homogeneous medium containing a population of cylindrical macropores of various diameters and lengths.

Homogeneous layers containing cylindrical regularly spaced macropores ; takes into account lateral flow between layers.

Similar to Anderson et ai. but considers macropore' size and spacing variability by a Monte Carlo simulation.

Webb and Anderson (1996) Preferential flow is reflected by the 3D variability of the hydraulic conductivity.

165

Models based on Philip (1968) Two-domain approach in which the flow in macropores is supposed to be stationary and is determined using the saturated hydraulic conductivity and the initial and final values of water content.

macropores' contribution to flow

Jayawiclcrama (1990)

Jarvis et ai. (1991)

Hosang (1993)

Mohanty et al. (1997)

Adaptation of Philip's model to compacted clays through a continuous approach considering the transient nature of the macropores' flow.

Two-domain approach where the macropores' flow is based upon Beven and Germann' equation introducing corrections in relation with dimensions of macropores.

Two-domain model considering that the flow is governed either by macropores or by micropores separately depending on the water supply rate.

Approach based on a modified Van GenuchtenMualem model for water retention and hydraulic conductivity taking into account preferential flow.

When reviewing the existent literature concerning macropores flow modeling, it appears that most of the models are based on the two-domain approach (macropores and micropores) with or without interaction term. Some of the authors consider that the flow occurs simultaneously in the two domains. Others however suppose that it takes place exclusively through one or the other domain depending on fixed conditions. This second hypothesis is certainly much more subjective because requiring to fix conditions for the predominance of the flow either in macropores or in the intact matrix.

To conclude this literature review, we can formulate the following recommendations which in our view, are important to develop a better model for macroporous flow:


• the two-domain concept with simultaneous flow in the two domains should be used combined to a realistic exchange mechanism;

• macropores should be defined upon their hydraulic contribution to flow rather than their geometry or their dimensional distribution, since for practical considerations, the determination of the model parameters would become much better adapted to measuring techniques.

In the following paragraph, the formulation of the presented model for macroporous flow with consideration to these recommendations is exposed.

3. Model formulation

3.1. FLOW MECHANISM THROUGH COMP ACTED LAYERS OF FINETEXTURED SOILS

Jayawickrama (1990) described the manner in which the infiltration occurs through initially unsaturated layers of compacted fine-textured soils when submitted to a constant hydraulic head at the soil surface. This mechanism (figure 1), first presented by Philip (1968), supposes that the flow begins first through a continuous net of macropores which present higher hydraulic conductivity values. As the wetting front progresses through one micropore, lateral adsorption of water takes place from the macropore to the soil intact matrix. As soon as the wetting front reaches the interface between two layers, the liquid tends to spread horizontally before the infiltration continues through the layer below. At the same time, an infiltration occurs from the surface within the intact matrix of the soil.

Figure 1. Infiltration mechanism into layers of compacted finetextured soils - Preferential flow paths.

TWO-DOMAIN MODEL FOR INFILTRATION 167

3.2. MATHEMATICAL FORMULATION

In this model, the infiltration into the intact matrix of the soil as well as through the macropores is supposed to be completely described by the Richards' generalized formulation of Darcy's law (equation 1) for unsaturated flow.

as = ~ a'¥ = -~(k(S) a'¥ J at a'¥ at a'¥ az

(1)

where: '¥ is the soil hydraulic potential; t is time;

k(8) is the hydraulic conductivity function, which may be kmacro(S) for the macropores

and kmicro (8) for the intact matrix, with 8 as the soil volumetric water content.

Considering that from a practical point of view, the water content at one point may not be decomposed, both of the porosity domains should present a unique characteristic water retention relation. The hydraulic conductivity should however have a higher value in the macropores' domain. For convenience, a linear proportionality is assumed to exist between the conductivity values in the two domains (equation 2).

Rk = kmacro ~ 1 kmicro

(2)

To completely describe the two-domain model, we should define an exchange term to reflect the flow between the two domains. Equation (3) presents the formulation used for this term, it reflects a mathematical mean for each depth step based on Darcy's law.

where:

S is the flux adsorbed by the soil intact matrix from the macropores; A'll is the pressure difference between macropores and the soil intact matrix; AX is representative distance related to macropores size and spacing.

(3)

In order to simulate the horizontal flow at the layer interface, the progression of the wetting front in the macropores is stopped and only the exchange mechanism remains activated until complete saturation of the layer.


3.3. RESOLUTION

Equation (1) submitted to limit and initial conditions is non linear. Its solution requires use of a descritization technique. Here, the Crank-Nicholson time-centered finite difference scheme is associated with an iterative algorithm. As described in the flow mechanism, for each time step, we first compute the pressure state in the macropores which will enable the estimation of the exchange term between the two domains allowing to correct the initial pressure in the intact matrix. The flow equation for the intact matrix is then solved to determine the final pressure state for the time step. The method flowchart is presented in figure 2.

4. Numerical simulation

In order to validate the presented model, a numerical simulation of a laboratory infiltration test (Amraoui, 1996) was undertaken. The test was performed on a 6-layer column of compacted Jossigny silt of 30 cm of height and 9 cm of diameter. Initially, the material was maintained at a uniform water content (Wj = 14.3 %) and then submitted to infiltration under a constant hydraulic head (here the top was maintained at atmospheric pressure). A limit condition of free drainage was applied to the lower boundary of the column. Complete model parameters for the soil are given in table 2.

Problem geometry Model Parameters

Initial and Limit Conditions

Figure 2. Flowchart for the two-domain model resolution.

Water content


TABLE 2. Model parameters for Jossigny silt.

Parameter S:tmbol Value Unit Dry specific weight Yd 17.05 kN/m3

Saturated hydraulic conductivity ks 6.5 10'9 mls Saturated volumetric water content Os 36.7 %

Residual volumetric water content Or 2 %

Van Genuchten parameter a 4.5 10,3 lIcm Van Genuchten Earameter m 0.2313

Finite Difference discretization parameters were investigated in order to determine whether their values could have an influence on the results. Figure 3 shows that there is no significant effect of the time step (increased from 10 to 100 min.) on the evolution of pore water pressure during infiltration. The influence of depth step however is shown to be much more important (figure 4). For the numerical simulation, a time step of 10 min and a regular depth step of 0.3 cm were used. The water retention and the hydraulic conductivity curves were described by the Van Genuchten (1980) - Mualem (1976) model. For comparison, three simulations were made: one with one-domain model for the case of a homogeneous soil without macropores and two with the two-domain model corresponding to two values of hydraulic macropore conductivity (Rk = 5 and Rk =10). The computation was continued until the complete saturation of the soil column.

As it was not possible to obtain an estimation of macropore hydraulic conductivity, the Rk coefficient was increased arbitrarily to match the simulated and the measured cumulated infiltration. Figure 5 shows that a value of Rk = 10 yielded a reasonably good agreement. We notice here that the final infiltrated water volume is the same for the three simulations. The preferential flow in the macropores, as simulated here, affects only the manner in which the wetting front progresses through the soil specially at the first stade of the infiltration.

Figures 6 and 7 compare simulated and experimental pore-water pressure variation during the infiltration respectively at 7.3 and 19.2 cm depth. The ability of the twodomain approach to reproduce measured data reasonably well in the higher part of the column is demonstrated (figure 6). In the bottom part however, a poor agreement was obtained (figure 7). This may be related either to the variation of hydraulic conductivity in the lower layers due to dissipation of the compacting energy, or to the occurrence under testing conditions of air entrapment in the soil pores restraining the wetting front progression. Indeed, the values of measured suction are higher than those obtained even by the simulation without macropores and then failure to match data can not be explained by the manner in which the macroporous flow is considered.

170

ti

o

ABDALLAH A. and MASROURI F.

20

Elapsed time [hI

40 60 80 0+-----~----~~----~----_+~~/7.--r------r----~--~~

:' ~ -20

-Without macropores Ot = 10 mn

-""'Without macropores Ot = 100 mn

-With macropores

-..... With macropores

Ot= 10 mn

Ot= 100mn -80 L ________ --=======================:..1

Figure 3. Time step influence on simulated variation of pore-water pressure at 7.3 cm depth.

-80 -60 Pore-water pressure [kPal

-40 -20

- With macropores Dz=0.3cm

...... With macropores Oz= 0.6 em

- Without macropores Oz=0.3 em

...... Without macropores Oz=0.6cm

o

10

~ .;; e-O

20

30

Figure 4. Depth step influence on simulated pore-water pressure profile after 12 hours infiltration.

TWO-DOMAIN MODEL FOR INFILTRATION

2,0 -r---------------------------,

~ 1,5

.~ fl ~ 1,0

] ~ 0,5 .. G·· Measured data

--Simulation without macropores

--Simulation with Rk = 5

--Simulation with Rk = 10 0,0 ;c::=----+-----I-----.:===+======l====~

o 20 40 60 80 100 Elapsed time [hI

Figure 5. Simulated and measured cumulated infiltration vs. elapsed time.

o 20

Elapsed time [hI

40 60 80

O+-------+------+--,-?----+-~------+

-20

~ ~ '" -40 ~

I ~ -60

• Measured data

--Simulation without macropores

--Simulation with Rk = 5

--Simulation with Rk= 10

-80 L _________ --...::====================::::J Figure 6. Pore-water pressure at 7.3 cm depth vs. elapsed time.

171

172

o

ABDALLAH A. and MASROURI F.

20

Elapsed time [h 1 40 60 80

O+------+------~----~----~------+_----_+------+_~--_+

~ -20

~ [ -40

i ~ =--60

..•.. Measured data

---Simulation without macropores

- Simulation with Rk = 5

---Simulation with Rk = 10

.........•.. .. _ ............................ .

.' .... .. '

.......

.... ......•..

....•. .'

... ....

.'

-80~--------------------------------------------------~

Figure 7. Pore-water pressure at 19.2 cm depth vs. elapsed time.

5. Conclusion

A two-domain approach was presented to simulate infiltration into unsaturated macroporous soils. The model takes into account the macropores contribution to flow through their hydraulic conductivity as a unique parameter. It doesn't need any other prior information concerning their geometry. Comparison of numerical simulation (made with an estimation of macropores' hydraulic conductivity) of an infiltration laboratory test with measured results was satisfying particularly in the higher part of the soil column.

The model should however be improved to consider the variation of macropores' hydraulic conductivity induced by air entrapment and/or compacting energy dissipation. An inverse scheme is currently developed by the authors for a better parameter estimation from measured results.

6. References

Amraoui, N. (1996), Etude de l'Infiltration dans les Sols Fins Non Satures, Doctorate Thesis, INPL, Nancy, 325 pages.

Beven, K.J. and Gennann, P. (1981), Water Flow in Soil Macropores: II. A Combined Model. Journal of Soil Science, (32), 15-29.

Chen, C., Thomas, D.M., Green, R.E., and Wagenet, R.J. (1993), Two-Domain Estimation of Hydraulic Properties in Macropore Soils, Soil Sci. Soc. Am. 1., (57), 680-685.

Edwards, W.M., Van der Ploeg, R.R., Ehlers, W. (1979), A Numerical Study of the Effects of Non capillarySized Pores upon Infiltration. Soil Sci. Soc. Am. J., (43), 851-856.


Jayawickrama, W.P. (1990), Uquid Transfer through Preferential Flow Paths in Compacted Clay, Ph.D. Dissertation, Texas A & M University, 144 pages.

Mua1em, Y. (1976), A New model for Predicting the Hydraulic Conductivity of Unsaturated Porous Media, Water Ressources Research, (12), 593-622.

Philip, J.R. (1968), The Theory of Absorption in Porous Media, Aust. J. Soil Res., (6),1-19.

Scatter, D.R. (1978), Preferential Solute Movement through Large Soil Voids: l. Some Computations using Simple Theory, Aust. J. Soil Res., (6), 257-267.

Shumacher, W (1864), Die Physik des Bodens, Berlin, In Jayawickrama (1990). Van Genuchten, M.Th. (1980), A Closed-Form Equation Predicting the Hydraulic Conductivity of

Unsaturated Soils, Soil Sci. Soc. Am. J., (44), 892-898.

Numerical study of heat and mass transfer in a cubical porous medium heated by solar energy

"Bou bnov-Galerkin method"

A . AL MERS ,A. MIMET and M. BOUSSOUIS

Laboratoire de Thermique, Energetique & Mecanique, Universite Abdelmalek Essaadi-FacuIte des Sciences B.P 2121.

Tetouan. Maroc

Abstract This paper presents The modelization of heat and mass transfer in cubical reactor of solar adsorption cooling machine. The reactor is heated by solar energy and contains a porous medium constituted of activated carbon reacting by adsorption with ammonia. From real solar data, the model computes the performances of the machine and shows the existence of the optimal dimensioning of the reactor. For the resolution of the equations describing the coupling between heat and mass transfer, we have adapted a "Boubnov-Galerkin" method combined to an iterative process, this method provides a continuous distribution of the temperature and adsorbed mass. The convergence of the method is discussed and the numerical results are compared with the results provided by finite-difference method. Considering the rapidity of convergence and the order of Algebraic system ( That is generally inferior to 1 0), the proposed method appeared to be very effective in solving such problem.

1. Introduction:

Solid adsorption cooling machines have been extensively studied recently [ 1 ], they constitute very attractive solutions recover important amount of industrial waste heat medium temperature and to use renewable energy sources such as solar energy. The development of the technology of this machines can be carried out by experimental studies and by mathematical modelization. This last method allows to save time and money because it is more supple to use to simulate the variation of different parameters.

The adsorption cooling machines consist essentially of an evaporator, a condenser and a reactor (object of this work) containing a porous medium, which is in our case the activated carbon reacting by adsorption with ammonia. The principle can be described as follows: When the adsorbent (at temperature T) is in exclusive contact with vapour of adsorbate (at pressure P), an amount m of adsorbate is trapped inside the micro-pores in an almost liquid state. This adsorbed mass m, is a function of T and P according to a divariant equilibrium m=f(T,P) . Moreover, at constant pressure, m decreases as T increases, and at constant adsorbed mass P increases with T. This makes it possible to imagine an ideal


176 NUMERICAL STUDY OF HEAT AND MASS TRANSFER

refrigerating cycle consisting of a period of heating/desorption/condensation followed by a period of cooling/adsorption/evaporation. Figure 1 sketches the ideal cycle of the machine, during some phases of which the refrigerant has interaction with solid. If the condenser and evaporator are considered perfect, desorption/condensation and adsorption/evaporation (respectively the transformation 2-3 and 4-1) accrue at constant pressures of condensation and evaporation Pc and Pev respectively, the sensible heating and cooling phases (respectively the transformation 1-2 and 3-4) are isosteric ( this is realised by closing the reactor). The cooling effect during one cycle is proportional to the desorbed mass during the desorption/condensation phase: LI m = m2 - m3 = m) - m4

Practically, the no homogeneity of the extensive variables (temperature, concentration) in the porous medium contained in the reactor, represents a major difficulty for the representation of the physical reality by mathematical modelization. Indeed, because of the weak thermal conductivity of the medium [2], when the temperature is modified to an extremity of the reactor, a strong inohomogeneity of temperature appears, That induces by the "adsorption-desorption" process a mass transfer and consequently a gradient of concentration. This mass transfer even affects the mechanism of heat transfer in the medium because of the exothermic (respectively endothermic) nature of adsorption (respectively desorption) phenomena

LnP

Pc

Pev

Tmax Tc Tmin Tev

Figure 1 : Adsorption thermodynamical cycle Tmax : minimal and maximal Temperature of adsorbent, Tc, Pc :

Temperature and pressure of the condenser, Tev, Pev : Temperature and pressure of the evaporator, m : adsorbed mass

liT

A. AL MERS et al. 177

The model presented in this study simulates a transitory behaviour of the reactor, it uses equations of balance of heat and mass transfer in the porous medium (adsorbentadsorbat). Giving the solar data and the ambient temperature as boundary condition on the reactor, the model computes the performances of the machine. The equations of the model describing heat and mass transfer are strongly coupled and non linear. In order to solve them, we propose a numerical algorithm based on the application of the "Boubnov-Galerkin" method.

2. Modelization of the reactor:

Porous medium (activated carbon-ammonia)

r-~------~~~L--4------------~'--I

glass x

.. ------.,.v-------.. ---- 1---- H ................... 0

L-____________________ /~~~----------------~I I~----------------~~------------~ I I 'I

steel I insulator

Figure 2: Sketch of the solar reactor studied

The solar reactor studied in this work is represented in figure (2), it is constituted of a transparent cover, a lateral and rear insulation and a steel adsorber containing the activated carbon.

The model of heat and mass transfer used in this study has been validated by experimentation in the initial work that was done by Mimet and al [3], [4] on a cylindrical reactor consisting of a double stainless steel envelope heated by thermal oil, where they are determined the equivalent thermal conductivity coefficient and the exchange coefficient between the metallic wall and the porous medium.

The porous medium is constituted by grains of activated carbon reacting by adsorption with ammonia. In such system three phases are present (Figure 3): a solid phase constituted by carbon grains (s), adsorbed phase of ammonia (a) and gaseous phase (g).

Under the action of solar radiation, the mixture of activated carbon-ammonia has a transitory behaviour, the heat transfer is made only in the transverse direction x of the reactor (the lateral gradient is neglected), The porous medium is characterised by an equivalent thermal conductivity and assumed to be at thermodynamical equilibrium [ 4 ]. Therefore, the bed pressure is uniform and the resistance of mass diffusion through the pores is neglected. The model behaviour is controlled by transient heat transfer with heat, mass and sorption equilibrium equations.


The following data are necessary for the model: • Thermodynamical equilibrium properties of the ammonia [ 5 ] • Adsorption isotherms: The B.E.T equation [ 3 ],[ 6 ] • solar and climatical data. [ 7 ]

steel

X+dX Adsorbate (a) ~--------------

~;e------~~I~)(s) x

Figure 3 : Sketch of the physical system (Porous medium) studied

2.1 Equations:

The general equation of heat and mass transfer in the porous medium is obtained by application of energy and mass conservation laws to a layer of thickness dx (Figure 3)[ 4].

(r\Or _ Q(r) = K 82r PCm r;}t e ox2

(1)

Where Pi and Ci are respectively the density and specific heat of the phase i, a the

volume fraction of the adsorbed phase, Papp the adsorbent apparent volume mass, E the

porosity of the activated carbon, rna the adsorbed masse Kg I Kg carbon), !J. H the heat

of sorption (KJ I Kg) and Ke is the equivalent thermal conductivity of the medium.

Ke =0.41WlrtfC, h=3165Wlm2°C [3]

The three terms of equation (1) represent respectively: - Gas, adsorbate and solid heating

A. AL MERS et al.

- Source term - conduction energy

the source term Q is constituted respectively by the three following terms: - gas elastic energy - adsorbate elastic energy - adsorption energy

Equation (l) is completed by boundary and initial conditions:

• Boundary conditions : i) Continuity of the heat flux at the interface adsorbent-metallic wall :

h( Tp - T(H)) = Ke oT ox X=H

179

(2)

Where h is the heat transfer coefficient between the metallic wall and the fixed bed and Tp is the metallic wall temperature.

ii) Energy balance on the metallic wall of the reactor:

oT 17oSG(f)=ULS(Tp-Ta)+MCp 0: +Sh(Tp-T(H)) (3)

G( t) is The solar flux (W / m2 ), Ta the ambient temperature, MCp the capacity of the

metal, U L the heat losses of the reactor ( W / m2 K) and 110 is the captation efficiency

of the solar collector.

iii) Adiabatic condition at the back of the solid adsorbent ( X = 0) :

• Initial condition :

oT =0

ox x=o

the bed is assumed to be at the uniform temperature.

TA . is the morning ambient temperature.

(4)

(6)

The relationship giving the amount rna of gas adsorbed on a mass unit of microporous

solid has been the subject of many theoretical works, B.E. T (Brunauer, Emmet and


Teller) [8] model is used in this work for it perfect description of the adsorption of ammonia on the activated carbon [ 6 ]. This model based on the statistical considerations uses the basis postulate of Langmuir by supposing in addition that the gas can be adsorbed on several layers from the surface (the Langmiur processing remains valid for the adsorption in one layer of molecules ). The adsorbed mass is calculated by the following expression (Equation of B.E.T):

m (T P)=m ex l-(n+l)xn+nXn+l 'X=_P-:---:-a' °l-X l+(e-l)X-eXn+l' Ps(T)

(7)

Where T is the carbon temperature, P the gas pressure, Ps ( r) the saturation pressure of

the liquid adsorbate at the temperature T and mo, C, n are the parameters of the

model, they have been established by Mimet and al[3]: rno = 0.271 , C = 3.2

n = 1.62

During the heating and cooling phases with closed reactor, the total mass of ammonia remains constant and is the sum of the total adsorbed mass and the free mass of gaseous ammonia. Therefore, the rate of change in ammonia mass over time can be represented as:

(8)

where rng is the mass of gaseous ammonia (Kg / Kg carbon).

The heating and cooling phases are not perfectly isosteric but the generating and the desorption phases are isobaric.

2.2 Numerical resolution: "Boubnov-Galerkin" method [9 J

The numerical method used for this type of problems by other authors are generally based on analytical methods (that remain applicable in some limited cases) [ 10] , [ 11 ] or on finite-difference methods[ 12 ]. The characteristic feature of this last methods is that values are sought and obtained for the unknown quantities (temperature, adsorbed mass, etc.) at the node points of a computational mesh. The variation ofthese quantities between node points is implied by the order of the approximations used, but in practice, no use is made of this knowledge. In Boubnov-Galerkin method, on the other hand, the assumption is made that the solution can be represented by an appropriate combination of analytical functions over the whole of solution region without passing inevitably (in some conditions) by the discretization of the domain[ 13 leas the case of finite element method).


To facilitate the task, we make the following change of variable to eliminate the non

homogeneity of equations (2) and (4): T* ;:;:: T - Tp

Equations (1), (2), (4) and (5) become:

oT" if2T* d T pc --K--Q-pc -P=O

m ot e ox2 m d t

oT*

ox

oT*

=0 X=Q

= -~ T*{H) ox X=H Ke

T*(X,O) = 0

By the Boubnov-Galerkin method the solution would be assumed by the form :

" N T (x,t) = . .E a;(t) lP;(x) 1=1

(9)

(10)

(11)

(12)

(13)

Where a;(t) are coefficients which must be evaluated during the solution process. And

qJ;(x) are the expansion functions or basis functions whose form might be chosen and

form a complete set of basis functions, at last in part, to be compatible with some anticipated features of the solution. The challenge is to choose appropriate function <Pi and then to find the coefficients a such that (13) satisfies equations (9), (10), (11)

and (12). In this problem we choose qJ;(x) of the form:

lP;{x) = COS mix (14)

These function must satisfy the boundary conditions: for ( X = 0 ) it is evident and for ( X = H) by replacing (14) in (11) we obtain:

(m;H) tg(m;H)=Bi ; i=l, ... N (15)

where Bi is the adimensional number of Biot: Bi;:;:: hH Ke

The pulsation co; (i;:;:: 1, .. N ) are solution of the transcendent equation (15) (Figure 4):


Bi/ro H

~

o ro2H • roH

Figure 4: Evaluation of the pulsation (tJj

The coefficient a;(t) are found by requiring that the scalar product of equation (9) and

the weighting function tp Ax) (j = 1, ... N) over the solution region is zero:

N N 2 dT. L d ai L Ke d tp; Q p -- < rn. rn. > - a· < ---- rn. > - < --+- rn. >= 0 '1'/' 'rJ / 2 • 'rJ ''rJ

;=1 dt ;=1 PCm dx PCm dt (16)

j = 1, ... N

Where: < u, v >= LH u(x)v(x) dx

On matrix notation we have: d [A]-[a] + [B][a] = [F] dt

(17)

This represents a system ofN differential equations strongly non linear, the resolution is mad by the implicit discretization. The coefficients in the discretization equations will themselves depend on T, we shall handle such situation by Gauss-seidel iterations combined with the following expression of relaxation:

A. AL MERS et a1. 183

[at = r[at* +(l-r)[at-l (18)

Where [a r is the approximation at the current iteration m, [a r- l the approximation

at the previous iteration , [a]m* the solution by Gauss-seidel method at the current

iteration m and r the relaxation factor varying between 0 and 2.

3. Results and discussion :

In our simulation, we have used the solar and climatical data measured in Tetouan (Morocco) for a clear type day of July 1991 [ 7 ] (Figure 5). The solar collector studied is normal with simple glazing of the captation efficiency of 0.75 and a thermal loss

coefficient to the external medium of 5.5 WI m20 C [ 12]. The different results of the purposed method are validated by comparing them with results obtained by the finite-difference method using the implicit scheme.

1000

900

800

700

N 600 E 500 ~ 400 (!)

300

200

100

0 6 8 10 12 14 16 18

Hours

Figure 5: Diurnal solar flux versus time


3.1 Convergence of the method

Table 1 : Convergence of the temperature profile relative to N (H = 6 em, Ke = OAIW / mK ,I = 360min)

Boubnov-Galerkin solution Finite-difference solution

X/H N=1 N=3 N=5 N=7 N=8 N=9

0.0 57.71 57.82 57.79 57.78 57.77 57.77 57.85 0.2 58.75 58.59 58.52 58.50 58.50 58.50 58.58 0.4 61.79 60.81 60.70 60.69 60.68 60.68 60.79 0.6 66.63 64.50 64.44 64.42 64.42 64.42 64.57 0.8 72.94 69.94 69.83 69.82 69.82 69.82 70.01 1.0 80.28 77.25 77.06 77.02 77.02 77.01 77.20

Table 1 illustrates in a given instant, the rate of convergence relative to N of the temperature profile (compared with the finite differences profile), we can see that the temperature profile converges rapidly to a unique approximate profile when N increases, this implies the good stability of the method. However, the calculation shows

that the parameter Ke/ H2 has a significant effect on the number of basis functions N

necessary to obtain a good approximation. In general for the studied problem where

Ke / H2 remains greater than 18.22 W / m3 K the minimal number of basis functions

N that insures the convergence is no more than 10, this number increases when

Ke/ H2 decreases ( For Ke/ H2 less than 10.25 W / m3 K N must be greater than

15). To avoid the divergence of the iterative process, the relaxation factor rmust be inferior than 1. Note that even the choice of r that insures the convergence during the heating and cooling phase of the closed reactor can cause the divergence during the two other phases, in order to avoid such situation, we have used tow value of r, the first during the isosteric evolution ( rs ) and the second during the isobaric evolution ( rb ).

Table 2 gives the optimal value of rs and rb (r* sand r;) that maximises the rate

of convergence ofthe iterative process calculated for different heights H of the reactor.

A. AL MERS et al.

Table 2 : Optimum relaxation factor according to H (Ke = 0.41W / mK)

H(cm) 2 4 6 8 10 12

r o 5 0.82 0.72 0.60 0.58 0.55 0.50

rho 0.42 0.42 0.40 0.40 0.40 0.40

185

By using r· 5 and r: ' results show that the present solution method yields converging

results very quickly. Generally at each step of the successive approximations, no more than 7 iterations are required for the solution of the non-linear matrix. Table 3 illustrates an example of the convergence rate of the temperature profile for the first six-iterations.

Note that the number of iterations and the optimal values rO 5 and r: remain

insensitive to the variation of N.

Table 3: The first six-order approximate temperature profile (at t=1080 min)

with r; =0.58, N=10, H=8cm, Ke = 0.41W / mK

XIH 1st 2nd 3rd 4th 5th 6th

0.0 53.715 53.688 53.677 53.676 53.676 53.676 0.2 53.352 53.323 53.313 53.312 53.311 53.311 0.4 52.269 52.234 52.227 52.225 52.225 52.225 0.6 50.491 50.448 50.442 50.440 50.440 50.440 0.8 48.069 48.021 48.015 48.012 48.012 48.012 1.0 45.090 45.056 45.050 45.048 45.047 45.047

3.2 dynamic behaviour of the reactor:

On figures (6), (7), (8) and (9), we present the results for a reactor of height H=6 cm containing 30 Kg of activated carbon ( S=lm2). The minimal temperature of adsorption (morning temperature ) is 24°C, that of condensation is 28°C and the evaporation temperature is fixed to O°C.


Temperature (OC) 100

90

80

70

60

50

40

30

20 6 9 12 15 18 21 24 3

Hours

Figure 6:Temperature evolution of solar reactor versus time • Finite-difference, _ Boubnov-Galerkin

6

During the heating phase (6-llh, Figure 6), because of the poor heat conduction and the absence of the fort convective current, the heat flux defuses badly inside the bed, this engenders an important gradient of temperature where is estimated to be 2.75°C/cm (average gradient) at the end of heating phase (llh). The difference between the wall and the bed temperature becomes more important during the desorption and is estimated to be 19°C at the end of the heating phase, this is due to the weak thermal conductivity of the medium, the latent heat of desorption (LJH = 1600 KJ / Kg) and the interface steel-bed resistance. The bed pressure increases from the initial value of 4.27 bar to 10.94 bar in 5h (Figure 7) and remains constant until the end of the heating whilst ammonia is condensed.

14

13

12

11

10 'C' ro e 9 D..

8

7

6

5

4

6

A. AL MERS et al.

.6 Finite-difference

-- Boubnov-G alerkin

9 12 15 18 21 Hours

24 3

Figure 7 : Variation of the pressure in the reactor during one cycle

187

6

The temperature gradient becomes less important during the cooling phase (17-6h), this is simply due to the poor heat exchange between the reactor and the external medium, the bed pressure decreases to the initial value (4.27 bar) in 9hours but the temperature stabilises at 37°C at the end of the cooling. This implies that the total adsorbed mass at the end of the cooling is less than that of the beginning of the cycle (Figure 9), this induces a decreasing of performance of the machine during the cycle that follow. Therefore the heat losses of the reactor must be increased during the cooling.


OJ ~ -.. 0)

~ ~

III III ro E "0 Ql

..c .... a III "0 «

0.24

1 X/H = 1 0.22 2 X/H=O.6

3 X/H=O.4

0.2 4 X/H=0.2 5 X/H=O

0.18

0.16

0.14

0.12

0.1

6 9 12 15 18 21 24

Figure 8: Concentration of adsorbed ammonia versus time Boubnov-Galerkin , ----- Finite-difference

3 6 Hours

The concentration gradient is inversely proportional to the temperature gradient (Figure 8), that is explained by the fact that during the heating phase the hottest external layers of the medium desorbe ammonia, which will be adsorbed in coldest internal layers, this tendency is inverted during the cooling phase.

Mass (K g)

7

6 .5 of

6 am man ia

5.5 total adsorbed

5

4.5

4

6 9 1 2 1 5 1 8 21 24 3 H 0 U rs

Figure 9: Variation of the ammonia mass contained in the reactor during one cycle Boubnov-Galerkin , ----- Finite-difference

6


The calculation of the total desorbed mass of ammonia L1 m during the desorption phase

for a different value of H (Figure 10) shows the sensitivity of the performance of the machine versus the geometrical parameter H of the reactor. L1 m presents a maximum

of 1.74 Kg/m2 for an optimal value of H (H=5 cm) corresponding to the best dimensioning of the reactor ( for the functioning conditions already mentioned ) , the quantity of cold that results is assumed to be 1982 KJ/m2 per day corresponding to a theoretical solar coefficient of performance of about 7.5 %. This coefficient of performance can be increased by decreasing the heat losses of the reactor to the ambience (by improving the internal conduction of the bed for example) that are directly linked to the wall temperature. For the studied collector the calculation shows that these losses represent about 1.17% of the total energy absorbed by the reactor during the heating phase for 1°C of difference between the wall and the ambient temperature. Therefore the great difference between the metallic wall and the porous bed can be considered among the essential causes decreasing the performances of the machine.

2.00

0.40 -+------.------.-------,-------,----.---.-----,-----.

0.00 0.04 0.08 H(m)

0.12 0.16

Figure 10: Cycled mass versus the reactor height

(770=0.75, UL =5.5Wlm2°C, Tev=O°C, Tc=28°C).

The good concordance between the numerical results provided by the proposed method and those obtained by the finite-difference method show the validity of the proposed method for studied such problem. Generally, the Boubnov-Galerkin solution converges to the approximate solution or diverge, depending upon the choice of the basis functions, the difficulty of this choice represents one of the constraints of the method.


4. Conclusion

In this work we have adapted the • 'Boubnov-Galerkin" method to studied the non linear problem of heat and mass transfer in a cubical reactor of solar adsorption cooling machine working with an activated carbon and ammonia. This method that gives a continuous distribution of temperature and the adsorbed mass appeared to be very effective in solving the such problem. The conclusions that may be drawn from the present study are as follow: • The poor heat conduction inside the porous medium and the resistance between the

metallic wall and the bed engender the important temperature gradient and a great difference between the metallic wall and the porous bed temperature, this is considered as the essential causes decreasing the performances of the machine.

• The model shows the sensitivity of the total desorbed mass versus the height (H) of the reactor. For a fixed conditions of functioning the total desorbed mass presents a maximum for an optimal value of H, this implies the existence of the optimal dimensioning ofthe reactor.

• The good concordance between the results provided by the finite-differences method and those of the purposed method allows to envisage the application of this method to study the bidimensional case of a reactor with fins that improve the internal conduction ofthe bed.

References

[ I ] L. Luo, M.Feidet, R.Boussehain, Etude Thermodynamique de machine a cycle inverse a adsorption, Entropie NO 183, PPJ-Il, 1994 [ 2 ] R.E.Critoph and L.Tumer, Heat transfer in granular activated carbon beds in the presence of adsorbale gases. Int.J. Heat Mass Transfer. Vo1.38, No.9, pp.1577-1585, 1995 [ 3 ] A. Mimet, These de Doctorat , Etude tMorique et experimentale d'une machine frigorifique a adsorption d'ammoniac sur charbon actif, FPMs, Mons (Belgique), 1991 [ 4 ] A.Mimet and J. Bougard, Heat and Mass Transfer in Cylindrical porous Medium of activated Carbon and Ammonia, Resent Advances in Problems of Flow and Transport in Porous Media, 153-163, Kluwer Academic publischer, J.M.Crolet and M.E.Harti (eds), 1998. [ 5 ] Institut International du Froid, Table et Diagrammes pour l'industrie du Froid, Proprietes thermodynamiques du R12, R22, R717, Paris 1981. [ 6 ] A.Mahaman, These de Doctorat, Etude de l'adsorption de vapeurs purs sur solides poreux, FPMs, Mons (Belgique), 1989 [ 7 ] H. Aroudam, Evaluation du gisement solaire dans la region de Tetouan. These de 3eme Cycle, Faculte des Sciences de Tetouan (Maroc),1992

A. AL MERS et al.

[ 8 ] 1. Fripiat, J. Chaussidon and A. Jeli, Chimie physique +des phenomenes de surface, Ed Masson (Paris) , 1971

191

[ 9 ] G. Marchouk, V. Agochkov, Introduction aux methodes des elements finies, Traduction Fran~aise Edition Mir . Moscou, 1985 [ 10 ] A. Adell, Distribution des temperatures dans un capteur solaire a adsorption solide, destine la refrigeration solaire. Resultats experimentaux en climat equatorial, Rev.Gen.Therm.Fr, NO 266, PP. 79-91, 1984 [ 11 ] Teh-Liang chen, James T. Hsu, Application of Fast Fourier Transform to nonlinear Fixed-Bed Adsorption problems, AICHE Journal, Vo1.35, NO 2, PP.332-334, 1989 [ 12 ] J.J.Guilleminot, F. Meunier and 1.Pakleza, Heat and mass transfer in nonisothermal fixed bed solid adsorbent reactor: a uniforme pressure non-uniforme temperature case, Int.J.Heat Mass Transfer, Vo1.30, No.8, pp.1595-1606, 1986 [ 13 ] Roger F. Harrington, Field computation by Moment Method, IEEE Press Series On Electromagnetic Waves, Donald G. Dudley Editor, 1993

CYLINDRICAL REACTOR PERFORMANCE EVALUATION FOR A SOLAR ADSORPTION COOLING MACHINE

EL H. AROUDAM and A. MlMET Energetic Laboratory, Abdelmalek Essaadi University, Faculty of Sciences P.O. Box 2121, Tetouan -Morocco-

Abstract

In this paper, a numerical simulation of solar adsorption cooling machine is presented for the region (Tetouan) real climate conditions. From the computed collected mass, we determine the produced cold quantity anct the performance coefficient for typical clear -sky daily global radiation for each month. The numerical results are in good agreement with experimental ones and have been used to design a solar installation producing cold.

1. Introduction

Adsorption solar cooling machines using solid I gas couples represent recently an important deal of solar energy conversion. The implementation of these machines is simple and offers several advantages owing to their main components, namely, a condenser, an evaporator and a reactor which is the motor organ of the cooling machine. The purpose of this work is to study the performances of the reactor containing a porous medium of active carbon and ammonia. The collector is subjected to a real solar radiation for typical clear days of each month. These are characterised by a sunshine fraction values cr ~ 0,9 and a nebulosity index KI < 0,2 [ 1 ] and are used to evaluate the machine operating conditions.

2. Nomenclature

'tv : Glass transmittivity <lac : Metallic tube absorptivity Vac : Volume of steel Ps : Solar Power UI : Loss coefficient hi : Heat transfer coefficient between the metal tube and the porous medium De : External diameter of the tube Di : Internal diameter of the tube

193 J.M. Crolet (ed.). Computational Methods for Flow and Transport in Porous Media. 193-201. © 2000 Kluwer Academic Publishers.

194 EL H. AROUDAM and A. MIMET

Tac : Temperature of the metal Ta : Ambient temperature T : Temperature of the porous medium P : Pressure in the reactor {; : Porous medium porosity ex. : Fraction of the adsorbed phase p : Volume mass C : Heat capacity r : Layer radius t : Time Mi..ds: Adsorption heat

3. Design

The adsorption solar cooling machine" Figure 1.", consists of the following elements: - A solar collectors in which there are cylindrical steel tubes containing activate carbon reacted by adsorbed ammonia" Figure 2." . - The dimensions of the studied reactor and the considered physical parameters are given in table l.

V2 VI

Solar collector

Receiver V3

Figure 1. Schematic solar refrigerant

PERFORMANCE OF SOLAR ADSORPTION REACTOR 195

Figure 2. Scheme of the reactor

Table 1. Reactor characteristics and parameters

Length Internal diameter Tube wall thickness

Glass transmittivity Steel absorptivity

Mass of the solid adsorbent Evaporating pressure

1m 6cm 2mm

0.9 0.7

1.4 kg 4.2 bar

Glass

Ammonia + Active carbon

Steel

Insulator

The optimum diameter value is obtained as a function of the collected mass per day in diurnal cycle. This mass is obtained subtracting the adsorbed mass at the end of heating ( afternoon) from that at the beginning of heating ( morning ). The adsorption cycle thermodynamics giving the variation of Log P = f m (Iff) is studied in detail by Mimet [ 2 ]. The characteristic points of the cycle are the evaporation temperature, the condensation temperature, the reactor maximal temperature and the adsorbate temperature. In this study , all the results are based on the computation of heat and mass transfer in porous medium beds taking into account the following assumptions; - the porous medium properties have a cylindrical symetry - the three phases are constantly in local thermal, mechanical and chemical equilibrium - the pressure is assumed to be uniform - the temperature distribution is radial - convection heat transfer can be neglected - conduction heat transfer within the medium is characterised by an apparent thermal conductivity coefficient Ae . These assumptions have been verified both numerically and experimentally [ 2 ].


The computation of the collected mass is deduced from the evolution of the daily adsorbed mass using the isotherms formula of Dubinin - Radusgufish [ 3 ] and taking into account the necessary thermodynamics properties for the reactive medium equilibrium :

activated carbon [ 4 ], ammonia gases [ 5 ] and adsorbed ammonia [ 6 ].

4. General equation of energy and mass transfer

The general equation of heat and mass transfer in a porous medium is obtained from the energy and mass balances concerning a porous medium layer discretised fictitiously in a certain number of layers of width dr :

8I' [(1-&)psCs + (&-a)pgCg + apaCaJ-

8 P -[(&-a)pg]- 8 P -(apa)-Ot Pg Ot pa

8 8 2T i1Hads(T,P)[-(a pa)] = Ae[-Ot iJr2

The five terms of equation ( I ) represent respectively: - gas, adsorbed and solid heating - gas elastic energy - adsorbed elastic energy - adsorption energy - conduction energy

Ot

(1)

From equation (1), written for all the layers contained in the cylinder, we obtained a partial derivative set of equations completed by non-linear boundary conditions.

4.1 INITIAL CONDITION

Uniform distribution in the porous medium T(r,O) == Ti ( r = 0, ... , R )

T( r,O) : Layer temperature at radius r ?nd instant t == ° Ti : initial temperature R : radius of the cylinder containing the porous medium

4.2 BOUNDARY CONDITIONS (r= 0, R)

r== 0, Since the system geometry is cylindrical, the boundary condition is a symetry condition expressed by :

r= R,


(or) - =0 or r=O

According to this boundary condition, the flux density is :

The energy balance of the stainless steel tube is given by :

The implicit flnite differences [7, 8 ] method is adopted in order to numerically resolve the obtained non-linear set of equations . The computed results are the temperature T (i , j), the pressure P( j ) and the adsorbed mass rna (i , j) which are computed at positions ri , and instants j. These parameters are used for the determination of the daily collected mass [ 9 ]. The thermal and solar performance coefflcients, COP and COPs respectively, are deflned as follows

COP =OF/Qc COPs = OF/Qs

Qc = Q.t + Qmu + QCA + Qdes

where: OF : Cold quantity depending on the collected refrigerant mass Qc : Heat quantity necessary to heat the reactor Qs : Daily global radiation Qst : necessary heat for the steel tube Qmu : necessary heat to increase the ammonia temperature from T ads to T max

QCA : necessary heat to increase the adsorbent temperature from T ads to T max Qdes : Heat for ammonia desorption [ 6 ]

5. Results and analysis

In this study, the adsorption temperature is equal to the ambient temperature and the condensation temperature is equal to the ambient temperature at the instant when the liquid desorption begins after certain heating of the reactor. The maximum reactor temperature, reached about 14h and ISh (LGT), illustrated versus typical days of each month in "flgure 3.". The maximum value is 98,8 °C (in June) and the minimum value is 55,5 °C (in January ). "Figure 4." shows the variation of the collected mass as a function of the temperature discrepancy obtained along with the reactor heating. It represents the difference between


the reactor maximum temperature and the adsorption minimum temperature. We can easily see that this dependence is a linear one and that the collected mass increases when II T increases. The variation of the daily-collected mass, for the typical clear - sky daily global radiation of each month, presents a maximum in the summer and a minimum in winter. It depends on the porous medium maximum temperature, hence on the necessaty solar flux to heat to maximum the reactor ''Figure 5.". We remark that the collected mass value for October is superior to that of September. This unpredicted behaviour can be explained considering the typical cloudless day characterised by the recorded ambient temperature which is favourable for the adsorption and the desorption phases. This behaviour is also observed in the cold quantity curve "Figure 6." considering the fact that this parameter depends on the collected mass and the evaporating temperature (0 °C). Extreme values are 193,4 Kj ( May) and 68,1 Kj (January). These values are favourable for several applications. "Figures 7. and 8." Show respectively the evolution of the thermal and the solar performance coefficients. The COP varies from 26,99 % to 36,84 % and which higher values are obtained during summer and the minimum ones during winter. From these two illustrations we conclude that the COPs cannot be used to optimise a solar cooling machine. These results confirm some experimental results [ 10 ].

