[deguan wang] mathematical methods for surface and(bookos.org)

Mathematical Methods for Surface

and Subsurface Hydrosystems

Series in Contemporary Applied Mathematics CAM

Honorary Editor: Chao-Hao Gu (Fudun University) Editors: P. G. Ciarlet (City University ofHong Kong),

Tatsien Li (Fudun University)

1.

2.

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4.

5.

6.

7.

Mathematical Finance - Theory and Practice

New Advances in Computational Fluid Dynamics - Theory, Methods and Applications

Actuarial Science - Theory and Practice

Mathematical Problems in Environmental Science and Engineering

Ginzburg-Landau Vortices

Frontiers and Prospects of Contemporary Applied Mathmetics

Mathematical Methods for Surface and Subsurface H yd rosyst e ms

(Eds. Yong Jiongmin, Rama Cont)

(Eds. F. Dubois, Wu Huamo)

(Eds. Hanji Shang, Main Tosseti)

(Eds. Alexandre Ern, Liu Weiping)

(Eds. HdimBrezis, Tatsien Li)

(Eds. Tatsien Li, Pingwen Zhang)

(Eds. Deguan Wang, Christian Duquennoi, Alexandre Ern)

Series in Contemporary Applied Mathematics CAM 7

M ~ t hematica M i t h o d s for Surface

editors

Deguan Wang Hohai University in Nanjing, China

Christian Duquennoi Alexandre Ern

CERMlCS ENPC, France

Higher Education Press

\: World Scientific N E W JERSEY * LONDON * SINGAPORE * B E l J l N G * S H A N G H A I * HONG KONG * TAIPEI CHENNAI

and Subsurface Hydrosystems

Deguan Wang Christian Duquennoi College of Environmental Science and Engineering Hohai University BP 44 1 Xikang Road Nanjing, 210098, France China

Le centre d’ Antony Parc de Tourvoie

92163 Antony Cedex,

H*aEG El (CIP) BtE &&a~-F7kE%@@J + ti%+&%*&=

Mathematical Methods for Surface and Subsurface Hydrosystems / ?%@’$%, <&) $k%% (Duquennoi,

C.), <a) (Ern, A.) %. -$& : $6%&@8 E$k, 2006.1 1

ISBN 7-04-020256-5

I.&. . . II.@E. . .@& . .@B. . . III.@%% ~ ~ - ~ ~ - ~ ~ 7 ~ - 7 ~ ~ ~ ~ - ~ ~ ~ ~ ~ ~ ~

3 c 0 ~ % t ; k ~ - ~ H - & ~ 7 ~ - 7 ~ ~ ~ ~ - ~ ~ ~ @%!!-%* IV.X52

Alexandre Ern Ecole nationale des ponts et chaussCes 6 et 8 avenue Blake Pascal 77455 Marne-la-VallCe, France

+BE$@#% CIP %%@? (2006) % 124198 3 Copyright @ 2006 by Higher Education Press 4 Dewai Dajie, Beijing 10001 1, P. R. China, and World Scientific Publishing Co Pte Ltd 5 Toh Tuch Link, Singapore 596224

All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission in writing from the Publisher.

ISBN 7-04-020256-5

Printed in P. R. China

Preface

The ISFMA Symposium on Mathematical Models for Surface and Subsurface Hydrosystems was held on September 13-17, 2004 at Hohai University, Nanjing, China. With the increasing awareness of the heavy burden placed on environmental resources and the need of industry and public institutions to cope with more stringent regulations, the scope of the Symposium was to focus on some specific, but very important, environmental problems, namely surface and subsurface hydrosystems. The purpose was to present state-of-the-art techniques to model such systems, to promote the exchange of scientific ideas between French and Chinese experts, and to foster new collaborations between France and China in this field. Approximately 70 participants, including five French representatives attended the Symposium.

The activities of the Symposium included five 3-hour keynote lectures elaborating from the basics to recent advances in hydrosystem modeling and several contributed presentations dealing with more specific problems. This volume collects the material presented in the keynote lectures and some selected contributed lectures. The topics covered include mixed finite element method, finite volume formulation, sharp front modeling, biological process modeling, red tide simulation, and contaminant transfer in coastal waters. As such, this volume should be useful to graduate students, post-graduate fellows and researchers both in applied mathematics and in environmental engineering.

As organizers of this Symposium, we would like to express our grati- tude to various institutions for their supports: National Nature Science Foundation of China, Mathematical Center of Ministry of Education of China, Hohai University, French Embassy in Beijing, Consulate General of France in Shanghai, ISFMA (Institut Sino-Franqais de Mathkmatiques Appliquhes) and SOGREAH. We also thank all the lecturers and participants for their contributions. Our deepest appreciation goes to Professor Li Tatsien for his support in launching this Symposium. Special thanks also to Matthieu Jouan for his instrumental help in the organization.

Deguan Wang, Christian Duquennoi, and Alexandre Ern October 2005

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Contents

Preface

Series Talks

P. Ackerer, A. Younes: A Finite Volume Formulation of the Mixed Finite Element Method for Triangular Elements ......... 1

Alexandre Ern: Finite Element Modeling of Hydrosystems with Fully Saturated, Variably Saturated, and Overland Flows ...... 19

Patrick Goblet: Sharp Front Modeling ............................ 60 Catherine Gourla y, Marie- Hk lk ne, Tusseau- Vuillemin:

Numerical Modeling of Biological Processes: Specificities, Difficulties and Challenges .................................... 75

Deguan Wang: Ecological Simulation of Red Tides in Shallow Sea Area ...................................................... 99

Ling Li: Subsurface Pathways of Contaminants to Coastal Waters: Effects of Oceanic Oscillations ....................... 126

Invited Talks

Tingfang Wang, Sixun Huang, Huadong DU, Gui Zhang: Studies on Retrieval of the Initial Values and Diffusion Coefficient of

Water Pollutant Advection and Diffusion Process ............ 174 Jing Chen, Zhifang Zhou: Application of Tabu Search Method

to the Parameters of Groundwater Simulation Models ........ 191 Xiaomin Xu, Deguan Wang: Several Problems in River

Networks Hydraulic Mathematics Model ..................... 201

viii Contents

Jue Yang, Deguan Wang, Ying Zhang: Study on the Character of

Equilibrium Point and Its Impact on the Changing Rate of Phytoplankton Concentration Using a Simple Nutrient- Phytoplankton Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Jae Zhou, Deguan Wang, Haiping Jiang, Xijun Lai: A Numerical Simulation of Thermal Discharge into Tidal Estuary with FVM 223

1

A Finite Volume Formulation of the Mixed Finite Element Method for Triangular

Elements

P. Ackerer, A. Younes IMFS UMR 7507 CNRS - ULP Strasbourg

2, *rue Boussingault 67000 STRASBO URG FRANCE

1 Introduction Numerous mathematical models are based on conservation principles and constitutive laws, which are formulated by,

dU dt

s-+v.q= f

where s is a storage coefficient, K is the flux related parameter and q is the flux of the associated state variable u. Equation (1.1) states for the conservation principle and (1.2) states for the constitutive law like Fourier’s law (u is the temperature), Fick’s law (u is the concentration of a solute), Ohm’s law (u is the electric potential) or Darcy’s law (u is the hydraulic head). The associated initial and boundary conditions are of Dirichlet or Neumann type,

u(x, 0) = uo(x) X € R

u(z, t ) = 91b, t ) (-KVu) . nan = g 2 ( z , t )

(z E dR1,t > 0 ) (z E dR2, t > 0)

(1.3)

where 0 is a bounded, polygonal open set of R2, 80’ and dR2 are parti- tions of the boundary dR of R corresponding to Dirichlet and Neumann boundary conditions and nan the unit outward normal to the boundary dR.

This system is often solved by finite volumes (FV) or finite elements (FE) methods of lower order (see LeVeque [l] and Ern & Guermond [2] among others). FV ensures an exact mass balance over each element and continuous fluxes across common element boundaries. FE ensures an exact mass balance on a dual mesh but leads to discontinuous fluxes at common elements edges. However, finite element is considered as

2 P. Ackerer, A. Younes

more flexible because of its high capacity of discretizing domains with complex geometry.

The mixed finite element method (MFE) keeps the advantages of both methods: accurate mass balance at the element level, continuity of the flux from one element to its neighbors one and mesh flexibility. Moreover, the method solves simultaneously the state variable and its gradient with the same order of accuracy (Babuska et al. [3], Brezzi & Fortin [4], Girault & Raviart [5] and Raviart & Thomas [S]). Therefore, MFE has received a growing attention and some numerical experiments showed the superiority of the mixed finite element method with regard to the other classic methods (Darlow et al. [7], Durlofsky [8], Kaaschieter [9] and Mod et al. [lo]).

However, its implementation leads to a system matrix with significantly more unknowns than FV and FE methods. When the lowest-order Raviart-Thomas space [6, 111 is used, which is very often the case, the resolution of (1.1) and (1.2) leads to a system with one scalar unknown per edge for the hybrid formulation of the MFE [6, 111.

Attempts to reduce the number of unknowns have been investigated by various authors. For rectangular meshes, mixed finite elements of lowest order reduce to the standard cell-centered finite volume method [ll] provided that numerical integration is used. Baranger et al. [12] provides similar results for triangles and Cordes & Kinzelbach [13] showed the equivalence between mixed finite element and finite volumes without any numerical integration. However, such equivalence is restricted to steady state and without sink/source terms inside the domain. Moreover, the mixed finite element method does not require a Delaunay triangulation [9] unlike a finite volume scheme.

We present here an alternative formulation of the MFE which leads to a system matrix with only one unknown per element without any approximation. In the first part, the steady state formulation is derived in details for the MFE, FV and the alternative formulation. In the second part, the main results concerning the general formulation for the elliptic PDE (steady state) will be described. The detailed developments can be found in Younes et al. [14, 151 and Chavent et al. [16]. The last part is dedicated to the parabolic PDE.

2 Resolution of the elliptic case with a scalar flux related parameter

Assuming that the system is in equilibrium, the storage term in (1.1) vanishes. The system of equations (1.1) and (1.2) leads to an elliptic

A Finite Volume Formulation of . . . 3

partial differential equation. The finite volume formuIation of yields, 3

2.1

In the lowest-order MFE formulation for triangular elements, the flux is approximated with vector basis functions that are piecewise linear along both coordinate directions. In any point inside element A, qA is approximated by (e.g. [9]):

The mixed finite element formulation

3

i=l

where Qi are the fluxes across the element edges Ai taken positive outwards and wi(L-') are the three vectorial basis functions for the element A (Figure 2.1) defined by:

l i f i = j

4 1 A { O i f i f j wi'nj = (2.3)

For a triangular element, the vectorial basis function is given by:

where (xi, yi) are the coordinates of the vertices of A and IAl is its area. ,~ In addition, they satisfy

1 (2-5) v.w.--

- I4 and, on the edge Aj,

where lAil is the length of the edge Ai.

flux's law (1.2) is written in a variational form Using properties (2.5) and (2.6) of the vectorial basis functions, the

/ ( K - ' q A ) . wi = -/(VU). wi A A


0.6' f f f 0 .6 - , I t I 0.5.. I t I 1.

0.4 - ' f f f f

0.3' 4 fl ,f r' f f

0.2' ' 1 # x I -T - - o.z-, t 4 I 4 t I 4

Vector W ; Vector w;'

0.7 I, + \

0.5. J - - J 0.5;- * + N \

0.4- 7 J J z J o,41- k I + \ \

0,J.T J J J J J J . 3 J

F l u (Qi= 2.0, Qj= -1.0, Q ) = -1) Vector w i

Figure 2.1 Vectorial basis function and related flux.

which can be written as

1 with BC = -s W? . wf, U A is the average value of the state variable

over A and uf is the average value of the state variable on element edge Ai. K A is a scalar and represents the average value of K on element A.

We define rij as the edge vector from node i toward node j and Lij as its length (Lij = IlrijII). As shown by Cordes & Kinzelbach [13],

applying the scalar product r i j r i k = -(L:j + L:k - L;,) leads to the

K A E

1 2

A Finite Volume Formulation o f . . . 5

following relation:

3

C B$ = (T$ + ~ ; k + r;i)/48KAIAI = L (2.9) j=1

The previous properties of the discretized flux law is used to build a system of equations with average edge value as unknown. The system of equations (2.8) gives

3 3

i=l i= 1

Equation (2.10) is inserted in (2.1), which leads to:

(2.11)

System (2.8) is inverted and u is replaced by the previous formulation (2.111,

T ~ ~ C [ T ~ ~ U ~ A A r k i u j + ~iju;] + 2 QA (2.12) K A Q4 = -~

a IAl 3 Using the continuity of the fluxes between two adjacent elements A and say B (Figure 2.2)

Q ~ + Q ; = o (2.13) leads then to the equation:

For continuity reasons, we have

(2.15) ui = ui

This equation is written for each edge of the mesh which is not a Dirichlet type boundary. If a Neuman boundary conditions is applied on edge i, (2.14) becomes:

A B

Equation (2.14) is the discretized form of equations (1.1) and (1.2) using the mixed finite element .method in its hybrid form.


Figure 2.2 Triangular element A and its three neighbors.

2.2 The finite volume formulation

The main idea of the finite volume formulation consists in defining the flux by

Qf = -KA(u$ - u i ) h (2.17)

where ug is the value of the state variable at the circumcenter of A, LA, is the length of the edge i, and L& is the distance from edge i to the circumcenter of the element A (Figure 2.3).

Writing flux and variable continuity at the common edge of element A and one of its neighbors noted B yields

L& 8

(2.18)

and there fore,

B ( ~ 2 - uC) = KAB(U$ - u:) (2.19)

where KAB is the harmonic mean of the flux related parameter multiplied by the length of the edge and is called the stiffness coefficient.

A Finite Volume Formulation of .. 7

Figure 2.3 A triangular finite volume and its neighbor.

The finite volume discretization of equations (2.1) and (2.2) is obtained by plugging (2.19) into (2.1):

K A B ( U c A - ug) + K A C ( U c A C - U C ) + K A D ( U 2 - ug) = &S (2.20)

This equation is written for each element. The finite volume requires significantly less unknowns than the mixed formulation (one per element against one per edge) and leads to a more sparse system matrix (4 non zero values per line against 5 ) .

2.3 The main idea of the reformulation is first to define a linear interpolate of the state variable by,

The re-formulation of the mixed finite element

3

U A = C7r$f (2.21) i=l

where U is the value of the state variable somewhere (in or outside the element A, not necessarily the average value over the element or the value at the circumcenter of the element), and second, to use a very generic finite volume formulation of the flux,

Qf = c t ( U A - uf) + rt (2.22)

Building an equation with unknowns U is then straightforward. The continuity of the fluxes between two adjacent elements and of the state variable on the common edge yields:

(2.23)


and therefore JAUA + [ B U B 'Yf + 'Y?

EA + t$ JA + s$ ui = + (2.24)

Equation (2.24) is plugged in equation (2.22) which leads to

Equation (2.25) is then plugged in the discretized mass balance equation, which leads to an equation with one unknown per element, if, of course, the values of 7rA, Jt and r t can be determined.

Replacing (2.21) in (2.22) and comparing with (2.12) allows the identification of these coefficients,

(2.26)

and the discretized form of (1.1) and (1.2) is then

r"? Ef + E?

Note that = KAB and therefore, the system equation (2.27)

of the mixed reformulation differs from the system equation of the finite volume formulation (equation (2.20)) only by the sink/source term. Without sink/source terms, both formulations are identical. Moreover, the variable U is then the state variable at the circumcenter since, from equations (2.26) and (2.21):

1 u=- [(TikTjk)(TijTkj)ui + (rijTik)(TikTjk)uj 4142

+ ( ~ i j r i k ) ( ~ i j ~ k j ) ~ k ] = U$ (2.28)

With sink/source terms, both formulations are different, except for equilateral triangles and homogeneous domain. Moreover, in that case, the velocity derived from the MFE approach varies linearly and therefore, the linear interpolation of the state variable is no more valid.


3 General 2D formulation for the elliptic case

We treat here the case where K is a full tensor. With the FV method, the computation of accurate fluxes with a full parameter tensor is a difficult issue, especially for discontinuous coefficient. For these methods, recent developments have been done to improve the flux computation by using locally additional constraints on the continuity of the state variable and fluxes [17-201.

With MFE method, the case of full parameter tensor is treated in an elegant way leading to a system with as many unknowns as the total number of edges. Reduction of the number of unknowns can be obtained for rectangular meshes when using appropriate quadrature rules with a variant of the MFE method, the “expanded mixed method” [21,22]. For general geometry, enhanced cell-centered finite difference method was obtained from a quadrature approximation of the expanded mixed method [23]. This method is improved by adding Lagrange multiplier for non smooth meshes or abrupt changes in K [23].

The parameter tensor K is generally symmetric [24]. It commonly arises from a rotation of the locally diagonal tensor from its principal axes with respect to the computational grid and is therefore always symmetric and positive definite.

K A is defined by

The principal components of K A are constant and positive over each element A, therefore

det(KA) = k$k t - (k$y)2 > 0 ( 3 4

3.1

We define now lij by l i j = r;(KA)-lrij where (KA)-’ is the parameter tensor defined over element A. The variational form of equation (1.2) leads now to:

The mixed finite element formulation


Written in a matrix form yields

3

BAQA 23 3 = uA - ut with B$ = / z u ~ ~ ~ ( I C ~ ) - ' W ~ (3.4) j=1 A

The matrix B is given by

with

where L A can be seen as the inverse of the parameter tensor scaled by the shape of the element. Therefore, we obtain the same equations than and (2.11), i.e.

,

3 3

L A ~ Q i = 3 u A - - ~ u t and i= 1 i= 1

The system (3.4) is inverted and (3.7) is used to obtain:

The final system of equations is obtained using continuity properties of the flux and state variable, which yields


and therefore, the discretized equation of system (1.1) and (1.2) is

=-+- 0% Q? 3 3

(3.10)

This equation is slightly modified when the edge belongs to the domain boundary.

3.2 The corresponding re-formulation

The same development as for the case of a scalar parameter K is used (see equations (2.21) and (2.22)). The coefficients .rrt, and 7 t are

Replacing this last relation in the balance equation for an element A surrounded by three elements B, C and D (Figure 2.2) leads to the equation

(3.12)


Equation (3.12) differs from equation (2.27) by the value of the stiffness coefficient. Notice that for both formulations, the discrete equation is obtained without any assumption on the way of approximating gradients or equivalent flux related parameters.

4 The parabolic case

The finite volume formulation of the balance equation (1.1) is now

We present here the development for K scalar parameter K .

4.1 The standard mixte formulation

The discretization of the flux equation is described in Section 2.1. Equa- tion (2.10) is inserted in the balance equation (4.1) which leads to

where S = 1AIL& and F = LQ,. Inverting equation (2.8) leads to:

where L l l = -IAIL-'/KA. Equation (4.2) is plugged in (4.3) which gives

KAb Q~ = C a i j u j - -

3

j=1 I4 (4.4)


where

K A I4 a33 = - - (T12T12 - r )

(4.5)

with

( F + Sun-l) (4.6) L11 LTl l c l c=--- and b = -

3 s+3 s+3 Again, the continuity of the fluxes and the average edge value of the

state variable is used to build the system of equations.

4.2 The same approach is used as in Section 2 but the flux formulation is modified by

The re-formulation of the MFE

Q i = <i(U - Piui) + ~i (4.7) By plugging (2.21) in (4.7) and comparing with the MFE scheme

(4.4), the coefficients ~ i , n-i, ti, and Pi are:

(4-9)


(4.10)

and (4.11)

K A

I4 71 = 7 2 = 7 3 = --b

where c and b are defined by (4.6). The 9 equations which allows the determination of the coefficients

q,&, and pi are not linearly independent. One parameter has been fixed a priori (PI) and taken constant over the domain [14,16].

The continuity of the fluxes and the state variable described by (3.9) is used to define up,

(4.12)

which is then inserted in (2.21). The discretized form of parabolic PDE is then

(4.13)

5 Comments and conclusion The mixed hybrid finite element method used for solving parabolic or elliptic PDE on triangular meshes can be reformulated in a finite volume


way by defining a new variable per element. This re-formulation exists for any kind of triangulation and for a full parameter tensor K . The reduction of the number of unknowns (one per edge to one per element) can be attractive in term of CPU costs. However, the attractivity of the new formulation depends on the properties of the system matrix, which is in all cases, more sparse than with the standard mixed method (4 non zero values per row for the new formulation, 5 per row for the standard).

For elliptic PDE, the system matrix is always symmetric but not necessarily positive definite. It depends on the shape of the triangle, the inverse of the parameter tensor and the discontinuity of the parameter from one element to the adjacent one. This can be checked with the following criterion [15, 161:

or, for a scalar flux related parameter,

> O 2

K A KD + cot 6A+D cot 6D-A

(5-2)

For homogenous domains, this condition is equivalent to the Delaunay- criterion [12].

Note that for the case of a uniform, anisotropic parameter tensor K, a transformation of coordinates can be performed leading to a scalar parameter K in the transformed space. In this case, the criterion for positive-definiteness of the resulting system of equations would be equivalent to the Delaunay-criterion of the grid in the transformed space [25].

Because the system matrix is not the same for both formulations, the accuracy of the solution might not be the same. The conditioning of the system matrix is not the same for both formulations, especially when one angle of an element tends to 7r/2 (see equation (5.2)).

For parabolic PDE, the coefficient matrix of the re-formulation is unsymmetrical and the use of an accurate solver is then an important issue.

Numerical experiments [14-161 show that the re-formulation reduces significantly the CPU time without significant loss in accuracy. This reduction of the CPU is less important for parabolic PDE but the new formulation is still more efficient.


Finally, the extension to a general 3-D tetrahedral discretization is not possible [15].

References

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[2] Ern A, Guermond J L. Theory and practice of finite elements. Vol. 159 of Applied Mathematical Series, Springer-Verlag, New York, 2004.

[3] Babuska I, Oden J T, Lee J L. Mixed-hybrid finite element method approximations of second-order elliptic boundary value problems. Part I, Comput. Methods Appl. Mech. Eng. 11, 175, 1977.

[4] Brezzi F, Fortin M. Mixed and hybrid finite element methods. Springer, Berlin, 1991.

[5] Girault V, Raviart P A. Finite element methods for Navier-Stokes equations: theory and algorithms, in Lecture Notes in Computa- tional Mathematics, 1986. Vol. 5 (Springer-Verlag, Berlin).

[6] Raviart P A, Thomas J M. A mixed finite element method for second order elliptic problems, mathematical aspects of the finite element method, Lectures Notes in Mathematics, Springer Verlag, New York, 1977; 606: 292-315.

[7] Darlow B L, Ewing R E, Wheeler M F. Mixed finite elements methods for miscible displacement problems in porous media, SPE 10501, SOC. Pet. Eng. J. 24 (1984), 397-398.

[8] Durlofsky L J. Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities, Water Re- sour. Res. 30, (1994) 965-973.

[9] Kaasschieter E F, Huijben A J M. Mixed-hybrid finite elements and streamline computation for the potential flow problem, Numerical methods for Partial Differential Equations, 1992; 8, 221-266.

[lo] Mos6 R, Siege1 P, Ackerer Ph, Chavent G. Application of the mixed hybrid finite element approximation in a groundwater flow model: luxury or necessity? Water Resour. Res., 30, (1994) 3001-3012.

[ll] Chavent G, Roberts J E. A unified physical presentation of mixed, mixed hybrid finite elements and standard finite difference approximations for the determination of velocities in waterflow problems, Adv. in Water Resources, 14 (1991), 329-348.

bridge Univ. Press, 2002.


[12] Baranger 3, Maitre J F, Oudin F. Application de la thkorie des 616ments finis mixtes & l'ktude d'une classe de schkmas aux volumes- diffkrences finis pour les probl6mes elliptiques, C. R. Acad. Sci. Paris, 1994; 319, Skrie I : 401-404.

[13] Cordes C, Kinzelbach W. Comment on <<Application of the mixed hybrid finite approximation in a groundwater flow model : luxury or necessity ?> by Mose et al., Water Resources Research, 1996;

[14] Younes A, Ackerer Ph, Mose R, Chavent G. A new formulation of the mixed finite element method for solving elliptic and parabolic PDE with triangular elements. Journal of Computational Physics,

[15] Younes A, Ackerer Ph, Chavent G. From mixed finite elements to finite volumes for elliptic PDEs in 2 and 3 dimensions. International Journal of Numerical Methods in Engineering, 2004, 59 (3), 365- 388.

[16] Chavent G, Younes A, Ackerer Ph. On the finite volume reform& lation of the mixed finite element method for elliptic and parabolic PDE on triangles. Computer Methods in Applied Mechanics and Engineering, 2002, 192(5-6), 655-682.

[17] Lee S H, Jenny P, Tchelepi H A. A finite-volume method with hex- ahedral multiblock grids for modeling flow in porous media, Com- putational Geosciences, 2002; 6: 353-379.

[18] Putti M, Cordes C. Finite element approximation of the diffusion operator on tetrahedra, SIAM J. Sci. Comput., 1998; 19 (4): 1154- 1168.

[19] Edwards M. M-Matrix flux splitting for general full tensor discretization operators on structured and unstructured grids, Journal of Computational Physics, 2000; 160: 1-28.

[20] Edwards M. Unstructured, Control-Volume Distributed, Full- Tensor Finite-Volume Schemes with Flow Based Grids, Computa- tional Geosciences, 2002; 6: 433-452.

[21] Arbogast T, Wheeler M F, Yotov I. Mixed finite elements for elliptic problems with tensor coefficients as cell centered finite differences, SIAM J. Numer. Anal., 1997; 34(2): 828-852.

[22] Wheeler M F, Roberson K R, Chilakapati A. Three dimensional bioremediation modeling in heterogeneous porous media, in Com- putational Methods in Water Resources IX, V01.2, Mathematical Modeling in Water Resources, Russell T F, Ewing R E, Brebbia C A, Gray W G, and Pindar G F, eds., Computational Mechanics Publications, Sout hampton, U. K., 1992; 299-3 15.

32(6): 1905-1909.

1999; 149: 148-167.


[23] Arbogast T, Dawson C N, Keenan P T, Wheeler M F, Yotov I. Enhanced cell-centered finite differences for elliptic equations on general geometry, SIAM J. Sci. Comput., 1998; 19 (2): 404-425.

[24] Durlofsky L J. Numerical calculation of equivalent grid block permeability tensors for heterogeneous media, Water Resources Research

[25] Russel T F, Wheeler M F. Finite element and finite difference methods for continuous flows in porous media, in the Mathematics of Reservoir Simulation, edited by Ewing R E, SIAM, Philadelphia, 1984, 35.

1991; 17(5): 699-708.

19

Finite Element Modeling of Hydrosystems with Fully Saturated, Variably Saturated,

and Overland Flows*

Alexandre Ern Cermics, Ecole nationale des ponts et chausse'es

6 et 8 avenue Blaise Pascal, 77455 Marne-la-Valle'e, France E-mail: ern@cermics. enpc.fr

1 Introduction The prediction of overland flow routing in hillslopes is of paramount importance during flood episodes. Owing to the complexity of the problem, in situ experiments are often difficult to perform and general conclusions are difficult to draw. This has prompted the development of numerical tools with the capability to analyze water transfer in virtual hillslopes subjected to various operating conditions. This approach has experi- enced a vigorous development over the past decades. Many advances have been reported in the specialized literature, concerning both the development of models describing the hydrological behavior of hillslopes through partial differential equations (PDEs) and appropriate boundary conditions (BCs) and the design of suitable numerical methods able to tackle such PDEs and BCs. Recent work in this direction includes, among others, Ogden & Watts [l], Cloke et al. [2], and Weiler & Mc- Donne11 [3].

An important aspect of hillslope hydrology during heavy rainfall episodes is the fact that subsurface flows can saturate the soil in various regions near the surface and, therefore, contribute to the production of overland flow. There is therefore a strong motivation to improve our understanding of subsurface flow dynamics in variably saturated porous media and, in particular, of the role played by the hydraulic state of the soil in storm hydrographs. As part of an ongoing research effort in that direction, the Cooperative Research Action DYNAS (Dynamics

*Part of this work has been carried in collaboration with H. Beaugendre (Cer- mics/Enpc), B. Augeard, I. Ginzburg, C. Kao (Cemagref), and T. Esclaffer, E. Gaume (Cereve/Enpc) in the framework of the Cooperative Research Action DY- NAS sponsored by INRIA.

20 Alexandre Ern

of Shallow water-tables) has been sponsored by INRIA since 2003; see http: //www-rocq. inria. fr/estime/DYNAS for a detailed description of the project. This paper discusses numerical tools, essentially based on finite element techniques, to discretize various PDEs relevant to hillslope hydrology models, and it will also present some selected results illustrating the interaction of subsurface flow dynamics with the ground surface.

Section 2 presents various finite element discretizations of Darcy’s equations. These equations can be used to model steady subsurface flows in fully saturated porous media. The emphasis in this section is set upon the fact that several finite element formulations can be designed, each leading to different physical properties at the discrete level. It is assumed that the reader is familiar with the basics of finite element methods; see, e.g., Ciarlet [4], Brenner & Scott [5], Quarteroni & Valli [6], or Ern & Guermond [7] for an introduction to this topic.

Section 3 presents a model often considered to describe unsteady flows in variably saturated porous media, namely Richards’ equation. Richards’ equation requires two algebraic relations to model the hydrodynamic behavior of the soil. These relations, which relate the unsaturated hydraulic conductivity, the volumetric water content, and the hydraulic head, are discussed in this section. Section 3 also addresses in some detail the case where the water-table (i.e., that part of the soil where saturation takes place) reaches the top boundary of the hillslope, i.e., the ground surface. Since the water-table position is an unknown of the problem, its intersection with the ground surface is a priori unknown. This leads to an unsteady obstacle-type problem.

Section 4 discusses the modeling of overland flow on the ground surface. Two simplified forms of the Saint-Venant equations are considered, namely the diffusive wave approximation and the kinematic wave approximation. Numerical algorithms coupling the overland flow with subsurface flows governed by Richards’ equation are also presented.

Finally, Section 5 presents some selected results on hillslope behavior during heavy rainfall episodes. It discusses the impact of phenomenological parameters on model predictions and compares two approaches to simulate the subsurface flow, one neglecting the height of the overland flow in the specification of BCs for Richards’ equation and the other taking it into account.

2 Finite elements for fully saturated steady flows

The goal of this section is to present various finite element approximations of the equations governing steady flows in fully saturated porous

Finite Element Modeling of Hydrosystems with . . . 21

media, the so-called Darcy’s equations. We stress the fact that several mathematical formulations of the problem at the continuous level can be envisaged, each one leading to a discrete setting with different features. We study the so-called (pressure) primal formulation as well as several (pressure-velocity) mixed formulations.

2.1 Darcy’s equations Let R denote a fully saturated porous medium with boundary dR. Let u be the flow velocity in field R and let p be the hydraulic head (or the pressure up to an appropriate rescaling) . Darcy’s phenomenological law states that

Here, K denotes the hydraulic conductivity (or permeability) tensor. Furthermore, mass conservation implies that

u = - K V p i n n . (1)

V . u = f i n R , (2) where f represents the volumetric water sources or sinks in R.

To enforce boundary conditions, we assume that the boundary 80 can be partitioned into 801 U 8R2 in such a way that the hydraulic head is imposed on dR1 and that a normal velocity is prescribed on 8R2. Using Darcy’s phenomenological law, the boundary conditions can be recast as follows:

p = g 1 o n a a 1 , (3) -n . K V p = g2 on 8 0 2 , (4)

where n is the outward normal to R and where g1 and 92 are prescribed functions on 8R1 and 2 2 2 , respectively. In other words, mixed Dirichlet- Neumann conditions can be imposed on the hydraulic head.

To pinpoint the key mathematical points, we shall make, in this section, the following simplifying assumptions:

0 The porous medium is homogeneous and isotropic; in other words, the hydraulic conductivity tensor takes the form K = 5I where 5 is a positive constant in R and I is the identity tensor.

0 Homogeneous Dirichlet boundary conditions are imposed throughout dR.

*

As a result, we shall seek the solution ( u , p ) to

V . u = f i n R ,

k - l u + ~ p = ~ i n R , p = O o n 8 R .

Henceforth, we assume that the data f is in L2(R).

22 Alexandre Ern

2.2 The pressure primal formulation

Taking the divergence of (6) and using (5) to eliminate V . u yields

- k A p = f i n o , p = O ondfl

This is the well-known Poisson problem whose weak formulation is

Seek p E Ht(R2) such that (10)

k V p * V q = f q , V q E Hi(R). (i, s, As an immediate consequence of the Lax-Milgram Lemma, we infer that the problem (10) is well-posed, i.e., it admits one and only one solution. Henceforth, this problem is referred to as the pressure primal formulation or simply the pressure formulation.

2.2.1 Conforming approximation

Let l h be a simplicia1 mesh of the domain R (the elements of '& are triangles in dimension two and tetrahedra in dimension three). To simplify, assume that the mesh is affine and that it covers f2 exactly, i.e., R is a polygon in two dimensions or a polyhedron in three dimensions. In the sequel, dh denotes the set of faces of the mesh (edges in two dimensions).

Consider the approximation space

P2.o = { P h E Co@) n Hi(R); VK E %, PhlK E pi } , (11)

consisting of those discrete pressure fields that are piecewise linear, continuous on n, and that vanish on do. The index c emphasizes the fact that P&, is conforming in &(a), i.e., that Pio c Hh(C2); the index 0 indicates that the functions in Pio vanish at the boundary. The conforming approximation of the primal formulation consists of the following problem:

Seek ph E P2,0 such that

Clearly, this problem is well-posed. Moreover, if the exact solution is smooth enough, namely p E H2(R) n @(a), the following error bound holds:

b - p h l l H 1 ( R ) 5 c h ~ ~ P ~ ~ H 2 ( 0 ) ~ (13)


2.2.2 Non-conforming approximation

In this section the pressure is approximated by piecewise affine functions that are continuous only at the midpoints of edges separating adjoining elements. Consider the so-called Crouzeix-Raviart finite element space

PAC,o = (2% E L2(a) ; VK E %, PhlK E va E .Ah, / [ P h ] a = o}, a

(14) where [Ph], denotes the jump of ph across the face a ( [ P h ] , = p h ( , is the face a is located at the boundary). The index nc emphasizes the fact that to a face a E Ah being affine, the condition J,[Ph]a = 0 is equivalent to the continuity of ph at the midpoint of a. Consider the discrete problem

@ Hi(s2). The restriction of a function ph in

Since ph $! Ht(R), the integral over in the left-hand side of (12) has been replaced by a sum of integrals over mesh elements.

A classical result is that the problem (15) is well-posed. Moreover, if the exact solution is smooth enough, namely p E H2(s2) n Hi(R) , the following error bound holds:

2.3

Once the pressure field has been obtained, the velocity field can be reconstructed in [L2(a2)ld by setting u = - k V p . This equation can be equivalently written in weak form by taking the scalar product with arbitrary test functions in [L2(R)ld. As a result, the problem (5)-(7) can be reformulated as follows:

Mixed formulation in [L2(R)Id x Ht(R)

Seek ( u , p ) E [L2(f2)ld x Ht(R) such that

( k - l u + Vp) . 2, = 0 , vv E [L2(O)]d , Is,

24 Alexandre Ern

or, in a more abstract form,

Seek ( u , p ) E [L2(fl)ld x H i ( 0 ) such that

a(u, w) + b(v,p) = 0 , vw E [L2((s2)ld, (18) i b(u, 4 ) = Z(q) , vq E H m 7

where

a(u, w) = s, L-lu. 21, b(v,p) = s, vp * 21, Z(q) = - s, fq. (19)

Problem (18) is one possible mized formulation of the problem (5)-(7). In (18), the mass conservation equation has been integrated by parts

' owing to the fact that u is a priori sought in a space with little regularity (u E [L2(Q)ld so that V * u is not a priori meaningful in L2(sZ)). This is at variance with Darcy's law where no integration by parts has been used. We shall see that in a discrete setting, these choices have important consequences: Darcy's law holds locally on each mesh element whereas mass conservation holds only in a weak sense.

2.3.1 Conforming approximation

The approximate problem (12) can be used to construct an approximation to the mixed formulation (18). The discrete velocity is sought in the space

[pold = {uh E [L2(0)ld;QK E '&, u h l ~ E [po ld ) , (20)

consisting of those discrete velocity fields that are piecewise constant on a. Consider the discrete problem

( Seek (uh ,ph ) E [Po]' x P2,0 such that

One readily verifies that this problem is well-posed. Moreover, if ph E PCl,o solves (12), the pair (-lcVph,ph) solves (21). Conversely, if (uh ,ph ) E [Pold x P:,o solves (21), u h = -kVph and ph solves (12). Finally, if the exact solution is smooth enough, namely u E [H1(Q)ld and p E H 2 (0) n H t (a), the following error bound holds:

IIu - uhll[L2(n)]d + IIP - PhllHl(n) I c h (ll4l[H'(n)]d + IlPllH.(n)>. (22)


To simplify, we consider a two-dimensional setting. Let Ne denote the number of elements in the mesh 7 h , N u the number of edges, and N s the number of vertices. Assume that the domain 0 is simply connected (meaning that there are no holes inside 0). Owing to the Euler relations,

Ne - N a + N s = 1 , 2Na - 3Ne = N a f ,

where N a f is the number of boundary edges, the following holds:

Ne = 2Ns - N a f + 2 M 2 N s , N u = 3 N s - N a f + 1 M 3 N s , (24)

in the practical limit where N s >> N a f . Using this notation, the approximate problem (21) is a linear system of size 2Ne + N s E 5Ns. Of course, this system is not inverted in practice; instead, the discrete pressure field ph is first recovered from (12) and then the discrete velocity field is reconstructed according to uh = -JcVph. Hence, the linear system to be solved is only of size Ns. Note that the above reconstruction of the velocity field implies that Darcy's law holds locally on each mesh element.

To investigate the discrete mass conservation equation resulting from ( 2 l ) , let us introduce some additional notations. Partition Ah into Ai U A: where A; and A: denote, respectively, the set of interior and boundary edges (interior means that the edge is not included in an). Similarly, let Sh denote the set of vertices in the mesh. Partition Sh into Sk U Sf where Si and Sf denote, respectively, the set of interior and boundary vertices. For K E 55, let 1K1 be its measure, SK the set of its vertices, and d K the set of its edges. For s E S K , let nK,, be the outward normal to K on the edge opposite to s and let hK,, be the distance of s to the edge in question. Let 1~ denote the characteristic function of K and let w, E P&, be the so-called hat function taking the value 1 at s and 0 at the other mesh vertices. Let 0, be the support of w,. For an interior vertex s E Si , let A, be the set of edges containing s. Finally, for an interior edge a E A;, let la1 be its length and [uh . n], be the jump of the normal component of uh across a defined as follows: if a = K1 n K2 where K1 and K2 are two distinct mesh elements with re- spective outward normals nl and 722, [uh.n], = ( ~ h l ~ ~ ) . n l + ( u h l K , ) . n 2 .

