a mass conservative numerical solution of vertical water flow and mass transport equations in...

Ann. Nucl. Energy, Vol. 20, No. 2, pp. 91-99, 1993 0306-4549/93 $5.00+0.00 Printed in Great Britain. All rights reserved Copyright © 1992 Pergamon Press Ltd

A MASS CONSERVATIVE NUMERICAL SOLUTION OF VERTICAL WATER FLOW AND MASS TRANSPORT

EQUATIONS IN UNSATURATED POROUS MEDIA

SEUNG CHEOL LIM and KUN JAI LEE

Department of Nuclear Engineering, Korea Advanced Institute of Science and Technology, Kusong Dong, Yusunggu, Taejon 305-701, Korea

(Received 2 June 1992)

Abstract--The Galerkin finite element method is used to solve the problem of one-dimensional, vertical flow of water and mass transport of conservative-nonconservative solutes in unsaturated porous media. Numerical approximations based on different forms of the governing equation, although they are equivalent in continuous forms, can result in remarkably different solutions in an unsaturated flow problem. Solutions given by a simple Galerkin method based on the h-based Richards equation yield a large mass balance error and an underestimation of the infiltration depth. With the employment of the ROMV (restoration of main varaible) concept in the discretization step, the mass conservative numerical solution algorithm for water flow has been derived. The resulting computational schemes for water flow and mass transport are applied to sandy soil. The ROMV method shows good mass conservation in water flow analysis, whereas it seems to have a minor effect on mass transport. However, it may relax the time-step size restriction and so ensure an improved calculation output.

INTRODUCTION

The prediction of simultaneous water flow and transport of solutes in unsaturated soils is an important problem in many branches of science and engineering. Likewise, in the nuclear industry, since the safe dis- posal of radioactive wastes has become an important issue, the analysis of the flow field and radionuclide transport in the unsaturated soil medium surrounding a waste repository has been the subject of considerable study. Fluid movement in unsaturated soils is gen- erally perceived to obey the classical Richards equation (Bear, 1975 ; Hirrel, 1980). This equation can be written in several forms, with either pressure head h[L] or moisture content 0[(La/L 3] as the dependent variable. The constitutive relationship between 0 and h allows for the conversion of one form of equation to another. Because the Richards equation is nonlinear, an analytical solution is not possible except for special cases. Therefore, numerical approximations are commonly used to solve unsaturated flow prob- lems. The typical approximations applied to the spatial domain are the finite difference and finite element methods. These are usually coupled with a simple one- step Euler time-marching algorithm. The equation governing the mass transport of solute is also nonlinear and more difficult to solve numerically than the flow of water.

General overviews of the literature for the numeri-

cal simulation of unsaturated flow and mass transport can be found in the recent works of Nielsen et al. (1986) and Paniconi et al. (1991). Most previous works used the h-based form of Richards equation for flow analysis. With this equation form, numerous finite difference and finite element solution techniques have been used (Narasimhan and Witherspoon, 1976 ; Haverkamp et al., 1977; Cooley, 1983; Huyakorn et al., 1984, 1986; Kaluarachchi and Parker, 1987). Recently, as the mass balance error which occurs in the discretization of the h-based equation has been under debate, peculiar numerical solution methods, different from previous works, have been proposed (Allen and Murphy, 1986; Celia et al., 1990). They used a mixed form of Richards equation to derive the numerical solution algorithms and demonstrated remarkable mass balance in their solutions.

Numerous finite element solutions of the mass transport equation have been presented in various publications (Pickens and Gillham, 1980; Yeh and Strand, 1982; Gureghian, 1983; Huyakorn et al., 1985). Most of them used the simple Galerkin approach for the accompanying water flow equation. In the finite element simulation of mass transport of solutes, Gureghian (1983) and Huyakorn et al. (1985) used the nonsymmetric weighting functions to eliminate the oscillations which occur when the convection of solute dominates the dispersion. But such a technique often yields large numerical dispersions.

91

92 SEUNG CHEOL Lira and KUN JAi LEE

In this paper, the equations of one-dimensional vertical unsaturated flow of water and mass transport of conservative-nonconservative solutes are solved simultaneously. With the employment of the ROMV (restoration of main variable) concept in the discretization step, the mass conservative numerical solution algorithm for water flow has been derived. The effect of mass conservation in the flow analysis of mass transport of solute seems to be less observable than in the case of the water flow analysis. However, it may relax the time-step constraint in the numerical simulation, and so is helpful in acquiring realistic concentration profiles and in computational time saving.