6. Conclusion

In this study, based on model of heat and mass transfer in porous medium, collected mass, thermal and solar performance coefficients are considered. Cold quantity with an interesting COP improve the quality of the adsorption machines. Moreover, they allow a design according to real operation condition, which is the main purpose of this study.

References

1. Aroudam, EI H. (1992) Evaluation du Gisement solaire dans la region de Tetouan, These de 3- Cycle, Facuhe des Sciences de Tetouan.

2. Mimet, A1991 Etude Theorique et experimentale d'une machine fiigorifique a adsorption d'ammoniac sur charbon actif, These de Doctorat d'Etat, Faculte Polytechnique de Mons, Belgique.

3. Bering, B. P. (1966) M.M. Dubinin. and V. V. Serpinsky. Theory of volume filling for vapor adsorption, J. Col. Int. Sci., 21,378-393.

4. Chemviron. (1988) Granular activated carbon, Chemviron, Bruxelles. 5. Institut International du froid. Tables et diagramr.les pour l'industrie du froid, Proprietes thermodynamiques

du R12, R22, R717, Paris. 6. Mahanlane, A (1988) Etude de I'adsorption de vapeurs pures sur solides poreux, These de Doctorat,

Universite de Mons, Belgique. 7. Smith, O. D. (1964) Numerical solution of partial differential equations, Oxford University Press. 8. Samuel, D. C. and Carl de Boor. Elementary Numerical Analysis. Mc Graw Hill inter Editions. 9. Aroudam, EI H. and Mimet A (1997) Etude numerique du transfert de chaleur et de masse dans un

reacteur cylindrique d'une machine fiigorifique solaire, 3lme congres de Mecanique, Facuhe des Sciences de Tetouan 22 - 24 Avril.

10. Critoph, R. E. An ammonia carbon soiarrefrigerator for vaccine cooling. Dept. of Engineering, University of Warwick, Coventry u.K.


110 it. maximal temp.rature

T( DC) f':,. condensation temperature

100 A A \l Adsorbate temperatura

A A

00 A

A A

80 A A

70 A

60

50

40

30 (:j (:j ~ ~ b

b f::, V

20 b ~ V V

.6. V V .6.

10 V

0 1 2 3 4 5 6 7 8 9 10 11 12

M:>nth Fig. 3. Temperature versus months

180 Me (g) 170 X 160 xX< 150

140

130 X X

120 X 110 X 100 X

90 X 80

70

60 X

X 50

40 45 50 55 60 65 70 75 80 Tmax" Tads

Fig. 4 Variation of the collected mass as a function of the temperature difference (Tmax -Tads)

200

180 170 160 150 140 130 120 110 100 90 80 70 60 50

EL H. AROUDAM and A. MIMET

Me (g)

A

401-~--~~~--~~--~~~--~-'

180 170 160 150 140 130 120 110 100

90 80 70 60 50 40

1 2 3 4 5 6 7 8 9 10 11 1 Month

Fig. 5 Collected mass versus month

Kj A

A A A

A

A

1 2 3 4 5 6 7 8 9 10 11 12 Month

Fig. 6 Variation of the cold quantity


0.40 COP

0.38

0.36 .6 .6

.6 .6 .6

0.34 .6

.6. 0.32

.6

0.30 .6

0.28 .6.

0.26

0.24 I ~

1 2 3 4 5 6 7 8 9 10 11 12 Month

Fig. 7 Evolution of the COP

0.15 COPs

A

0.14 A

0.13

A A

A 0.12 A

A

0.11 A .6.

A .6.

0.10

1 2 3 4 5 6 7 8 9 10 11 12 Month

Fig. 8 Evolution of the COPs

RETRASO, a parallel code to model REactive TRAnsport of SOlutes

I. BENET LLOBERA1,2, C. AYORA1 and J, CARRERA2

Iinstitut de Ciencies de la Terra. C.S.l.c., Barcelona.

2Departament d'Enginyeria del Terreny, E. T.S.E. C. c. P. B., u.P.c., Barcelona.

Abstract

Reactive transport of solutes in porous media has received an increasing attention due to a growing of the social sensibilitation on environmental and health problems caused by contamination of solutes. It is important, then, the characterization of reactive transport in order to predict accurately the behavior of solutes.

RETRASO is a code capable to simulate REactive TRAnsport of SOlutes. The code solves the reactive transport problem by substituting the chemical equations into a source/sink term of the transport equation leading to a system of non-linear partial differential equations. This system is discretized by applying the finite element method and the obtained discretized system is solved with the Newton-Raphson method (NR). For interesting cases, this size could be huge so, large computing time is required.

A parallel version of RETRASO has been developed to reduce the simulation time. The method used to parallelize is a SPMD (Single Program Multiple Data) with message passing communication for distributed memory architecture. As a result of the CPU profiling analysis; parallelization was focused on the following most consuming CPU time (more than 90%) parts: 1) the building of the Jacobian matrix of the NR linear system, and 2) solving the system itself. As communication between processors should be optimized for message passing models, a specific algorithm that minimizes the communication needs was designed for part (1). For solving the system, a linear solver module was developed at CEPBA (Centro Europeo de Paralelizaci6n de Barcelona, UPC).

The performance of the parallelized version of RETRASO was checked in a SGI ORIGIN 2000 machine with a PVM version based on sockets.


204 I. BENET LLOBERA et al.

1. Introduction

Reactive transport of soluts allows to consider simultaneously multiple solut reactions and the interaction between the different phases of the medium. This has been the objectives of different researchers during the last decades. Saaltink et al. (1998) have an extensive compilation of them.

The different formulations of solving reactive transport of soluts are basically: the Sequential Iteration Approach (SIA) and the Direct Substitution Approach (DSA). The first one (SIA) consist on solving the transport equation for each species where the chemical equations are represented as a source/sink term iteratively updated (Yeh and Tripathi, 1989; Samper and Ayora, 1993). The second one (DSA) consist on solving simultaneously the transport and chemical equations by applying a non lineal system solving method wich leads to big sizes of system equations (Steefel and Lasaga,1994; Saaltink et al., 1996). Saaltink et al. (1995) comparing both formulations conjeture that DSA is more robust (for precipitation/dissolution in kinetics conditions) and converges in less iterations (even they are more expensive in terms of computational time), so that, it can be competitive. Saaltink et al. (1998 b) give details about the test of this conjecture comparing DSA and SIA methods. They conclude that SIA requires generally more iterations than DSA and particularly gives problems in the cases of high kinetics rates and/or with a high number of flushed pore volumes.

Attending to the DSA method to simulate reactive transport in groundwater, the obtained system to solve has the following characteristics: (1) big needs of memory because of the big sizes of the system and, (2) high computing time required.

A new code is beeing developed (RCB) at the Universitat Politecnica de Catalunya resulting of the integration of CODE_BRIGHT, a multiphase flow code (Olivella et al., 1996) and RETRASO, a reactive transport solute code (Saaltink et al., 1997). At this moment we have a sequential version of RCB and a parallel prototype of the reactive transport module.!n the present paper are presented the first results of a prototype of RETRASO code developed for multiprocess machines in order to solve the above needs.

2. Numerical approach

2.1 Reactive transporl

2.1.1 Multiphaseflow and conservative transport

Acording to the formulation of Olivella et al. (1994), the multiphase flow in a porous medium can be written as a function of: (a) continuity or mass balance equation (1) and, (b) the Darcy's law (2) describing groundwater flow in terms of pressure:

a a/Pa¢J Sa)=V(PaqJ+fa

qa =-Ka (Vpa-pag)

(1)

(2)

RETRASO, A PARALLEL CODE 205

where a refers to the considered phase and being Pa the density, ¢ the porosity, Sa the saturation degree, qa the Darcy flow,fa the source/sink term, Ka the permeability tensor, Pa the pressure and g the gravity.

The transport of dissolved solutes in water (liquid phase a=l) is described by the following balance equation:

() a (PI fjJ SI c) = -V(PI qc)+ flce (3)

being c the concentration, Ce the external concentration and qc the mass flow composed of the advective flow and the dispersive contribution plus de diffusive one (4):

(4)

Combining equations (3) and (4) with the continuity equation (l), the transport equation (5) is obtained:

eX p,¢s/ a =L(c)

L(c) = V(p/DVc)-p/q/1c + flce-c)- fvc

withj;, the condensation term.

2.1.2 Chemical equations

(5)

(6)

Chemical reactions can be considered in equilibrium or kinetics depending on the consideration of sufficiently fast» or «insuffiently fast». For this reason two cases have been considered: equilibrium and kinetics.

Chemical equilibrium

In the same way of Saaltink et al. (1996) the action mass law (minimum of Gibbs) is considered as:

Se10gc + Se10g y(c) = logk (7)

where Se is the stoichiometric matrix, of NrxNc (N, reactions and Nc chemical species), k the equilibrium constants array, c the concentration array and rthe activity coefficients array.

Kinetics

The «slow» reactions can be characterized by the reaction rate (rk) being it a function of the reactives and the products. In the same way of Saaltink et al. (1998), the reaction rate is considered as a function of the concentration array (c). One example of kinetics is the mineral precipitation/dissolution expression (Saaltink et aI., 1998):


rk = ~ k (Y In () -11" (8)

where is ~ equal -1 for dissolution and + 1 for precipitation, k is the kinetic rate constant, (J'

the specific reactive surface of mineral, Q the saturation index of the solution respect to the mineral and, e and 17 empiric parameters. Another example is the specific kinetic terms for suspended particles (Perez-Paricio et aI., 1998):

(9)

being ka the atachment coefficient, kd the detachment coefficient (including both the effects of interception, sedimentation, inertia, Brownian diffusion, Van der Waals and electrokinetic) and a the concentration of the retained particles.

2.1.3 Reactive transport equation

Reactive transport equations are based on conservative transport equations (5). They can be written as (Saaltink et aI., 1996):

a a a PI rjJSI aUa + a (PI rjJSIUs(Ua»+ a (PI rjJSIUp)=4Ua)+USirk(Ua)

(10) Ua = u(C) ,. C = (C" cicl» (11)

being U a the total aqueous concentration, Us the total sorbed concentration, up the total concentration of equilibrium mineral, U the component matrix, Sk the stoichiometric matrix for kinetic reactions, c] the primary concentrations (unknowns) and C2 the secondary concentrations which are a function of the primary ones through the chemical equations.

2.1.4 Numerical solution of reactive transport equation

The solving method used in the code is the Direct Substitution Approach (DSA) consisting in the substitution of the chemical equations into the transport equations previously described. This leads to a non linear system of partial differential equations. This system is discretized by the finite element method obtaining a big size system of equations (number of unknowns = number of primary species x number of nodes) which is solved applying the Newton-Raphson method (equation 12):

d k k+l k k J fCc) (Cl - c) = - f(cl) (12) Cl


where! is the array of transport equations with the chemical equations substituted and k the iteration number of the iterative process.

2.2 Parallelization

2.2.1 General aspects

The numerical solution resulting from the Direct Substitution Approach (DSA) of Saaltink et al. (1998) leads to a big size systems with two essentials needs: (1) size and, (2) CPU time. The first one is a direct consequence of the geometry discretization (nodes of the finit element grid) and the number of chemical species simultaneously considered (the number of unknowns per node). A more computational cost is required because the iterative resolution procedure resulting of the non lineal system to solve. For this reason parallel techniques have been analysed in order to reduce CPU time and improve future simulations in real problems. The main goal of these techniques is to accelerate the solving procedure substituting the single process by various processes in order to obtain a better relation between the work done and the time spent. The measure for testing the parallel code is the speed-up or the relative increment. A speed-up of 2 means that the answer time has been reduced in a half. Two parallel computer architectures are possible: shared memory and distributed memory. The first one only has one memory where all the variables of all processes are stored. In the second one, each process has its own memory and when one of them needs information of the other one, a communication must be established. There are two programming models based on these two architectures: with shared memory or with message passing. The last one can be used in both architectures but must be implemented trying to mlllUllize communication as much as possible in order to avoid retarding the execution time. Distributed memory has been the selected system, The reasons are basically: (1) portable to different platforms, since PCs nets to workstations, (2) accesible in terms of money, nowadays a net of PCs is easier to obtain than a powerful multiprocess machine, (3) scalable, that means the facility of increasing the number of processes and (4) suitable in both architectures.


no lime tc

Code jlowc//(In

Reactive transport module

Figure I. Code structure

2.2.2 Analyses of the sequential code

The code is basically divided in the following modula (Figure 1): flow and transport data input and initialization, flow solving and transport solving. The transport module is divided in: (1) an initial block of conservative transport and, (2) reactive transport with two main parts which are computation and assembling the system matrix and solving the lineal system resulting.

AOO DEOO

examples

I_no RT·I DR.T.

10-30% O L5.

R. T. module Figure 2. CPU analyses (% total; R.T. = Reactive Transport, L.S. = Lineal system solving): CPU profiles od ACID and

DEDO examples and CPU values percentage in the reactive transport module.


The correct application of the selected parallel technique implies to characterize the most consuming CPU time parts of the code in order to obtain a real acceleration in the execution time. For this reason the CPU time of some examples were studied. The profiles evaluated showed (Figure 2) that more than 90% of CPU time was used by the reactive transport module. So that, the parallel technique has been developed in this module in the following parts:

(1) Compute and assemble the system matrix and the residual term. (2) The solving of the lineal system as a result of applying the Newton-Rapshon

method.

2.2.3 Description of the parallel prototype developed

The parallel code prototype is a SPMD model (Single Programm Multiple Data) with message passing for communication between processes of distributed architecture.The parallel technique applied has been focused on:

Q

0 ~ II

Q

0 ~

(1) Developing an specific algorithm that minimizes communication between processes (because communications must be optimized in the message passing programming model) for the residual term and system matrix computation and assembling.

(2) Coupling a resolution module developed by Cela et al. (1996) for solving the linear system.

Matrix

~ II

i~

.. i

tj

-Rfrsidlwf ten"

+ + i -

Y

R, R, R, R,.

Figure 3. The residual term and system matrix building where Ri is the residual term contribution of process i. and akiikiiij the reactive transport coefficient of colunmj. row i and the chemical species ii andjj

210 1. BENET LLOBERA et al.

Parallelism of the residual term and the system matrix

As it can be seen in Figure 3, parallelism has been focused on the node loop because of the independence of chemical nodal information and the conservative transport matrix coefficients already computed. The building of the different subcolumns (number of primary species) associated to each node j of the system can be done in different processes without communication between them. The residual term requires to be computed in each process because for a selected column all the rows can be updated. Any updating of the residual term implies a sum to the previous residual coefficient, so that, the residual term is evaluated as a sum of the partial residual contributions computed individually in every process (Figure 3).

Parallelism of the linear system solving

The module for solving the lineal system with the selected parallel technique has been developed at CEPBA (Centro Europeo de Paralelismo de Barcelona) during the European project RET ACO.

RETRASO, A PARALLEL CODE

o ~

* i = O,j = 1

yes

Compute precond. j 0=1,2)

GMRES

* i=i+l

yes i= 0,

j=j+l

yes

Figure 4. Flowchart of the linear solver algorithm.

211

This module is based on using the preconditioned GMRES method. The algorithm of the parallel linear solver (Cela et aI., 1997) tries initially (Figure 4) to use the unpreconditioned GMRES. In the case of no convergence, an iterative procedure which computes the preconditioner begins. In each iteration (no convergence of the solver with the preconditioner) the parameters of the preconditioner are modified by an adaptative algorithm. The most important parameter is the fill-in parameter. It has two bounds: maximum and minimum values. For a bigger fill-in parameter, a better preconditioner is obtained and the GMRES method needs less iterations to reach convergence. By the other hand, the drawback is that more memory and time is spent computing preconditioner. The adaptative algorithm modifies the parameters of the preconditioners in order to obtain the best relation between the memory/time spent computing the preconditioner compared with the time required to reach convergence.

The parallel solver has two types of preconditioners: SPAI and RILUT (Cela et aI., 1996). The first one use random communications and the second one requires deterministic communications. For this last situation, the matrix structure must to be known because communication needs depend on it.


Domain a ocialoo

Old structure New structure

Figure 5. Change in the grid numbering.

The optImIzation of communications leads to a selected matrix structure with the minimum communications required. The connectivity graph of the matrix is partitioned (Figure 5) into two sets of subgraphs named domains and boundaries. Cela et a1. (1997) have been used the domain decomposition in order to distribute the work between processes obtaining well balanced domains and minimum boundaries. After doing that, a block matrix structure is obtained (Figure 5) with communications between processes totally restricted to the boundary rows/columns.

This type of matrix structure implies a numbering of the nodal grid depending on the number of processes considered. For this reason, a module with an algorithm for renumbering the grid nodes has been added. It has been implemented at the beginning of the code once the connectivity graph of the matrix is known. This procedure has been incorporated in the code at this level for the following objectives: (1) to avoid pre-post process external work depending on the number of processes to be considered in the next execution. (2) Only input data is affected and consequently, output information where the numbering is removed to the initial one. With this, by one hand, for the user a single numbering exits and by the other hand, the programming of the code excluding input/output module remains unchanged.


3. Applications

3.1 Test description

The behaviour of the parallel code has been studied with two examples: ACID and DEDO. Both of them represent a one dimensional column (Figure 6) with steady state flow conditions and submitted to a known concentration inflow. ACID test is a calcite domain washed by an acidic water subsaturated to calcite, dissolving the calcite. DEDO test represents a dolomite column with a water infiltration rich of Ca that causes a replacement process of dolomite dissolution and calcite precipitation (Ayora et aI., 1998).

3.2 Numerical performance

The performance of the parallel version of code has been checked in a SGI ORIGIN 2000 machine with a PVM version based on sockets.

A family of both test has been created using a numerical model with a two dimensional grid of triangular elements. Both families have different sizes of the exactly the same geometrical domain and the hydrogeochemical problem but are generated with different criterion. The ACID family has a fixed horizontal discretization (~x) modifying exclusively the vertical one (~y) resulting different sizes that have different relation (~y/~x). By the other hand, DEDO family has been generated keeping a constant relation (~y/l1x).

The main reasons to consider two test families were: (1) Analyze different grid size for the same conceptual problem in order to check the

code scalability. (2) Study the influence of the size of the problems and the relationship between the

number of nodes and the number of chemical unknowns per node. (3) To check the influence of the (l1y/l1x) relation.

speed-lip 14 TEST (unknowns) Processes Speed-up

12 ~------~--~~~~ 10 M-----::---j . - __ -1

O ACIDl ACID I (6804)

ACID 2 (13440)

DEDO I (4059)

DEDO 2 (9900)

2 4 6 8

processes

Figure 7 Values of speed-up

2

4

2

4

2

4

2

4

6

5.04

6.08

6.29

9.54

2

3

2.99

4.57

6,88


Figure 7 summarizes the problem sizes considered and the obtained results in terms of speed-up. From this figure the most important feature is the extremely high speed-up reached. A detailed analyses of the speed-up in the reactive transport module is graphically shown in

20

15

10

5

0

5

4

3

2

1

o

2 4 6

~~ ~ .. f--

I --

2 4

.ACID1 (T.R) 40

30 o ACID1 (L.S.)

20

D ACID1 (J.+R.) 10

0

.DEC01 (T .R) 15

o DEC01 (L.S.) 10

o DEC01 (J.+R.) 5

I 0

• ACID2 (T.R) 1'1

... - 1'1 o ACID2 (L.S.) I--

- r-- o ACID2 (J.+R.)

~ l I, ~ IJ 2 4 6 8

DED02 (T.R)

o CED02 (L.S.)

o DED02 (J.+R.)

2 4 6

Figure 8. Analyses of reactive transport speed·up versus number of processes.

Figure 8. As it can be seen there is a good relationship between the reached speed-up in the building of system matrix and the residual term and, the number of processes. A well working has been obtained in this point. By the other hand, the solver block analyses show important increases in the relative increment overtaken. This effect seems to be caused by the behaviour of the iterative solving method of lineal system because the number of iterations required is different depending on the selected number of processes.

4. Conclusions

The experiments done allow us to say that the parallel code behaviour shows that the linear system solving is the most important part to affect in the execution time reduction.

Finally, it can be seen that, the future main goals of the code are summarized in the following points: (1) Continue with an exhaustive study with examples, the behaviour of that allows to check

the powerful tool developed.


(2) Analyze the needs of a parallel extension to other parts of the reactive transport and/or the flow system.

(3) Check in more detail the solving module behaviour in terms of: (a) general convergence of the iterative method and (b) convergence sensibility when changing the relation between the size of the problem and the number of process.

Acknowledgments

This work has been performed with the findings of CEE through RET ACO project and under contract with ENRESA. The work of the first author has also been supported by the «Direcci6 General de Recerca» (Generalitat de Catalunya).

References

AYORA, c., TABERNER, C., SAALTINK, M.; CARRERA,1. (1998). A discussion on textures and reactive modelling. Journal of Hydrology, 1998, in press.

CELA , J.M; Alfonso, 1.M. (1996). Parallel Linear Solver in RETACO project developed at CEPBA (Centro Europeo de Paralelismo de Barcelona).

CELA,1.M; JORDANA, S., BENET, I. (1997). '»RETACO, Development and parallelisation of reactive transport codes of use in environment management strategies.» PCI-II, Third deliverable.

OLIVELLA, S.; CARRERA, 1.; GENS, A., ALONSO, E. (1994). Nonisothermal Multiphase Flow of Brine and Gas Through Saline Media. Transport in Porous Media 15:271-293.

OLIVELLA, S.; GARCIA, A. (1996). CODE_BRIGHT: User's guide. Universitat Politecnica de Catalunya, Barcelona (Spain).

PEREZ - PARICIO, A., CARRERA, 1. (1998). A conceptual and numerical model to characterize clogging. Accepted in TISAR'98 (Third International Symposium on Artificial Recharge of Groundwater).

SAALTINK, M.; A YORA, c., BENET, 1. (1997). RETRASO: User's guide. Universitat Politecnica de Catalunya, Barcelona (Spain).

SAALTINK, M.; A YORA, C., CARRERA, 1. (1995). Comparacion de metodos para la simulacion de transporte reactivo multisoluto en el agua subterranea. VI Simposio de hidrogeologia, tomo XIX, 845-856.

SAALTINK, M.; CARRERA, J., AYORA, C. (1996). On the numerical formulation ofreactive transport problems. Computational methods in water resources. XI International Conference.

SAALTINK, M.; AYORA, C. , CARRERA, 1. (1998). A mathematical formulation for reactive transport that eliminates mineral concentrations. Water Resources Research, 34 (7), ppl649-1656.


SAALTINK, M.; AYORA, C. , CARRERA, J. (l998b). A comparision of two alternatives to simulate reactive transport in groundwater In these proceedings.

SAMPER, J., A YORA, C. (1993). Acoplamiento de modelos de transporte de solutos y de modelos de reacciones qufmicas. Estudios geol6gicos, 49, 233-251.

STEEFEL, c.1. , LASAGA, A.C. (1994). A coupled model for transport of mUltiple chemical species and kinetic precipitation/dissolution reactions with application to reactive flow in single phase hydrothermal systems. American Journal of Science, 294, 529-592.

YEH, G.T., TRIPATHI, V.S. (1989). A critical Evaluation of recent Developments in Hydrogeochemical transport models of reactive multichemical components. Water Resources. Reseach., 25(1),93-108.

A NUMERICAL STUDY OF THE COUPLED EVOLUTIONS OF MICROGEOMETRY AND TRANSPORT PROPERTIES OF SIMPLE 3D POROUS MEDIA

D. BERNARD (1), G. VIGNOLES (2) ( 1) LEPT, CNRS-ENSAM-Univ. Bordeaux 1 esplanade des mts et metiers, F 33405 TAlENCE Cedex bernard@lept-ensamu-bordeaux·fr (2) LCTS, CNRS-SNECMA-CEA-Univ. Bordeaux 1 3 allee La Boilie, F 33600 PESSAC [email protected]

Simulating, understanding and predicting the evolution during mineral diagenesis of porous rocks physical properties is a very complex problem. When properties like effective diffusivity, formation factor or permeability are considered interest can be mainly focussed on the coupled evolutions of micro-geometry and transport properties. This approach is theoretically justified using the volume averaging method: This now classical method yields differential problems at the microscopic scale. Their resolution permits the computation of the macroscopic transport properties. For the properties listed above, the results are completely determined by the microscopic geometry. The principles and the main properties of the numerical programs used to solve those 3D closure problems are presented. For well-sorted granular porous media, random closed packings (RCP) of spheres with realistic porosity are good first approximations of an initial micro-geometry. After a short presentation of the algorithm used to generate RCP of spheres for any granulometry, the different methods used to modify the micro-geometry are exposed. The differences observed between the effects of a purely geometrical evolution and geochemically governed evolutions suggested that both the history and the depositional conditions influence the properties of natural porous media.

1. Introduction

I a general way, the saturating fluid of geological structures contains dissolved species that are able to react with the minerals constituing the porous rock matrix when the thermodynamic conditions (temperature, pressure, stress, ... ) vary. These variations can be induced by fluid displacement, sedimentation, erosion, tectonic events, etc ... Geochemical reactions modify the micro-geometry of the porous rock and, consequently, its transport properties. Transport properties modifications will then induce new variations of the thermodynamic conditions. This feedback between large scale and microscopic phenomena is far from being completely


218 D. BERNARD AND G. VIGNOLES

understood even if it is of prime importance for several problems (hydrocarbon reservoir formation, nuclear waste disposal performance, etc.). In a simplified approach, it can be assumed that microscopic phenomena are in a quasi-steady state and isotropic. The large-scale phenomena will then control the velocity of evolution through the module of the local fluxes of the different species. The directions of the fluxes do not affect the evolution as a consequence of the assumed isotropy. If several species and minerals are to be taken into account, their spatial distributions will obviously affect the transport properties evolution mainly for two reasons:

1. All parts of the pore space do not have the same importance for transport. Then, depending of its position, the displacement of a portion of the fluid-solid interface may have very different effects.

2. Mineral textures can be extremely different (think about quartz and illite for instance) and, for the same change in porosity consequences will also be extremely different from one mineral to another.

The results presented in this work illustrate the first point and demonstrate that, even for simple porous media, characterising the porosity evolution is not sufficient to understand and predict the response of other physical properties to micro-geometry changes.

2. Microscopic models and volmne averaging

2.1 DIFFUSION AND REACTION

We consider a fluid phase (P) within which the product a is transported by diffusion. This product can react with the solid phase (a) at ~, the fluid-solid interface (figure 1). The equations governing those phenomena are the following ones:

dCa =V(DVC ) at . a

- n~cr . (DVCJ = k Ca

in the ~ phase

at the interface A ~

(1)

(2)

where Ca (mol m~3) is the concentration, t (s) the time, D (m2 S~l) the tracer diffusion coefficient of a in the solvent fluid, n~ the normal vector from the p phase to the a phase and k (m S~l) the first-order reaction rate coefficient. The fonn of the boundary condition (2) is obtained assuming several simplifications of thc interfacial phenomena: surface diffusion is negligible, the reaction is first order and irreversible (it is the case when surface concentration is quasi-steady and adsorption and desorption are linear with constant rates, that is, when no surface saturation occurs). Then we apply the volume-averaging operator (defined for the concentration in equation 3) to the previous equations.

(Ca )=! f Ca dV Vv

~

(3)

where Vis the volume of the REV (Representative Elementary Volume) represented by a circle onfigure 1.

MICRO-GEOMETRY EVOLUTION OF 3D POROUS MEDIA 219

To average the diffusion equation, integrals and derivatives must be correctly inverted. The rules for this are provided by the theorems of transport. The averaged form of equation (1) is the following [1]:

£ a(ca/ =V.[D(£V(Ca)~ +! JnJ}GCa dAJ~-av k(Ca )~ -! JkCa dA (4) at VA VA ~ ~

where £ is the porosity, av (m· l ) the specific surface and where the classical GRAY's decomposition of the concentration has been used:

Figure 1 :Considered geomelly: In the fluid phase (in white) dissolved species can diffuse and react with the solid phase at the fluid·solid interface.

(5)

This decomposition splits the microscopic concentration into a macroscopic part, the averaged concentration varying at a scale larger than the REV, and a microscopic part, the concentration fluctuation varying within the REV. Equation (4) is not strictly a macroscopic equation because some microscopic terms are still present (even if they are into integrals over the fluid-solid interface). To close the approach, we need a representation of the concentration fluctuation. It can be shown that the following one is appropriate [1]:


(6)

where the vector f (m) and the scalar s are solutions of two specific closure problems (partial differential problems posed within the REV):

• Forf:

• For s:

V2 f=0

-n~CI . Vf =n~ at A~CI

+ Periodicity at the limits of the REV

V2 S =_ a v k ED

k -n~ .Vs=- atA~

D + Periodicity at the limits of the REV

Equation (4) can now be written under its classical form:

where the effective diffusion tensor Deff is given by:

Deff =D(I+ ~ hofdA I A", )

(7)

(8)

(9)

(10)

Knowing the geometry of AtJcr within a REV and solving the closure problem (7), we are now able to compute the effective diffusion tensor. The ratio DeBf'D is an intrinsic property of the porous medium only depending of the micro-geometry. Deff is not modified by the existence of a reaction at the fluid-solid interface. The third tenn of equation (9) is the only one taking into account the existence of the surface reaction. In its complete fonnulation, the averaged fonn of equation (4) includes two extra tenns linked to this surface reaction:

1. A wnvective Uke lenn: V. [ e D[ ~ L"" S dA } C.)' 1 that can be proved to be generally

negligible compared to the others [I].

2. A second surface averaged tenn containing the concentration fluctuation: ~ J k Ca dA . VA

Ilo

This tenn is neglected because it is generally admitted and generally correct that

Ca «(Ca)~ every where. If this were not the case, then the representation (6) would be

invalid and the whole averaging procedure impossible. Direct simulation at the pore-scale


would then be an alternative for configurations where relevant boundary conditions can be specified at the limits of the computation domain.

2.2 PERMEABILITY

In this work convective transport is not taken into account but nevertheless the effects of diffusion and reaction on penneability can be calculated. When the fluid flow at the pore scale can be described by Stokes equations (1), applying the volume averaging method in a similar way that in the previous chapter [2, 3] yields a macroscopic equation, i.e. Darcy's law (12), and a closure problem (13), to be solved within the REV in order to compute the penneability tensor K (m2).

- VP p + IIp V2Vp = 0

V.Vp =0

Vp =0 at Ape-

+ BC at the other boundaries

(Vp)= __ l K.[V(Pp)P -ppg] IIp

-Vd+V2D=I

V.D=O

(d)P =0

(1)

(12)

(13)

where d (m) is a vector and D (m2) a tensor. The closure problem is completed by periodic boundary conditions at the limits of the REV and by the condition D = 0 at the Apcr interface. The permeability tensor can then be computed using:

(14)

3. Numerical models

3.1 RANDOM CLOSE PACKINGS OF SPHERES

Random close packingss (RCP) of spheres are often considered as a good models for unconsolidated granular porous media Experimentally, large RCP of spheres have a porosity of 0.3634 ± 0.0005 [4]. This value is difficult to obtain by computer simulation; a lot of methods are limited to porosity values around 42% [4]. It is not the case for the algorithm proposed by Jodrey and Tory [5]. Using the implementation proposed by Bargiel and Moscinski [6], generating several RCP of 10000 spheres with porosity values of 0.364 ± 0.002 is an easy task. Smaller values and large number of exact contacts between spheres are more difficult to obtain [4] but these aspects are not relevant here because we always use discretized approximations of the real geometry to solve the closure problems.


The following steps can resume the algoritlnn: 1. Random generation of Ns points within a cube. They will be the initial position of the

centres of the Ns spheres. 2. Computation of all the distances between the centres. Din is the minimum and E the porosity

of the packing composed of the non over-lapping spheres of diameter Din centred on the Ns previous points.

3. Definition of DOlt , a distance large enough to be never reached by the spheres diameter. For instance, the following expression can be used:

where V is the volume of the cube.

DOlll =V 6V N s1t

4. Definition of a contraction rate coefficient 'to 5. Contraction loop: Do while Din < Dout:

(15)

• Displacement of the nearest spheres (distance = Din) along the line linking their centres to a distance equal to Dolt.

• Computation of the new distances between centres and of the new value of Din. • Computation of the new porosity E.

• Computation of the new value of DOlt as a function of t, Ns and OE, the variation of porosity in the iteration.

6. End of the process, storage of the result and post-processing.

A generalisation of the algoritlnn to non-equal spheres is actually under test (figure 2 a) and comparison with 2D experiments is presently going on (figure 2 b) to evaluate the realism of the generated pac kings. We want especially evaluate the importance of the periodic boundaty conditions for large packings and find practical ways to introduce other kinds of boundaty conditions (like solid boundaries, pre-existing spheres, free surface, etc ... ).

3.2 PERMEABILITY

The closure problem (13) is similar to three Stokes problems with three different body forces. Several numerical methods are available to solve this kind of problem. The objectives being to deal with very complex large-sized 3D geometries, we chose a method as simple as possible [7]: • Regular grid. • Finite volume spatial discretization using staggered marker-and-cell meshes. • Pseudo transient iterative algoritlnn including an artificial compressibility coefficient. • Fully explicit scheme for time integration with stability criteria extended to 3D from [8].

On the fluid portion of the REV boundary, we use periodic boundaty conditions (BC) even for random porous media. This kind of BC is not absolutely required by the volume averaging method but it simplifies the form of the closure problem [3] and it is a relatively weak BC It is generally considered that the effect of the BC on ~ have much more influence on the final result than periodicity (the classical approach used here requires this condition even for radom media Specific methods are under development as presented in this conference). A definite demonstration of this point is not available but some 2D numerical tests that we performed [9] clearly showed that the periodic Be influence vanishes before the REV size has been reached


a) b)

Figure 2: Random close packings of spheres obtained nwnerically :; a) 2000 spheres with two diameters, 80% of the spheres have a diameter of OJ d and 20% a diameter of d.

The porosity is equal to 34.47%. The packing seems to be less compact on the boundary because the periodic spheres are no represented.

b) 2000 spheres in 2D. The periodic spheres have been added in this representation. Well-packed domains are visible as in experimental2D packings.

3.3 DIFFUSION

The closure problem (7) is simpler to solve than the previous one. A simple finite volume spatial discretization has been adopted along with a preconditioned conjugate gradient linear solver. As the expression of Deff (equation 10) contains an integral over the interface ~c" the control volumes have been defined in order to calculate f on this interface. This choice improves the precision of the computation of Deff but it also complicates the program by introducing irregular control volumes and multiple values on some nodes. Indeed, when two fluid voxels are in contact only by a summit, there is no diffusion from one to the other. If f has the same value for each voxel on the common node, this condition is not verified. Several cases, some much more complicated than this one, had to be considered.

3.4 REACTION

Random walk is a rather popular approach for transport simulation at the different scales encountered in porous media studies. At the pore scale, it provides an efficient way to compute effective diffusivities in complex media through the use of Einstein's formula in the so-called mean-square displacement method [10] and, in principle, introduction of fluid flow and reactions is simple [11]. Any heterogeneous chemical reaction is simulated by a sticking coefficient S assigned to the fluid-solid interface. S is the probability that an impinging particle is incorporated by reaction. For a first order reaction we have:


A S=y-A -+1 2

with A=~ D

(16)

where yis a coefficient depending of the rule used to handle particles bouncing on AtJcr [11] and 8 (m) is the size of the voxels used for simulation. Here we see that in this case, the sticking coefficient does not depend on any knowledge of the concentration in the bulk of the fluid phase. This is not ttrue in any other case (except for order zero). The number of particles stuck on each element of ~cr divided by the simulated time interval is an approximation of the reactive flux. The spatial repartition of this reactive flux can be rather heterogeneous (figure 3) even for simple porous medium.

Figure 3: Spatial distribution of the reactive flux for a simple porous medium: In red we have the most reactive surface elements (> 80% of the maximum) and

in blue the less reactive surface elements « 20% of the maximum). The green elements correspond to the intersection of the solid phase with the REV boundaries.

Geometry evolution can be simulated by transforming fluid (solid) voxels into solid (fluid) voxels when a given number of particles have been stuck to their solid boundary elements.


Figure 4: Initial geometry. Porosity = 30.2%.

1) 2) 3)

Figure 5: Details of three intermediate states of the three considered geometry evolutions. Porosity around 18%.


4. Considered geometry evolutions

Three different geometry evolutions are considered here. The initial geometry is the same for the three cases (figure 4): It is a cube (5d x 5d x 5d) extracted from a large random packing of spheres (diameter d, porosity about 36%). Before discretization in 100 x 1.00 x 100 cubic voxels, a smaIl initial consolidation has been imposed by increasing the spheres diameter d to 1.1 d in order to obtain a porosity of 30.2 %.

The considered geometry evolutions are:

1. Uniform by increasing the spheres diameter. 2. Controlled by diffusion/reaction using the random walk approach for precipitation. 3. Controlled by diffusion/reaction using the random walk approach as previously but with

25% of the spheres that can be dissolved with the same reaction rate coefficient that precipitation on the other spheres.

The computation process is then:

• Modification of the geometry • Percolation test • Computation of the permeability tensor and of the effective diffusion tensor by

solving the closure problem (7) or by the random walk approach (both methods give similar results.)

During the geometry evolutions the creation of non-connected porosity is common. This portion of the porosity do not participate to transport but, if not removed, would cause numerical difficulties (very slow convergence) during transport properties computation. We preferred to work only with the percolating fraction of the porosity even if some exotic phenomena are possibly eliminated (for instance, reopening of a closed zone by dissolution in the third kind of evolution). As shown in figure 5, the morphology of the pore space is very different from one kind of evolution to another. Pore locations are the same but pore walls are more and more irregular going from evolution 1 to 3. Pore volumes are notably changed but it is the modification of the pore connectivity that strongly affects the transport properties evolutions.

5. Transport properties evolutions

The consequences of the differences in mOlphology of the pore space are illustrated by figures 6 to 9. In a log-log diagram (figure 8) it is obvious that the main differences are between the geometric evolution 1 and the diffusion/reaction-controlled evolutions 2 and 3. The curves are almost linear in the first parts of the evolutions: permeability can then be described as a power law of the percolating porosity:

(17)

with m ~ 3.5 for evolution 1 and m ~ 5.2 for evolutions 2 and 3.

MICRO-GEOMETRY EVOLUTION OF 3D POROUS MEDIA

6. III

5.0 _

4. III _

3. III _

2. III _

1.1ll_

Ill. III

Ill. 0 I

Ill. 1 I

Ill. 2 I

Ill. 3

Ill. 15

0.10 _

0.05 _

1ll.1ll1ll

Figure 6: Evolutions of permeability with percolating porosity in the three studies cases (from top to bottom:I,2,3)

I I I I I I

0. 0 Ill. 1 I I

Ill. 2

I I I Ill. 3

I

Figure 7: Evolutions of the effective diffusivity with percoIating porosity in the three studied cm;es (from top to bottom:I,2,3)

227

Ill. 4

0. 4


101~ ______________________________________________ ~

.

Figure 8: Evolutions of the permeability with percolating porosity in the three studied cases (from top to bottom: 1,2,3)

Figure 9: Effective diffusivity versus permeability for the three considered evolutions


If we now plot in a log-log diagram the effective diffusivity versus the permeability for all the geometries considered, we obtain the figure 9. Deff can also be approximated as a power law of K with almost the same exponent for the three evolutions.

D K2/3 eff oc (18)

6. Conclusions

The results presented in this work clearly demonstrate that the way micf(}-geometry evolution is taken into account greatly influences the transport properties evolution with dissolution/precipitation. The two kinds of evolutions considered here, i.e. geometrically uniform or controlled by diffusion/reaction, can be seen as two extreme physical cases: uniform evolution is related to a case where the local concentration is almost constant and the other ones to cases where the local concentration is governed by the interfacial reaction (in the form used here, the random walk approach is directly linked to the closure problem (8) verified by s). Some further developments are necessary to provide results really applicable to geological problems:

• Introduction of non-constant reaction rate coefficient in the volume averaging theory. • Introduction of multi-mineral matrix and of realistic geochemical systems. • Development of the theory to take into account local anisotropy (linked to convective

transport for instance) and to represent, in a workable way, the feedback between largescale fluxes and microscopic sources terms.

• Evaluation of the influence of the REV size on the results. • Validations with simplified real examples.