Using the above notations, the discrete pressure and velocity fields can be written as

26 Alexandre Ern

Figure 1 0,.

A mesh element K E 7j, a vertex s E SK, and the domain

Proposition 1. The solution ( u h , p h ) to (21) satisfies the discrete mass conservation equation

and the local velocity reconstruction formula

Proof. Let a E Ah and let g be an integrable function on a. Recall that the distribution 6, @ g da is defined as follows:

where D ( n ) denotes the set of infinitely differentiable functions that are compactly supported in 0. Using this definition and classical results of distribution theory, it is readily inferred that for all uh E [Pold,

Equation (26) is then a direct consequence of the above equation by applying it to the hat functions w, for all vertices s E Si. Furthermore, (27) results from the fact that for K 6 Th, UK = - k V p h I ~ and that for s E SK, VW,IK = - ( l / h ~ , s ) n ~ , s .

Finite Element Modeling of Hydrosystems with . . 27

2.3.2 Non-conforming approximation

To construct a non-conforming approximation to the mixed formulation (18), one can proceed as in the previous section. The discrete velocity field is again sought in [Pold. Since the discrete pressure is not in H ; ( a ) , the bilinear form b has to be modified into

This leads to the approximate problem

Problem (31) is well-posed. Furthermore, if ph E solves (15), then, letting uh E [Pold be such that for all K E z, U ~ J K = - k V p h J ~ , the pair (uh ,ph ) solves (31). Conversely, if (uh ,ph ) E [Pold x P&, solves (31), then, for all K E '&, U ~ I K = - k V p h I ~ and ph solves (15). Finally, if the exact solution is smooth enough, namely u E [H1(R)ld and p E H2(R) n H;(s1), the following error bound holds:

(32) Let wa be the function in P2c,o taking the value 1 at the midpoint of

the edge a and the value 0 at the midpoint of all the other edges. Let R, be the support of w, (0, consists of the two triangles of 7 h sharing the edge a). Using the above notation, the discrete pressure and velocity fields can be written as

uh = U K 1~ and p h = p a w a . (33) KETh aEAh

Proceeding as in the proof of Proposition 1, one easily verifies the following:

Proposition 2. The solution (uh,ph) to (31) satisfies the discrete mass conservation equation

28 Alexandre Ern

and the local velocity reconstruction formula

where nKqa is the normal to the edge a pointing outwards K .

2.4

We now consider a mixed formulation in which the velocity field is sought in a space with more regularity than [L2(R)ld, namely in

Mixed formulation in H d i v x L2(R)

Hdiv = { U E [L2(R)ld, V * U E L2(R) } ,

IlUIlHdiv = l l ~ l l [ ~ 2 ( n ) l d + 11' * ~ l l ~ ' ( n ) *

(36)

(37)

equipped with the norm

Consider the following problem:

Seek (u,p) E Hdiv x L2(R) such that

a(u, V ) + b ' ( ~ , p ) = 0 , VV E H d i v , (38) b ' b , 4) = l ( 4 ) 9 v4 E L2P) >

where r

b ' ( v , p ) = - ]nPv.v . (39)

A comparison with the mixed formulation (18) is noteworthy. In (38), Darcy's law has been integrated by parts owing to the lack of regularity of the pressure ( p is sought a priori in L2(R); hence, V p is not meaningful in [L2(fl) ld) whereas the strong form of the mass conservation equation has been used. In the discrete setting, these choices will imply that mass conservation holds exactly on each mesh element while Darcy's law holds only in a weak sense.

Using classical arguments, it is shown that (38) is well-posed and that its unique solution ( u , p ) verifies (5)-(6) a.e. in R and (7) a.e. on dR. F'urthermore, one readily verifies that if ( u , p ) solves (38) and only if (u,p) solves (18).

2.4.1 The Raviart-Thomas finite element

To simplify, consider a two-dimensional setting. The idea is to seek an approximation for the velocity that is uniquely determined on each element K E 7 h by its normal fluxes across the three edges a E d K . On K , introduce the polynomial space

RTo(K) = [Po(K)I2 + (.,Y)'PO(K). (40)


This space is generated by the polynomials

where (xi, y i ) are the coordinates of the vertex i of K . The polynomials {WK,1 , WK,2, W K , ~ } satisfy many important properties. In particular, V . W K , ~ = 1/JK1 for all 1 5 i 5 3, and for all 1 5 i , j 5 3,

W K , ~ . n ~ j = &j > (42)

where Sij stands for the Kronecker symbol. Consider the approximation space

RT:c = { u h E [L2(fi2)ld; YK E %, UhlK E RTO(K)} . (43)

The index nc emphasizes the fact that RT;, is not conforming in Hdiv,

i.e., RT& @ &iv. A well-known result is the following:

Proposition 3. The approximation space RT; = RT;, n Hdiv as equal to

RT: = { u h E [L2(O2)ld; VK E %> UhlK E RTO(K);

Ya E k h , [Uh * n], = o} . (44)

Since the discrete velocity Uh E RT: is affine on each element K E %, the last condition in (44) is equivalent to the continuity of the normal component of Uh at the midpoint of the edges a E di. Hence, RT: is a vector space of dimension N u whereas RT;, is a vector space of dimension 3Ne. In other words, dim(RT:) M f dim(RT2,).

Consider the discrete problem

The degrees of freedom for the discrete velocity are the normal fluxes across the edges and the degrees of freedom for the discrete pressure are the mean-value at the mesh elements; see Figure 2.

Using standard arguments, one can verify that the problem (45) is well-posed. Moreover, if the exact solution is smooth enough, namely

E [H1(a)ld, V .u E H1(R), and p E H1(R), the following error bound holds:

30 Alexandre Ern

2

Figure 2 Degrees of freedom for the standard Raviart-Thomas approximation: normal fluxes across edges for the velocity and mean-value at elements for the pressure.

For a function f E L2(f2), let I Ih f be its L2-orthogonal projection onto Po, i.e., IIhf is the piecewise constant function such that

Using this notation, the discrete mass conservation equation resulting from (45) simply takes the form

The discrete Darcy's law takes a more complex form that will not be detailed here; on each element K E 7 h 7 the velocity can be expressed using the pressure in K and the pressure on the three triangles sharing an edge with K .

2.4.2 Hybrid finite elements

The discrete problem (45) is a linear system of size N u + Ne M 5Ns. It is possible to express explicitly the solution of this system in terms of the solution of a smaller linear system. The idea is to relax the continuity constraint of the normal velocity component across edges by introducing a Lagrange multiplier on each edge.

Introduce the vector space spanned by functions defined on the edges a E and constant on each edge, i.e.,


Consider the approximate problem

(50) Then, one can show that the above problem is well-posed. Moreover, if

(uh,ph,Xh) E RT;, x Po x A' solves (50), then uh E RT: and (uh ,ph ) solves (45).

The advantage of the discrete problem (50) is that the first two equai tions are purely local. Indeed, on each element K E 7 h , let WK be the vector formed by the three components of the velocity in the local basis of RTo(K), i.e., WK = ( W K , ~ , wK,2, W K , ~ ) ~ where

Let BK be the 3x3 matrix with entries B ~ , i j = sK W K , ~ . W K , ~ . Let AK be the vector whose three components are the three Lagrange multipliers associated with the three edges of K (the numbering of the edges is such that the ith edge of K is the edge opposite to the ith vertex). Using this notation, the discrete Darcy law can be written locally in the form

where U = (1,1, l)T. Furthermore, the local mass conservation equation takes the form

where (., .) denotes the scalar product in It3. Owing to the elementary properties of the matrix BK (see Lemma 4 below), it is readily inferred that

p ( K = i ( A K , u) -k ak-'&nKf 7 (54)

and hence,

32 Alexandre Ern

The quantity p~ (having dimension of a length) denotes the gyration radius of K defined as

P”KKl = / llrK(x)I12 dx 7 (56)

where 7 r ~ ( x ) = (z1- gK, l , . . . , xd - gK,d), ( 2 1 , . . . , xd) are the Cartesian coordinates, and QK is the barycenter of K.

The implementation of the hybrid method is as follows. On each edge, the continuity of the normal velocity component is enforced using (55) leading to a linear system of size N a where the only unknowns are the Lagrange multipliers at the edges. Once the Lagrange multipliers have been evaluated, the formulas (54) and (55) are used locally to evaluate the velocity and the pressure. To sum up, the hybrid mixed finite element method requires only the solution of a linear system of size N u M 3Ns; this means a saving of 40% with respect to the original size of problem

K

(45).

Lemma 4. Let K E 77. Then, the matrix BK is invertible with

a2 +a3 -a3

B-’ = -a3 a1 +a3 1:; ) +‘U@U, 3 1 ~ (57) K i -a2 -a1 a1 +a2

where ai = 2 c o t Q ~ , i , 1 5 i 5 3, O K , ~ being the angle at the ith vertex of K , and 1~ = $&, where p~ is the gyration radius of K defined in

(56).

2.5

The motivation for introducing the following non-standard mixed formulation is to use test functions for both the velocity and the pressure that can be localized at the element level, i.e., test functions in [L2(f2)ld and L2(1;2), respectively. Such schemes bear a close relationship with finite volume schemes and are often termed finite volume box schemes. The results presented hereafter are borrowed from [8].

The starting point of the derivation is to seek the velocity and the pressure in spaces that yield sufficient regularity to avoid integration by parts when setting the weak formulation. Consider the following mixed formulation:

Mixed formulation in Hdiv x Ht(R)

Seek ( u , p ) E Hdiv x Hi(f2) such that

U ( U , V ) + b(w,p) = 0 , VV E [L2(1;22)ld, (58) b’(u, q) = -Z(q) , vq E L2(1;2) .


Then, one can show that the above problem is well-posed. Moreover, its solution ( u , p ) solves (5)-(6) a.e. in R and (7) a.e. on dR. Finally, (u,p) solves (58) if and only if it solves (18).

The terminology non-standard is employed for the mixed formulation (58) because the space where the solution (u,p) is sought is dzfferent from the space for the test functions (v,q).

2.5.1 The discrete setting

The lowest-order finite volume box scheme consists of seeking the discrete velocity in the conforming Raviart-Thomas finite element space RT; and the discrete pressure in the non-conforming Crouaeix-Raviart finite element space Test functions for both the Darcy law and the mass conservation equation are taken to be piecewise constant. This leads to the following problem:

One can show that the problem (59) is well-posed. Moreover, if the exact solution is smooth enough, namely u E [H1(R)ld, V . u E H1(R), and p E H2(R), the following error bound holds:

5 c h (II4[H'(n)]d + IIV . UlIHl(n) + IlPllHyn)) . (60)

The discrete problem (59) is a linear system where the number of unknowns is

dim RT: + dim = N u + ( N u - N a f ) , (61)

and where the number of equations is

dim[P0Id + dim Po = ( d + 1)Ne. (62)

The equality between the number of unknowns and the number of equations results from the Euler relations (23). This equality is of course only a necessary condition for the well-posedness of (59); to prove well- posedness, a discrete inf-sup condition must also be established; see [8].

34 Alexandre Ern

2.5.2 Some properties of the finite volume box scheme

Proposition 5. The solution (uh,ph) to (59) is endowed with the following properties:

1. Local mass conservation

V K E T h , V * ' U h l ~ = n K f , (63)

or, in other words, v * uh = n h f . 2. Local velocity reconstruction:

VK E lh UhlK = -kvPhIK + i ( n K f ) r K ( z ) 7 (64)

where T K ( X ) is defined below equation (56). 3. Equivalence with the non-conforming approximation of the pressure

primal formulation: ph is the unique solution of (15) with right- hand side f replaced by & f .

Proof. (1) Equation (63) is obtained by taking qh = 1~ in the second equation of (59). (2) Since uh E RT:, the following holds:

VK E lh, U ~ I K = n ~ u h + i ( V * U ~ ) ~ K ~ K ( Z ) . (65)

Taking Vh = 1Kei where ei is the i-th vector of the canonical basis of Rd in the first equation of (59), it is inferred that I I K u ~ = - k V p h I ~ . Equation (64) then results from (63) and (65). (3) Let qh E Since V q h E [Po]', it is inferred that

and the sum of boundary terms vanishes since qh E Hence,

and uh E RT;.

(67) The proof is complete. 0

Statement (3) in Proposition 5 has important practical consequences on the optimal way to solve (59). Firstly, the non-conforming approximation of the pressure primal formulation is considered (with data IIh f


instead o f f ) ; the discrete pressure is obtained by solving a linear system of size N u M 3Ns. Secondly, the velocity field is reconstructed locally from (64). This approach yields a saving of 50% in the size of the linear system to be inverted.

2.6 Summary Table 1 summarizes the various approximations to Darcy's equations investigated in this section. The third and fourth lines in the table indicate whether Darcy's law and the mass conservation equation, respectively, are satisfied in a strong or in a weak sense.

Table 1: Finite element approximations to Darcy's equations.

primal conforming

strong weak

primal non-conforming

strong weak

[L2((n)ld x H,'Q) [ P O I d x P n l C . 0

mixed standard

weak strong

Hdiv X L2(Q) RT; x PO

mixed non-standard

strong strong

3 Finite elements for variably saturated flows This section focuses on Richards' equation to model variably saturated flows. In particular, some emphasis is set on the finite element formulation in the case where the water-table reaches the ground surface, thus leading to an obstacle-type problem for the hydraulic head.

3.1 Richards' equation In an unsaturated porous medium, the fluid phase contains water and air. The most general approach to model flows through such media is to address the two-phase problem by considering mass and momentum balance for each of the two phases; see, e.g., [9]. However, a simpler approach is feasible if the gas-phase always remains connected so that a single air pressure can be considered. This is the first important assump tion underlying Richards' equation. Note that entrapped air pockets are excluded from the present configuration.

Let 19 be the volumetric water content and let cp be the (total) hydraulic head. Set

c p = + + z , (68) where z is the vertical coordinate (oriented upwards) and + is the so- called matrix potential. This quantity is related to the difference between

36 Alexandre Ern

air pressure and water pressure in such a way that $ < 0 in unsaturated regions and $ > 0 in saturated regions. The water-table position can be located by the isoline {$ = 0).

The second assumption underlying Richards’ equation is that a generalized Darcy’s law still relates the water flow velocity to the hydraulic head gradient through the concept of relative hydraulic conductivity. In other words, it is assumed that

u = -Ic,lc,(e)vp, (69)

where k, is the hydraulic conductivity at saturation (i.e., the hydraulic conductivity pertinent for Darcy’s equation) and kT(8) is the so-called relative hydraulic conductivity that depends on the volumetric water content 8. For the sake of simplicity, the medium is assumed to be isotropic so that a single scalar hydraulic conductivity can be considered instead of a full tensor. As a result, the water conservation equation ate + V . u = f takes the form

ate - v . (klJ%(e)(v$ + e z ) ) = f 1 (70)

where e, is the unit vector oriented upwards and where f represents the volumetric water sources or sinks in Q.

To close the problem, it is necessary to assume that an algebraic relation links the volumetric water content 8 to the matrix potential $, say $ H 8($). The graph of the function $ H 8($) is often called the soil water retention curve. Equation (70) then yields a PDE where the sole dependent variable is the matrix potential $, namely,

ate($) - V . (kSkT(8($)>(W + e,)) = f . (71)

This PDE can be written in mixed form by introducing the generalized Darcy velocity u($),

(72) ate($) + v . 4.111) = f 7 { u($) + WG(~($))(V$ + e,) = 0 .

To enforce boundary conditions for (71) or (72), assume that the boundary dR can be partitioned into 801 U 8 0 2 in such a way that the hydraulic head is imposed on dQ1 and that a normal velocity is prescribed on dR2. This yields

$ = 91- x on dR1, (73) - k3kr (q$) ) (V$ + e z ) 12. = g2 on dR2, (74)

where n is the outward normal to 0. In other words, Dirichlet and (nonlinear) Neumann conditions are imposed on $. A more complex


situation where the partition dR1 udR2 is not known a priori is discussed in Section 3.4.

Finally, (71) or (72) is supplied with an initial condition specifying the initial value of the matrix potential, namely

$I = $I0 at t = 0 and in R . (75)

3.2 Soil hydrodynamic functions To close (71) or (72), two algebraic relations between 8, IC,, and $J must be specified.

In soils, the volumetric water content does not vary between 0 and 1, but between a minimum value 8, called the residual volumetric water content and a maximum value Bs called the saturated volumetric water content. Define the reduced volumetric water content

which, by definition, varies between 0 and 1. A model often considered in the engineering literature to specify the

soil water retention curve is that derived by van Genuchten in 1980 [lo]

with m = 1 - i. Equation (77) involves two parameters, a and n. It should be noted that if the exponent n is less than 2, the function $I H $($I) is only of class C1 at $I = 0 and its second-derivative explodes as $I -+ 0-, whereas for n 2 2, the function $I H g($I) is of class C2 at $I = 0. This fact can be important in numerical approaches where the volumetric water content is chosen as the main dependent variable for

A model to express the relative hydraulic conductivity has been pro- (71).

posed by Mualem in 1976 [ll],

where 1 is the pore connectivity parameter (generally set to 1 = $) and T H $(T) denotes the reciprocal function of y!~ tt $($I). Combined with (77), the Mualem relation yields

2 lc,(i?) = $+ (1 - (1 - 8 q m ) . (79)

38 Alexandre Ern

Equations (77) and (79) are referred to as the Van Genuchten-Mualem (VGM) model. In the framework of the VGM model, a soil is described by five parameters: k,, a, n, O,, and Or.

Recently, it was observed that relatively small changes in the shape of the soil water retention curve near saturation could significantly affect the numerical performance of variably saturated @ow simulations [12]. As a result, a modified form of the VGM model was proposed to account for a very small, but non-zero, minimum capillary height, $, < 0, in the soil water retention curve. This modification leads to less non-linearity in the hydraulic conductivity function near saturation and, hence, to more stable numerical solutions. The modified VGM model takes the following form:

where the parameter ,6 is defined as ,B = (1 + ( - Q $ ~ ) % ) - ~ in such a way that g($,) = 1. Furthermore, the relative hydraulic conductivity dependency on g is modified so that its dependency on $ is unchanged if (80) replaces (77). This leads to

[l- (l- ,6-k)m]-2. (81)

Equations (80) and (81) reduce to the original VGM model when $, = 0 (and, hence, ,6 = 1). Equation (81) bears some similarity with the well-known Brooks-Corey model [13] relating the relative hydraulic conductivity to the matrix potential in the form

(82)

where $, is the minimum capillary height and e is a suitable parameter. Although the approach leading to Richards’ equation is attractive

since it reduces the variably saturated flow problem to a single PDE, it requires two algebraic relations for its closure, and this presents two drawbacks. Firstly, phenomenological parameters must be supplied to specify the soil hydrodynamic functions. For instance, five parameters are required in the VGM model and six in the modified VGM model. These parameters are often difficult to determine experimentally, but can have a significant impact on model predictions. This issue will be further discussed in Section 5. The second drawback is that field experiments indicate that the functions $ H 8($), $ H kT($ ) , and 8 H k r ( e ) are not single-valued. Instead, an hysteresis behavior is often observed


depending on whether the water content in the soil is decreased or increased. Hence, the use of the above models implicitly assumes that the volumetric water content is monotone during the simulation.

3.3 Space and time discretization

Many strategies to approximate Richards’ equation are feasible depending on the choice for the primal unknown, of the non-linear iterative solver, and of the space and time discretization scheme; see, e.g., Celia et al. [14].

In the present work, we focus on some specific choices. Firstly, the matrix potential $ is retained as the primal unknown. Secondly, the so- called method of lines is employed for the time-space discretization: the problem is first approximated in space using (for instance) finite element methods, thereby yielding a system of coupled ordinary differential equations (ODEs) where the time is the only independent variable. Then, a time approximation is constructed by using the vast theory of solution techniques for ODEs; see, e.g. [15] for a thorough review. Finally, New- ton’s method is used as the iterative solver to obtain an approximate solution to the non-linear discrete equations.

In this section, we briefly describe two finite element approximations to Richards’ equation, namely a conforming approximation of Richards’ equation in primal form and a non-conforming approximation of Richards’ equation in mixed form.

3.3.1

Consider the primal form (71) of Richards’ equation. Up to an appropriate lifting of the boundary data g1 and a modification of the right-hand side f , we can assume that a homogeneous Dirichlet boundary condition is prescribed on dR1. Set

Conforming approximation in primal form

and let UR be the form (non-linear in $, linear in 4):

Then, a possible weak formulation of (71) is the following:

( Seek $ E C1([O, TI; Van,) such that, for all t 2 0,

40 Alexandre Ern

where C'([O,T];Van,) is the space of functions in time with values in Vanl that are of class C1 in time, and T is a fixed simulation time.

To approximate (85), consider conforming, piecewise linear finite elements. Assume that the mesh 7 h is compatible with the partition 6'01 U dR2, i.e., that 6'01 n dR2 consists of mesh vertices (and edges in three dimensions). Introduce the finite element space

P,,anl = { 4 h E co(a); VK E $, $h(K E Pi; & = 0 On 6'a1} . (86)

The conforming approximation of Richards' equation in primal form is the following:

( Seek $h E cl([O, T]; P,',,,,) such that, for all t 2 0,

To approximate (87) in time, we restrict the discussion to the implicit Euler scheme. Let 6t be the time step and define

Let +fl E P,',an, be a suitable approximation to the initial value $0,

e.g., the L2-orthogonal projection of $0 onto Pianl or its Lagrange in- terpolant. Then, the time-marching scheme consists of generating the sequence of approximations $; E PkaS2,, n 2 1, where $; is an approximation of $ at time tn = n6t, by solving sequentially in time the following problems: For n 2 1,

( Seek $; E P:,an, such that

3.3.2

Consider the mixed form (72) of Richards' equation. This section briefly describes the extension of the finite volume box scheme presented in Sec- tion 2.5 to Richards' equation in mixed form. For the sake of simplicity, assume g2 = 0, i.e., no-flux conditions are imposed on 8 0 2 .

Non-conforming approximation in mixed form

Define the finite element spaces .I


and

R T $ ~ , = { u h E [ L ' ( ~ ) ] ~ ; v K E Z, U ~ I K E RT~(K);

Va E d:, [uh . n], = 0;Va E n do', uh . n1, = 0 ) . (91)

Let 4: E be a suitable approximation to the initial value $0. Then, the time-marching scheme to generate a sequence of non-conforming approximations (UE,$;) E RTtanz x P2,,an1, n 2 1, consists of solving sequentially in time the following problems: For n 2 1,

( Seek (uE,$z) E RT,qan2 x such that

Proposition 6. For all n 2 1, the solution (u;, $:) to (92) is endowed with the following properties:

1. Local mass conservation 1

VK E z , nK ( t ( ~ ( + ; ) - e($;-l)) + v . u; - f) = 0 . (93)

2. Local velocity reconstruction: For all K E 7h,

U E I K = - h n ~ ( b ( $ E ) ) ( V $ E k + e,)

+ : W f - &(e($z) - e($;-l)))..K(x). (94) 3. Equivalence with the non-conforming approximation of the primal

formulation: $; can be evaluated by solving sequentially in time the following problems: For n 2 1,

( Seek $: E Piclanl such that

[ = 6t J, H h f $ h + s, I w ( $ ; - l ) ) $ h , W h E P&,anl .

Proof. Similar to that of Proposition 5. 0

3.4 Water-tableground interact ion and the obstacle problem

To investigate water-tableground interaction, we consider a simplified problem: the domain R is a quadrangle, no-flux conditions are imposed

42 Alesandre Ern

on the le&, bottom, and right boundaries, and the top boundary, aQ,, is exposed to a constant rainfall with velocity v, = -ie, where i > 0 is the rainfall intensity; see Figure 3. In this setting, a%E, coincides with the ground surface. Henceforth, it is wsumed that i 5 k,, i.e., that the rainfall intensity never exceeds the infiltration capacity of the soil.

$ 5 0 and u s n = v, . n

/'

Figure 3 Setting for water-tableground interaction.

A s s m e that initially the water-table does not intersect the ground surface. Then, the boundary condition is u . n = v, . n throughQut 8Q,, Le. , a non-homogeneous Neumann boundary condition expressing the fact that She rainfall i d h a t e s inside the soil. As a result, the v o ~ u ~ ewater content in the soil increases, and the water-table moves upwards until it rexhes the ground surface. Once the water-table ~n&ersect~ the ground surface, the following holds on ail,:

$I 5 0 , u n 2 v, 'n, so that the ground surface can be divided into two regions

+(u n - v, * n) = 0 , (96)

an; corresponds to the non-saturated part of 8 2 , : 11, < 0 and u , n = v, . n;

corresponds to the saturated part of 82,: + = 0 and u . n >

An import an^ ~ s ~ ~ p t ~ o n in this model is that the height of the overland flow above the ground surface is neglected so &hat the ~ o ~ d a r y ~ ~in 8Sa; is indeed $ = 0. A model accounting for the coupling between the water-table dynamics and the overland flow will be discussed in Section 4.

Figure 4 presents an example of profiles for the matrix potential and the normal flow velocity along the ground surface. %he point where the water-table intersects the ground surface is clearly visible and corresponds to a longitudinal abscissa of a p p r o ~ ~ a S e ~ y 0.55 m. It is also observed that the saturated part can be further divided into two subregions,

v, " 12.

one part still allows for some infiltration: + = 0 and v, .n < u ~ n < 0;

Finite Element Modeling of Hydrosystems with + . . 43

Figure 4 the ground surface.

Matrix potential (solid) and normal flow velocity (dash) along

a the other part corresponds to the exfiltration region: $ = 0 and u . n > 0.

To investigate the formulation of Richards’ equation (71) together with the obstacle boundary condition (96), we consider first a steady setting and then an unsteady setting. For the sake of brevity, only the conforming approximation to the primal form of Richards’ equation is presented; the extension to the non-conforming approximation in a mixed setting is straightforward.

3.4.1 Steady obstacle problem

Given dRg c dR,, let P:,anz be defined as in (86) and let

Note that working with the space Van; implies $ = 0 on the saturated part an:.

The discrete steady obstacle problem consists of seeking a pair {an:, $ h } such that

dR9+ c dR, , (98)

$h E P:,an; 7 (99)

ax?$ ($h’ 4 h ) = 0 W J h E P:,a,$ ’ (100)

$h 5 0 on Xl;, (101) u($h) . n 2 w, . n on d~;2g+. (102)

The well-posedness of (100) requires that table has reached the ground surface.

# 0, i.e., that the water-

44 Alexandre Ern

An approximate solution {an:, $h} of (98)-(102) is sought using Newton's method embedded into a fixed-point iteration to determine the intersection of the water-table with the ground surface. The following iterative algorithm is proposed to solve the problem:

1. choose an initial dR:; 2. solve problem (100); 3. check whether (101) and (102) are satisfied; 4. if (101) is satisfied and (102) is not, decrease dR$ by one or more

5. if (102) is satisfied and (101) is not, increase dC$ by one or more

6. if both (101) and (102) are satisfied, then the current pair {do:, $h}

mesh cells; go to step 2;

mesh cells; go to step 2;

is the desired solution; one can refine the mesh and go back to step 2.

Owing to the maximum principle, both (101) and (102) can not be vi- olated simultaneously. However, in numerical approximations, this can happen. In this case, we still consider that the water-table has been cor- rectly positioned. With this "loosened" convergence criterion, the final position of the water-table depends from whether the converged has been approached from below or above. The two resulting values give lower and upper bounds for the water-table position (typically differing from one or two mesh cells at the most).

3.4.2 Unsteady obstacle problem

The unsteady version of the above obstacle problem can be implemented using an implicit Euler scheme. For n 2 1, given (aCl$)n-l and $,"-', seek { (afl:)", $;} such that

(an;)" c an,, (103)

$2 E P;(an;)- 7 (104)

s , ( B ( i i t ) - Wh"-')) 4 h + a(an;,-($r, 4h) = 0 W h E P1 .,(an;)- 7

(105) $;I 0 on (an;). , (106)

u($,hn) . n L w, . n on (107) This problem can be solved using the same iterative algorithm as for the steady obstacle problem. In step 1, the initial choice is =

Note that in the unsteady case, problem (105) is well-posed even if the water table has not reached the top boundary yet.


4 Overland flow This section briefly presents some models that can be used to describe overland flow over hillslopes. The coupling with Richards’ equation is also discussed.

4.1 Model formulation Transient flow of shallow water can be modeled by the Saint-Venant equations. To simplify, we consider a two-dimensional hillslope model; the overland flow then reduces to a one-dimensional problem governed by the following equations:

where y is the water depth, V the flow velocity, w the source term, g the gravity, S the river bed slope, and Sf the energy line slope.

We assume that in the momentum equation (log), the first two terms, i.e., the inertia terms, can be neglected in comparison with the last two terms. This yields the so-called diffusive wave approximation

aty + az(Yv) = w , a,y - Sf + s = 0 .

This model is widely used to describe flood routing; see, e.g., [16].

evaluate the energy line slope Sf [17,18], namely The Manning-Strickler uniform flow formula is usually chosen to

v = K~R:S), (112)

where KS is the Strickler coefficient of roughness and R is the hydraulic radius defined as the ratio between the cross-sectional flow area, A, and the wet perimeter, x. Assuming that the overland flow occurs as a thin layer with a wide rectangular section of width B yields the relation y << B. Hence, A = By, x = B + 2y, and

Let g(y) = yV. Using equations (111) and (113) in (112) yields

g(y) = K s y f (&y + S)i . (114)

The continuity equation then becomes the so-called diffusive wave equation

aty + a, (Ksy%(&g + S)i) = w . (115)

46 Alexandre Ern

One advantage of the diffusive wave approximation is to reduce the Saint- Venant equations to a single PDE. However, by doing so, the differential order of the PDE is increased, and it is necessary to supply two boundary conditions to close (115).

A further simplification with respect to the diffusive wave model consists of assuming that 8, y << s. This yields the so-called kinematic wave equation

&y + 8, (KsSiyg) = w . (116)

Equation (116) is a first-order nonlinear hyperbolic PDE. Analytical solutions can be obtained by using the method of characteristics; see, e.g., [19].

4.2 Coupling with Richards’ equation This section briefly discusses the coupling between Richards’ equation (71) and the diffusive wave equation (115) in the framework of the water- table-ground interaction described in Section 3.4.

Firstly, observe that (115) is posed only on afig since y = 0 on afi;. Taking into account the mass transfer from the subsurface flow and the rainfall into the overland flow yields

20 = u($) * n - ?Jr * n , (117) where u($) is the generalized Darcy velocity in the soil and n the outward normal to f2 along 80:. Furthermore, assuming the vertical pressure distribution in the overland flow to be hydrostatic, the following holds:

$ = y onaR:, (118) where $ is the matrix potential solving Richards’ equation (71) and y is the overland flow height solving (115). Equation (118) replaces the boundary condition $ = 0 on 80: which was previously enforced for the obstacle problem.

Finally, boundary conditions must be enforced for (115). In a two- dimensional hillslope setting, af2: is a segment, say [xmin,xm,], and assume, without loss of generality, that the point with abscissa zmin is located uphill and that the point with abscissa z,,, is located at the hillslope outlet. The following conditions can be enforced:

y = 0 at xmin ,

aZy = o at xmax *

Condition (119) is natural since it expresses the fact that the overland flow originates at zmin and then flows downstream. Condition (120) is reasonable, but is harder to motivate physically; it amounts to the balance of gravitational effects and energy losses at the hillslope outlet.


4.3 Numerical method Since the value of the matrix potential is now unknown over the whole ground surface, the finite element space in which the discrete matrix potential is sought is

At each time step, the discrete form of Richards' equation is, for all $h E p:'

Using (117) yields

$'h% . n . (123) - _ - la.;,- 4hW - laa;)n

JaaZp 4hW = ian;)n (at$h -t azq($h))dh ,

To approximate the first term in the right-hand side, use the coupling condition (118) and the diffusive wave equation (115) to infer

(124)

where the function y H q(y ) is defined in (114).

Figure 5 Space discretization at the ground surface.

A finite volume method can be used to discretize in time and space the right-hand side of equation (124); see Figure 5. The unsteady term is approximated by the implicit Euler scheme and a mass lumping in space, i.e., for the vertex Ni located on (aflT)n,

48 Alexandre Ern

where #h,i is the nodal basis function of P: associated with the vertex Ni. The second term is approximated by an upwind scheme. Assume that the nodes on dR, are numbered downwards. Define the numerical flux

Then,

Equation (127) is used for all the nodes in the interior of At the outlet node, say Nio, equation (127) can still be used by conventionally setting

QZ,io+l = Ks($,"(Ni,))gSt - (128) With this choice, the boundary condition a X y = 0 is weakly enforced at the hillslope outlet.

To sum up, the numerical method for approximating the coupling between Richards' equation and the diffusive wave equation consists of solving sequentially in time the following problems: For n 2 1,

Seek $; E P: such that

(e(?%> - e($;-l)) #h -k a(an$)-. ($;, #h) = R($;, $:-'; #h) 7

if there exists avertex Ni on (aR$)n such that #h = #h,i , and R(@, $$-'; # h ) = 0 otherwise.

5 Applications This section presents two applications of the above finite element techniques to simulate hydrosystems, namely infiltration into a one-dimensional soil column and a two-dimensional hillslope exposed to heavy rainfall.


5.1 One-dimensional infiltration One-dimensional stationary and unstationary infiltration problems with analytical solutions are simulated to validate the subsurface flow code.

5.1.1 Stationary test case

Consider a one-dimensional steady infiltration under a constant rainfall rate, 21, = -ie,. Mass balance readily yields

i = h k T ( $ ) (&$ + 1) (131)

Assuming $(zL) = $L, the solution to (131) is

The Brooks-Corey model (82) is used to express the relative hydraulic conductivity in terms of the matrix potential. Three types of soils are considered. Table 2 lists the saturation hydraulic conductivity, k,, and the two Brooks-Corey model parameters, $s and e, for each soil. The left plot in Figure 6 displays the relative hydraulic conductivity as a function of matrix potential near saturation.

Table 2 Saturation hydraulic conductivity and Brooks-Corey model parameters for the three soils considered in the stationary infiltration test case.

3.3 5.64 11.88

Using (82) in (132) yields (see, e.g., [20])

where G is the hyper-geometric function G(x) = zF l (1 , l ; 1 + 3; x). Simulations are performed with the the same infiftration rate, i =

1.224 x lo-* m/h, for the three soils. The boundary condition is $L = 0 at ZL = 0. The right plot in Figure 6 displays simulation results. The agreement with the analytical solution is excellent.

50 Alexandre Ern

1

._ 0.8

3

-$ 8 0.6 .- - 2 3 ._ J

d

TJ 0.4

c _m 0.2

0

3.5

3

2.5

2

1.5

1

0.5

0

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 Pressure head h (cm)

Silt: !!lsniy =

0 0.5 1 1.5 2 2.5 3 3.5 Dimensionless capillary tension

Figure 6 Stationary infiltration test case. Top: relative hydraulic conductivity kT versus matrix potential for the three soils. Bottom: simulated and analytical profiles of capillary tension, &+(-$-), for the three soils.

5.1.2 Unstationary test case

The following test case is discussed in Barry et al. [21]; see also [20]. Assume that the relative hydraulic conductivity is defined by the Brooks- Corey model, equation (82), with parameters qS and @. Let k: denote the derivative of the function + H kr($). Assume that the soil water retention curve takes the following form


with parameter

Set G = and

A(t) = 1 + W [- exp(tL/y - l)] , (137)

where W is the lower portion of the Lambert function. Then, one can show that an analytical solution to Richards’ equation, $la(t, z ) , such that $la@, 0) = 0 for all t , is, for z < 0,

z &(t, z ) = -- . A@)

- - N

0 0.2 0 4 0.6 0 8 1 1.2 1.4 Effective water content

-45 -4 3 5 -3 -2 5 -2 -1.5 -1 -0.5 0 Pressure head (m)

Figure 7 Unstationary infiltration test case. Simulated and analytical vertical profiles for the water content (top) and the pressure head (bottom) at the times t = 10.1, 20.1, 30.1, 40.1, 50.1, 60.1, 70.1, 90.1, 100.1, 130, 170, 200, 500, 1000 h.

52 Alexandre Ern

Simulations are performed with the clay soil considered in the previous section (13, = 0.04, OS = 0.432). The simulation domain is the interval z E [-1.6, -0.61. The boundary conditions enforce the value of the analytical solution at both ends of the interval. The initial condition is the analytical solution at t = 0.1 h. The saturation front reaches the top boundary at about t = 60 h, and the domain is completely saturated at about t = 290 h. Results are displayed in Figure 7. The agreement with the analytical solution is very good.

5.2 Two-dimensional hillslopes Consider the two-dimensional hillslope configuration shown in Figure 8. Define &rain = iL(e, en). Let Qin(t) be the (time-dependent) infiltration flux and let Qnotin(t) be the “direct runon” flux (the water that never infiltrates); hence,

Qrain = &in@) + Qnotin( t ) . (139)

Let Qexf( t ) be the exfiltration flux and let Qrunoff(t) consist of the exfiltration and the direct runon fluxes

Qrunoff(t) = Qexf(t) + Qnotin(t) = Qexf(t) + Qrain - Qin(t) . (140)

If the rainfall intensity is kept constant in time, the time to reach equilibrium, Te, is conventionally defined as the first time in the simulation for which

IQin (Z) - Qexf(Te)l I 0 - 0 0 5 Q i n ( Z ) . (141) At equilibrium Q i n ( m ) = Qexf (m) and, hence, Qrunoff(m) = Qrain. Note that at a given time t, Qrunoff(t) is not the instantaneous water flux at the toe of the slope but the instantaneous water flux into the overland flow.

A (035) B (1014.8) C (39.85:ll) D (54.8530.8)

F (39.85;8) G (1011.8)

L = L t L 1 2

A E (54.857.8)

H

0

E

Figure 8 Hillslope geometry (length unit: meter)


0.002

0.0015

0,001 c S

0,0005

0

5.2.1 Influence of the soil hydrodynamic parameters

....... .:. ..... -1. .... ..;. ........... 4.. ..

........,...................... ....... J . . . .

....... ~i.. .... -1. ..... .: .....

........I. ..... . I . . ... ..I.. .

For all the test cases presented in this section, the initial condition is the steady-state solution corresponding to a rainfall intensity of & = 0.4%. The initial condition is presented in Figure 9. Furthermore, the obstacle model is used to simulate the water-table dynamics once it has reached

15

14

13

10

9

a 7

. . . . . . . . -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5

Psi (m)

0.1

........ I . . ..... ............. ............

-0.8 0 10 20 30 40 50

x (m)

-0.0005 I 0 10 20 30 40 50

Figure 9 Initial condition for the hillslope test case. Top : vertical profile of matrix potential at various points located by their horizontal coordinate 5. Meddle: matrix potential along the ground surface. Bottom: normal flow velocity along the ground surface.