GOVERNING EQUATION

The one-dimensional vertical flow of water in unsaturated porous media can be decribed by the Richards equation

Oh O / Oh Ch ~ = Ozz ~Koz - - K ) , (1)

where h is the pressure head [L], Ch is the specific moisture capacity [l/L], K is the hydraulic con- ductivity [L/T], t is the time and z denotes the vertical dimension, which is assumed positive downward.

The corresponding equation of mass transport of radionuclides can be expressed as

Ou 0 [ Oc'\ O = Oz ~OD O z ) - Oz (qc) - 2u, (2) at

where 0 is the volumetric water content (L3/L3], c is the radionuclide concentration [M/L3], D is the dispersion coefficient [L2/T], q is the volumetric flux or Darcy velocity [L/T], 2 is the decay constant [l/T] and the total radionuclide per unit volume, u, is given by

u = Oc+ps, (3)

where p is the soil bulk density and s is the adsorption per unit mass, a function of radionuclide concentration (and possibly also water content). It is assumed that a linear equilibrium isotherm can be used to relate the adsorption s and the radionuclide concentration c.

Equation (2) can be converted to a more convenient form using the continuity equation of water flow

0q 00 - Oz = ~ " (4 )

After substitution of equations (3) and (4) into equation (2), one can obtain

0c O [ 0c'~ 0c OP~ di = Oz ~OD~zJ--q oz --20RdC, (5)

where the retardation coefficient, Rd, is given by

• pKo Rd = 1 + ~ . (6)

Here Ks is the distribution coefficient. Note that the retardation coefficient Rd is a function of the volumetric water content, 0.

The dispersion coefficient, D, in equations (2) and (5), according Bear (1972) and Freeze and Cherry (1979), can be expressed as

D = D0z+alvl", (7)

where Do is the molecular diffusion coefficient [L2/T], z is the tortuosity factor, ~ is the dispersivity [L], n is an empirical constant and v ( = q/O) is the average pore-water velocity [L/T].

In the case of one-dimensional flow and transport in porous media, the initial and boundary conditions are as follows :

initial condition,

h(z, O) = h i or O(z, 0) = 0i, (8a)

c(z, 0) = cl ; (8b)

boundary condition at the surface,

Oh - K ~ z + K = q0, z = 0, t > 0 , ( 9a )

0c -OD~- z+qc=qoco, z = 0 , t > 0 , (9b)

or

h(O,t)=ho or 0(0, t ) = 0 o , (10a)

c(O,t) = Co; (10b)

and

boundary condition at the bottom,

h(1,t)=hl or O(1, t)=O~, ( l l a )

0c c(1,t)=cl or ~z(1, t ) = 0 . ( l i b )

NUMERICAL SIMULATIONS

Numerical approximation of the water flow equation

The finite element approximation is usually coupled with a simple one-step Euler time-marching algorithm to solve equation (1). But the solution using the standard h-based Richard equation and a backward Euler time discretization is shown to produce a large mass

Solution of vertical water flow and mass transport equations 93

balance error for many example calculations. The reason for the poor mass balance lies in the time derivative term: because O0/Ot and Ch(ah/Ot) are mathematically equivalent in the continuous partial differential equation but their discrete analogs are not equivalent. To accommodate this point, we use the ROMV concept in the discretization stage. That is, we use same iteration level for h and 0 which is restored from the nonlinear coefficient Ch. Then we get the following mass conservative linear equation for h,+ l,m+ 1, in terms of the increment in the iteration, 6h = h "+ I'm+ I--h"+ Á,m.

0.+' ,"__0. At , (12)

where the superscripts n and m indicate the time and iteration levels, respectively.

The finite element approximation is usually gen- erated using the piecewise linear basis functions to approximate h and 0 as well as the nonlinear coefficients Ch and K :

and

h(z, t) -~ ~ hj(t)Nj(z), (13a) j = 0

O(h) ~- ~ O(hj)Nj(z) = ~ OjNj(z), (13b) 1=0 j=O

K(h) ~- ~ K(hj)Nj(z)= ~ KjNj(z) (13c) j = 0 .i=0

Ch(h) ~ ~ Ch(hj)Nj(z)= ~ C(h)jNj(z), (13d) j = 0 j = 0

where Nj is the standard piecewise linear basis function, nn is the number of nodes in the solution domain and hi(t) is the nodal value of pressure head at time t.