References

[I] Whitaker S" Transport p= with heterogeneous reactions, Chemical reactor analysis: Concepts and design, Whitaker, S. And Cassano, AE. Eels, 1986, 1-94, Gordon Brea:h, New Yolk

[2J Whitaker S.,FIowin porousmediaT: A theoretical derivation of Darcy's law, T1W1Sp?J1 inp?musmetiia, 1 (1986),3-2'1 [3] Barrere J., Gipouloux 0., Whitaker S., On the closure problem for Darcy's law, TranspoJ1 in p?rous media, 7 (1992),

209-222 [4] Zinchenko AZ., Algorithm for random close pocking of spheres with periodic bomdaIy conditions, 1. Camp. Phys.,

114 (10994), 298-307 [5] Jodrey w'S., Tory EM., Computer simulation of close random packing of equal spheres, Phys. Rev, A, 32 (1985)

2347-2351 [6J Bargiel M, Moscinski J., C-Ianguage progrnm for the irregular close packing of hard spheres, Camp. Phys. Comm, 64

(1991),183-192 [7] Anguy Y., Bernard D., The local change of scale method for modelling flow in natural porous !redia (1): Numerical

tools,Adv. Water Res., 17 (1994), 337-351 [8] Anguy Y, Application de la prise de moyenre volumique a I' etude de la relation entre Ie tenseur de pe~ilite et la

micro-geornetrie des milieux poreux naturels, These U niv. Bordeaux 1, 1993 [9] Bernard D., Using the volume averaging technique to perfonn the first change of scale for natural random porous

media, Advanced methods for groundwater pollution control, Gambolati & Verri Eels, 1995,9-24, Spinger-Verlag, New Yolk

[10] Vignoles G" Modelling binary, Knudsen and transition regime diffusion inside complex porous media, 1. Phys, Iv, Colloque C5, 5(1995), 159-165

[II] Salles J, Thovert J F., Adler PM., Deposition in porous media and clogging, ChenL Eng. Sci., 48 (1993), 2839-2858

Pore-scale modelling to minimize empirical uncertainties in transport equations

J.P. DU PLESSIS

Department of Applied Mathematics, University of Stellenbosch Private Bag X1, 7602 Matieland, South Africa

Abstract As computer methods and the underlying numerical procedures improve dramatically with time the demands on transport equations to reflect the real physical conditions are also on the increase. Empiricisms in transport equations cannot always satisfactorily describe the extremal physical conditions enforced onto them by numerical strains and the present work is aimed at minimizing the need for empirical expressions in favor of simple but physically plausible models of the various processes taking place in porous media. Pore-scale modeling is used for closure of volume averaged transport equations. The model addresses the interstitial geometry and the physics of the particular phenomena to provide closure for the general volume averaged equations.

1. Introduction

Transport equations for multi phase phenomena are normally subject to quite a large number of empirical coefficients of which the physical origins are not clear and the magnitudes not known. These coefficients are then used as tuning parameters to get good comparison between computational results and experimental data. This presentation reports on analytical modelling efforts to minimize empiricisms in the transport equations. The model addresses the interstitial geometry and the physics of the particular phenomena to provide closure for the general volume averaged equations. The result is a set of transport equations with a minimal number of empirical parameters. The same basic set of equations are then applicable to a wide variety of different applications which stimulates cross-fertilization among different scientific disciplines and enhances confidence in numerical results.

Recent applications have shown that the simple deterministic model is capable of very accurate quantitative predictions for flow phenomena in various porous media. The wide range of applicability of the model is demonstrated by predictive results obtained for non-Newtonian flow through fixed beds and high porosity metallic foams as well as percolation effects in low-porosity sandstones.

Let us assume for simplicity that both the porosity, t, of the microstructure and the density, p, of the traversing fluid are temporally and spatially constant so that fluid momentum transport within the interstitial fluid phase, Uj, is governed by the following transport equation

av P -= + p\l . v v - \l'_T at -- Pfl - \lp.

231 1.M. Crolet (ed.), Computational Methodsfor Flow and Transport in Porous Media, 231-235. © 2000 Kluwer Academic Publishers.

(1)

232 IP. DU PLESSIS

Here P is the pressure field and .Q the interstitial velocity of the fluid phase. Time is denoted by t, r is the viscous shear stress within the fluid phase and 9 the body force per unit mass. -

2. Volume averaged equations

Volume averaging of the interstitial velocity .Q over a representative elementary volume, Uo ,yields the superficial or Darcy velocity q of which the local direction defines the streamwise direction field, an important parameter in the modelling procedure. The transport equations like (1) are also averaged over a representative volume consisting of an ensemble of both solid and fluid phases. These averaged equations are generally applicable to all porous media and open in the sense that they contain integral expressions to be evaluated for particular porous medium microstructures and flow conditions. Being phased averaged, these equations are expressed as a function of the porosity of the porous medium, e.g.

aq P at + ~'V' 9..9.. - 'V. (I:) = €PfJ.. - €\lPi

P€'V· \Y.Y.) f + ~o II (J1 . ~ - J1p) dS.

Sfs

(2)

Here (.) denotes a phase average, (.) f an intrinsic phase average over the fluid o

phase, Pi == (P}f, (.) specifies deviation above the intrinsic phase average and Sfs the fluid-solid interface.

The first line of equation (2) represents the convective, diffusive and source terms of the macroscopic flow through the porous medium. The term 'V . (~) is commonly referred to as the Brinkman term which for Newtonian flow reduces to The second line represents the contribution of interstitial drag to the source and may conveniently be designated by 9..F for the present discussion, yielding

aq P P at + ~ 'V . lJ.. 9.. - 'V. (r:.) = € PfJ.. - € 'V P i - lJ.. F. (3)

In this form, where F is considered part of the source term, the transport equation for flow in a porous medium closely match those for viscous flow in an unobstructed domain and any computational fluid dynamics code could easily be adapted for possible inclusion of porous media in the flow domain.

It is also evident that up to this point F is the only undetermined variable and its quantification is the subject of the following sections. From equations (2) and (3) it follows that F consists of three terms which somehow resembles an interstitial momentum transport equation and modelling of the actual flow conditions in the interstices of a particular porous medium may thus provide quantitative information on the F-field.

3. Closure modelling

Various approaches are nowadays followed for closure modelling, by which the specifics of the particular porous medium is introduced to evaluate the

PORE-SCALE MODELLING ... 233

integral expressions present in the parameter F. In the present work this is effected by the introduction of a simple rectangular representation of the porous medium microstructure to which interstitial physical processes may be applied. In addition to the closure of other terms, the modelling must also yield the dependency of the hydrodynamic permeability on porosity and microstructure.

3.1 Porous medium microstructure

A rectangular representation for the microstructure is adopted to facilitate quantification of parameter influences. This also allows exact modelling predictions for porosity values close to zero without possible overlaps of the solid and void regions. In case of isotropic microstructures the RUC's are cubes and modelling is facilitated by positioning one cube side normal to the general flow direction.

3.1.1 Morphology

Three basic morphologies can easily be analysed in the rectangular mode, namely consolidated spongelike media (Du Plessis and Masliyah, 1988), granular media (Du Plessis and Masliyah, 1991) and unidirectional fibre beds (Du Plessis, 1991), the latter being the two-dimensional analogue of the granular system for flow normal to the fibre bed. For each of these maximal pore connectivity and maximal staggering of solid parts are assumed to ensure fluid convection in all void sections.

3.1.2 Length Scale

The rectangular RUC'representations may be used for any microstructural length scale, d, according to the particular porous medium to be modelled. This freedom allows cross-fertilization among different fields of application, which leads to confidence in the industrially applied numerical computations based on the particular modelling.

3.2 Fluid properties and How phenomena

3.2.1 Tortuosity

As approximation to the tortuosity, X, the streamlines are assumed to be piecewise straight, of equal length and filling al interstitial voids, hence yielding a unique tortuosity being the ratio of the average streamline length through the RUC and the streamwise displacement through the RUC. This eliminates the notorious use of tortuosity as tuning parameter to disguise inexplicable discrepancies between experimental measurements and modeling predictions. The tortuosity, being uniquely determined by the porosity and the particular microstructure, necessitates discrepancies to be resolved only on physical grounds. Also, since the porosity dependency of tortuosity is uniquely determined by the microstructure, it may be used to discriminate between the different morphologies. We thus have

x = X(t, microstructure) (4)

234 IP. DU PLESSIS

by which interstitial velocity, Q, may be linked to the superficial velocity I{ by means of the Dupuit-Forchheimer relationship (Carman, 1937),

X Q - q. (5) c -

3.2.2 Viscous flow

The particular arrangement of rectangular solid blocks in the RUC presents flow regions which closely resemble flow between parallel plates. This allows the determination of shear stresses on the fluid-solid interface as a function of fluid viscosity, p, and fluid velocity, which in turn leads to prediction of the factor F as

F = F(c,X,d,p). The hydrodynamic permeability, J{, follows directly as

, c R. =-.

pF

(6)

(7)

The Blake-Carman-Kozeny equation for the hydrodynamic permeability of packed beds is therefore analytically predictable.

4. Particular generalizations

The basic model overviewed above serves as a core structure of modelling of which the assumptions may be relaxed for more general application or more restricted for specialized use. Some of the successful adaptions are discussed below.

4.1 Inertial flow

Inertial flow is modelled by assuming pressure drops due to internal recirculation on the lee side of all solid particles and with drag coefficient Cd, yielding:

F = F(C,X,Cd,d,p,Re). (8) This provides, without any undetermined coefficient, a deterministic expression for the so-called Forchheimer term (Du Plessis, 1994). Inclusion of the inertial effects in this manner thus allow a very successful quantitative prediction of the empirical coefficient in the Burke-Plummer equation for inertial flow in packed beds. Asymptotic matching of the analytical expressions obtained from equations (6) and (8) provides a fullyu determined expression for the Ergun equation for packed beds.

4.2 Anisotropy of the microstructure

Although the model was developed for isotropic media, generalisation to some geometrical anisotropic microstructures is fairly easy as was shown by Diedericks and Du Plessis, 1996. This necessitates a tortuosity dyadic with three principle tortuosities and three RUC side lengths so that

(9) Also this extension of the model was tested against experimental results over a wide range of microstructures and porosities.

PORE-SCALE MODELLING ... 235

4.3 Percolation effects at low porosity

The geometric model is also well suited for the modelling of pore closure due to different physical effects. Introduction of a percolation threshhold at some critical Reynolds number, Ree , derived through modelling or by experiment, then yields

F = F(E, d, X, p, Rec). (10)

Inclusion of a properly determined percolation limit Rec in this manner allows the application of the model in numerical computations where some subdomains may be impermeable or may change of permeability due to pore closure processes.

4.4 Fluid rheological properties

The simple structure of the modelling procedure allows the easy introduction of many different kinds of rheological behaviour by considering the flow fully developed flow of such fluids between parallel plates (Smit and Du Plessis, 1997). In cases where the rheology depends on a yield stress, T y , and the shear rate, I, thus follows

F = F(E, d, x, Re, Ty, 1'). (11)

5. Conclusions

The porous medium modelling procedure presented provides a sound basis for the numerical computation of flow phenomena in porous media. The model was tested severely and over a wide range of parameters against experimental work. It provides plausible solutions in both the limits of high and low porosities and due to its simplicity, further refinements are possible within the framework of the model. Since the same modelling procedure is used for different microstructures, porosities, etc., confidence in the modeling results is enhanced considerably. This in itself is of particular importance in numerical computations for which robustness of transport equations is sought.

The model also allows a linkage between computational work which normally utilizes the full transport equation and efforts to provide the particular parameter depedencies of the different coefficients.

One other very important, but often neglected, aspect of computational modeling is the correct specification of boundary conditions which are evidently strongly dependent on the peculiarities of the microstructure involved. Also in this respect special closure modeling for boundary conditions may be effected with the model presented here.

The references included are mainly those describing aspects of the particular model discussed. Credit is however due to all contributers to the vast amount of literature available on the subject and many authors whose whose work had an influence on

MODELING CONTAMINANT TRANSPORT AND BIODEGRADATION IN A SATURATED POROUS MEDIA

Abstract

S.A.KAMMOURI*, M.EL HATRI*, J.M.CROLET** * Laboratoire de Calcul Scientifique, Ecole Superieure de Technologie, Fes, Maroc **Equipe de Calcul Scientijique, Universite de Franche-Comte, Besanr;on-France

The present paper describes a numerical model, which allows to compute solute transport and biodegradation in a saturated porous media. Mathematical formulation of such processes leads to a set of non- linear partial differential equations coupled to ordinary differential equations. The transport equation is approximated by a finite volume scheme whereas biodegradation equations are treated separately as a system of ordinary differential equations. Numerical results for biorestoration using Monod kinetics are presented.

I-Introduction

Microbial biodegradation is one of the most promising technique for groundwater decontamination. It is a natural process that can be accelerated by the injection of certain nutrients such as dissolved oxygen, nitrates, and acetate. Biological decontamination is physically and chemically complex involving transport of substrates, nutrients, microorganisms and interaction of components between the aqueous and the solid phase through adsorption and biodegradation. In this study, we simulate biorestoration process in an homogeneous medium (a saturated aquifer). The mathematical model is a system of non-linear differential equations [2] that couple unstructured microbial growth kinetics with the transport of bioactive components in groundwater systems. The numerical method implemented here is based on a splitting technique, which allows as to treat separately the different physical and chemical processes. The approach decouples the transport portion of the equations from the reaction portion, by first solving the transport problem which is approximated by a finite volume scheme. The concentrations obtained from this step are then used as the initial concentrations to solve the reaction equations which are treated as ordinary differential equations and are


238 S.A.KAMMOURI, M.EL HATRI, J.M.CROLET

solved with a second -order, explicit Runge-Kutta method with time steps that are generally much smaller than those used for transport.

2- mathematical model

The general transport and biodegradation model for a single phase and incompressible flow is described by coupling non-linear partial differential equations. Here, for a solute undergoing linear instantaneous adsorption, we obtain the following system of equations [4]:

Transport-diffusion- and reaction of substrates:

Development ofbacteries:

Darcy's law:

The continuity equation:

For incompressible flow and if a source is present in the medium:

VV=q

Where we denote:

c= The concentration of various substrates in solution. Nj= The concentration of nutrients. B=The concentration of various bacterial species. K=The permeability tensor of the porous medium.

(1)

(2)

(3)

(4)

(5)

MODELING CONTAMINANT TRANSPORT

P=The pressure. p= The viscosity of the mixture. ~ The fluid density. R.=The retardation factor due to adsorption. Dc,=The diffusion/dispersion tensor of substrates. Dni=The diffusion/dispersion tensor of nutrients. flo=The maximum substrate utilization rate per unit mass of microorganisms. K-i=The substrates half saturation constant. Kni= The nutrients half saturation constant. K,FThe substrates constant of decay.

239

In this model, the flow is governed by Darcy's law (4) and the continuity equation (5). A coupled system of parabolic advection-diffusion-reaction (1)-(2) describes adsorption. transport and removal of substrates C (contaminants) and nutrients N. Bacterial transport is neglected, microorganism growth and decay are then simulated by a set of coupled differential equations.

3- Numerical method

The numerical resolution of the coupled system of equations (1)-(5) is achieved by a splitting procedure [3] which can be described in the following way. The combination of Darcy's law (4) with the continuity equation (5) gives the following elliptic equation known as the pressure equation:

-v(~(vp -P(C)eJi = q jl(C) )

(6)

So in the first step, knowing C at time t, we compute the pressure P on the center of all cells with the classical finite volume scheme. we consider a rectangular cell (control volume) Qij of8xx8y size with 8Oij=C+1/2jUC_1/2,jUli,j+1/2u1i,j_ld"Figurel.").

u

v

P,C,K

o

v

u

Figure 1. Representation of a control volume Qij.

240 S.A.KAMMOURI, M.ELHATRI, lM.CROLET

By integrating equation (6) on the cell nij and by green's formula, we get the discretized equation for pressure: (we denoted a=K/fl(C) )

ex ex & (q+I/2,/P;+LJ -P;)-q-II2,/F:,J -P;-I)+ & (q,J+1I2 (F:,J+I -F:,)-q,j-1/2(F:,J -F:,J-I»

(7)

Equation (7) with the boundary conditions can be written as a system of equations of the form: AP"=L. The resolution of this system enables as to determine the profile of pressure at each time. Afterwards, Darcy velocities are calculated on the edge of the cells by utilizing equation (4). In a second step, advection-diffusion-reaction equations (I) and (2) are decoupled by time splitting method. A finite volume scheme is used to treat the transport equation. so, by integrating equation (1) without the kinetic term on the cell nij, and by green's formula we can write: (we assume that C is linear on each r k)

f d ll ac dy = ~ dll .(Cn+l . - C n+l ) (8) ;}., Ox l+ II 2,} l+ I,} I,}

[i+1 1 2,J vy

fd l2 ~ dy = di~1I2,J+1/2(Ci:~1/2,j+I/2 -Ci:~1/2,J+IIZ> (9) [;+1/2,J vy

d l2 (C n+1 Cn+1 ) + i+1I2,J-1/2 i+1I2,J - i+1I2,J-1I2

Where d'! are the coefficients of dispersion which are discretized on the control volume as it is shown in ("Figure2.").

Figure2. Discretization of dispersion coefficients

MODELING CONTAMINANT TRANSPORT 241

By utilizing the same procedure for the flux H i+1/2"j, H~j+I/2 and H~j-I/2 , we get the following numerical scheme:

C;:;I ; (dJ;lI~j(G:~~j -G:;I)~~I/~iC;:;I--e:~)

I ; (tf:+II2(c,:"!1 -G:;1)-df:-1/2«(~t ---q;~l»

2~ (dJ~/~j+II2(G:~j+I-G:;I)~:I/~j-1I2(c,:"I--G'~:j-i»

+ 2:& (~;1/~j+l/2(~j_I-G:;I)~~/~j+II2(C;:;I--e:~j+I»+8g(c;,t)=C;:;1/2 (10)

The resolution of equation (10) as a system of equation ofthe form: AC=B enables as to obtain the profile of concentration at time t+8t12.Finally, solving biodegradation equations by a forth order Runge-Kutta method using several small time steps gives the concentrations C, N, and B at time t+81.

4- APPLICATION

In this section, we consider a rectangular mesh in a two dimensions space (a simplified aquifer) and we simulate biorestoration of a single organic component C=C} and of dissolved oxygen O=N} by microbes B=B} in such porous media. We suppose that the domain is homogeneous and isotropic, the flow is one-dimensional and steady state. The initial conditions: C (x, y, 1=0)=0 on the domain Boundary conditions :C(O, y, 1 >0) = Co if /y I<a aEfll. The domain is semi -finite. C (x, y, I) is only defined down stem the sources.

Initially, non dissolved oxygen is present in the aquifer and a constant microbes concentrations of 1O-4mgll is assumed. The initial distribution of substrate concentration at time t=500 days is shown in( "Figure 3.").

242 S.A.KAMMOURI, M.EL HATRI, lM.CROLET

6

2

1~~---L--~--~--~~~~--~--~

1 2 3 4 5 6 7 8 9 10 x·50

Figure 3: The initial distribution of substrate concentrations at time t=500 days

The biodegradation processes are then simulated by injecting 9 mgll of dissolved oxygen at the inflow boundary. The biodegradation and transport parameters used in the simulation are [1]: v=1.24m1d, R=1.5, D/=50m2/d Dt=5m2/d, Kc=O.07mg//, Ko=2mg//, flo=O.6(d'I), Y=O.2, F=1.5, Kd=O.l(d'I). In "Figure 4" , we plot the substrate distribution after 500 days ofbiorestoration.

6

2

2 3 4 567 x·aJ

8 9 10

Figure 4: The distribution of substrate concentrations after 500 days ofbiorestoration

MODELING CONTAMINANT TRANSPORT 243

The second test consists on stimuling the microbial activities, so we inject in the domain from the inflow boundary 40 mgll of oxygen. "Figure 5." Show the substrate concentrations after 500 days of pollution.

4- Conclusions

6

2

11~--~2--~3--~4---5~~6--~7--~8---9~~10

x·50

Figure5: The distribution of substrate after 500 days of biorestoration with the injection of 40 mg of oxygen.

The whole numerical method using splitting technique enables us to treat separately the different physical and chemical processes. The discretization of the transport equation by a finite volume scheme is very efficient as it is shown on the various difficult tests presented. The biodegradation equations are treated as a system of ordinary differential equations using a forth order-Runge-Kutta method, which is too restrictive for time step. The biodegradation needs to be approximated on mush smaller time scale than the advection and the dispersion.

5- References

[I] MACelia, J.S.Kinred, and 1.. Herrera. (1989) Contaminant transport and biodegradation. 1. A numerical model for reactive transport in porous media, Water resour. Res.25, pp.1141-1148.

[2] C.H.Chiang, C.N.Dawwson, and M .F. Wheeler. (199 1) Modeling of in-situ biorestauration of organic compounds in Groundwater,Transport in porous media, 6, pp.667-702.

[3] C.N.Dawson, and M.F. Wheeler. (1991) Time-splitting methods for advection-diffusion-reaction equations arising in contaminant transport, ICIAM.

[4] B.D. Wood, C.N.Dawson, and G.P.Streile. (1994) Modeling contaminant transport and biodegradation in a layered porous media system, Water resources research, Vol. 30, NO.6, pp.1833-1845.

Water Phase Change And Vapour Transport In Low Permeability Unsaturated Soils With Capillary Effects

S. Olivella, A. Gens and J. Carrera

Geotechnical Engineering and Geosciences Department

Universidad Politecnica de Catalunya

Barcelona, Spain

Abstract

A discussion of water phase change in unsaturated soils that develop capillary effects is first carried out in the paper. A distinction between the GR (geothermal reservoir) and the NUS (nonisothermal unsaturated soil) approaches is performed. Several aspects concerning advective and non advective fluxes of vapour are described secondly and some relationships concerning the case of mass motion in a closed system subjected to temperature gradients derived. Since the structure of unsaturated clays changes with moisture content, in order to correctly simulate the coupled phenomena induced by temperature gradients a model for intrinsic permeability as a function of humidity is required. A preliminary version of the model is presented and applied to interpret a laboratory test by means of a numerical simulation using CODE_BRIGHT.

1. Introduction

The main purpose of this paper is to review some aspects related to water phase change into vapour form and its transport in the context of unsaturated soil behaviour. Also a model for intrinsic permeability that can explain some features observed in clays is presented. A number of problems require a good knowledge of the vapour phase change and migration in unsaturated soils. This problem of water phase change and vapour transport has been treated in different ways depending on the area of interest.

In the context of radioactive waste disposal low permeability soils are used for building isolating barriers. These clays are initially unsaturated, have swelling properties and are subjected to wetting - drying / heating cooling cycles. Depending on the possibility of structure change induced by chemical effects, temperature may be limited (typically 100 DC). At 100 DC the vapour pressure is 0.1 MPa according to the phase diagram of pure water. Although surface tension effects reduce vapour pressure in equilibrium, the total gas pressure (Pg=Pa+Pv, i.e. gas pressure equals air pressure plus vapour pressure) can not, in general, be considered constant.

Philip and De Vries (1957) early investigated thermal effects in unsaturated soils. Later, Milly (1982) extended their work to heterogeneous and hysteretic medium. The theoretical approach followed by these soil scientists assumes that air can be neglected which is an adequate assumption in a number of situations. The soil scientists' approach

245 J.M. CroZet (ed.), Computational Methods/or Flow and Transport in Porous Media, 245-272. © 2000 Kluwer Academic Publishers.

246 S. OLIVELLA et al.

is based on matric head and temperature as state variables and therefore it is not suitable for simulating the process of soil desaturation caused by heating. Of course, modelling this process was not an objective of these authors when the formulations were developed.

These formulations were later extended to include the presence of air. For instance, Pollock (1985) presents a multiphase approach incorporating the balance of air. In this work liquid saturation, gas pressure and temperature are used as state variables. Of course, liquid saturation is not an adequate choice for saturated zones of the porous medium but it can be easily changed by liquid pressure. If saturation occurs, one only needs to check the condition of Pg-p[>O in order to know if the soil is unsaturated. Since gas pressure is composed by vapour pressure plus air pressure, a temperature increase induces vapour pressure increase and this may induce a desaturation of an initially saturated soil.

Another area of research related to thermal effects in geological media is the field of geothermal energy. A reference work by Faust and Mercer (1979) contains the basic formulation for modelling geothermal reservoirs. Although there are points in common with the work of the above mentioned soil scientists, the approaches used in these two investigation fields are conceptually different. Following the geothermal reservoir approach, it is possible to handle the problem of phase change induced by heating. However, in this approach capillary effects and the possible presence of air are neglected.

More recently the interest in this topic has increased also due to the environmental problems related to organic compounds. A relevant work has been presented by Falta et al (1992) with the objective of modelling the process of steam injection for the removal of NAPL (non aqueous phase liquids) in contaminated soils. One aspect that makes this approach more complex than the preceding ones is the flow in three phases and with three components (water, air and NAPL).

Bear and Gilman (1995) have investigated the migration of salts in the unsaturated zone induced by heating. A related work is presented in Olivella et al (l996a) where porosity variations in saline media caused by temperature gradients were investigated. In both works, it is shown that the liquid flux induced by vapour migration is able to transport dissolved salts from the cold toward the hot side.

In this paper, the theoretical aspects of phase change are presented first with special attention to the case of phase change under unequal phase pressures. It is discussed why the geothermal reservoir approach is not applicable to unsaturated soil behaviour. On the other hand, it will be shown that the approach that takes into account capillary effects and presence of air is more general and reduces to the geothermal reservoir approach when applied to a soil that develops small suctions.

In the third section, the theory for vapour flow in unsaturated soils is reviewed with special attention to the reasons for flux enhancement due to air immobility and the

WATER PHASE CHANGE AND VAPOUR TRANSPORT 247

pressure gradients induced by temperature gradients. The widely used approximation in which the equation of air balance is not solved may cause vapour flux enhancement.

Liquid and gas permeability is the subject of the fourth section. It has been observed that for clays, even at constant porosity, it is not possible to consider that the intrinsic permeability remains constant with respect to water content. In fact, the structure of the soil changes with water content. A model to take into account this effect is proposed. By means a very simple double structure approach it is possible to model the variation of intrinsic permeability with degree of saturation. The model may simulate flow in clays that show a maximum permeability before full saturation is achieved.

Finally, an example of application of this model will be presented in which water flow induced by temperature gradients takes place. The example is calculated using CODE_BRIGHT (Olivella et al 1996b), which is a finite element code for solving coupled thermo-hydro-mechanical problems (mechanical effects are beyond of the scope of this paper). The application that is presented is related to problems of radioactive waste disposal in underground openings including barriers of low permeability porous materials. It is shown that the proposed model allows a more suitable modelling of the drying process induced by temperature gradients than the usually adopted one that considers intrinsic permeability only a function of porosity.

2. Phase change of water. Importance of capillary effects

2.1. Phase change of water without capillary effects. The GR approach.

The Phase change diagram for pure water is represented in Figure 1 (see for instance Faust and Mercer, 1979) which displays that there are several regions. Depending on the pressure of water (p, MPa) and the enthalpy per unit mass of water (h, Jlkg) 3 regions are distinguished: liquid phase region, two phase region and gas phase region.


100

-CIS Il. :E 10 -II) ... m 180°C II) ... Co ...

@] @] .! 100°C CIS

0.1 3: Q.

0.01

O.E+OO 1.E+06 2.E+06 3.E+06 4.E+06

h, water enthalpy (J/kg)

1 Single phase region (liquid water)

2 Two phase region (liquid water+vapor)

3 Single phase region (heated vapor)

Figure 1. Pressure-enthalpy dIagram for pure water. Two Isotherm hnes (l00 °C and 180°C) are represented.

The phase diagram displayed in Fig. 1 represents also the behaviour of water in a soil if it can be assumed that capillary forces are negligible, in other words, surface tension is assumed zero. This implies that water and vapour pressures are equal, and then capillary pressure is equal to zero.

The Geothermal Reservoir (GR) approach (Faust and Mercer, 1979) uses this diagram and, among other laws, the equations representing it are introduced in the mass and energy balance equations. For water, density and enthalpy are defined as:

and

h = hiP lSI + ~Pv(l- S/)

P

(1)

(2)


where SI is degree of saturation, i.e. volume of liquid per unit volume of voids; Pv is density of water vapour; PI is density of liquid water; hv is enthalpy of water vapour, and hi is enthalpy of liquid water.

Following the geothermal reservoir approach, the state variables that are usually selected are the pressure (P) and the enthalpy (h). From these variables the others are calculated. Several empirical functions allow to calculate density, enthalpy of liquid water and vapour and temperature (see for instance Roberts et al. 1987). Degree of saturation is obtained from (2). In the formulation for GR it is assumed that air is not present in the soil. Therefore, voids are filled with liquid water plus pure water vapour (region 2) or only one of them (regions 1 and 3). As vapour progressively condenses the soil progressively saturates and vices versa.

Following this diagram, the variation of temperature and degree of saturation as a function of enthalpy for a saturated soil with negligible capillary forces (e.g. coarse granular soil) that is heated at atmospheric pressure (0.1 MPa) have been calculated (Figures 2 and 3). Since pressure is constant, this heating process is represented in Fig. 1 by a horizontal line that crosses the three regions from the left to the right. Figure 2 shows that temperature is constant during the evaporation of water, i.e. when the pressure/enthalpy point falls in region 2. Figure 3 shows the process of desaturation that begins at 100°C and finishes when degree of saturation tends to zero. As soon as all water is evaporated temperature starts again to increase.

It should be pointed out that the assumption of negligible capillary forces implies that for a given pressure and enthalpy, degree of saturation is unambiguously determined using only the phase change diagram. As explained below, this is clearly in contrast to what happens in a soil where capillary effects are not negligible.

240 220 -

q 200

6 180 0 160 -Q) 140 ... ::l - 120 nJ ... Q) 100 Co E 80 Q)

60 I-

40 20

0

/ f

/ I / I

/ I I

/ '- - ._- I --Heating at 1 MPa I_

I - Heating at 0.1 MPa 1_

==l··-- .--... ~.-

O.E+OO 1.E+06 2.E+06 3.E+06 4.E+06 water enthalpy, h (J/kg)

Figure 2. Temperature vs water enthalpy for heating a granular soil at 0.1 MPa (atmospheric pressure) and at 1 MPa.

250 s. OLNELLA et al.

1 T---r-~-r--------~------'-------~

~ c 0.1 o ;:I

t! ::s -

--Heating at 1 MPa

-Heating at 0.1 ~

II 0.01 +----+--+__----'~-+__---+__--______l

'0 CI) CI)

g, 0.001 +------+----""o~--+_-__T-___t----__j

~

0.0001 +-___ +-___ -+-~_....l...._+_---__!

O.E+OO 1.E+06 2.E+06 3.E+06 4.E+06 water enthalpy, h (J/kg)

Figure 3. Degree of saturation vs enthalpy for heating a granular soil at 0.1 MPa (atmospheric pressure) and at 1 MPa.

2.2. Phase change of water with capillary effects. The NUS approach

When capillary effects can not be neglected water phase change in a soil incorporates different features. Liquid water and water vapour have different pressures. The implications are the following:

• Degree of saturation can be obtained from a retention curve that relates capillary pressure (gas pressure minus liquid pressure). The shape of the curve can, however, change with temperature. In fact, a model for retention curve should incorporate a dependence on surface tension so the temperature influence on the meniscus radius is properly accounted (Milly, 1982).

• Vapour density is modified by capillary pressure according to psychometric law (Edlefson and Anderson, 1943). The driest the soil, the lower the relative humidity with respect to the case in planar surface state.

• Surface tension is a decreasing function of temperature. Therefore at high temperatures the behaviour of phase change should tend towards the situation without capillary effects.


The Nonisothermal Unsaturated Soil (NUS) approach is used for modelling the nonisothermal multi phase flow of water and gas through a soil. This approach takes into account capillary effects and also it considers the possible presence of air in the soil. Philip and De Vries (1957) first established this approach. With this background, several approaches have been developed and used for modelling unsaturated soil behaviour (among others: Milly, 1982; Olivella et aI, 1994). The formulation developed by Olivella et al (1994) has been used to build CODE_BRIGHT (Olivella et aI, 1996b) which is a finite element code that solves coupled thermo hydro mechanical problems. The formulation implemented in this code contains also features such as mechanical effects and dissolution I precipitation of salt. In fact, it was first established for saline media.

In order to see the influence of capillary effects on phase change a sensitivity analysis using CODE_BRIGHT, which is essentially based on the NUS approach, has been performed. The retention curve of van Genuchten (1980) has been chosen because it is widely used to represent unsaturated soil behaviour. This law can be written as:

( 1/0-"-)J-"-SI - Smin ( P g - PI) Se = = 1 +

Smax - Smin P

(3)

This equation contains the parameters "- and P. The first (A) essentially controls the shape of the curve while the second (P) controls its height, so this latter can be interpreted as the capillary pressure required to start the desaturation of the soil. P can be scaled with surface tension. If Kelvin's law is recalled:

2a p - p. =-

g I r (4)

where 0' is surface tension and r is the curvature radius of the meniscus and if Eq. (3) is written as:

then an analogy between (4) and (5) leads to conclude that:

aCT) p=p--

o a(TJ

(5)

(6)

where Po is the corresponding parameter at temperature To. This means that P will decrease with temperature because surface tension also does. Consequently suction will decrease with temperature for a given degree of saturation.


Surface tension is calculated as:

aCT) = (1- O.625a )(0.2358a 1.256)

aCT) = O.0019106exp(O.05(360 - T))

a= 374.15- T

647.3

and, vapour density is calculated, as a first approximation, as:

Pv = R(273 + T)

( -5239.7) (- (Pg - ~)MwJ P)T, Pg - ~) = 136075exp 273 + T exp R(273 + T)PI

(7)

(8)

Using the retention curve given above, different values of Po have been chosen (10, 10-3,

10-6, 10-7 MPa) to study the sensitivity of phase change to different soil types_ Since liquid and gas pressure will not be equal only one can be prescribed (when capillary pressure is neglected, constant vapour pressure implies constant water pressure and vice versa). If liquid pressure is prescribed, then vapour pressure will change. For the calculations performed here liquid pressure is assumed constant and equal to 0_1 MPa and the soil is considered saturated before heating. This means that air is not present in the soil. A sufficiently small amount can be considered initially dissolved in water as an artefact.

Figure 4 shows the temperature variation as a function of the enthalpy. It can be seen that temperature does not remain constant after 100°C but it increases with a slope, which in fact is lower as capillary effects decrease (lower Po). In order to understand better this temperature variation, the diagram of degree of saturation is shown in Figure 5. Due to capillary effects, for a given water enthalpy, an amount of water has not evaporated (degree of saturation is higher than the corresponding value for the phase diagram of water). Consequently, heat is used to increase temperature instead of being used to evaporate the amount of water that cannot evaporate. The curves show a tendency towards the curve for zero capillary pressure as capillary effects reduce (i.e. for lower values of Po)

Figure 6 shows the variation of vapour pressure as a function of water enthalpy. In this plot, the curves that define the regions in the phase diagram for water (Fig. 1) are also included. Since temperature increases after 100°C and liquid pressure is maintained at 0.1 MPa, vapour pressure increases in accordance with the phase diagram for water.


2.3. Consequences

From Figs 4 to 6, it can be concluded that the NUS approach tends to the GR approach when capillary effects decrease if the soil does not contain air. It should be mentioned, however, that the NUS approach still calculates degree of saturation as a function of capillary pressure, therefore it may become less robust than the GR approach when this dependence is lost. When temperatures are high, the NUS approach for a soil without air tends also to the OR approach because the capillary effects decrease when surface tension decreases.

The presence of air is naturally handled in the NUS approach. In fact, in most of cases soils are unsaturated due to the presence of air but not due to water phase change. On the contrary, the OR approach cannot handle the presence of air. If it is assumed that vapour and air behave as perfect gases, the gas phase pressure is: Pg=Pa+Pv. In absence of capillary effects, the gas phase pressure should be equal to the liquid phase pressure. Then, the vapour pressure cannot be equal to the liquid water pressure as assumed in the phase diagram. Or, from another point of view, if vapour pressure is equal to liquid pressure then gas pressure is higher than liquid pressure and this implies surface tension effects.

400

350

300

6 0 250 -Q) ... :::s - 200 as ... Q) c.. E 150 Q) I-

100

50

0

O.E+OO 1.E+06

-tr-pO=O.OOOOO01 MPa ~ FU=O.000001 MPa """¢- FU=O.001 MPa .....0- FU=1 0 MPa - No capillary effects

2.E+06 3.E+06

water enthalpy, h (J/kg)

4.E+06

Figure 4. Temperature vs water enthalpy for heating soils with different retention curves.


....jz- RJ=O.0000001 MPa -0- RJ=O.000001 MPa ~ RJ=O.001 MPa .....o-RJ=10 MPa - No capillary effects

0.1 +---~---+--------~~~--~------~ r:::: o ;:; l! ::::I -IX _ 0.01 o CI)

~ C)

~0.001

0.0001

O.E+OO 1.E+06 2.E+06 3.E+06 4.E+06


Figure 5. Degree of saturation vs water enthalpy for heating soils with different retention curves.


100.-------~--------,_--------r_~~--~

~ 10 +-----------+-~~------r_------~~~--------_1 ca D.. :::::!l -~

m ~ Co L.

:::s o Co ca

1

> 0.1

0.01 O.E+OO 1.E+06 2.E+06

--&- A)=O.OO00001 Iv'Pa .....a- A)=O.OOOOO1 MPa ----¢- A)=O.001 MPa -0- A)=1 0 MPa -No capillary effects

3.E+06


4.E+06

Figure 6. Vapour pressure vs water enthalpy for heating soils with different retention curves.

3. Vapour flow in an unsaturated soil

In Wa'il Abu-EI-Sha'r and Abriola (1997) the existing approaches for modelling gas flow in porous media are reviewed. According to these authors the DGM (dusty gas model) approach should be used for modelling flow of gas mixtures in porous media. Since this approach is not yet widely used, as agreed by these authors, we will still apply the usual formulation based on First Fick's law.

3.1. Mechanisms oJ vapour flow

The mechanisms of vapour flow in an unsaturated soil can be classified as advection, diffusion and dispersion. These fluxes can be expressed as follows:


Advective flux:

(9)

Equation that is generally considered valid either for a gas phase composed by pure vapor (Wgw=l) or either by a mixture of air and vapour (wgw<l). Generalised Darcy's law (Bear, 1972) is usually used to calculate the volumetric phase flux. Of course, the viscosity of the phase can be considered a function of its composition and temperature.

The above-described advective flux has been written under the assumption of laminar flow. When the pores are small and the pressure is also low, the mean free path of gas molecules may be comparable to the pore sizes. When this happens, slip takes place and the assumption of purely laminar flow is no longer valid. A simple modification of Eq. (9) to account for Knudsen diffusion consists in using an apparent intrinsic permeability with the following form (Wa'il Abu-EI-Sha'r and Abriola, 1997):

k = k + I Dk I-l g a p

g

(10)

where Dk is the Knudsen diffusion coefficient.

Diffusive flux:

(iW) = _(Dw) Vro w

g diffusion g diffusion g

(DW) = 't<j>DIS P g diffusion g g

(11)

which has been written as a generalised form of Pick's law for molecular diffusion in porous media.

Dispersive flux:

(i W ) = -(D W ) V (0 W

g dispersion g dispersion g

(12)

also using Pick's law but with the corresponding tensor to model dispersion of a component in a phase that is flowing in a porous medium (Bear, 1972). Both diffusion and dispersion vanish, as the gas phase becomes a single component phase. This is well


represented by Eqs. (11) and (12) because mass fraction (ffigW) becomes constant and equal to 1 and, consequently, the gradient of mass fraction becomes zero.

Coupled processes (e.g. advective flux due to temperature or concentration gradients; non advective fluxes due to pressure gradients) are not considered here, so only the fluxes in the diagonal of the Onsager's diagram are taken into account.

Finally the total flux of vapour can be written as:

j W = (i W ) + (i W ) + (i W ) = w W p q _ D W V w W

g g advection g diffusion g di.'persion g g g g g

(13)

Where, as usual, the diffusive and the dispersive terms have been put together.

A similar expression holds for air flux:

ja =(ia) +(ia) +(ia) =Wap q -DaVwa g g advection g diffusion g dispersion g g g g g

(14)

The gas phase is considered a binary mixture of two species (vapour and air) in which an influence of each species motion is exerted on the other species motion. According to Bird et al (1960) in a binary system the mass fluxes of species with respect to the mass averaged velocity of the phase are zero. If it can be assumed that the mass averaged velocity coincides here with qg then it should follow that Dg w=D/. On the contrary, if Darcy's flux (qg) gives the volume (or molar) averaged velocity (Bear and Bachmat, 1986) then mass fractions can be substituted by molar fractions and the same result is obtained (i.e. D/=D/). In this latter case, if mass fractions are still used in calculating (11) and (12) then it is required to use D/#D/. The ratio between these two coefficients is equal to the ratio between the molar mass of air (28) and the molar mass of vapour (18) approximately 1.5. Bear and Gilman (1995) still use D/=D/ in a mass fraction based formulation, so here it is also considered to hold.

3.2. Low vapour concentration versus high vapour concentration.

If the fluxes in equation (13) are split then:

'W W Dwn W Jg=WgPgqg- gYW g =

-Ol;P, k:>"p, -( d,lq,II+(d, -d,) i~:r }, VOl; -(~<I>s,D1)p, VOl; (15)

258 s. OLIVELLA et al.

Where, without loss of generality, the gravity term in the Darcy's law has been neglected assuming that the flow is horizontal. In this equation,

• The third term is dominant in low vapour concentration situations if there are no external pressure gradients. This is the case of an unsaturated soil subjected to a moderate temperature and temperature gradient.

• The first one is dominant when the gas phase is pure vapour. In this case the second and the third vanish because ())gw=l. For the case of low vapour concentrations, the first term may become important if an external pressure gradient is imposed.

• The second term is only important if both mass fraction and pressure gradients are considered. However, the high molecular diffusivity of vapour in air usually leads to neglect this dispersion term.

Equation (15) contains 4 material parameters that control the process of vapour transport. The two dispersivities (dt and dt) enter in the second term and since vapour molecular diffusion is very efficient, specially if temperature is high, dispersion will be neglected. The other two material parameters are intrinsic permeability (k) and tortuosity (t).

3.3. Flux enhancement due to air immobility

When a nonisothermal flow problem is simulated, simplifications can be performed in order to avoid solving the air mass conservation equation. The following two possibilities considered here are (a) air immobility and (b) constant gas pressure. In any case, dissolved air is neglected, i.e. air is only present in the gas phase.

a) Bear and Gilman (1995) have shown that the assumption of air immobility implies an enhancement of vapour diffusion. This is explained in the following way considering that the total air flux is zero:

(16)

Which can be introduced in Eq. (13) to obtain:

(17)


In other words, the gas phase flux generated to balance the diffusion of air is able to induce a transport of vapour.