54

soil T, [h]

Alexandre Ern

YLC I n = 1.31 n = 3.9 a = 0.3 a = 8.0 2034 I 865 2925 71 3298

the ground surface. We first investigate the impact of the parameters a and n in the

VGM model. The soil chosen as a reference is a so-called Yo10 Light Clay (YLC) soil; its VGM parameters are 8, = 0.23, Os = 0.55, a = 3.6 m-l, n = 1.9, and k, = 0.018 m/h. Then, the parameters a and n are varied, and a new steady-state solution is computed. In all the simulations, the soils are subjected to a constant rainfall intensity of t = 2%.

0.3

0.2

0.1.

Table 3 Time to reach equilibrium.

. f ' /'

YLC - ,' .

^ _ , n=1.31-- . - n=3.9 7 ' .

Times to reach equilibrium are reported in Table 3. The first line indicates the parameter that has been varied and the corresponding value. The impact of both a and n cn Te is evident. Figure 10 presents the hydrodynamic functions $ H O ( $ ) and $ H lc,($) for the various soils considered in the simulations. A more specific investigation of the impact of the parameter n and that of the parameter a is presented in Figure 11.

0.wol c ." YLC -

n=1.31- - n=3.9- -

4.7 4.6 0.5 4.4 4.3 4.2 0.1 1 6 4 5 ' ' ' ' ' . .

1 :;--."--.----- 0.9 . 0.8 . 0.7 .

I b

,I

YLC - . / ' 0.3 . 0.2-

0 . 1 . ' ' " ' - oL=0.3 - -

a1=8.0 7 - I __ . - .

4.7 4.6 4.5 4.4 4.3 4.2 4.1 0 + (m)

1

0.1

0.01 C

Y

0.w1

o.wo1

1 6

YLC - 01=0.3 - - a=s.o .- . -

0.7 4.6 -0.5 0.4 4.3 4.2 0.1

Figure 10 Hydrodynamic functions for various values of n (left) and a (right). Top: $ H g($). Bottom: $ H k, (+) .

Finite Element Modeling of Hydrosystems with ... 55

1

0.9

0.8

0.7

0.6 0.5 0.4

0.3

0.2 0.1

time trre

t 1 ---I exf.

h j h h , I 0 0.2 0.4 0.6 0.8 1 1.2 1.4

time trre

YLC - : 0.003

-0.0005 0 1 0 m 3 0 4 0 5 0

e a

n

1

0.6

time VTe

0

-0.0005

Figure 11 Left column: influence of n on simulation results; right column: influence of o on simulation results. Top: saturated length L, normalized by length L (see Figure 8) as a function of time normalized by T,; middle: runoff and exfiltration fluxes normalized by Qrain as a function of time normalized by T,; bottom: normal velocity flux 'U . n along the ground surface.

As a second test case, we investigate the impact of the modified VGM model on simulation results. The reference soil is again taken to be the YLC soil. Two additional simulations are performed by setting the minimum capillary height to $, = -2 cm and to $, = -10 cm in the modified VGM model, Figure 12 presents the corresponding hydrodynamic functions 4 H e($) and $ H kT($) . A more specific investigation of the impact of the modified VGM model on simulation results is presented in

56

Figure 13.

Alexandre Ern

1

0.1

c &?

0.01

0 . 7 0.6 0 . 5 0.4 0.3 0.2 0 .1 0 0.w1' ' ' ' ' ' ' '

-0.7 0.6 0 . 5 -0.4 -0.3 -0.2 0.1 0

vj (m) vj (m)

Figure 12 values of $s. Left: $ ++ t9($). Right: $ A kT($) .

Modified VG-M model: hydrodynamic functions for various

100 1

0.9

0.8

0.7

8 6o @ 0.6

3 0.5 0.4

20 0.3

0.2

0 0.1

80

c

a - - 5 4 0

w 0

time (h) time (h)

0 10 20 30 40 50 x (m)

Figure 13 Impact of the modified VGM model on simulation results. Top left: saturated length L, normalized by length L (see Figure 8) as a function of time normalized by T,; top right: runoff and exfiltration fluxes normalized by Qrain as a function of time normalized by T,; bottom: normal velocity flux 2 1 . n along the ground surface.


5.2.2

Consider the two-dimensional hillslope configuration shown in Figure 14 and investigated previously in [l]. A no-flow boundary condition is imposed at the bottom and left surfaces, simulating an impermeable layer. The right boundary corresponds to a stream in which the hydraulic head is prescribed. The initial condition is a horizontal water table located at the toe of the slope; see Figure 14. The problem geometry is characterized by the slope So = lo%, the depth to the impermeable layer D = 1 m, and the length L = 50 m. Simulations are run with a constant rainfall intensity such that $ = 0.6%. The soil hydrodynamic properties are Or = 0.069, Os = 0.435, a = 0.326 m-l, n = 3.9, and k, = 5.0 m/h; they correspond to the sand considered in [l].

Influence of the overland flow

water table

Figure 14 overland flow model.

Hillslope geometry to compare the obstacle model with the

Figure 15 Comparison of the obstacle model with the model accounting for overland flow. Left: relative saturated area. Right: exfiltration flux.

Figure 15 compares the results obtained with the obstacle model to those obtained with the model accounting for overland flow. The Strickler coefficient of roughness is set to Ks = 10 Both models

58 Alexandre Ern

yield very similar results, especially for the time evolution of the relative saturated area. Slight differences in the exfiltration flux are observed for sufficiently long times; these differences arise mainly near the toe of the slope.

References [l] Ogden F L, Watts B A. Saturated area formation on non convergent

hillslope topography with shallow soils: a numerical investigation, Water Resour. Res., 36(7), 1795-1804, 2000.

[2] Cloke H L, Renaud J P, Claxton A J, McDonnell J J, Anderson M G, Blake J R, Bates P D. The effect of model configuration on modeled hillslope-riparian interactions, J. of Hydrology, 279, 167- 181, 2003.

[3] Weiler M, McDonnell J. Virtual experiments: a new approach for improving process conceptualization in hillslope hydrology, J. of Hy- drology, 285, 3-18, 2004.

[4] Ciarlet P. The Finite Element Method for Elliptic Problems. North- Holland, Amsterdam, 1978.

[5] Brenner S, Scott L R. The Mathematical Theory of Finite Element Methods. Vol. 15 of Texts in Applied Mathematics, Springer-Verlag, New York, 1994.

[6] Quarteroni A, Valli A. Numerical Approximation of Partial Differ- ential Equations. 2nd edition, Vol. 23 of Series in Computational Mathematics, Springer-Verlag, New York, 1997.

[7] Ern A, Guermond J L. Theory and Practice of Finite Elements. Vol. 159 of Applied Mathematical Series, Springer-Verlag, New York, 2004.

[8] Croisille J P. Finite volume box schemes and mixed methods, Math. Mod. Numer. Anal., 31(5), 1087-1106,2000.

[9] Baer J, Bachmat Y. Introduction to Transport Phenomena in Porous Media. Kluwer Academic Publishers, 1990.

[lo] van Genuchten M Th. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. SOC. Am., 44,

[ll] Mualem Y. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res., 12, 513-522, 1976.

[12] Vogel T, van Genuchten M Th, Cislerova M. Effect of the shape of the soil hydraulic functions near saturation on variably-saturated

892-898,1980.


flow predictions, Advances in Water Resources, 24(6), 133-144, 2001.

[13] Brooks R H, Corey A T. Hydraulic properties of porous media, Hydrol. Pap., 3, Colorado State University, Fort Collins, p. 27, 1964.

[14] Celia M A, Boulotas E T, Zarba R L. A general mass-conservative numerical solution for the unsaturated flow equation, Water Resour. Res., 26(7), 1483-1496, 1990.

[15] Lambert J. Numerical Methods for Ordinary Differential Systems. Wiley, New York, 1991.

[16] Moussa R, Bocquillon C. Criteria for the choice of flood routing methods in natural channels, J. of Hydrology, 186, 1-30, 1996.

[17] VanderKwaak J E, Loague K. Hydrologic-response simulations for the R-5 catchment with comprehensive physics-based model, Water Resour. Res., 37(4), 999-1013,2001.

[18] Liu Y B, Gebremeskel S, De Smedt F, H o h a n n L, Pfister L. A diffusive transport approach for flow routing in GIS-based flood modeling, J. of Hydrology, 283, 91-106, 2003.

[19] Borah D K, Prasad S N, Alonso C V. Kinematic wave routing incor- porating shock fitting, Water Resour. Res., 16(3), 529-541, 1980.

[20] Ginzburg I, Carlier J P, Kao C. Lattice Boltzmann approach to Richards’ equation, Proceedings of the CMWR Conference, North Carolina, 2004.

[21] Barry D A, Parlange J P, Sander G C, Sivaplan M. A class of exact solutions for Richards’ equation, J. of Hydrology, 142, 29-46, 1993.

60

Sharp Front Modeling

Patrick Goblet Centre d’Informatique Gkologique

Ecole Nationale Supkrieure des Mines de Paris 35, rue Saint-Honor6

77305 Fontainebleau, France E-mail: [email protected]

1 Introduction This paper discusses some recent orientations in the modeling of the concentration fronts resulting from the solution of the solute transport equation (dispersion equation). The objective is to show the main appo- raches and to discuss their advantages and shortcomings. It is beyond the scope of this paper to describe in detail the techniques, and the reader is referred to the bibliography for a detailed description of each particular approach.

2 Sharp front problem statement The dispersion equation describes the movement of a solute (tracer) in an aquifer. This movement is the result of three main mechanisms ( [lo]):

0 Advection is the global movement of a volume of water containing tracer at the average pore water velocity.

0 Molecular diffusion is the spreading of tracer in immobile water under a concentration gradient (Brownian motion)

0 Dispersion is an additional spreading mechanism due to water velocity fluctuations around the mean value. It is ususally described by a diffusive (fickian) model.

The dispersion equation writes:

dC V(DVC - UoC) = W Z

where

C is the solute concentration (tracer mass/unit volume)

Sharp Front Modeling

D =

61

a L 0 0

0 0 w r 2 0 Q T ~ 0

D is the dispersion tensor (m2/s)

UD is the Darcy velocity (m/s)

w is the kinematic porosity

t is the time (s)

The D tensor includes the molecular diffusion coefficient and the dispersion coefficient. It writes, in axes aligned with the local velocity:

IUD~ + doI

Q T ~ and Q T ~ are transverse dispersivities (m)

do is the tracer diffusion coefficient in the porous medium

I is the unit tensor

Equation (1) is made of an advective and a diffusive term. The relative importance of each term depends on the value of the dispersivity tensor. As dispersivity is in fact a global representation of local velocity heterogeneity, it can be considered as reflecting our ignorance of the actual velocity field. The role played by dispersion is therefore reduced as the knowledge of the velocity field is refined. For this reason, in a well characterized Aow field, advection may become the main transport mechanism.

On the other hand, solute transport is not sufficient to describe the fate of chemical substances in an aquifer: most solutes are subject to chemical interactions with the immobile medium as well as with other species. The dispersion equation is now merged in more general models which describe the movement of a chemical system in groundwater. This leads to source and sink terms in the transport equation of each species, which often create sharp transition zones (redox front are a typical example).

The numerical simulation of sharp concentration fronts is a problem faced in many field studies. Most classical numerical techniques (Finite Volumes, Finite Elements) perform well when the dispersion front is spread over several grid blocks. This constraint can be expressed in terms of a Peclet Number:

UD AX Pe = - 5 2 D (3)

Patrick Goblet

This constraint may not be met in practical applications involving large scale models, because it would require either inacceptable values of the dispersivity tensor, or an unpractically fine mesh.

As a consequence, better approximation techniques are continuously looked for.

3 Classification of approaches Most numerical techniques used for the solution of the advection- dispersion problem fall into one of the following categories:

0 Eulerian techniques: the transport equation is solved in a fixed coordinate system. This means that the advective and dispersive terms must both be discretized. This can be done by Finite Ele- ments, Finite Differences, Finite Volumes, . . .

0 Lagrangian techniques: the advective term is treated in a moving coordinate system. The dispersive term can be treated by various techniques.

0 Eulerian-Lagrangian techniques: both terms are generally discretized in a fixed coordinate system. However, some techniques are borrowed from the lagrangian approach to facilitate the treatment of the advective term.

Particle techniques are a particular class of approaches which do not necessary rely on a discretization of the equation, although they may use one. These techniques are interesting by themselves, but also because they bring some ideas which can be used in the Lagrangian or Eulerian- lagrangian approaches.

In this paper, we shall first present a few ideas pertaining to the particle approach technique. Then we shall present approaches which aim at improving the performance of Eulerian techniques. Lagrangian approaches will then be briefly presented. Finally the more recent developments in the Eulerian-Eagrangian field will be discussed.

4 Particle techniques The general idea behind so-called “particle techniques” is to follow the movement of a small volume of water. This volume carries a certain mass. It is tracked along the flow lines at pore water velocity, which accounts for the advection mechanism. As the velocity field in the general case is not uniform, but rather varies continuously along the flow path, time must generally be discretized and the velocity field must be

Sharp Front Modeling 63

approximated. The position of a particle at time t + At is obtained from its position at time t by the following relations:

~ ( t + At) = ~ ( t ) + V,At y ( t + At) = y ( t ) + VgAt (4)

The precision of this scheme depends on the time step, but also on the precision of the velocity field. When, as is generally the case, no analytical expression is available for the velocity field, it must be derived from an approximate solution of flow, obtained e.g. by Finite Elements or Finite Volumes. The precision on the velocity field can then be increased in two ways:

0 Increasing the order of representation of the velocity. Mixed Fi- nite Elements are a classical approach to achieve this goal without increasing the degree of approximation of the hydraulic head.

0 Formulating the flow problem in terms of stream function rather than in terms of hydraulic head. This approach is only possible in steady state, and when no hydraulic source terms are present.

Both approaches yield a velocity field which has a continuous normal component accross element interfaces. This property is very valuable to allow the tracking of particles in strongly varying velocity fields.

The description of dispersion in the framework of particle techniques can be done in two ways, which can be related to the main “historical” techniques:

0 Method Of Characteristics (MOC, see e.g. [B])

0 Discrete Parcels Random Walk Technique (DPRW, see [l]).

The MOC technique is based on a fixed grid. Each time step is split into an advective and a diffusive step. At the end of an advective step, particles are counted in each grid block. The obtained mass is then averaged on the grid block, yielding a concentration value. The dispersion operator is then solved on the fixed grid. The new concentrations are spread again in particles, and a new advective step is performed.

In the DPRW technique, no mesh is required for the description of dispersion (a mesh may however be required to describe a non uniform flow field). The redistribution of particles in space at the end of each advective step is done according to a Brownian motion, which describes the diffusive nature of dispersion.

Particles techniques can be found in the literature under various flavours (see e.g. [14] for an object-oriented implementation in fracture networks). They are implemented in several classical water management softwares. MOC methods often suffer from numerical diffusion, due to

64 Patrick Goblet

the repetitive mechanism of scattering-gathering of particles. This drawback does not exist in DPRW methods, because each particle retains its identity throughout the whole simulation.

A large number of particles is necessary to achieve a sufficient precision, especially in 3 D simulations. On the other hand, particles are only requested in the regions of the studied domain where tracer is effectively present. This contrasts with classical discretized equations, for which the equations must be solved even in tracer free regions. This makes particle techniques well suited for the simulation of short and localized contaminations.

Two important ideas of particle techniques have been retained in the Eulerian-lagrangian techniques which will be discussed later: the splitting of operators (solving advection and dispersion in two separate steps), and the development of accurate techniques to track water volumes along characteristic lines of flow.

5 Improving eulerian approaches As stated earlier, eulerian techniques perform well when the ratio between advective and dispersive flux permits the development of a relatively smooth front, i.e. typically a front spread over several grid blocks. When this condition is not met, a problem arises, due to the difficulty to describe appropriately the sharp concentration variation around the front: spurious oscillations (overshoot-undershoot) develop and are prop agated (see Figure 1). The solutions proposed in the literature to this problem try to tackle it according to various strategies:

by an optimization of the definition of the upstream concentration, either in Finite Element Schemes (Streamline Upstream schemes, e.g. SUPG ([13]), space-time approaches (SUFG, [12])), or in Fi- nite Volume schemes (Monotone Upstream Centered Schemes for Conservation Laws, MUSCL, [IS]) by an increase of the degree of approximation, either in a Finite Element Framework, or in a Spectral Element framework ([S]) by discontinuous approximations of the concentration field.

We shall first discuss in this section the basic principle of upstream weighted approaches and show examples of this approach. Spectral elements will then be presented and their use illustrated on an example.

5.1 Improving a 1D Finite Difference Scheme A standard approximation of the advection-dispersion equation on a 1D Finite Difference grid for a fixe grid block size Ax and a time step At


140

120

100

80

60

40

20

0 0 20 40 60 80 100 120

Distance (m)

Figure 1 Instable propagation of a dispersive front.

(see Figure 2) writes:

where Ci is the value of concentration at node i for some time between tn and tn+l, and Ci-1/2 (resp. Ci+1/2) are concentration values at some point between points i - 1 and i (resp. i and i + 1).

I A x i- 1 i i+ 1

I I

i-112 i+1/2

Figure 2 tion.

1D Finite Difference Approximation of the Dispersion equai

One simple choice, which gives the best accuracy for this scheme, is a centered approximation in space as well as in time:

Ci-1+ Ci 2

2

Ci-1/2 =

ci+1/2 = Ci + Ci+l

66 Patrick Goblet

and

This scheme gives a second order accurate approximation, which is stable under the condition:

U o AX Pe= - 5 2 D

Pe is the Peclet Number. When this condition is not met, numerical solutions exhibit oscillations.

5.1.1 First order stabilizing scheme

The classical cure to this problem is to replace the centered spatial approximation of velocity in the advection term (eq. 6) by a so called “upwind” approximation: the concentration is taken upstream of the cell boundary.

This scheme is stable and monotonous, but only first order accurate. It can be shown simply that it is equivalent to introducing into the centered scheme an extra diffusion term (numerical diffusion term) equal to

As a general rule, it can be shown that a linear, monotonous stabilizing scheme cannot be 2nd order accurate. Note that this upstream scheme is classically used in classical Finite Difference softwares. These can therefore be expected to introduce a numerical diffusion with a longitudinal component equal to half the grid block size.

5.1.2 Higher order stabilizing schemes

The basic idea of higher order schemes is to adapt the scheme to the smoothness of the concentration profile: far from the front, a centered scheme can be adopted. Close to the front, a variable formulation, intermediate between a fully upstream and a fully centered formulations, is used. This type of scheme, which depends on the concentration profile, is by nature non linear. A classical example of such approach is the family of MUSCL schemes (see [16]). Discussing this type of scheme is beyond the scope of this presentation. Let us just show a simple example of a “quasi-second order” scheme.


Front propagation in an homogeneous medium - Pe = 100 Variable upstreaming scheme

Figure 3 upstream scheme.

Solution of the 1D dispersion equation using an adaptative

We start by defining an indicator which is related to the concentration gradient variation: for a node i, this indicator is the ratio between the gradient upstream of the node and the average gradient at the node:

ai is close to 0.5 in smooth concentration zones, close to 1 on the top edge of a front, and close to 0 at the foot of a front. We use ai as a variable weighting factor to interpolate the upstream concentration:

Ci-l/2 = Qi-lCi-1 + (1 - ai-l)Ci

Ci+1/2 =z aiCi + (1 - ai)Ci+l (12)

This yields a centered scheme in smooth concentration regions, an upstream scheme on top of the front, and a “downstream” scheme at the foot of the front.

Figure 3 shows the propagation of a dispersive front calculated on a 120 grid blocks mesh according to this scheme.

The solution is compared to a reference solution obtained on a 12000 grid blocks mesh. The apparent (real+numerical) dispersivity for this scheme is 3 cm, showing a numerical dispersivity of 2 cm, a very low value compared to the grid block size of 1 m.

68 Patrick Goblet

5.2 Improving a 1D Finite Element Scheme

The abstract nature of Finite Elements does not allow to introduce the same kind of “intuitive” upstreaming as in Finite Differences or Finite Volumes. The same type of result can be obtained using asymmetric test functions for the advection term. This idea was first introduced, at least for solute transport problems, by Huyakorn ([7]). Figure 4 shows an example of an asymmetric test function. The degree of asymmetry can be related to the Peclet Number.

Figure 4 Asymmeric test function.

Variants of this type of scheme are widely applied in simulators. The asymmetric test functions can be generalized in 2 or 3 space dimensions. Upstreaming is then performed along the velocity vector (Streamline Upstream Petrov Galerkin, SUPG, see e.g. [13]).

Basically, the stabilization introduced by these schemes may be in- terpreted in terms of numerical diffusion. A more general approach has been proposed e.g. in [12], in which time is treated as an extra dimension (Space-Time Finite Elements). The advantage of this approach is that upstreaming is performed not along flow lines, but along characteristics, which drastically reduces the spatial numerical diffusion. This technique is termed SUFG (Streamline Upstream Full Galerkin).

While upstream appproaches provide often a very efficient way to simulate sharp fronts, they are generally subject to a Courant-Friedrich- Levy (CFL) condition on the time step:

c=- Ax < 1 VAt -

This condition can be severely limited for long term environmental as- sessments.


5.3

An alternative approach to refine the modeling of a sharp front is to increase the degree of the basis functions in a Finite Element framework. Spectral Elements provide an appealing alternative to classical Finite Elements .

The term “spectral approach” refers to a decomposition of the approximate solution of an equation into a truncated Fourier series. This technique can give highly precise results when boundary conditions give a periodic or pseudoperiodic character to the solution. The same denom- ination is used for techniques using polynomials of arbitrary order as approximating functions. No constraint of periodicity is then required. This approach has been applied successfully to the simulation of shock waves in fluid mechanics ([9]), and more generally to the solution of differential equations describing a variety of physical processes: Navier- Stokes equations ( [ 5 ] ) , non-newtonian fluid flow ([15]), electromagnetism ([4]). . . One important advantage of this approach is to provide a general formalism for arbitrarily high order approximations of the unknown, which permits a very flexible adaptation of the degree to the problem at hand without code modification.

Spectral elements can be included into a Finite Element code, because they use the same basic approach (assembling elemental coefficients into a global matrix). An application of the spectral element formalism to the dispersion equation can be found in [6]. Figure 5 illustrates the application of the spectral element approach to the propagation of a moisture front (Richards equation). The spectral solution is computed on a 10 elements mesh. The degree on each element is dynamically adapted to the local smoothness of the solution. The average total number of nodes is 70, to be compared to the 500 nodes necessary to achieve the reference solution with a classical linear Finite Element scheme.

Higher order schemes: spectral elements

6 kagrangian approaches

In purely lagrangian approaches, the advection term is removed from the dispersion equation by using a moving grid. For instance, in [MI, grid nodes are moved along the characteristics. This permits the exact solution of purely advective problems. For the general dispersion equation, the problem reduces to a purely diffusive one.

A problem specific of the moving grid approach is the treatment of boundaries, where new nodes must be introduced (inlet boundary) or removed (outlet). While this technique is very efficient in lD , it becomes cumbersome in 2 or 3D, because of the difficulty to deal with the distortion of elements in a non uniform flow field, and because of the

70

-1

-2

-3

-4

-5 E .- 5 -6

-7

-8

-9

-10

-1 1

,-. - "

Patrick Goblet

Propagation of a moisture front Comparison of the adaptative spectral calculation to a Finite Element calculation

0 0.2 0.4 0.6 0.8 1 Elevation(m)

Figure 5 spectral element scheme.

Solution of the 1D Richards equation using an adaptative

boundary description problem.

propose solutions to both problems. Eulerian-lagrangian approaches, to be discussed in the next section,

7 Eulerian-lagrangian approaches

These approaches use a standard Finite Element or Finite Volume formalism for the dispersive operator, and a characteristic-type formulation for the advective operator.

Many variants have been proposed (see e.g. [ll]). The most widely used class of Eulerian-lagrangian methods is the

ELLAM (Eulerian Lagrangian Localized Adjoint Methods) family (see

The principle of ELLAM approach is to remove the advective term in the weak formulation of the dispersion equation by using test functions which move at water velocity. This approach yields a consistent framework for Eulerian-Eagrangian techniques, in Finite Elements as well as in Finite Volumes.

We shall briefly derive the ELLAM formulation in 1D to make its principle clearer. This development is inspired by [3].

The weak formulation of the dispersion equation uses a space-time

e.g. PI, 131).


weighted integral:

- $ (Dg)) Wi(x,t) d x d t = 0 (14)

Note that the weighting function depends on x and t. Equation 14 is integrated by parts in space and time, to yield:

( 5 ) ( 6 )

By a proper choice of the W(x7 t ) function:

aw dW d X 6%

u- +w- = 0

(which means that the W function moves with pore water velocity), terms (2) and (4) cancel. Terms (3) and (5) will support boundary conditions. This leaves us with:

d C dW (CW)n+l - (CW)") dx + D-- dxdt = O (17) n, nt ax

which is a purely diffusive equation. The main technical difficulty in the implementation of the ELLAM

formalism is the evaluation of integrals involving the moving test function: except in very simple 1D cases for which an exact analytic integration is possible, the integration must be done numerically. The difficulty lies then in the fact that the position of a test function does generally not coincide with element limits, either at the beginning or at the end of a time step. Depending on the implementation, either the Jnx w(CW)"+' dx or the In w(CW)" dx involves an integration on a domain intersecting element Loundaries. Two methods are proposed to treat this problem:

0 If the test functions are built so as to coincide with element boundaries at the beginning of the time-step (Figure 6) , the integration will be problematic at the end of the step. In this case, integration points at the end of the step are back-tracked7 and the value of the test function at the foot of the characteristic line is interpolated.

72 Patrick Goblet

W(x) at time level n+l W

I

I I ~ time level n

X

Figure 6 ELLAM test function defined at the beginning of a time step.

0 Forward tracking is used instead if the tests functions coincide with element boundaries at the end of the time step (Figure 7).

W W(x) at time level n+l

I ~ time level n

Figure 7 ELLAM test function defined at the end of a time step.

ELLAM approaches present several other interesting features:

0 They are by construction mass conservative, at least up to the precision of the integration scheme.

0 All types of boundaries can be properly accounted for, through a careful description of the movement of test functions through the boundaries.

0 The test functions do not have to be piecewise linear as in the examples shown above. They can also be “top hat” functions. In this case, a Finite Volume version of the technique is obtained.

0 Finally, the CFL constraint is absent from this formulation. One may even argue that long time steps are more favourable, because


they reduce the number of interpolation steps required to integrate the moving test functions.

8 Conclusion

As general conclusions, one may say that:

0 No “perfect” numerical technique is presently available to solve the whole range of applications.

0 Many techniques for sharp front modeling (SUPG, SUFG, d’ iscon- tinuous FE) give satisfactory results when the CFL condition is met.

0 Eulerian-Lagrangian techniques seem to be the most promising tool, since they produce practically oscillation-free solutions without a CFL constraint.

References

[l] Ahlstrom S W, Forte H P, Arnett R C, Cole C R, Serne R J. Mul- ticomponent mass transport model: theory and numerical implementation (Discrete-Parcel-Random-Walk Version), BNWL 2127 May 1977.

[2] Binning P J, Celia M A. A forward particle tracking eulerian La- grangian localized adjoint method for solution of the contaminant transport equation in three dimensions Advances in Water Resour.

[3] Celia M A, Russell T F, Herrera I, Ewing R E. An Eulerian- Lagrangian localized adjoint method for the advection-diffusion equation. Advances in Water Resources 13(4), 187-206, 1990.

[4] Rhabi E L, Mohammed. Analyse numrique et discrtisation par lments spectraux avec joints des quations tridimensionnelles de l’lectromagntisme, Thse de Doctorat en Mathmatiques Appliques, Universit Pierre & Marie Curie, 2002.

[5] Giraldo F X. Strong and weak Lagrange-Galerkin spectral element methods for the shallow water equations, Computers and Mathe- matics with Applications 45 (2003) 97-121, 2003.

[6] Goblet P, Cordier E. Solution of the flow and mass transport equations by means of spectral elements Water Resources Research,

25, 147-157, 2002.

~01.29 (9), 3135-3144, 1993.

74 Patrick Goblet

[7] Huyakorn P S, Nilkuha K. Solution of transient transport equation using an upstream weighted finite element scheme Appl.Math.Modelling, 3: 7-17, 1978.

[8] Konikow L F, Bredehoeft J D. Computer model of two-dimensional solute transport and dispersion in groundwater U.S.Geologica1 Survey Techniques of Water Resources Investigations, book 7, chap. C2, 90, 1978.

[9] Maday, Yvon, Patera, Anthony T. Spectral element methods for the incompressible Navier-Stokes equations. in State-of-the-art surveys on Computational Mechanics, edit par A. K. Noor, 71-143, American Society of Mechanical Engineers, Fairfield N. J., 1989.

[lo] de Marsily G. Quantitative hydrogeology: groundwater hydrology for engineers, Academic Press, 1986.

[ll] Neuman S P, Sorek S. Eulerian-Lagrangian methods for advection Dispersion Proc. of 4th Int. Conf. on Finite Elements in Water Resources, Hannover, Germany, June, 1982.

[12] Noorishad J, Tsang C F. Streamline Upstream/Full Galerkin Method for Solution of Convection Dominated Solute Transport Problems. Transport in Porous Media, vo1.16, 53-74, 1994.

[13] Noorishad J, Tsang C F, Perrochet P, Musy A. A perspective on the numerical solution of convection-dominated transport problems. A price to pay for the easy way out. Water Resources Re- search, vo1.28, 551-561, 1992.

[14] Thorenz, Kolditz, Zielke. A “method of characteristics” concept for advective tracer transport in fracture networks, (Oral presentation at ModelCARE’99, Zurich/Switzerland, accepted for publishing in IAHR redbook), 1999.

[15] Van 0s R G M, Gerritsma M I. A variable order spectral element scheme applied to the velocity-pressure-total-stress formulation of the upper convected Maxwell model J. Non-Newtonian Fluid Mech. 108 (2002) 73-97, 2002.

[16] Van Leer B. Towards the ultimate conservation difference scheme, V: a Second Order Sequel to Godunov’s Method, Journal of Com- putational Physics, Vo1.32, 101-136, 1979.

[17] Varoglu E. A finite element model for the diffusion-convection equation with application to air pollution problems, Adv.Water Resour. 5, 35-41, 1982.

[18] Varoglu E, Finn W D L. A finite element method for the diffusion- convection equation, Finite Elements in Water Resources, Proc. of 2th Int. Conf. on Finite Elements in Water Resources, London, July 1978, Brebbia, Gray & Pinder, ed., 4.3-4.20, 1978.

75

Numerical Modeling of Biological Processes: Specificities, Difficulties and Challenges

Catherine Gourlay, Marie-Hklkne, Tusseau-Vuillemin Cemagrei Unit& Qualit& et Fonctionnement Hydrologiques des

Systkmes Aquatiques. Parc de Tourvoie - P B 44 - 92163 Antony Cedex, France

E-mail: [email protected]

1 General statements

1.1 How do biological processes differ from others (physical, chemical-. - )?

The modeling of biological processes covers very large research themes, from molecular level biochemical processes within the cell, up to large scale ecological evolution. Mathematical methods involved are also very diverse: statistical algorithms, conceptual models, or numerical resolution of non-linear differential equations.

The main difficulty for the biological modeler consists in the inability for determining and using universal laws, contrarily to mechanical or physical processes. Several reasons can be raised:

- Firstly, biology and ecology sciences are relatively young sciences, and have long been dedicated to qualitative observations. First attempts to give logical quantitative formalisms are quite recent: the theory of heredity and genetic by Gregor Mandel was proposed in 1866, the theory of evolution by Darwin in 1859. Cellular and molecular biological mechanisms have been explicated quite later: the DNA structure in 1954, the photosynthesis process in 1961.. .

- Secondly, the complexity of all environmental phenomena make the observations difficult to be reliable. Elemental processes are all interrelated, and the various effects of environment factors, with non quantifi- able variables, on the biota are not easily identifiable and measurable separately.

- Finally, the more important reason may be the adaptability of the biota, which definitely defies universal laws. Genetic laws are depen- dant on erratic mutations of genes. No prediction is available for such phenomena. An example in aquatic ecosystems is given by algae adaptation: the same algae species is able to grow with various rates and

76 Catherine Gourlay, Marie-HBlBne, Tusseau-Vuillemin

kinetic, depending on the available nutrients in the ecosystem or the past environmental conditions.

However, the need for prediction of biological processes is fundamental. It implies to develop models for biological processes. Therefore, due to the inability for descriptive and mechanistic models, the modeling of biological processes requires macroscopic parametrisation. Contrarily to classical physical or mechanical phenomena, predictive models for biological processes may consequently contain some numerical parameters that need to be calibrated to fit experimental values, without having a mechanistic signification.

1.2 1.2.1 Enzymatic kinetic

The Michaelis- Menten formalism (1911)

In 1911, Michaelis and Menten proposed a formalism for the enzymatic formation of a product from a substrate. The enzymatic process is represented with a simple two-steps reaction:

E + S +ES S E + P

with E the enzyme, S the initial substrate and P the final product. Several hypothesis are implicitly made: - the enzyme combines directly with the substrate - the binding is not always productive, the combination ES can dis-

- the enzyme is not consumed by the process. Supposing a stationary state, a simple manipulation of the kinetic

equations leads to the establishment of the famous relation of the product formation kinetic:

sociate

(1.1) k2 + k3 where V,,, = k3[E], K, = -

kl dP S - = Vrnax- d t K,+S

is the maximum rate of the reaction and depends on the enzyme concentration. Ks is called the Michaelis-Menten constant, or half- saturation constant (homogeneous with a concentration).

Figure 1.1 shows the formation speed of the product, depending on the substrate concentration. Under high concentrations of the substrate, all active enzyme sites are saturated. The rate of the reaction is maximum and depends only on the enzyme concentration and formation kinetic of P from the enzyme-substrate complex. When the substrate is less concentrated, the formation kinetic is slowed and depends on the substrate concentration. The formation of the product is limited by the formation of the enzyme-substrate complex.

Numerical Modeling of Biological Processes: . . 77

V,x=lO, KS=7 0 ’ 50 100 150 200 250 v S

Figure 1.1 concentration.

Kinetic of product formation as a function of the substrate

Assuming lc3 is much less than kz, then KS is equal to the dissociation constant of the ES complex. Ks is thus a useful measurement of the affinity of the substrate for the active site.

In this case, the three parameters correspond to thermodynamic well established constants. They can be assessed from molecular thermodynamic study. The Michaelis-Menten formalism allows to properly describe most enzymatic reactions, and is widely applied in molecular biology.

1.2.2 Generalisation to biological processes with limiting factors

Many other biological processes have been showed to follow this kind of kinetics, which makes the Michaelis-Menten equation so famous and powerful. In 1942, Jacques Monod proved that the bacterial growth could be modeled empirically by the same equation as a function of the substrate. The Michaelis-Menten formalism was consequently extended to macroscopic observations, for which constants do not represent thermodynamic molecular phenomena, but integrate complex unknown molecular phenomena that lead to the limitation of the growth by the substrate availability.

For example, Lancelot et al. (1991) chose to describe the kinetics of the grazing (i.e. “eating”) of bacteria by heterotrophic nanoflagellates with such a model, as well as the proteins synthesis of phytoplankton as a function of nitrogen (Figure 1.2). It is interesting to note that those two rates are expressed respectively as “per flagellates biomass unit” and “per protein unit”, which means that the global rate of the reaction is obtained by multiplying the Michaelis-Menten kinetic by either the flagellates biomass or the proteins concentration.


. P O / , , , ,

i 0 o 1 2 3 4 -- bacteria, 1 09. 1-’

Relationship between bacteria uptake by heterotrophic nanofagellate and food concentration

e a ‘ 10 20 30 40 horg.N, pmole 1-’

Relationship between specific protein synthesis and inorganic nitrogen

Figure 1.2 Menten kinetics. (from Lancelot et al, 1991)

Examples of biological processes described with a Michaelis-

1.3 Principles of a model construction: calibration- validation-extrapolation

As explained above, the kinetics of biological processes are based on a few simple types of equations (Michaelis-Menten, first order kinetics. . . ) that imply some numerical parameters. On the contrary to chemical constants that should be assessed by a theoretical thermodynamic con- siderations, macroscopic parameters cannot be assessed a priori. Mo- roever, because living organisms are adaptatable to their environment, biological processes cannot so far be described with a set of “universal” parameters.

The calibration of these parameters is necessary because the calibration is achieved on the basis of a set of experimental data, that the modeler tries to reproduce by adjusting the numerical parameters. This means:

- One assumes that the shape of the equations is correct, only the parameters need to be adjusted.

- No biological modeling can be reasonably achieved without data. - The estimation of the parameters includes the uncertainty linked

to the data. - Both the extent and the quality of the set of data can limit the

identifiability of the set of parameters (which is not guaranteed!). The validation of the parameters consists in performing a priori

simulation with the model and the calibrated parameters that fit another independent set of data. The validation is absolutely needed if one wishes to use the model in a predictive way. The parameters are considered validated under well-defined environmental conditions, that are those under which the calibration and validation data sets were obtained.

The extrapolation consists in performing simulations with a vali-

Numerical Modeling of Biological Processes: . . 79

dated set of parameters, under environmental conditions out of the strict domain of validation.

The calibration-validation procedure is common to all fields for which numerical modeling is used. An illustration in the field of biology in aquatic ecosystems is given in Figure 1.3 that represents the dose- response curve of an small aquatic crustacean (Duphniu mugnu) to copper. After calibration and validation of the toxicity, the effect of a copper ligand (EDTA) can be assessed.

.

Calibration@lack)and validation(white) Extrapo1ation:in the presence of a ligand(EDTA)

l:::]

I . . \ c u 6 n o ~ B w 4

L

Figure 1.3 Dose-response curve of Copper toxicity to Daphnia magna: lethal effect (% of died population) against the concentration of copper in the solution. Left: calibrated curve from data (black triangles) and validation with another set of data (white triangles). Right: extrapolation: evaluation of the role of EDTA on copper toxicity. From Tusseau-Vuillemin et al. (2004)

1.4 Numerical difficulties: non linear phenomena Among more classical examples stands the limitation of algal growth by available nutrients. This is widely used id the biogeochemical models that predict the oceanographic primary production as coupled with the geochemistry of nitrogen (Tusseau et al. 1997).

However, despite its being a crude simplification of phenomena, the simple Michaelis-Menten equation may present numerical difficulties. In- deed, the striking point is that, even such a very simple process is described with a highly non linear coupled set of equations that needs to be numerically processed. A very simple model describing the evolution of phytoplankton in a closed batch, where only nitrate is the limiting factor, can be as follows:

80 Catherine Gourlay, Marie-Hklkne, Tusseau-Vuillemin

p being the stoechiometric ratio between [Phy] and [NO3]. Three simulations are shown below (Figure 1.4), the nutrient limitation being more or less severe ( K ~ 0 3 varying from 5 mmol/L to 0.2 mmol/L).