The Galerkin approach requires that the residual Lw(h), obtained by substituting equation (13) into equation (12), be orthogonal to the selected weighting functions

olL,(h)N~(z)dz=O (i = 0, 1,2 . . . . nn), (14)

where N~ is the weighting function of node i and the integration is performed over the solution domain.

Substituting equation (12) into equation (14), using Green's theorem to remove the second derivatives and applying integration by parts to the spatial derivatives, leads to the matrix equation

([AiA+[BJ) {6hi} = {El}, (15)

where

f, [A, A = ~ ~tC(h)kN, NyNk dz, (16a)

[B~A = Z f K~ dNi dNj

¢ ~e ~ ~-z Nk dz (16b)

and

with

dN~ dNj (Fi} ~" q0--ql--~ Kk~- z -~z Nkdz

l" dN~ "t,;+"m+ ~ JeKSd-zz Njdz

(16c)

[ E J ~ +[Gtj+M~j+Ro]{ej} = {Hi}, (19)

[E'A = ~e f OP~N, Njdz, (20a)

(K ~h -K). (17) q = - \ Oz

Thus, we obtain nn + 1 algebraic equations. The iter- ative process continues until a satisfactory degree of convergence is obtained.

Numerical approximation of the mass transport equation

As in the case of the water flow equation, the dependent variable e of the mass transport equation (5) is approximated by the linear basis function

c(z,t) _~ ~ cj(t)Nj(z). (18) j=O

Where cj(t) are the nodal values of the radionuclide concentration at time t. Similarly, substituting the mass transport equation (5) into the relationship which minimizes the residual Lr(c), using Green's theorem to remove the second derivatives, leads to

94 SEUNG CHEOL LIM and KUN JAI LEE

and

[Miil = ~ f qNi ~ d z , (20c)

[Rij I = ~ f 20RaUi Nj dz (20d)

(OD oc'] (OD ac'] (20e) {14,} = \ ~z/, - \ ~Zjo

The integrals of the matrix equations (20) are evalu- ated by approximating 0, Re D and q by relationships of a form similar to function (18).

The time derivative of equation (19) can be approximated by the general two-time level finite difference discretization. With superscript n denoting the time step and time weighting on the relaxation parameter a, we get

[EiA.+ ~ {c j}"+ ' - {cj} ° At

+ao+Mii+Riil"+~{ej} "+" = {HI.} "+' , (21)

where

{cj} "+~ = a{ej} " + ' + (1 - ~){cj} °. (22)

When, a = 1/2, a time-centered Crank Nicolson type algorithm is obtained ; and when ~ = 1, a fully implicit type algorithm is obtained. Thus, we can write equation (22), with ~ = 1, in the form

[Pij]"+'{cj} "+' = [Qij]"+'{cj}"+{Hi} "+l, (23)

where

and

1 [ P j = [Gij + M, j + Rijl + At [Eijl (24a)

1 [QiJ] = At [Eij]. (24b)

NUMERICAL APPLICATIONS

Two different cases of solute t ransport-- infi l t rat ion in a sandy soi l - -were examined : (1) no decay and no adsorpt ion; and (2) radioactive decay and sorption. In the applications, the data used by Antonopoulos and Papazafiriou (1990) were cited for soil water characteristic functions.

No radioactive decay and no sorption In this application, nonreactive solute water infil-

trated a column of sandy soil with an initial soil moist-

ure content of 0~ = 0.1 cm3/cm 3 and zero solute concentration. The experimental data and the functions relating h, 0 and K (see the Appendix) are taken from Antonopoulos and Papazafiriou (1990). The two sets of boundary conditions at the soil surface are: (1) continuous infiltration of solute water at a constant rate of 3.39 cm/h and a constant solute concentration of 1000 rag/l; and (2) infiltration of solute water at a constant rate for a period t~ followed by fresh water at the same rate :

initial conditions,

O(z, O) = O. 1 cm3/cm 3,

c(z, O) = 0 ;

boundary conditions at the soil surface,

q(0, t) = q0 = 3.39cm/h,

c(0, t) = Co = 1000 mg/1,

or

c(0, t) = Co = 1000 mg/1 for t ~< 0.5 h,

c(0, t) = 0 fo r t > 0 .5h;

and

boundary conditions at the soil bottom,

0(1, t) = 0,l cm3/cm 3

c(1, t) = 0.