If an unsaturated soil closed to mass transfer is considered, air immobility only takes place at steady state regime. Therefore this assumption will not give exact results during the transient phases.

b) Constant gas pressure assumption. In this case <)g is neglected following the assumption that gas pressure gradient is zero then the total vapour flux is:

(18)

In this case no enhancement is obtained because no gas phase flow (considered as a whole) is taken into account. It is not necessary to calculate the air balance p,quation because gas pressure is assumed known.

3.4. Gas pressure gradient induced by vapour migration in a mass closed system.

If no assumption regarding gas phase mobility is considered, combination of (13) and (14) leads to:

(19)

Eq. (19) reduces to (17) if j/=O (case a) and, as mentioned above, this happens only at steady state conditions. During transient phases, jg a is not zero in general. The following result can be obtained from the condition of air immobility (dispersion is neglected and the soil is assumed isotropic):

(20)

This dependence of gas pressure gradient on vapour mass fraction gradient is an important point. From (20), it can be seen that a factor controlling the gas pressure gradient is the ratio between the material parameters tortuosity ('t) and intrinsic permeability (k).

260 S. OLIVELLA et aI.

3.5. Liquid pressure gradient induced by vapour migration in a mass closed system.

In a similar way as described in 3.4., liquid pressure gradient induced by vapour migration can be obtained as a function of vapour mass fraction. This is derived from the condition of mass of water conservation at steady state regime (dispersion is neglected and the soil is assumed isotropic):

(21)

where the vapour flux (jg') has been calculated from Eq. (19) plus the condition of air immobility (jga=O at steady state regime). It can be seen that liquid pressure gradient is proportional to the ratio between tortuosity ('t) and intrinsic permeability (k) but it develops with opposite sign.

In Section (5) these results (Eq. 20 and 21) will be used to obtain an important result for vapour migration in low permeability soils.

4. Intrinsic permeability in clays

4.1. Measured intrinsic permeability in clays

A very low intrinsic permeability is encountered in clays when this parameter is measured under water saturated conditions. In contrast much higher values are observed when, under unsaturated or dry conditions, the permeability to gas is measured. Figure 7 shows examples of measurements in Boom clay and Febex clay.

In the clays considered for these permeability determinations, the dry density ranged from 1.65 to 1.7 glcm3. Assuming that the variations of porosity are small (and this is difficult to be guaranteed because wetting induces very relevant swelling of the soil) it is clear that a single intrinsic permeability can not be used for these clays. If the value of intrinsic permeability for water was used, the gas permeability that one would obtain is several orders of magnitude smaller than the actual values. This is because in these plots


the curves for gas permeability would be lowered until the value for SI=O of gas permeability would be the same as the value for SI=1 of liquid permeability.

4.2. New model for intrinsic permeability of clays.

In order to maintain the concepts of intrinsic permeability and relative permeability a dependence of intrinsic permeability on degree of saturation should be introduced. This dependence can be explained by the change in the structure that takes place when clays are wetted. This can be achieved by introducing a double porosity structure. Then the intrinsic permeability would be considered a function only of the macro porosity. In this case the micro-macro porosity relationship depends on the water content of the clay. If M stands for macro and m for micro, the following relationships are proposed:

<P=<PM + <Pm (22)

<P M = <P exp( - ~Sl )

where an exponential function has been introduced for the dependence on degree of saturation. It is assumed that when the medium is nearly dry, practically all porosity is a macro porosity while when the medium tends to saturation, all porosity is micro. Using this definition, intrinsic permeability will be calculated as:

k = k(<p M) = k(<pexp(- ~Sl)) (23)

In equation (23) any available function of intrinsic permeability on porosity can be used as a first approximation, for instance, Kozeny's relationship.

262

~ :s

S. OLIVELLA et al.

Boom Clay. Estimated values from experimental data

from Volkaert et al. 1994. lE-l2,----------------,

lE-l3

lE-l4

lE-1S ~ :s

FebexClay. Values from experimental data from Villar

1998. lE-l2,----------------,

~ lE-l3

lE-l4

lE-1S

~ lE-la I lE-l6

! i ~ lE-17 ~ lE-l7 c

~ c

~ lE-la lE-l8

lE-la lE-l9

lE-20 lE-20

lE-2l ~-~--~-~-~-----4 lE-2l +---~-_--...,..,.'--_-----l o 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

Liquid degree of saturation Liquid degree of saturation

Figure 7. Compiled data on intrinsic permeability of clays to gas and water (experimental data from Volkaert et aI, 1994 and Villar, 1998).

4.3. Other enhancement phenomena of gas flow

Knudsen diffusion or Klinkenberg effect can also explain a higher permeability of the gas phase when pores are small. In equation (10) a simple way to consider this process has been included. In order to see the relative influence of this process, i.e. if it can explain that the permeability of the gas phase increases by orders of magnitude, values of Dk in equation (10) are included in Figure 8. A number of experimental tests have been carried out by Wa'il Abu-EI-Sha'r and Abriola (1997) on high permeability soils and the obtained Knudsen diffusivities (Figure 8) show a good correlation with intrinsic permeability. Here, Knudsen diffusivities for a much lower intrinsic permeability soil have been incorporated. These latter have been calculated from experimental results on tests described in Volkaert et al (1994). Although the clayey soil that has been investigated by Volkaert et al (1994) show a much lower intrinsic permeability to water, for gas the value is around 10-15 m2 (see Fig. 7) which seems still a high value for this clay (Boom clay).

In any case, the relative increase of intrinsic permeability due to Knudsen diffusion is much less than an order of magnitude (of the order of 1.5-2 times for pressure between 0.125 and 0.3 MPa). Therefore, it can not be the cause that explains the higher mobility of gas phase. Of course if the Knudsen diffusivity was calculated on the basis of the


intrinsic permeability found under saturated conditions for this clay, the values encountered would have been much higher. This would be also inconsistent because the tendency in Figure 8 would be lost.

1.E-tOO ,---,---,---,---,---,---,---,---

1.E-01

~ ... E 1.E-02 -:: :~ 1.E-03 ::::J

:a:: =c 1.E-Q4 r:: 5: -g 1.E-05 r:: ~

1.E-06

• Wa'il Abu-EI-Sha'r and Abriola, 1997

• Boom day (tests from Volkaert et ai, 1994)

'---------,-1 T_rend __ line--"c--__ ,--, ----,Ic-------': : _____ L _____ L _____ L _____ L _____ L_____ _ __ ~~-----

I I I I I I ~ I I I

I I I I ••• : -----,-----,-----,-----,----- --.--,-----,-----I I I I I

I I I I -----r-----r-----r----- -----r-----T-----T-----

I I 1 I I I

-----~-----~-~--- -----~-----~-----~-----~-----I • I I I

_____ l _____ ! _____ ! _____ l ____ _ I I I

1.E-Q7 +---_~-_____,__--____,__-----,----,___-____r--_____,_--__1

1 E-17 1 E-16 1 E-15 1 E-14 1E-13 1E-12 1 E-11 1E-10 1E-09

Intrinsic permeability under dry conditions (m;

Figure 8. Knudsen diffusivity as a function of intrinsic permeability.

5. Modelling drying induced by temperature gradients

In this section, we want to show experimental evidence of the capabilities of the model proposed. For this purpose a heating test carried out at CIEMA T laboratories has been chosen (Villar et aI, 1997). In the test chosen here (CTFl) a temperature gradient (9 °C/cm) is imposed on a sample composed by four layers of clay (2.5 cm each). There is no water flow through the boundaries and the initial dry density is 1.65 glcm3• After 14 days, water content and dry density were measured in the four different layers. From these measurements, the experimental degree of saturation curve in the sample has been obtained (see Figure 11). Although a change of dry density from the initial value to 1.48-1.60 glcm3 (higher value near heater) the mechanical effects are ignored in this work. Figure 9 shows the water fluxes induced by the temperature gradient.

264

Constant temperature (hot side). No mass flow

S. OLIVELLA et aI.

Vapour flux

Evaporation

Liquid flux

Condensation ~ Constant temperature (cold side). No mass flow

Figure 9. Schematic representation of fluxes of water induced by a temperature gradient.

The simulation of the test has been performed using the variables and parameters that are presented in Table 1. In Figure 10 the permeability for each phase (in m2) is plotted together with the intrinsic permeability curve. The parameters chosen lead to values which are in agreement with the experimental results shown in Figure 7 for FEBEX clay. The results of the simulation are shown in Figure 11. It can be seen that a quite good agreement with the measurements is obtained with this analysis, except for the driest zone. It is important to mention that the high mobility of the gas phase is a crucial point for the drying to occur. In fact, if the intrinsic permeability of the medium were considered constant (of the order of 10-19 m2) vapour diffusion would have induced a high gas pressure gradient (according to Eq. 19). In such case, the steady state regime obtained would not have shown this profile of degree of saturation but a nearly constant value indicating that the drying is prevented.


Table 1. Variables and parameters used for the simulation of the CIEMAT test.

Description

Sample length:

Clay porosity:

Initial gas phase pressure (Pg ):

Initial liquid pressure (PI):

Initial temperature (n:

Specific heats (cs, C'" ca):

Thermal conductivity:

IOcm

0.44

0.1 MPa

-75.0 MPa

30°C

1100,4180, 1000 J/kgK

'I _ 'I S. 'I S, II. - II. dry II. sal

Adry =O.5W/mK Asal = 1.28 WI mI<

Liquid phase relative permeability:

Gas phase relative permeability:

Vapour molecular diffusivity (D):

Retention curve parameters (Po, A,):

Boundary temperatures (Tj, T2):

Intrinsic permeability:

krl=S/2

krg=Sg (T 0 C)2.3

5.9 X 10-12 ' m2 / s (Pg ,MPa)

18 MPa, 0.38

120,30°C

k(n. ) = Ik (l-Ij>of Ij>~ 'I'M 0 n. 3 ( )2

'1'0 l-Ij>M

k = 10-13 o

266

lE-12

lE-13

lE-14

lE-15

f lE-16

~ lE-17

:s lE-18 III ~ lE-19 ... 8!. lE-20

lE-21

lE-22

lE-23

lE-24

0

S. OLIVELLA et al.

Proposed model for clay permeability

----r---'----~----r----r---'--

1 1 1

- krl*k (liquid intrinsic permeability)

- krg*k (gas intrinsic permeability)

- ;;; ;; - 1- - - - -1- - - - k (intrinsic - .. - .. 1 permeability) - - - -1- - - - -j- - - - -+ - - - - t- - - - - _ ...... _j.- - ~~~~~~~~~

I I I I I til ....... 1 I I

----1-----1-----+----+-----1-----1-- --r ........ -~----I_----I I I .. ,..... I

- - - -1- - - - -I - - - - -+ - - - - +- - - - - 1- - - - -I - - - - --I - - - - - '!!. - oIr- - - - -I I I I t I ......

- - - -1- - - - --l - - - - ....j. - - - - .j.... - - - - -

I I I I

-~----~---~----~----~----~-- -I I I I I

____ 1 ____ --I _ _ _...1 ____ l.- ____ 1 _____ 1 ____ ...J ____ J... ____ I-- ___ _

ttl I I I ____ 1_ _ _ _ _ ___ .! ____ L ____ 1 _____ 1 ____ -I ____ L ____ L ___ _

I I I I I I I I I

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Liquid degree of saturation

Figure 10. Curves of permeability to gas and liquid phases according to the proposed model.

0.7

0.6 c 0 ~O.5 ... ::::J -; 0.4 II) -00.3 (1) (1)

t;,0.2 (1)

C 0.1

-+-Calculated. liS~: 0 - Calculated. li : 0.5 -+-CaIculated. li : 2 --*-CaIculated. li : 14 -Ii. -Experimental: lirre (d):14

0 0 0.02 0.04 0.06 0.08 0.1

Distance from hot side (m)

Figure 11. Interpretation of a lab-heating test performed in eIEMAT. Profiles of degree of saturation for different times during the calculation and experimental results for 14 days. Averaged values are shown with symbols while the full profile is shown in grey.


In order to understand better the effect of permeability on drying, a relationship will be derived from Eqs. (20) and (21). These equations have been introduced in the gradient of mass fraction split in terms of the state variables considered here (Pg, PI and 1), i.e.:

am w am w am w

Vm; =~VPg +~V~ +~VT oPg o~ oT

am; T am; T am; Vmw =---AVmw ----BVmw +--VT

8 aPg k 8 a~ k 8 aT am W

_-8 VT Vm W = aT

8 ( Tam; Tam; J l----A+---B k aPg k a~

(24)

Where A (>0) and B (>0) have been defined in Eqs. (20) and (21), respectively. Using the ideal gases law, mass fraction of vapour can be written as:

(25)

where Pv is the vapour pressure and Pa the air pressure. If psychrometric effects are neglected CdooloPI = 0) and from (25) it follows that o(JioPg < 0 and, from Eq. (24), this implies that the gradient of vapour mass fraction is proportional to intrinsic permeability. Therefore the lowest the intrinsic permeability, the lowest the mass fraction gradient and the lowest the drying induced by a temperature gradient.

In order to show the influence of this effect on drying a sensitivity analysis to the parameters ko and f3 has been performed for the modelling of the experiment of heating. Table 2 shows the values adopted and Figure 12 the profiles of degree of saturation at 14 days for each case. It is clearly shown that gas mobility plays an important role. In the three cases compared in Figure 12 the intrinsic permeability to liquid phase is the same but the intrinsic permeability to the gas phase has been reduced. It can be seen that as gas mobility decreases the drying effects also reduce.

268

Ko

10-13

10-15

10-18

0.7

0.6 I: 0 :;: 0.5 co ... ::::s 1; 0.4 U) -00.3 (1) (1) ... ~0.2 c

0.1

0

0

S. OLIVELLA et al.

Table 2. Parameter values for sensitivity calculations.

~

4.75

3.2

1.0

0.02

k(gas) (m2)

10-13

10-15

10-18

0.04

k(liquid) )

(m2)

2. 11 X 10-20

2.17x10-2O

2.05xlO-W

~CASEO. Time(d): 14 ~CASE 1. Time(d): 14 ---CASE 2. Time(d): 14

Case 0

Case 1

Case 2

-k- Experimental: Time (d):14

0.06 0.08 Distance from hot side (m)

0.1

Figure 12. Sensitivity to intrinsic permeability function. Profiles for degree of saturation at 14 days for different functions of permeability. Averaged values are shown with

symbols while the full profile is shown in grey.

6. Conclusions

The main objective of this paper was to investigate water evaporation and vapour transport in the context of unsaturated soils and this has been motivated by several research works related to radioactive waste disposal in geological media. In fact, the maximum temperatures permitted are in some cases above 100 DC, but even in the case


that temperature was limited to 100 °C there is a need to treat in an appropriate way the gas phase.

First we have discussed the two main existing approaches for modelling thermal effects in unsaturated porous soils and the main differences have been highlighted. It has been shown that when the NUS (nonisothermal unsaturated soil) approach is used to model an initially saturated soil that develops small capillary pressures, phase change tends to behave in the same way as one would obtain with the GR (geothermal reservoir) approach. In this case the phase change diagram of water controls desaturation.

Secondly, transport processes have been discussed. The relative importance of terms that appear in a general formulation has been explained. Also, the question of vapour flux enhancement has been treated when the cause is related to air immobility. Particularly the problem of mass transfer induced by temperature gradients has been treated. Gas and liquid pressure gradients are developed by a thermal gradient in an unsaturated soil due to water evaporation-migration-condensation.

Finally, a model for intrinsic permeability for clays that change its structure when wetting-drying occurs. This model is necessary for modelling the moisture content profiles induced by temperature gradients in low permeability soils. The proposes model explains at the same time the higher gas mobility encountered in clays compared to water mobility and the process of drying induced by temperature gradients. In order to show the capabilities of the model and formulation an experimental test has been modelled and the calculated results show a very good agreement with the measurements. Also some sensitivity calculations have been carried out to demonstrate the necessity of the dependence of intrinsic permeability on liquid water content introduced.

Appendix. Balance equations

Mass balance of water

Water is present in liquid an gas phases. The total mass balance of water is expressed as:

(A.l)

wherer is an external supply of water, ~ is porosity, OOai is the mass fraction of species i in phase a, Pa is the density of phase a., and Sa is the degree of saturation of phase a.

Mass balance of air

Air is present in liquid an gas phases. The total mass balance of air is expressed as:


:t (ffi~P1Sl<l> + ffi:PgS g<l» + V· (j~ + j:) = fa (A2)

Internal energy balance for the medium

The equation for internal energy balance for the porous medium is established taking into account the internal energy in each phase (Es, E/, Eg):

(A3)

where ic is energy flux due to conduction through the porous medium, the other fluxes (iEs, jEt. jEg) are advective fluxes of energy caused by mass motions and f is an internal/external energy supply.

Acknowledgements

The support of ENRESA and ANDRA through research grants is gratefully acknowledged.

References

Bear, J. (1972): Dynamics of fluids in porous media, American Elsevier Publishing Company, inc ..

Bear, J. and A Gilman, (1995), Migration of Salts in the Unsaturated Zone Caused by Heating, Transport in Porous Media, 19: 139-156.

Bird, R. B., W.E. Stewart and E.N. Lightfoot (1960): Transport Phenomena, John Wiley, New York, 1960.

Edlefson, N.E. and AB.C. Anderson, (1943): Thermodynamics of soil moisture. Hilgardia, 15(2): 31-298.

Falta, R.W. , K Pruess, I Javandel, and P.A Witherspoon, (1992): Numerical Modelling of Steam Injection for he Removal of Nonaqueus Phase Liquids from the Subsurface. 1. Numerical Formulation. Water Resources Research, vol. 28, No 2, 433-449.


Faust, c.R. and J.W. Mercer, (1979): Geothermal Reservoir Simulation: 1. Mathematical Models for Liquid- and Vapour- Dominated Hydrothermal Systems, Water Resources Research, vol. 15, No 1,23-30.

Milly, P.C.D. (1982): Moisture and Heat Transport in Hysteretic, Inhomogeneous Porous Media: A Matric Head-Based Formulation and a Numerical Model, Water Resources Research, vol. 18, No 3: 489-498.

Olivella, S., J. Carrera, A Gens, E. E. Alonso (1994): Non-isothermal Multiphase Flow of Brine and Gas through Saline media. Transport in Porous Media, Vol 15: 271-293.

Olivella, S., J. Carrera, A Gens, E. E. Alonso (1996a): Porosity Variations in Saline Media Caused by Temperature Gradients Coupled to Multiphase Flow and Dissolution/Precipitation. Transport in Porous Media, Vol 25: 1-25.

Olivella, S., A Gens, J. Carrera, E. E. Alonso (1996b): Numerical Formulation for a Simulator (CODE_BRIGHT) for the Coupled Analysis of Saline Media. Engineering Computations, Vol. 13, No 7,87-112

Philip, J.R. and D.A. de Vries, (1957): Moisture Movement in Porous Materials under Temperature Gradients, EOS Trans. AGU, 38(2):222-232.

Pollock, D. W. (1986): Simulation of Fluid Flow and Energy Transport Processes Associated With High-Level Radioactive Waste Disposal in Unsaturated Alluvium. Water Resources Research, Vol. 22, no.5 :765-775.

Roberts, P.J., R.W. Lewis, G. Carradori and A Peano (1987), An Extension ogf the Thermodynamic Domain of a Geothermal Reservoir Simulator, Transport in Porous Media, Vol 2,: 397-420.

van Genuchten, R., (1980): A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J.: 892-898.

Villar, M.V., AM. Fernandez and 1. Cuevas, 1997, Full Scale Engineered Barriers Experiment in Crystalline Host Rock, Caracterizacion Geoqufmica de bentonita compactada: efector producidos por flujo termohidniulico. Informe 70-IMA-M-0-2, Ciemat, Enresa.

Villar, M. V. (1998), Ensayos para el proyecto FEBEX, CIEMAT-report 70-IMA-L-5-51, prepared for ENRESA

Volkaert, G, L. Ortiz, P. De Canniere, M. Put, S.T. Horseman, J.P. Harrington, V. Fioravante and M. Impey, (1994), Modelling and Experiments on Gas Migration in Repository Host Rocks, MEGAS Project, Final Report, Phase 1,.

272 s. OLIVELLA et al.

Wail Abu-EI-Sha'r and L. M. Abriola, (1997), Experimental assessment of gas transport mechanisms in natural porous media. Parameter estimation. Water Resources Research, vol 33, no 4, pp 505-516.

BEHA VIOUR OF INFILTRATION PLUME IN POROUS MEDIA.

Adequacy between Numerical Results and a Simplified Theory

C. OL TEAN and M.A. BuES Laboratoire Environnement, Geomecanique et Ouvrages Rue du Doyen Marcel Roubault, BP 40 F - 54501 Vandoeuvre-les-Nancy

Abstract - The proposed numerical code simulates the movement of a fluid as well as the transport of a non-reactive pollutant into a saturated porous media (2D configuration). The model uses a combination of the mixed hybrid finite element method and the discontinuous finite element method. Coupling between flow and transport is carried out by an equation of state. In the mixing zone, the density is assumed to vary as a function of concentration. In a saturated media, the transport of an incompressible fluid is described by a set of initial and boundary conditions and by a system of equations constituted by Darcy's law, the continuity equation, Fick's law and the advectiondispersion equation. Precision in estimating the velocity field, which determines pollutant propagation, is essential. Results obtained by classical numerical methods (conforming finite element method or classical finite difference methods) are often not very satisfactory due to the diffusive character of these methods. In order to compensate for these disadvantages, a combination between the mixed hybrid finite element technique and the discontinuous finite element technique has been implemented. When applied to the problem under consideration, this technique makes it possible to simultaneously estimate the pressure field and the velocity field (hydrodynamic module) as well as the dispersive flux and concentration field (mass transport module). Furthermore, application of these methods makes it possible to preserve the mass balance at the scale of each element and to ensure the continuity of the normal components of the velocity and dispersive flux from one element to another. In order to analyse the infiltration of a salt solute punctually injected into a porous medium, a comparison between a simplified theory and numerical simulations is presented. The density contrast between the two miscible fluids, as also the injection flow rate, play an important role. Studies carried out on 2D physical models have shown the existence of a steady-state regime located in the middle of the mixing zone. With such observations, the equations describing the transport phenomenon can be modified in order to lead to a simplified analytic solution. This result experimentally established is bounded by numerical verifications.

1. Introduction

The development of prevention measures to avoid degradation of underground water resources relies on numerical modelling that makes it possible to estimate the migration


274 C. OLTEAN AND M. A. BOOS

of harmful products through porous media. These products can present different physical properties (e.g.: density, dynamic viscosity) from the fresh water that constitutes the natural aquifer. When these differences become significant the pollutant can no longer be considered as a tracer. Its propagation should be studied by coupling momentum transfer concepts with mass transport concepts using equations of state. In order to simultaneously resolve both problems, most packages use either the conforming finite element method - CFEM - or the finite difference method - FDM-.

The application of one of these methods to the resolution of the hydrodynamic module requires pressure field estimation in order to determine the velocity field by differentiation of the pressure field. Frind and Matanga (1985) have shown that in certain cases (e.g.: aquifers with a low-pressure gradient), the velocity field calculated with the CFEM can give less than satisfactory results. Furthermore, the normal component of the velocity field is discontinuous from one element to the next. Cordes and Kinzelbach (1992) have proposed a solution using the velocity field determined with the CFEM that makes it possible to obtain a new velocity field where the normal component is continuous at the interface between two adjacent elements. Thus, this technique improves the velocity field accuracy close to the imposed flux boundaries (especially in the case of impermeable boundaries) but does not eliminate the error due to numerical differentiation of the pressure.

The mixed finite element method - MFEM - [Meissner, 1973; Chavent and Jaffre, 1986] or the mixed hybrid finite element method - MHFEM - [Chavent and Roberts, 1991] provide an answer to the problems described above.

The streamlines as well as the fluxes calculated with the MFEM have been compared with those estimated using the control volume finite element method - CVFEM -on a squared area composed of porous heterogeneous media [Durlofsky, 1994]. The conclusions reached as a result of this comparison show the advantage of the MFEM when the system being studied is highly discontinuous or heterogeneous. In this case, the MFEM leads to better estimation concerning the flow variables than the CVFEM with the same number of unknowns. The efficiency of the mixed hybrid finite element method - MHFEM - has been shown by Mose et al. (1994). These authors have compared the results obtained by the MHFEM with those obtained by Cordes and Kinzelbach (1992). The comparison was carried out in the form of streamlines constructed using a semi-analytical method based on the resolution of the streamline function [Frind and Matanga, 1985; Pollock, 1988; Cordes and Kinzelbach, 1992]. Their conclusions are similar to those reached by Durlofsky (1994), i.e.: a more accurate estimation of the velocity field for highly heterogeneous systems (high ratio between the lowest and highest hydraulic conductivity).

The application of classical methods to the resolution of the transport equation frequently introduces artificial dispersion in the concentration front estimate (porous system subjects to a pulse or stepwise profile) as soon as the advective phenomenon becomes dominant. Several methods have been developed to limit or avoid these effects. Amongst these methods, one can point out the random walk method that was successfully used in tracer test simulation experiments carried out in situ at Twin Lake in which the 3D configuration was taken into account [Ackerer et at., 1990].

The upstream decentralized scheme incorporated in the finite difference or finite element methods [Richtmeyer and Morton, 1967; Sun and Yeh, 1983; Zienkiewicz, 1986] also makes it possible to obtain an approximation that is stable, but has a nonnegligible numerical diffusion at the front level. To compensate for this problem, more

BEHAVIOUR OF INFILTRATION PLUME IN POROUS MEDIA 275

precise schemes of a higher order have been developed which are non-oscillating at the front level [Harten, 1983; Chakravarthy and Osher, 1985; Van Leer, 1985]. Advective fluxes are obtained by resolving Riemann's problem at the discontinuity point. A slope limiter making it possible to redistribute the concentrations in an element to suppress the oscillations stabilizes these methods. Putti et al. (1990) have developed and applied this scheme to triangular elements with a finite volume approximation to resolve the transfer equation in porous media. Even though finite volume techniques have been in use for quite some time [Peyret and Taylor, 1983], Putti et al. (1990) were among the first in the water resource community to implement them.

Finally, another numerical technique that has been tested by experiments carried out on two laboratory physical models [Oltean et al., 1994; Oltean, 1995; Siegel, 1995] should be noted. This technique, known as the discontinuous finite element method (DFEM), was also tested by some analytical solutions [Siegel et al., 1997]. Used for the resolution of the advectif term only, it provides accurate results when this term is predominant. The dispersive term is solved by using the MHFEM.

As a result, in order to resolve the mass transport equation taking into account the influence of density contrast, we have developed and tested these last techniques [01-tean and Bues, 1998] (combination between the DFEM and MHFEM) on Henry's problem [Henry, 1964]. These methods make it possible to ensure continuity of the normal component of the dispersive flux from one element to another and to preserve the mass solute balance at the scale of each element.

2. Numerical methods

The motion of a solute into a saturated porous medium can be described by a set of partial differential equations expressed by the fluid and solute mass balances, Darcy's and Fick's laws and the state equations:

a(pE) +V.(pV)= Q at p

a.

a(PEC m ) (- ) at + V.I,pVC m + J = QpC mo b.

v = _k(vp+pgVz) ~

c. (1)

J = - pEDV(C m ) d.

p=p(Cm ) e.

where V denotes the gradient, V. the divergence, t [T] the time, p [ML-3] the density, V

[Lrl] Darcy's velocity of a fluid particle, E [dimensionless] the effective porosity, Qp

[ML-3r 1] flow source/sink term, Cm [MIM] the solute mass fraction, Cmo [MIM] the

solute mass fraction in the source/sink fluid, J [ML -2r 1] the dispersive solute flux, k

[L2] the permeability tensor, g [Lr2] the gravity acceleration, P [ML-lr2] the hy-


draulic pressure, ~ [ML -1 r 1] the dynamic viscosity, z [L] the elevation and D [L 2r 1 ] the general hydrodynamic dispersion tensor which is written as:

(2)

with Dm [L 2r 1] the molecular diffusion coefficient, aL and aT [L] the longitudinal

and transversal dispersivities, respectively, U=VIE [Lrl] the real flow velocity and Oij

Kronecker's symbol. If the porous medium is considered as non-deformable (oEloP = 0 ), the fluid mass

balance equation can be written in terms of the dissolved mass fraction:

op oC m () E----+V. pV =Q OCm at p

(3)

With this consideration, the solute mass balance becomes [Voss, 1984; Herbert et al., 1988]:

These two equations are coupled by the equation of state described in our model by the relationship:

(5)

where Po and Co are reference density and mass fraction concentration respectively

(generally Co = 0). Y is the experimental constant computed by using the data given by

Weast (1981). The hydrodynamic equations (La and I.c) were solved using the mixed hybrid fi

nite element method. Theoretical aspects as well as the numerical technique, applied to 2D flow of incompressible and homogeneous fluids into saturated porous medium, were widely developed by Chavent and Roberts (1991) and Mose et al. (1994). During the displacement of two miscible fluids with different physical properties - contrast of density and/or dynamic viscosity - the application of this technique involves a new formulation of previous equations.

Let V, a vector having the properties of Raviart-Thomas's space [Thomas, 1985]

and QKi its volumic flux through the edges "i' of each element 'K'. The variational

form of Darcy's law ensures the link between this volumic flux and the change in total pressure appertaining to each 'K' element. By putting this equation into (l.a), the pres-


ence of the term V.(pV) involves the development in a non conservative form for the continuity equation, i.e.:

when TPKi represents the approximation of the mean value of the density on the edges Ai' Due to this variable, the linear system in TP Ki obtained by hybridation is modified to a non-symmetric system. So, the advantage of the hybridation (symmetric, positive definite linear system in the unknowns TP only) is not justified.

On the other hand, the conservative form of the continuity equation is preserved if

the initial vector V (resp. QKi) is replaced by vector q == p V (resp. QKi its mass flux

through the edges 'i' of each element 'K'). So, the development of V.(pV) == V.(q) does not introduce the variable Tp and the system in TP remains a symmetric, positive definite linear system.

By combining (l.a) and (l.b), the equation describing the transport process of a concentrated solute in porous media (l.b) can be expressed in the following form:

pE aC m = -v.(p ve m )-v.(J)+ Cm [ V.(p V )]+ Qp (C mo - Cm ) (7) at

where V.(p VC m ) represents the advection term, Cm V.(p V) the accumulation term,

V.(J) the dispersion term and Qp(Cmo - Cm) the source/sink term.

In order to apply the MHFE technique, some modifications of the dispersive term should be carried out. In accordance with the properties of the Raviart Thomas space, the dispersive term on each element 'K', JK == -PKEDKV(CmK) == -DKV(CmK) can be

* * written as J K == QKi 8i where QKi represents the dispersive flux of JK that crosses

the 'K' edges and 0i the components of the unit vector. The components of the DK

tensor are:

(8)

where qKi and qKj are the components of 'Darcy's flux'. Under these considerations,

the dispersive term can be expressed and solved in the same way as Darcy's flux. Like the mass balance equation, the transport equation is also a parabolic equation.

Nevertheless, when the dispersive term, in comparison with the advective term, is negligible, the transport equation becomes a hyperbolic equation. Under these conditions, to apply the MHFE technique is not advisable. It introduces numerical dispersion and spurious oscillations. In order to avoid these phenomena, the advective term was solved by using the DFEM. The theoretical approach of this method for the tracer case and in one and two-dimensional configurations is largely described by Gowda and Jaffre (1993) and Siegel et al. (1997). In spite of the density variations, by using Darcy's flux

278 C. OLTEAN AND M. A. BuES

in the transport equation, the advective term can be transformed into a similar expression used by the authors mentioned above, i.e.:

aCm ( ) p£--= - v. qCm at (9)

By using the divergence theorem and an explicit time dicretization scheme, the variational form of this hyperbolic equation can be written as:

Cn+1 Cn f Pn£ mK- mK'\'} =f (qncn }V(t} )-f cinoroutqnt} n (10) K K ~t KKK mK K oK mKe KKK

Nevertheless, the calculated solution oscillates. To stabilize it, Gowda and Jaffre (1993) propose to use a slope limiter and a second-order time discretization scheme (by introducing an intermediate time step in which the solution is calculated at the time tn+ 112 = (tn + tn+ 1)12 by means of a local calculation).

In order to solve the transport equation, the idea of Siegel et al. (1997) was to combine the DFEM - used for the resolution of the advective term - with the MHFEM -used for the resolution of the dispersive term -. Consequently, the dispersive, the accumulative and the source/sink terms are added to the equation (10), in order to obtain the complete transport equation.

With these methods, the solute mass balance is preserved over each element, the advective fluxes are uniquely defined at the interface between the adjacent elements and the dispersive flux is continuous between two adjacent elements. Moreover, as proved by Gowda and Jaffre (1993) the solution obtained by the discontinuous finite element method associated with the slope limiter is such that its averages over the discretization intervals are total variation diminishing (TVD). This property excludes the existence of non-physical oscillations.

Based on these variational equations, our numerical code executes the coupling of fonctionnal blocks constituted by the hydrodynamic module and the mass transport module.

3. Simplified Solution and Study Domain

The study conducted by Triboix et al. (1975), concerns the salt solution infiltration into a 2D homogeneous, isotropic and non-deformable porous medium.

Figure 1 presents the experimental configuration. Through a thickness "e", the pollutant of "Co" concentration is injected in "0" with a mass flow rate "m". The mass

flow rate is estimated in order to assure a mechanical dispersion regime (V y dpllDI > 500 where dp represents the grain diameter, Vy the magnitude of the velocity vector in

the "y" direction and IDI is the matrix norm of the dispersion tensor). By visualization, the authors note that in the region OAB called "core" the flow becomes steady-state. Moreover, the opening angle "a" becomes rapidly and remains constant. With these experimental observations and with some simplifications (the third order of C, the lon-


gitudinal dispersion and the molecular diffusion in the transport equation were been neglected), as well as the Darcy's law and the transport equation can be simplified in order to obtain an analytical solution written as:

(11)

x

y(t)

Figure 1. Scheme of the mixing zone

Moreover, if the pollutant mass dispersed to the extremity of the "core" is neglected, the mass balance for the time "t" becomes:

yo{t) +00

m.t = f dy f p£Cedx

o -00

(12)

The integration of this last expression permits to the authors to establish the advancing of the "core" versus time:

(13)

where:

280 C. OLTEAN AND M. A. Elms

a=[(L1P)o j0.4[ 25g ]0.4 [..!...j0.4[m]0'4 =A[m]0,4

PoCo 16~8n!loE2 aT e e

and (L1p)o = PI - Po .

These expressions (11 and 13) have been verified on the experimental set-up. In order to numerically test the hypothesis and results proposed by Triboix et al. (1975), the experimental set-up is divided in 15x100x2 triangular elements. The numerical study carried out on the half of the domain. The initial and boundary conditions are represented by: (i) hydrostatic distribution of pressure in steady-state flow and (ii) a constant pressure imposed to lower part of the domain with the help of a weir attached to the setup by an evacuation tank (S) and a mass flow rate injected in the upper part of the domain (figure 2).

weir no flow x .... ....::::::::::::::::::.. .... ~it--"JI\--1~ Porosity: 0.405

................... . .. ................... ................... ................... ................... ................... .................... ....... . .................... ....... . .................... ....... . .................... ....... . .................... ....... . .................... ....... . :::::::::::: 0,18m :::: ............................. ............................. .. .............................. . .............................. . .............................. ,., .................................

::::::::::::: Po

1,5 % Initial concentration Co:

3% Grain diameter: 0,3 cm

Pressure Po: Pogh with h = 1,0 m

Dispersivity a,.: 3,0 10.3 m

Dispersivity 0,.: 3,0 10-4 m

Permeability: 8,3 1O-2 m/s

Figure 2_ Domain, boundary conditions and characteristics of porous medium

4. Results and Discussions

Initially, the existence of a steady state must be verified. This step will be carried out numerically by using the complete equations describing the two processes linked together, i.e. : flow and transport. Although some other physical size may be taken into account (velocity, pressure), it will be supposed that the state steady is reached when the evolution of concentrations in the field described previously is independent of time.

Let us note, however, that the arrival of the pollutant in the evacuation tank will provoke a modification of the boundary condition in y = h. In order to avoid consider-


ing a temporal variation of this condition, it will be advisable to limit simulation time to

* * a value t ,such as: C(x,h,t ) = O. The evolution of the concentrations is represented on figure 3, for certain points lo

cated in the centre section of the field (i.e.: x = 0). Although this figure relates only to the numerical results obtained for an injected

mass flow rate m = 7.5 10-5 kg/s, it can be regarded as representative of the other mass flow rate used. Thus, it is noted that, for a given ordinate "yi", the concentration becomes constant as from a time noted "ti". This observation, expressed as ac/at = 0, demonstrates the existence of a flow state comparable with a steady state. Nevertheless, this observation is valid only for the ordinates located in the higher section of the field (y = Ym of about 0.5 m). For the other ordinates (y > Ym)' a slight variation in the con-

centration is observed according to time. Let us also note that the estimate of t *, obtained by following-up qo, h), is about 2000 s.

In order to explain the second experimental observation (a = constant), let us look at the distribution of the velocity field represented both on figure 4 and on figure 5. Figure 4 represents a total evolution with 15 and 30 minutes of the distribution of the velocity field in the whole domain studied. Figure 5 is limited to the distribution of velocity according to time for certain points co-ordinate (Xi, 0.01 m) located in the upper section

of the domain. Thus, the analysis of figure 4 shows the appearance and development of a convective cell located at the limit of the mixing zone between fresh water and salt water. This cell follows the advance of the polluting front whilst constantly maintaining the same rotation direction, i.e.: clockwise. The fresh water is thus pushed towards the centre section of the domain. As velocity quickly becomes constant (figure 5), the opening of the angle also becomes constant.

In parallel, our study deals with the concentration distribution on the axis of symmetry (x = 0). For a given mass flow rate injected, the expression (11) is reduced to the following form :

c2(0 y)_ Ilo m 1 Ilo m 1 B2 y-O.5 (14) , - kg (i1p )0 e ~21taT y kgy e ~21taT y

Co

. ap (i1p)o 2 Ilo m 1 With y=-=--=ct.andB =--- =ct.

ac Co kgy e ~21taT

The equation (14) is also written as:

c(o, y)= B y -0.25 (15)

which, in logarithmic co-ordinates, represents the equation of a line. Thus, the variation in concentration in the central section depends only on the ordinate "y" of the considered point. In figure 6, both the theoretical (eq. 15) and numerical variations of the concentration in agreement with the ordinate "y" are drawn. Whatever the mass flow rate used, the concentrations obtained using the simplified solution over-estimate those

282

~ = .. .. ~ u co .. u

C. OLTEAN AND M. A. BvES

Iojected Concen tratioo Co = 1.5 'l> I.()~

\' 0.7

• I 0.90 •.....•...................... ~ ....... 0.6

0.75 • 0 o I 0.5 0

OOOOO •• ii~OOOOOOOOOOO~OOOOOOO 0.6() 0.4 . ..~... ~~ ~ ~

US • YI",0.105m I

OJ

0.30 0 Yl= 0.305 m 0.2

• Y J= 0.505 m O.IS 0 Y,'" 0.705 m

9':1

0 0.00 ... _ .•.....

() 500 1000 1500 2000 2500 Time (s)

Figure 3. Distribution of the concentration in some points located in the middle section of the domain (x '" 0)

1.0 ...------,

0.8

~ 0.6 .. o iii

~ ..., 0.4

0.2

o .0 L...L..;L-1-1..J....JL...LJ

0.00 0.09 0.18 Longueur (m)

0.00 0.09 0.18 Longueur (m)

\ = 15 min \ '" 30 min Figure 4. Distribution of the concentration and the velocity field

0.90

0.60

0.45

O.lO

0.15

.. ~ !::!. " .~ " .. v = 0

U

1: g .... .. '"

.. u

.~ o

U >

BEHAVIOUR OF INFll..TRATION PLUME IN POROUS MEDIA 283

3.0E-4

2.0E-4

J.OE-4

O.OE+O

~

" .S <;; !:: " " u

" 0 u

~ " . ~ " " u

" o U

1.0

0.9

0.8

0.7

0.6

0.5

1.0

0.9

0.8

0.7

0.6

0.5

0

• Injected Concentration Co = 1.5 %

• 0······································ o o • 0000000000000000000000000000000000000

•••••••••••••••••••••••••••••••••••••• • 000000000000000000000000000000000000000

0 • x j =0.015m • x J= 0.035 m

0 x 2= 0.025 m 0 X 4= 0.045 m

500 1000 1500 2000 2500 Time (s)

Figure 5.Distribution of the velocity field in some points located in the same perpendicular section to the mean flow (y = 0.0 I)

- Theoretical distribution

• S im ulation distribution

0.1 0.2 OJ 0.4 0.5 0.6 Travelled Distance - Y (m)

Figure 6. The variation of the concentration in the central section

•• •• •• ••• •• - Theoretical distribution

• Simulation distribution

0.1 0.2 OJ Travelled Distance - Y (m)

0.4 0.5

Figure 7. The variation of the concentration in the central section

0.6


calculated by numerical simulations. The differences noted between the two solutions, expressed in relative errors, vary between 5 and 15%.

Although the mode of flow is steady until ordinate Ym' the variation of the simu-

lated concentration, contrary to the theory, is not approached by a line. As our code was validated using certain classical problems recommended by Voss and Souza, this divergence of behavior enables us to make three comments : (i) simplifying the advection dispersion equation provokes a development of the

mixing zone which does not reflect reality. In fact, the mass of pollutant neglected at the end of the core is necessarily underestimated by this theory thereby leading to an over-estimation of longitudinal core extension. Although it was possible to formulate an estimate of core advance by using the numerical solution, a comparison with expression (13) is not adequate.

(ii) the finite boundaries of the physical model (as of the numerical model) is in disagreement with the theoretical assumptions of infinite extension of the domain.