Figure 1.4 Simulations of phytoplankton biomass and nitrate concentrations as derived from the simple model above, with three different values of half-saturation constant. (Vmax=0.5 mmol L-l. h-l; p=l; kgraz= 0.05 % h-l)

These simulations have been obtained in a simplistic manner, with an Eulerian scheme and a 0.1 h time step. More powerful techniques (typically 4th-order Runge-Kutta algorithm, with fxed or variable time-step) would be necessary in order to accurately simulate the sharp variations of [Phy] and [NO;] (when kinetics are very rapid, due to the lack of limitation for instance).

2 Aquatic biogeochemical modeling 2.1 Definitions and objects of the modeling The main process that converts inorganic compounds (COa, NOS,. . . ) into organic matter is photosynthesis. The solar energy allows the building of complex, organic molecules based on very basic inorganic compounds. Photosynthesis allows the primaryproduction of organic matter.

Organic matter then propagates through the trophic chain (Figure 2.1). At each step of this chain, mortality and wastes produce detritic organic matter. This organic matter is used by heterotrophic bacteria, that catalyse its reduction into the initial inorganic compounds and use the gained energy for their metabolism. This process is called the biodegradation of organic matter. Respiration consists in the final degradation of organic matter into COa.

Photosynthesis and biodegradation are of importance in aquatic systems because those two processes control the dissolved oxygen level in water. Basically, oxygen is produced by algal photosynthesis and atmospheric exchanges and consumed by the ecosystem respiration, including

Numerical BAodeling of Biological Processes: . . ~ 81

the m~n~ralisation oE organic matter by heterotrophic bacteria (Figure 2.1).

Figure 2.1 systems.

A simplified view of the organic matter cycle in aquatic

In ~ ~ w - . ~ ~ r t u r b e d rivers, these processes contribute to an overdi balanced oxygen budget. Conversely, the anthropic pressure leads to a considerable input of organic matter to rivers that Aow thro~gh urban- ized and industrialised regions. Treated wastewaters or direct sewer ove~~ows during rain events account for a sign~ficant part of it. Con- seque~t~y, oxygen consumpt~on due to organic matter deg~adat~on in the ecosystem may exceed its formation by photos~thes~s, leads to the desoxygenation of the ecosystem and affect all living o r g ~ ~ s m s ~

The high discharge of nutrients in anthropic media may also unbalance the global Carbon and oxygen cycle, by leading to an excess of primary production. This process is called eutrophication of aquatic ecosystems.

The b~o~eochemical models generally aim at representing process@s schemed in Figure 2.1 and axe extensively used for various purposes. These models allow to better understand the oceanic ecosystems (algal bloom dynamics, organic carbon cycles) and the freshwaters dpslmics. They have also been applied as management tool for pred~ct~ng eutrophication as a response to an excess of nutrients, or the impact on the e ~ o ~ y s t e ~ s of anthropic pressure, through specific planning for instance: wastewater tre~tment plant construction, industrial eBuents discharge, hyd~au~ic arra~~ements. . They have been used also to emhate the effect of the climate change on the aquatic ecosystems.

2.1.1 Biogeochemical model purposes

82 Catherine Gourlay, Marie-HQlQne, Tusseau-Vuillemin

A wide variety of such models can be found in scientific literature (Arhonditsis and Brett 2004). They greatly vary in terms of number of variables and processes described, time and space scales. . However, they are all based on two majors processes: (1) the phytoplankton dynamic and the primary production and (2) the organic matter biodegradation. In the following sections we will present basics of the modeling for each process.

2.2 Modeling of phytoplankton growth Molecular phenomena involved in photosynthesis are now well established. Photosynthesis is divided into two steps, the solar energy if transformed into chemical energy that is stocked in ATP and NADPH molecules, then, the chemical energy allows to build organic matter from CO2. this process is described by the Calvin cycle.

The global macroscopic chemical equation for primary production is therefore:

Although cellular processes are well known the modeling of phytoplankton dynamics remains very difficult. First because the difficulty to understand the complexity of biological phenomena within an algal cell makes an extended model of microscopic phenomena impossible. A macroscopic approach is therefore used, based on the main factor that in- fluences the dynamics of the phytoplankton. Usually, light and nutrient availability are the two processes that govern the growth of phytoplankton. As shown above, the limitation of the growth of phytoplankton by the nutrient availability is modeled by Michaelis-Menten equations (Fig- ure 1.2). The influence of light is commonly modeled by an exponential equation:

growth rate = - d A = p = pm--e Ed (1-3)

d t E: Steele equation(Even et a1.1998)

Secondly, phytoplankton species are of great diversity: size, shape, elemental composition, behaviour in media. Consequently, each phytoplankton species reacts differently to nutrients availability, to light, temperature, etc- . . In modeling, the difference among species is represented by using different numerical values for parameters: Michaelis-Menten parameters, as well as others, can be adjusted for each algae species in order to express the diversity of species. As an example, parameters can be

Numerical Modeling of Biological Processes: . . . 83

measured in the laboratory using radiotracer techniques for specific algae species (the method was first developed by Steemann-Nielsen, 1952)

However, it is impossible to explicitly describe each species in an aquatic ecosystem: in a lake about one to two hundreds different species co-exist. In a classical global water quality modeling (marine of fresh waters), several major classes of species are defined that correspond to various ecological groups:

- Diatoms. Their particularity is their need for silica for the building of their test. They are highly limited by silica and nutrients, but do not need high levels of light.

- Chlorophycae (or green algae). They usually grow optimally under higher light exposure and are less sensitive to nutrients than diatoms.

- Cyanobacteria. This class usually does not represent a large amount of algae, but may be of great concern in some media because of their toxic potential.

Table 2.1 Values of some parameters involved in phytoplankton dynamic modeling in the RIVE model developed for the modeling of the river Seine quality. (From Even et al. 1998)

Parameter Diatoms Chlorophycae Photosynthesis max rate Pmaxp- l) 0.25 0.5

N-limitation KN(pgN.L- 1 ) 70 70 P-limitation Kp(pgp,L-l) 15 46 Si-limitation KSi(rngSiOz.L-i.

(Steele model) Growth ' max growth rate Vrnax(,- 1) 0.06 0.12

constants

_ - 0.42 h-l) Lysis mortality rate(h-') 0.004 0.004 Sedimentation 0.004 0.0005 velocity(m.h-l)

On Table 2.1 are shown some parameters used on the RIVE model (Even et al. 1998) to simulate the phytoplankton dynamic in the river Seine (France). The phytoplankton compartment is divided into diatoms and chlorophycae.

2.3 Usual modeling of organic matter biodegrada- t ion

2.3.1

Organic matter (OM) is commonly quantified by chemists through its carbon content (mg C/L). However, because one of its main impact on the environment is the consumption of oxygen for its oxidation, other ways of quantification have been developed, such as COD (Chemical

Organic matter quantification and biodegradation


Oxygen Demand) and BOD (Biological Oxygen Demand). COD represents the amount of oxygen necessary to achieve the chemical oxidation of the sample. BOD represents the amount of oxygen that is biologically consumed (or biodegraded) in the sample if left alone in the dark for a given period of time (usually five days). Indeed, not every organic molecule can undergo biodegradation. Those compounds that can undergo biodegradation are called biodegradable compounds and the BOD measurement is an evaluation of the organic matter that is biodegradable within five days.

According to microbiologists, only small monomeric substrates (such as sugars) can be directly assimilated by bacteria and thus truly biodegraded. Most of other large organic molecules, such as proteins, first undergo hydrolysis usually catalysed with extracellular enzymes. That is the reason why one can distinguish between slowly and rapidly biodegradable compounds, based on the kinetics of the microbial process of biodegradation. Therefore, the description of biodegradation processes usually implies to distinguish between the organic substrates, i.e. to consider not only a global pool of organic matter, but several compartments.

2.3.2

The first approaches to model the biodegradation of organic matter are derived from the Streeter and Phelps (1925) model. In this model, the rate of organic matter degradation is simply assumed to be proportional to the organic load:

Streeter and Phelps model (1925)

In order to take into account the different susceptibilities to bacterial attack of various classes of compounds making up the overall organic matter, several authors suggested to consider several organic matter fractions, [OM],, each with its own first-order degradation constant.

This simple approach leads to consider the degradation as a chemical property of the organic matter, without taking into account the bacterial activity.

2.3.3 River water quality models

Based on a complete experimental study of the degradation of proteins, Billen (1991) proposed a modeling framework that reconciles bacterial


ecology and geochemistry. Two types of macromolecules are considered (HI and H z ) , either rapidly or slowly hydrolysable, plus a pool of monomers (S) that are directly assimilable by bacteria (B). The system is simply described with the following set of differential equations, that reflect mass transfers within the OM compartments (all expressed as mg C/L). This leads to the following equations (Billen, 1991):

S - = Y * bmax- dB d t K , + S B - kdB

Bacterial hydrolysis of each type of organic matter is described by Michaelis-Menten kinetic, in which B acts as enzymes. The bacterial growth is also described by a Michaelis-Menten equation, in which the substrate is the limiting factor (Monod Equation). P1 and P2 represent the potential sources of organic matter to the system. ei m, represent the maximal rates of hydrolysis of H1 and H2, K H ~ and K H ~ being the corresponding half-saturation constants. b,, is the maximal bacterial growth rate, Ks is the half-saturation constant for the growth, Y is the yield of growth ( Y ~ 0 . 3 ) and kd is the mortality rate.

These parameters have been experimentally determined in case of pure protein substrates by means of radiolabelled tracers (Billen, 1991). This is however not always possible for complex and unknown mixtures. A two-steps calibration-validation procedure is then needed (see e.g. (Tusseau et al., 1997).

2.3.4 Activated Sludge Models (ASM)

A similar view comes from the wastewater treatment field. The main cost of the most common biological treatment (activated sludge) is linked to the providing of dissolved oxygen to bacteria for metabolising the organic matter. Therefore, the question of how much and how oxygen has to be brought to the biological treatment has long been studied. Similar models as those used in microbial ecology have been developed, with all variables expressed as Chemical Oxygen Demand (mgOz/L) rather than mgC/L. Because organic matter and bacteria are much more concentrated in the aeration basin of an activated sludge wastewater treatment plant than in the natural waters, the processes are not described exactly

86 Catherine Gourlay, Marie-HdBne, Tusseau-Vuillemin

in the same way and the parameterisation is usually much coarser.

dxs ~ = - K H X S , example after Henzeetal (1987) dt d x H SS XH - b H X H

In this case, Ss represents the directly assimilable substrate, X s , the hydrolysable substrate, and X H the bacterial biomass. This type of models is mainly used for the optimisation of wastewater treatment plant. They will be further developed in the following section.

ss + Ks = P H

2.4 Complex biogeochemical modeling of aquatic ecosystems

The global aquatic biogeochemical models aim at representing the dynamic of organic matter, nutrients and oxygen in the aquatic ecosystems. Numerous models have been developed in the last fifteen years, based on the mechanistic description of interrelated processes (such as described schematically in Figure 2.1). They cover a wide range of complexity, ecosystems type (lake, estuaries, oceans- . . ). . . An interesting compar- ative evaluation of these models is proposed in Arhonditsis and Brett (2004).

Biogeochemical processes are usually also coupled with hydrological models, in order to reproduce spatial and temporal features (see for instance Tusseau-Vuillemin et al. (1998) for 3D-marine ecosystem modeling, or Even et al. (1998) for river ecosystem modeling, Figure 2.2).

The biogeochemical model presented above is composed of twenty- six state variables. Coupled with a hydrodynamic module, it allows to simulate the evolution of the water quality along the river, and the impact of some anthropogenic activities, such as waster water treatment plant effluents or sewer overflows during rain events. Various examples of application to the Seine river of this model could be found in Billen et al. (1994), Garnier et al. (1995), Even et al. (1996) or Even et al. (1998).

An application of this model is shown in Figure 2.3. The dissolved oxygen in water has been measured downstream the Paris town, at the location of a combined sewer outlet. During dry weather, the daily photosynthesis-respiration cycle was observed. It could be simulated , although with a less stringent amplitude. When a sewer overflow took place during a storm event, high amounts of biodegradable organic mat-


Air-water interface

Reaeration Reoxygenation

I

Phytoplankon,

Lysis I t t t

Dissolved, particulate degradable, refacto.y Excretion, lysis

Uptake Sedimentation

Uptake / Mineralization

Respiration Nitrifying bacteria

w & Sedimentation Sedimentation Water sediment interface Denitrification Aerobic respiration Mineralization

Figure 2.2 Conceptual scheme of the biogeochemical model used for the modeling of the river Seine (France) ecosystem. (from Even et al, 1998)

Figure 2.3 0 2 concentration in the river Seine downstream Paris : daily simulation and impact of a combined sewer overflows during storm event. Measurements and simulations. (from Even et al, 1996)

88 Catherine Gourlay, Marie-HGne, Tusseau-Vuillemin

ters in the river were discharged, which induce bacterial oxygen consumption. Consequently, the oxygen concentration felt dramatically. The model was able to properly simulate such an impact, as well as the recovery of the ecosystem after the rain event.

3 Modeling of wastewater biodegradation

3.1 ASM1: a widely used wastewater treatment plant model

Recently, thanks to the increased public concern for the protection of the environment, wastewater treatment plants have been required to meet more stringent effluent standards to reduce the pollution load on the receiving water bodies. The main pollution from treated wastewaters are organic matter and nitrogen discharge. This, in turn, has challenged the efficiency of existing treatment plants and necessitated the upgrade of the existing biological carbon and nitrogen removing plants to achieve a better performance that will ensure an improved effluent quality. In order to meet these challenges, mathematical models have been used widely and shown to be an effective tool not only in upgrading but also in control and optimization of the biological treatment for nitrogen and organic carbon removing plants.

In this context the activated sludge model number 1 (ASM1) of Henze et al. (1987) has become rather popular in the wastewater treatment field and applied successfully for dynamic simulations of the treatment plants. This model has been the first of a growing “family”, developed under the auspices of the International Water Association. These models are called “white box” models, in that they are based on first engineering principles, the equations being derived from general mass balance equations applied to mass and other quantities, exactly as those described above. In addition to the activated sludge model, a hydraulic model describes tank volumes and hydraulic behaviour within the plant (for a review, see Gernaey et al. (2004)).

3.1.1

The removal of carboneous and nitrogen pollution from wastewaters is achieved mostly by applying biological treatment: as exposed above, heterotrophic bacteria use organic carbon as substrate for their growth and produce COz. Under aerobic conditions, 0 2 is used to provide oxygen. Under anoxic conditions, some heterotrophic bacteria are able to use nitrates to provide oxygen to biodegrade organic carbon and produce gaseous Nz. This process is called denitrification.

Some basics on wastewater treatment plants

N ~ ~ ~ e r ~ c a l ~ o d e l ~ n g of Biological Processes: . . 89

ianrrmoaiiumn-oxidizing bacteria (AOB) uptake (N nitrite (NO,). Nitrite is then oxidized into nitrate (NO,) by nitrlte- Q x i d ~ ~ ~ g bacteria (NOB). OAB and HOB are called ai~to~roph~c bacte-

carbon (CQz) and produce energy fr01-11 ). Both processes, which occurs generally

s ~ x ~ ~ ~ ~ a ~ e o u s l y in aerobic cond~tions, are called n ~ t r ~ ~ c a ~ ~ o ~ , The most common technology employed for nitrogen and carbon re-

~ o ~ ~ n ~ i s based on simul~aneous carbon ox~dat~on~ nitri~cation and den- ~ t ~ ~ ~ ~ ~ t i Q ~ ~ ~ o c e s s ~ s in a single sludge stream contain~ng all types of ~ a c t ~ ~ i a a5 i l ~ t ~ s t r ~ t ~ d in the Figure 3.1 r e ~ r e s e n t ~ ~ ~ the Neaux ( ~ a n c e ) ~ a ~ t e w a t ~ ~ t ~ e a ~ ~ e ~ ~ plant.

1-Anaerobic tank 2-Anoxic tank

Figure 3.1 The wastewater treatment plmt of Meaux ( ~ a n c e ) ~ One ~ ~ ~ s t ~ n g u i ~ h e s the Marne river (top right), which receives the eBuent. The plant is c o ~ s t ~ t ~ t e d of two inde~endent files, each with three circular tanks (anaerobic, anoxic and aerobic ones), and a settling tank.

3.P,3

The ASMl model describes the Chemical Oxygen demand (COD) and nitrogen removal along the biological treatment of an effluent based on ~ e c ~ a n ~ s t i c models for bacteria dynamics. The treatment is described by eight processes. A conceptual scheme of the model is shown in Figure 3.1.

The model if coniposed of thirteen state variable and nineteen parameters, the whole model and the coupled kinetics of the processes are usually formalised within a matrix, where the components are listed as columns, and the rate at which they are transfor~ed or created from or to other components are displayed in the lines that correspond each to a ~ ~ r t ~ c ~ l a r process. The matrix for the ASMl model is presented in Table 3.1 below. Let us take the example of the hydrolysable products

~ ~ ~ ~ ~ ~ t ~ t ~ ~ ~ of the model and ~~m~~~~~~ solving

90 Catherine Gourlay, MarieHQlQne, Tusseau-Vuillemin

Slow1 Rapidly eterotr ophic

Again, this example illustrates the complexity of the numerical solving of such kind of problems, as the equations are non linear and non linearly coupled.

The numerical solving of such systems requires powerful algorithms. For example, the GPS-X software (www.hydromantis.com) proposes several solutions, depending on the problem to solve. Very stiff problems require implicit or semi-implicit schemes if one wishes to avoid tiny time- steps. Runge-Kutta-Fehlberg algorithms (adaptive time-steps) provide also good results, especially when the simulation is highly dynamics and possibly discontinue (Press et al. 1992).

Heterotr ophic bio&%c&ble

Heter otroph c biomass

Autotrophic xm sm- biomass

Assimilation

Particulate Soluble xBH organic + organic

Hydrolysis Ammonification Nitrification x,, o2 7f coz

V

+biodesadable -+’ biomass + biomass SS xBH XP

Figure 3.2 Scheme of the ASMl model, including coupled carbon and nitrogen removal processes.

3.1.3 Practical application

These models allow to simulate the functioning of wastewater treatment plants, as a response to the quality and quantity of the influent and the


operating conditions. Therefore, it provides a way of a pr ior i dimension- ing the plants and optimising the treatment and the cost. The inputs to the model are:

- The composition of the influent: a time series of the fluxes of the different compartments identified in the model: X s ; Ss , X B H X P in COD, X N D , SND, S N H , X P A .

- The hydraulic characteristics (volume of the basins, hydraulic fluxes between them).

- The operating conditions (sludge extraction, aeration sequences, temperature, etc. . . ). The outputs of the model are:

basin of the plant. - The full simulation of all the compartments of the model, in any

- The composition of the effluent in terms of COD and nitrogen forms. These models rely on an important set of parameters, that need ei-

ther to be experimentally determined, or numerically calibrated and validated. Because the activated sludge models have been continuously developed and tested for about fifteen years, the default parameter set can be used with confidence. However, it is sometimes necessary to adjust some of the parameters and then validate this calibration (see example in Figure 3.3). Again, this adjustment absolutely requires an experimental support.

3.2 The estimation of the composition of wastewater samples through inverse modeling

As mentioned above, the wastewater treatment plant models are forced with boundary fluxes for all the components. Therefore, the composition of the influent in terms of readily biodegradable (Ss ) , slowly biodegradable ( X S ) or refractory ( X I ) organic matter is needed. Although some default values are available in wastewater treatment plant models, it is of interest to determine the composition for the specific wastewater coming into the plant (which may change, during rain event for example), in order to better simulate the wastewater treatment and optimise each treatment step.

Only indirect methods allow to experimentally determine those fractions of organic matter. Respirometry is a coupled experimental and numerical technique that allows to asses the composition of organic matter in water in terms of components used in the models.

3.2.1 Principles of respirometry

Respirometry consists in incubating a sample to analyse and continuously recording the dissolved oxygen concentration that tracks the

92 Catherine Gourlay, Marie-HQlkne, Tusseau-Vuillemin

default parameters 1st campaign data

Calibration: calculated parameters

E . 1st campaign data x

4

3

2

1

0

y:

0 6 12 18 24 30 0 6 12 18 24 30 Time(hours) Time@ours)

Validation: calculated parameters 2d campaign data

4

Time(hours)

Figure 3.3 Nitrate concentrations (mgN/L) as a function of time (hours) in the aerated tank of the wastewater treatment plant of Meaux (Figure 3.1) Crosses: experimental data; line(top left): simulation with default parameters (ASM1); line (top right): simulation with calibrated parameters; line (bottom): simulation with calibrated parameters with another set of experimental data. After Lagarde et al., submitted to Environmental Modeling and Software.

biodegradation by bacteria. The fractions of organic matter corresponding to the various kinetics of degradation are obtained by optimising the initial conditions (i.e. the composition of the sample in terms of biodegradability fractions) and the model’s parameters on experimental data. The model is selected to reproduce the bacterial degradation dynamics occurring during the biotest. Modeling makes it possible to reproduce the oxygen uptake rates (OUR), i.e. the derivative of oxygen versus time. The equation below shows an example of OUR expression related to some of the components of the system and to some of the parameters. This equation corresponds to the oxygen consumption as


modeled by the ASMl model (Table 3.1), when only carbon biodegradation is taken into account (no nitrogen removal).

An example of an experimental respirogram and of the corresponding

Based on this protocol, it is possible to assess the composition of simulation after calibration is given in Figure 3.4.

various types of wastewater, as illustrated in Figure 3.5.

3.2.2 Identifiability and optimisation problem

The problem to solve here consists in optimising the initial components (Ss, X S , X B H , X I ) under a mass constraint, so that the simulated respirogram fits the experimental one. This is inverse modeling and allows to determine the composition of the sample. This is complicated by the fact that there are practically no “default” parameter values for these systems, that are wastewater and not activated sludge. Because they are highly variable in time and space, wastewater is difficult to characterize with a unique parameter set. Therefore, the problem is not only to optimise the initial components, but also the kinetic parameters of the model. Typically, seven parameters and four fractions need to be opt imised .

Two types of difficulties are encountered for solving this problem. First, the set of equations is very complex, non-linear and coupled, and the number of parameters of initial values of components to optimise is rather high. The theoretical identifiability of such a system is not straightforward established, nor guaranteed.

Moreover, because the technique is not very easy to handle either, the data may be noisy, not continuous (the system must be re-aerated otherwise it goes to anoxia), not homogeneous (this is inherent to wastewater and sludge). . . And this brings difficulties that are related to practical identifiability. This means that, beyond the fact that the problem might be theoretically solvable, the quality of the experimental data may in practice make the solving impossible. Moreover, because non-linear constrained optimising is not very easy to handle either, the difficulty encountered for solving the problem might be linked to a bad estimation of the gradients of the cost functions used for the optimisation.

Those questions are of great interest and mix experimental work with theoretical applied mathematics. These subjects are still in the field of researchers. A first approach can be found in Dochain et al. (1995).

Table 3.1 Matrix of ASMl model for wastewater biodegradation, includmg nitrogen treatment.

r. - C.V. . 1 - I LIP1

l1

1

1 14 -

Numerical Modeling of Biological Processes: . 95

600 5002 4008 300 oil

200# 100 0

0 4 8 12 16 20

Time(bours) Time(hours)

Figure 3.3 Example of a respirogram. Left: the respiration data md s~mulatio~ ( - ~ ~ ~ / ~ ~ ) . Right: the corresponding simulation of all the components of the model. (from Lagarde et al., submitted)

Combined sewer so

40 a 0 6) 3 30 i? CM

g 20

10

0

Rapidly Slowly Heterotrophc Inert bioepdable bioepdable biomass .fraction

Figure 3.5 Composition of wastewaters from combined and separate sewers, identified by means of respirometry, following the ASMl model. (from Lagarde et al., submitted)

Biogeochemical models developed for the description of aquatic ecosystems or wastewater biodegradation models used operationally ;ts man. agement tools have been presented here. They both illustrate the classi-

4 Strategies for a better modeling of bio-

logical processes in aquatic systems.

96 Cat her ine Gourlay, Marie- H616ne, Tusseau-Vuillemin

cal mechanistic approach developed in biological aquatic modeling. This approach is relatively recent in the field of aquatic sciences, and has already been proved to be efficient for various purposes. The ASM model for instance is now widely used by engineers for the design and the management of wastewater treatment plants.

The examples cited here also point out the main difficulties encountered by “biological” modelers.

Firstly, aquatic biological models are mainly macroscopic and need a calibration/validation procedure before being applied. This procedure is based on experimental observations. Consequently, no model can be achieved without experimental data. Moreover, the validity of the model would always be dependent on the uncertainty of experimental data and on the extent of available observations.

Secondly, the examples cited in this paper show the mathematical difficulties encountered by biologists: biological models need appropriate numerical methods because (1) the processes involved are highly non linear and (2) they depend on numerous parameters (most often more than ten), with many state variables, which may involve some theoretical or practical identifiably issues. The optimisation of a non-linear cost function depending on several parameters is a very hard task that can be tackled with numerous strategies. Among these, it proves interesting to estimate the cost function gradients with special care. For example, Lagarde et al. (submitted) used a secondary piece of code generated by automated differentiation based on Odysse software, described by Faure and Papegay (1998). The optimising algorithm was based on the L-BFGS-B low-memory quasi-Newton algorithm proposed by Zhu et al. (1997). It is based on the gradient projection method and uses a limited memory BFGS matrix to approximate the Hessian of the objective function. These types of powerful algorithms are necessary for the resolution of complex non linear models developed in biological modeling, as some of them were briefly presented here.

Finally, the modeling of complex biological systems is mainly an inter-disciplinary task. To improve existing models, biologists and ecol- ogists need to develop more collaboration with applied mathematicians. This is fundamental in order to be able to improve the robustness, the validity, the efficiency, and finally the practical utility as management tools of aquatic biological models.

Cited References [l] Arhonditsis G B, Brett M T. “Evaluation of the current state

of mechanistic aquatic biogeochemical modeling.” Marine Ecology Process Series 271: 13-26, 2004.


[a] Billen G. Protein degradation in aquatic environments. Microbial enzymes in aquatic environments. R. J. Chrost, Springer-Verlag:

[3] Billen G, Garnier J, Hanset P. “Modeling phytoplankton development in whole drainage networks: the Riverstrahler model applied to the Seine river system.” Hydrobiologia 289: 119-137, 1994.

[4] Dochain D, Vanrolleghem P A, Van Daele M. “Structural identifiability of biokinetics models of activated sludge respiration.” Water Research 11: 2571-2578, 1995.

[5] Even S, Poulin M, Garnier J, Billen G, Servais P, Chesterikoff A, Coste M. “River Ecosystem Modeling: application of the PROSE model to the Seine river.” Hydrobiologia 374 - 375: 27-45, 1998.

[6] Even S, Poulin M, Mouchel J M, Billen G. Simulating the impact of CSO’s from greater Paris on the Seine river using the model Prose. Seventh International Congress on Urban Drainage Storm Water, IAWQ. 1996.

[7] Faure C, Papegay Y. Odysee User’s guide. Version 1.7., INRIA. 1998.

[8] Garnier J, Billen G, Coste M. “Seasonnal sucession of diatoms and chlorophycae in the frainage network of the river Seine. Observations and modeling.” Limnology and Oceanography 40: 750-765, 1995.

[9] Gernaey K, Van Loosdrecht M C M, Henze M, Lind M, Jorgensen B. “Activated sludge treatment plant modeling and simulation : state of the art.” Environmental Modeling and Software 19(9): 763-783, 2004.

[lo] Henze M, Grady C P L J, Gujer W, Marais G V R, Matsuo T. “A general model for single-sludge wastewater treatment systems.” Water Research 21(5): 505-515, 1987.

[11] Lagarde F, Tusseau-Vuillemin M H, Lessard P, 6duit A H, Dutrop F, Mouchel J M. (Submitted). “Variability estimation of urban wastewater biodegradable fractions by respirometry.” Water Re- search.

[12] Lancelot C, Billen G, Barth H. “The dynamics of Phaeocystis blooms in nutrient enriched coastal zones,.” Water research pollution reports n”23, EC. 1991.

[13] Press W H, Teukolsky S A, Vetterling W T, Flannery B P. Numer- ical recipes, the art of scientific computing, Cambridge university press. 1992.

[14] Steemann-Nielsen E. “The use of radioactive carbon (14’) for measuring organic production in the sea.” Journal du conseil parmanent international pour l’exploration de la mer 18: 117-140, 1952.

123-143, 1991.

*


[15] Tusseau M H, Lancelot C, Martin J M, Tassin B. “1D coupled physical-biological model of the north-western Mediterranean Sea.” Deep Sea Research I1 44(3-4): 851-880, 1997.

[16] Tusseau-Vuillemin M H, Gilbin R, Bakkaus E, Garric J . “Perfor- mance of diffusive Grandient in Thin Films in evaluating the toxic fraction of copper to Daphnia magna.” Environmental Toxicology and Chemistry 23(9). 2004.

[17] Tusseau-Vuillemin M H, Mortier L, Herbaut C. “Modeling nitrate fluxes in a open coastal environment (Gulf of Lions): transport versus biogeochemical processes.” Journal of Geophysical Research

[18] Zhu C, Byrd R H, Lu P, Nocedal J . “Algorithm 778: L-BFGS- B: Fortran subroutines for large-scale bound-constrained optimization.” Acm Transactions on Mathematical Software 23(4): 550-560, 1997.

103: 7693-7708,1998.

Some Recommended References 1. Michaelis-Menten formalism A very interesting bibliography on Michaelis-Menten formalism and applications in biological processes modeling in available in:

http://math.fullerton.edu/mathews/n2003/michaelismenten /MichaelisMentenBib/Links/MichaelisMentenBiblnk~.html

2. Biogeochemical models For a review of most of most of the biogeochemical models developed in the past fifteen years, one should refer to:

Arhonditsis, G. B. and M. T. Brett (2004). Evaluation of the current state of mechanistic aquatic biogeochemical modeling. Marine Ecology Process Series 271: 13-26.

3. Wastewater treatment plant models One may find some more information about the Activated Sludge Models (ASM) on the International Water Association (IWA) website:

http://www.iwahq.org.uk/

99

Ecological Simulation of Red Tides in Shallow Sea Area

Deguan Wang College of Environmental Science and Engineering, Hohai University,

1 Xzkang Road, Nanjing, 210098, China

1 Introduction

Among the thousands of species of microscopic algae at the base of the marine food chain are a few dozen which produce potent toxins. These species make their presence known in many ways, ranging from mas- sive “red tides” or blooms of cells that discolor the water, to dilute, inconspicuous concentrations of cells noticed only because of the harm caused by their highly potent toxins. The impacts of these phenomena include mass mortalities of wild and farmed fish and shellfish, human intoxications or even death from contaminated shellfish or fish, alter- ations of marine trophic structure through adverse effects on larvae and other life history stages of commercial fisheries species, and death of marine mammals, seabirds, and other animals. “Blooms” of these algae are commonly called red tides, since, in some cases, the tiny plants increase in abundance until they dominate the planktonic community and change the color of the water with their pigments. The term is misleading, however, since non-toxic species can bloom and harmlessly discolor the water; conversely, adverse effects can occur when algal cell concentrations are low and the water is clear. Given the confusion surrounding the meaning of “red tide”, the scientific community now prefers the term “harmful algal bloom”, with HAB as the obligatory acronym. This new descriptor applies not only to microscopic algae but also to benthic or planktonic macroalgae which can proliferate in response to anthropogenic nutrient enrichment, leading to major ecological impacts such as the displacement of indigenous species, habitat alteration, or oxygen depletion.

HAB phenomena take a variety of forms. One major category of impact occurs when toxic phytoplankton are filtered from the water as

100 Deguan Wang

food by shellfish such as clams, mussels, oysters, or scallops, which then accumulate the algal toxins to levels which can be lethal to humans or other consumers[l]. Typically, the shellfish are only marginally affected, even though a single clam can sometimes contain sufficient toxin to kill a human. These poisoning syndromes have been given the names paralytic, diarrhetic, neurotoxic, and amnesic shellfish poisoning (PSP, DSP, NSP, and ASP). Another type of HAB impact occurs when marine faunas are killed by algal species that release toxins and other compounds into the water, or that kill without toxins by physically damaging gills or by creating low oxygen conditions as bloom biomass decays. Farmed fish mortalities from HABs have increased considerably in recent years, and are now a major concern to fish farmers and their insurance companies.

The nature of the HAB problem has changed considerably over the last two decades in world. Figure 1.1 marks occurring locations in the United States. The maps show the red tide outbreaks known before (top) and after (bottom) 1972. This is not meant to be an exhaustive compilation of all events, but rather an indication of major or recurrent HAB episodes. We can see formerly a few regions were affected in scat- tered locations, now virtually every coastal state is threatened, in many cases over large geographic areas and by more than one harmful or toxic algal species. The types of resources affected and the number of toxins and toxic species have all increased dramatically in recent years around the world [21[31[41. The reasons for this expansion come from both natural and human-assisted, including: 1) species dispersal through currents, storms, or other natural mechanisms[5] ; 2) nutrient enrichment of coastal waters by human activities, leading to a selection for, and proliferation of, harmful algae[3]; 3) long-term climatic trends in temperature, wind speed, or insolation[6]; 4) introduction of fisheries resources (through aquaculture development) which then reveal the presence of indigenous harmful algae in waters formerly “free” from HAB problems[7]; 5 ) dispersal of HAB species via ship ballast water or shellfish seeding activities[’]; and 6) increased aquaculture operations which can enrich surrounding waters and stimulate algal growthlg1.

Recent years, recurrent HABs are a serious threat to the success of mussel rafts. To control them, the debate over the relative value of practical or applied versus fundamental research has heated up considerably in recent years. Someone with marine interests have always focused their resources on practical problems, but those with a traditional commit- ment to basic research have increasingly had to fight to maintain their freedom to fund quality science without regard to practical applications. Within this context, it is instructive to combine both sides. In this paper we studied the basic character of differentiation equations of phytoplankton. The results indicated that under certain environment conditions phytoplankton concentration meets with the equilibrium point at the

Ecological Simulation of Red Tides in Shallow Sea Area 101

Per-1972

W P R

c Post-1972

W H I

Figure 1.1 tom) 1972 in the U.S.

The HAB occurring locations before (top) and after (bot-

end and the differences between the equilibrium point and the concentration of phytoplankton determine the changing rate of phytoplankton concentration. Based on this, we applied water hydrodynamics model and ecological model to Pearl River estuarine and testify the results were correct. These conclusions can guide us find more efficient methods to control HABs. The objective is to highlight fundamental ecological, water dynamics, physical, biological, and chemical oceanographic question that must be addressed if we are to achieve the practical goal of sci-

102 Deguan Wang

entifically based management of fisheries resources, public health, and ecosystem health in regions threatened by toxic and harmful algae.

2 The character of equilibrium point and its impact on the changing rate of phyto- plankt on concentrat ion

According to related study on HAB, the fundamental mechanism is well understood, although it still needs improvement. Factors related to harmful algal bloom are as follows:

1. Underwater light climate (instant irradiance, attenuation coefficient, etc.)

2. Nutrients (nitrogen, phosphorus, trace elements, etc.) 3. Nitrogen / phosphorus ratio (important to species composition

and succession) 4. Hydrodynamics (nutrient transportation, stratification, upwelling,

turbulence, etc.) 5. Physiology (division rate, vertical migration, half saturate values,

etc.) Some of the factors (two, three or more) are involved in the phyto-

plankton or algae models. By observation, monitoring, calculation or analysis, the HAB could be predicted. But there is still one question that although we have known the changes of some factors when red tide bursts out, we do not know what on earth the effects of all these factors on the view of ecology.

Theories on ecological system and model have already well proved. And’we know that HAB is an ecological phenomenon and the population of algae exponentially increases when red tide breaks out.

The simplest density-dependent model of phytoplankton is:

d u a d t b - = - u ( b - U )

where u denotes the density of population for the phytoplankton, a denotes the intrinsic rate of natural increase, and b denotes the carrying capacity in certain environment.

From equation (2.1), the positive equilibrium point u1 = b can be d u d u

retrieved. If u > b, then - < 0, if u < b, then - > 0. That is to say, d t d t

u + b when t + 00. It can be seen in Figure 2.1. Now, the question is how to determine the value of ‘b’ when red tide

breaks out.


t

Figure 2.1 Integral line of u for equation (2.1).

From equation (2.1) it can be found that the larger ( b - u) is, the

larger - is. Imagining that ‘b’ is very large and the initial ‘u’ is small, what would happen to the population of phytoplankton? Apparently,

is very large at the beginning, and it will drop when the value of d t ( b - u) decreases.

The factors influencing ‘b’ value are exactly those mentioned above. The nutrients (pollutants) in the water are the main reasons that lead to algae over-grow. The equilibrium point of the phytoplankton concentration is the value when phytoplankton concentration is unchanged with time. Under certain environment conditions phytoplankton concentration meets with the equilibrium point at the end and the differences between the equilibrium point and the concentration of phytoplankton determine the changing rate of phytoplankton concentration. The equilibrium point is essentially dependent on the environmental factors. This is used to explain the phenomenon of HAB.

d u d t

3

3.1 Physical/Biological Coupling

Red tide ecological dynamic model

Despite the diverse array of HAB species and the many hydrographic regimes in which they occur, one common characteristic of such phenomena is that physical oceanographic forces play a significant role in bloom dynamics. Furthermore, the interplay or coupling between the physics and biological “behavior”, such as respiration, photosynthesis, physiological adaptation, or prey-predator relation holds the key for un-

104 Deguan Wang

derstanding red tide outbreaks. This physical/biological coupling can occur at both large and small scales. And the biological and physical mechanisms that account for enhanced phytoplankton biomass are diverse, as they must be accounted for the different physical and chemical gradients associated with the regions. Many suggested mechanisms relate to differences in nutrient availability and light. High nutrient concentrations are often found at or below the pycnocline. Physical processes which can influence nutrient exchange across a front include inter- facial friction and Ekman pumping[lO][ll], baroclinic eddies[l2I, and tidal ~tabilization/destabilization[~~]. On the other side, the growth and accumulation of individual harmful algal species in a mixed planktonic assem- blage are exceedingly complex processes involving an array of chemical, physical, and biological interactions. So the practical implications of the basic research on these aspects are large with respect to our ability to understand and predict the patterns of blooms.

There is thus a clear need to develop realistic physical models for regions subject to HAB events, and to incorporate biological behavior and population dynamics into those simulations. Models of red tide algae blooms have been developed from several different perspectives. Kamyk~wsk i [~~] [~~] examined the response of a swimming dinoflagellate to internal waves and showed that accumulation of motile and non-motile cells may occur due to an internal wave field, with the accumulation of vertically migrating cells being most significant. These models consider only the physics of the wave field and the swimming behavior of the phytoplankton, without regard to the phytoplankton response to nutrients or light. Others have examined the response of phytoplankton to the flow field of Langmuir cell^[^^][^^] or to 2-dimensional, cross-frontal circu1ation[l81, to name just two of many physical systems.