A grid size of Az = 1 cm and three time intervals of At = 0.001, 0.01 and 0.05 h are used.

Figure 1 depicts the moisture fronts computed by the simple Galerkin method and Galerkin method combined with the R O M V concept. It can be found from the graphs based on the simple Galerkin method that the total water content is not conserved, as At increases. On the contrary, the mass conservation is preserved in the solution of the R O M V method, though the numerical dispersion increases as At increases. Actually, the mass conservation can be easily demonstrated from equations (15) and (16). Namely, if we sum up the all finite element equations for all nodes i, after evaluation of the integrals, the only terms that do not cancel are the boundary fluxes and the change in total mass over the time step At. Therefore, the set of equations based on the R O M V concept possesses the conservative property.

Figures 2 and 3 show the solute concentration profiles for continuous infiltration of solute water at a constant rate and infiltration of solute water at a constant rate for a period of time tc = 0.5 h followed by fresh water applied at the same rate. In both cases, the water flow is calculated by the simple Galerkin

.-.

0.3

0 o 0

.2

1.0

" ~

=

= m

~ dt is O.O01h

o ~

) \

oocc

o dt is O.O01h

(~} ~"

~~

,5

~

=r..-

-r.=

dt

O.O

lh

~ (a

=~

===

dt

O.0

1h

±~.

:--"

±d

t O.

05h

h ~

L2

h

....

. dt

O.

05h

ID ~ o.5

[

~o

ra 0

.1

..

..

.

= ~

o~

O.Sh

ioo

, i

, i

, i

, oo

k

.L i

• l

• "0

.0

20.0

40.0

80.0

80.0

0.0

20~0

-

40~0

-

6o~o

-

solo

Dept

h z(

cm)

Dept

h z(

em)

(a)

(a)

o.,:-

.

..

..

.

,.oTT

.- -

- =

-

cc

c~

o

dt

is

O.O

01h

oo

oe

e

dt

is

O.O

01h

O.0

5h

~

~"

" "-

'-"

" dt

O

.05

h

I "~

01

..

..

° f,

. ~o

2h

I I

J I

~ 0.

0 0

.0 0.

0 20

.0

40.0

60

.0

80.0

D

ep

th

z(c

m)

(b)

Fig.

1. M

oist

ure

prof

iles

bas

ed o

n th

e (a

) si

mpl

e G

aler

kin

and

(b)

RO

MV

met

hods

in

a s

andy

soi

l, un

der

a co

nsta

nt i

nfil

trat

ion

of 3

.39

cm/h

.

0.0

20.0

40

.0

B0.

0 80

.0

De

pth

z

(em

)

(b)

Fig.

2.

Dis

trib

utio

n of

sol

ute

conc

entr

atio

n in

a s

andy

soi

l ba

sed

on t

he f

low

an

alys

is b

y th

e (a

) si

mpl

e G

aler

kin

and

(b)

RO

MV

met

hods

, un

der

a co

nsta

nt

infi

ltra

tion

of

3.39

cm

/h.

o C

O

~D

o e~

C

96 SEUNG CHEOL LIM and KuN JAI LEE

and ROMV methods. As regards the step release case, there seems to be no distinct difference in the concentration profiles. But in the case of the impulse release mode, the concentration based on the simple Galerkin water flow analysis tends to be underestimated compared to that based on the ROMV method as At increases. So the mass conservation in the water flow analysis using the ROMV concept is expected to yield a more accurate estimation of solute transport.

Radioactive decay and sorption

As a second application, we considered the radioactive nuclide Sr-90 which has a half-life of 27.7 years and experiences adsorption during the transport process. The soil physical and dispersion related parameters are given in Table A2. The initial conditions are the same as in the previous application, but the Dirichlet boundary condition is used at the soil surface :

boundary conditions at the soil surface,

0(0, t) = 0.28 cm3/cm 3,

c(0, t) = 1000 mg/1.

Sr-90 has a distribution coefficient ranging (Onishi et al., 1981) from a few hundred to the order of a thousand. However, in this application, to explore the effect of adsorption phenomena, three hypothetical distribution coefficients-q), 0.1 and 1.(~-are used.