(iii) therefore, the fact that the theory of Triboix and the results of Trinh Thieu are in good agreement can be justified only by the uncertainties regarding the experimental measurements of the concentrations.

With regard to the second comment, let us look again at figure 4. It is apparent that the advance of the convective cell provokes a heavy disturbance on the velocity field located close to the end of the domain (East lateral limit). Reciprocally, this limit can influence both the development of this convective cell and the development of the mixing zone. Thus, a new simulation has been undertaken on a more extensive domain by distancing this limit by 0.5 m compared to the centre section.

The new concentration distribution for an injected mass flow rate is represented on figure 7. Although the difference between the new numerical solution and the simpli-fied solution (qo, y) = 5.473 10-3 y -0.25 ) increased by approximately 5%, its variation at the core is linear. It can be represented by a single line expressed in the follow-ing form: qo, y) = 4.306 10-3 y -0.298. One can thus note that, compared to the simplified solution, the numerical solution presents a smaller coefficient B and a larger power coefficient.

Finally, the last part of this study relates to the advancing of the 'core'. By using the simplified solution, the advancing could be expressed by the expression (13). Nevertheless, the numerical simulations highlighted that the steady state does not relate to the entirety of the mixing zone. Its development is strongly influenced by the existence of the convective cell. Thus, in the adjacent sections to the centre of rotation of the convective cell, the flow in the mixing zone cannot be compared to a steady state flow. Consequently the equation (12), expressing the mass balance inside the core by using a linear variation - in logarithmic co-ordinates - of the concentration, is not valid any more with the results obtained by numerical simulations. Although an estimate of advancing of the core by using the numerical solution could be carried out, a comparison with the expression (13) is not adequate.

References

Ackerer, Ph., Mose, R., and Sernra, K. 1990, Natural tracer test simulation by stochastic particle tracking method, in: G. Moltyaner (ed.), Transport and Mass Exchange Processes in Sand and Gravel Aquifers, 595-604.


Chakravarthy, S., and Osher, S. (1985) Computing with high resolution: upwind schemes for hyperbolic equations, Lect. Notes Appli. Math., 22, 57-86.

Chavent, G., and Jaffn:, J. (1986) Mathematical models and finite elements for reservoir simulation, North Holland, Amsterdam.

Chavent, G., and Roberts, J.E., (1991) A unified physical presentation of mixed, mixed hybrid finite elements and standard finite difference approximation for the determination of velocities in water flow problems, Adv. Water Resour., 14, 329-348.

Cordes, C., and Kinzelbach, W., (1992) Continuous groundwater velocity field and path lines in linear, bilinear, and trilinear finite elements, Water Resour. Res., 28, 2903-2911.

Durlofsky, LJ., (1994) Accuracy of mixed and control volume finite element approximation to Darcy velocity and related quantities, Water Resour. Res., 30, 965-973.

Frind, E.O., and Matanga, G.O., (1985) The dual formulation of flow for contaminant transport modeling. 1. Review of theory and accuracy aspects, Water Resour. Res., 21, 159-169.

Gowda, V., and Jaffre, J., (1993) A discontinuous finite element method for scalar nonlinear conservation laws, Rapport de recherche INRIA 1848.

Harten, A., (1983) High resolution schemes for hyperbolic conservation laws, 1. Comput. Phys., 49, 357-393.

Henry, H.R., (1964) Effects of dispersion on salt encroachment in coastal aquifers, U.S. Geol. Surv. Water Supply Paper, 1613-C, C71-C84.

Meissner, U., (1973) A mixed finite element model for use a potential flow problem, Int. 1. Numer. Methods Eng., 6, 467-473.

Mose, R., Siegel, P., Ackerer, Ph., and Chavent, G., (1994) Application of the mixed hybrid element approximation in a groundwater flow model: lUXury or necessity?, Water Resour. Res., 30,3001-3012.

Oltean, c., (1995) Comportement du deplacement d'unfront d'eau douce/eau sa/ee en milieu poreux sature: modelisations physique et numerique, PhD Thesis, Universite Louis Pasteur, Strasbourg, France, pp. 198.

Oltean, C., and Bues, M.A. (1998) Transport in saturated porous medium, in J.M. Crolet and M. E. Hatri (eds), Recent Advances in Problems of Flow and Transport in Porous Media, Kluwer Academic Publishers, Dordrecht, pp. 101-115.

Peyret, R. and Taylor, T.D.: 1983, Computational methods for fluid flow, Springer-Verlag, New York, pp. 358.

Pollock, D.W., (1988) Semianalytical computation of pathlines for finite difference models, Ground Water, 26,753-750.

Putti, M., Yeh, W.W.-G., and Mulder, V.A., (1990) A triangular finite volume approach with high resolution upwind terms for the solution of groundwater transport equations, Water Resour. Res., 26, 2865-2880.

Raviart, P.A., and Thomas, J.M., (1977) A mixed finite method for the second order elliptic problems, in Mathematical Aspects of the Finite Element Method, Lect. Notes Math., Springer-Verlag, New York, 292-315.

Richtmeyer, R.D., and Morton, K.W., (1967) Difference methods for initial-value problems, WileyIntersciences, New York.

Siegel, P., (1995) Transfert de masse en milieu poreuxfortement heterogene: modelisation et estimation de parametres par elements finis mixtes hybrides et discontinus, PhD Thesis, Universite Louis Pasteur, Strasbourg, France, pp. 185.

Siegel, P., Mose, R., Ackerer, Ph., and Jaffre, J., (1997) Solution of the advection-diffusion equation using a combination of discontinuous and mixed finite elements, Int. 1. for Numer. Methods in Fluids, 24, 595-613.

Sun, N.-Z., and Yeh, W.W.-G., (1983) A proposed upstream weight numerical method for simulating pollutant transport in groundwater, Water Resour. Res., 19, 1489-1500.

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Triboix, A.,Trinh Thieu, B., and Zilliox L. (1975) Infiltration d'un liquide miscible dans un milieu poreux sature d'eau au repos, C. R. Acad. Sc. Paris, t.280, 1713-1716.

Van Leer, B., (1985) Upwind-difference methods for aerodynamic problems governed by the Euler equations, Lect. Notes Appl. Math., 22, 327-336.

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A comparison of two alternatives to simulate reactive transport in groundwater

MAARTENW. SAALTINKi ,2, JESUS CARRERAi AND CARLOS AYORA2

1 Dep. d' Enginyeria del Terreny i Cartogrdjica, ETSECCPB, Universitat Politecnica de Catalunya, CI lordi Girona 1-3, MOdul D-2, 08034 Barcelona, Spain.

2 Institut de Ciencies de la Terra (laume Almera), Consejo Superior de Investigaciones Cientijicas, CI UUIS

Sole i Sabaris sin, 08028 Barcelona, Spain.

Abstract

Numerical simulation of reactive transport in groundwater (that is, transport of species undergoing chemical reactions) requires the solution of a large number of mathematical equations, which can be highly non linear. This can cause many problems of numerical nature. Therefore, the choice of a method to solve these equations is important. Two types of methods exist: The Direct Substitution Approach (DSA), based on Newton-Raphson, and the Picard or Sequential Iteration Approach (SIA). The advantage of the DSA is that it converges faster and is more robust than the SIA. Its disadvantage is that one has to solve simultaneously a much larger set of equations than for the SIA. We applied both methods to several examples and compared computational behaviour. Results showed that, for chemically difficult (that is, highly non linear) cases, the SIA often requires very small time steps leading to excessive computation times, whereas the DSA does not show this inconvenience due to its robustness. On the other hand, for chemically simple cases but with grids of many nodes, the DSA tends to be less favourable because of the size of the set of equations to be solved.

1. Introduction

The use of numerical models can greatly help the performance assessment of waste disposal facilities, the study of groundwater contamination and the understanding of the groundwater quality in natural systems and the processes undergone by rocks. These models should consider the concentrations of various species and should be able to simulate both solute transport processes, such as advection and dispersion, and chemical reactions, such as complexation, adsorption and precipitation. This requires the solution of a large number of mathematical equations, which can be highly non linear.


288 MAARTEN W SAALTINK et al.

For complex problems this may easily lead to excessive computation times. Therefore, the choice of an approach to solve these mathematical equations is important. Several approaches are available. However, one can consider them to be variants of two.

The first is the Picard that includes the Sequential Iteration Approach (SIA) and the Sequential Non Iteration Approach (SNIA). It consists of separately solving the chemical equations and the transport equations. The difference between the SIA and the SNIA is that the first iterates between these two types of equations, whereas the second does not. The SIA has been used by, among others, Kinzelbach [1991], Yeh and Tripathi [1991], Engesgaard and Kipp [1992], Simunek and Suarez [1994], Zysset et al. [1994], Morrison et al. [1995], Schafer and Therrien [1995] and Stollenwerk [1995]. The SNIA has been used by, among others, Liu and Narasimhan [1989a], McNab and Narasimhan [1994], Walter et al. [1994], Engesgaard and Traberg [1996]. Valocchi and Malmstead [1992], Miller and Rabideau [1993] and Barry et al. [1996] discussed some limitations of the SNIA and proposed solutions.

The second approach is the Newton-Raphson, one-step, global implicit or Direct Substitution Approach (DSA). It consists of substituting the chemical equations into the transport equations and solving them simultaneously, applying Newton-Raphson. It has been used by, among others, Valocchi et al. [1981], Steefel and Lasaga [1994], White [1995] and Grindrod and Takase [1996].

The disadvantage of the DSA is the large set of equations that one has to solve simultaneously, leading to high computational costs per iteration. On the other hand, the SIA and SNIA generally show slower convergence and are less robust and more stiff. This may require finer temporal discretisations than the DSA, leading to a larger number of iterations. Reeves and Kirkner [1988] and Steefel and MacQuarrie [1996] compared the different approaches by applying them to a number of cases of small onedimensional grids (number of nodes ~ 100?). Both reported more numerical problems for the SIA and/or SNIA than for the DSA. The first found generally smaller computation times for the DSA, whereas the latter for the SIA and SNIA. Nevertheless, in both articles the computation times for the different approaches were always of the same order of magnitude. In an article, which has had great impact, Yeh and Tripathi [1989] compared the different approaches for larger grids of one, two and three dimensions. They concluded that the DSA leads to excessive CPU memory and CPU times of realistic two- and three-dimensional cases, due to the very large set of equations that one has to solve for the DSA in these cases. However, they made their comparison on a theoretical basis without in fact applying them and measuring CPU times. Therefore, they could not take into account the fact that the DSA may require fewer iterations. We conjecture that for some cases this may be important and that hence the DSA could become more advisable than stated by Yeh and Tripathi [1989].

The objective of our work is precisely to test this conjecture. To do so, we first formulate several cases of varying chemical complexity, number of nodes and dimensions, second, solve them with the SIA and DSA and, third, compare required temporal discretisation, number of iteration and CPU time.

We start by explaining with more detail on implementation of SIA and DSA. Then, we give a short description of the cases that we used for the comparison. The next section discusses the results of the comparison between the SIA and DSA. Finally, the last section contains some conclusions.

A COMPARISON OF TWO ALTERNATIVES TO SIMULATE 289

2. Numerical Approaches

2.1. Sequential Iteration Approach (SIA)

Saaltink et al. [1998] discussed six mathematical formulations for reactive transport. For the SIA we used their fourth formulation, which is the most suitable one for this approach:

Oha Ohs Ohm t Jt+-at+Jt:::: L(uJ + USkrk(cJ

logc2 ::::Sa(logc1 +logyJ-IogY2 +logka

logcd :::: SAlogc1 + logy I) -logy d + logkd

O=Sm(logc1 +logYI)+logkm

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Equation (1) is the overall reactive transport equation. Equations (2), (3) and (4) express the chemical reactions in equilibrium for respectively aqueous complexation, sorption and precipitation-dissolution. Vectors Ca, Cd and Cm contain concentrations of respectively aqueous, sorbed and precipitated species. We divide vector Ca into two vectors C] and C2 of concentrations of primary and secondary species. We define them in such away that the second can be expressed as an explicit function of the fIrst by means of equation (2). Vector y has the same subscripts as C and refer to the activity coefficients. Matrices Sa, Sd and Sm contain the stoichiometric constants of respectively the equilibrium aqueous complexation, sorption and precipitation-dissolution reactions and Sk of all kinetic chemical reactions. Vectors U a, Ud and Um contain the total concentrations of a chemical component in respectively dissolved, sorbed and mineral form. Vector rk contains the rates of the kinetic reactions, which one can normally express as functions of all concentrations. Equations (6), (7) and (8) relate the concentrations of the species with the total concentrations; Ua, Ud and Urn being called the component matrices. L is a linear operator for the convection, dispersion and prescribed sink-source terms. For the sake of simplicity, we considered only transport in a single, aqueous phase. Then, L is given by:

1 L(c)=- ¢ V·(qc)+V.(DVc)+m (9)

290 MAAR TEN W SAALTINK et al.

where q is the water flux, ¢ the volumetric water content, D the dispersion tensor and m sources and sinks.

The SIA consists of first solving the transport equations (1) with the total aqueous concentrations of every chemical component (vector ua) as unknowns. It treats the concentrations of sorbed species, minerals and kinetic reactions as source-sink term (vector n, computed by the previous iteration:

OUi . . 1 (10) Jta = L(u~) + fl-

where subscript i refers to the iteration number. We used finite elements for spatial and finite differences for the temporal discretization. This leads to linear equations that one can solve for every component separately. We used LV decomposition of a banded matrix to solve these linear equations. As the matrix of this system only changes if the time increment changes, one can take advantage of the decomposed matrix to solve the systems of all components and even of previous time steps as long as the time increment is not changed.

In the second step one updates the source-sink terms. As there is no explicit expression for f as a function of U m one has to calculate first the concentrations of the primary species (Cl) and the minerals in equilibrium (cm) from the total aqueous concentrations (ua) by means of the chemical equations:

u~+u~-I+U~I=Ua( C~i J+UdCd(C~)+UmC~ C2 (CI )

o = Sm(logc~ + logy I) + logkm

(11)

(12)

Note, that we have substituted equations (2) and (3) into (6) and (7). Because these equations are non linear, we applied a Newton-Raphson scheme for its solution (not to be confused with the Newton-Raphson applied to the DSA to solve the whole set of equations). One can do this for every node separately. From Cl and Cm one calculates new source-sink terms:

f i U tXd(C~) tXm st i i = d Jt + Um a + U krk (Cl'C2 (CI» (13)

One repeats this process until some convergence criteria is reached.

2.2 Direct Substitution Approach (DSA)

For the DSA we use the third formulation of Saaltink et al. [1998]. This formulation eliminates the concentration of the minerals in equilibrium by multiplying the reactive transport equation by the kernel matrix of the component matrix of the minerals in equilibrium (Um):

OUa(C li ) OUs(C li ) t E Jt + E Jt = EL(ua(cIJ) + EUSkrk(clJ (14)

where E is the kernel matrix. This formulation substitutes all equations for chemical equilibrium including those for precipitation-dissolution into the transport equations. The unknowns (Cli) are a subset ofthe concentration of primary species (CI):


CI = (C li ] Cld

(15)

Vectors Cli and Cld are chosen in such a way that one can calculate Cld from Cli by means of the equation for equilibrium precipitation-dissolution (4).

As for the SIA, we used finite elements for spatial and finite difference for the temporal discretization to solve equation (14). This leads to a non linear set of equations that one has to solve for all unknowns at all nodes simultaneously. We applied NewtonRaphson for their solution and LU decomposition of a banded matrix for the set of linear equation that one has to solve for every Newton-Raphson iteration.

2.3 Time increment control Both the SIA and DSA can fail to converge if one uses a too large time increment.

On the other hand, CPU times can become unnecessarily large if one uses a too small time increment. As our conjecture is, that the SIA requires smaller time increments than the DSA, it is important to work with the optimal time increments for both approaches. However, it is difficult to estimate a priori an optimal time increment. Therefore, we developed an algorithm that changes automatically the time increment during the simulation, according to the following scheme:

IF (failed to converge) THEN Decrease time step L1T by factor F D

Go back to repeat calculations ELSE

IF (number of iterations < minimum threshold THRM1N) THEN Increase L1T by a factor FI IF (L1T> maximum time increment L1TMAX) THEN

L1T = L1TMAX

ENDIF ELSE IF (number of iterations> maximum threshold THRMAX) THEN

Decrease L1Tby factor FD ELSE

ENDIF ENDIF

Maintain L1T

Do next time step

If convergence cannot not be reached, it repeats the calculations with a smaller time step. In case of a successful convergence, it reduces the time increment for the next time step, if the number of iterations is large, and it increases it, if this number is small.

3. Case Descriptions Table 1 shows a summary of the cases used for the comparison. The first set of

examples (CAL) is the most simple one. It treats the dissolution of calcite in a one dimensional domain. Initially the water is saturated with calcite. Infiltrating water, that is


subsaturated to calcite, dissolves the calcite. This case consists of several subcases: one assuming equilibrium dissolution of calcite (CAL-E) and four with kinetic dissolution, with various calcite dissolution rates (CAL-l to CAL-4, the first having the slowest rate and the last the fastest). We also calculated a case without calcite (CAL-O).

The next set (WAD) contains cases of the flushing of saline water by fresh water in the Waddenzee (the Netherlands) in a one dimensional domain described by Appelo and Postma [1994]. They include dissociation of water, carbonate reactions, cation exchange and dissolution of calcite. Likewise the calcite dissolution cases, we calculated cases of equilibrium, kinetic and no calcite dissolution.

The third set (DEDO) simulates the replacement of dolomite with calcite, which is driven by the infiltration of Ca rich water, called dedolomitization [Ayora et al., 1998]. We used a two dimensional domain, which includes a fracture with a 100 times higher water velocity than in the surrounding rock. Note that a high number of pore volumes are flushed for this case. The number of flushed pore volumes is the volume of water that enters the domain during the simulated time divided by the volume of water in the domain.

The last case (OSA) is chemically the most complex one. It models the deposition of uranium resulting from infiltration of oxygenated, uranium bearing groundwater through a hydrothermally altered phonolitical host rock at the Osamu Utsumi uranium mine, P090S de Caldas, Brazil [Lichtner and Waber, 1992]. As for the DEDO cases, a high number of pore volumes are flushed.

Table 1. Characteristics of the cases, used for the comparison.

Case No. of No. of No. of No. of No. of No. of Flushed name nodes primary compi. adorbed minerals minerals pore

sQecies sQecies (eguiI.) (kin.) volumes CAL-O 21 3 5 1.0 CAL-l 21 3 5 1 1.0 CAL-2 21 3 5 1 1.0 CAL-3 21 3 5 1 1.0 CAL-4 21 3 5 1 1.0 CAL-E 21 3 5 1.0 WAD-O 21 6 3 3 37.5 WAD-l 21 6 3 3 1 37.5 WAD-2 21 6 3 3 1 37.5 WAD-3 21 6 3 3 1 37.5 WAD-4 21 6 3 3 1 37.5 WAD-E 21 6 3 3 1 37.5 DEDO-E 15x15 7 8 2 22704.6 DEDO-K 15x15 7 8 2 22704.6 OSA 101 13 29 8 80000.0


4. Comparison

4.1 Basic grids For the cases described above we measured the number of required time steps, the

number of required iterations and the CPU times (figure 1). Number of time steps and iterations may depend on the parameters that control the time increment and convergence criteria and the CPU time on the programming style. So one should interpret these figures with care. Nevertheless, one can observe some clear differences between the two approaches. For the cases with a small number of flushed pore volumes (CAL and WAD), the SIA and the DSA behave similarly, when the mineral is in equilibrium or absent. The number of time steps and iterations and consequently the CPU time of the SIA rise with higher dissolution rates, when one assumes a kinetic dissolution. On the other hand, dissolution rates do not seem to have much influence on the numerical behaviour of the DSA. A bigger kinetic rate makes the non-linear sourcesink term f in equation (10) to be more important causing more numerical problems for the SIA. The DSA does not show these problems thanks to its robustness.

For the cases with a high number of flushed pore volumes (DEDO, GSA) the SIA requires really excessive number of time steps, iteration and CPU time (centuries) also for cases that assume equilibrium dissolution-precipitation (in fact, we have estimated these figures from runs that took about one day). The high number of flushed pore volumes makes that fulfilling the Courant condition would lead to a very high number of time step. This condition states that the solute cannot go over an element during a simple time step:

I1t ::; ¢/1x q

(16)

where fu and M are the element size and the time increment. It seems that the Courant condition is important for the SIA, whereas it is not for the DSA.

fr 10 ,-----------------,------------------,-------, -(I) 8 .. -~

8 • DSA 1------+---------------

oSIA 6+-~~--~------r_------------~~

--------------f-------------Lf-

9 ::; N :3 3 UJ 0 N M ... UJ ~ .... ~ ~ 6 6 6 6 6 6 « « « « « « « « « « « « 0 u u u u u u ~ ~ ~ ~ ~ ~ Cl

"' Cl

Case

« '" 0


<:IJ 10 = 0 ... .... 8 co

'"' - .DSA r-- r--

~ .... ... 6 c...

0

oSIA r--

'"' ~ 4 .c - r-- r--

5 = 2 = ClI 0

...:l 0

..n

1 r- • • r-I III I If I I 9 ::; :) :; 3 Ul 9 6 ~ ":' "'i 6 ;2 < ..J ..'l Q Q Q Q V)

~ < < < < < < < ~ < ~ < 0 0 u u u u u ~ ~ ~ ~ Q

Ul Q

Case

10

---<:IJ .DSA "e 8 = 0 OSIA c.I

6 ~ <:IJ '-" ;;;;J

4 ~ U ClI 2 0

...:l

0 3 ::; :) :; 3 Ul <;> ;:; 3 2; 6 "! < < ..'l Q Q ..: V)

< < < < < < < < < < < < 0 0 u u u u u u ~ ~ ~ ~ ~ ~ Q

Ul Q

Case

Figure 1. Number of time steps, number of iterations and CPU times for the cases with basic grids. The values for DEDO and OSA cases calculated by the SIA have been derived by extrapolating those for a small

number of time steps. Note the log scale.

4.2 Influence of number of nodes

To study the influence of the number of nodes and the dimensions, we carried out comparisons for some of the cases of table 1 with finer grids of more nodes. We used two-dimensional structured quadratic grids. That means, that the nodes are located according to rows and columns with the number of rows equal to the number of columns (see figure 2). We converted the one-dimensional cases of table 1 into two-dimensional cases by using the same initial and boundary condition along the axes of the new dimension.


Grid with smaller number of nodes Grid with larger number of nodes

Figure 2. Examples of two-dimensional structured quadratic grids.

To interpret the results, we used a model that calculates the required CPU time for each given case as a function of the grid characteristics. For the SIA we assumed that the chemical calculations of equations (11) and (12), the LU decomposition of the matrix and solution of the linear system of equation (10) spend most of the CPU time. Further, we assumed that the number of required time steps and iterations do not depend on the grid. Then, one can calculate the CPU time as follows:

CPU SIA == CPU SIA + CPU SIA + CPU SIA chem dec sol ( 1 7)

CPU SIA kSIA N chem - chem nod (18)

(19)

SIA SIA SIA ( ) 1.5 CPUsol == ksol NbanNnod == ksol N nod (20)

where the subscripts chern, dec and sol refer to respectively the chemical calculations, the decomposition and solution of the linear system; k are constants that only depend on the case; Nnod is the number of nodes and Nban the semi-bandwidth. As we work with quadratic grids, Nban is proportional to ...JNnod'

For the DSA we assumed that the CPU time was mainly spent by chemical calculation (Cld, C2, Ua, Us, and rk as a function of Cli and their derivatives) the construction of the jacobian matrix and the LU decomposition of this matrix:

CPU DSA == CPU DSA + CPU DSA + CPU DSA chem Jac dec (21)

CPU DSA kDSA N chem - chem nod (22)

(23)

CPU~;A == k~;A(NbanY N nod == k~;A(NnodY (24)

where the subscript jac refers to the construction of the jacobian matrix and Ncon is the number of nodes connected to a node, including itself. It equals 7 for two-dimensional structured grids of figure 2.

We obtained the constants k by fitting the measured CPU times of the various parts (chemistry, LU decomposition, etc.) for several grids. Figure 3 shows the measured and calculated CPU times. We could obtain good fits except for the DEDO-E case. One can see, that the DSA grows linearly (slope equals one on a log-log scale) for grids of few


nodes and grows quadraticaly (slope equals two on a log-log scale) for grids of many nodes. The chemical calculations and the construction of the jacobian matrix dominate at the linear part, the LU decomposition at the quadratic part. Also the SIA has a linear (chemistry) and a quadratic part (LU decomposition). However, the quadratic part starts to dominate at a bigger number of nodes than for the DSA. Actually, only for the CAL cases one can notice a quadratic part or domination of the LU decomposition. The reason is that for the SIA LU decomposition has to be done fewer times and for smaller matrices than for the DSA. Moreover, the chemical part of the SIA requires the solution of a non linear set of equations, whereas that of the DSA can be calculated almost straightforward.

I 5 J

I g J

8

6

4

2

o

10

8

6

4

2

o

12

10

8

6

4

2

o

12

10

8

6

4

2

o


DEDO-K DEDO-E

2 3 4 5 log number of nodes

2 3 4 5 log number of nodes

Figure 3. Calculated (continuous lines) and measured (dots) CPU times as function of nodes for cases with

two-dimensional structured quadratic grids. The squares refer to CPU times extrapolated from short

simualtions. Note the log scales.


5. Conclusions and discussion

The results show that indeed the SIA requires generally more iterations than the DSA. The SIA particularly gives problems for two types of cases: cases with high kinetic rates and cases with a high number of flushed pore volumes. The DSA does not show these problems thanks to its robustness. On the other hand, the CPU times of the DSA start to grow quadratically with the number of nodes earlier than the SIA. This means that for two and three-dimensional grids with large number of nodes the DSA tends to be less favourable. Particularly for the chemically "simple" cases (CAL-E and W AD-E) the SIA starts to spend less CPU time then the DSA from already a small number of nodes. For the chemically more difficult cases this number is much higher. As the CPU times of this turning point are for some cases so high, one could argue that for these chemically complex cases the DSA practically always has to be preferred. However, one should be aware of the hardware development. Cases that would take years of CPU time on present computers, may require more reasonable times in the future.

We solved the linear system by means of LU decomposition of a banded matrix. This may not be the most appropriate method for grids with a large number of nodes and with large bandwidths. In that case a better option would be to use an iterative method (such as the conjugate gradient method and GMRES) for sparse matrices, which only grows with an order of about 1.5. That would diminish the disadvantage of the DSA of solving a large set of equations.

Another disadvantage of the DSA is that the computer memory simply may not be big enough to store the matrix. Also here one should be conscious of the hardware development. Computer memories tend to grow.

Acknowledgement

This work was funded by ENRESA (Spanish Nuclear Waste Management Company).

References

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Ayora, C., C. Taberner, M.W. Saaltink and J. Carrera, the origin of dedolomites. A discussion on textures and reactive transport modeling, Journal of Hydrology, 1998, in press.

Barry, D.A., K. Bajracharya and C.T. Miller, Alternative split-operator approach for solving chemical reactiOn/groundwater transport models, Advances in Water Resources, 19(5),261-275, 1996.

Engesgaard, P., and K.L. Kipp, A Geochemical Transport Model for RedoxedControlled Movement of Mineral Fronts in Groundwater Flow Systems: A Case of Nitrate Removal by Oxidation of Pyrite, Water Resour. Res., 28(10),2829-2843, 1992.

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Kinzelbach, W., W. Schafer and I. Herzer, Numerical Modeling of Natural and Enhanced Denitrification Processes in Aquifers, Water Resour. Res., 27(6), 1123-1135, 1991.

Grindrod, P. And H, Takase, Reactive chemical transport within engineered barriers, Journal of Contaminant Hydrology, 21,283-296, 1996.

Liu, C.W., and T.N. Narasimhan, Redox-Controlled Multiple-Species Reactive Chemical Transport, 1. Model Development, Water Resour. Res., 25(5), 869-882, 1989a.

Liu, C.W., and T.N. Narasimhan, Redox-Controlled Multiple-Species Reactive Chemical Transport, 2. Verification and Application, Water Resour. Res., 25(5), 883-910, 1989b.

Lichtner, C.L. and N. Waber, Redox front geochemistry and weathering: theory with application to the Osamu Utsumi uranium min, Poc;os de Caldas, Brazil, Journal of Geochemical exploration, 45,521-564, 1992.

McNab Jr., W.W., T.N. Narasimham, Modeling reactive transport of organic compounds in groundwater using a partial redox disequilibrium approach, Water Resour. Res., 30(9),2619-2635, 1994.

Miller, C.T., and AI. Rabideau, Development of Split-Operator, Petrov-Galerkin Methods to Simulate Transport and Diffusion Problems, Water Resour. Res., 29(7), 2227-2240, 1993.

Morrison, S.T., V.S. Tripathi and RR Spangler, Coupled reaction/transport modeling of a chemical barrier for controlling uranium(VI) contamination in groundwater, Journal of Contaminant Hydrology, 17,347-363,1995.

Saaltink, M.W., C. Ayora and I. Carrera, A mathematical formulation for reactive transport that eliminates mineral concentrations, Water Resour. Res., 34(7), p.1649-1656,1998.

Schafer, W., R Therrien, Simulating transport and removal of xylene during remediation of a sandy aquifer, Journal of Contaminant Hydrology, 19,205-236,1995.

Simunek, J., and D.L. Suarez, Two-dimensional transport model for variability saturated porous media with major ion chemistry, Water Resour. Res., 30(4), 1115-1133,1994.

Steefel, C.L, and AC. Lasaga, A Coupled Model for Transport of Multiple Chemical Species and Kinetic PrecipitationiDissolution Reactions with Application to Reactive Flow in Single Phase Hydrothermal Systems, American Journal of Science, 294,529-592, 1994.

Stollenwerk, K.G., Modeling the effects of variable groundwater chemistry on adsorption of molybdate, Water Resour. Res., 31(2),347-357, 1995.

Valocchi, AJ., R.L. Street, and P.V. Roberts, Transport of Ion-Exchanging Solutes in Groundwater: Chromatographic Theory and Field Simulation, Water Resour. Res., 17(5),1517-1527,1981.

Valocchi, AI. and M. Malmstead, Accuracy of Operator Splitting for AdvectionDispersion-Reaction Problems, Water Resour. Res., 28(5), 1471-1476, 1992.

Walter, A.L., E.O. Frind, D.W. Blowes, C.J. Ptacek, and J.W. Molson, Modeling of multicomponent reactive transport in groundwater, 1. Model development and evaluation, Water Resour. Res., 30(11),3137-3148,1994.

300 MAAR TEN W SAALTINK et aL

White, S.P., Multiphase nonisothermal transport of systems of reacting chemicals, Water Resour. Res., 31(7),1761-1772,1995.

Yeh, G.T., and V.S. Tripathi, A Critical Evaluation of Recent Developments in Hydrogeochemical Transport Models of Reactive Multichemical Components, Water Resour. Res., 25(1),93-108, 1989.

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Part ill

Case studies

MODELING OF ORGANIC LIQUID ENTRAPMENT AND SURFACTANT ENHANCED RECOVERY IN HETEROGENEOUS MEDIA

Abstract

LINDA M. ABRIOLA Department of Civil and Envi'f'Onmental Engineering University of Michigan 1351 Beal, Ann Arbor, MI48109-2125 USA

This paper provides an overview of modeling investigations designed to explore the influence of textural heterogeneities on the migration, persistence, and surfactant enhanced solubilization of organic liquid contaminants in the subsurface. Multiphase flow and transport simulators are used to model the entrapment and surfactant enhanced recovery of perc hi oro ethylene from saturated media. A laboratory-based linear driving force expression is used to represent solubilization. Simulation results illustrate the influence of local permeability variations and mass transfer limitations on remediation performance. Additional model simulations of a controlled laboratory sandbox experiment are used to highlight the strengths and weaknesses of current modeling approaches.

1 Introduction

The failure of traditional pump-and-treat technologies to remediate sites contaminated by nonaqueous phase organic liquids (NAPLs) is now well-documented in the literature [14]. Particularly intractable are sites contaminated by chlorinated solvents, also know as dense NAPLs (DNAPLs). Due to their high densities and low solubilities, DNAPLs often migrate deep into the saturated zone of a formation, posing a long term threat to potable water supplies. Unfortunately, such DNAPL contamination is widespread in the United States. The treatment of DN APL source regions is currently one of the most challenging problems facing environmental engineers in the United States.

Surfactant enhanced aquifer remediation (SEAR) offers a promising alternative to traditional pump-and-treat technologies for the recovery of NAPLs. Surfactants of interest in aquifer remediation are amphiphilic molecules with a hydrophilic "head" and a hydrophobic "tail." At a critical concentration in aqueous solution (the critical micelle concentration (CMC)), surfactant molecules or monomers will assemble to form thermodynamically stable structures known as micelles. The collection of tail groups form the hydrophobic "core" of the micelle, which can incorporate molecules of nonpolar organic compounds within its structure, increasing


304 LINDA M. ABRIOLA

the apparent solubility of the compound. Surfactant monomers will also accumulate at aqueous/NAPL interfaces, reducing the interfacial tension and associated capillary entrapment forces. The removal of organic liquids in SEAR may, thus, take place through two mechanisms: enhanced solubilization and/or entrapped residual mobilization. Although mobilization is a potentially more efficient recovery mechanism, there are substantial concerns that mobilization of DNAPLs could lead to uncontrolled downward migration of contaminants to less accessible regions of the subsurface [4, 17]. In this paper, the selection of surfactants is based upon the desire to minimize mobilization.

SEAR has been shown to be highly efficient at solubilizing nonaqueous phase liquids (DNAPLs) in laboratory soil columns [9, 11, 15, 16]. Results from these studies have provided the basis for the formulation of a conceptual model of micellar solubilization in porous media, leading to the successful development and verification of a simulator for NAPL solubilization in one-dimensional columns [3]. In multi-dimensional systems, however, the efficiency of surfactant enhanced solubilization will be affected by soil texture variations, which will influence both the delivery of surfactant solution and NAPL solubilization and recovery. This paper provides an overview of experimental and numerical studies designed to explore the influence of soil textural variations on SEAR performance.

2 Modeling Micellar Solubilization

The modeled scenario is the remediation of an entrapped DNAPL spill in the saturated zone. Perchloroethylene (peE) is selected as a representative DNAPL. A description of the surfactant flushing process requires determination of the multiphase flow field in conjunction with the transport of the organic and surfactant constituents. Here the entrapped NAPL phase will be treated as immobile, thus assuming sufficient time has passed for the saturation to be reduced to residual levels. Under such conditions, solution of the flow problem requires description of a single mobile (aqueous) phase. The aqueous phase fluid mass balance equation is given by:

a [ ] [kkrw ] at nSwpw = V· Pw---;:;-' (V Pw -'Yw Vz) (1)

where the subscript w denotes the aqueous phase, n is porosity, Sw is the fluid saturation, Pw is the aqueous density, k is the intrinsic permeability tensor, krw is relative permeability, J.lw is the dynamic viscosity, P w is pressure, 'Yw is the specific gravity, and z is the vertical coordinate. Note that a modified Darcy law expression has been used to represent the aqueous velocity, and the influence of solubilization mass exchange on the flow field is incorporated by considering the variation of aqueous phase density with composition.

The transport of aqueous phase constituents, including solubilized organic and surfactant, is described with a component balance equation [IJ:

S aCi aQi vc V S D VC E WO n w at + Pb at + q w . i - . n W hi' i = i (2)

MODELING ORGANIC LIQUID MIGRATION AND RECOVERY 305

where i denotes constituents of the aqueous phase (s=surfactant, o=organic), Ci

is the aqueous phase concentration of constituent i, Pb is the bulk soil density, Qi is the sorbed phase concentration of i, qw is the Darcy velocity, and Dhi is the hydrodynamic dispersion tensor (assumed to be Fickian). Here it is assumed that the solid is water wetting, such that the organic and solid phases do not contact; mass exchange between the solid and organic phases is neglected.

A mass balance equation for the immobile organic phase is also required:

(3)

where Egw represents the organic-aqueous interphase mass exchange of organic. The partitioning of surfactant into the organic phase is neglected (E~W = 0), an appropriate assumption for POE(20) sorbitan monooleate (Witconol 2722), the nonionic surfactant considered in the simulations presented herein.

The above equations represent a system of four coupled nonlinear partial differential equations in four unknowns: Pw , cs, Co, and So. A number of constitutive expressions are required to close this system. Among these are models for flow and transport properties (saturation-capillary pressure, relative permeability, and dispersion). The treatment of these parameters is standard to most multiphase flow models and is not described in detail herein. A complete description may be found in [7].

Sorption of Witconol 2722 to the low organic carbon sands considered herein has been shown to conform to a Langmuir isotherm [15]:

Q _ QmsbCs s - 1 + bCs (4)

where Qms represents a maximum sorption capacity and b is a constant characteristic of the steepness of the sorption isotherm. For the simulations presented here, organic sorption does not greatly influence the predicted DNAPL recovery, and sorption will not be discussed further. A more complete description of organic and surfactant sorption treatment in the model can be found in [7].

3 Quantification of Interphase Mass Exchange

A critical relation required for the closure of the system (1)-(3) is the expression of mass exchange between the organic and aqueous phases. Results of a number of SEAR column experiments suggests that solubilization is often rate limited [3, 11]. Rate limited dissolution of entrapped NAPLs in columns has traditionally been modeled by a linear driving force expression [12, 13, 18]. Although the linear driving force model perhaps over simplifies the solubilization mass transfer process, treating it as a single diffusion step, this expression has been shown to adequately reproduce the behavior of surfactant flushing at the column scale [3]. Using the linear driving force model, the flux of a constituent across the interface can be represented by

(5)


where Geo represents the apparent equilibrium solubility of organic in the surfactant solution and Co is concentration in the bulk phase. Here k f is a mass transfer coefficient (L/t). This quantity is multiplied by the specific interfacial area for mass transfer, ao (I/L), which represents contact area between the two phases. In soil column mass exchange studies, an expression of the type (5) is frequently cast in terms of a lumped mass exchange coefficient, keff = kfao, due to the difficulty of quantifying the interfacial contact area between phases.

The equilibrium concentration of organic, Ceo, is typically observed to be a linear function of the surfactant concentration, Gs . Batch solubilization experiments exploring the solubilization of peE with Witconol 2722 produce the following solubilization relationship:

Ceo = 2.4 X 10-4 + 0.891Gs (6)

in which concentrations of the organic and surfactant are given in gig [17]. As noted above, the linear driving force expression has been used successfully

to model surfactant enhanced NAPL solubilization in column experiments with entrapped dodecane [3]. In that work, based upon laboratory data, the lumped mass transfer coefficient was expressed as a function of the hydrodynamics of the system:

b A

keff = aq + ko (7)

where q is the local Darcy velocity, ko is the mass transfer coefficient under conditions of no flow, and a and b are empirical fitting parameters.

A systematic set of laboratory experiments has recently been undertaken to investigate the influence of soil texture on the parameters a and b in equation (5) [5]. Here a representative NAPL, decane, was entrapped in a series of sandy porous media of varying grain size distribution. A surfactant solution of Witconol 2722 was flushed through the columns and measured effluent concentrations were used to estimate initial lumped mass transfer coefficients for these systems. From these data, a correlation expression was developed for the dimensionless lumped mass transfer coefficient as a function of normalized mean grain size, uniformity index, and Reynolds number. These experiments revealed that surfactant enhanced solubilization rates are more than an order of magnitude lower than those for simple aqueous phase dissolution. They also demonstrated that the dependence of interphase mass transfer on soil textural variations is quite similar to that observed for dissolution. The exponent b was found to be approximately 0.5 for all investigated sands, quite similar to that observed for aqueous dissolution (0.6) [19]. These experimental observations suggest that boundary layer diffusion may not be the sole rate limiting mechanism in the surfactant systems, but that hydrodynamics plays a similar role in both dissolution and solubilization. Note that solubilization rates will also be strongly dependent upon the selected surfactant/organic system. Thus, the magnitude of a in (5) cannot be directly determined from this decane study. Further experiments will be required to explore the dependence of surfactant enhanced solubilization on organic and surfactant molecular structure.

The mass transfer coefficient given by (7) quantifies the initial rate of solubilization. As the entrapped NAPL is solubilized, however, the specific interfacial


area will change, lowering the mass transfer coefficient and reducing interphase mass transfer. To incorporate this time and space dependence of the mass transfer coefficient in the model, the entrapped NAPL is assumed to be distributed as spherical blobs with varying characteristic radii. This approach is similar to that of [18], and has been shown to calibrate well with column dissolution data [20]. Given an initialcharacteristic blob radius, which may be soil texture and/or saturation history dependent, the number of NAPL blobs in a specific element can be calculated from the initial saturation. If the total number of blobs at a point is held constant throughout the simulation, the loss of organic to the aqueous phase at a point results in a diminishing saturation of organic (according to equation 3) at that location and a corresponding decrease in the average blob radius and specific interfacial area. Further details of this algorithm can be found in [7].

4 Numerical Implementation and Example Simulations

The mathematical formulation described above was implemented in a two dimensional (areal or cross-sectional) numerical model entitled MSURF2D [6, 7]. The simulator allows for anisotropy and heterogeneity in permeability and dispersivity values, and permits spatial variation in most other aquifer properties including capillary properties, sorption coefficients, and blob geometry characteristics. This numerical model is an extensive modification of the USGS SUTRA code [26]. The model has been verified through comparisons with analytical solutions and other model simulations and by verification with laboratory data for surfactant enhanced solubilization in soil columns [3]. The model also employs a global mass balance check for the fluid phases and solute components.

As an example of the influence of soil texture on the distribution and recovery of a DNAPL, consider a scenario in which 75 L of PCE is allowed to leak into and redistribute within a 5m by 10m region of an aquifer. A schematic of the cleanup scenario is presented in Figure 1. The domain is saturated, with zero-flow boundary conditions enforced at the top and bottom of the model domain, and hydrostatic pressure conditions at the left and right hand sides. The PCE is introduced at a constant volumetric rate for a period of five days and then permitted to redistribute until immobile. MVALOR [2], a two dimensional immiscible flow simulator, is employed to create this spill event. The resulting residual PCE distribution is then used as an input to MSURF2D. To simulate SEAR performance, a 4% by weight solution of Witconol 2722 is introduced along the left side of the model domain under a 2% hydraulic gradient, typical of pump-and-treat remediation scenarios. The simulated remediation time is sufficient for all of the organic to be removed from the domain and the organic plume to exit the downstream end. All model input parameters are summarized in Table 1.