The primary method for exploring the details of the interactions of HAB populations with coastal circulation should be through the incorpo- ration of biological and physical field data into circulation models. The physical dynamics of the model can be constrained by the equations used to describe the flow and by physical data gathered in field programs. Nu- merical experiments can then examine the distribution and fate of cells under a variety of forcing mechanisms. HAB cells can initially be treated as passive tracers, but it will most likely be necessary to include factors such as grazing or physiological adaptation in the models to accurately simulate observed cell distributions. It would then be possible to show, for instance, how cell division or predator behavior will cause cell accumulation and blooms, or to evaluate the relative importance of physical losses (advection, sinking) and biological losses (grazing, cyst formation) in bloom termination. The flexibility of such models allows certain aspects of the field data to be explored in a way that would be impossible through direct sampling. Another advantage is that the behavior and


physiology of a single HAB species can be better understood and simulated than that of an entire community. This is an area long neglected in HAB research, and one that should be strengthened in the coming years. The insights to be gained from modeling studies will do much to advance our general understanding of the dynamics and consequences of HABs. In this section, we established a set of red tide ecological model, which include hydrodynamic simulation, water quality simulation and phytoplankton-zooplankton ecological simulation.

3.2 Governing equations

3.2.1 Hydrodynamic simulation

A two-dimensional, depth averaged, hydrodynamic mathematical model was applied for simulation of hydrodynamics of water body. The conservative (divergence) form of the coupled two-dimensional shallow water equations and advection-dispersion equation is given by

where q is conserved physical vector; f (q) and g ( g ) are flux vectors in the 5 and y direction, respectively and S(g) is source/sink term vector. They can be expressed respectively as:

ar,, dY

where,& = 0,272 = f v h - gh(soz + sfz) + (% + -) , S3 = f u h

+ gh(sOy - sfv) + (% + h) . h is water depth; u and v are depth-

averaged velocity components in the x and y directions, respectively; g is aY

gravitational acceleration; sox and sfzare bed slope and friction slope in the x direction respectively; soyand sfv are bed slope and friction slope in the y direction respectively.

3.2.2 Three-component ecological simulation

This section presents an overview of the three-component ecological model. Although we look phytoplankton and zooplankton as one of water quality constituents at first, in the loading terms we take their behavior into account. The equations for various water quality constituents are based on the key principle of the conservation of mass. This principle requires that the mass of each water quality constituent being investigated must be accounted for in one way or another. We trace each water

106 Deguan Wang

quality constituent from the point of spatial and temporal input to its final point of export, conserving mass in space and time.

A mass balance equation for dissolved constituents in a water body must account for all the material entering and leaving through direct and diffuse loading; advective and dispersive transport; and physical, chemical, biological transformation and prey. These aspects are concerned in loading terms.

Mathematical Representation of Physical System Input: Estuary or Lake Geomorphological Data Output: 2-D Finite Volume Network

2-D Hydrodynamic Simulation Input: Hydraulic/Hydrologic Data and Boundary Conditions Output: Velocity. Depth. and Water Surface Elevation

Water Quality Simulation Ecological Simulation: Input: water ~ ~ l i t y ~ ~ t a and Boundary Conditions Distributions. Output: Constituent Concentrations

Input: Initial Spatial and Temporal

Output: Spatial and Temporal Distributions of PhytoplanMon Population Estimates

Figure 3.1 Red tide ecological model system.

We use a three-component to represent concentrations of nutrient ( N ) , phytoplankton ( P ) and zooplankton (2) in a physically homogeneous oceanic mixed layer. Phytoplankton, aquatic plants that are mostly unicellular, take up nutrients from the water in order to photosyn- thesise. The phytoplankton are grazed upon by the animal zooplankton, which in turn provide sustenance for the higher trophic levels of the food web. Figure 3.1 illustrates the structure of the model, whereby arrows indicate flows of matter through the system, and arrows not starting or not ending at a compartment represent the input to and the losses from the modelled system.

The general mass balance equation is given by

d d - -u-(C) - .-(C) + dC - _ at dX OY

where C is concentration of the water quality constituents. Ex and Eg are diffusion coefficients in the x and y directions.


When the explicit functional forms are included, the model becomes:

d d - = -u-(N) - v-(N) + at dX dY

dN

d d - = --(UP) - -(wP) dP at d X dY

d a - = - - (UJ) - -(U,Z) at dX dY dZ

d dZ d d Z +EZ [z (z) +& (&)I +sz

(3.3)

(3.4)

(3.5)

S N , Sp and Sz is loading term. It can be expressed as:

z - (s + k ) P (3.7) XP2

P - r P - - sp 1 -- N a e + N b + c P p2 + P2 d P 2

p2 + P2 sz = - Z - dZ"

where E N , Ep and EZ are diffusion coefficients on the assumption of isotropic.

The parameter definitions of above equations are given in Table 3.1, together with the value of each parameter originally used by Steel and Henderson[lgI.

3.3 General description of numerical methods The sets of equations listed above are highly non-linear, one can hardly obtain the analytical solution for those equations unless simplified under several reasonable assumptions. The numerical solution technique is the only method to obtain the approximate solution to those sets of equations. There are many numerical methods that can be applied for this purpose. The general frame of the numerical solution procedure and the types of numerical methods are shown in Figure 3.2.

In order to give some impression about numerical model, two examples are given in following sections.

108 Deguan Wang

Table 3.1 Parameter definitions

3.3.1 Finite element model[20][21][22]

In the Auid dynamics, the finite element method based on the Galerkin weighted residual method is mostly used method in descritizing the set of equations.

1. Governing equations Equation (3.1) can be expanded into following form:

(3.9)

(3.10)

(3.11)

where, U = ha, V = hv;

Ecological Simulation of Red Tides in Shallow Sea Area

Fundamental Conservation Laws

109

Differential Formulation Integral Formulation

Selection of Solution Method

.1 .1 Analytic Solution Method

Program Code

Finite Difference Method Finite Volume Method

Figure 3.2 types of numerical methods.

General frame of the numerical solution procedure and the

Finite Element Method

After some modifications, above expressions can be reduced into following forms,

P

P

Tzxb = - pgU @Tv TZYb = C2h2 pgV J iEF T~~~ = c,-wJwI Pa sino

P

(3.16)

(3.17)

(3.18)

c2h2

Uniform Grid Non-uni form Grid Grid

Method Transfor- mation

(3.19) Pa T,,, = C,--WlWl COSO

P

Boundary Variations 1 Element Residual Method Method Method

110 Deguan Wang

where c is the Chezy coefficient; c, is wind drag coefficient; pa is the density of the air; w is the wind speed; Q is the wind direction.

2. Finite element representation of the governing equations (1) Trial function

(3.20) (3.21)

where U , V , h are the unit width discharges and water depth at any point in the element; U I , VI, h I are the unit width discharges and water depth at the nodes of the element; (PT and QT are the shape functions for the unit width discharge and water depth respectively, for triangular element with 6 nodes, they are as

Q,=

Q =

(3.22)

(3.23)

where <I, &, 6 are triangular coordinates. (2) Garlerkin integral expression Continuity equation

kg - ( E U + E V ) ] dA = - lv Q(Udy - Vdx) (3.24)

Momentum equation

Pa - - E ~ ~ - + - E ~ ~ - dA - Q,c,-wIwIsina

P2 d@ au dQ, au dx dx dy dy

+@- d m - Q,.fVdA = 0 p2c2 h2

(3.25)


a(h + .a) dY

Jn@-+@- - + @ - a - vv +@gh aV at ax a (uhv) a y ( h )

+@&dw+ fVdA = 0 (3.26)

(3) Element matrices The finite element equation can be established by applying above '

equations to each element. The equations are given as follows.

Ah1 + BIUI + BzV' = R M ~ I + CUI + PI(hI + Z b I )

(3.27)

(3.28)

(3.29)

EIUI - wi +TUI - KVI = 0

MVI + Cvl + P2(h1+ Z ~ I ) + E2vl- W2 +TVI + KUI = 0

. ahI , auI . avl where, hi = -, UI = - Vl = - I = 1 , 2 ,. ' . , n, n the number of nodes of the element;

at a t ' d t '

e

B~ = IJ g + T d x d y

B2 = JJ E Q T d x d y

e

e

R = lu Q(Udy - Vdx)

112 Deguan Wang

e

WI = / J ~ c ~ p 2 w / w l Pa sinadxdy

e

~2 = JJ ~ c , $ w l w l cosadxdy e

e

K = JJ ~ 2 w sin !PQTdxdy

Equations (3.27), (3.28) and (3.29) can rearranged as following form,

(3.30) (3.31) (3.32)

A ~ I + B ~ U I + BzK = R M U I + D ~ U I + 4 ( h I + 261) = KVI + W1 MQI + D2VI + P2(hr + Z ~ I ) = -KVI + W2

where DI = C + El +T, 0 2 = C + E2 +T. (4) Global equations The global equations can be derived by assembling the element ma-

trices and by applying the boundary conditions. The general form of the global equations are as following.

A a h ~ + B,IUI + B,~VI = R, Ma01 + DaiUi + Pai(hI + % I ) = KaVI+ Wal

(3.33) (3.34) (3.35) + Da2VI + Pa2(h1+ 261) = - K a h + Wa2

( 5 ) Solution procedure The time derivative can be descritized with explicit or weighted im-

plicit finite difference scheme. The resultant system of algebraic equations may be solved by any method for solving system of algebraic equations.


3.3.2 Finite volume model (FVM)

By integrating Eq. (3.1) over an finite element R, the basic equation of FVM obtained using the divergence theorem is given by Z h a ~ [ ~ ~ ] :

11 g d R = - (P(U)n, + G(U)ny) dS + 11 S(U)dR (3.36) R R

Where, S is the boundary of the element R under consideration; n,, ny are the direction of the outer normal of the boundary S. If the element is a concave polygon with m boundary sides, above equation can be rewritten as:

m // E d R = C(E(U)L + G(U)j,)Lj + // S(U)dR (3.37) R j=1 R

where E(U)i, G(U)i represent the normal components on the side j , Lj is the length of the side j. Assuming the value of U is constant over the element, then equation (3.38) can be reduced as:

(3.38) dU A- = C ( E ( U ) i + G(U)i)L’ + AS(U)

j=1

where A is the area of the element. According to the definition, E(U)A, G(U)A have following relations

E(U)A = E(U) cos 0 G(U)A = G(U) sin8

(3.39)

where Ois the angle between outer normal of the,side and the x-coordinate. Substituting this relation into equation (3.38), it yields

(3.40) dU A x = x ( E ( U ) cos0 + G(U) sin0)Lj + AS(U)

j=1

Let Fn(U) = E(U) cos 0 + G(U) sin 8,

(3.41)

According to Spekreij~ef~~] , Fn(U) satisfies following relation

T(O)F,(U) = F(T(6’)U) = F(U) (3.42)

114 Deguan Wang

where = T(O)-lU; T(O)-l is inverse matrix of T(O), that is,

1 0 0 (3.43)

-sine cos0

(3.44) 0 sin0 cos0

Therefore, one obtains the final form of the equations for FVM,

(3.45)

or

A(U"+l- Un) = At T(0)-lF(U)jLj + AS(U) (3.46) j = 1

The most important part of the work is calculation of the normal flux F,(U). There exist several methods available. For example, TVD, FVS, FDS, FCTOsher schemes. The readers are referred to

4 Practical work The causes for occurrence of red tide in Pearl River Estuary have been both human and natural. The most important human-induced components of environmental change over the last 20 years have been water diversion and runoff, brush and clearing, human population increases and pollution. Extensive agriculture has reduced the total body of water ecosystems. The simulated area is Peal River Estuary extended to near sea of bathymetric -40m. The area includes Lindingyang, Hong Kong islands, Dapeng bay and related sea area. The simulated domain is shown in Figure 4.1. The results from simulation are given in following paragraphs.

4.1 The governing equations

The governing equations are for hydrodynamic process, nutrient dynamics and the nutrient-algae-dynamics. They are described as follows respectively.


113"30' 114'00' 114"30' E

Hydrodynamic process

= o (4.1)

8 Y d2U d2U

= Ez- + E y T 8x2 8 Y

d t dX dY

& + d H u d H v +- dt d x dy

+ g H - - f H v + g +- -+- at dX

- -

d H u dHu2 dHuv dc UdiZ-3- C2 dX

(4.2)

dHv dHuv dHv2 dc 7 I d W + g H - + f f H u + g c2 -+-+-

= Ez- + Ey- dY

(4.3) d2V d 2 V

8x2 aY2

where, C is water level; H = h+[, h is depth in reference to mean water level; u, v are velocity components in the x- and y-direction respectively; f is Coriolis coefficient; Ex, E1/ are vortex coefficients in the the x- and y-direction respectively; c is Chezy coefficient; g is gravity acceleration.

116 Deguan Wang

Nutrient dynamics

dHC dHuC dHvC d ac a dC -+- +-=- ( (D~H-)+- ( (D ,H- ) -~~~HC+SI , at ax a y dx ax dy aY (4-4)

where, C is the concentration of constituents of the water quality indices; Dx and D, are the dispersion coefficients; lcc is degradation coefficient; SI, is the sink or source. Sk has different form for each constituent of the water quality indices. Nutrients-algae-dynamics, Jingrong, Lee et al.[34],

* = r1 cl(E0 - E ) - ~

Em+E aES 1 " [ dt

" [ EaES = r 2 -(y + c2)S + ~ - b(1 - eDs)N] (4.6) dt Em+E

(4.5)

dN - = 7-2 [-(S + C ~ ) N + ~ g b ( 1 - e p ' ) ~ ] dt (4.7)

where, E = Concentration of nutrient+ g/L; S = Concentration of plankton, cell/L; N = Concentration of algae causing red tides, cell/L; EO = Concentration of loading nutrients, kg/m3; Em = Half-saturated parameter of increasing rate of nutrients, p

c1, c2, c3 = Dissipation rates of E , S , N , l/sec; r1,rz,r3 = Relative increasing rate of E , S , N , respectively, dimen-

a, b = Increasing rate of plankton, algae causing red tides,l/sec; y, 6 = Death rate of plankton, algae causing red tides,l/sec; ~ 1 , ~2 = Absorb and transformation rate of plankton, algae causing

,B = Saturated growth level of plankton, %. The conditions for the occurrence of red tides are derived from above

mol/l;

sionless;

red tides,dimensionless;

equations.

&la > y + c2

~ 2 b > 6 + ~3

Eo > E* > EB

4.2 The numerical solution method The equations are solved numerically by different suitable schemes. The schemes are as follows.


Hydrodynamic process The equations are discretized by finite volume (FVM) method.

Nutrient dynamics The equations are discretized by finite volume (FVM) method.

Nutrients-algae-dynamics The equations are solved by Runge-Kutta method. If there exist two

time instants, t 2 > tl > t o , and the inequality, N ( t ) 2 fi, exists for tl < t < t 2 . The red tides may occur.

4.3 Results

4.3.1 Results from hydrodynamic simulation

Measuring point 1

1.5j I

-2.0 135 140 145 150 155 160

Time(hour)

-2.0 135 140 145 150 155 160

Time(hour)

Figure 4.2 Water level results.

Two examples of the results from hydrodynamic simulation are shown in Figs.4.2-4.3. It can be seen that the model simulates the hydrodynamic process in the Peal River Estuary properly.

4.3.2

The parameters were determined from experiments. The re-aeration constant was 0.12, 0.20 and 0.08. Four examples of results are given in Figs.4.4-4.7. The results are well agreed with the field data at the water quality surveying point. The calculated results could be used for estimating the possibility of the occurrence of the red tides.

Results from Nutrient dynamics simulation

118 Deguan Wang

Mesuring point 1

1 .0

-0.5

-1.0

-1.5 135 140 145 150 155 140 165

~ ~ ~___

Figure 4.3 Velocity results.

Measuring point 9 0.9 i

I i

0 5 I0 15 20 25 10 T ~ e ( h o ~ ~

Figure 4.4 Results €or C

~ ~ ~ ~ n g point 9 0.03 ;

,esults for NH3-N.

The typical values of ~utrients in the Peal River Estuary from the nutrient ~ l ~ d e l i n g are listed in Tables 4.1 and 4.2.

4.3.3 Results for nutrients-algae-dynamics


Lindingyang

High flow Low flow Items

DIN Pg/L 420.0 400.0

I 0.37 L.--L _I_i--i ~ - - 2

30 35 40 45 50 55 60 Time(hour)

i- ...............................................................................

Dapeng bay

High flow Low flow

450.0 600.0

Figure 4.6 Results for N03-N.

DIP Pg/L

Plankton pg/L

Temperature "c

0.01

2 % z

0.005

8.0 17.0 5.0 15.0

1.7 0.8 2.5 1.05

25.3 19.8 25.87 19.9

Measuring point 9

35 40 45 50 55 60 Time(hour) ..............................................................................

Figure 4.7 Results for Pod-P.

It is seen that the nutrients in the estuary is rich, especially, the ammonia nitrogen. The most suitable condition for growth of algae is DIP>,5.4pg/L ,DIN>57.lpg/L. The ratio of the N and P in the algae body is N:P = 1:16. Therefore, the nutrient environment in the Peal River Estuary is favorable to the algae growth.

120 Deguan Wang

Table 4.2 Constants used in the calculation of biota

70000 X000~ ~ 1OOOOO 1 Loo00 12oooO 13oooO 14OWJ 15oooO 1- 170M10

Figure 4.8 Distribution of plankton amount.(before red tide occur)

Figure 4.9 D~~tribution of red tide algae arnount.(before red tide occur)

~ ~ o l o g ~ ~ a ~ S i ~ u l a t ~ o n of Red Tides in Shallow Sea Area 221 - - u - " A - - - I .

Figure 4.10 ~ i s ~ ~ i b u t i o n of plankton amount~(after red tides occur)

Figure 4.11 ~ i s t ~ ~ b ~ t i o n of red tide algae amount.(a~er red tides cpccur)

The constants used in the calculation of the amount of biota in the estuary are given in Table 4.1. Some of the results from the c a l c u ~ a tare shown in Figures 4.8 and 4.9. Figures 4.10 and 4.11 give the a ~of algae when the red tides occur.

122 Deguan Wang

These figures indicate that the red tide algae amount closely to the plankton amount. The plankton grows fast when the nutrient supply is large and the red tide algae grows fast accordingly. Before and after the red tides occur the amount of red tide algae is high. The amount of red tide algae is higer in the Dapeng bay, hence the occurrence of the red tides has higher possibility than in Shengzheng bay and Hong Kong area. This situation also presents in Table 4.3.

Period

Growth period

Water area Average rate of growth

Lindingyang Dapeng bay bay

Nutrient (pg/L . day) 1.2 2.6 20.8

4.6 Plankton (lo4 cell/L . day) 0.1 0.3 ,

Red tide algae(104 cell/L . day) o.35 1.61 4.4

Decay period

Nutrient (pg/L . day)

Plankton. (lo4 cell/L day)

Red tide algae(104 cell/L . day)

-0.4 -0.7 -15.6

0.0 0.0 -3.9

0.0 0.0 -4.0

Measuring point 9

4.9992 4.999

4.9988 4.9986 4.9984 4.9982 4.998

4.9978 4.9976 4.9974 ~

30 35 40 45 50 55 60

I Tirne(houi-s)

Figure 4.12 Time variation of nutrient concentration for initial N=0.1 and No=O.Ol.

4.3.4

We calculated the equilibrium point of algae at measuring point 9. The time variation of it is showed in Figure 4.12. From Figure 4.12 we can

The equilibrium point of algae trend at, measuring point 9

Table 4.3 Nutrient, Planktom and Red tide algae pattern in different


see the time variation of equilibrium point of algae is unsteady. It drops at first and increases later. Based on the conclusions in section 2, we can infer at measuring point 9 the rate of time variation of algae is large at first, then it drops and turns larger again. The instability is of significance. If the rates in the area exceed the critical value and last for enough time, red tide will break out.

4.3.5 Analysis of results

The formation of the red tides depends not only on the nutrient condition, and also the hydrodynamic condition in the considered area. The flow from four east mouths of the Peal river has larger velocity and brings large quantity of nutrients to the estuary. The tide current is strong in the area near Hong Kong and also brings the nutrients to the Dabong bay and Shengzheng bay. The tidal current in Dabong bay and Shengzheng bay is slow that enriches the nutrients in these two bays and develops the environment for the red tides to occur.

References

[l] Shumway S E. A review of the effects of algal blooms on shellfish and Aquaculture, J. World Aquacult. SOC., 21, 65-104, 1990.

[a] Anderson D M. Toxic algal blooms and red tides: a global perspective, in Red Tides: Biology Environmental Science and Toxicology, edited by T. Okaichi, D. M. Anderson and T. Nemoto, 11-16, Else- vier, New York, 1989.

[3] Smayda T. Novel and nuisance phytoplankton blooms in the sea: Evidence for a global epidemic, in Toxic Marine Phytoplankton, edited by Graneli El Sundstrom B, Edler L, Anderson D M. Elsevier, New York, 1990.

[4] Hallegraeff G M. A review of harmful algal blooms and their apparent global increase, Phycologia, 32, 79-99, 1993.

[5] Anderson D M, Kulis D M, Orphanos J A, Ceurvels A R. Distri- bution of the toxic red tide dinoflagellate Gonyaulax tamarensis in the southern New England region, Estuarine, Coastal, and Shelf Science, 14, 447-458, 1982.

[6] Reid P C, Lancelot C, Gieskes W W C, Hagmeier El Weichart G. Phytoplankton of the North Sea and its dynamics: A review, Neth. J. Sea Res., 26, 295-331, 1990.

[7] Anderson D M. Toxic algal blooms and red tides: a global perspective, in Red Tides: Biology Environmental Science and Toxicology,

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edited by Okaichi T, Anderson D M, Nemoto T, 11-16, Elsevier, New York, 1989.

[8] Hallegraeff G M, Bolch C J. Transport of diatom and dinoflagellate resting spores via ship’s ballast water: implications for plankton biogeography and aquaculture, J. Plankton Res., 14, 1067-1084, 1992.

[9] Gowen R J, Bradbury N B. The ecological impact of salmonid farm- ing in coastal waters: A review, Oceanogr. Mar. Biol. Ann. Rev., 25, 563-575,1987.

[lo] Garvine R. Dynamics of small-scale oceanic fronts,

[ll] Garrett C J R, Loder J W. Dynamical aspects of shallow sea fronts,

[12] Simpson J H. The shelf-sea fronts: Implications of their existence

[13] Demers S, Legendre L, Therriault J C. Phytoplankton responses to vertical mixing, in Tidal Mixing and Plankton Dynamics. Lecture Notes on Coastal and Estuarine Studies 17, edited by Bowman M J, Yentsch C M, Peterson W T, 1-40, Springer-Verlag, New York, 1986.

[14] Kamykowski D. The growth response of a model Gymnodinium splendens in stationary and wavy water columns, Mar.Biol., 50, 28S303, 1979.

[15] Kamykowski D. The simulation of a southern California red tide using characteristics of a simultaneously-measured internal wave field, Ecol. Model., 12, 253-265, 1981.

[16] Evans G, Taylor F. Phytoplankton accumulation in Langmuir cells,

[17] Watanabe M, Harashima A, Interaction between motile phytoplankton and Langmuir circulation, Ecol. Modelling, 31, 175-183, 1986.

[18] Franks P J S, Anderson D M, Alongshore transport of a toxic phytoplankton bloom in a buoyancy current: Alexandrium tamarense in the Gulf of Maine, Marine Biology, 112, 153-164, 1992.

J. Phys. Oceanogr., 4, 557-569, 1974.

Phil. Trans. R. SOC. Lond., 302, 563-581, 1981.

and behavior, Phil. Tranc. R. SOC. Lond., 302A, 531-546, 1981.

Limnol. Oceanogr., 25, 840-845, 1980.

[19] Steele J H, Henderson E W. A simple plankton model. Am Nat.,

[20] Dutt P, Singh A K. The Galerkin-Collocation Method for Hyper- Journal Computational

117, 676-91, 1981.

bolic Initial Boundary Value Problems, Physics, 1994.


[21] Gueremont G, Ghuabashi W, Kotiuga P L. Finite Element solution of the 3D Compressible Navier-Stokes Equation by a Velocity- Vorticity Method, Journal Computional Physics, 1993.

[22] King I P, Norton W R, Iceman K R. A Finite Element solution for Two-Dimensional Stratified Flow Problems, Finite Element in Fluides, 1975.

[23] Zhao D H, Shen H W, Tabios I11 G Q, Lai J S, Tan W Y. “Finite- Volume Two-Dimensional Unsteady-Flow Model for River Basins”, Journal of Hydraulic Engineer, ASCE, Bol. 120, No. 7, 863-883, 1994.

[24] Spekreijse S P. Multigrid solution of steady Euler equations. CWI Trac 46 Amsterdam, 1988.

[25] Roe P L. Some contributions to the modeling of discontinuous flows. AMS-SIAM Seminar, Sam Diego, June 1983.

[26] Var Leer B. Towards the uitimate conservative difference scheme. Journal Computational Physics. 14, 363-389, 1974.

[27] Osher S, Solomone F. Upwind Difference Scheme for Hyperbolic Systems of Conservation Laws, Mathematical computation, vo1.38,

[28] Osher S. Shock modeling in transonic and supersonic flow recent advances in numerical methods in fluids. Vo1.4, W. G. Habashi Ed.

[29] Yee H C, Warming R F, Harten A. Implicit total variation diminishing (TVD) schemes for steady-state calculations. AIAA Paper

[30] Yee H C. Construction of explicit and implicit symmetric TVD schemes and their applications. Journal Computational Physics,

[31] Pan D, Cheng J. Incompressible flow solution on unstructured triangular meshes. Int. J. Numer. Methods Fluids. 16, 1079-1098, 1999.

[32] Anastasiou K, Chan C T. Solution of the 2D shallow water equations using the finite volume method on unstructured triangular meshes. Int. J. Numer. Methods Fluids. 24, 1225-1245, 1997.

[33] Zhao D H, Tabios 111 G Q, Shen H W. RBFVM-2Dmode1, River Basin Two-Dimensional Flow Model Using Finite Volume Methods, Program Documentation, University of California, Berkely, Califor- nia, USA, 1995.

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68, 151-179,1987.

126

Subsurface Pathways of Contaminants to Coastal Waters: Effects of Oceanic

Oscillations

Ling Li Centre for Eco-Environmental Modeling, Hohai University, Nanjing, 21 OU98, China

Abstract

Submarine groundwater discharge (SGD) has been identified by the International Geosphere-Biosphere Programme as an important contamination source for coastal marine and estuarine environments. Inputs of land-derived contaminants associated with SGD cause serious threats to the near-shore ecosystem. Water exchange driven by oceanic oscillations such as tides and waves at the shore contributes to SGD significantly. Moreover, the oceanic oscillations modify the near-shore groundwater flow and induce mixing of fresh groundwater with seawater. These processes affect not only the transport but also the chemical reactions of the contaminants in the near-shore aquifer prior to the discharge. In this chapter, we first review studies of costal groundwater responses to tides and waves, focussing on analytical solutions of the groundwater table fluctuations. We then discuss the effects of the groundwater fluctuations on the fate of chemicals in the near-shore aquifer and chemical fluxes to coastal water. These discussions are based on several on-going studies aiming to improve the understanding and quantification of subsurface pathways and fluxes of chemicals to coastal environments.

1 Introduction

Coastal/estuarine water pollution is becoming an increasingly serious global problem largely due to input of land-derived contaminants. For example, nutrient leachate from the sugar cane production areas of North-East Queensland is causing great concern for the Great Barrier Reef in Australia (Haynes and Michael- Wagner, 2000). The resulting

Subsurface Pathways of Contaminants to Coastal Waters: . . - 127

degradation of coastal resources affects significantly economic and social developments of coastal regions. Traditionally, terrestrial fluxes of chemicals to coastal water have been estimated on the basis of river flow alone. However, recent field observations indicate that contaminants entering coastal seas and estuaries with groundwater discharge (submarine groundwater discharge, SGD) can significantly contribute to coastal pollution, especially in areas where serious groundwater contamination has occurred (e.g., Cable et al., 1996; Church, 1996; Moore, 1996; Burnett et al., 2001; Portnoy et al., 1998). The International Geosphere- Biosphere Programme (IGBP; Buddemeier, 1996) has identified submarine groundwater discharge as an important but rather unknown source of contamination for coastal marine and estuarine environments. As the groundwater contamination problem worsens, the SGD may become a dominant source of coastal pollution in certain areas.

SGD consists of both groundwater flow from upland regions and water exchange at the aquifer-ocean interface (Simmons, 1992). While the upland groundwater flow can be estimated based on the aquifer recharge (Zelcster and Loaiciga, 1993), it is difficult to quantify the rate of water exchange across the seabed, which is influenced by near-shore processes ( L i et al., 1997a; Turner et al., 1997; La and Barry, 2000). Large rates of SGD, derived from geochemical signals of enriched natural tracers (e.g., 2zsRa; Moore, 1996) in coastal seas, have been found excessive and cannot be supported by the aquifer recharge (Younger, 1996). This suggests that water exchange at the interface may have constituted a large portion of the SGD. A theoretical model of SGD has been developed to include tidally oscillating groundwater flow and circulation due to wave set-up (i.e., on-shore tilt of the mean sea level; Figure 1.1). These two local processes were found to cause a large amount of water exchange across the interface.

Although the exchanging/recycling water is largely of marine origin, it mixes and reacts with groundwater and aquifer sediments, modify- ing the composition of the discharging water. The exchange processes can reduce the residence time of chemicals in the mixing zone of the aquifer, similar to tidal flushing of a surface estuary (L i et aL, 1999). As a result, the rates of chemical fluxes from the aquifer to the ocean increase but the exit chemical concentrations are reduced (dilution effects). The exchange can also alter the geochemical conditions (redox state) in the aquifer and affect the chemical reactions. It has been shown numerically that the exchange enhances the mixing of oxygen-rich seawater and groundwater, and create an active zone for aerobic bacterial populations in the near-shore aquifer. This zone leads to a considerable reduction in breakthrough concentrations of aerobic biodegradable contaminants at the aquifer-ocean interface (Enot et al., 2001; Li et aL, 2001).

In essence, the water exchange and subsequent mixing of the recy-

128 Ling Li

Figure 1.1 A simple model of SGD consisting of inland fresh groundwater flow (Dn) and seawater recycling (water exchange due to wave set-up D,, and due to tides Dt), i.e., total outflow from'the aquifer to coastal water. The mixing of the recycling water with fresh groundwater results in the near-shore salinity profiles as schematically shown by the thin dot-lines (in contrast with the traditional saltwater wedge view shown by the thick dot-dashed line).

cling water with fresh groundwater, driven by the oceanic oscillations, lead to the creation of subsurface estuary (subsurface analogue to surface estuary) as suggested by Moore (1999). The role of a subsurface estuary in determining the terrestrial chemical input to the sea may be compared with that of a surface estuary. Most previous studies of coastal groundwater, focussing on large-scale saltwater intrusion in aquifers, have ignored the dynamic effects of tides and waves on the flow and mixing processes in the near-shore area of the aquifer (Huyakorn et al., 1987). Beginning with the work of Badon Ghyben and Herzberg in the late 1800s, researchers have carried out an enormous amount of work to quantify the extent of saltwater intrusion and to predict the encroach- ment of saline water into fresh water aquifers (e.g., Henry, 1959; Volker and Rushton, 1982; Huyakorn et al., 1987). Despite some early work on coastal groundwater flow and discharge to the sea (Cooper, 1959; Glover, 1959; Kohout, 1960), it was not until the 1980s that researchers began to investigate the environmental and ecological impacts of groundwater discharge (Bokuniewicx, 1980; Johannes, 1980).

Globally, the fresh groundwater discharge has been estimated to be a few percent of the total freshwater discharge to the ocean (Zekster and Loaiczga, 1993). Recently, Moore (1996) conducted experiments on 226Radium enrichment in the coastal sea of the South Atlantic Bight. From the measurements, he inferred, on the basis of mass balance, that groundwater discharge amounts to as much as 40% of the total river flow into the ocean in the study area. This estimate contrasts with previous figures that range from 0.1 to 10%. Younger (1996) demon-

Subsurface Pathways of Contaminants to Coastal Waters: . . 129

strated that the recharge to the coastal aquifer could only support 4% of the estimated discharge. A theoretical/conceptual model of submarine groundwater discharge that includes recycling/exchanging water across the seabed has been found to predict the excessive discharge rate (L i et

Since the exchanging water is largely of marine origin, its impact on the fate of chemicals in the aquifer and chemical fluxes to coastal water depends on its mixing with groundwater. Laboratory experiments have revealed large tide-induced variations of flow velocities and salinity in the intertidal zone of the aquifer (Boufadel, 2000). This suggests that the mass transport of salinity (the product of the salinity and flow velocity) be affected by tides significantly. The mixing of the tide-induced recycling water with fresh groundwater results in a salinity profile of two saline plumes near the shore (Boufadel, 2000), as schematically shown in Figure 1.1. The mixing zone is in contrast with the traditional saltwater wedge. Field measurements also showed fluctuations of salinity near the shore in response to tides and waves (Nielsen, 1999; Cartwright and Nielsen, 2001). Tidal effects on salinity distribution in the aquifer have also been demonstrated by numerical investigations ( Ataie-Ashtiani et al., 1999; Zhang et al., 2001). These results suggest the existence of a mixing zone of the coastal aquifer that behaves much like an estuary. In this subsurface estuary, flow and mass transport/transformation are affected by both the net groundwater flow and water exchange/mixing induced by oceanic oscillations, particularly the tides (Robinson and Gal- lagher, 1999).

The first part of the chapter is on the effects of tides on costal groundwater, focussing on the water table fluctuations in shallow coastal aquifers. Various analytical solutions of the tide-induced groundwater table fluctuations under different conditions will be presented. The groundwater response to other oceanic oscillations such as storm surges will also be discussed. The second of the chapter is on the effects of the groundwater fluctuations on the fate of chemicals in the near-shore aquifer and chemical fluxes to coastal water. These discussions are based on several on-going studies aiming to improve the understanding and quantification of subsurface pathways and fluxes of chemicals to coastal environments.

al., 1999).

2 Tide-induced groundwater oscillations in coastal aquifers

Groundwater heads in coastal aquifers fluctuate in responses to oceanic tides. Such fluctuations have been subject to numerous recent studies (e.g., Lanyon e t al., 1982; Nielsen, 1990; Turner, 1993; Baird

130 Ling Li

and Horn, 1996; Li et al., 1997a; Nielsen et al., 1997; Baird et al., 1998; Raubenheimer et al., 1999; La et al., 2000a,b; Li and Jiao, 2002a,b; Jeng et al., 2002). In unconfined aquifers, such responses are manifested as water table fluctuations. These fluctuations are attenuated as they prop agate inland, while the phases of the oscillations are shifted (Nielsen, 1990; Li et al., 1997a). Modeling of tidal groundwater head fluctuations are often based on the Boussinesq equation assuming negligible vertical flow (e.g., Nielsen, 1990; Baird et al., 1998),

dh d2h - D- at 8x2

_ -

where h is the groundwater head fluctuation ( H - H , H is the total head and H is the mean head) as shown in Figure 2.1; 2 is the inland distance from the shore; t is time; and D is the hydraulic diffusivity, = T / S ( S and T are the aquifer’s storativity/specific yield and transmissivity, respectively). Note that (2.1) is a linearised Boussinesq equation. Although it is applicable to both confined and unconfined aquifers, the application to the latter requires that the tidal amplitude be relatively small with respect to the mean aquifer thickness (Parlange et al., 1984). The effects of non-linearity will be discussed later.

Figure 2.1 water table fluctuations in an unconfined aquifer.

Schematic diagram of tidal conditions at the beach face and

Analytical solutions for predicting’ tidal groundwater head fluctuations are available, for example (Ferris, 1951),

h = Aoexp(-kx) cos(wt - kx) (2.2a)

where A0 and w are the tidal amplitude and frequency, respectively; lc is the rate of amplitude damping and phase shift, and is related to the tidal frequency and the aquifer’s hydraulic diffusivity,

(2.2b)

Subsurface Pathways of Contaminants to Coastal Waters: . . . 131

The solution can be presented in an alternative form,

h = Re[Aoexp(iwt - kx)], (2.3a)

with k = g , (2.3b)

where i = fl and k is the complex wave number. The solution assumes that the seaward boundary condition of the groundwater head is defined by the tidal sea level oscillations, i.e.,

h(0, t ) = A0 cos(wt). (2.4)

Far inland (x -+ oo), the gradient of h is taken to be zero (the tidal effects are diminished), i.e.,

This simple solution is also based on several other assumptions as

(1) Ao/H(for unconfined aquifers) small, i.e., negligible non-linear

(2) vertical beach face; (3) negligible capillary effects; (4) no leakage exchange between shallow and deep aquifers; (5)negligible vertical flow effects (Hk small). In the following, we discuss relevant effects in situations where these

listed below:

effects;

assumptions do not hold.

2.1 Non-linear effects Parlunge et al. (1984) examined the non-linear effects. Their analysis is briefly presented here. The head fluctuation is governed by the non- linear Boussinesq equation as follows,

d h K d - dt n e x - - - - [ ( H + h ) g ] . -

The seaward and landward boundary conditions are described by (2.4) and (2.5). A perturbation technique is applied to solve (2.6). The solution of h is sought for in the following form,

h = 1? L E h l + ~ ~ h 2 + O(c3)] , (2.7)

where E is the perturbation variable, = Ao/1?; it is less than unity under normal conditions. Substituting (2.7) into (2.6) gives the following

132

perturbation equations, O(E):

Ling Li

(2.8a)

and hi (0, t ) = cos(wt), (2.8b)

O ( E 2 ) :

dh2 - KR [ d2h2 1 d2ht ] + -- dt n, dx2 2 ax2 -

(2.8~)

(2.9a)

h2(0, t ) = 0 (2.9b)

(2.9~)

The solution to (2.8) is simply,

hl = exp(-kz) cos(wt - kx), (2.10)

where k is defined by (2.2b). The solution to (2.9) can then be obtained,

1 ha = -[exp(-hkx) cos(2wt - h k z ) - exp(-2kz)

2 1 . cos(2wt - Zk:r)] + - [I - exp(-2kx)]. 4

(2.11)

The water table fluctuation affected by the non-linearity is thus described by,

h = Aoexp(-kx) cos(wt - kx)

+-[exp(-d%x) A; cos(2wt - 6) - exp(-2kz) (2.12) 2H

cos(2wt - 2kx)] + -= A; [l - exp(-2kz)]. 4H

The non-linear effects as shown by (2.12) lead to the generation of a second harmonics (the second term of the RHS with frequency 2w) and a water table overheight (increase of the mean water table height; the third term of the RHS). The superposition of the second harmonics and the primary signal gives rise to the asymmetry between the rising and falling phases of the water table fluctuations, often observed in the field (Figure 2.2). The above approach can be extended to obtain O ( E ~ ) or higher terms. These solutions, including harmonic waves of higher frequencies, are of smaller amplitudes.


Figure 2.2 solution, exhibiting asymmetry between the rising and falling phases.