Figure 4 shows the moisture content profiles computed by the simple Galerkin and ROMV methods. As in the case of the first application, the computational

output by ROMV shows good mass conservation, whereas the output of the simple Galerkin method undergoes a large mass balance error as At increases. Figure 5 depicts the concentration profiles when the distribution coefficient is assumed to be zero. In the graphs in Fig. 5 it can be seen that oscillations increase as the time-step becomes smaller. Hence, it is desirable to increase the time-step slightly to acquire smooth and realistic concentration profiles. However, increas- ing the time-step has an undesirable effect on the water flow analysis, when the simple Galerkin method is used, and the induced error can be propagated to transport analysis. Hence, proper calculation of the water flow using the ROMV method leads to a great improvement in the solute transport analysis. Figure 6 shows the effect of the adsorption distribution coefficient. As we know, intuitively, the increase in the distribution coefficient greatly retards the solute transport.

DISCUSSION AND CONCLUSIONS

The Galerkin finite element method was applied to solve the one-dimensional vertical water flow and mass transport equations [equations (1) and (5)]. To eliminate the large mass balance error which occurs when the simple Galerkin method is used for the standard h-based Richards equation, the ROMV concept is used in the discretization step of the water flow equation. In the case of mass transport, the standard discretization technique was used. In both cases, the linear one-dimensional basis functions were employed.

1.0 ~eoee dt O.O01h, simple G.

o A mav~ dt O.05h. simple G. ] \ ee:e--" dt O.O01h. ROMV

"~ / I =====dr O.05h, ROMV

o ~ 0.5 © ] ~I elapsed time is 2h 0 O ~ ~ ~ release duration is 0.5h

0.0 = - - • 1

0.0 20.0 40.0 60.0 80.0 Depth z(cm)

Fig. 3. Distribution of solute concentration in a sandy soil, under the impulse release mode.

~ 0.3

~ 0.2

~ 0.

1

~I- ~-~-'-'-" dt

is

O.01h

0.0

i 1

, I

, t

I 0.0

20.0

40.0

80.0

80.0

Dep

th z(cm)

(a)

A

0.3

0 c,~

o 0

.2

o 0

.1

I=

- (b)--"~ll__ e

eeee

dt is

O.O

01h

==

==

= dt is O.O05h

cccz

c d

t is

0

.01

h

0. l

h

0.5

h

i .

..

..

--

0.

0 t

I I

0.0

2

0.0

8

0.0

i I

i l

40

.0

80.0

D

epth

z(cm)

(b)

Fig.

4. M

oist

ure

prof

iles

bas

ed o

n th

e (a

) si

mpl

e G

aler

kin

and

(b)

RO

MV

met

hods

in

a s

andy

soi

l, un

der

a co

nsta

nt m

oist

ure

cont

ent

of 0

.28

cm3/

cm 3

at t

he s

oil

surf

ace.

o ~.) o 0 I:I

o

1.2

0.8

0.4

0.0

0.0

0.1h

20

.0

eeee

o dt i

s ....

. dt i

s -~-'~A dt

is

(a)

0.5h

40

.0

Dep

th

z(cm

)

(a)

O.O01h

O.O05h

O.Olh

8o.-o

-

80-0

1.2

0 o 0 o

0.4

-

0.0 0.0

eeee

o

dt

is

O.O

01

h

c===

= dt

is

O

.O0

5h

..

...

dt

is

O.0

1h

(b)

O. lh

~5h

20.0

40.0

80.0

80-0

De

pth

z(cm

)

(b)

Fig.

5.

Dis

trib

utio

n of

sol

ute

conc

entr

atio

n in

a s

andy

soi

l ba

sed

on t

he f

low

an

alys

is b

y th

e (a

) si

mpl

e G

aler

kin

and

(b)

RO

MV

met

hods

, un

der

a co

nsta

nt

moi

stur

e co

nten

t of

0.2

8 cm

S/cm

~ at

the

soi

l sur

face

.

O

e-*

=u

O

ga

O

98 SEUNG CHEOL LIM and KUN JAI LEE

1.Z

o ~ ~ e e e e o Kd is 0 tac~==n Kd i s 0.1 . . . . . Kd is 1.0

o~ 0.8 -

e l a p s e d t i m e i s 0 . 5 h dt is 0.001h

o 0 . 4 -

o . o ~ _ _ .t ~ .t

0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 80-.0 D e p t h z(em)

Fig. 6. Solute concentration profilcs in a sandy soil, under various adsorption distribution coefficients.