Figure 1 also depicts the heterogeneous permeability field used in the model simulations. Here light areas represent zones of higher conductivity. This field was generated using the turning bands approach of [25], which reproduces permeability fields with well defined spatial correlation lengths and prescribed mean and variance of the log(K). The permeability distribution shown in Figure 1 represents

308 LINDA M. ABRlOLA

Figure 1: Simulation scenario and permeability distribution.

an aquifer with geostatistical parameters similar to that of the Borden aquifer [27J, but with a moderate increase in the variance of the In(K) distribution (=1.0, K in cm/sec). This variance is similar to that reported by [10J for a Swiss formation (Jussel formation) having a similar mean conductivity and correlation length to the Borden site. Capillary parameters were scaled to permeability variations using a Leverett scaling approach (see [7] for further detail) .

Figure 2(b) illustrates the influence of formation heterogeneity on the distribution of entrapped PCE. For comparison to the heterogeneous case, the predicted entrapment under homogeneous conditions in given in Figure 2(a). Comparison of the figures reveals that the presence of heterogeneity tends to increase lateral spreading of the organic and to decrease spill penetration depth. These effects are due to the tendency of the PCE to pool on lower permeability layers, prior to penetration. An increase in heterogeneity also results in an increase in the maximum entrapped saturation in the domain.

Figure 3 graphically illustrates the surfactant enhanced removal of the entrapped PCE distribution illustrated in Figure 2(b) . Here saturations and aqueous phase concentrations of organic are plotted as grayscale maps, with organic saturation contours at 0.005, 0.025, and 0.05; and concentration contours at 0.01, and 0.005 gig. Comparison of Figures 2(b) and 3(a) reveals that the presence of higher permeability lens-like regions in the upper portion of the formation lead to good PCE recovery in those zones after injection of only 5,000 L. After 10,000 L have been injected, however, isolated regions of organic still remain in zones of relatively low permeability, where PCE was initially pooled. This isolated PCE acts as a significant and persistent source of organic to the aqueous phase. For this scenario, 25,800 L (24.3 days) were required for the complete removal of the


Table 1: Simulation input data.

Parameter Value Units Reference Water:

Density 0.999 g/cm3

Viscosity 0.0114 g/cm s Organic:

Density 1.625 g/cm3 [6] Diffusion coeff. 6.56 x 10-6 cm2/s [6] Solubility 240 mg/L [6]

Surfactant: Density 1.08 g/cm3 [15] Diffusion coeff. 1.48 x 10-6 cm2/s [6]

Solubilization rate parameters: units for q are cm/hr, and for keff are l/hr a 0.0250 [3] b 0.192 [3] ko 0.0109 [3]

Porous Medium: Porosity 0.34 [24] Hyd. Conductivity: cm/sec

variance(ln(K) ) 0.24 [27] In(K)mean -4.62 [27] horiz. corr. length 5.1 m [27] vert. corr. length 0.21 m [27]

Dispersivity (h,v) 61.0,1.0 cm [24] Anisotropy Factor 0.5 [6]

Grid spacing: ~x 0.5 m [6] 6.z 0.1 m [6]

organic liquid. To appreciate the efficiency of SEAR for NAPL recovery, this injection volume and remediation time should be contrasted with that required for NAPL dissolution during a traditional pump-and-treat operation at the same injection rate (1,180,000 Land 1,410 days).

Inspection of the aqueous phase contours in Figure 3 reveals interesting behavior of the solubilized PCE plume. Note that in the areas of highest organic concentration, the trajectory of the plume has a downward component, due to the relatively high density of the plume in contrast with the injected surfactant solution. The solubilized organic plume is observed to migrate downward, out of the low permeability region, and then to the right, as a region of higher flow velocity is encountered.

Figure 4 plots flux-averaged organic concentration of the recovered solution as a function of flushing volume. To explore the full concentration range of interest in


Figure 2: Distribution of entrapped PCE: a. Homogeneous formation, b. Jussel formation.

remediation, a log scale has been employed. For comparison, effluent for the homogeneous case is also plotted. Here the influence of heterogeneity is very marked in the long tailing behavior of the concentration profile. Approximately three times the flushing volume is required to reduce concentrations to regulatory levels in the heterogeneous case. Perhaps surprisingly, for the heterogeneous case an additional 329,000 L (305 days) of surfactant flushing is required to remediate the formation, subsequent to complete recovery of the NAPL. This is contrasted with an additional 81 days for remediation of the equivalent homogeneous formation. Also note the decrease in concentration peak for the heterogeneous case. (Note that this decrease would be more apparent on a normal scale plot.) The decreased peak concentration value is attributed to the PCE entrapped in less accessible regions and to increased dilution effects which result from the smaller penetration depth of the spill. The observed long tailing behavior is likely due to slow, diffusion-limited transport of organic from regions of low permeability. Recall that transport of solubilized PCE into such low permeability zones is facilitated by the density plunging of the plume.


Figure 3: Jussel formation simulation results showing organic saturation and PCE concentration after flushing with: a. 5000 L, b. 10000 L, c. 18000 L.

5 Laboratory Sand Box Experiments

:rhe example simulations presented above illustrate the profound influence soil texture variations may have on D N APL recovery. These simulations, however, were generated with parameters and models derived from centimeter scale, one dimensional, laboratory data. In an effort to further validate the modeling approach and explore the effect of macro-heterogeneties on SEAR performance, a series of bench-scale, two dimensional sand tank experiments was undertaken [23, 21] . A laboratory sand tank (32 x 62 x 1.5 cm) was packed with a medium sand under saturated conditions with intermittent vibration. During emplacement , four horizontally oriented rectangular sand lenses of smaller size fraction and lower permeability were embedded in the medium sand. Figure 5 illustrates the tank configuration, and sand properties are listed in Table 2. Both ends of the sand tank are screened over the entire vertical distance and are connected to adjacent fluid chambers (32 x 1 x 1.5 cm) to facilitate fluid flushing .


M 10.1

E 10.2 --homogeneous formation

~ ......... Jussel formation c: 10.3 0

~ 10-4 1: Q) 10.5 CJ c: 0 () 10'6 .!:! c: 10.7 til ~

10.8 0 0 100000 200000 300000

Volume Flushed (l)

Figure 4: Effect of heterogeneity on removal of the organic plume.

Approximately 38 ml of oil-red-o dyed PCE was injected into the box near the top-center using a syringe pump. The PCE was injected over a period of about 2 hours and was allowed to redistribute overnight. Surfactant enhanced solubilization was initiated by flushing a, 4% Tween 80 (same structure as Witconol 2722) surfactant solution through the box at approximately 4 ml/min. Aqueous effluent was collected over nonuniform collection periods and analyzed for PCE and Tween 80 using gas chromatography. Nearly 8 pore volumes of surfactant solution were flushed through the box. In addition, the presence of nonequilibrium mass transfer was assessed through a series of three flow interruptions, each lasting approximately 15 hours. A complete description of the sand box experiments is given in [21].

Visual observation of PCE infiltration and redistribution in the 2D box revealed that, as anticipated from the earlier 2D modeling work, macro-heterogeneties considerably influenced the PCE migration pathway and final distribution. The PCE drained vertically downward but did not enter the low permeability zones, instead

Table 2: Soil properties of sands used in 2D tank experiments.

Ottawa Sand F -70 Sand porosity 0.33 0.41 mean grain size (mm) 0.71 0.2 permeability (m2 ) 3.9x 10-10 8.19x 10-12

PCE-water retention parameters (Brooks-Corey Function): entry pressure (em H20) 5.6 33.1 pore size index 3.46 3.3 residual water saturation 0.07 0.189 residual PCE saturation 0.12 0.21

Wurtsmith Sand 0.33 0.35

4.2xl0-11

17.7 6.1

0.125 0.15


20-30

nee (em)

Figure 5: Configuration of the laboratory sand tank.

flowing around and pooling over the sand lenses. After cessation of drainage, pools of high peE saturation (::::;20-50%) were evident on top of the lenses and at the bottom of the tank, and regions of lower residual peE saturation (::::;10-20%) were observed along migration pathways above the pools.

Measurements of PCE effluent concentration are plotted in Figure 6(a). These results share qualitative similarities to those from earlier dodecane solubilization experiments in ID soil columns [15]: (1) NAPL effluent concentrations are enhanced by orders of magnitude over the aqueous solubility limit (::::;150 mg/l) , but are substantially below the equilibrium value (27,000 ppm) measured in batch solubilization experiments; and (2) effluent concentrations are sharply elevated following elution of resident fluids during flow interruptions. A definite downward trend in concentration is evident following a peak concentration at one pore volume. This behavior can be attributed to changes in the NAPL/aqueous contact area. In the 2D system an initial large NAPL/aqueous contact area exists as surfactant is flushed through regions ofresidual NAPL (Le. flow paths between NAPL pools). These zones of residual NAPL saturation, however, comprise a small percentage of the total NAPL entrapped volume. As they are gradually removed, the contact area diminishes correspondingly, achieving a low level as flow bypassing occurs around the N APL pools.

Simulation of the sand box solubilization experiment was performed with an approach similar to that described above. The PCE infiltration and redistribution event was simulated first with M-VALOR and the resulting peE distribution was then used as an initial condition for simulation of the surfactant flush using MSURF2D. In both simulations the sand box domain was discretized as 2600 cells, ranging in size from 0.6-1.0 cm. The injection and extraction wells in the sand box system were approximated as a high porosity, high permeability porous medium with a pore volume identical to the experimental system. The simulated injection rate corresponded to measured average discharges within effluent collection periods.

The majority of soil parameters and their distribution were measured or could be reasonably estimated (Table 2), and all pertinent fluid properties were measured independently [21]. Standard parametric models (e.g. see [22]) were employed for


12000

c: 10000 0

~ 8000 C Q) 0 6000 c: 0 0

W 4000 0 a.

2000

0 0

(a) measured effluent concentrations

? (15 hrs. 10 min)

oS I % Vee r;. 0 o£,

o t ~ ~ (14 hrs. 40 min)

~ flow interrupt ~ ~~ (14 hr •• 50 min) -,....,

: " o~ 2345678

pore volumes

(b) predicted effluent concentrations 20000,--------------,

.§ 15000 i§ C ~ 10000 c: o o W o 5000 a.

o

,.. --measured ......... peE correlation

- - - - . dissolution correlation

2345678 pore volumes

Figure 6: Measured and simulated effluent concentrations from a PCE solubilization experiment in the 2D laboratory sand tank.

representation of the retention functions and estimation of the relative permeability relations. The longitudinal and transverse dispersivity coefficients used in transport simulations were evaluated by fitting to tracer test data, obtaining values of 0.1 cm and 0.001 cm, respectively. These values are consistent with values observed in column experiments [15]. Mass transfer relationships for PCE solubilization with Tween 80 were obtained from analysis of data from column experiments, similar to the approach of [3] (Keff = 0.04qO.50 + 0.011).

Measured and simulated PCE effluent concentrations are compared in Figure 6(b). A reasonably good match was obtained between measured and simulated effluent concentrations, suggesting that column-measured mass transfer rates can be used for the prediction of larger scale experiments. The most apparent discrepancies were failure to match the height of the first shoulder and under prediction of the peak height following the second flow interrupt. These discrepancies are thought to be related to the PCE distribution, which is not accurately known but is determined here by numerical simulation. In other simulations (not shown) the predicted effluent concentrations were observed to be sensitive to the initial PCE saturation, which in turn is sensitive to the parametric representation for the retention and relative permeability relations [8, 22].

Figure 6(b) also compares measured and predicted results using a mass transfer correlation developed for simple dissolution (Keff = 1.49qo.61) [19]. Here results are compared only at early time because the dissolution correlation was developed only for flowing systems. As expected, mass transfer coefficients are substantially over predicted by the dissolution correlation, demonstrating the importance of rate-limited mass transfer at this scale.

6 Conclusions

A conceptual and numerical model for the simulation of surfactant enhanced solubilization of NAPLs in saturated systems was presented. Important physical pro-


cesses incorporated in the model include the compositional dependence of aqueous phase fluid density and rate limited interphase mass exchange. Application of the model to a hypothetical field scenario reveals that heterogeneities will playa dominant role in NAPL entrapment and SEAR performance. Moreover, the potential for density plunging of the solubilized organic should also be considered in SEAR design. Additional model simulations of a controlled laboratory sandbox experiment suggest that column-measured mass transfer rates can be used successfully for the prediction of larger scale experiments. Accurate estimation of NAPL recovery, however, appears to be strongly dependent on the knowledge of the initial NAPL entrapment configuration.

Acknowledgments

Funding for this research was provided by the Great Lakes and Mid-Atlantic Center for Hazardous Substance Research under Grant R-819605 from the Office of Research and Development, U.S. Environmental Protection Agency. Partial funding of the research activities of the Center was also provided by the State of Michigan Department of Environmental Quality. The content of this publication does not necessarily represent the views of either agency.

References

[1] Abriola, L.M., Modeling migration of organic chemicals in groundwater systems - A review and assessment, Envir. Health Perspectives, 83,117-143,1989.

[2] Abriola, L.M., K Rathfelder, S. Yadav and M. Maiza, VALOR: A PC Code for Simulating Subsurface Immiscible Contaminant Transport, Electric Power Research Institute (TR-101018), Palo Alto, California, 1992.

[3] Abriola, L.M., T.J. Dekker and KD. Pennell, Surfactant-enhanced solubilization of residual dodecane in soil columns. 2. Mathematical modeling, Environ. Sci. Techno!., 27(12),2341-2351, 1993.

[4] Abriola, L.M., KD. Pennell, G.A. Pope, T.J. Dekker and D.J. Luning-Prak, Impact of surfactant flushing on the solubilization and mobilization of dense nonaqueous-phase liquids. in: Surfactant-Enhanced Aquifer Remediation - Emerging Technologies, ACS Symposium Series 594, 11-23, 1995.

[5] Abriola, L.M., W.E. Condit and M.A. Cowell, Investigation of the influence of soil texture on rate limited micellar solubilization, accepted for publication in J. Environmental Engineering, 1999.

[6] Dekker, T.J., An Assessment of the Effects of Field-Scale Formation Heterogeneity on Surfactant Aquifer Remediation, Ph.D. Dissertation, The University of Michigan, 1996.

[7] Dekker, T.J. and L.M. Abriola, The Influence of Field-Scale Heterogeneity on the Surfactant-Enhanced Remediation of Entrapped Nonaqueous Phase Liquids, accepted for publication in J. of Contaminant Hydrology, 1999.

[8] Demond, A.H., K Rathfelder and L.M. Abriola, Simulation of two-phase flow using estimated and measured transport properties, J. of Contaminant Hydrology, 22(3-4), 223-239, 1996.


[9] Fountain, J.C., A. Klimek, M.G. Beikirch and T.M. Middleton, The use of surfactants for in situ extraction of organic pollutants from a contaminated aquifer, J. Hazardous Materials, 28, 295-311, 199L

[10] Jussel, P., F. Stauffer and T. Dracos, Transport modeling in heterogeneous aquifers: 1. Statistical description and numerical generation of gravel deposits, Water Resources Research, 30(6), 1803-1817, 1994.

[11] Longino, B.L. and B.H. Kueper, Use of upward gradients to arrest downward dense nonaqueous phase liquid (DNAPL) migration in the presence of solubilizing surfactants, Can. Geotech. J., 32, 296-308, 1995.

[12] Mackay, D., W.Y. Shiu, A. Maijanen and S. Feenstra, Dissolution of non-aqueous phase liquids in groundwater. J. of Contaminant Hydrology, 8, 23-42, 199L

[13] Miller, C.T., M.M. Poirer-McNeill and A.S. Mayer, Dissolution of trapped nonaqueous phase liquids: mass transfer characteristics, Water Resources Research, 26(11), 2783-2796, 1990.

[14] National Research Council, Alternatives for Ground Water Cleanup, National Academy Press, Washington, D.C., 1994.

[15] Pennell, KD., L.M. Abriola and W.J. Weber, Surfactant-enhanced solubilization of residual dodecane in soil columns. L Experimental investigation, Environ. Sci. Technol., 27(12), 2232-2340, 1993.

[16] Pennell, KD., M. Jin, L.M. Abriola and G.A. Pope, Surfactant enhanced remediation of soil columns contaminated by residual tetrachloroethylene, J. of Contaminant Hydrol. 16, 35-53, 1994.

[17] Pennell, KD., G.A. Pope, and L.M. Abriola, Influence of viscous and buoyancy forces on the mobilization of residual tetrachloroethylene during surfactant flushing, Environmental Science and Technology, 30(4), 1328-1335, 1996.

[18] Powers, S.E., C.O. Loureiro, L.M. Abriola and W.J. Weber, W.J., Theoretical study of non equilibrium dissolution of nonaqueous phase liquids in subsurface systems, Water Resources Research, 27(4), 463-477, 1991.

[19] Powers, S.E., L.M. Abriola and W.J. Weber, An experimental investigation of nonequilibrium dissolution of nonaqueous phase liquids in subsurface systems, Water Resources Research, 28(10), 2691-2705, 1992.

[20] Powers, S.E., L.M. Abriola, J.S. Dunkin and W.J. Weber, Phenomenological models for transient NAPL-water mass transfer processes, Journal Contam. Hydrol., 16, 1-33, 1994.

[21] Taylor, T.P., KD. Pennell, L.M. Abriola and J.H. Dane, Surfactant enhanced recovery of tetrachloroethylene from a porous medium containing low permeability lenses 1. Experimental observations, in preparation, for submission to J. Contam. Hydrol., 1999.

[22] Rathfelder, K and L.M. Abriola, The influence of capillarity in numerical modeling of organic liquid redistribution in two-phase systems, Advances in Water Resources, 21(2),159-170,1998a.

[23] Rathfelder, KM., T.P. Taylor, L.M. Abriola, and KD. Pennell, Simulation of the surfactant-enhanced solubilization of PCE in bench-scale laboratory studies, in Proc Remediation of Chlorinated and Recalcitrant Compounds, 1(2):91-96, Batelle Press, Columbus, 1998b.


[24] Sudicky, E.A., A natural gradient experiment on solute transport in a sand aquifer: Spatial variability of hydraulic conductivity and its role in the dispersion process, Water Resources Research, 22(13), 2069-2082, 1986.

[25] Tompson, A.F.B., R. Ababou and L.W. Gelhar, Implementation of the threedimensional turning bands random field generator, Water Resources Research, 25(10), 2227-2243, 1989.

[26] Voss, C.l., USGS SUTRA manual, U.S. Geological Survey National Center, Reston, Va., 1984.

[27] Woodbury, A.D. and E.A. Sudicky, The geostatistical characteristics of the Borden aquifer, Water Resources Research, 27(4), 533-546, 1991.

APPLICATION OF THE BACK-TRACKING METHOD TO THE DEFINITION OF SANITARY ZONES OF CATCHWORK PROTECTION FOR DRINKING WATER SUPPLY

MARC BONNET AND FRAN~OIS BERTONE*

Abstract

The implementation of sanitary zones of catchwork protection for drinking water supply has become a requirement stated by the legislator for years. However, this is a complex and technical issue which has often led to inadequate or questionable-and questionedrecommendations or implementations of sanitary zones. Sometimes, their implementations were even delayed.

However, new systems of mathematical groundwater flow modeling allow to determine these sanitary zones, at reasonable cost and in any case objectively, while referring to widely recognized normative criteria. This is particularly true for travel times to the catchwork - from the boundaries of the sanitary zone - that most of European laws laid down at 50 days.

This paper presents a method to calculate time contours around a catchwork by backtracking pathways. This method, known as the back tracking method can easily adapt to all kinds of flow mathematical models. Thereafter, we analyze and discuss examples of results obtained in real situations.

Context

Sanitary zones of catchwork protection, an agreed requirement in France

In France, the Article L.20 of the Public Health Code, enhanced by the Article 13 of the Water Act of 1992 lays down the implementation of sanitary zones of groundwater catchwork protection around for drinking water supply.

The obligation to seek advice from a geologist for a drinking water supply project has been in force since 1900. The notion of sanitary zone of protection for well catchwork was introduced in 1902 and was extended to all kinds of groundwater catchwork in 1924. "A general sanitary zone of catchwork protection to be set up on the whole or part of the capture zone" already had to be considered in the economical requirements of towns (LALLEMAND-BARRES et ai, 1989).

The implementation of sanitary zones of catchwork protection became compulsory in 1935. At this time, there was no particular method set to determine the sanitary zone extent. *HYDROEXPERT Orsay, France


320 M. BONNET and F. BERTONE

In 1962, the Article L.20 of the Public Health Code specified that the official geologist would be responsible for determining the sanitary zone of catchwork protection. The Implementation Decree of 1967 specified that "sanitary zones are established according to the geologic survey and according to the velocity relation between the recharge area and the sampling sites to be protected".

Therefore, French legislation gives full scope to the certified hydrogeologist to determine this sanitary zone according to his basic objective which is "to efficiently protect the catchwork from underground migration of polluting agents". However, such a scope may become harmful to the community. We won't come back over the fact that the implementation of sanitary zones which are questionable for hydrodynamic reasons, and most of the time impossible to implement for financial reasons.

A factual harmonization in Europe

In most European countries, the standard chosen to determine the sanitary zone of catchwork protection is a travel time of at least 50 days. The extent of the sanitary zone of catchwork protection must not allow each water table's element outside this sanitary zone -limited on the map by the time contour at 50 days- to reach the catchwork in less than 50 days.

The 50-day limit mainly allows to protect the catchwork from bacterial germs. This is the time necessary to eliminate most of bacterial germs. As far as chemicals are concerned, this is not a question of protecting completely the catchwork but rather to have enough time to implement enforcement measures.

German legislation states that the protected area (zone II) includes all points from which the travel from the surface to the catchwork is less than 50 days. Therefore, it does not strictly refer to the time contour at 50 days calculated in the water table because the calculation must take travel times into account in the unsaturated zone.

As there is no common standard set in France, in Spain or in the UK, most scientists recommend to determine the sanitary zone of catchwork protection on the time contour at 50 days (COROMINES I. BLANCH, 1997).

The back tracking method

Most of the time, the certified hydrogeologist determines a sanitary zone of catchwork protection from a preliminary hydrogeological survey and a hydrogeological map of low and high waters. He is responsible for calculating water travel time and direction into the water table, without any particular tool.

Application of the back-tracking 321

In straightforward situations -homogeneous infinite environment, unifonn natural flow, continuous pumping- there are analytical fonnulations which allow to calculate the time contour around a catchwork. In more complex situations, that is to say when an analysis of heterogeneities into the water table needs to be carried out -thickness, direction of flow travel, preferential supply, impact of a well nearby-, a hydrodynamic model, completed by a back tracking calculation must be considered.

Mathematical models to calculate groundwater flow have been existing since 1960. In contrast with analytical models, they are perfectly adapted to calculate groundwater flow especially in heterogeneous environment. Most of these tools were of confidential use. Now, they are available to everyone. (BONNET, 1998).

A hydrodynamic model

The first step to implement a mathematical method to calculate time contours around a catchwork is to build a hydrodynamic model.

As financial constraints have to be considered, this is better to do the calculations in steady states. However, considering the fluctuation ofthe flow conditions (especially for the "stream aquifer" or for subsurface alluvial sheets) and therefore, considering the movements of the catchwork's area of influence through the time, the steady flow of high waters and of low waters will have to be considered separately. In that case, two "limit" positions of the area of influence will have to be calculated.

Back tracking method principles

Calculating a time contour around a catchwork remains a complex problem. The most common method is to position in the space, a group of "particles" which take a time t to reach the catchwork : the particles which take 50 days to reach the catch work will line up to fonn the time contour at 50 days. The back tracking method principle consist in positioning a discrete group of "particles" just around the catchwork and consists in calculating, step by step, the position of the particles by backtracking the velocity vectors. It is then essential to know the velocity vectors at any point of the space.

Darcy's velocity can be expressed as follows:

(1) VDarcy = -Khgrad ( H)

The velocity of a water particle, known as the interstitial velocity Cnterslilia/) is, on one hand a function of Darcy's velocity, and on the other hand, a function of the effective porosity, :

322

(2)

M. BONNET and F. BERTONE

VDarcy Vinterslicia/= --

a In the case of a numeric model and whatever the measurement methodology used

(BONNET, 1998), the average hydraulic head, H, is known for each cell. On each side of the cell, thus, it is possible to calculate the vector of the normal velocity, which is a function of the hydraulic conductance and of the heads' difference between one cell and another nearby. As normal velocities are known for each side of the cell, the velocity vector is then calculated at any point by linear interpolation along every axis (RAMAROSY, 1998).

To determinate very precisely the pathway of the particles, their progression along the straight velocity vectors must be finely discretized. There are two methods: the first one consists in determining a maximum time of progression along the velocity vector, the second method consists in determining a maximum distance to be covered (i. e. according to the length of the cell's sides). In both cases, a new velocity vector will be calculated as soon as the particle enters a new cell.

An exclusively convective calculation

The method presented above only proposes a convective calculation. As far as the transport of solutes in the groundwater are concerned, three different phenomena have to be considered:

Dispersion. This phenomenon that preserves the mass, tends to spread the "migration front" of an element. It depends on the nature of the considered product but also on the aquifer material. It has an insignificant effect on very transmissive aquifers (when Darcy's velocity is high). It becomes even more significative as the permeability is weakened.

Adsorption. This phenomenon also preserves the mass. Its main action is to delay the progression of the considered element, in a nearly linear relation with the importance of the phenomenon. On this account, its effect is extremely favourable, especially when this mechanism works in the same time with degradation (which is frequent).

Possible degradation, whatever the origin, intrinsic to the product, biological or reactive with the environment. Of course, this phenomenon is linked to the nature of the product. Therefore, each case will have to be treated accordingly. This treatment is not really part of the hydrogeologist's skills.

The fact that the adsorption and degradation terms are not taken into account in the calculation, underlines the radical approach of travel times. Therefore, it takes into account security around catchworks. However, this approach does not allow to apply


the maximum constraints, including for products that are degradable and can be absorbed.

If the case arises, the hydrogeologist can use specialised tools. There are successful calculation models in the s<?lute transport. They feature the convection but also the dispersion, the adsorption and the degradation. However, they are more difficult to implement than models presented above and require important additional investigations. Moreover they must be implemented for each polluting agent whereas an exclusively convective calculation does not depend on the nature of the solute. We can see that it is not realistic to apply it to define sanitary zones of catchwork protection for drinking water supply.

Examples of application

Protecting drillings in an alluvial sheet of former terrace (drinking water supply of Perpignan (66)- Drilling sites of Millas/Saint Feliu)

In this first example, the established model is a management tool of alluvial resource. Thanks to the importance of investigations realised for the resource recognition, many data had been integrated into the model. This model is a two-layer model that corresponds to interactions between the alluvial sheet caught and the "underlying sheet of Pliocene sands" of regional importance.

The hydrodynamic calculation was realised with the MODFLOW code (U.S. Geological Survey), thanks to Winflow, pre and post processor developed by HYDROEXPERT. For the calculation of time contours, the PATH3D code (S.S. PAPADOPOULOS) was used.

Simulations were made in steady flow of low waters. For other hydrodynamic conditions, the directions of flows were kept : only gradients were reduced. The calculation in steady flow of low waters then gives the most critical situation regarding catchwork protection. Three maps were proposed to the hydrogeologist responsible for elaborating sanitary zones of catchwork protection. These maps show calculations about effective porosity at 8, 10 and 12%, homogeneous on the whole modelled field.

Picture 1 gives some cartographic results of the survey (respectively for simulations at 8, 10 and 12% of effective porosity). Particles located at 10, 20, 30, 40 and 50 days from a sampling site form the time contours around this catchwork. The following pictures give the position of time contours for two catchworks that work simultaneously.

The certified hydrogeologist then has some quantitative elements that allow him to assess the risks concerning catchworks' contamination and financial constraints linked to the sanitary zone extent he proposes.

324 M. BONNET and F. BERTONE

Protecting a drilling catching a "stream aquifer" (drinking water supply of Canet d'Herault (34))

The survey realised did not consist of a mathematical modelling strictly representative of groundwater flows of the Herault "stream aquifer", since there were little piezometric information available. This survey aimed at evaluating time contours around a new catchwork from an assessment of runoffs in steady flow thanks to a simplified groundwater flow model.

The calculation code used is the TALISMAN code developed by HYDROEXPERT in collaboration with the Laboratory of Numeric Analysis of Orsay University.

The numeric model built is based on a typical conceptual model: "stream aquifer". The boundaries of the modelled zone were adapted to observations about geologic outcrop.

Calculations were made considering various hypothesis: first, depending on the existence or not of the "stream aquifer", laterally, from a former terrace; then according to the effective porosity value, that can be evaluated between 5 and 15% for this type of formation.

Extracts presented on picture 2 give some results for an effective porosity at 15%, for the two hydrodynamic situations tested. They permit the certified hydrogeologist to evaluate the possible movement of the capture zone. In a non-steady state, we can suppose than the two supply modes can become preponderant according to the stream flood velocity or according to the importance of precipitation that refill perched groundwaters of former terraces.

Prospects

The significance of the results obtained thanks to mathematical models depends on the model capacity to show the heterogeneity of the represented aquifer field. Most of the time, the operator will have very few transmissivity measures and some piezometers that will permit him to adjust permeabilities for parts of the area bigger than twenty acres in many cases.

Many doubts remain about the dispatching of aquifer parameters in the space and even the groundwater recharge that can change the groundwater route.

KINZELBACH (1996) proposes to elaborate a stochastic method in order to underline the heterogeneity that exists when refining space discretization.

"For a transmissivity and a groundwater recharge known at the level of a "study field", KINZELBACH proposes to generate a distribution of values applicable to a smaller


space, by a Monte Carlo simulation (conditioned or not). A reverse calculation permits to find an equivalent permeability for each distribution of permeabilities". Effective porosity parameter will be easily treated in the same way.

It will be possible to calculate again the velocity field for each simulated distribution. Thus, it will be possible to link a probability to include or not each point of the space in a period of 50 days.

References

BONNET M. (1998) : La modelisation mathematique des systemes aquiferes : historique, methodologie, situation actuelle et perspectives, Colloque Eau 50, Nancy France, octobre 1998.

COROMINES I. BLANCH J. (1997) : Perimetre de proteccio per ales captacions municipals d'aigua potablede la comarca del Garraf, Estudis i monographies 19, Diputacio de Barcelona.

EYMARD R., HERBIN R., HILORST D., RAMAROSY N. (1998) : Error estimates for a finite volume scheme for an advection-reaction-diffusion equation, a paraitre.

LALLEMAND-BARRES A. et ROUX J - C. (1989) : Guide methodologique d'etablissement des perimetres de protection des captages d' eau souterraine destinee a la consommation humaine, Manuel et Methodes n019, Edition du BRGM.

KINZELBACH W. et al. (1996) : Determination of capture zones of wells by Monte Carlo simulation, Calibration and Reliability in Groundwater modelling (Proceedings of the ModelCAKE 96 held at Golden, Colorado, September 1996) IAHS Publ. no. 237.

RAMAROSY N. (1998) Application de la methode des volumes finis sur des problemes d'environnement et de traitement d'image, These de doctorat, Universite de Paris XI.

EXPERIMENTAL AND NUMERICAL INVESTIGATION OF POROSITY VARIA TIONS IN SALINE MEDIA INDUCED BY TEMPERATURE

GRADIENTS

S. Castagna, S. Olivella, A. Lloret and E.E. Alonso

Geotechnical Engineering and Geo-sciences Department

Universidad Politecnica de Catalunya

Barcelona, Spain

Abstract

Preliminary experimental results and a numerical interpretation of porosity variations in saline media induced by temperature gradients are presented in this paper. The experimental results show a pattern of porosity variation with the same features as obtained theoretically in Olivella et al (1996a). An interpretation of the experimental results using CODE_BRIGHT (Olivella et aI, 1996b) has been carried out. Since it is possible to reproduce the experimental results with the numerical approach it can be concluded that the mechanism for porosity changes induced by temperature gradients proposed in a preceding work effectively takes place.

1. Introduction

The study of porous salt aggregates as a material for backfilling openings in radioactive repositories in salt rocks, has motivated thermo-hydro-mechanical investigations. A quite extensive work has been performed in the field of crushed salt compaction (e.g. Spiers, 1990: Korthaus, 1996) while less effort is dedicated to investigate its hydrothermal behaviour. Although saline media remain apparently dry, small contents of moisture are the cause for several processes.

Olivella et al (1996a) have shown from a theoretical point of view that temperature gradients may induce porosity variations in wet saline media caused by dissolutionprecipitation of salt. Although the moisture contents in field situations may be very low, in radioactive waste schemes temperature gradients are large. For instance in TSS in situ test (Droste et aI, 1996), the temperature on the surface of a simulated canister is near 200°C while in the drift walls falls below 100°C. The above mentioned theoretical work is summarised in section 2.


328 S. CASTAGNA et al.

An+ experimental framework has been designed in order to measure porosity variations in a sample of crushed salt with an initial brine content and subject to a temperature gradient. The testing equipment is described in section 3.

It is foreseen to perform a series of tests in order to see the influence on porosity variations of the different variables, for instance initial brine content or temperature gradient among others. At present, however, only one of the preliminary tests has given results that can be interpreted theoretically. The test and the results obtained are described in section 4.

Finally, in section 5, a calibration of a set of parameters was performed in order to reproduce the experimental results. This calibration should be considered preliminary because only the results from one test are available. It is expected that more results will allow a more objective calibration.

2. Porosity changes induced by temperature gradients.

The main objective is to study the sealing/unsealing phenomena due to precipitation of salt, which is transported through the liquid phase.

Heel in

Vapour flu. (d1fiUBlon)

¢>

- - - - - - - -5011 lIuw - - - - - - - -_ - _ - _ -(diffu,i;mldi,p~.,iQn) Qno (gd,o't'~ign}-_ - _ -_

~ ----------~-----------~--------i -----------------Co ------ - _-_-Liquid rluw (.,c:t.,oeE:lir.nr_ - _-_ -_-_-

~ =-=-=-=-=-=-=-=-~=-=-=-=-=-=-=-= 0-::::i

- 5011- - - - - - - - - - - - -5a~ -~r • 'I t' n - - - .Heat fluw_ - - - d'-=o! I' -ec ':.' 0 '0 _ (conduction) Qnd (oc:t.'I!dl en) _ ~ .)} '9!'

--\[----9--- -.-Q---n--- --- --- ------ --- ---- --- --

Heal out

Figurel. Coupled phenomena induced by a temperature gradient (from Olivella et al. 1996a).

EXPERIMENTAL AND NUMERICAL INVESTIGATION 329

Figure 1 shows a schematic representation of the phenomena involved in a horizontal one-dimensional sample subjected to a temperature gradient. In this problem the medium is assumed rigid to avoid the effect of the deformability on porosity changes.

Solubility of salt in water is a function of several variables, one of the most important ones is temperature. In fact, migration of brine inclusions in the solid phase may be caused by temperature gradients. We believe that temperature gradients may induce important healing phenomena in wet unsaturated saline media.

First, temperature dependence of solubility originates salt concentration gradients. Second, temperature vapour concentration originates gradients of vapour pressure and, consequently, migration of water vapour. When water evaporates from saturated brine, the concentration of solute increases above the equilibrium solubility and salt precipitates. In the same way water condensation induces dissolution of salt. The vapour, transferred from a hot region to a cold one, generates a brine motion. Motion of salt in dissolution may be advective, liquid tends to flow in order to compensate for vapour migration, or nonadvective (diffusion plus dispersion), induced by gradients of salt concentration. (Olivella et al. 1993)

Under unsaturated conditions, it seems that the second mechanism of solute transfer (i.e. induced by vapour migration) will be dominant with respect to the other one (i.e. induced by temperature - concentration differences). Of course, this relative importance of transport mechanisms depends on the transport properties of the medium. But it must be said that the diffusion of vapour in air is much more effective than the diffusion of salt in brine. The dominance of solute migration induced by concentration differences would take place in saturated conditions because vapour migration is not possible in such a case.

A formulation for nonisothermal multiphase flow of brine and gas in deformable saline media has been developed in Olivella et al (1994). More details on the problem described here, including a sensitivity analysis to parameters like permeability and retention curve can be found in Olivella et al (1996a).

3. Testing equipment

A testing equipment has been specifically developed in the UPC for this investigation. The test operation can be divided into three parts: the compaction operation, the actual test operation, and the measurement operations.


Figure 2. A sample during the compaction phase and schematic representation of the testing apparatus.

The compaction operation (Fig. 2) is very important because the initial conditions of the tests should be obtained in this phase. On the other hand, this process should be repetitive because the porosity of the sample at a given time is obtained by a destructive technique. Therefore, it is necessary to repeat the experiment several times in order to obtain different isocrones of porosity and brine content.

Also a basic condition is the sample homogeneity, because, if there was heterogeneity, this heterogeneity would have to be known and there would be implications in the numerical modelling. In order to improve the homogeneity of the sample, the compaction operation is performed using two pistons, one for each side of the sample


(Fig 2). Heterogeneous samples could be used in the future but for the time being homogeneity facilitates interpretation.

The variables, measured before starting the test, are porosity and degree of saturation. These are only the initial conditions for the numerical analysis.

The second operation is the application of the temperature gradient to the sample. Figure 3 shows the scheme of the equipment. There are two iron heads in which the water flows at constant temperature. These heads keep the temperature constant during the test in the two sides of the sample. The sample is a cylinder with a height of 100 mm and with a diameter of 52 mm, and it is isolated with respect to mass transfer. There is no possibility of deformation except for the deformability of the container.

The apparatus can be divided into three parts: the cooling-heating system, the heating system and the sample unit with the two iron heads. The two heating units work in the same way. There is a liquid, normally water or a mixture of water and alcohol, and a thermostat keeps the temperature constant in the deposit. The liquid is pumped through the iron head and it goes back to the deposit.

The two systems, one for each head, are slightly different. One can reach temperatures between -45 DC and 130 DC while the other one from 25 DC to 70°C. So one of the ends the system is not only a heating system but a cooling system.

I' lha.,-t'ooIIIre .,01'" I} Ih~"",,,,

liP" -~IN1' _ N SWI ~8' .".,. fllhIlIw ofIIII ...


Figure3. Apparatus for the application of Temperature Gradient in soil samples and sample prepared to be tested.

4. Preliminary test to assess the experimental procedure.

Several preliminary tests have been performed to calibrate the equipment and to check its performance. A number of problems were solved before the system worked. In principle, the apparatus can be used for carrying out series of tests.

A test that lasted 10 days was carried out and the final density in the sample measured. The initial porosity in the sample was of the order of 33% and the initial degree of saturation of 40%. The external temperature gradient had the extreme values of 50 and 15°C.


2XO~------~--------~------~--------~-------, I I I I I I

10c0 - ~1741~ - -11"r.!i~ - -117a;~ ~ - - - - - - - - ~ - - - - - - - - -:- - - - - - - --I I I

-"'E 1EiX)

~ -~ 1400 - - - - - - - - i - - - - - - - - ..,. - - - - - - - - --, - - - - - - - - -1- - - - - - - - -

I I "iii c Q)

C 1200 - - - - - - - - r - - - - - - - - ,- - - - - - - - - -, - - - - - - - - -I

I

1CXXl -1 -0- ~rrentaI. lirre(d): 10 1- -! ---------j- -OCO+---------+---------r---------~' --------~'--------~

0.10 0.00 0.00 0.04 0.02 0.00 [lstance fran cold side (m)

Figure 4. Measured average density for the 7 sub-samples in which the sample was cut after the temperature gradient had been maintained during 10 days.

One of the main difficulties of the test has been the determination of the final porosity. Several solutions have been considered. The first one considered, was the possibility of using image tomography. Secondly, the injection of resins and after using images to determine the porosity. Finally the classical method to determine porosity, density and water content by destructive test, i.e. by cutting the sample in small parts and determining its weight before and after oven drying.

Due to the presence of different phases in the sample and to practical reasons it was seen that the two first methods were not suitable, at least for the time being. The use of injection of resins do not allow the determination of brine content because liquid phase is removed. Determination of porosity using images is not possible because calibration did not succeeded. So the last method considered, i.e. weighting the sample after cutting it seems the most adequate at present. We are working to improve the methodology.

Although the number of difficulties encountered, the results from this preliminary test seem quite satisfactory. The main reason is that the pattern of porosity measured (porosity was measured in 7 sub-samples) shows similar features as the patterns numerically calculated in Olivella et al (1996a). This motivated a first interpretation of the results although we must recognise that a single test is not enough for drawing general conclusions.


5. Numerical interpretation of the results using CODE_BRIGHT

A preliminary numerical simulation of the test described in the preceding section is included here. This numerical calculation has been performed using CODE_BRIGHT (Olivella et aI, 1996b) which is a finite element program that can be used to solve the type of problem that is described in this paper. First of all, it should be said that a number of constitutive laws are required for the analysis. Only in a few cases the constitutive law and parameters have been investigated independently. Intrinsic permeability as a function of porosity has been measured for samples compacted at different porosities. Also an investigation is currently in progress to determine retention curves for crused salt. However, no information exists on these properties for porosities developed in processes of dissolution and precipitation of salt. Here, as a first step in this investigation, a trial and error procedure has been used to determine a set of parameters that were able to reproduce in a qualitative way the experimental results. A 2-dimensional representation has been used to simulate the test. This is necessary because the capillary pressures are relatively low in this material and therefore the degree of saturation is not constant along a vertical section of the sample.

Table I. Set of parameters used for the numerical simulation.

PARAMETER VALUE COMMENT RETENTION Po=0.005 MPa A=O.4 Obtained from CURVE preliminary lab-

tests INTRINSIC

k = k <1>3 (1-<1>0)2 Reference value

PERMEABILITY <I>~

for initial o (1- <1»2 porosity.

ko=5xlO·15 m2, <1>0=0.33 DISPERSIVITIES uI=O.l <It=0.05 Guessed values

based on sample length.