Tidal water table fluctuations predicted by the non-linear

2.2 Slope effects Nielsen (1990) reported the first analytical investigation on the slope effects. He derived a perturbation solution for small amplitude water table fluctuations based on the linearised Boussinesq equation by matching a prescribed series solution with the moving boundary condition. Later, Li et al. (2000b) presented an improved approach, which is described below. To focus on the effects of the moving boundary, only small amplitude tides are considered, as modelled by the linearised Boussinesq equation (2.1) subject to the boundary conditions defined by (2.13) and (2.5).

As shown in Figure 2.3, tidal oscillations on a sloping beach create a moving boundary:

h [X(t),tI = d t ) and X ( t ) = cot(@)rl(t), (2.13)

where X ( t ) is the 2-coordinate of the moving boundary (the origin of the x-coordinate is located at the intersection between the mid-tidal sea level and the beach face), @ is the beach angle, and q(t) represents tide-induced oscillations of the mean sea level. By introducing a new variable z = 2 - X ( t ) , equations (2.1), (2.13) and (2.5) are, respectively, be transformed to,

d h d 2 h d h - = D- - w(t)-,and at 822 dz2

(2.14a)

134 Ling Li

tides

Figure 2.3 aquifer subject to tidal oscillations at the sloping beach face.

Schematic diagram of water table fluctuations in a coastal

(2.14~)

where v(t) = -- dX(t) = Aw cot(@) sin(wt). (2.14d)

The moving boundary problem of (2.1) is thus mapped to a fixed boundary problem of (2.14). A perturbation approach is adopted to solve (2.14), i.e.,

h = ho + €hi+ O ( E ~ ) , (2.15)

where E = Akcot(@) and k is given by (2.2b). Physically, E represents the ratio of the horizontal tidal excursion to the wavelength of primary mode tidal water table fluctuations (l /k). Substituting (2.15) to (2.14), the following perturbation equations are obtained,

dt

O ( E 0 ) :

(2.16a)

O(E1):

(2.16~)

(2.17a)

hl(0,t) = 0. (2.17b)


(2.17~)

The solution to (2.16) is simply,

ho = A exp(--Kz) cos(wt - ~ z ) . (2.18)

The solution to (2.17) is

hl = { 1 + &exp(-&kz) cos (2wt - h k z + :) - fiexp(-kz) [cos (2wt - kz + T ) 4 + cos ( k z - ;)I} (2.19)

Therefore,

A& hl = A exp(-kz) cos(wt - k z ) + { 1 + h e x p ( - h k z )

+ cos ( k z - ;)I} + 0 ( & 2 ) . (2.20)

To obtain the solution in the x-coordinate, one can substitute z = z - Acot(,B) cos(wt) into (2.20). The approach can be extended to obtain O(E’) or higher terms. These solutions, including harmonic waves of higher frequencies, are of smaller amplitudes. The slope effects are qual- itatively similar to those caused by the nonlinearity of finite amplitude tides, i.e., generation of the sub-harmonics and water table overheight.

2.3 Capillary effects

Parlange and Brutsaert (1987) derived a modified Boussinesq equation to include the capillary effects,

(2.21) dh K H d 2 h BH d3h +-- at n, ax2 n, atdx2’ - - -- -

where B is the average depth of water held in the capillary zone above the water table. Barry et al. (1996) solved this equation subject to the boundary conditions described by (2.4) and (2.5),

h = aoexp(-klx) cos(wt - kax), (2.22a)

with

136 Ling Li

and

0.8

The capillary effects cause the difference between the damping rate ( k l ) and the wave number (k2). In Figure 2.4, kl and k2 are plotted as functions of w (the values of ne,K,B and I? are set to be 0.25, 0.001 m/s, 0.1 m and 20 m, respectively). Also plotted in the figure is the behaviour of k with w as given by the solution without capillary effects, (2.2b). The results suggest that capillary effects are only important for high frequency oscillations. Under normal conditions, the effects of unsaturated flows on the tidal water table fluctuations are probably minor. It is interesting to note that at high frequencies, the water table fluctuations, affected by the capillarity, become standing waves (k2 approaches zero). More detailed discussion on the capillary effects can be found in Li et al. (1997b).

a: damping rate -with capillary effects - ----without capillary effects -

o,8

-- 0.6

-Y 0.4

0.2

0

h

7 0 . 6 6 .E ,

& 0.4 / * - *

..-- 0.2 __..----

0 10-2 10-1 10-4 W(Rad/s)

1 - . , I

- -with capillary effects - - - -without capilliuy effects b: rate ofphase shift (wave number)

.* - - '

*.*+*

- -.*+ / * - * _.--

___..--* - -

" ' J

Figure 2.4 Comparison of the damping rate and wave number predicted by the solutions with and without capillary effects.

2.4 Leakage effects In the above solutions, the bottom boundary of the aquifer is assumed to be impermeable. In reality, it is not uncommon to find composite

~ubsurface Pathways of C o n t ~ ~ n a n t s to Coastal ~ a t e r s : . ~ I 137

aquifer system^ such as the one shown in Figure 2.5: an ~ c o n f iaquifer overlying and separated Horn a confined aquifer by a thin seml- permeable layer. The groundwater heads fluctuates in both the confined and the phreatic aquifer. The two aquifers interact with each other via leakage th~ough the sem~-per~eable layer. The governing equations of the head ~~c tua t ions in both aquifers are (e.g., Bear, 1992):

Figure 2.5 Schematic diagram of a leaky confined aquifer with an overlying phreatic aquifer.

(2.23b)

where hl and ha are the heads in the confined and the phreatic aquifers, respectively; TI and T2 are the transmissivities of these two aquifers, respectively; 91 is the specific yield of the phreatic aquifer and s2 is the s to ra t~v~~y of the confined aquifer; and L is the specific leakage of the $ e ~ i - p e r ~ e a ~ l e layer. inea ear is at ion has been applied to the gov@rning ~ ~ u a t i o n of the phreatic aquifer, (2.23a). The boundary conditions are

(2.24b)

138 Ling Li

where hMSL is the averaged mean sea level. Note that h used here is the total water head in the aquifer.

In reality, the damping of the tidal signal in the unconfined aquifer is much higher than that in the confined aquifer (since s1 >> SZ). Usu- ally the fluctuations of hl become negligible 100 meters landward of the shoreline while the tides propagate much further inland in the confined aquifer. Jzao and Tang (1999) solved (2.2313) subject to (2.24) assuming that hl is constant, i.e., neglecting the tidal fluctuations in the unconfined aquifer. Their solution shown below suggests that the leakage reduces the tidal signal in the confined aquifer significantly, i.e., the damping rate increases.

h2 = hMSL + Aoexp(-kLlz) cos(wt - k ~ 2 2 ) , (2.2 5a)

and (2.25~)

Jeng et al. (2002) solved the coupled equations (2.23a) and (2.23b),

h2 = hMSL+Re {Ao[baexp(-k~lz) + b4exp(-k~2z)]exp(iwt)}, (2.26a)

(2.2 6b)

(2.26~)

(2.26d)

with

w2s1s2 iwL - (s1 + .2>,

and

(2.26e)

(2.26f)


(2.26i)

This solution also demonstrates the reduction of tidal signal in the confined aquifer due to leakage. However, the extent of the reduction is less than predicted by (2.25). The solution also shows that the water table fluctuation in the unconfined aquifer is enhanced as a result of the leakage. Li and Jiao (2001a,b and 2002a) presented a study on this problem including the storage effects of the semi-permeable layer and Li et al. (2002a) examined the damping effects due to the phreatic surface based on a vertical flow model.

2.5

Spring-neap tidal water table fluctuations The above solutions consider only one tidal constituent. In reality,

tides are more complicated and often bichromatic, containing oscillations of two slightly different frequencies (Godin, 1972). For example, a semi- diurnal solar tide has period TI = 12 h and frequency w1 = 0.5236 Rad h-l while T2 = 12.42 h and w2 = 0.5059 Rad h-' for a semi- diurnal lunar tide. As a result, the spring-neap cycle (i.e., the tidal envelope) is formed with a longer period, T,, = 27r/(wl- w2) = 14.78 d. Recently, Raubenheirner et al. (1999) observed water table fluctuations of period Tsn. These fluctuations (called spring-neap tidal water table fluctuations, abbreviated as SNWTF) occurred much further inland than the primary tidal signals (i.e., diurnal and semi-diurnal tides). While one may relate this long period fluctuation to the spring-neap cycle, the cause of such a phenomenon is not readily apparent.

Low frequency and other oscillations

Spring-neap tides are bichromatic signals as described by

~ ( t ) = A1 cos(w1t) + A2 cos(wzt - b) , (2.27)

where A1 and A2 are the amplitude of the semi-diurnal solar and lunar tide, respectively, and S is the phase difference between them. Only two primary forcing signals exist at the boundary. If they propagate in the aquifer independently (as would occur in a linearised model assuming a vertical beach face), the water table response will also be bichromatic and simply described by Alexp(-klz) cos(wlt-klz)+A2exp(-k2z) cos(w2t- S - k2z). Both k l ( d m ) new1 and k 2 ( d m ) new2 are high damping rates

corresponding to the'semi-diurnal freqhencies. A slowly damped spring- neap tidal water table fluctuation is not predicted. However, the beach face is sloping and creates a moving boundary as discussed in 2.2. The moving boundary induces interactions between the two primary tidal signals as they propagate inland. Such interactions lead to the generation of the SNWTF.

140 Ling Li

La et al. (2000b) reported an analytical study on this problem. The same approach as described in 2.2 was adopted to solve the Boussinesq equation subject to the bichromatic tides. Based on the same transformation, z = z - X ( t ) , the governing equation becomes

dh d2h dt a22 - _ - D- - [Alwl cot(/?) sin(w1t)

(2.28) dh +A2w2 cot(/?, sin(w2t - a)] -. dz

Again, a perturbation solution to (2.28) is sought for, i.e., assuming

(2.15) with E = Alkl cot(/?) and kl = @. The perturbation equations

are then obtained: O ( E 0 ) :

ah0 - = D 3 , ho(O, t ) = Al cos(wlt) + A2 cos(w2t - S), dt dz2

O ( d ) :

(2.29)

ah0 sin(w2t - S)] -, dz

hl(0,t) = 0. (2.30)

The solution to (2.29) is simply

ho = Alexp(-klz) cos(w1t - k l z ) +Azexp(-kzx) cos(w2t - k2z - 6). (2.31)

The solution to (2.31) is,

hi =$i(z,t) +$2(z , t ) + $ ~ ( z , t ) +$4(z,t), (2.32a)

where

"> ( 111 $1 = $ { 1 - JZexp(-k1z) [cos (2w1t - k1z + - + cos k1z - - A 4

. cos 2W2t -k2z-26+ - +cos k 2 z - - " 7 4 ( 31


+ h e x p ( - h k z z ) cos (2wzt - h k z z - 26 + ">> , (2.32~) 4

f i A 2 f i [exp(-kzx) cos w3t - k2.z - 6 + - $3 = - 2 ( "> 4

- exp(-ksx) cos w3t - k3x - 6 + - ( ">I 4

- JZrzAl [exp(-kla) cos w3t - l ~ l x - 6 + - 2 ( 7 4

- exp(-ksz) cos w3t - k3z - 6 + - ( "I 4 , (2.32d)

WZ Az w1 A1

with w3 = wl+wz, k3 = = J-lcl, r1 = - = 0.9662 and - =

rz *

a A z J T T [exp(-kzz) cos w4t + kzz + 6 - - "> 2 ( 4

( 71 [exp(-klz) cos ( w4t - k1.z + 6 + 4)

(

+4 = -

- exp(-k4z) cos w4t - k4z + 6 - - 4

-

- exp(-k4z) cos w4t - 1c4z + 6 + 4)] ,

li-

2

n- (2.32e)

with w4 = w1 - wz,k4 = 6 = d-lcl . The solution in the x-

coordinate can be obtained by substituting x = x - [A1 cot(p) cos(w1t) + A2 cot(,@ cos(w2t - S)] into (2.31) and (2.32). Expansions of these equations in z lead to

with

ho = Alexp(-klz) cos(wlt - klx) + A2exp(-k2z) cos(w1t - kzx - 6),

~ I O = -(A + r z f iAz) ,

(2.33b)

(2.33~) 1 2

142 Ling Li

+ & exp(-l~z) cos wQt - 6 - 1632 + z) , and 4 (2.33f)

2 h13 =

h14 = mA2exp(-164z) 2 cos(w4t - S - Ic4z + 6), (2.33g)

where .

tidal system, the mov- (hloe), and additional

The solution indicates that, in a bichromatic ing boundary condition generates an overheight . harmonic waves of frequency 2wl(hlle), 2wz(h12e),w1 + W2(hl3&) and w1 -wz(h14&). The oscillation of w1 -w2 represents the spring-neap tidal water table fluctuations. Since the damping rate, k4, is much smaller than 161, 162 and 163, the SNWTF propagates much further inland. The damping distance (l/lc4) for the SNWTF is five times larger than those for the primary mode water table fluctuations. In Figure 2.6, we plot the simulated water table fluctuations at z = 20,50,100 and 200m (parameter values used in the calculation are A1 = 0.25m, A2 = 0.75m, 6 = ORad, K = 0.002m/s,ne = 0.2, tan@) = 0.15, and = 2m). The results clearly show that the SNWTF occurs much further inland than the semi- diurnal tides, consistent with the experimental findings.

,

Storm-induced water table fluctuations Most coastlines around the world are exposed to frequent, generally

seasonal, offshore storms during which the wave height increases. The mean shoreline is elevated by the order of 0.4HT,, (root of mean square wave height) above the tide level as a result of wave set-up and runup (Figure 2.7; Hanslow and Nzelsen, 1993). As the storm passes, the wave height decreases and so does the elevation of the mean shoreline. Thus the passing storm produces a pulse in the mean shoreline level (Figure 2.7). Depending on the duration of the storm, this pulse can propagate far inland, further than tidal oscillations; its effects on the water table and coastal groundwater dynamics are therefore important.

L i et al. (2003) presented an analytical solution to this problem based on the linearised Boussinesq equation. They considered the water table fluctuations in the aquifer, assuming a vertical beach face subject to a pulse in the mean shoreline level, i.e.,

h(z = 0 , t ) = ho(t), (2.34)

where ho represents the pulse signal. Far inland, the oscillation effects are diminished and so equation (2.5) applies. Since the focus is on the propagation of the pulse, the initial condition assumes zero water table oscillations, i.e.,

h(z, t = 0) = 0. (2.35)


0.4 0.3

2 0.2 g 0.1

0 -0.1 -0.2 t

0 5 10 15

0.25

0.2

g0.15

0.1

0.05

0

h

.r

1

0.15 -

h .s 0.1 - -r:

5 10 15 0.05

0

0.131 0.125,

0.11 0.1058

Figure 2.6 Predicted total water table fluctuations at (a) z = 20m, (b) 2 = 50m, (c) x = loom, and (d) z = 200m. Solid lines are from analytical solutions and circles are numerical predictions. SNTWF propagates much further inland than the semi-diurnal oscillations.

The solution to (2.1) subject to the boundary conditions as described by (2.34) and (2.5) and the initial condition of (2.35) is (Carslaw and Jaeger, 1959),

(2.36)

144 Ling Lj

change of deep water wave height during a storm

oscillations of mean shoreline elevation (zJ during a storm

Figure 2.7 Schematic diagram of a coastal aquifer subject to waves and mean shoreline oscillations (IWS: instantaneous water surface; SWL: still water level; MWS: mean water surface; MSL: mean shoreline; Harms: root of mean square deep water wave height).

where erfc() = 1 - erf() is the complimentary error function. Based on the field data, the following equation can be used to approximate ha,

ho = A exp [-G(t - , (2.37)

where A is the maximum increase of the mean shoreline elevation (representing the magnitude of the oscillation), t o is the time when the maximum occurs, and G is a time factor (l/a reflects the duration of the storm). Substituting (2.37) into (2.36) gives

t

h(z , t ) = -2AB (r - t0)exp [-G(r - t o ) 2 ] erfc J -W

x 1

(2.38) For the purpose of generality, (2.38) is rewritten in a non-dimensional

form,

h * ( ~ * , t*) = -2 T*exp [ - ( T * ) ~ ] erfc -m 7

Subsurface Pathways of Contaminants to Coastal Waters: . . 145

X . From (2.39), h A 2 d D m

where h* = -,t* = (t - to)&? and x* =

one can calculate the local maximum of h*(z*;t*) and the time of the maximum occurrence. These two quantities represent respectively the damping and time lag of the oscillation as it propagates in the aquifer. The solution suggests that the amplitude of the storm-induced water table fluctuations decreases more gradually than the exponential damping of the tidal water table fluctuations. Near the shore (z* < 5.82), the pulse signal is predicted to travel faster than the tidal signal. Such a trend is reversed further inland (x* > 5.82).

The analytical solution was applied to a field dataset. The details of the field site and measurement are given in Cartwright et al. (2003). The analytical solution predicted reasonably well the observed water table response to the storm-induced pulse in the shoreline (Figure 2.8).

1

0.8

0.6

0.4

0.2

0

I 0 1 2 3 4 5 6 1 8 9 10 -0.2 I

t (d)

Figure 2.8 the analytical predictions.

Comparison of the observed water table fluctuations with

2.6 Vertical flow effects (intermediate depth)

The validity of the Boussinesq equation depends on the shallowness of the aquifer, i.e., n,wHIK small (Parlange et al., 1984; Nzelsen et al., 1997). For aquifers of intermediate depths, the vertical flow effects become considerable] in which case the Boussinesq equation needs to be

-

146 Ling Li

expanded to include high-order terms, e.g., (Parlunge et al., 1984)

+ h ,dhd3h -- + -h3- 1 a3h1 . (2.40) dxdx2 3 8x3

In a linearised form, (2.40) can be rewritten as

(2.41)

The solution to (2.41) subject to the usual boundary conditions, i.e., (2.4) and (2.5), is given by

h = Aoexp(-lc,Ix) cos(wt - lc,zx), (2.42a)

with

.

(2.42b)

(2.42~)

Note that the damping rate and wave number can be combined to form a complex wave number lc = k,l + ilcv2, and the solution can be expressed in an alternative form,

h = Re [Aoexp(zwt - kx)] (2.42d)

with

(2.42e)

The relation expressed by (2.42d) is called the wave dispersion. The behaviour of kvl and k,, is different from that predicted by the Boussi- nesq solution. The vertical flow effects lead to difference between the damping rate and wave number (the rate of phase shift). In particular, the signal appears to propagate faster than predicted by the Boussinesq solution, i.e., smaller phase shifts.

Using a Rayleigh expansion of the hydraulic potential function in terms of the aquifer depth, Nzelsen et al. (1997) derived a groundwater wave equation, which includes an infinite number of high order terms to account for the vertical flow effects,

a t n (2.43)


where h is the fluctuation of the water table, and tan H - is an

infinite order differential operator as defined by Nzelsen et al. (1997). The solution of water table fluctuations based on this governing equation is

(-L>

, (2.44a) 1 sin(5 H ) cos( I C ~ H ) exp( - ~ c j x)exp (iwt) 2kjH + sin(2kjH) j=1

where k is the complex wave number; and j denotes wave mode. The solution is a summation of an infinite number of different wave modes. For each mode, the following dispersion relation holds,

inwd lcd tan(lcd) = - K - (2.4413)

The behaviour of the wave numbers is displayed in Figure 2.9. Only the first two modes are shown. As the frequency increases, the wave

7r 3T numbers approach asymptotic values: - for the first mode and - for

2d 2d the second mode. More generally,

(2.45)

This suggests that the water table fluctuations become standing waves as w increases. However, substituting (2.45) into (2.44a) gives h = 0, i.e., no water table fluctuations.

To examine the combined effects of vertical flows and capillarity, Lz et al. (2000~) derived another groundwater wave equation based on (2.42),

_ - (2.46)

where B is the thickness of the capillary fringe. Based on (2.46), the dispersion relation is given by

inwd K + ZwB’ kd tan(kd) = (2.47)

This dispersion relation is different from that of Nzelsen et al. (1997) as described by (2.44b). The extra term in the denominator is due to the inclusion of the capillary term in (2.46). The solution for the water table fluctuations in the coastal aquifer subject to the oceanic oscillation is of the same form as (2.44a) with the new dispersion relation that determines K for each mode. As the oscillation frequency w increases, the right hand side of (2.47) approaches a finite number, nd/B, predicting standing-wave-like water table fluctuations, different from the behaviour of (2.44b) as shown in Figure 2.9.

Ling Li 148

1.5

1

8 A 0.:

Re (kd)

Figure 2.9 Plots of the complex wave number (with d = p). Dotted lines are for Nielsen et al.’s solution (1997) and solid lines from Li et al. (2000c).

2.7 Density effects The above solutions ignore the density effects due to seawater intrusion in the aquifer. Wang and Tsay (2001) investigated this problem based on a sharp interface approach and derived a governing equation for h including the density effects,

dh - K d d t ne dx - - -- [(671+~)g] , (2.48)

where 77 is the height of the saltwater-freshwater interface from the base of the aquifer ( H is the height of the water table also from the base of the aquifer) and b is given by

(2.49)

where 7 f and ^/s are the specific weight of the freshwater and seawater, respectively; and uf and us are the kinematic viscosity of freshwater and seawater, respectively. Taking yf = 1000kg/m3, ys = 1020kg/m3, vf = 1.01 x 1OF6rn2/s, and vs = 1.06 x 10-6m2/s, S is calculated to be -0.028. The ratio of 677 to H is at the maximum (near the shoreline where 77 is at the maximum, being close to H ) -0.028. Equation (2.48) can therefore be approximated by the Boussinesq equation. In other words, the density effects on the water table fluctuations are negligible.


On the other hand, the interface fluctuates in conjunction with the water table. The governing equation of the interface fluctuation is

(2.50)

where q$s is the hydraulic potential in the saltwater layer and can be approximated by -h; K, is the hydraulic conductivity in the saline zone,

= -K(ufandu, are the kinematic viscosities of the fresh and seawater,

respectively). Assuming lq - f j l << f j where f j is the mean interface height (i.e., the magnitude of the interface oscillation is small relative to the local saline layer thickness), equation (2.50) can be reduced to

aq 0.93K 8

Yf Ts

Vf

7 s

(2.51)

These equations have been used to model observed water table and interface fluctuations in response to storm surges, and were found to predict well the data (L i et al., 2003; Cartwright et al., 2003).

2.8 Seepage face effects In reality, the occurrence of seepage faces is commonplace, in which

case the exit point of the water table at the beach face is decoupled from the tidal signal (Figure 2.1). The boundary condition is, in that case, defined by the movement of the exit point rather than the tidal level. Based on the concept of Dracos (1963), it can be shown that the formation of seepage faces reduces the primary forcing signals (semi-diurnal solar and lunar tides) and results in a third (additional to the two primary tides) forcing oscillation on the boundary, which has the period of a spring-neap cycle. The propagation of this signal in the aquifer leads to the SNWTF too.

Modeling seepage faces Dynamics of seepage faces at a beach subject to tidal oscillations

are not well quantified. Turner (1993) developed a model of dynamic seepage faces based on Dracos's concept, which was found to replicate field observations to some extent. This model was also shown to compare well with numerically simulated seepage faces for large beach slopes (hi et al., 1997a). In the Turner/Dracos model, the movement of the exit point is described by

(2.52a) K '&

Phase l(coupling) : ze = z, for Kide 2 -- sin2 (p),

150 Ling Li

K K 72, ne

Phase 2(decoupling) : z, = zep-- sin2(p)(t-tep)for Kid, < -- Sin2(p)

(2.52 b) where z , and z , are the elevations of the exit point and shoreline, respectively; n,, and K are the effective porosity and hydraulic conductivity of the beach sand, respectively; p is the beach angle; Kid, is the tidal velocity; t,, is the instant when decoupling commences; and zep is the elevation of the exit point at time t,.

0 2 4 6 10 12 14 t (4

Figure 2.10 Calculated elevations of the sea level (thin solid line) and the exit point (thin dashed line). Thicker solid and dashed lines show the 25-h averaged elevations of the sea level and the exit point, respectively.

Spring neap tides are described by equation (2.27), i.e., z, = A1 cos (wlt) + A:! cos(w2t - 6 ) . From (2.27), &d,(dzs/dt) can be calculated and then used in (2.52) to calculate the tidal seepage faces (i.e., the elevations of the exit point). As an example, Figure 2.10 shows the calculated seepage face over a spring-neap cycle using the above model with the following parameter values: A1 = 0.75m, AS = 0.25m, tan(@) = 0.1, K = 0.0005m/s, n, = 0.2, and H (mean aquifer thickness) = 2 m. The long period (of Tsn) oscillations are clearly evident in the exit point’s movement. The fluctuations of the exit point are analysed further in the frequency domain. The magnitudes of their Fourier transformations are shown in Figure 2.11. Compared with the tidal signals, the exit point oscillations display additional forcing frequencies apart from those of semi-diurnal solar and lunar tides. In particular, large oscillations occur


~ 1200

d 800 2

600

5 400 200

0

5 1000

at the spring-neap frequency while the amplitudes of the semi-diurnal oscillations are reduced by a factor of 0.4.

(a) fides Q

-

-

. !

1200-

% 1000 $ 800

(b) exit point oscillations

Figure 2.11 Magnitude of the Fourier transformation of the tidal signal (a) and the exit point’s oscillation (b). Symbols indicate the forcing frequencies at which considerable oscillations occur: square for the spring-neap oscillation, star for semi-diurnal lunar oscillation and circle for semi-diurnal solar oscillation.

Approximate solution fo r spring-neap Oscillations of the exit point

oscillations can be described approximately as For spring tides (which have the largest tidal range), the sea level

Z, = (A1 + A2) cos [ ( w 1 ; w 2 ) t ] ~ 7

with a tidal velocity,

(2.53a)

-(A1 + A2) (7) w1 +w2 sin [ ( T) w1 +w2 t] . (2.5313)

For neap tides (with the smallest tidal range), the approximation is similar; however, the ampli tude is given by IAl - A21 instead. To quantify the seepage face during a tidal cycle at either a spring or neap

152 Ling Li

tide, it is necessary to determine the times when the decoupling and coupling occur, respectively, i.e., t, and t,. Both times depend on the ratio of the exit points falling velocity to the maximum tidal velocity, R = - sin2 (/3) where A = A1 + A2 for a spring tide and JA1- A2 1 for

neAW a neap tide, and w = (WI + w2)/2. ForR < 1, seepage faces are formed; and the decoupling and coupling times are defined by

sin(wt,) = R, (2.54a)

and (2.5413)

K ne

zep - - sin2(/3)(tep - tep) = A cos(wt,),

where zep = A cos(wtep). The solution to (2.54a) is simply

arc sin(R) tep =

W (2.55a)

To obtain a simple solution to (2.54b), we approximate the RHS of the equation, A cos(wt,), by A(wt, - 3 ~ / 2 ) , a first order Taylor expansion around 3 ~ / 2 . This approximation is based on the fact that coupling occurs during the rising tide (i.e., between phases of T and 2 ~ ) . Thus,

(2.55b) 2(1+ R)w

Once t, and t, are determined, we can compute the elevations of the exit point. Subsequently, the averaged (over a spring or neap tidal cycle) elevation of the exit point can be calculated,

3~ + 2 cos(wtep) + 2Rwtep tep =

AT -%A [sin(wt,) - sin(wteP)] - -R(t:p T2 - tb). (2.56)

Clearly, the averaged elevation of the exit point over a spring tide is higher than that over a neap tide. The difference between these two averaged elevations gives an estimate of the amplitude of the exit points spring-neap oscillations (Asn), so,

(2.57)

where Rs and Rn are the ratios of the exit point’s falling velocity to the maximum tidal velocity for the spring and the neap tide, respectively. Therefore, the spring-neap oscillations of the exit point (Ze- sn)


are described approximately as

(2.58)

For the purpose of simplicity, 6 has been taken to be zero (i.e., time starts when the spring tide occurs).

Propagation of the spring-neap tidal oscillations in the aquifer The mean elevation of the spring-neap oscillations on the boundary

is nonzero but equals (ZeS + zen)/2 and hence the propagation distance is z - z o where z o = ( Z e s + Zen)/[2tan(P)] is the averaged distance of the exit point from the mean shoreline. The resulting spring-neap tidal water table fluctuations are thus described by

qsn = AsnexP[-ksn(z - XO)] COS[Wsnt - ksn(z - ZO)], (2.59a)

with (2.5913) newsn

ksn =

2.9 Two-dimensional tidal propagation The above solutions consider only the cross-shore propagation of tidal signals based on the one-dimensional Boussinesq equation. Sun (1997) considered the aquifer’s responses to tidal oscillations in an estuary, in which case the propagation of the tide in the aquifer becomes a two- dimensional problem because the tidal loading varies along the estuary, i.e., a non-uniform boundary condition,

h(z, 0, t ) = Aexp(-k,+) cos(wt - k, i z ) , (2.60)

where h(z, 0, t ) is the fluctuation of the water level in the estuary (the seepage face at the aquifer-estuary interface has been neglected); A and w are the tidal amplitude and frequency, respectively; and x is the distance along the estuary from the entry (Figure 2.12d). Ice, and k,i are the amplitude damping coefficient and wave number of the tidal wave in the estuary, respectively (Ippen and Hurleman, 1966)- A two-dimensional analytical solution was obtained for the resulting tidal head fluctuations in the aquifer (Sun, 1997),

h(x, y, t ) = Re{Aexp[-(k,,. + ik,i)y]exp[iwt - (a,. + ikei)x]}, (2.61)

where h(z, y, t ) is the head fluctuation; y is the distance from the aquifer- estuary interface (Figure 2.12d). k,,. and k,i are the amplitude damping coefficient and wave number of the tidal head fluctuations in the aquifer,

154 Ling Li

respectively; both are constants (see equations (2.14) and (2.15) of Sun (1997) for the formulae).

The analytical solution of Sun (1997) describes only the aquifer’s response to the tidal loading in the estuary. In reality, the oceanic tides along the coastline also influence the aquifer and thus interfere with the transmission of the estuarine tides in the aquifer (Figure 2.12), leading to more complex patterns of tidal head fluctuations in the aquifer than predicted by (2.61). Since the damping of both oceanic and estuarine tidal oscillations increases with the distance inland, their interactions are weakened as either the distance from the shore or that from the estuary increases. However, the interaction zone (the area where the tidal wave interaction remains significant) can be very large at a natural coast. For a confined aquifer, this area may be of several hundred square meters.

(2000a) examined the propagation of both oceanic and estuarine tidal oscillations in the aquifer based on the two-dimensional groundwater flow equation (Bear, 1972),

Li et al.

S d h d2h d2h T dt dx2 dy2

+ -. - _ _ - - (2.62)

Equation (2.62) is applicable to both confined and unconfined aquifers. In the latter application, the equation has been linearised and hence is applicable when the tidal amplitude is relatively small with respect to the mean aquifer thickness. For a confined aquifer, S is much smaller than that for an unconfined aquifer. This leads to a much-enhanced inland propagation of the tidal waves in the confined aquifer compared with that in the unconfined aquifer.

The boundary conditions along the coastline vary with the ocean tide, i.e.,

h(0, y, t) = A cos(wt). (2.63)

The beach slope and seepage face dynamics are also ignored. Along the coastline, the tidal amplitude and phase vary much less than those of tidal waves in the estuary and thus have been assumed to be spatially constant in (2.63). The boundary conditions along the estuary are specified by (2.60). Far inland, the tidal effects are diminished and so,

lim h(x, y, t) = 0. 1’00

(2.64)

Away from the estuary, the effects of the estuarine tide become negligible and thus the boundary condition there is determined by the cross-shore propagation of the ocean tide alone, i.e., the one-dimensional solution to the Boussinesq equation,

lim h(x, y, t) = Aexp(-k,,,x) cos(wt - kaiOx), (2.65a) Y’W


' interaction zone ocean

a. interaction zone for a small 8

estuarine tides

--__..--__

oceanic tides

interaction zone ocean

c. interaction zone for 8 z 90"

3 estuarine tides

' interaction zone ocean

b. interaction zone for 8= 90"

d. definition of the coordinates

Figure 2.12 in the ocean, estuary and aquifer. Case b is considered here.

Schematic diagram of coastal aquifers, and tidal oscillations

where k,,, and kaio are the amplitude damping coefficient and wave number of the one-dimensional tidal head fluctuations, given by

kar0 = kaio = g. (2.65a)

Because of the interaction between cross- and along-shore head fluctuations, the separation-of-variables method used by Sun (1997) is not applicable here. Instead, the Green's function method is used to solve equations (2.62)-(2.65). First the following decomposition is taken,

h(z,lJ,t) = ho(z,t) + h1(z,y,t), (2.66a)

with ~o(z, t ) = Aexp(-k,,,z) cos(wt - kaioz), (2.66b)

(2.66~)

156 Ling Li

hl(z, 0, t) = Aexp(-k,,z) cos(wt - k,iz) - ho(z, t), (2.66d)

(2.66e)

(2.66f)

lirn hl(z, y, t) = 0,

lim hl(z, y, t) = 0. X+CQ

Y - + a

Both ho and hl satisfy (2.62). Using the Green’s function method, the solution of hl, involving by a double integral, is found,

dG t o o

hl(z,y,t) = D / / h l ( zo ,O , to ) - ( z , y , t ; zo ,O , to )d~od to , (2.67a) 0 0 dY0

and

where D = T / S and G is the appropriate Green’s function. The initial head fluctuation has been assumed to ho(z, 0), i.e., hl(z , y, 0) = 0. Ex- pressing h1(z0,0, to) as Re{Aexp(iwto+klzo) -Aexp(iwto+k2zo)} with kl = -(Ice, + keii) and k2 = -(kaTo + kaioi), one can reduce the above solution to a single integral, i.e.,

hi(%, y,t> = ARe{ [f(ki , z) - f(h, -z>

(2.68a) l

- m a , z) + f(k2, -z)ldto}, where

(2.68b)

e-l/(t-to) = 0, the integrand is non-singular and Because lim

the integral can be easily evaluated via any standard quadrature scheme. Physically, the solution is expected to become periodic as t increases. In other words, the effects of an initial condition on the solution are diminished after a certain time (memory time, tc) elapses. Li et a2. (2002b) and Li and Jiao (2002b) reported further studies on two-dimensional tidal water table fluctuations.

1 to+t (t - to)3/2

2.10 There have been many other studies on the tidal water table fluctuations, some of which are listed Table 2.1.

Recent developments and other effects

Subsurface Pathways of Contaminants to Coastal Waters: 157

2003 Li and Jiao, 2001a

Main contributions Jeng, Teo, Seymour,

aquifers bounded bv rhythmic shorelines. Tide-induced groundwater fluctuation in a coastar leakv confined aquifer system extending under the sea. Analytical studies of groundwater-head fluctuations in

Li and J im, 2001b

Li, Jiao, Luk and Che-

a confined aquifer overlain by a semi-permeable layer with storage. ‘ride-induced groundwater level fluctuation in coastal

2000 Townley, 1995 Trefry, 1999

3 Implications for contaminant transport and transformation in tidally influenced coastal aquifers

I P lex P orosit y . The effects of landward boundary conditions. Periodic forcing in composite aquifers.

As demonstrated above, the tides affect greatly the coastal ground-water. The water table fluctuation is the manifestation of such effects in the shallow unconfined aquifer and has been studied extensively. The fluctuation leads oscillating groundwater flow in the near-shore area of the aquifer, enhancing the water exchange and mixing between the aquifer and coastal sea/estuary. In the following, we will illustrate the importance of these local flow, exchange and mixing processes for chemical transport and transformation in the near-shore aquifer and the associated chemical fluxes to coastal water.

3.1 Tide-induced flushing and dilution effects on chemical transport processes

Li et al. (1999) developed a model of SGD that incorporates the net groundwater discharge, and the outflows of the tide-induced oscillating groundwater flow and wave-induced groundwater circulation (Figure

Table 2.1

I58 Ring Li

Ll) , i.e., D S G D = B, -i- L>w + L)t.

Since a large portion of the high rate SGD (a, and D,) is of marine origin, questions remain as to how much the discharge contr~bute~ to the transfer of ~an~-derived pollutants to the ocean. Using the "box" model described below, Ei et a,?. (1999) examined the i ~ ~ p o r t aof SGD, especially Dw and Dt, on the process of cliemical transfers horn the aquifer to the ocean.

(3.1)

Figure 3.1 A box model of chemical transfer from the aquifer to coastal sea.

The model includes three water bodies: coastal sea (CS), brackish aquifer (BA), and freshwater aquifer (FA). Chemical transfers occui between the water bodies as shown by arrows in Figure 3.1. The chemicals are assumed to be strongly absorbed by sand particles in fresh groundwater and LO desorb into brackish round water. Phosphate, a ~ mand cadmium are important constitue~~ts with this behavior.

The mass balance for FA can be described by

FFA-zrb = FFA-out if 8 =I Seq,

dS F F A - ~ ~ ~ = 0 and VFA-- = FFA-~, if S < Seq, d t

(3.2a)

(3.2b)

FFA-zn = DnCm and (3%)

Seq = kilolczn, (3.24

where F F A - ~ ~ and F F A - ~ ~ ~ : are the input and output mass flux for FA, respectively (the alon~-shore distance of the water body is taken to be a unit meter); S is the amount of absorbed chemical and the ~ u ~ s c ~ i p t eg denotes the ~ q u ~ l i ~ r ~ u ~ n state; V'A is the effective volume of the FA; is the di~tribL~tion coefficient (the dimensions of 6' have been cha~ged to ME-" and so K d is ~ o ~ - ~ i ~ i e n s i o n a l ) ; and Czn is the input chemical concentration. Equation (3.251) expresses an e ~ u ~ l i b r i ~ state where the


maximum adsorption has been reached and hence the output flux equals the input flux. At a non-equilibrium state, the output flux is reduced to zero due to adsorption and S increases at a rate equivalent to the input mass flux.

The chemical input to BA includes FFA-ozlt, Fcs (mass flux due to the incoming seawater) and FSI resulting from seawater intrusion. The chemical adsorbed on sand particles tends to desorb in seawater. Thus, seawater intrusion produces an input flux to BA, and the magnitude of this flux is related to the speed of seawater intrusion and the amount of adsorption S. The output mass flux is due to SWGD. The governing equations are listed below:

(3.3a)

FSGD = (D , -I- D, + Dt)CBA, (3.3b)

Fcs = (D , + D, + Dt)Ccs and (3.3c)

(3.3d)

where VBA is the volume of BA and CBA is the chemical concentration in BA. VSI is the volume of intruded seawater. Ccs is the chemical concentration in the ocean and, for the contaminants considered, is usually small compared with CBA and can be neglected.