The resulting computational schemes were applied to sandy soil to explore the water flow and mass transport of conservative-nonconservative solutes. Dirichlet and Neumann type boundary condi t ions were used for the flow analysis, whereas step and impulse type release modes were used to examine the solute transport. In the case of moisture infiltration, the simple Galerkin analysis showed the large mass balance error for both boundary conditions as At increases. But the combinat ion of the R O M V concept with the Galerkin approach preserved a good mass balance in both cases. The mass conservation in water flow seemed to have a minor effect on the solute transport. But when large At is used, the solute concentration profile based on the simple Galerkin analysis of water flow is somewhat underestimated compared to that based on the R O M V method. Some- times the larger time-step is desirable for saving calculation time and for a nonoseillating, realistic solute concentration profile. In such cases, the use of the R O M V for a concept in the numerical approximation ensures the mass conservation and proper calculation for the water flow and solute transport.

REFERENCES

Allen M. B. and Murphy C. L. (1986) Wat. Resour. Res. 22(11), 1537.

Antonopoulos V. Z. and Papazafiriou Z. G. (1990) J. Hydrol. 119, 157.

Bear J. (1975) Dynamics of Fluids in Porous Media. AmeriCan Elsevier, New York.

Celia M. A., Bouloutas E. T. and Zarba R. L. (1990) Wat. Resour. Res. 26(7), 1483.

Cooley R. L. (1983) Wat. Resour. Res. 19(5), 1271. Freeze R. A. and Cherry J. A. (1979) Groundwater. Prentice-

Hall, Englewood Cliffs, NJ. Gureghian A. B. (1983) Report ONWI-465. Haverkamp R., Vauclin M., Touma J., Wierenga P. and

Vachand G. (1977) Soil Sci. Soc. Am. J. 41, 285. Hayhoe H. N. (1978) Wat. Resour. Res. 14(1), 97. Hirrel D. (1980) Fundamentals of Soil Physics. Academic

Press, New York. Huyakorn P. S., Thomas S. D. and Thompson B. M. (1984)

War. Resour. Res. 20(8), 1099. Huyakorn P. S., Mercer J. W. and Ward D. S. (1985) Wat.

Resour. Res. 21(3), 346. Huyakorn P. S., Springer E. P., Guvanasen V. and Wads-

worth T. D. (1986) Wat. Resour. Res. 22(13), 1790. Jayawardena H. W. and Kaluarachchi J. J. (1986) J. Hydrol.

84, 231. Kaluarachchi J. J. and Parker J. C. (1987) J. Hydrol. 90,

269. Narasimhan T. N. and Witherspoon P. A. (1976) Wat.

Resour. Res. 12(1), 57. Nielsen D. R., van Genuchten M. Th. and Biggar J. W.

(1986) War. Resour. Res. 22(9), 89S. Onishi Y., Serne R. J., Arnold E. M. Cowan C. E. and

Thompson F. L. (1981) Report NUREG/CR-1322. Paniconi C., Aldama A. A. and Wood E. F. (1991) War.

Resour. Res. 27(6), 1147. Pickens J. F. and Gillham R. W. (1980) War. Resour. Res.

16(6), 1071. Willis R. and Yeh W. W-G. (1987) Groundwater System

Planning and Management. Prentice-Hall, Englewood Cliffs, NJ.

Yeh G. T. and Strand R. H. (1982) Report ORNL/TM- 8104.

APPENDIX

The soil-water characteristic functions of sandy soil used by Antonopoulos and Papazafiriou (1990) are:

Solution of vertical water flow and mass transport equations 99

and

ct O(h) = (0s--0r) ~ + 0 r

A K(h) = gsath+lhlY.

Table A 1. The values of the physical and fitted parameters

0s 0r Ks,, ~ /~ 7 A

0.29 0.075 33.9 1.611E6 3.69 4.74 1.175E6

The functions of the dispersion coefficient in the first application are :

D = 0.0306cm2/h forq/O ~ 0.52cm/h, D = 0 . 0 6 5 (q/O)llScm2/h forq/O>O.52cm/h.

Table A2. The soil physical and dispersion related parameters in the second application

Parameter Value

Soil bulk density 1.6 g/cm 3 Molecular diffusion coefficient 0.0018 cm2/h Tortuosity factor 0.7 Local dispersivity 0.001 cm

ANE 20:2-B

a mass conservative numerical solution of vertical water flow and mass transport equations in...

Documents