RELATIVE krl = Sl4 krg = 1- krl Power law. PERMEABILITY THERMAL A = A (1-$) A «PSI A $(I-SI) Geometrical CONDUCTIVITY s I g mean including

Ag=O.Ol, AI=1.0, As=5.734-1.838xlO·2T porosity and +2.86xl0·5r-1.51xlO·8r' all in degree of

W/mK saturation dependence.

SOLID PHASE Ps=2163 kg/m3 cs=874 Jlkg/K Values for NaCI PROPERTIES

VAPOUR ((273 + T)23) Tortuosity has

DIFFUSIVITY D:ap = T5.9 X 10-6 Pg

been used with a large value to

m2/s accelerate the

where p. is the gas pressure in Pa. process.


't=15 SALT

-4 (-24530) Arrehnius law. DIFFUSIVITY Dm = 1.0 x 10 exp R(273 + r)

m2/s INITIAL 0.33 From test. POROSITY INITIAL DEGREE 0.43 From test. OF SATURATION

The results obtained from the finite element calculation in terms of density are shown in Figure 5. It can be seen that the pattern obtained experimentally is also reproduced by the calculations. Essentially, precipitation of salt takes place near the hot side while dissolution of salt occurs near the cold side. Among the number of parameters that have been used for the calculation (Table 1) the most surprising value is the vapour diffusion coefficient that has been increased by a factor of 15. This is necessary in order to accommodate the time of the calculated profile with the measured one.

The density variation calculated near the hot side is higher than the measured one. On the other hand, there is a relevant density reduction near the cold side according to the experiment while the numerical calculation also shows this tendency. However, in this case, an important variation is concentrated in the last element. The fact that these variations occur in a concentrated way (in the case of the cold side) should be investigated in the future. This may be revealing that the process is not well represented near the cold side by the numerical approach. A first explanation is that near the cold side an amount of vapour tends to condensate on the iron head and in this case it would not dissolve salt.

336 S. CASTAGNA et aI.

c

2000r---------r-------~--------_r--------~------_,

1800

1600

1400

1200

1000

- -1- - - - - - - - - - t" - - - - - - - - - -; - - - - - - - - - -

1

_________ ~ __________ 1 __________ .L ______ _

1

I

1 1 I _________ J __________ 1 __________ L _________ -.!

I I I I

-+-Initial value ...... Calculated: Time(d): 10 -a- Experimental: Time(d): 10

I 1 I --,----------,---

800+---------~--------~--------~------~~------~

0.10 0.08 0.06 0.04 0.02 0.00 Distance from cold side (m)

Figure 5. Measured and calculated densities in the sample.

1.00 __ ----.,....----.,....----.,.....----.,.....---__.

0.90 -+-Initial value ----~----------r---------

0.80 ...... Calculated: Time(d): 10 ----1----------~---------

:8 0.70 I

---------4----------~---------1----------~---------I 1 1 1

f! ::J

1t 0.60 ---------4--- _______ ~----------~--4-~--~-~--~----~---1 , ___

I

'5 0.50 _________ ~ ___________________ ~ __________ L ________ _

1

e 0.40 _________ 1______ __L _________ J __________ L ________ _

1 I 1 tJ) I

~ 0.30 _________ L _________ J __________ L ________ _

I I I

0.20 I I I --- ------f---------l----------~---------

I

0.10 I I -------r---------l----------~---------

0.00 -4-1~ __ ....... ~----+-----+------I----____'

0.10 0.08 0.06 0.04 0.02 0.00 Distance from cold side (m)

Figure 6. Calculated degree of saturation in the sample.


6. Conclusions

A dissolution/precipitation of salt is induced by a thermal gradient in crushed salt containing brine. This process is induced by vapour migration from the hot towards the cold side. The patterns obtained numerically and experimentally are rather similar. The variations however are different. Only a single experiment has been presented in this paper but an extensive work will be performed in the laboratory of the UPC. A full calibration of the model is not possible until more experimental results are available, but the preliminary parameters obtained here can be used for predicting the test durations. It should be mentioned that the porosity and degree of saturation are measured once at the end of the test by a destructive technique and, so, it is important to feed back continuously the tests with numerical predictions.

Acknowledgements

This work has been carried out on the framework of a project funded by ENRESA (Empresa Nacional de Residuos, Spain) and the European Community Commission.

References

Droste, 1. H.K. Feddersen, T. Rothfuchs & U. Zimmer, 1996, The TSS Project: Thermal Simulation of Drift

Emplacement, GRS Report.

Korthaus, E., 1996, Consolidation and Deviatoric Deformation Behaviour of Dry Crushed Salt at

teemperatures Up to ISO 0c. Mechanical Behaviour of Salt IV, Trans Teeh Publications, 365-379.

Olivella, S.; A. Gens, 1. Carrera & E. E. Alonso, 1993: Behaviour of Porous Salt Aggregates. Constitutive and

Field Equations for a Coupled Deformation, Brine, Gas and Heat Transport Model. Mechanical

Behaviour of Salt III, Trans Tech Publications, 269-284.

Olivella, S., 1. Carrera, A. Gens & E. E. Alonso, 1994. Non-isothermal Multiphase Flow of Brine and Gas

through Saline media. Transport in Porous Media, IS: 271-293

Olivella, S., 1. Carrera, A. Gens & E. E. Alonso, I 996a, Porosity Variations in Saline Media Caused by Temperature Gradients Coupled to Multiphase Flow and Dissolution/Precipitation, Tran!>port in Porous Media, 25: 1-25.


Olivella, S., A. Gens, J. Carrera, E. E. Alonso, 1996b, Numerical Formulation for a Simulator

(CODE_BRIGHT) for the Coupled Analysis of Saline Media. Engineering Computations, Vol 13, No 7,

87-112

Spiers C.l., P.M.T.M. Schutjens, R.H. Brzesowsky, C.l. Peach, l.L Liezenberg, and H.J. Zwart, 1990,

Experimental determination of constitutive parameters governing creep of rocksalt by pressure solution,

Deformation Mechanisms, Rheology and Tectonics, Geological Society Special Publication no 54;

215:228.

CLOG: A CODE TO ADDRESS THE CLOGGING OF ARTIFICIAL RECHARGE SYSTEMS

Abstract

A. PEREZ-PARICI01, I. BENETI, M.W. SAALTINKI,2, C. AYORA2

& J. CARRERA I 'Technical University of Catalonia (UPC). Dep. of Geotechnical Engng.

& Geosciences. Module D-2. e! lordi Girona 1-3. 08034 Barcelona, Spain. [email protected].

2CSIC. Earth Sciences Institute -Institut laume Almera. e! Luis Sate i Subarfs sin. 08028 Barcelona, Spain

Clogging of groundwater Artificial Recharge systems is a very ubiquitous problem that affects numerous recharge facilities and can have dramatic technological and economic impacts. A quantitative approach to clogging is presented in this paper by describing a new comprehensive numerical model.

The current version of this code, termed CLOG, has been obtained by using two existing codes, one for multiphase flow and the other for reactive transport, and by adding specific clogging subroutines. As a result the model is capable of treating the basic clogging processes: transport of particles, bacterial growth (attached to the medium), chemical reactions (homogeneous and heterogeneous kinetics, with biocatalysed paths), gas flow, and compaction. Therefore, the code is integrating the clementary processes, as determined by numerous experiences.

This paper summarises the numerical structure of CLOG, focusing on the implementation of the clogging-related issues. A real example is enclosed with the aim of remarking its capabilities and, also, discussing which are the main limitations. Finally, a rapid discussion of future trends and modifications is done.

1. Introduction

Clogging consists of the reduction of porosity or infiltration area. This implies that the resistance to liquid flow increases, and, in practical terms, that either the infiltration capacity diminishes for constant pressure conditions or either there is an extra pressure build-up when the infiltration velocity is held constant. These adverse effects should not minimised, because they can be responsible for serious economic losses.

In particular, clogging is one of the main limitations in most Artificial Recharge systems -such as wells and basins for incrementing the natural recharge to groundwater.

339 J.M. Crolet (ed.), Computational Methods for Flow and Transport in Porous Media. 339-351. © 2000 Kluwer Academic Publishers.

340 PEREZ-PARICIO ET AL.

This is why clogging has received a notable attention during the last decades, as there are no quantitative tools that can reliably predict the response under field conditions. A detailed description of the most significant quantitative efforts is not within the scope of this text, but they can be classified as follows (Perez-Paricio, 1998): • Use of laboratory-developed indexes. These are based on the behaviour of water

flowing under standard conditions through soil columns or commercial membranes. They suffer from strong limitations when extrapolated to field scales, even though laboratory-obtained indexes provide a very valuable reference from the point of view of prevention.

• Semi-empirical models. They are generally deduced from correlation between the time evolution of a key magnitude (infiltration velocity, liquid pressure, and piezometric head) as a function of a few basic parameters -which must be adequately identified beforehand. These models are often obtained from field observations, but are completely site-dependent.

• Theoretical models. They perform mass balances across the aquifer medium and try to find suitable relationships for the elementary processes. These models are very important, given the fact that they must supply a consistent conceptual model of actual systems.

From the previous comments it is evident that various satisfactory tools can be found. However, all of them are strongly limited when clogging is caused by more than one mechanism. In other words, the existing methods can achieve excellent results when clogging is only due to a single process. For instance, a recharge water that has been disinfected but has a significant concentration of suspended solids will lead to clogging by a single process (physical clogging in this case), provided that there is no chemical incompatibility between that water and groundwater (Osei-Bonsu, 1996). Alternatively, there are examples in the literature where bacterial growth alone accounts for all the clogging problems (Perez-Paricio, 1998).

The fact that clogging is a complex combination of physical, biological and chemical processes, forces us to adopt a different approach. The previous discussion explains why we have devoted a considerable work to the implementation of a comprehensive numerical model that includes the basic mechanisms:

I. Attachment and detachment of suspended particles carried by the recharge water 2. Growth and die-off of bacterial populations 3. Precipitation and dissolution of minerals 4. Generation of gas, air entrapment; 5. Compaction of the clogging layer due to an excess of hydraulic load. This paper is structured as follows: first, there is a section devoted to the

mathematical formulation, where special attention is paid to the clogging terms and equations. Second, we explain the outstanding aspects of the numerical implementation, focusing again on the clogging-related changes. Third, an example with laboratory data of clogging by suspended particles is shown to discuss some relevant topics. Finally, we summarise the basic achievements of CLOG and which should be the future perspectives and trends.

CLOG: A CODE TO ADDRESS CLOGGING 341

2. Mathematical Formulation

The mathematical formulation of this comprehensive clogging model has already been described (Perez-Paricio and Carrera, 1998). Therefore, it is not the interest of this paper to rewrite all the model equations. Instead, we give a simple qualitative description of the main governing equations, namely multiphase flow and reactive transport, and restrict here to the clogging formulae.

2.l. FLOW OF LIQUID AND GAS

As stated above, a multiphase flow scheme has been adopted. Three phases are considered: liquid, gas and solid. The former is composed by water, air and liquid species (suspended particles, bacteria and solutes). The gas is a binary mixture of water and air. The solid is formed by the matrix and attached bacteria.

A mass balance is made for both the water and the air species. So, water in the liquid and water in the gas and water in the solid (biomass) must be considered for the water balance. For each species, we consider the advective and diffusive fluxes, plus the storage variations, plus the variations caused by the solid matrix deformations.

2.2. TRANSPORT

Once that the distributions of liquid and gaseous pressures are known, the respective flux velocities are calculated by means of Darcy's law. It is then possible to perform a mass balance for each species carried by the liquid. This means that one equation is written for the suspended particles, another one for each bacterial species and another balance for each dissolved species (or solutes). The number of bacterial and solutes is completely adjustable, as it only depends on the conceptual understanding of the actual aquifer -and the numerical limitations.

Equation (1) expresses a mass balance for the species i in the liquid (subscript I), with i=p,b,s referring to particles, bacteria or solutes, respectively:

where w is the mass fraction of i species in phase I (dimensionless); p is the phase density (::ML-3); ¢ is the aquifer porosity (dimensionless); t is time (::T); D is the diffusion coefficient (: :MT1);j is the water recharge (::ML-2T 1) at a fixed mass fraction given by the subscript i/o; and r (::ML-2T 1) is the sink/source term that governs the kinetic evolution of a certain species.

2.3. KINETIC SINK/SOURCE TERM

The sink/source term is given by r in Equation (1). Consequently, we just need to provide an adequate mathematical relationship between the different inter-dependent species. Basically, the problem consists in finding a' suitable expression that represents


the target phenomenon. Perez-Paricio and Carrera (1998) deepen in the precise form of these terms; we will refer to their generic form hereafter.

2.3.1. Kinetic term for solutes Equation (2) below is a generalisation of the typical formulae for kinetic precipitation/dissolution of minerals (Saaltink et al., 1997):

(2)

where A(T) is a exponentially decreasing function of temperature, T; j is an index accounting for the number of reaction paths; i is an index running over the whole set of aqueous species; k mj is a reaction rate coefficient for the m mineral related to the j catalytic path; ai is the activity of the i-th solute; Pmji is the exponent for a given mineral m, a given catalytic path j and a given solute i; and f( Q) is a function of the saturation ratio Q for a mineral, which in turn depends on the activities of the solutes.

2.3.2. Kinetic term for particles and bacteria The following general equation is applicable to both particles and bacteria:

ay -=f(x)x-g(x)y at

(3)

where y denotes the concentration of attached particles or bacteria; f(x) is the retention term that is a function of the concentration x of mobile particles or solutes, respectively; g(x) is the decolmatage term, which also depends on the concentration of flowing particles or solutes. Functions f(x) and g(x) are conceptually analogous to the sorption and desorption functions.

Equation (3) is a generalisation of Equation (2). Therefore, it is a means of writing all the kinetic relationships for the so-called generic minerals, which include true minerals, attached bacteria and deposited particles.

2.4. POROSITY CHANGES

As a result of all the physical, biological and chemical processes porosity is reduced, i.e. the clogging evidence. A simple linear relationship has been adopted in CLOG to quantify the extent of those modifications, so that porosity varies proportionally to the change of concentration of retained particles, p; attached bacteria, b; and minerals, m. Equation (4) shows this relationship:

a <I> = - I lBp a p + Bb a b + Bm a m J p.b.m

(4)


where we make a sum over all the concentrations of non-mobile (added to the matrix) species. Some proportionality factors, denoted here by B, indicate the influence of variables such as density and volumetric fractions.

2.5. INTRINSIC PERMEABILITY CHANGE

The final consequence of porosity reductions is that a diminution in the intrinsic permeability of the aquifer takes place; this is the definite observation in Artificial Recharge systems. There exist numerous relationships linking both porosity and intrinsic permeability, all of which are based on semi-empirical evidences. As a first approach we have opted for a Kozeny-Carman equation:

(5)

where ko is the initial intrinsic permeability tensor (::L2); k is the current tensor; ¢Jo is the initial porosity value; and ¢J is the current porosity value.

1. Input

RETRASO

Figure 1. Schematic diagram of CLOG. (l=changes at the input level; 2=integration between the two existing codes; 3=c1ogging specific subroutines).

3. Numerical Implementation

The object of this paper is to explain how the numerical implementation of CLOG has been done. As commented above, the task has consisted of integrating two existing codes, CODE_BRIGHT (Olivella and Garda-Molina, 1996) and RETRASO (Saaltink


et al., 1997), and adding the specific clogging parts. The general set-up is shown In

Figure 1, while the next sections are devoted to each of the main groups of routines.

3.1. MULTIPHASE FLOW: CODE_BRIGHT

CODE_BRIGHT, acronym of COupled DEformation with BRIne, Gas and Heat Transport, was produced at the Technical University of Catalonia with the aim of dealing with saline environments. This is a 3-D finite elements code for unsaturated flow of both liquid and gas phases. Table 1 summarises the basic balances and constituents.

Table 1. Description of CODE_BRIGHT. (From Olivella et al., 1994)

The Problem

Phase / Species

Liquid Gas Solid

Equation

Water

X X X

Balance

Mass of water Mass of air

Mass of halite II III IV V VI

Mass of water in inclusions Momentum

Internal energy

Constitutive equations

Darcy's law Fick's law Inclusion migration law Fourier's law Retention curve Mechanic constitutive model Phase density Liquid viscosity Gases law

Equilibrium restrictions

Solubility Henry's law Psychrometric law

Air

X X

Halite (salt)

X

X

State variables

Liquid pressure Gas pressure

Porosity Mass fraction of water in solid

Solid velocity Temperature

Adevective flux of liquid and gas Non-advective flux of vapour and salt

Noon-advective flux of inclusions Conductive heat flux

Liquid degree of saturation Stress tensor

Liquid density Liquid viscosity

Gas density

Mass fraction of dissolved salt Mass fraction of dissolved air

Mass fraction of vapour

As for the numerical solution of the system of equations, CODE_BRIGHT adopts a Newton-Raphson scheme to update the state variables (Table 1) at each node per each time step.

After the spatial and temporal discretization it is possible to write:


d(Xk+l) d(Xk) reX k+l ) = ~ + A (X k+E) X k+9 + b (X k+E ,X k+9) = 0 (6)

ilt

where r is the vector of residuals at time k+ 1 that depends on the vector of state variables, X, both at times l (previous step) and l+l; d are the storage terms, A the conductance terms, and b the sink/source terms plus boundary conditions. Two intermediate points, l+e and l+fJ, are taken between the initial and the final times for a given time step in order to evaluate the conductance and boundary terms and the vector of unknowns.

Finally, the Newton-Raphson scheme for this non-linear set of equations yields:

ar(X k+1) _--":""--:--:-;"'(Xk+1,1+1 _Xk+l,l) = _r(Xk+1,1)

a X k +1

where I indicates iteration during the calculation of the residuals at time tk+l.

Table 2. Summary of chemical reactions as considered by RETRASO. (E=equilibrium; K=kinetics)

Reaction type

Homogeneous

Heterogeneous

Others

Type

-Aqueous complexation -Acid-base -Redox

-Precipitation/dissolution -Gas dissolution

-Adsorption a) Electrostatic models

Comments / Models

-surface electric potential -capacitance models -triple layer model -diffuse layer model -constant capacitance

Approach

E E E

E,K E

E

b) Non-electrostatic -linear isotherm E models -non-linear (Freundlich)

-Langmuir model -Cation exchange -Gaines-Thomas E

convention -Gapon convention

3.2. REACTIVE TRANSPORT: RETRASO

(7)

RETRASO, acronym of REactive TRAnsport of SOlutes, was also generated at the Technical University of Catalonia (Saaltink et aI., 1997) in order to cope with equilibrium and kinetic reactions for non-conservative dissolved species. This was originally a 2-D finite element code for saturated conditions, as represented in Table 2. RETRASO is based on the direct substitution approach (DSA) to solve the coupled transport processes and chemical reactions. The chemical equations are substituted as a


sink/source term in the transport equations that are solved at the same time. The solution of these non-linear partial differential equations is carried out by a Newton-Raphson iterative procedure, so that the chemical source term is updated after convergence. DSA needs to solve for Ncomp x Nnode equations, that is, the number of components times the number of nodes. However, a new mathematical formulation (Saaltink et aI., 1998) allows for a reduction in the number of components leading to a system where only the thermodynamic degrees of freedom must be directly computed. The rest of species depend on the concentrations of the main components, termed reduced primary species.

The final system is (Newton-Raphson):

JI'(UI+l-ul)=-rl (8)

where u is a vector containing the reduced primary concentrations at each node (consecutively). Jij is the Jacobian matrix relating nodes i and j, such that

_ dfi J .. -1J de.

J

(9)

and where f is given by the spatial and temporal discretization of the advection-diffusion equation with reactive sink/source term. Saaltink et al. (1997) should be consulted for further information.

There exist two convergence criteria, one based on the maximum relative difference between two consecutive iterations (l and 1+ 1), and the other one based on the maximum value of the residual:

U I+1 _ u 1

max 1----:--1 ::; E I U I+1

maxi rl::; E2

3.3. CLOG AND RCB (OR THE JOINT MODEL)

(10)

RCB (Benet et al., 1998) is the result of integrating both CODE_BRIGHT and RETRASO. In this context, CLOG is an extension of RCB, where the clogging-related changes have been included. This means that CLOG is a research code that is being applied to synthetic and real data and after a thorough validation will be definitely implemented in the final version of RCB.

In this section we present a brief summary of the modifications required by CLOG in order to incorporate the required calculations. As depicted in Figure 1, three main tasks had to be done: 1. Adaptation of the input files and subroutines to accommodate the new options and

variables. 2. Link between CODE_BRIGHT and RETRASO.


3. Clogging subroutines: addition of the kinetic terms for particles and porosity and intrinsic permeability update based on the changes in all the concentrations of speCIes.

We will concentrate here on the third task, as it condenses the essence of CLOG from the point of view of clogging. Next sub-sections compile a brief summary of some numerical aspects of the clogging subroutines, based on the previous theoretical presentation. Before doing this, we enclose a rough sequence of CLOG's performance for time step k+ I (Figure 2):

FLOW EQUATION WATER & AIR

PARTICLES

BACTERIA

ay - = f (x) x - g(x) y at

k CHANGE

REACTIVE TRANSPORT EQUATION

MINERAL PRECIPITATION

ay - = f (x) x - g(x) y at

POROSITY CHANGE

Fij;ure 2. Schematic diagram of CLOG calculations.

• The chemistry is initialised through the composition of groundwater, recharge water and the boundary waters.

• A value for porosity at time step k is used for the calculations at time step k+ 1. This means that a time lag is adopted, because porosity is only updated at the end of the reactive transport calculations -when the new concentrations of retained particles, attached bacteria and precipitated minerals are known.

• Liquid pressure (and the gaseous pressure) is calculated by the CODE_BRIGHT module. Darcy's law yields the flux velocity field that is subsequently used by the


RETRASO module. Temperature and matrix displacements can be obtained too by CODE_BRIGHT.

• Based on some chemical and mathematical constraints, RETRASO selects a base of primary reduced species at each node that are the true unknowns of the system. The rest of species (non-reduced primary species, secondary species, gaseous species, minerals under equilibrium reactions, adsorbed species, retained particles and attached bacteria) can be written as some functions of the primary reduced ones.

• The rest of species are calculated for each Newton-Raphson iteration. Particles, bacteria and minerals are linked to the primary species through Equations (2) & (3).

• If there is convergence, then all the concentrations are updated. Porosity is accordingly adjusted by means of Equation (3), and the intrinsic permeability is also corrected (with Equation (4)). Finally, a new time step (k+2) begins.

3.3. I. The kinetic term First, Equation (3) is a linear first-order partial derivative equation, where the time dependency of y is approximated by a finite difference approach. The problem, though, consists of the presence of y in the right hand side of the equation. This forces us to adopt a certain scheme to remove the dependency at time l+J. There are several alternatives, depending on the value of O. So, if 0=0 we are assuming an explicit scheme for y; on the other hand, 0=1 would imply a fully implicit scheme. However, it has been considered convenient to generalise the formulation and a Crank-Nicholson approach for l has been selected:

(II)

With this choice, it is straightforward to come to the final discretised expression for Equation (3) where the concentration of mobile species, x, is evaluated at time l+J because they are the only unknowns of the system:

r(x)=y -y = [f(x)x-g(x)l] k+l k { 1 }

~t 1 + g(x)8~t (12)

All the Jacobian components or derivatives have been directly obtained from Equation (12), by deriving that expression with regards to the concentrations of the reduced primary species (Saaltink et aI., 1998). After rearranging all the terms, we come to the following expression:

a rex) \ = =

ax J

= f(x)~-l + I Xi + I -yr -8 ~t r(x)i {I }[ ax. af(x) ag(x). ( )]

I + g( x) 8 ~t a x j a x j a x j

(13)


The derivatives of f(x) and g(x) are quite complicated from the point of view of calculus, but they do not present serious conceptual inconveniences.

3.3.2. Porosity update Porosity is updated, at the end of a given time step, in accordance with Equation (4). The resulting discretised formulae is:

3.3.3. Intrinsic permeability update Prior to CODE_BRIGHT calculations, the intrinsic permeability tensor is adjusted with the use of Equation (5).

(IS)

4. Application to laboratory data

The present section offers an approximation to some of the CLOG capabilities for a 2-D vertical medium, with axisymmetric flow. Water is recharged at a borehole located at the origin of the medium, whilst there is a prescribed piezometric head at the outer boundary. Figure 3 shows the basic conditions and parameters.

Data were obtained by Osei-Bonsu (1996), who performed several laboratory experiments by means of a 90°-sector reproducing field condition. Potable water was injected at a constant rate, and fixed concentrations of clay particles were conveniently added in order to compare among different circumstances. The objective of the runs was to draw conclusions about clogging evolution based on the existence or not of a gravelpack, the influence of the concentration of suspended solids, the impact of redevelopment and the mechanisms of filtration.

11.52 nn bUl-UP i m'/h -.

II t 1.3 m

1 T = 10 m2/d . 0.5 m

t < ~

1.3 m well radius: 0.05 m

FiMure 3. Geometry and basic parameters of the example. A front and a plan view are plotted in the left and right pictures, respectively.


Only one of these simulations is described here. It consists of the case without gravel-pack, that is, where clogging occurred at the borehole face. Injection rate was 11.52 m3/h, suspended solids concentration was 5 mg/l, while the initial transmissivity of the aquifer medium was 10 m2/d. Given the high flowrate and small radial extension of the model sector, steady state was very rapidly achieved (the addition of particles was always done after establishing a steady state flow).

EXPERIMENT A-4 (5 mg/l)

1.20 r-----~-----~----~-----~----__,

1.00

E Q 0.80 ..-r: ~ :r: u Q;: 0.60 E-~ ::; o [;;j OAO

s:

0.20

. . - ,-•

-. _.- _.- t • • •

• • • • • --.-..---0 • .-----

0.00 -1--------r------r--------,-------.--------I o 5 10 15 20 25

TIME (h)

Fi/iure 4. Measured (points) and calculated (line) additional piezometric head due to clogging at the injection well face.

This example only concentrates on one of the possible clogging mechanisms (physical clogging), because there is neither chemical incompatibility nor biological growth under the prevailing experimental conditions. Furthermore, the relevant attachment mechanisms are interception and inertia, due to the size and velocity of the suspended particles, that minimises the importance of surface forces, Brownian diffusion and sedimentation.

Figure 4 depicts both the measured and calculated temporal evolutions of piezometric head (caused by clogging, as the starting point was already in steady state) at the well radius. Apart from the excellent agreement between CLOG and the empirical data, it is important to note that just two parameters had to be adjusted in order to obtain these results, so that it was not necessary to make additional assumptions. However, the validity and sensibility of CLOG are currently being assessed and more applications will be needed before coming to definitive conclusions.


5. Conclusions

A thorough previous research has conducted to the implementation of a numerical model, whose current working version is known as CLOG, that is capable of satisfactorily providing an integrated approach to the clogging of Artificial Recharge systems. The most important result of this is that a new quantitative tool is available dealing with all the basic processes. It has been demonstrated by previous attempts that an integrated approach is essential when dealing with inter-related complex phenomena.

This paper contains a summary about the most interesting aspects of the numerical implementation of CLOG, apart from giving a taste of the sustaining conceptual model. It focuses on the specific clogging calculations, as other aspects (such as the coupling of two existing models and the detailed mathematical formulations of these) can be consulted in different publications.

Finally, an example with laboratory data on physical clogging serves to understand some of the possibilities of CLOG. Because clogging can result from a combination of completely different processes, there are still many uncertainties and predictive limitations. Needless to say, further research is needed at the laboratory and field scales, but specially, at the modelling level. On-going work with synthetic, laboratory and field data will determine the validity of CLOG and, still more important, its real predictive value. This, as well as improving our understanding of clogging, would be very useful to improve the current design guidelines of Artificial Recharge plants.

6. Acknowledgements

Thanks are due to the European Union for funding the 3-year Project Artifj'cial Rechar!:e t!f" Groundwater. The first author wishes to acknowledge the Generalitat de Catalunya (Spain) and the Centre filr Groundwater Studie.~ (Australia), for supporting a 6-month stay at Adelaide (South Australia).

7. References

Benet, I.; C. Ayora, and J. Carrera. (1998). RETRASO, a parallel code to model REactive TRAnsport of SOlutes. In these Proceedin!:s.

Olivella, S.; 1. Carrera; A. Gens, and E.E. Alonso. (1994). Nonisothermal multiphase flow of brine and gas through saline media. Transport in Porous Media. 15: 271-293.

Olivella, S. and A. Garcfa-Molina (1996). CODE_BRIGHT: User's Guide. Technical University 01' Catalonia (UPC), Barcelona (Spain).

Osei-Bonsu, K. (1996). Clogging by sediments in injected fluid flowing radially in a confined aquifer. A Thesis presented/ill' the De!:ree 01' Ph.D. at the School of' Earth Sciences, Flinders Universityof' South Australia. Adelaide (South Australia). 253 pp.

Perez-Paricio, A. (1998). Clogging of Artificial Recharge systems: fundamental aspects. A Thesis presented filr the De!:ree t!lM.Sc. (It the Technical University 01' Catalonia. Barcelona (Spain). 170 pp.

Perez-Paricio, A., and 1. Carrera. (1998). A conceptual and numerical model to characterise clogging. Third International Symposium on Artificial Rechar!:e, T1SAR, 21-25 September 1998. Amsterdam (the Netherlands). Edited by J.H. Peters et aI., Balkema. pp 55-60.

Saaltink, M.W.; C. Ayora, and I. Benet. (1997). RETRASO: User's Guide. Technical University of Catalonia. Barcelona (Spain).

Saaltink, M.W.; C. Ayura, and 1. Carrera. (1998). A mathematical formulation for reactive transport that eliminates mineral concentrations. Water Resources Research, v 34, n 7, pp 1649-1656.

GROUNDWATER FLOW MODELLING OF A LANDSLIDE

Abstract

1. Rius, 1. Mora and A. Ledesma Dept. of Geotechnical Engineering & Geosciences. ETSECCPB. UPC. ct. Jordi Girona, 1-3. Campus Nord. 0-2. 08034 Barcelona. Spain.

The development of pore water pressures is an important issue in the behaviour of landslides under rainfall infiltration. Thus, the understanding of groundwater flow is an important achievement. The modelling of these rainfall events has been attempted so as to develop a system to predict large movements in landslides triggered by rainfall.

The aim of this paper is the presentation of the results obtained in the groundwater flow modelling of Vallcebre landslide (Eastern Pyrenees) under both saturated and unsaturated conditions. This is a traslationallandslide with two clay layers sliding over a limestone bed. The movements are concentrated in the deepest clay layer. There are some scarps and extension zones along the landslide. Some high intensity rainfall events cause a fast raising of water levels ~d a reactivation of movements. This behaviour has been verified in the monitoring devices: inclinometers, wire extensometers and piezometers in the landslide zone and a rain gauge in the basin of Vallcebre. A GPS survey device is also monitoring surface movements.

The landslide has been modelled by the finite element method and some rainfall events have been tested to calibrate some flow parameters like permeability or storativity. Also, an inverse analysis has been performed in order to estimate in a systematic manner those parameters. The model has been attempted in both two-dimensional and three-dimensional analyses. The effect of preferential flow paths produced by tension cracks has been included in the model as well. The results of the analyses are consistent with the measurements, and show the importance of preferential flow paths and boundary conditions on the simulation of the landslide behaviour.

1. Introduction

The understanding of groundwater flow in landslides under slow movements reactivated by rainfall events is important to explain their mechanical behaviour because of the development of pore water pressures. The Department of Geotechnical Engineering and Geosciences of the Technical University of Catalonia (OPC) has been

353 J.M. Crolet (ed.;, Computational Methodsfor Flow and Transport in Porous Media, 353-370. © 2000 Kluwer Academic Publishers.

354 1. RIUS et al.

working to achieve this objective with a test site, the landslide of Vallcebre in the Eastern Pyrenees. This work has been done under the framework of the European Commission research project: "NEWTECH".

The objectives of the present work concerning the hydrological modelling can be stated as the following:

- Assessment of a general method to predict raising of water levels in landslides during rainfall events. This requires an understanding of the landslides behaviour. It means knowledge of its geomorphology, mechanical and hydrological properties and availability of data from monitoring survey to calibrate the models. This general method is the first step in the aim of a wider one to predict movements in landslide areas for a given inflow during rainfall events. This approach could be very useful in the future, when alarm systems activated from rainfall data could be available for dangerous landslide areas.

- Development of the present tools to model groundwater flow in the subject of landslides, such as the finite element code used in this case, TRANSIN [1], checking its advantages and drawbacks.

The understanding of the Vallcebre landslide has been attempted by means of a hydrological and mechanical model. In this paper, only the work about hydrological models is presented. Nevertheless, some general aspects concerning geomorphologic and mechanical properties will be presented so as to make the landslide behaviour more comprehensive.

The hydrological modelling has been performed using TRANSIN, a finite element code for solving non-linear flow analysis in both two dimensional (2D) and tbreedimensional (3D) conditions. First, a 2D saturated analysis was used to reproduce the actual water table fluctuations. The importance of the cracks and preferential flow paths has been highlighted and reproduced with the 2D model. Secondly, the analysis was attempted by means of a 2D-unsaturated model with transient flow to simulate some rainfall events. The last step was the analysis with a 3D model to improve the simulation of the cracks and preferential flow paths.

Only as a comprehensive explanation of the whole work of the NEWTECH project, the next stage of this work is summarised. This was a mechanical model based on a limit equilibrium analysis performed using the computer code ST ABL. The results show that safety factor for some parts of the landslide are close to 1, depending on the water table position. Indeed that explains the activity of the landslide that is clearly related to the rainfall intensity. Velocities decrease in dry periods, whereas wet seasons can produce up to 30mm/week of surface displacement.

GROUNDWATER FLOW MODELLING OF A LANDSLIDE 355

2. The test site of Vallcebre

The landslide ofVallcebre [2] is located in the Eastern Pyrenees in the upper Llobregat river basin, 140 Km North of Barcelona, Spain. The valley is shaped on the Garumnian limestones and red clayey siltstones (Upper Cretaceous - Lower Palaeocene age) and affected by two thrust faults and some associated folds. The landslide extends on the eastern slope of the torrents of Vallcebre and Llarg (figure 1). The landslide is a thick translational slide of approximately 1200 m long and 600 m wide and is developed in the red clayey siltstones that form the core of a gentle syncline. This syncline has a WNW-ESE trend and is inclined towards the WNW, more or less parallel to downslide direction.

Three main morphological units have been recognised in the slide. The units have decreasing thickness towards the landslide toe. Each unit includes a relatively flat surface that is bounded by a traverse scarp of a few tens of meters high. This gives a stair-shaped profile to the landslide though the slope is only of about 10° on average. Most of the evidences of the surface deformation appear at the edges of the slide units as distinct shear surfaces and grabens. At the toe of each traverse scarp, the ground surface and the trees are tilted backwards due to the development of a graben. Inside the units, the ground surface is only disturbed by few cracks, small scarps of short length (less than 50 m) and by cracking of the walls of farmhouses standing on the landslide. The direction of both, the traverse scarps and grabens, is suggestive of a movement towards the Northwest. A secondary direction of movement, towards the Torrent Llarg, is also observed in the upper slide unit.

The most active area is the lower unit, bounded on its northeastern side by a shear surface and, at the southwestern side, by the torrents of Vallcebre and Llarg. The landslide toe reaches the torrent bed and overrides the opposite slope. The torrent, in tum, undermines the landslide foot during episodic floods, increasing therefore the instability.

The field and laboratory works performed are the following:

Geophysical investigations carried out in March 1996. Ground penetrating radar was used in order to identify the failure surface in the landslide that was supposed to be at the contact with the limestone bed. Vertical electrical sounding and one electrical continuous profile were also performed. The results of these investigations were not conclusive because of reflections produced by the presence of gypsum lenses and lost of reflectivity by the presence of water. These results were taken as estimation. However, the location of boreholes drilled later was decided based on the interpretation of the geophysical survey.

356

+ G18

G17

G14

GI

J. RIU S et al.

GU1

G13 #

N--8 o 250m

LEGEND n SCARP

UPSLOPE DIPPING

SURFACE

G2 CPS ~ONITORING

+ POINT

52 BOREHOLE +

~ TORRENT

1SJ CRACKED HOuSE

GRABEN

r:] DIRECTION OF

OISPl..AC£I.IENl

Figure 1. Location and geomorphologic sketch of the Landslide ofVallcebre.

- Mechanical investigations carried out between 1996 and 1997. Two series of boreholes were drilled in order to identify the materials involved. The landslide consists of a series of slid materials, from the bottom to the top: a) laminated and densely fissured clayey siltstones 1 to 6 m thick, b) gypsum lenses, up to 5 m thick and some tens meters long, c) clayey siltstones rich in veins and micronodules of gypsum. In addition to these layers, in the extension zones located at the toe of the scarps it may be found colluvium composed by gravel with silty matrix. The graben acts as a trap that may accumulate a thick wedge of colluvium (figure 2).

Laboratory tests carried out on undisturbed samples during boring at different layer materials to determine parameter and soil characteristics to be used in different numerical analysis. Basic identification tests and shear tests have been performed. Also, undisturbed samples have been tested in triaxial and oedometer apparatus and the water retention curve of one of them was also obtained. Tests to obtain shear strength were carried out predominantly on the fissured clayey siltstone unit where the slip surface is located. Oedometer tests were carried out in both fissured clayey siltstone and clayey siltstone with gypsum. Permeability (= hydraulic conductivity)


values of these materials obtained among other parameters for different values of load related with different depths can be seen in table 1 [3].

FAULT

FAULT

4 GRAVEL WITH SILTY W"'TRIX (COLLLMUU) 50 100 m

J CLAY Sil T'STONE WITH SCATTERED MICRONODULES OF GYPSUN

2 LAI.tINATEO AND F1SSUR£O CLAY SILTSfONE

1 lIUESTONE

Figure 2. Profiles assessed from boreholes and prior information (location in figure 1).

Table 1. Permeability values estimated from oedometer tests [4].

Layer Clayey siltstone with gypsum Clayey siltstone with gypsum Clayey siltstone with gypsum Clayey siltstone with gypsum Clayey siltstone with gypsum Clayey siltstone with gypsum Laminated clayey siltstone Laminated clayey siltstone

Depth (m) 10 15 20 25 34 44 4 9

Permeability (m/s) 5.10-7 _ 3.10-9

1.10-7 _ 2.10-9

5.10-7 _ 3.10-8 8.10-9 _ 9.10-11

3.10-8 _ 1.10-10

6.10-8 _ 2.10-10

2.10-7 - l.1O-8

7.10-8 _ 8.10-10

Water retention curve of a sample of clayey siltstone with gypsum taken 4 m depth, above the water level was obtained and it was tested in an oedometric cell. Matrix suction was applied through the axis translation technique. The air entry value of porous stone is 500 KPa. Vertical load of 100 KPa was applied during the test by means of pressurised air. Water change in the specimen is calculated measuring the water volume that crosses the porous stone.

358 J. RIUS et al.

Starting from natural conditions, an initial suction of 350 KPa was imposed. Successive steps of suction decrease was applied until reach a null suction. Figure 3 shows the retention curve obtained, suction (MPa) - degree of saturation (%). Time necessary to obtain stationary conditions in the sample after each suction change was about 10 days. During each step, evolution in time of water volume in the sample was recorded in order to be able to estimate the unsaturated water permeability associated to average suction applied in the step.

- Monitoring devices were also installed in the boreholes cases. Inclinometers were installed all along the landslide to locate the shear surface at different places. It was clearly shown that the shear surface was placed at top of the laminated clayey siltstone layer. Wire extensometers were also located to know the rate of movements at different places of the slide. A GPS survey is systematically performed to monitor the surface displacements. A set of piezometers was installed with datalog devices to monitor in a continuous manner the fluctuations of water table all around the slide.

1000

~ ~

"-.

~ 1\ \

100

ii IL

!. " .2 'll " OJ

10

1

50 60 70 BO 90 100 110

Degre. of salurallon ('HoI

Figure 3. Water retention curve obtained from laboratory test (wetting path).

Piezometric records of borehole S2 are presented in figure 4. Piezometers show a very fast response to the rainfall. This fact suggests that water infiltration is controlled by fissures or macropores rather than by soil porosity. It is also obselVable that there is practically a simultaneous response of the piezometers. Two basic types of responses to rainfall are obselVed depending on the location of the piezometers. The piezometers located in tension zones, as S5, show smaller variations of the groundwater level (ranging between 0.5 to 2 m) and quicker drainage compared to the piezometers placed out of this zone, i.e. S2. The latter experienced changes up to 5 m and a slower rate of lowering of the groundwater level. The behaviour of the piezometer S5 is consistent


with the presence of a very pervious zone. Consequently, it is assumed that cracks act as a preferential flow path within the landslide body.

A clear relationship between raising of water levels and landslide activity is observed with piezometric and wire extensometer records [4]. From those records it seems that the landslide is very sensitive to the rainfall, since the highest rates of displacement take place in coincidence with the rainy periods.

,-..,

120 0 e ~

.? 100 <U

::0 "" $ N 2 .... .= 80 .s ca e 3 ::: g 60 "0

4 c: ::l

;§ 40 5 e bI) c: '+-< '0; 20 6 0

P::: ~ .... 0 7 0..

<U Cl

21-Nov-96 20-Jan-97 21-Mar-97 20-May-97 19-Ju\-97 17-Sep-97

Figure 4. Piezometric records from boreholes (lines) and rainfall from rain gauge (bars).

- Finally, some in-situ tests were carried out to obtain more realistic values of in-situ permeability of the ground where cracks and preferential flow paths are widespread. Those tests were falling head tests (Lefranc tests) performed in the boreholes before the installation of the monitoring devices. Values of the permeability of different layers obtained from falling head tests are greater than those obtained from laboratory tests, that could be explained by the fissures that can be found in the test site [5]. These values can be compared with those obtained in the laboratory tests (table 2).

Table 2. In-situ values of permeability, from falling head tests.

Layer Clayey siltstone Clayey siltstone with gypsum Colluvium Fissured clayey siltstone

Permeability (m/s) 8.8 '10-7

7.9 '10-7

3.4'10-7

2.4 '10-6

360 1. RIUS et a1.