Transfer fluctuations of land-derived pollutants to the ocean due to D, and Dt

Chemicals such as phosphate and ammonia are land-derived pollutants, for example, as a result of nutrient leaching from the agricultural fertiliser. Sediments in the fresh water aquifer, as a temporary storage for these chemicals due to high adsorption, become the immediate source of chemical to the brackish aquifer when seawater intrusion occurs and the chemical desorbs into the brackish groundwater from the sediment. In this section, we present a simulation to illustrate how the local groundwater circulation and oscillations affect the transfer of land- derived pollutants. In the simulation, the distribution coefficient, K d ,

was assumed to be 400. The FA is assumed to be in an equilibrium state and seawater intrusion occurs between t = 0 and 10 d. The saltwater front retreats shoreward between t = 10 d and 20 d. The salt wedge moved at a constant rate, equal to 5% of the net groundwater discharge rate. The net groundwater discharge rate (Dn) was set to be 3.75 m3/d/m, and the sum of D , and Dt is 90 m3/d/m (24xD,). The inland chemical concentration (Ci,) was 1 kg/m3. The volume of the BA is 500 m3/m. During seawater intrusion, the output mass flux from

160 Ling Li

the FA is described by (3.3a) and during the retreat of the salt wedge, FFA-out is given by (3.3b). The time that it takes for the FA to reach the equilibrium state after the retreat of the salt wedge can be estimated by P S I t S I / F F A - o u t '

Figure 3.2 the ocean.

Simulated rates of the transfers of land-derived chemicals to

The simulated rate of chemical transfer to the ocean is shown in Figure 3.2. Also plotted in the figure are the results from a comparison simulation with D, and Dt neglected. It is clearly shown that a large increase of the transfer rate resulted from the seawater intrusion and the local groundwater circulation/oscillating flows. The first factor (i.e., seawater intrusion) contributes to an extra and excessive source of the chemical. The second factor (i.e., the local groundwater circulation and oscillating flows) provides the mechanism for rapid flushing of the BA, resulting in increased chemical transfer to the ocean. Without the second factor, the large impulse of chemical input to the ocean would not occur as demonstrated by the comparison simulation (dashed curve in Figure 3.2). The increase of F ~ G D is very substantial, more than 20 times as high as the averaged rate. As the salt wedge retreats, the transfer rate decreases to zero since the inland chemical is all adsorbed in the FA. The local processes do not change the total amount of the chemical input to the ocean, which is determined by the inland source.

The tide-induced flushing effect is further illustrated by the following simulation based on a simple one-dimensional mass transport model,

dc d2C dc - = Dc- - V-, at ax2 ax (3.4a)

~ u b s ~ ~ r ~ a c e Pathways of Contaminants to Coastal Waters: - - 161

with

'I/ = Ki,, -i- d2KkAoexp( -kx) cos w t - kx + - , (3.4b) ( "> 4 where the first term of the RHS is the net groundwater flow rate and the second term represents the oscillating flow induced by tides (based on the analytical solution, ( 2 . 2 ) ) . The initial concentration is specified according to an existing plume as shown in Figure 3.3. The b o u n ~condit~ons for the chemical transport are: c = 0 at the inland boundar~, and c = 0 for V > 0 and dc/dx = 0 for r/ < 0 at the seaward boundary. The following parameter values are used in the simulation: i, (regional hydraulic gradient) = 0.01, A0 = 0, 1 and 2 m, T (tidal period) = 0.52 d, K == 20 m/d, = 10 m, rze = 0.2, a = 3 m (L) , = aV)? L (distance of the landward boundary from the shore) = 150 xn.

Figure 3.3 Tidal effects on transport of a contam~ant plume.

The results displayed in Figure 3.4 show that residence time of the chemical in the aquifer decreases due to tidal oscillations (LHS panel of Figure 3.4). The tidal effects also lead to dilution of the exit chemical concentration significantly (RHS panel of Figure 3.4). Such dilution may reduce the impact of chemicals on the beach habitats.

In this section, we will address how the fresh groundwater discharges to the ocean. Previous studies, neglecting the tidal effects, predict that the freshwater overlies the intruded seawater and discharges to the ocean with little mixing with the saltwater. The limited mixing, driven by the density effects, occurs along the saltwater wedge. A s~mulation was conducted using Sea Wat (htt p: //.rat er .usgs.gov/o~/seawat /; by Wei- xing Guo and ~hrist ian D. Langevin) to examine the tidal effects on the freshwater discharge. ~ensity-dependent groundwater flow in a coastal aquifer subject to tidal oscillations was simulated with a set of parameter values re~resenting the shallow aquifer conditions. The simulation grid is shown in Figure 3.5.

3.2 Tide-induced mixing of fresh groundwater andseawater

16% Ling Li

I00

90

80

70

t./ \= 60 c4

3 P B R? rfo * 3 30

20

10

0 0 50 100 150 0 70 40 60 80 100

time (d) Dist from the shore (m)

Figure 3.4 Tidal effects on chemical transport in a coastal aqu~fer. LHS panel: mass ~ e n ~ a ~ n i ~ g in the aquifer (in percentage of the initial m ~versus time under different tidal conditions. 8 panel: chemical con- cent~ation profiles in the aquifer at different times (dark for the results shortly afier the sirn~lation started, light a bit mediaq medium much later, cyan near the end of the simulation). ashed lines are for results with tidal effects ( A = 2 rn) and solid lines for results w ~ t ~ i o ~ ~ t tidal effects.

Figure 3.5 The grid used in the sirnulation.

The s i ~ ~ ~ ~ a t ~ o n was run first without the tidal oscillationas until t?,

steady state was reached. The result of the salinity d~s t r ibu t~~n in the aquifer shows the traditional view of the groundwater discharge as discussed above (the top panel of Figure 3.6). The tide was then ~ ~ t r o dinto the simula~~on~ which continued to run for 100 tidal cycles and reached a quasi-§~ead~ state. The result shows a very ~ ~ ~ ~ rsalinity ~ ~ s t r ~ b u t i o n horn that without the tidal eEects (middle panel of Figure 3.6). First, a saline plume wias formed in the upper part of the beach. The freshwater dischar~ed to the sea through a ~ube/channel between


this upper saline plume and the intruding saltwater wedge. Secondly, the freshwater discharge tube contracted and expanded as the tide rose and fell (shown in the attached animation)) suggesting considerable mixing activities. Such mixing is also indicated by the salinity gradient shown in the bottom panel of Figure 3.6. These simulated salinity profiles are consistent with recent results from laboratory experiments.

Fresh water

70 12 74 16 78 80 82 84 86 88 90 Nm)

Figure 3.6 Salinity distribution in the near-shore area of the aquifer. Top panel: without tidal effects; fresh groundwater discharging to the sea without much mixing with underlying seawater. Middle panel: tidal effects leading to the formation of the upper saline plume and the freshwater discharging, and considerable mixing between the freshwater and seawater.

In analysing the simulated flow and mass transport process, we are particularly interested in (a) how the mean (advective) transport of salinity is affected by the oceanic oscillations, and (b) whether the oceanic oscillations (water exchange) cause diffusive/dispersive transport of salinity. This diffusive transport represents the local, small scale mixing. For this purpose, the following decomposition is taken,

u(z , z , t ) = U ( z , 2) + UI (Z ) 2, t ) , (3.5a)

c(z, z , t ) = C(z, z ) + cyz, 2, t ) , (3.5b)

where u and c are the raw data of flow velocity and salinity; U and C are averaged flow velocity and salinity over the tidal cycle (24 hrs); and u' and c' are the tidally fluctuating flow velocity and salinity. The total

164 Ling Li

mass transport of salinity can then be determined,

The first term represents the transport due to the mean flow (advection). The second transport component is the dif€usive/dispersive flux. In Figures 3.7 and 3.8, we show calculated mean transport flux and diffusive flux. It is interesting to note that the two fluxes exhibit different patterns; and the magnitude of the mean transport flux is one order of magnitude larger than the diffusive flux. Based on the calculated difh- sive/dispersive flux, one can estimate the apparent diffusion/dispersion coefficient (the local mixing intensity parameter),

(3.7)

where VC is the mean salinity gradient that can be estimated from the measured salinity data.

These results show that the tidal effects affect the near-shore groundwater flow and transport processes significantly, leading to increased exchange and mixing between the aquifer and the ocean. Such effects can also alter the geochemical conditions (redox state) in the aquifer and affect the chemical reactions. As shown numerically below, the exchange enhances the mixing of oxygen-rich seawater and groundwater, and create an active zone for aerobic bacterial populations in the near-shore aquifer. This zone leads to a considerable reduction in breakthrough concentrations of aerobic biodegradable contaminants at the aquifer-ocean interface.

, . , , , . . . . . ,

I I . , . . . I 3.5t]; t ;,; f 1 ; ; : : : : : : : . , . . . . .

, . . . _ . . , . . I

70 72 74 76 78 80 82 84 86 88 90 x (m)

Figure 3.7 The mean mass transport flux.