A pumping test is going to be performed to have another independent measurement of permeability.

3. Governing equations

This section contains a brief description of the generic flow equation solved by the finite element code TRANSIN [1]. It adopts a compact generic form of the flow equation:

JC 811& = V(fK Vh) + q on il (1)

where h[LO is piezometric head (h = p 10 + z ; P [FIL2 0 is water pressure, OO [FIL3 o is specific weight and z[LO is vertical position from a reference level. K is hydraulic conductivity tensor, Cis storativity,jis a dimension factor and q[TI ,LIT, L2/T0 is an instantaneous recharge per element size (length for I-D elements, area for 2-D elements and volume for 3-D elements. K, C and J are dependent upon the type and dimension of the problem [1]. In this code it is possible to solve the equation (1) in terms of pressure head (0 = p I D), This state variable was used in the unsaturated conditions in our case and the former variable in the saturated ones.

Initial conditions are specified, when piezometric head is used as state variable, as:

h(x,O) = ho(x} (2)

where x is the position vector, ho is an arbitrary function, a solution of a previous problem in our case. In terms of pressure head, initial conditions are given by:

rp(x,O) = CPo(x) (3)

where Do has an analogous meaning as hrr

Boundary conditions are written as:

(fK V(h))n = a(H-h) +Qo on r (4)

where 0 is the boundary of 0, n is the unit vector normal to 0 and pointing outwards, H is an external head, Q is prescribed flow and 0 is a coefficient controlling the type of boundary condition; that means: 0 = 0 for prescribed flow, 0 = 0 for prescribed head and 0 0 0, 0 for mixed condition, being 0 in this case the leakage coefficient. The left-hand side in (4) represents water flux entering (if positive) or leaving (if negative) the medium.


The numerical model assessed from these governing equations is formulated within the framework of the finite element method. The site was modelled as a continuum with a finite elements grid. Triangular elements with one Gauss integration point were used in the 2D analysis while triangular prisms with six Gauss integration points were used in the 3D analyses.

The boundaries of the landslide were modelled with the following conditions: prescribed flow condition for bedrock outcrop, prescribed head condition for torrent and stream boundaries and mixed condition for torrent boundary over bedrock outcrop.

Initial conditions were assessed with pore water pressures obtained performing a previous analysis with the water levels measured with the piezometers and the levels of the torrents and streams in dry periods.

The method used by the code to calibrate the model is performing an inverse problem defining a set of parameters to be calibrated and an objective function to minimise the differences between computed and measured available data:

n

F= :LIZi -z;1 (5) i=l

where F is the objective function, Zi is computed data, Zi' is measured data, n is the total number of available data. The set of selected parameter values to be calibrated that best fit the objective function is the result of the inverse problem.

4. Hydrological modelling

The main objective of the hydrological modelling has been the simulation of the water table behaviour during rainfall events. Two kinds of work have been undertaken to achieve this objective.

The first one has consisted of checking hydrological conditions starting from information provided by field work that included geometry conditions (i.e. position of the impervious boundary) and soil properties (i.e. permeability values). The second task starts from the groundwater flow model developed in the first task. It has consisted of simulating the water table behaviour during rainfall events. Information of several piezometers installed in the test site was available (i.e. continuous data of depth of water table) and also data of rainfall from a rain gauge at Vallcebre basin. These measurements have been useful to solve the inverse problem as well (i.e. calibration of soil properties from measured data), Finally, the simulation of some rainfall events has been reproduced from all data available and from calibrated parameters.

The hydrological model of Vallcebre landslide has been treated in different ways: two dimensional (2D) models under saturated and unsaturated conditions of a

362 J. RIU S et al.

representative profile and a three-dimensional (3D) model under saturated conditions. All geometrical information has been obtained from boreholes drilled in the test site.

4.1 TWO-DIMENSIONAL SATURATED ANALYSIS

The first case analysed has been a two dimensional model implemented on a finite element mesh that was defined for the cross section passing through boreholes SI to S7 (figure 2). Four materials were considered as representative of the flow problem, each of them having different properties: limestone bedrock, fissured clayey siltstone, clayey siltstone with gypsum and gravel with silty matrix (colluvium). This 2D mesh has been used to perform preliminary analyses and to reproduce actual water table fluctuations in a simple way. The importance of cracks and preferential flow paths has been also highlighted and reproduced using this 2D model. That is, in some examples, outflow values were introduced in particular points in order to represent loss of water through the cracks.

A trial an error procedure to reproduce the water table with respect to the permeability values has been performed. This is because cracks and tension zones can change more than one order of magnitude the permeability value obtained from laboratory tests. As usually, samples for laboratory tests were small and they were obtained from undisturbed zones and, therefore, permeability values obtained from laboratory tests are smaller than the actual ones when a bigger area is considered.

The computations performed in this 2D model indicate that a loss of water is necessary to explain the measured position of the water table. Otherwise water would go out and different sources on the lower part of the landslide should appear. Even for different ranges of permeability values, it has been necessary to set up this outflow in order to reproduce the measured pattern of water levels at the instrumented boreholes. Using the trial and error procedure, the permeability values found to be more realistic, are shown in table 3.

Table 3. Values of permeability after trial an error procedure in 2D model.

Layer Bedrock Fissured clayey siltstone Clayey siltstone with gypsum Gravel + silty matrix (colluvium)

Permeability (m/s) 10-13

10-4

10-5

10-5

The loss of water required by the model to explain the position of the water table is consistent with the cracks found in one of the scarp areas, separating the lower part from the intermediate unit of the landslide, i.e. fault zone (figure 2). This is a tension zone and the openings are expected to be preferential flow paths. A value of q = 10-5

m3/s (per meter of length of the crack) for this outflow has been found by trial and


error, by means of the hydrological 2D model, in order to reproduce the actual water table position. This analysis has been performed assuming saturated conditions and a steady state situation.

4.2. TWO DIMENSIONAL UNSATURATED ANALYSIS

This 2D model is a transient one that considers rain infiltration in order to relate water table fluctuations with rainfall intensity. The response of the lower unstable zone of Vallcebre landslide has been modelled with a surface inflow simulating some rainfall events observed between November 1996 and February 1998. Figure 5 represents the computed pore water pressure values along the lower zone profile in a high rising of water table during the rainfall event between 17/8/97 and 19/8/97. Note that in the lower part there is outflow indicating flood of this part, as can be seen in the test site during some rainfall events.

________ -.-- o. - --

________ -- \u.----·· . ____ ~ 20 - ---

_____ ._ .. ,0 _ 40 - - -

_.----- ;0 - ----- __________ -- 60 --

Figure 5. Computed pore water pressures values (in meters) along the lower zone profile for an unsaturated analysis.

The model under unsaturated conditions needs some more parameters than the saturated one: porosity, storativity coefficient and relative permeability (capillary pressure, minimum and maximum saturation degree). They have been estimated considering the laboratory tests results. Since this analysis is performed for the lower part of the slide, where maximum rate of displacements, take place, only three materials are involved in this unsaturated case (table 4):

364 1. RIDS et al.

Table 4. Values of different parameters in the numerical model for different layers.

Layer Porosity Storativity Capillary Saturation degree (%) pressure min max

(m) Bedrock 20 0.00001 Fissured clayey 20 0.001 5.8 0.65 0.95 siltstone Clayey siltstone with 20 0.001 5.8 0.65 0.95 gypsum

The results of the simulation shows that the behaviour of the water table can be reproduced with an accuracy of the order of 1 meter with the values of the parameters calibrated in the saturated case. The rising and dropping of the water levels at the observation point (i. e. piezometers) can be gently reproduced. Figure 6 shows these measured and computed movements of water table at boreholes S2, S4 and S9 during and after the rainfall event between 17/8/97 and 19/8/97. The inverse problem in this 2D unsaturated case has been useful to confirm the values of the model parameters as well. The objective function has been defined with the saturated permeability of the layers and the surface inflow. In the first step the permeability of the layers was calibrated to confirm the values obtained in the saturated case. In a second step, the surface inflow was calibrated to achieve a quantitative relationship with the rainfall values. This relationship is, a percentage of the rainfall depending on every case simulated, and therefore, an estimation of the infiltration. An average value of the percentage has result in between 75 and 95%.

I

I I -II ..

of\: , • f----

0

:~ . "'? .. . , -1-t'T f::

+-r-.r-' ~ r-- .-

I

+- I

~.t---- -+--+--t

:Jf~-+--·~-+--+-------t ~ ot----+-l-~ :-j----t -~-+--

-1--,,---+---1-+-_+-1_+---+ , ,I I

I .~,----+--+--+

, "

, l':< J.

TlMt(IU)")

COtdPlJT~D PRESSl'RE IN OBSERVATION POINT SI COMPnED PRESSURE IK OBSERVATIO~ POINT .11 roMPnED PRESSURE I~ OBSERVA no~ POINT 54

Figure 6. Measured (dots) and computed (lines) movements of water table at boreholes S2, S4 and S9.


4.3. THREE-DIMENSIONAL SATURATED ANALYSIS

All the analyses described above are not capable to reproduce so good some of the hydrological processes that take place in the Vallcebre landslide. In particular, the water loss throughout the cracks is moving actually out of the plane considered in the 2D analyses. So a three dimensional model has been developed to better understanding of the 3D effects affecting geological properties of the materials involved. These properties range from thickness of the strata to physical properties of materials. A new profile of the site was performed from a second series of boreholes carried out, and some more information in order to make the implementation of the 3D model has been obtained. This second profile passes through boreholes S8 to S14 (figure 2).

The advantage of this 3D model has been the possibility of simulating some of the processes involved in the landslide that were not considered in a 2D analysis, such as the consideration of lateral flow through the extension zones. Now, this extension area has been modelled with a thin set of surface elements with particular properties (i.e. higher permeability, anisotropy). Figure 7 shows the geometry modelled and computed piezometric levels. By means of a sensitivity analysis and using all the information available, the permeability values of the different layers have been estimated once again (table 5).

Table 5. Values of permeability after sensitivity analysis in 3D model.

Layer Permeability (m/s) Bedrock 10-13

Fissured clayey siltstone 10-5

Clayey siltstone with gypsum 10-6

Gravel + silty matrix (colluvium) 10-6

Extension zone 10-2

This extension zone has been useful to simulate a lateral flow that takes place in the Vallcebre slope (outflow in the 2D model). In this case a better simulation is done with an anisotropy degree of the permeability (i. e. more permeable in the torrent direction) of Kx 0 25'Ky . However, the permeability values obtained are significantly different from those derived in the 2D analysis. These values are more similar to those obtained in the in situ tests for fissured clayey siltstone and clayey siltstone with gypsum than those obtained with 2D analyses. The permeability of the colluvium layer is not determinant on the simulation of piezometric records.

366 1. RIUS et al.

PRES, (in Pl)

N 180 M 170 L 160 I< 150 J 140 I l30 H 120 G 110 F 100 E 90 D 80 C 70 E 60 A SO

Figure 7. Computed piezometric levels (in meters) in the 3D geometry modelled (lower scarp and extension zone, figure 2).

4.4 PREDICTION OF WATER LEVELS

Finally, an attempt to relate computed water levels and rainfall data from a rain gauge has been done in order to assess the quoted method to predict raising of water levels during rainfall events. This approach has been focused on the lower part of the slope, where the major displacements have been recorded during the development of the project. The idea was to select the main aspects of the problem in order to avoid difficulties not derived from the fundamental parameters governing the landslide. Therefore, only a two dimensional analysis of this lower part has been considered.

For this analysis, the full capabilities of the TRANSIN code have been used, i.e. unsaturated flow and inverse analysis. This inverse procedure has been used to calibrate the flow parameters. That is, the key parameters of the problem (permeability and storativity of the two types of clayey siltstone) have been obtained from field measurements of the water table increments.

As a result of this analysis, a computed relation between rainfall and water table increment at different locations has been reproduced. Note that this relation has also been measured in some piezometers, but only for the rainfall events produced during the period recorded. The advantage of this procedure is that, as the measured events have been reasonably reproduced, other situations not yet produced (i.e. dangerous


rainfall events) can be simulated. This is in fact an extrapolation of the known behaviour, which should always be performed carefully, but it is considered as the more consistent procedure to predict future situations.

The inverse analysis increases its accuracy when much more data is available. On the other hand, it is difficult to analyse rainfall activity due to the difference consequences of each event: the effect of two similar storms on the flow slope could be totally different if the initial ground water table is different. Also, the characterisation of a rainfall event can not be performed in terms of a single quantity. However, usually both, the intensity and the rainfall volume are needed to define in a reasonable way the event. Analysing the Vallcebre rainfall data recorded during the project period, a repeated pattern has been observed: most of the events correspond to 1 to 3 days rainfalls, and in most cases they act on a similar initial ground water table.

Figure 8 shows the increment of ground water table measured in borehole S2 related with the volume of rainfall for different recorded events. Numbers in the plot represent the duration of the event in days. It can be seen that for short rainfalls (1 to 2 days), the volume is a reasonable quantity to be related in a direct way to the water level increment.

50.00

'E + 5

S + 3

Q) 25.00 + 2 E + 4 + 2 + 3 ::J +ur .. ;1

+ 1 1= ~ "0 + 1 + 1 + 3 > +,3

0.00

-0.05 0.00 0.05 0.10 0.15 S2 W. I. incr. (m)

250.00

--- 7' + 13

E 200.00 .s 150.00 Q)

E 100.00 + 1

-= + 5 0 50.00 + 8

> 0.00

-1.00 0.00 1.00 2.00 3.00 4.00 5.00 S2 W. I. incr. (m)

Figure 8. Increment of ground water table measured in borehole S2 related with the volume of rainfall for different recorded events (numbers refer to the rainfall duration in days).

Thus the increase of the water level at boreholes S2, S4 and S9 can be directly related to the rainfall event. This applies for short duration events and assuming a particular initial ground water table that is the most common in this area. In fact, field

368 1. RIUS et al.

measurements show that fluctuations of the water table are very rapid in this case and a typical base water table is reached after the rainfall event.

Using TRANSIN code, the same increase of the water table has been reproduced in boreholes S2, S4 and S9 by applying an infiltration of 1, 2 or 3 days. The amount of infiltration has been backanalysed in order to reproduce the measured increments of piezometric heads. Using this calibration procedure, a numerical relation between rainfall and computed water table has been found. This numerical relation is quite in agreement with the measured values available and it is, in fact, an average of the relation represented in figure 8.

Now it is possible to predict the position of the water table for any rainfall event of short duration for this area of the Vallcebre landslide as a relationship between water level increments and rainfall for different location of piezometers (table 6). Furthermore, these predicted positions of water table have been related to a safety factor (strength forces / acting forces) of the slip surface by means of a limit equilibrium analysis. This analysis shows a good agreement between raising of water table and dropping of safety factor. Nevertheless, since the landslide has not stopped sliding during last two years, and the limit equilibrium analysis for the base water table results in stability conditions, some kind of viscous behaviour could be expected.

Table 6. Relationship between rainfall and raising of water level (in metres) for different boreholes.

Event Rainfall S2 S4 S9 (mm) Computed Computed Computed

Measured Measured Measured 5/6/97 104.1 2.42 - 4.22 2.21 - 1.74 3.54 - 2.50

10/7/97 - 1/7/97 20.2 0.44 - 0.35 0.08 - 0.16 0.70 - 0.28 17/8/97 - 9/8/97 36.1 1.01 - 1.19 0.88 - 1.01 1.63 - 1.35 18/1/97 - 5/1/97 62.9 4.10 - 4.28 1.55 - 1.58 3.78 - -

5. Concluding remarks

The concluding remarks that can be assessed after the groundwater modelling procedure are the following.

- Hydrological modelling by means of TRANSIN code allows reproducing variations of groundwater table at any point from volume of rainfall. This has been the case in Vallcebre, but in other test sites the use of rainfall intensity or any other parameter or combination of parameters could be necessary.


- A relationship between rainfall and raising of water level (and furthennore global safety factor) is obtained, and an idea of the threshold value that can either generate instabilities or increase dramatically the speed of the landslide can be achieved.

- The parameter values in this test site were evaluated after different works: in situ tests, laboratory test and sensitivity analyses in different models. This gives an idea of the variation that can be obtained and the wide range of these possible values. Nevertheless, a value has to be selected for every different parameter and this has been done with a combination of the different sources.

- The role of cracks and preferential flow paths has been reproduced in the model and these elements seem to have a definite influence in the landslide behaviour. This work is part of an ongoing research program, and a new 3D-model coupling groundwater flow and defonnational analysis is now under development.

Acknowledgement

Part of this work has been undertaken within an E.C. research project NEWTECH (ENV4-CT96-0248) and within additional support from the Spanish CICYT (AMB96-2480-CE). Within the UPC group we thank J. Ramon, C. Wiegel, A. Matilla, D. Gomez, X. Falomir, T. Perez, 1. Alvarez, L. Vives, G. Galarza, A. Lloret, 1.A. Gili, 1. Corominas and J. Moya for the assistance in field, laboratory and modelling work. We also thank the School of Civil Engineering of the UPC (Escola Tecnica Superior d'Enginyers de Carnins, Canals i Ports de Barcelona, ETSECCPB) which has supported economically our participation in this event.

References

1. Galarza G., Medina A. and Carrera J. "TRANSIN-III Fortran code for solving the coupled non-linear flow and transport inverse problem". User's guide. E.T.S. d'Enginyers Camins, Canals i Ports. Universitat Politecnica de Catalunya. 1 vol, 256 pp, 1996.

2. Ledesma A., Corominas 1., Rius J., Moya J., Gili 1.A. and Lloret A. "Utilizacion con junta de diversos dispositivos de auscultacion en el deslizamiento de Vallcebre" IV Simposio Nacional sobre Taludes y Laderas Inestables. Granada (Spain). Vol. 1, pp 311-324, 1997.

3. Ramon 1. "Estudi geomecaruc de l' esllavissada del vessant de Vallcebre". Graduation project. E.T.S. d'Enginyers de Carnins, Canals i Ports. Universitat Politecnica de Catalunya. 1997.

370 J. RIUS et al.

4. UPC Group. "NEWTECH: New technologies for landslide hazard assessment and management in Europe". Final Report. European Commission. 1998.

5. Mora J. "Analisis del flujo de agua subterranea en la ladera inestable de Va1lcebre". Graduation project. E.T.S. d'Enginyers de Camins, Canals i Ports. Universitat Politecnica de Catalunya. 1998.

TRACE GAS ABSORPTION BY SOIL

Simulation Study on Diffusion Processes of Trace Gases, CO, H2} and CH4 in Soil

S. YONEMURA, M. YOKOZAWA, S. KAWASHIMA and H. TSURUTA National Institute of Agro-Environmental Sciences, Kannondai 3-1-1, Tsukuba Ibaraki, Japan

Abstract

A two-layered diffusion model was applied to the uptake process of trace gases as CO, H2 , and CH4 which are utilized by soil microorganisms or enzymes assuming that its uptake obey first-order kinetics about its concentration. Analytical solutions for mono-layered model exhibit that the physical property as gas diffusivity in soil is more important for uptake process than emission process. The numerical simulation shows that the deposition of CO, H2, and CH4 are limited by the combination of transport process and the localization of the soil uptake zone, within O.06cm 5- 1 for CO and O.lcm S-1 for H2, respectively, which are in reasonable consistence with field measurements.

1 Introduction

Soil as one of porous media can emit and absorb many kinds of gases. Some trace gases as sulphur dioxide (S02) and ozone (03), are physically destroyed or deposited to the surface of soil and/or plants. The deposition velocity of these gases [7] are usually more than 0.5 em S-1 (Table 1).

On the other hand, the trace gases as methane (CH4), carbon monoxide (CO), hydrogen (H2), and nitrous oxide (N20) are utilized by soil microorganisms or enzymes [1, 2, 5, 11]. In contrary to the gases physically deposited to soil, these gases show deposition velocities under O.lcm S-1 (Table 1). However, these gases play important role in the global atmosphere. CH4 acts as a green house gas inducing a secondly radiative heating force. CO is known as a serious atmospheric pollutant and plays an important role in the atmospheric chemistry. Soil is one major global sink of these gases in the atmosphere [10]. As yet, relatively limited works have been reported from the point of view of diffusion processes in soil.

In this study, we examine the characteristics of the uptake process of these trace gases in soil based on a diffusion model. We introduce the data taken in the field experiments and simulate the transport processes of CO, H2 and CH4 in soil, by using numerical simulations.

371 J.M. Crolet (ed.), Computational Methods/or Flow and Transport in Porous Media, 371-38l. © 2000 Kluwer Academic Publishers.

372 S. YONEMURA et al.

Table 1: The deposition velocities (em s-l) of trace gases and particles. Data is refered from the results of [3,5,7] and our measurement results.

Destruction at soil surface S02 0 3

Particles Uptake by soil microorganisms

N20 CO H2

CH4

2 Diffusion Equation in Soil

0.4-1.2 0.6-1.1

0.6

0-0.012 0-0.05 0-0.10

0-0.004

Gas transport process in soil can be described by Fick's law and diffusion equation.

F - D OCM. - s oz ' (1)

OCM 0 ( OCM) at = oz Ds--a;- + pPin-situ - Vin-situCM, (2)

where F is the flux of the gas (ng cm-2 S-l), Ds is the gas diffusivity in soil (cm2 S-1), CM is the concentration on the mass basis per unit volume of soil space (ng cm-3 ), Z is the soil depth from surface (em), t is time (s), p is the density of the pure gas (g cm-3), Pin-situ is the in-situ production rate on the spatial base (S-1), and Vin-situ is the in-situ uptake rate on the spatial base (S-l),

The concentration of the soil air, CM, is related to the volume mixing ratio of the gas in the soil air, C(ppbv), as follows:

CM = paC, (3)

where a is the air-filled porosity (em3 em-3 ). We assumed the vertical uniformity of DB, p, and a. Then, the above equations can be written using C as follows:

OC F = paDs oz;

oC Ds 02C lin-Situ at = -;; OZ2 + a - Vin-situ C ,

(4)

(5)

TRACE GAS ABSORPTION BY SOIL 373

The term (lIin-situC) represents the first-order uptake process; the term (Hn-situ/a) is zero-order production process about the concentration, which was supported for various gases [1, 2]. lIin-situ and Pin-situ in eq.(5) are given by

(6)

(7)

where dp is the particle density of the soil, Va (cm3 cm -3) is the ratio of solids in volume, lis (cm3 g-l s-l) and Ps (cm3 g-l S-l) is the in-situ soil uptake rate and the in-situ production rate on the mass basis of dry soil, respectively.

The gas diffusivity in soil Ds can be given as follows [8, 9]:

(8)

DA = Do 1013.25 (T + 273.15)1', Patmos 273.15

(9)

where ¢ (cm3 cm-3 ) is the porosity (the sum of liquid and gas phase in volume), Patmos is the atmospheric pressure (hPa), T is the soil temperature (0G). Molecular diffusivities, Do and 'Y for CO, H2 , and CH4 were fitted from the data of [6J (Table 2). H2 can permeate easily into soil since its molecular weight is light. The values of diffusivity for CO and CH4 are almost the same, then their diffusion behaviors are also similar in soil.

Table 2: Molecular diffusivity of CO, H2 and CH4 These values are obtained from [6] by exponential fitting.

Molecule Do 'Y cm2s-1

CO 0.186 1.70 H2 0.611 1.75

CH4 0.185 1.46

Surface net deposition velocity lid is obtained by

Fo lid = - ,

pCatmos (10)

where Fo is the flux at the soil surface.


Whether the gas is emitted from soil or deposited to soil is determined by the balance of the production and the uptake in soil. If no transport in soil, the compensation concentration in soil, Cequiv, is determined from the balance between the production and the uptake processes, as follows:

C . _ Pin-situ eqUlv -

allin-situ (11)

If Cequiv > Catmos(atmospheric concentration), the gas is emitted from soil; if Cequiv < Catmos, the gas is absorbed by soil. In most cases, the absolute value of the activation energy of the production is higher than that of the uptake. Thus, the Cequiv increases with the temperature.

3 Steady State Solutions for Mono-layered Model of Emission and Uptake Processes

To make clear the basic characteristics of the emission and the uptake processes by soil, we consider the steady state of eq.(5), assuming that the vertical uniformity of soil.

If the concentration is controlled only by diffusion and production processes, eq.(5) is written as:

Ds d2C ~n-situ 0 -;; dz 2 + a =. (12)

This can be solved analytically. We define the thickness of the production layer as d (cm):

C - C 2Pin-situ Fin-situ 2. - atmos + Dsd Z - Ds z, (13)

F = P~n-situd. (14)

This shows that the concentration increases with the soil depth from surface and is large when soil diffusivity is low. The flux values cannot be changed by physical properties of soil such as Ds and a, if the in-situ production rate Ps is not lowered by the higher concentration of the relevant gas.

If the concentration is controlled by only diffusion and uptake processes, eq.(5) is written as:

(15)


This can be also solved mathematically as:

C = Catmos exp ( - (16)

(17)

The concentration exponentially decreases with the soil depth from surface and lid increases with the square root of air-filled porosity a, soil diffusivity D s , and in-situ soil uptake rate lIin-situ.

Thus, soil physical property such as soil diffusivity is more important for the uptake process than for the emission process.

4 Observational Features of CO and H2 Uptake Processes

We have measured CO and H2 deposition velocities by chamber methods in fields and laboratory. In Fig.I, we show the results of CO and H2 deposition using dynamic chamber methods at a crop field in National Institute of AgroEnvironmental Sciences [11]. The soil type was an Andisol Hydric Hapludands (Agency for International Development, United States Department of Agriculture, Soil Conservation Service, Soil Management Support Services 1992) named Kuroboku in Japan. As the air-filled porosity increases, the deposition velocities become large. Clearly, you can see the gas transport limitation is limited by lower air-filled porosity. Previous studies [2, 5] and our results showed that the deposition velocity of H2 is usually higher than that of CO.

In the forest area in NIAES, slight positive dependence of CO and H2 deposition velocity on air-filled porosity was found. This slight dependence was due to slight changes of air-filled porosity of the forest soil. At the same time, slight dependence of H2 and CH4 deposition velocities on the soil temperature was also found. This was within the range expected by the increases of molecular diffusivity due to high temperature.

5 Two-layered Model for Emission and Uptake Processes of Trace Gases

We propose a two-layer model for the description of CO, H2, and CH4 uptake profiles which reflect microbial activity to absorb CO, H2 , and CH4 • Upper layer cannot absorb the gases; we hereafter call it the inactive layer. On the other hand, lower layer can absorb the gases; we call it the active layer. We define da as an index of the depth of active layer (i.e. the thickness of the inactive layer ). This


0.06

(a) CO • • 0.05 .. • • • •• ,-. ... • •• 'Ill • = 0.04 tJ '-' • • • • ~." • 0.03 • 'C ~ • • • • ~ ...

0.02 ~ III

,I:J • 0 0.01 • , • • 0.00

20 30 40 50 60

Air-fIlled porosity

0.10

(b) H2 •• • • ,-. 0.08 • •

":'Ill •• •• e • • •• • •• ~ 0.06 ~ ... • • •

'C • ~ ;.. 0.04 • • ... ~ • • III

,I:J • 0 0.02 •

• • • •

0.00 20 30 40 50 60

Air-fIlled porosity

Figure 1: The measured relationship between the air-filled porosity and the deposition velocities for (a) CO and (b) H2. The measurements were done during the summer season in 1995 in a crop field in NIAES.


concept of two-layered model is derived from the fact that the microbial activity in the surface soil within several mm depth is lowered due to the stresses of water drought and high temperature. Numerical calculation is needed because analytical solution cannot be obtained for this two-layered model.

For simplicity, we considered the horizontal uniformity, where temperature, soil moisture, air-filled porosity, and uptake rates depended only on the soil depth. Soil layer was divided into l(mm) grids to 1000(mm) depth from surface. Time step was set at 0.01 (sec). Particle density dp was set at 2.67 g cm-3 . Patmos was set to 1013.25 (hPa). The volume of solid state was set at 0.2, which corresponds to the case of well-cultivated field. No transport to deeper soil was assumed at the bottom (1000mm). Iterative calculations were conducted until the soil concentration profile converged to steady state.

Simulations were done using a SX4B computer (NEC Corporation, Japan) equipped with vector processors.

6 Simulation Results and Discussions

6.1 Validation of Numerical Model

An example concentration profile of the simulation results is shown in Fig.2. In the inactive layer, the flux is constant. The inactive layer acts as a diffusion resistance. In the active layer, the gas concentration drops exponentially with the soil depth as is shown by analytical solution.

We compared the results of the model with those observed in the field.

InJl~ti.e 12ye~ (Linear zone)

o~ r-------~----~

so 100 1.50 100

CO (ppb.)

Figure 2: An example of simulated profile.

378 s. YONEMURA et al.

Numerical calculations were conducted with changes in air-filled porosity using the production and uptake function determined from laboratory experiment. The siIIlulated relationship is shown in Fig.3. Largely, the coincidence is found between the measured deposition velocities (Fig.I) and the simulated deposition velocities (Fig.3). H2 has larger deposition velocity compared to CO. This is attributed to the difference of molecular diffusivities (Table 2).

6.2 Sensitivity Analysis due to Vin-situ and da

With changes in /lin-situ, /ld was calculated (Fig.4). When da=O.Ocm, /ld increases linearly with the square root of /lin-situ as is shown in the analytical solution. Namely, /ld does not increase in proportion to the in-situ uptake rate vin-situ because the uptake zone in soil was restricted in the shallow soil zone when the /lin-situ was larger.

The /lin-situ values of CO and H2 are less than 0.2s-1 from our laboratory experiment. The /lin-situ value of CH4 of the arable soil in NIAES was in the order of O.OOOls- l . Under these values in Fig.4, /ld values of CO, H2 , and CH4 are less than 0.06cm S-I, O.Icm S-1 and O.OOIcm S-1, respectively, which is consistent with the measured values of Table 1 [3,5,l1J.

In Fig.4, there are limit values dependent on da when /lin-situ = 00. The values show the limits caused by the diffusion resistance of the inactive layer. The existence of the inactive layer (Le. localization of microbial uptake) also eases the dependence of /ld on /lin-situ. Furthermore, our laboratory experiments show that the dependence of /lin-situ on the soil temperature is week and did not double with the soil temperature increment of 10°C. From these results, CO, H2 and CH4 uptake by soil (Le. /ld) is not sensitive to the in-situ soil uptake rates (/lin-situ) in comparison to the emission as CO2 .

Figure 4 also shows that the gases physically destroyed in the soil surface are destroyed within Imm depth. Other diffusion resistances as the surface resistance in the atmospheric boundary layer are important factors which control the deposition velocity.

6.3 Further Investigations

In ecosystems, the uptake process of the trace gases is complex and controlled by many factors. Experiments and numerical modeling are necessary in the further understanding of the complex behavior of the trance gas uptake by soil.

The soil diffusion process in soil is basically driven by molecular diffusivity. This assumption could be applicable to the water evaporation or CO2 emission whose emission is done in the deeper soil layer. However, it should be noted that the diffusivity in surface soil very near to the atmosphere «Icm) is affected by the atmospheric turbulence [4J. Hence, it may be also plausible that CO and H2 soil uptake are promoted by the atmospheric turbulence. These effects should be incorporated into the diffusion process in future studies.


0.06

(a) CO 0.05

V =0.2 s

0.04 4" '" e 0.03 (J '-'

~ ... 0.02

0.01

0.00 0.2 0.3 0.4 0.5 0.6

Air-fIlled porosity 0.10

(b) H2 0.08 V =0.2

s

,-.. 0.06 "':'",

e (J '-' ~ ... 0.04

0.02

0.00 0.2 0.3 0.4 0.5 0.6

Air-fIlled porosity

Figure 3: The simulated relationship between the air-filled porosity and the deposition velocities for (a) CO and (b) H2. The in-situ uptake rates and production rates used for the simulation were obtained in laboratory experiments.

380 S. YONEMURA et aL

1 -d=O.Ocm · - -d=O.lcm

(a) CO and CH4

· ---d=0.2cm · 0.1 -----d =0.5cm ,-.. · "':'1Il ·····d =1.0cm • 5 ---d=2.0cm · - ---·-d =5.0cm ... '" · ------

0.01 .-.-._ ...... - .... _ ....•.. _. __ ... .

0.001 L...&. ............ ~~...-........ -:t'~---........ ~~-................ 't--.-...I 0.001 0.01 0.1 1

,-.. ... '1Il

1

0.1

... ... 0.01

-d=O.Ocm · - -d=O.lcm · ---d =0.2cm · -----d =0.5cm · -- ---d =1.0cm · ---d =2.0cm ·

V (S-l) In-situ

.........

.-------

... -.-... -.-._.-._ ..... _ ..... _ .

0.001 ................ ~~ ............... ~~ ...................... ~--'-....... ~-.............. 0.001 0.01 0.1 1

V (S-l) In-situ

Figure 4: The simulated relationship between the in-situ uptake rates (lIjn-situ) and the deposition velocities (Vd) for (a) CO and (b) H2.


References [1] Bender M. and Conrad R., Microbial oxidation of methane, ammonium and carbon monox

ide, and turnover of nitrous oxide and nitric oxide in soils, Biogeochemistry, 27,97-112, 1994.

[2] Conrad R. and Seiler W., Influence of temperature, moisture, and organic carbon on the flux of H2 and CO between soil and atmosphere: Field studies in subtropical regions, J. Geophys. Res., 90, 5699-5709, 1985.

[3] Dor H., Katruff L., and Levin 1. , Soil texture parameterization of the methane uptake in aerated soils, Chemosphere , 26, 697-713, 1993.

[4] Ishihara Y., Shimojima E., and Harada H., Water vapor transfer beneath bare soil where evaporation is influenced by a turbulent surface wind, Journal of Hydrology, 131, 63-204, 1992.

[5] Liebl K.H. and Seiler W., CO and H2 destruction at the soil surface. In Microbial production and utilization of gases (ed. Schlegel H.G., Gottschalk G., and Pfenning N. ) , Akademie der Wissenschaften, Gttingen., pp.215-229, 1976.

[6] Marrero T.R. and Mason, E.A. , Gaseous diffusion coefficients, J. Phys. Chern. Ref. Data, 1, 3-118, 1972.

[7] Mcmahon T.A. and Denison P.J., Empirical atmospheric deposition parameters - a survey, Atmos. Envir., 13, 571-585, 1979.

[8] Millington R.J. and Quirk J.M. , Permeability of porous solids, Trans. Faraday Soc. , 57, 1200-1207, 1961.

[9] Sallam A., Jury W.A., and Jetey J., Measurement of gas diffusion coefficient under relatively low air-filled porosity, Soil Sci. Soc. Am. J., 48, 3-6, 1984.

[10] Seiler W. and Conrad R. , Contribution of tropical ecosystem to the global budgets of trace gases, especially CH4, H2, CO and N20. In The Geophysiology of Amazonia (ed. Dickinson RE.), Wiley, New York, pp.133-162, 1987.

[11] Yonemura S. , Kawashima S., and Tsuruta H. , Continuous measurements of CO and H2 deposition velocities in an andisol by open flow method; uptake control by soil moisture, Tellus, in print, 1999.

List of Contributants

Abdallah, A. Ecole Nationale Superieure de Geologie 163 Vandoeuvre-les-Nancy, France

Abriola, L. University of Michigan 303 Ann Arbor, USA

Al Mers, A. Universite Abdelmalek Essaadi 175 Tetouan, Marrocco

Alonso, E.E. Universitat Politecnica de Catalunya 327 Barcelona, Spain

Aroudam, EI H. Universite Abdelmalek Essaadi 193 Tetouan, Marrocco

Ayora, C. Institut de Ciences de la Terra, C.S.I.C. 203,287, Barcelona, Spain 339

Benett Llobera, L. Universitat Politecnica de Catalunya 203,339 Barcelona, Spain

Bernard, D. Universite de Bordeaux I 217 Talence, France

Bertone, F. HydroExpert 319 Orsay, France

Bonnet,M. HydroExpert 319 Orsay, France

Boussouis, M. Universite Abdelmalek Essaadi 175 Tetouan, Marrocco

Bues,M.A. Ecole Nationale Superieure de Geologie 273 Vandoeuvre-Ies-Nancy, France

Carrera, J. Universitat Politecnica de Catalunya 53,203, Barcelona, Spain 245, 287,

319

Castagna, S. Universitat Politecnica de Catalunya 327 Barcelona, Spain

383

384

Crolet, J.M. Universite de France-Comte 107,237 Besan~on, France

Du Plessis, P. University of Stellenbosch 231 Matieland, South Africa

El Hatri, M. Universite Sidi Mohamed Ben Abdellah 237 F es, Marrocco

Eymard, R. Ecole Nationale des Ponts et Chaussees 3 Marne la Vallee, France

Firdaouss, M. UPMC Parix VI 29 Orsay, France

Gabbouhy, M. Universite Ibn Tofail 39 Kenitra, Morocco

Gens,A. Universitat Politecnica de Catalunya 245 Barcelona, Spain

Herbin, R. Universite de Provence 3 Marseille, France

Heredia, J. INGEMISA 53 Madrid, Spain

Hilhorst, D. Universite de Paris Sud 3 Orsay, France

Hoffmann, M. Wageningen Agricultural University 83 Wageningen, The Netherlands

KaCur, J. Comenius University 93 Bratislava, Slovakia

Kammouri, A.S. Universite Sidi Mohamed Ben Abdellah 237 F es, Marrocco

Kawashima, S. National Institute of Agro-Environmental Sciences 371 Tsubaka Ibaraki, Japan

Ledesma, A. Universitat Politecnica de Catalunya 353 Barcelona, Spain

385

Lloret, A. Universitat Politecnioa de Catalunya 327 Barcelona, Spain

Masrouri, F. Ecole Nationale Superieure de Geologie 163 Vandoeuvre-Ies-Nancy, France

Medina Sierra, A. Universitat Politecnica de Catalunya 53 Barcelona, Spain

Mghazli,Z. Universite Ibn Tofail 39 Kenitra, Morocco

Mimet, A. Universite Abdelmalek Essaadi 175, 193 Tetouan, Marrocco

Mora, J. Universitat Politecnica de Catalunya 353 Barcelona, Spain

Murea, C. University of Bucharest 107 Bucharest, Roumania

Olivella, S. Universitat Politecnica de Catalunya 245,317 Barcelona, Spain

Oltean, C. Ecole Nationale Superieure de Geologie 273 Vandoeuvre-Ies-Nancy, France

Panfilov, M. Moscow Lomonosov University 115 Moscow, Russian

Perez-Paricio, A. Universitat Politecnica de Catalunya 339 Barcelona, Spain

Ramarosy, N. Universite de Paris Sud 3 Orsay, France

Rius, J. Universitat Politecnica de Catalunya 353 Barcelona, Spain

Saaltink, M. W. Universitat Politecnica de Catalunya 287,349 Barcelona, Spain

Tchijov, A. Moscow Lomonosov University 115 Moscow, Russian

386

Tran, P. UPMC Paris VI 29 Orsay, France

Tsuruta, H. National Institute of Agro-Environmental Sciences 371 Tsubaka Ibaraki, Japan

Vignoles, G. Universite de Bordeaux I 217 Talence, France

Yokozawa, M. National Institute of Agro-Environmental Sciences 371 Tsubaka Ibaraki, Japan

Yonemura, S. National Institute of Agro-Environmental Sciences 371 Tsubaka Ibaraki, Japan

Zolotukhin, A. Stavanger College 141 Stavanger, Norway

Theory and Applications of Transport in Porous Media

Series Editor: Jacob Bear, Technion - Israel Institute of Technology, Haifa, Israel

1. H.I. Ene and D. Polissevski: Thermal Flow in Porous Media. 1987 ISBN 90-277-2225-0

2. J. Bear and A. Verruijt: Modeling Groundwater Flow and Pollution. With Computer Programs for Sample Cases. 1987 ISBN 1-55608-014-X; Pb 1-55608-015-8

3. G.I. Barenblatt, V.M. Entov and V.M. Ryzhik: Theory of Fluid Flows Through Natural Rocks. 1990 ISBN 0-7923-0167-6

4. J. Bear and Y. Bachmat: Introduction to Modeling of Transport Phenomena in Porous Media. 1990 ISBN 0-7923-0557-4; Pb (1991) 0-7923-1lO6-X

5. J. Bear andJ-M. Buchlin (eds.): Modelling and Applications of Transport Phenomena in Porous Media. 1991 ISBN 0-7923-1443-3

6. Ne-Zheng Sun: Inverse Problems in Groundwater Modeling. 1994 ISBN 0-7923-2987-2

7. A. Verruijt: Computational Geomechanics. 1995 ISBN 0-7923-3407-8

8. V.N. Nikolaevskiy: Geomechanics and Fluidodynamics. With Applications to Reser-voir Engineering. 1996 ISBN 0-7923-3793-X

9. V.I. Selyakov and V.V. Kadet: Percolation Models for Transport in Porous Media. With Applications to Reservoir Engineering. 1996 ISBN 0-7923-4322-0

lO. J.H. Cushman: The Physics of Fluids in Hierarchical Porous Media: Angstroms to Miles. 1997 ISBN 0-7923-4742-0

11. J.M. Crolet and M. El Hatri (eds.): Recent Advances in Problems of Flow and Transport in Porous Media. 1998 ISBN 0-7923-4938-5

12. K.C. Khilar and H.S. Fogler: Migration of Fines in Porous Media. 1998 ISBN 0-7923-5284-X

13. S. Whitaker: The Method of Volume Averaging. 1999 ISBN 0-7923-5486-9

14. J. Bear, A.H.-D. Cheng, S. Sorek, D. Ouazar and I. Herrera (eds.): Seawater Intrusion in Coastal Aquifers. Concepts, Methods and Practices. 1999 ISBN 0-7923-5573-3

15. P.M. Adler and J.-F. Thovert: Fractures and Fracture Networks. 1999 ISBN 0-7923-5647-0

16. M. Panfilov: Macroscale Models of Flow Through Highly Heterogeneous Porous Media. 2000 ISBN 0-7923-6176-8

17. J.M. Crolet: Computational Methods for Flow and Transport in Porous Media. 2000 ISBN 0-7923-6263-2

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