~ubsu~face Pathways of Conta~n~nants to Coastal ~ ~ t e r s : I . . 165

0.5

I

1.5

2.5

3

Figure 3.8 The fluctuating mass transport flux.

~~~~~~~~~

As d~s~ussed above. ocea71 tides induce local oscillating ~ r o ~ ~ dflows in the near -shore aquifer. These flows enhance exc~aiiges and mixing between oxygen-r~ch seawater and gro~dwater , and create an active zone for aerobic bacterial ~opulation in the aquifer, leading to s ~ g n i f i creduction of biodegradable contam~nants.

OW and PHT3D were used to model contaminar~t transport radation in coastal aquifers affected by tidal oscillations. Two

mobile chemicals were included in the s i ~ u ~ a t ~ o n : oxygen as the electron tor and toluene as a representative b~odegradable contamina~t. An

aerobic bacterium was included as an immobile phase. The biodegradation process was oxygen-~imited (i.e., sufficient substrate) a The inland ~ontaniinant source was specified at the cells near the water table. Other c~7id~t~ons and the parameter values used in the simu~ation are shown in Figure 3.9 and listed and in Table 3.1.

,--->

Figure 3.9 ditions ~ ~ ~ ~ e r n e n t e d in the simulation.

Schematic diagram of the model set-up and boundary con-

The s ~ ~ u l a t i o n was run first without tides until a steady state of chemical conce~itrations was reached. Tides started after that. An ani-

3.3 Tidal effects on chemical reactions

166 Ling Li

Fable 3.1 Parameter values used in the...simtdatioB-ParameterLongitudinal dispersivity(m)Transversal dispersivity (in)Horizontal hydraulic conductivity (m/d)Vertical hydraulic condiictivity (m/d)Effective porosityTidal range (in), diurnal tide

Value0.010.00116.88.40.252 m [-1 1]

Toluene

0.05

Oxygen

Aerobes

Figure 3.10 Image plots of the steady state concentrations for toluene(top panel), oxygen (middle panel) and bacteria (bottom panel).

mation of the simulation results is attached on the CD. The image plotsof steady state concentrations for toluene, oxygen and bacteria are shownin Figure 3.10. Due to the lack of oxygen, little degradation of tolueneoccurred except in the smearing diffusive layers. Correspondingly, littlegrowth of bacteria can. be observed. The chemical concentrations at ahigh tide after 5 tidal cycles were shown in Figure 3.11. The tidal effectis clearly evident: first it created an oxygen-rich zone near the shoreline,which led to biodegradation of toluene. Secondly, it enhanced the mix-ing process. The smearing layer was thickened. The results at the low

Subsurface Pathways of Contaminants to Coastal Waters: 167

tide show similar patterns and changes in the chemical concentrations,In short, the simulation demonstrates that tidal oscillations lead to theformation of an oxygen-rich zone in the near-shore aquifer area. Aerobicbacterial activity sustained by the high O2 concentration in this activezone degrades the contaminants. These effects, largely ignored in previ-ous studies, may have significant implications for the beach environment.

Toluene

Oxygen

Aerobes

10 SO 60x(m)

Figure 3.11 linage plots of the concentrations for toluene (top panel),oxygen, (middle panel) and bacteria (bottom panel) at the high tide afterfive tidal cycles.

Coastal water pollution is a serious environmental problem around theworld. Most contaminants are believed to be sourced from the land. Todevelop sound strategies for coastal water pollution control, we must beable to quantify the sources, pathways and fluxes of the contaminants tothe coastal zone. Traditionally, terrestrial fluxes of chemicals to coastalwater have been estimated on the basis of river flow alone. However,recent field observations indicate that contaminants entering the coastal

4 Conclusions

168 Ling Li

sea with groundwater discharge can significantly contribute to coastal marine pollution, especially in areas where serious groundwater contamination has occurred.

To determine the fluxes of chemicals to coastal water, it is important to quantify both the transport processes and chemical reactions on the pathway. There has been a large amount of research work conducted on how the chemicals may be transformed during the transport along the surface pathway, i.e., the role of a surface estuary. In contrast, little is known about the chemical transformation in the near-shore area of a coastal aquifer prior to chemicals’ discharge to coastal water.

In this chapter, we first reviewed a large volume of work on tide- induced groundwater oscillations in coastal aquifers, focussing on various analytical solutions of the tidal water table fluctuations. In the second part, we discussed the effects of tides and other oceanic oscillations on the chemical transport and transformation in the aquifer near the shore, drawing an analogy to the surface estuary-subsurface estuary. The discussion, based on several on-going studies, illustrated the important role that a subsurface estuary may play in determining the subsurface chemical fluxes to coastal waters. Although the tidal influence on the water table dynamics has been subjected to numerous studies, the effects of tides on the fate of chemicals in the aquifer have not been investigated adequately. Quantification of these effects is clearly needed in order to

0 provide better understanding of the pathway of land-derived nutrients and contaminants entering coastal waters, leading to improvement of strategies for sustainable coastal resources management and develop ment ;

0 provide useful information for integrating the management of u p land and lowland catchment areas; and

0 develop sound risk assessment methods and mitigation plans for coastal and estuarine pollution.

Acknowledgement A shorten version of this paper has been published by CRC Press, i.e., Li, L., D. A. Barry, D.-S. Jeng, and H. Prommer, 2003, Tidal Dynamics of Groundwater Flow and Contaminant Transport in Coastal Aquifers, in Coastal Aquifer Management - Monitoring, Modeling, and Case Studies, A. Cheng and Driss Ouazar (Edi), CRC Press, 2004.

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174

Studies on Retrieval of the Initial Values and Diffusion Coefficient of Water Pollutant

Advection and Diffusion Process I

Tingfang Wang Chengdu Meteorological Center of Airfore,

Chengdu, 61 0041, China

Sixun Huang, Huadong Du Institute of Meteorology, PLA Universty of Science and Technology,

Nanjing, 21 1101, China

Gui Zhang College of Science, PLA Universty of Science and Technology,

Nanjing, 21 11 01, China

Abstract

In this paper, the variational adjoint method with Tikhonov regularization principle is applied to retrieving the initial concentration filed and the diffusion coefficient of a 3-D water pollutant model. The adjoint models are deduced for the complete and incomplete observation of water pollutant concentration respectively. Then a particular algorithmic scheme is designed for 1-D model as an example. Finally, a series of ideal numerical tests are performed to show the validity of the method. The numerical r+ sults show that the adoption of Tikhonov regularization principle can be used to eliminate oscillation in retrieval and the difficulty of short of information for the incomplete observations can be overcome efficiently by introduction of the background field of diffusion coefficient.

1 Introduction Since the Industrial Revolution, the environment has been destroyed se- riously by human activities. Release of all kinds of pollutants causes both air and water environments to be worsen and worsen. In water environment, not only river but also groundwater has been polluted according to the observations in the last several decades. Therefore the advection

Studies on Retrieval of the Initial Values and . . . 175

and diffusion problem of water pollutant is attracting scientists to research and the hydrodynamic models need to describe transport process of water pollutant more accurately. The problem includes the transport process and the initial concentration distribution of the pollutant. The transport process can be divided into two sides, i.e. advection and diffusion. When change of water current velocity is small, diffusion is mainly produced by turbulence-a scientific problem far to be resolved (Hu 1995), but it is mainly due to shear dispersion for the inhomogene- ity of velocity in current section when the change is obvious, and the diffusion is just about the total effects of them in this paper. On the other hand, it is worth determining initial distribution of pollutant, such as pollutant source of groundwater and so on. Consequently, the present studies are concerned with two aspects, i.e. to jointly retrieve the d i h - sion coefficient and initial pollutant concentrations by a transport model with certain observations. We discuss the problem in the two cases of complete observations and incomplete ones obtained at the end time respectively. For lack of observational data, the 3-D retrieval model about the two cases are deduced theoretically, but numerical tests are only for l-D ideal ones.

The variational adjoint method is applied in this paper (F. X. Le Dimet, J. Q. Yang, 2002). The retrieval of multi-parameters is usually so ill-posed that yields failures in calculation (TIKHONOV A N et al., 1977; KIRSCH A., 1996), which mainly results from short information, not from number of dimensions. Therefore, the Tikhonov regularization principle is applied to diminishing oscillations in the retrieval (Huang sixun et al., 2001, 2003, 2004; Xiao Tingyan et al., 2003). In the case of incomplete observations, a stabilized functional about the background field is adopted to remedy the short of observation information. Finally, the validity of our method is verified by a set of ideal numerical tests.

2 The 3-D model of water pollutant advection and diffusion process

The 3-D model of water pollutant advection and diffusion process with fixed boundary conditions is

[ g + ui- = - ( ka- ::) + f(x, t ) , d X i axi

176

where C is the concentration of pollutant, time t E (O,T), x, U are 3-D vectors of position and water current velocity, k, I are diffusion coefficient and length of space domain, C = (0, ZI) x (0,Zz) x (O,Z3),ni is the 2-D position vector obtained from x by deleting the component xi, and f is a source or sink of pollutant. The Einstein sum stipulation is adopted in the above equation and the following text, and i ranges from 1 to 3 (mentioned in the following text anymore).

It is worthwhile to mention that this model can describe almost all kinds of the processes of advection and diffusion including those about air, water even silt in river.

Tingfang Wang, Sixun Huang, Huadong Du, Gui Zhang

3 Retrieval in the case of complete obser- vat ions

The present problem is to determine the initial distribution of pollutant Co(x) and diffusion coefficient k(x) such that the following functional is minimal,

2

J[Co, k] = - (C - Cobs)2dx + $ J ( k i g ) dxdt = min! (3.1) 2 's R R

here Cobs is observations of distribution of pollutant. Retrieval of multi-parameters is usually an ill-posed problem which

leads to application of the regularization principle. The second term in J is referred to as stabilization functional and y1 is the regularization parameter. As the first order norm of C , the stabilization functional can overcome ill-posedness and diminish oscillation in calculation. Next the adjoint model, adjoint boundary conditions of the tangent linear model (TLM) and gradient formula of the functional about ki, Co are derived which are divided into four steps as follows.

Step 1 Derivation of TLM

disturbed to &, 60 in the following way Suppose that Ki, CO with the corresponding solution C of (3.1) are

with the corresponding solution 6. Define

i& - ki ,. ZI-C ki = lim ~ , C = lim -. a+o a a-0 a

Studies on Retrieval of the Initial Values and .

Then, the TLM is obtained as follows,

177

Step 2 The first order variation of J From the definition, we have

J'[k, CO; E , CO] = 1 Vc0 J . &dx + / VkiJ kidxdt. (3.3) c $2

On the other hand, according to (3.1), the first order variation of J is

= / ( C - Cobs) . kdxdt + 71 / 12EZ axi axi + kiki ( g)2] dxdt

= 1 [ (C - Cobs) - 71- (ki -)] . edxdt

R n

d ,dC axi axi

n

here ki, C be zeroes in C as (3.2).

Step 3 Derivation of the adjoint equations

able), and integrating in 0, yields Multiplying the first equation of (3.2) by P (called as adjoint vari-

178 Tingfang Wang, Sixun Huang, Huadong Du, Gui Zhang

and

J $ . Pdxdt = P(x , 0)Codx - C-dxdt, n J c n s T

J U i g . Pdxdt = - s C-(UiP)dxdt, Li n n

- J& ( k i g ) Pdxdt = - J C- ( ki- E) dxdt, n n -1; (his) . Pdxdt = J k i z z d x d t , A d P d C

n n

here let P be zero on the boundary and at the end time. Thus (3.5) may be written as

compared with (3.4), the adjoint equation is

g + & p i p ) + & @a%) = (C - Cobs) - Y 1 L axi (k=), 2 axi

P I t 9 = 0, (3.7)

P lxi=0 = P IXi& = 0.

0 = c x (0,T) I

Step 4 The formula for the gradients of J It follows from (3.3), (3.4) and (3.6) that

0 = Vc, J Codx + J Vki J . hidxdt - J P(x , O)Co(x)dx n c

c J

Then the gradients of the functional J with respect to I c , ~0 are

( vc, J = P(X, O),


The next task is to seek the optimal parameters through iteration algorithm. Let R" = (ka, Con), then the iterative scheme is

4 Retrieval in the case of incomplete observations

The present problem is to determine the initial distribution of pollutant Co(x) and diffusion coefficient k ( ~ ) from an observation Cobs(x) at end time T , such that the following functional is minimal,

+- (ki - k!)'dxdt = min! 2 s

n

For the short of observations information, the third term of (4.1) is introduced. It is a stabilized functional about the background field restriction of the diffusion coefficient k.

Steps 1 and 2 are very similar to the case of complete observations, i.e. the form of TLM and definition of the first order variation of J are (3.2) and (3.3) respectively, but the expression of J includes an additional term of constraint on k from the background field. Thus


Multiplying the first equation of (3.2) by P (called as adjoint variable), and integrating in R, yields

and here

P(x , T)CT(x)dx - n c c

- 1 C c d x d t , at n

here let P be zero on the boundary and at the end time. Thus (4.3) may be written as

1 P(x, T)CT(x)dx - P(x , O)Co(x)dx + J { - C [ % c J c n

Compared with (4.2), the adjoint equation is

Studies on Retrieval of the Initial Values and . 181

It follows from (3.3), (4.2) and (4.4) that

0 = Vc, Jeodx + Vki J . kidxdt - P ( x , O)Co(x)dx c s R s c s

Then, the gradients of the functional J are

Figure 4.1 is the flow chart of retrieval. In addition, since in both two cases, an original model and the as-

sociated adjoint model are mixed problems of PDE of advection and diffusion with forcing, and with t and U reverse, the adjoint model becomes same as the original model. Therefore, a difference scheme from Keller box scheme with second order accuracy and stability (Lu Jinfu et a1.,1987) is given below for l-D original model.

The l-D original model in a rectangle domain of time and space can be written as

a(1, T ) = { ( x , t ) l O I x I 1 , O 5 t 5 T }

The discrete grids in R(1, T ) are taken as follows

- . (4.8) ~j = j h , j = 0, N x , N X . h = 1

t , = nr,n = 0, N t , N t . r = T

Introducing the notations


k, Co are new guessing values Is error of k, Co small than expected value?

caculate V , J , Vco^J from (3.8) or (4.6)

t * caculate k, Co from (3.9)

1 Is functional fiom (3.1) or (4.2) decreasing?

I

Yes

k, C, are inversion output, caculation is finished

Figure 4.1 Flow chart of adjoint retrieval.

then the original equation turns to

(4.11)

Then the original model is transformed into one composed of (4.10) and (4.11), with the Keller box scheme as follows


The time distribution of C at T can be obtained through iteration using initial and boundary conditions, and the adjoint variable P is computed to get the gradient of J.

5 Numerical tests To test the effect of the above theory and algorithm, a series of tests are performed. The model used here is as follows, which has been used in the literature(Min Tao et a1.,2003).

[ g +U(x,t)- = - ” ( k(x) - E) +f(.,t), ax ax C(5,O) = 5 3 ,

C(0, t ) = 0, C(1, t ) = e t .

R = ((5,t)IO 5 z 5 l , o 5 t 5 l},

1 3 here U ( z , t ) = - ( x 2 + t2), f(x, t ) = -z2(x2 + t2 )e t .

10 10

If diffusion coefficient k(x) = A x 2 , the model has the analytic solution C(x , t ) = x3et , and then the observation at t = 1 is C(x , 1) = ex3.

In tests, we disturb k(x) by its 10 percent as an initial guess, and the corresponding integration of the model is taken as an observation with error, but the observation C(x, 1) at the end time is assumed to be true.

Test 1: The complete observation C in the whole temporal and spacial domain is provided. In test, the regularization parameter y1 is


set to be y1 = 0, i.e. the regularization method is not used. From Figure 5.1 it can be seen that the profile of the retrieved diffusion coefficient approaches that of the true value with small amplitude oscillation. From Figure 5.2 it can be seen that the error between the prediction of C from the retrieved initial value and the true value is three orders of magnitude less than that between the prediction of C from the initial guess and the true value. For sufficient observation, the retrieval is satisfactory.

0.09 I I

X

Figure 5.1 fusion coefficient in Test 1.

Comparison of the true, guess and retrieved profiles of dif-

Test 2: The observation C in the whole temporal and spacial domain is still provided as in Testl. But the regularization parameter y1 is set to be y1 = From Figure 5.3 it can be seen that oscillations of the error of diffusion coefficient C is eliminated, and the profiles of retrieved and true diffusion coefficients fit well. From Figure 5.4 it can be seen that the error between the prediction of C from the retrieved initial value and the true value is four orders of magnitude less than that between the prediction of C from the initial guess and the true value.

Test 3: Only the observation C at the end time t = 1 is provided. Again the regularization parameter is set to be y1 = 10V6, while the background parameter 7 2 = 0. From Figure 5.5 and Figure 5.6 it can be seen that the retrieval results are unsatisfactory being short of observation.

Test 4: The test is the same as Test 3 except 7 2 = 1. From Figure 5.7 and Figure 5.8 it can be seen that the retrieval results are amelio- rated greatly. The profile of the retrieved diffusion coefficient rather

Studies on Retrieval of the Initial Values and . - 5 185

0.02

25 12

Figure 5.2 field of pollutant on cent ratio^ in Test 1.

Comparison of the errors between the guess and retrieved

0.07

0.06

-

-

8 10 12 14 16 18 20 X

Figure 5.3 dispersion ~ o e ~ ~ i e ~ t in Test 2.

~ o ~ ~ a r i s o ~ of the true, guessing and retrieved profile of

a ~ ~ r o ~ c h e ~ that of t.he true value, and the error between the prediction of C from the retrieved initial value and the true value is two orders of

4

2

0

-2 25

Copt- color x i 0 4

0 02

0

-0 02

-0 04 25

Figure 5.4 ~ ~ ~ l ~ ~ t ~ n t concentrat~on in Test 2 .

~ o ~ ~ a r i s o n of the errors of guess~ng and r e ~ ~ ~ e ~ e ~

Figure 5.5 ~ o ~ ~ a r i s o n of the true, guessing an retrieved profile of d ~ s ~ e ~ s ~ o n coeEcient in Test 3.

~ a g n i t u ~ e less than that between the ~ ~ e ~ i c t i o n of C from the initial guess and the true wlue.

Studies on Retrieval of the Initial Va1u.es and . I

C.-.C&

1.137

0.02 -

'-12

Figure 5.6 ~ Q m ~ a r i s o n of the errors of guessing md re t r~e~ed field of ~ o l ~ ~ t a n t con6entration in Test 3.

0.08

0.07

0.05

0.05 .c-:

0.04

0.03

0.02

0.01

' 0 2 4 6 8 10 12 14 16 18 20 X

F.igure 5.7 ~ o ~ ~ a r ~ s o ~ of the true, ~ e s s i n g and retrieved profile of ~ ~ ~ ~ e ~ s ~ o n coefficient in Test 4.

The above retrieval results are compared in Table 5.1. The error in the table is root-mean-square error(^^^^)^ For ~ o n ~ e n ~ ~ a t ~ o n , i t is root

188 Tingfang Wang, Sixun Huang, Huadong

c-cobs

4

2

0

-2 25

002 - 0

-0 02

-0 04,

Figure 5.8 ~ o ~ p a r i s o n of the errors of guessing and retrieved field of ~ o l l ~ t a n t co~centrat~on in Test 4.

mean square of errors at the grids in the whole temporal and spatial do- sion coefficient, it is root mean. square of errors at the grids

in the whole spatial domain; The table shows that retrievals with complete ob~ervations are successful, and retrievals with incomplete observations in the presence of background are ako successfuI, while retrieval with i n c o ~ ~ l e t ~ observations in the absence of background is unsuccess- ful. The retrieval results also demon st rat^ that for inverse problem, not only the r e ~ ~ ~ a r ~ z a t i o n but also introduction of background is i ~ ~ ~when the informat~on is sparse.

Table 5.1 ~ o ~ p a r i s o n of the error nornis in all retrieval tests

I error norm of retrieved field I in Test 1 0.049 I 0.0019 I error norm of retrieved field in Test 2 ._....._-.__I- error norm of retrieved field

in TeBt 4


6 Conclusions

In this paper, the variational adjoint method with the Tikhonov regularization principle is applied to retrieving the initial concentration filed and the diffusion coefficient of a 3-D water pollutant model. The adjoint models are derived for the complete observation of water pollutant concentration and incomplete observation at the end time respectively. Then a particular algorithmic scheme is designed for 1-D model as an example. Finally, a series of ideal numerical tests are taken to show the validity of the method. The main conclusions are as follows:

(1) Tikhonov regularization principle can be used to overcome ill- posedness of the problem and diminish the oscillation in retrieval especially.

(2) Retrieval of complete observations is very successful and accurate due to abundance of information.

(3) Retrieval of incomplete observations are very difficult due to lack of information; but the difficulty of short of information can be overcome efficiently by introduction of the background field on diffusion coefficient.

Numerical tests are only for 1-D problem in present paper, and multi- dimensional problems will be studied later.

Acknowledgement

This work was supported by the National Natural Science Foundation of China (Grant Nos. 40175014 ) and the Shanghai Association Fundation For Science & Technology (Grant Nos. 02DJ14032 ). The authors wish to express their appreciation for the careful and thoughtful review by the reviewers and comments by the editor.

References [l] Du H D, Huang S X, Shi H Q. The Theoretical Analysis and Nu-

merical Experiments in the Retrieval for the 1-D Semi-geostrophic Shallow Water Model. Journal of Hydrodynamics Ser. A, 19(1): 38- 45, 2004.

[2] Fang H X, Huang S X, Wang T F. Retrieval Study of 1-D Drop- sonde's Motion. Journal of Hydrodynamics Ser. A, 19(1): 53-60, 2004.

[3] Le Dimet F X, Yang J Q. Models and Data for the Water Cycle. Mathematical Problems in Environmental Science and Engineering. Beijing, Higher Education Press, 41-65, 2002.


[4] Hu F. Turbulence, Intermittence and Atmospheric Boundary Layer. Beijing, Science Press, 1995.

[5] Huang S X, Han W, Wu R Sh. Theoretical Analyses and Numer- ical Experiments of Variational Assimilation for One-Dimensional Ocean Temperature Model. Science in China, Ser. D, 9: 903-911, 2003.

[6] Huang S X, Wu R Sh. Physical Mathematical Problems of Atmo- spheric Science. Beijing, Chinese Meteorological Press, 2001.

[7] Kirsch A. An Introduction to The Mathematical Theory of Inverse Problems. Springer-Verlag, 1996.

[8] Lu J F, Guan Zh. Numerical Solutions of Partial Differential Equa- tion. Beijing, TsingHua University Press, 1987.

[9] Min Tao, Zhou X D. An iterature method of the inverse problem for the dispersion coefficient in water quality model. Journal of Hy- drodynamics Ser. A, 18(5): 547-552, 2003.

[lo] Tiknonov A N, Arsenin V Y. Solutions of Ill-posed Problems. V. H. Winston and Sons, Washington D.C., 1977.

[ll] Xiao T Y, Yu Sh G, Wang Y F. Numerical Method of Inverse Prob- lems. Beijing, Science Press, 2003.

191

Application of Tabu Search Method to the Parameters of Groundwater Simulation

Models

Jing Chen, Zhifang Zhou College of Civil Engineering, Hohai University,

Nanjzng 21 0098, China

Abstract

Tabu search method, a meta-heuristic approach, with which simulated annealing and genetic method axe extensively used in combinational optimization. This paper successfully applied Tabu search method to the inverse analysis of hydrogeological parame ters and validate its effectiveness in the simulation of the groundwater field affected by cutoff wall in Hanjiang dike. This method, which is based on human memory, has shown the predominance of random search.

1 Introduction The word Tabu (or Taboo) comes from Tongan, a language of Polynesia, where it was used by the aborigines of Tonga island to indicate things that cannot be touched because they are sacred. According to Webster‘s Dictionary, the word now also means “a prohibition imposed by social custom as a protective measure” or of something “banned as constitut- ing a risk”. These current more pragmatic senses of the word accord well with the theme of Tabu search. The risk to be avoided in this cme is that of following a counter-productive course, including one which may lead to entrapment without hope of escape. On the other hand, as in the broader social context where “protective prohibitions” are capable of being superseded when the occasion demands, the “Tabus” of Tabu search are to be overruled when evidence of a preferred alternative becomes compelling.

Engineering and technology have been continuously providing examples of difficult optimization problems. The roots of Tabu search go back to the 1970’s; it was first presented in its present form by Glover (1986); the basic ideas have also been sketched by Hansen (1986). When we come to its application in groundwater, it must be emphasized that

192 Jing Chen, Zhifang Zhou

Zheng and Wang (1996) applied both Tabu Search method (TS) and Simulated Annealing (SA) to determine parameters for a hypothesis one dimensional problem for the first time. Chung and Chun(2002) focus on the application of Tabu Search to groundwater parameter zonation and they had got a good result.

2 General description of tabu search

[4]Tabu Search was applied to optimize ground water parameters in this study. TS allows moving to the worse solution among the neighborhood. The better solution of recent iterations are recorded in a Tabu list to reject moving back to these solutions again within a given number of iterations, which can prevent being trapped by local optimum.

Tabu Search seeks the optimal solution by moving from the current solution to the best solution within its neighborhood. If the best solution within the neighborhood is not listed on the Tabu list, the current solution is replaced with the new selected solution. The new solution could be worse than the previous solution. TS keeps moving from one solution to its neighborhood to find the optimal solution until the stop criterion is met. The main components of TS include initial solution, movement, Tabu list, aspiration criteria, and principles to stop searching. These components are briefly introduced as follows.

2.1 Initial solution

The optimization process improves solutions by moving from a current solution to its neighbor solutions. Thus, an initial solution should be given before the searching process. Different initial solutions may lead to different local optimal solutions in a nonlinear problem. However, it is difficult to have the best guessed initial solution. An initial solution may be randomly selected. Thus, a robust algorithm needs to result in minimum deviation.

2.2 Neighborhood and movement The process of optimization is to move from one solution(X) to another solution within its neighborhood, N ( X ) . Different definitions of neighborhood may be made by different analysts or for different problems. After defining neighborhood for a problem, the neighborhood for a s e lution can be easily determined, and then the best solution among the neighborhood is selected as the next stop. Tabu Search records the optimal solution among the previous explored solution space. If the new

Application of Tabu Search Method to the Parameters of . . . 193

selected solution is better than the recorded solution, the latter is replaced with the new one. When the algorithm stops searching , the final recorded solution is the found optimal solution.

2.3 Tabu list Tabu list records better solutions in recent iterations and prohibits moving back to these solutions to avoid being trapped by local optimums. Thus, the length of the list cannot be too short. Otherwise, the searching process may fall into a cycle and may not have chances to find a global optimum. On the other hand, the Tabu list should not be too long. Otherwise, the searching process could become inefficient and may be limited to a small feasible space and thus reduce the chance to find a global optimum. There is no universal principle to determine the length of Tabu list. Glover(1990) suggested a magic number of 7, which can be the first guess.

2.4 Aspiration criteria

The Tabu list records better solutions of previous iterations, and it is nor- mally designed to record only parts of decision variables. For example, a solution set contains two decision variables, X and Y . When changing X1 to X2 with Y = Y1 finds the best solution within the neighborhood of N ( X 1 , YI), The Tabu list will record that X = X I is Tabued. However, when Y changing to Y2 with X = X I has the better value of objective function, the Tabu is relaxed and the solution is allowed to move to ( X I , Yz).

2.5 Principle to stop searching

Different principles to stop searching process can be applied. These may include: (1)the recorded optimal solution reaching required level; (2)given maximum iteration; and(3)the number of successive iterations on which solutions are not improved. There is no common practice for the use of the above criterion. It is dependent.

3 Where can we use tabu search method? (1) Application of Tabu Search to Parameter Structure and Parameter Values

It has been mentioned above that Zheng and Wang (1996) applied both Tabu Search method (TS) and Simulated Annealing (SA) to determine parameters for a hypothesis one dimensional problem for the first


time. Chung and Chun(2002) focus on the application of Tabu Search to groundwater parameter zonation and they had got a good result. They all have proposed a combinatorial optimization model that identifies the best parameter structure. But they indicate that this method can be extended readily to include both parameter structure and values. This paper just applied it to the optimization of parameter values.

(2) A n Inverse-Simulation Approach to Determine Optimal Strategies for Developing Public Water-Supply System

How to find the best strategies for managing groundwater resources is often accomplished with the method of optimization. For example, the approach to find best pumping strategies in a coastal aquifer, well location and pumping rate are optimized with respect to (1)minimizing impacts to nearby surface water; (2)preventing saltwater intrusion contamination due to overpumping, etc. If the pumping rate is assumed to be known and unchanged, the well placement becomes our goal of optimization. Typically “trial and error” approach is used for finding best strategies, but the chances to get what we need diminish because of large number of potential option. By using Tabu Search method, the better solution of recent iterations are recorded in a Tabu list to reject moving back to these solutions again within a given number of iterations, which can prevent being trapped by local optimum. Figure 3.1 shows the well placement Changjiang estuary where the problem of optimization can be solved with Tabu Search method. Figure 3.2 shows that the definition of neighborhood set associated with the grid of estuary.

Figure 3.1 The Well placement of Changjiang estuary.

Application of Tabu Search Method to the Parameters of . 195

Figure 3.2 Definition of neighborhood set.

4 Application of tabu to the inverse analysis of hydrogeological parameters

4.1 Evaluation model

The mathematical model for the problem is given as follows:

the object function is,

where hi,t and h$ are calculated and observed hydraulic heads at time t ant at grid i that has a ground water observation well; n is the number of ground water observation wells; Kh is the hydraulic conductivity; s3 is the specific storage coefficient; and w is the net vertical,srecharge, respectively. In this study, ground water heads and Kh are all variables S3w are assumed to be known. By using Tabu search method, the better Kh is continually hunted within its scale. The hydraulic heads can be solved by FEM program. After several iterations, the value of object function draws near zero.


4.2 Discrete variables and neighbors

Because Tabu Search method comes of problems of discrete path networks(such as TSP problem), continuous parameters can be searched only after it becomes discrete. Thus far we have used the term neighborhood but not given precise definition. For a discrete variable we define its surroundings as its neighborhood. For example, given a certain searching radius, a one-dimensional variable may have two neighbors while a two-dimensional variable may have four or eight neighbors. A three-dimensional variable may have six or 26 neighbors. Of course, this is not the only way to define a set of neighbors. The neighbors of a two-dimensional variable may be circularity while a three-dimensional variable has a sphere as its neighbors.

4.3 Tabu search procedure

Step1 (Initialization) Select a starting solution X ; Evaluate the object function, Set the neighbor of X , N ( X ) ; Set the Tabu T empty;

Step2 (Choice and termination) Evaluate the object function of neighbor, Determine the best solution X ’ of N ( X ) ;

Step3 (Update) Find the N ( X ) , and update Tabu list T ; Let X = X’; Go back to Step 2 to continue the search until a stop criterion is satisfied.

4.4 result

Cutoff wall, as a seepage control measure, has a good effect on the improvement of safety condition of dikes. But whether the cutoff wall has a bad effect on the groundwater environment or not has been widely concerned. In the flood season, the cutoff wall effectively prevents seepage and protects dikes, while in the dry season it gets in the way of the natural hydraulic relation of both sides of dike. Now we study the procedure of Tabu search with the simulation of the groundwater field affected by cutoff wall in Hanjiang dike. The hydraulic conductivity of aquifer, the hydraulic conductivity of cutoff wall and the specific storage coefficient associated with certain region are to be optimized.

The region we study is shown in Figure 4.1, where there are a cutoff

Application of Tabu Search Method to the Parameters of . . . 197

coutoff wall

cross secti

cross section No.5

Figure 4.1 The sketch map of placement of wells.

wall and 25 wells, GK1-GK25. Cutoff wall, as a seepage control measure, has a good effect on the improvement of safety condition of dikes. It is made of concrete with thickness of 30cm. The value of hydraulic conductivity is assumed to be 1.0 x 10-7cm/s - 9.9 x 10-7cm/s while the hydraulic conductivity of the aquifer is assumed to be 5.1 x 10W4cm/s - 7.28 x The specific storage coefficient is 0.05 - 0.25. TABU method “searches” the best solutions in the range of these parameters. We choose the heads of GK12, GK13, GK14, GK15 as observation heads, the water level of Hanjiang and the heads of another wells are assumed to be known. After the initial parameters are given, the FEM program simulates the hydraulic heads in the study region. The value of object function can be calculated. Then we determine the neighborhood of the current solution, and also, the objective value of neighborhood can be calculated. TABU method is “searching” the best solution in the range of these parameters and the calculated heads are drawing near the observed heads. When the stop criterion is satisfied, the best solutions are got. The hydraulic conductivity of the aquifer is K = 6.96 x 10W4cm/s. The hydraulic conductivity of the cutoff wall is 1.02 x 10W7cm/s. The specific storage coefficient is 0.021. The value of object function is shown in Figure 4.2. Figures 4.2 and 4.5 show the optimal simulated and observed hydraulic heads of No.14 No.16 weIls, from which we can see the optimal simulated heads are almost the same with observed heads.


- Quadrangled mesh - Triangular mesh

01 0 100 200 300 400 500 600 700

Figure 4.2 dure.

Variation of object function in the optimal searching proce-

365 -

36 -

355 -

35

345 -

-

36.5

36

35.5

35

34.5

Figure 4.3 Optimal simulated and observed heads of No.16 well.

~~

35.5 -

35 -

--Optimal Simulated Head

34.5 -

34 -


Application of Tabu Search Method to the Parameters of . . 199 36.2 36.5

36 35.8 36 35.6 35.4 35.5

35.2 35 35

34.6 34.5 34.8

34.4 1 i


5 Summary

(1) This paper applied Tabu search method to the inverse analysis of hydrogeological parameters. The result shows that Tabu search method is well suited for solving the inverse problem.

(2) The soul of Tabu search method is its "protective prohibitions". The better solution of recent iterations is recorded in a Tabu list to reject moving back to these solutions again within a given number of iterations, which can prevent being trapped by local optimum.

(3) Tabu search method is well suited for solving the proposed combinatorial optimization model. It has shown its predominance in solving TSP problem, Job-shop problem, etc. For the area of groundwater modeling, we think it has more extensive foreground.

References

[l] Zheng C, Wang P. Parameter Structure Identification Using Tabu Search And SimulatedAnnealing. Advances in Water Resources. V01.19, No.4, 215-224, 1996.

[2] Ching-Pin, Chun-An Chou. Application of Tabu search To Ground Water Parameter Zonation. Journal of The American Water Re- sources Association. vo1.38. No.4. 1115-1125.

[3] Alain H, Eric T. Dominique de Werra. A Tutorial On Tabu Search. Report.

[4] Glover F. "Heuristics for Integer Programming Using Surrogate Constraints," Decision Sciences, Vol8, No.1, 156-166. Seminal work on tabu search and scatter search. 1977.


[5] Glover F. “Tabu Search - Part I,” ORSA Journal on Comput- ing, Vol.1, No.3, 190-206. First comprehensive description of tabu search. 1989.

[6] Glover F. “Tabu Search - Part 11,” ORSA Journal on Computing, V01.2, No.l,4-32. The second part of this comprehensive description of tabu search introduces additional mechanisms such as the reverse elimination method. 1990.

[7] Glover F. “Tabu Thresholding: Improved Search by Nonmonotonic Trajectories,” ORSA Journal on Computing, Vol. 7, No.4, 426-442. A description of a specialized form of tabu search known as tabu thresholding. 1995.

20 1

Several Problems in River Networks Hydraulic Mat hemat ics Model

Xiaoming Xu Department of Applied Mathematics, Hohai University,

Nanjing, 21 0098, China

Deguan Wang College of Environment Engineering, Hohai University,

Nanjing, 21 0098, China E-mail: [email protected]

Abstract

The U.S. National Weather Service (NWS) FLDWAV model has capacity in simulating variety of hydraulic phenomena in river systems. However it is originally designed for simulating one- dimensional unsteady flow in dendritic channel systems and is not full suitable for river networks. The modified relaxation iterative method was proposed when simulating the unstedy flow and the water quality in river networks of Shanghai with more than 400 named rivers. The advantage of this improved model is that it can satisfy for any complex river networks. The convergence of Netwon iterative method using in this model for nonlinear systems was also theoretically proved and Netwon down-hill method was applied to the case of unconvergence situation in original numerical method. A combined Gauss elimination with partial pivoting method with compress-storage (GEPP-CS) technique was developed to solve the linear systems.

1 Introduction

Shanghai, the most economically developed city in China, is located in Yangtze delta area with complicated river networks. The total river number is about 33000 with various scales and more than 170 gates, locks and pumping stations on the river systems. The rivers are strongly influenced by tides, typhoon and heavy rains. More than 400 named channels were simulated in this paper.

202 Xiaoming Xu, Deguan Wang

For the purpose of managing water resources in the Shanghai river networks, a proper water quantity numerical simulation model must be selected. The NWS FLDWAV hydraulic model (F'read and Lewis, 1998) was employed as the basic models. Although the FLDWAV hydraulic model has the capacity of simulating variety of hydraulic phenomena in the river systems which includes lots of waterworks. However this model was originally designed for simulating l-D unsteady flow in tree-type river systems, so it was not full suitable for looptype river networks such as Shanghai river networks.

In order to establish Shanghai river networks hydraulic mathematical model, substantial improvements were made to strengthen the FLD- WAV hydraulic model capacity. These improvements include: (a) the relaxation iteration method of branch flow in tree-type river systems was extended to the loop-type river networks, (b) the convergence of Newton iterative method was theoretically proved for nonlinear systems and two methods were proposed for Newton iterative method to deal with unconvergence situation in numerical computation, (c) new solution technique with small rounding off and high stability was set up for the linear systems.

2 Saint-Venant equations and relaxation iterative method

2.1 Saint-Venant equations and discretization

Preissman weighted four-point implicit numerical scheme is used to solve the Saint-Venant equations of l-Dunsteady flow (Read, 1985)

d A dQ - + - - q = o at dx dQ d Q2 dh - + -(P-) + gA( - dX + S f ) = O dt dx A

where A is the cross-section area of flow, Q is discharge, t is time, x is the longitudinal mean flow-path distance measured along the center of the water course, q is the lateral inflow or outflow per lineal distance along the watercourse, 0 is the momentum coefficient for nonuniform velocity, g is the gravity acceleration constant, h is the water level, and S f is the boundary friction slope. Nonlinear algebraic equations with hS+', &I+', hy;: and 9::; as unknown variables for a river have the

Several Problems in River Networks Hydraulic . . . 203

follow form:

here i = 1 ,2 , . . . , N - 1 (N is the number of cross sections), superscript n + 1 means time tn+l, there are 2 N - 2 equations with 2 N unknown variables. Adding to upstream and downstream boundary conditions, nonlinear systems with 2 N variables is formed:

F ( x ) = 0 (2.5)

where F(.) is a function from R2N into RZN.

2.2 Newton iterative

Newton’s method for nonlinear systems is generally expected to give quadratic convergence. Suppose upstream and downstream conditions are water level time series, then the Newton’s equations of step lc has the form:

(2.6) A(k)z = b ( k )

where A(k) is the Jacobian matrix of equations (2.5) (superscript n + 1 is omitted).

L O

Initial approximation value take as at time tn’s. The solutions of linear systems (2.6) are cross sections water levers and discharges at time tn+l.


2.3 River networks relaxation iterative method

Since Saint-venant equations is only suitable for a single river. For tree- type river systems ( Figure 2.1), the tributary flow at each confluence is treated as lateral flow q which is estimated by solving equations (2.1 and 2.2) for each river. Relaxation iterative method about junction discharges is proposed

where gi is the new estimated confluence lateral flow for the next iteration, qii is the computed discharge at the confluence in the previous iteration, qE:w is the previous estimated confluence lateral flow, Q is a weighting factor(0 < Q < l), Ic is branch river number. Convergence is attained when Iqii - qEil < EQ.

/

Figure 2.1 Tree-type river systems.

However when the relaxation iterative method in the FLDWAV was applied to looptype water networks (Figure 2.2), such as Shanghai water networks (Figure 2.3), the discharges at the confluence of loop-type rivers can not be balanced. For solving this problem, improved iterative method (Xu et al., 2001) was proposed. In the new relaxation iterative method the iterative must be operated for every confluence of the river networks. Therefore the problem of the discharge unbalance was well solved.

Several Problems in River Networks Hydraulic .

/

Figure 2.2 Loop-type water networks.

205

Figure 2.3 Shanghai water networks.

3 The convergence of Netwon iterative method

Netwon’s iterative method is usually to give quadratic convergence, hence it is applied in the FLDWAV model to solve the nonlinear systems (2.5). However, there are two problems about Netwon’s iterative method. The first problem is the convergence of Netwon’s iterative method. According to convergence theorem of Netwon’s iterative method, all of the Jacobian


matrix A(k) for each time step and for each river are required continuous and nonsingular at z*, where x* is the solution of equations (2.5). Xu and Wang (2001) proved the matrix A@) meeting, therefore the Net- won’s iterative method was convergent in theoretical for equations (2.5). The second problem is the local convergence of the Netwon’s iterative method. That is the each time step initial approximation dn) must in the convergence area N6 of (n + 1) time step solution z*:

However N6 is unknown. In the FLDWAV model, the solution of n time step is taken as the initial approximation of (n+l ) time step. When the solution of n time step is not in the area of N6, the iterative series may unconvegence. To solve the problem in numerical computation, two methods were proposed in the modified model. One is reduction time step, At/2, At/4,. . . is taken as new time steps until convergence. Another method is Netwon’s down-hill method:

where [F’(zk)]-’ is W k is relaxation factor (0 < W k < l), Wk

will be chosen as 1, 1/2, 1/4, - ” , until 11 F(Z(’+’)) II<II F ( z @ ) ) 1). When w k 1, it is Netwon’s iterative method.

4 The new solution method of linear systems

For each time step, for each river and for each step of Netwon’s iterative method, there is a linear systems (2.6). So there are large number of linear systems to be solved. The solution method is very important for numerical simulation. It will directly determine the precision and success or fail of simulation. A lot of division operation is done in the FLDWAV model (Read, 1971). The solution procedure will fail if the one of the divisor is zero. The FLDWAV model does not supply a method to avoid division by zero. This situation appeared in Shanghai river networks hydraulic computation by using of this model. In order to solve the problem, a combined Gaussian elimination with partial pivoting method with compress-storage technique was proposed (Xu and Wang, 2001). The new solution technique had the advantage of small rounding off and high stability for the linear systems.

Several Problems in River Networks Hydraulic . . . 207

4.1 Compress-storage Suppose the 2N x 2N coefficient matrix of linear systems (2.6) is

A =

all a12

a21 a22 a23 a24

a31 a32 a33 a34

a43 a44 a45 a46

a53 a54 a55 a56

. . .

a2N-2,2N-3 a2N-2,2N-2 a2N-2,ZN-1 a2N-2,Zh

aZN-l,ZN-3 a2N-1,ZN-2 a2N-1,ZN-1 a2N-l,2h

1 a2N,2N-1 a2N,2N

(4.1) By defining matrix (4.2) to reduce the storage from 2 N x 2N to

2N x 4.

A =

where

0 0 d13 d14

d21 d22 d23 d24

d31 d32 d33 d34

d4l d42 d43 d44

d5l d52 d53 d54

... ... ... . . .

dzN-n ,~ dzN-z,z ~ z N - z , ~ ~ z N - z , ~

dzN-i, i d z ~ - i , a d z ~ - i , 3 dz~--1,4

.dzlv,i ~ Z N , Z 0 0

ai,j+i-2 if i = edd, j = 1,2,3,4

ai,j+i-3 if i = even, j = 1 ,2 ,3 ,4 (4.3)

208 Xiaoming Xu, Deguan Wang ,

4.2 Combined Gaussian elimination with partial pivoting method with compress-storage

Gaussian elimination with partial pivoting method has the benefit of high stable and low round off error. In addition, this method can avoid divide by zero in the procedure of elimination only if the matrix A is nonsingular. However if the normal Gaussian elimination with partial pivoting method is applied to the matrix A in the process of the inter- change of the two columns of the matrix A, the new nonzero element (fill-in) will yield, hence 2N x 4 array for storage is not enough. To avoid this problem, the fill-in element is stored in the position of the element is zero which is eliminated in the last step. According to this method, the eliminating with partial pivoting can be performed in 2 N x 4 array.

5 Case study and conclusions

Figure 5.1 shows river networks of the one example to verify the FLD- WAV model. It consists of 286 channels. Among them 19 are branched channels and the others are net channels. The main channel, Huangpu river, is inflected by the tide at downstream. The roughness coefficient of the main channel varied from 0.020 to 0.031 and adjusted automatically according to the water level and flow direction. The roughness coefficients of all other channels were 0.025. The time step was 0.2 hours. The total simulation time was 30 days. The parts of results from the simulation are shown in Figures 5.2 and 5.3. The computed water levels are reasonable agreed with the measured data.

Figure 5.1 The structure of river networks.

Several Problems in River Networks ~ y d r a u ~ c . . 209

After successful employing improved relaxation iterative method to iterative each confluence of the river networks, the discharges of confluence reached balance and the Row co~putation of treet-type river systems was extended to the looptype river networks. On the other hand, the convergence of Newton iterative method was proved theoretically for ~ o ~ ~ ~ n e a ~ systems a d two methods were proposed for ~ e ~ oitera, tive method to deal with unconverg~nce case in numerical co~putation.The new GEPP-CS solution technique with small rounding of2 and high stability was set up for the linear systems.

Mishidu gauging station 4.5

4

3.5

3

2.5

7 -

I___- _ . _ . ~

Figure 5.2 station.

Comparison of measured and computed stage at Mishidu

Suzhouhe gauging station 5

4.5 4 t-. 8 3.5

3 3

B z 1 2.5

1 .5 1 480 504 628 662 676 600 624 648 672 696 720

Figure 5.3 station.

Comparison of measured and computed stage at Suzho&e

fl] Eead D L. Discussion of implicit Rood routing in natural charmels by M. Amein and C. S. Fang. J. Hydraulic Div., ASCE, 97 (IXY7), 1156--1159, 1971.

References


[2] Fread D L. ‘Channel Routing’, Hydrological Forecasting. Edit by M. G. Anderson and T. P. Burt, 437-503, 1985.

[3] Fread D L, Lewis J M. NWS FLDWAY MODEL, Hydrologic Re- search Laboratory, Office of Hydrology, National Weather Service, NOAA, Nov. 28, 1998.

[4] Xu X M, He J J, Wang D G. Nonlinear Method on Large Scale River Networks, J. of Hydrodynamics, Ser. A, Vo1.16, No.1, 18-24, 2001.

[5] Xu X M, Wang D G. Proof of Convergence of Newton-Raphson Iterative Method in Simulating Unsteady Flow in River Networks, J. of Hydrodynamics, Ser. A, Vo1.16, No.3, 319-324, 2001.

[6] Xu X M, Wang D G. A Method for Solving Large Size Linear Alge- braic Equations, J. of Hohai University, Vo1.29, No.2, 38-42, 2001.

211

Study on the Character of Equilibrium Point and Its Impact on the Changing Rate

of Phytoplankton Concentration Using a Simple Nutrient-Phytoplankton Model

Jue Yang, Deguan Wang, Ying Zhang College of Environmental Science and Engineering, Hohai University,

1 Xzkang Road, Nanjing, 210098, China

Abstract

This study is focused on the equilibrium point, which is the value when phytoplankton concentration is unchanged with time, of phytoplankton concentration. The changing rate of p h y b plankton population and the character of equilibrium point were studied using simple Nutrient-Phytoplankton mathematical model. The results indicated that under certain environment conditions phytoplankton concentration meets with the equilibrium point at the end and the differences between the equilibrium point and the concentration of phytoplankton determine the changing rate of phytoplankton concentration. The equilibrium point is essentially dependent on the environmental factors. In this paper, it is used to explain the phenomenon of red tide. The environmental factor, nutrient, was the only factor concerned in the work.

1 Introduction

Red tide (Harmful Algae Bloom): Small, single-celled algal bloom where the cell densities are high enough to change water color, often to red. Algae is an important primary producer, therefore, an important part of the food chain. And the red tide algae often belongs to phytoplankton.

According to related study on red tide, the fundamental mechanism is well understood, although it still needs improvement. Factors related to harmful algal bloom are as follows:

1. Underwater light climate (instant irradiance, attenuation coefficient, etc.)

212 Jue Yang, Deguan Wang, Ying Zhang

2. Nutrients (nitrogen, phosphorus, trace elements, etc.) 3. Nitrogen / phosphorus ratio (important to species composition

and succession) 4. Hydrodynamics (nutrient transportation, stratification, upwelling,

turbulence, etc.) 5. Physiology (division rate, vertical migration, half saturate values,

etc.) Some of the factors (two, three or more) are involved in the phyto-

plankton or algae models. By observation, monitoring, calculation or analysis, the red tide could be predicted. But there is still one question that although we have known the changes of some factors when red tide bursts out, we do not know what on earth the effects of all these factors on the view of ecology.

Oceanographers and Limnologists have studied the water ecological system, ecological model and their characters. Theories on ecological system and model have already well proved. On the other hand, we know that red tide is an ecological phenomenon and the population of algae exponentially increases when red tide breaks out. Then, how can we explain the phenomenon using the ecological theories?

For example, the simplest density-dependent model of phytoplankton is:

du a d t b _ - - -u(b - u)

where u denotes the density of population for population, a denotes the intrinsic rate of natural increase, and b denotes the carrying capacity in certain environment.

From equation (l.l), the positive equilibrium point u1 = b can be du du

retrieved. If u > b then - < 0, if u < b, then - > 0. That is to say, dt d t

u --t b when t -+ 00.

Now, the question is what the value of ‘b’ is when red tide breaks out.

Let us study the equation again. The fact can be found that the larger ( b - u) is, the larger % is. Imagining that ‘b’ is very large and the initial ‘u’ is small, what would happen to the population of phytoplankton? Apparently, $ is very large at the beginning, and it will drop when the value of ( b - u) decreases.

The next question is about the factors influencing ‘b’ value. These factors are exactly those mentioned at the beginning of the paper. The nutrients (pollutants) in the water are the main reasons that lead to algae over-grow. So in the following, the impacts of environmental factor (nutrient) on phytoplankton population will be studied using simple ecological model. In section 2, nutrient concentration in water environment is assumed constant. Based on the information in section 2, the

Study on the Character of Equilibrium Point and . . . 213

temporal variability of nutrient concentration is discussed in section 3.

2 Phytoplankton population model with constant nutrient concentration

2.1 Mat hemat ical model Based on that of Steele and Henderson (1981), one-component model was used to represent concentrations of phytoplankton (u) in a physically homogeneous oceanic mixed layer. Phytoplankton and aquatic plants that are mostly unicellular, take up nutrients from the water for photosynthesis. When a constant nutrient concentration is assumed, the model becomes:

d u N a d t e + N b + c u

u - ( s + k)u - - -- -

where u denotes the density of population for population and N denotes the concentration of nutrient. The parameter definitions and the value are given in Table 2.1, together with the value of each parameter originally used by Steele and Henderson (1981). The equations of the model are described in detail (for linear and quadratic closure) by Edwards and Brindley (Edwards and Brindley, 1996; Edwards and Brindley, 1999); here brief outline of the model is given. A physically homogeneous mixed layer is assumed, within which volumetric concentrations of N and u are uniform. Units of nutrient ( N ) and u are gCm-3, with time t measured in days, and all parameters are positive. The ratios to convert back into units of nitrogen or chlorophyll, as used by Steele and Henderson (1981), are as follows:

Table 2.1 Parameter definitions and default values Parameter Symbol Default value

a / b gives maximum u growth rate light attenuation by water u self-shading coefficient

half-saturation constant for N uptake

1~ sinking loss rate

0.2 m-

0.03 gCm- cross-thermocline exchange rate 0.05 day-

0.04 day-1

1 g carbon 20 mg chlorophyll = 10 mmol nitrogen The mixed-layer depth is kept fixed at 12.5 m, and it is an au-

tonomous dynamical system. The water below the mixed layer is assumed to have zero phytoplankton.

The next step is to analysis the formulation of the model.


At firstly, the steady state ( d u l d t = 0) is assumed. The following solution can be retrieved (when u f 0):

(2.2) N u - b(s + k) (e + N )

c(s + k ) ( e + N ) u1=

Equation (2.1) becomes:

d u -(s + ~ ) C U [ U - U I ] - - - d t b + c u

du d t

If u1 < 0, - < 0 when initial value of u is greater than zero. That is to say, if there's enough time the phytoplankton will be extinct. So the

condition N > be(s + Ic) can be regarded to maintain certain phyto-

plankton population. It can be explained that only when the nutrient concentration is greater than a critical value in ocean environment could phytoplankton exist.

Secondly, the positive equilibrium point u1 is discussed. If u < u1

then - > 0; if u > u1 then - < 0. So u1 is steady (Figure 2.1).

u - b(s + Ic)

d u d u d t d t

I * t

Figure 2.1 Integral line of u for equation (2.1).

Thirdly, note the limit value of u1, equation (2.2) can be rewritten as follows:

b

,c - _ U

u1= c(s + k ) ( R + 1)


From equation (2.4), it can be found that u1 +

when N + 00.

2.2 Results and discussion

In this section, the positive equilibrium point was achieved and the processes of phytoplankton population were observed along a linear nutrient gradient using four-order Runge-Kutta method. The nutrient concentrations ( N ) 0.001, 0.01, 0.1 and 1 were chosen for simulation. For

du N=0.001, phytoplankton population is dropping because - < 0. Its d t dropping line is shown in Figure 2.2. For N=0.01, 0.1 and 1, - > 0, their increasing lines are shown in Figure 2.3. For the four N values, the equilibrium points are -0.32079,0.88889, 3.7735 and 4.89374, respectively. For N=0.001, the phytoplankton concentration drops to 0. For other three N values, the phytoplankton concentrations reach their own equilibrium points at last. Apparently, the phytoplankton concentrations increase very fast at first, and then slow down when they approach their equilibrium points (see Figure 2.3). The dropping rate of phytoplankton concentration for N=0.001 is shown in Figure 2.4 and the increasing rates for other three N values are shown in Figure 2.5. These results showed that the larger N is, the faster phytoplankton concentration increasing rate is.

du d t

0'012 0.01

a 0.008 8

8 f 0.006

k 0.004 c a 0.002

0

-

1 2001 4001 6001 8001 10001 12001 time

Figure 2.2 Phytoplankton concentration dropping line for N=0.001.


N= 1

N=0.1

N=0.01

1 2001 4001 6001 8001 10001 12001 14001 16001 18001 time

Figure 2.3 Phytoplankton concentration increasing line for N=0.01,0.1 and 1.

0.0007

0.0006

0.0005

f4 0.0004 % & 'E. 0.0003

0.0002

0.0001

0 1 2001 4001 6001 8001 10001 12001 14001 16001 18001

time

Figure 2.4 Phytoplankton concentration dropping rate for N=0.001.

0.25 I

0.2

0.15

0.1

0.05

0 1 2001 4001 6001 8001 10001 12001 14001 16001 18001

time

Figure 2.5 0.1 and 1.

Phytoplankton concentration increasing rate for N=0.01,

' Study on the Character of Equilibrium Point and . . . 217

3 Phytoplankton population model with non-constant nutrient concentration

3.1 Mathematical model In this section, the phytoplankton model in section 2 was used, but N value varies as described in equation (3.1):

N a u + NO - N ) - --___ - dN

dt e + N b + c u d u N a d t u - ( s + k ) u - -___ -

e + N b + cu

where NO de'notes N concentration below mixed layer. Other parameters and conditions are the same as in section 2.

The equilibrium point and its stability are consistent with the results discussed in section 2. The initial N and u value are set to be 0.01. NO is set to be 0.01, 0.1, 1 and 10.

3.2 Results and discussion Both the phytoplankton concentration and the equilibrium point are changing with time for NO = O.Ol(see Figure 3.1 and Figure 3.2). It is indicated that when the equilibrium point becomes smaller the phytoplankton concentration begins to drop. At last they meet at 0.00389. For other three NO values, the change logarithm lines of nutrient, equilibrium point and phytoplankton concentration are showed in Figure 3.3, Figure 3.4 and Figure 3.5, respectively. The change lines of the increasing rate are showed in Figure 3.6. Apparently, the changing trend of equilibrium point is consistent with that of nutrient. The phytoplankton concentration and the equilibrium point meet at the end. In Figure 3.5, before the point A1, A2 and A3, the changing rates are very fast and after them the rates decrease. The characters can be observed easily in Figure 3.7, which includes the nutrient concentration, the equilibrium point and the phytoplankton concentration logarithm line for No=l. A is the critical point for u. Before it, the line of phytoplankton u is steeper than the line after it.

For detailed study, we set the initial N=0.1 and use N0=0.01 to simulate. The line of phytoplankton concentration-u is shown in Figure 3.8. The nutrient line is shown in Figure 3.9.


0.014 I

0.002

0 1 1001 2001 3001 4001 5001 6001 7001 8001 9001 1000111001 12001

time

Figure 3.1 Time variation of phytoplankton concentration €or No=O.Ol.

1

0.8

0.6 .- B 3 0.4 z! ." 0.2

2 0

-0.2

-0.4 time

Figure 3.2 Time variation of equilibrium point for No=O.Ol.

10

1

0.01

0.001

No= 10

No=O.l

1 2001 4001 6001 8001 10001 12001 14001 16001 18001

time

Figure 3.3 . Time variation of nutrient concentration for N0=0.1, 1, 10.

Study on the Character of Equilibrium Point and - .

0.01

10 No= 10

No=O.l

" " " " ' ~ " I , ' " ' /

219

Figure 3.4 Time variation of nutrient concentration for N0=0.1, 1, 10.

10 No= 10

No= 1

No=O.l

' ( ~ ~ I ' ~ I I S l I I 8 8 ,

I 2001 4001 6001 8001 10001 12001 14001 16001 18001 time

Figure 3.5 1, 10.

Time variation of phytoplankton concentration for No=O.l,

0.3

0.25

0.2 3 ' 0.15 .s" .- 8 g 0.1

0.05

0 1 2001 4001 6001 8001 10001 12001 14001 16001 18001 1

-0.05 time

Figure 3.6 Time variation of nutrient concentration for N0=0.1, 1, 10.


Nutrient N

0.0011 ~ " " " " " " " " " " " " " " " " " " " " " " " " " ' " " ' " ' " " ' ~ 1 501 1001 1501 2001 2501 3001 3501 4001 4501 5001 5501 6001 6501 7001 7501

time

Figure 3.7 Time variation of u, N , Equilibrium point for No=l. 0.09 I

0.08

0.07

0.06

0.05

8' 0.04

E4 0.03

0.02

0.01

!3 - B

O L " " " " " " " " " ' 1 2001 4001 6001 8001 10001 12001 14001 16001 18001

time

Figure 3.8 N=0.1 and No=O.Ol.

Time variation of phytoplankton concentration for initial

0.12

0.1

0.8

0.04

0.02

0 1 2001 4001 6001 8001 10001 12001 14001 16001 18001

time

Figure 3.9 Time variation of nutrient concentration for initial N=0.1 and No=O.Ol.


4 Conclusions The character of equilibrium point has been investigated using a simple Nutrient-Phytoplankton model. The equilibrium point is used to represent the phytoplankton concentration when its changing rate is zero. For phytoplankton, nutrient is the only factor concerned. The nutrient concentration ( N ) is constant in section 2 and varies with time in section 3. From the results, some conclusions can be made:

1. The changing trend of equilibrium point is determined by the change of nutrient;

2. The phytoplankton concentration approaches the equilibrium point at the end;

3. The larger the difference between the equilibrium point and the phytoplankton concentration is, the faster the changing rate of phytoplankton u is.

In terms of red tide, if the algae’s steady positive equilibrium point is made very high by some environmental sources (e.g. nutrient), the difference between the concentration of algae in water and the equilibrium point could be very large, and then the increasing rate of the algae could be very large. This is the first phase of red tide. When the sources are poorer, the steady positive equilibrium point of algae is lower, and then the increasing rate of it is lower. This is the secondphase of red tide. The equilibrium becomes smaller than the practical concentration of algae because the sources are exhausted. So the algae concentration drops. This is the third phase of red tide. At last, the algae’s concentration meets the equilibrium point, and red tide disappears. The process can be seen in Figure 3.9.

In practical environment, factors affecting red tide are very complicated. When hydraulic conditions are concerned, the variability of the equilibrium point could become very complicated. This study only deals with the key environmental factors such as nutrient. From the results, the phytoplankton concentration and the changing trend are determined by the equilibrium point. Further work needs to be done on the impacts of other environmental factors on the equilibrium point.


References [l] Edwards A M, Brindley J. Oscillatory behaviour in a three-

component plankton population model. Dyn Stab Syst., 11: 347-70, 1996.

[2] Edwards A M, Brindley J. Zooplankton mortality and the dynamical behaviour of plankton population models. Bull Math Biol., 61:

[3] Steele J H, Henderson E W. A simple plankton model. Am Nat., 303-39, 1999.

117: 67691,1981.

223

A Numerical Simulation of Thermal Discharge into Tidal Estuary with FVM

Jie Zhou, Deguan Wang Environmental Engineering and Science School,

Hohai University, China

Haiping Jiang Pearl River Water Resources Commission,

Ministry of Water Resources, China

Xijun Lai Nanjing Institute of Geography and Limnology,

Chinese Academy of Sciences, China

Abstract

The two dimensional shallow water equations with heat convection and diffusion equation is proposed to simulate thermal pollution of Guangdong Nansha Thermal Power which is planed to be built. Finite volume method (FVM) and flux difference splitting(FDS) scheme is applied, and the result shows that this method can keep high precision when it is applied to simulate heat transferring in estuary.

Thermal pollution, which is environmental pollutions because of thermal waste from modern industry and life, take a great deal effect on atmosphere and water. For instance, thermal waste water from firepower or steel factory do great harm to water environment. Since water temperature becomes higher, there is less dissolved oxygen in water. Water body-will be in an oxygen-lack status, and fishes and other biology in water will be hard to live on. If temperature of water in estuary makes an ascent, it will hold back the fishes from going upstream to spawn. Then distribution of biotopes will change, and the balance of old system will break.

Spatial distribution of thermal pollution beyond tolerance criterion in water, and its effects on biological environment are the key point

224 Jie Zhou, Deguan Wang, Haiping Jiang, Xijun Lai

of national environmental management and conservation. At present, there is no easy way to forecast the pollution distribution. Its general method is numerical simulation and laboratory experiment. As a numerical method, finite volume method (FVM) is an efficient, high precision, well-fitted boundary method. The method is applied to simulate distribution of the thermal pollution from to-be-built Nansha firepower of Gangdong province.

1 Two dimensional shallow water model to simulate thermal discharge

1.1 Main idea When hot wasted water from the power discharge into estuary, it im- mediately dispersion away from the port with the force of tide flow. Generally, heat transport depends mainly on convection effect. So hot waste transport law is governed by tidal force. Precision of tidal hydre dynamics scheme becomes the key point in the model. Two dimensional shallow water equation will be applied to simulate the tidal flow with FVM .

Since in the study that focus on far or middle thermal field, tidal force take more effects on the velocity field much than thermal waste does, so thermal waste’s effect will be ignored. Thermal waste transport includes several procedures, such as convection and dispersion, exchange with atmosphere, radiation of short waves or long waves, etc. Convection and dispersion is main procedure, exchange must be taken into account because of large area of estuary. But radiation can be ignored.

1.2 Governing equations 1.2.1 Hydrodynamics equations

Conserved form of two dimensional non-static shallow water equations is used to describe the flow. The vector equations can be written as:

where q = [h, hu, hvIT , h, hu, hv are conserved physical variables respectively, f (q) = [hu, hu2 + gh2/2, huvIT are fluxes in x-direction; g ( q ) = [hu, huv + hv2 + gh2/2IT are fluxes in y-direction; h is depth; u and v is vertical depth-averaged velocity in x and y direction; g is gravitational acceleration. b(q) is source/sink term, written as:

b(Q) = 10, gh(S0z - Sfz), gh(S0, - S,,)lT (1.2)

A Numerical Simulation of Thermal Discharge into . . . 225

where S,, and S f , is bed slop and friction water head slop in x direction respectively, while So, and Sf, is in y direction.

1.2.2 Thermal waste transport governing equation

d ( h A T ) d ( h u A T ) d ( h v A T ) d d A T = --(oixh=) dX

8 Y + d t + ax

d A T KAT + Si +-(Di,h-) - - d 8Y aY PCP

where AT is thermal waste temperature increment reference to environmental water body; Di,, Di, is dispersion coefficient in x and y direction respectively; C, is specific heat of water; p is density of water; Si is source/sink term,

K = 15.7 + [0.515 - 0.00425(TS - T d )

+0.00005l(T~ - Td)2](70 + 0.7W:)

where K is heat diffusion coefficient at water surface, whose evaluation is recommended by “technical rules on surface water environmental evaluation (HJ/T 2.3-93)”, T, is temperature at water surface; T d is dew point temperature; W, is wind velocity at 10m above water.

1.3 Numerical methods Since the Riemann invariant remains constant along their corresponding characteristic curves and flux will not change when rotate it, two- dimensional problem can be solved in a one-dimensional system. Flux difference scheme-an approximate solver proposed by Roe[’] is applied to evaluate the fluxes, such as flow quantity, momentum, temperature increment across the edge of elements. Zhao et a1.[’] used this method to calculate the normal flux of sediment transport and experiential formula of sediment transport to evaluate deposition and erosion successfully. This scheme is tried here to solve the heat transport problem in shallow water system. Discretization can be found in reference [2].

1.4 Boundary conditions 1.4.1 Scope for calculation

Based on measured tides and the locations of port to fetch fresh cold water and heat outfall port, upstream boundary is decided to be at estuary of Dongjiang river, downstream boundary locate at Dahu cross section


t N

Figure 1.1 Scope and mesh.

of Pear river estuary. Local refined irregular quadrilateral element mesh is applied, and the control volume of FVM is the element itself. Coarse mesh elements locate at far field where there is less effects of thermal drainage, whose spatial step is about 600m. While refined elements are at near area of thermal waste outfall port and fetch water port, whose spatial step is between 25m and 80m. There are 1359 nodes and 1133 elements in mesh, see Figure 1.1.

1.4.2 Boundary conditions

There are five boundary conditions for hydrodynamics system, which are measured tidal head and discharges at Dasheng, Zhangpeng, Sishengwei, Sanshakou, Dahu hydrology station respectively. When calculating heat field, radiation open boundary conditions is applied at outflow boundaries. Since upstream boundary is far away enough so that the thermal

A Numerical Simulation of Thermal Discharge into . . 227

discharge can not effect its temperature, so temperature increment at upstream boundary is taken to be zero.

2 Validat ion of hydrodynamics model

Measured data at hydrology stations of low water period from year 2000 to year 2001 validates the hydrodynamics model. To validate the model, downstream boundary is extended to Shajiao temporally. Tides and discharge validation is taken at Dahu cross section. Results of validation are shown as Figure 2.1 and Figure 2.2. Calculated water head and discharges agree with the measured ones very well. So the parameters of the model are well adjusted, and the calculation method is trusty. And FVM method is fit for calculation at Pear river estuary.

-calculated 0 measured

-1.5 timeihour

Figure 2.1 Water head validation at Dahu station.

-calculated o measured

timeihour

Figure 2.2 Discharge validation at Dahu station.


3 Case study

stage tide type waste discharge (rn3/s)

Different tide types and different power capacity cases are calculated. The power station will be built in two stages. At first stage, equipped with 2x200MW power engines, thermal waste water discharge will be about 15.75 m3/s. At second stage, equipped with 4x200MW power engines, discharge will be about 31.51m3/s. Cases listed as Table 3.1. Total simulation time is 14 tidal periods. Then distribution of temperature increment at different cases during the 14 tidal periods is simulated. Most extent of contours in the same case is obtained to evaluate the effect scope of waste discharge. And averaged contours during all periods are to evaluate the effect degree, shown as Figure 3.3 and Figure 3.4. More attention is taken to the temperature increment near the fetch cold fresh water port to cool the power engines. Tide is the main factor that takes effect on the temperature. Higher tide, higher temperature. The highest temperature increment at fetch port is about 3"C, shown as Figures 3.2 and 3.3.

waste temperature (oC)

Table 3.1 Cases to be calculated

low water period

floodwater period stage

15.75 10.0

15.75 10.0

second stage

low water period 31.5 10.0

floodwater period 31.5 10.0

24 48 72 96 120 144 168 I - timehour

Figure 3.1 period at first stage.

Temperature increment near fetch port during low water

A Numerical Simulation of Thermal Discharge into . . - 229

24 48 72 96 120 144 168

timehour

Figure 3.2 Temperature increment near fetch port during floodwater period at first stage.

Figure 3.3 Averaged contours of temperature increment during 14 floodwater tide periods.

Figure 3.4 Averaged contours of temperature increment during 14 low water tide periods.


4 Conclusions Rather than finite element method and finite difference method, FVM can ensure matter transport keeping conserved and high precision. From the calculated results, it is noticed that effect scope of the waste discharge is governed by which is stronger between tidal flow and runoff flow. At low water period, tidal flow is stronger than runoff is, and the waste goes upstream more far away than it does at floodwater period.

References [l] ROE P L. Approximate Riemann solvers, parameter vectors and

difference schemes [J]. Computational Phys. 43, 1981. [a] Zhao D H, Shen F X, Yan Z J, Lu G H, A 2D sediment transport

model based on the FVM with FDS for tidal rivers [J]. Hydrody- namics, Ser. A. 19: 98-103, 2004. (in Chinese)

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