Environmental Health PerspectivesVol. 83, pp. 97-115, 1989

Applications of Numerical Methods toSimulate the Movement of Contaminants inGroundwaterby Ne-Zheng Sun*

This paper reviews mathematical models and numerical methods that have been extensively used tosimulate the movement of contaminants through the subsurface. The major emphasis is placed on thenumerical methods of advection-dominated transport problems and inverse problems. Several mathemat-ical models that are commonly used in field problems are listed. A variety of numerical solutions for three-dimensional models are introduced, including the multiple cell balance method that can be considered avariation of the finite element method. The multiple cell balance method is easy to understand andconvenient for solving field problems. When the advection transport dominates the dispersion transport,two kinds of numerical difficulties, overshoot and numerical dispersion, are always involved in solvingstandard, finite difference methods and finite element methods. To overcome these numerical difficulties,various numerical techniques are developed, such as upstream weighting methods and moving point meth-ods. A complete review of these methods is given and we also mention the problems of parameter iden-tification, reliability analysis, and optimal-experiment design that are absolutely necessary for construct-ing a practical model.

Introduction

In recent years, mathematical models and numericalmethods have been used extensively to simulate themovement of contaminants to groundwater. Manybooks and review papers have contributed to this topic(1-8). The distributed parameter model describing sol-ute transport in porous media generally includes an ad-vection-dispersion equation, a groundwater flow equa-tion, and state equations (1). Analytical solutions canbe derived only for a few classical models that are notsuitable for complex situations normally encountered inthe field. Consequently, the development of numericalsolutions is required.During the last two decades, a variety of numerical

approaches were presented, such as the finite differencemethods (FDM), the finite element methods (FEM), andother alternative methods (2-14). However, at leastthree important problems have not been solved per-fectly:

* The movement of contaminants in groundwater isan inherently three-dimensional (3-D) process. Unfor-tunately, most simulations of field problems are now in2-D because of the high expense of 3-D models. There-fore, 3-D numerical models that can decrease the de-

*Environmental Science Center, Shandong University, The Peo-ple's Republic of China.

Present address: Civil Engineering Department, UCLA, Los An-geles, CA 90024.

mands both on human and machine resources need tobe developed.

* When using the regular numerical methods men-tioned above to solve advection-dominated transportproblems, two types of numerical errors always occur:the oscillation around the computed profile (overshoot)and the smearing of the concentration front (numericaldispersion). To decrease these numerical errors, manyspecial techniques have been presented. Each has itsown advantages and disadvantages in comparison withthe others. Therefore, further work in this area is re-quired.

* Another difficulty in using mathematical models tosimulate mass transport in porous media is the deter-mination of model parameters. Because it is impossibleto measure directly the pore velocity and dispersivitiesin the whole flow region, one has to use an indirectmethod to determine these input parameters accordingto observation data of heads and concentrations, i.e., tosolve the inverse problem of the simulation of ground-water quality. The inverse problem ofgroundwater flowsimulation has been extensively studied (15-24). A re-view paper on this topic was presented recently by Yeh(25). In comparison, the body of published research forparameter estimation for mass transport problems isstill small (26-34). In the solution of inverse problems,the primary obstacle is their ill-posed nature, i.e., theexistence, uniqueness, and stability of the solution maynot be satisfied. However, if the observation data arecollected through a reasonable experiment design, theill-posedness may be controlled under certain restrictedconditions.

N-Z. SUN

In this paper, we focus our attention on the numericalmethods with respect to the previously mentioned threeproblems. The distributed parameter models of masstransport in porous media are given in the second sec-tion. This paper does not deal with stochastic models(35-38).The third section is devoted to 3-D numerical models

developed in recent years and mentions the mixed finitedifference-finite element method (FD-FE), the multiplecell balance method (MCBM), and the alternating di-rection Galerkin method (ADG). The MCBM (13,39-41)combines the idea of local mass balance coming fromthe integral FDM (42), and the idea of basic functionscoming from FEM (43,44). It is very suitable for thesolution of field problems.

In the fourth section, various numerical methods arereviewed. These have been developed to deal with theadvection-dominated transport problems. We mentionthe upstream weighting FEM, the upstream weightingMCBM, and the method of characteristics (MOC) andits modified forms. A new method, the advection-controlMCBM, recently presented by Sun and Liang (45), willbe also introduced. It displays the ability of eliminatingthe overshoot and decreasing the numerical dispersionefficiently.

In the last section of the paper, the concepts andnumerical methods of inverse problems are introduced.Here we discuss extended identifiability and its appli-cations to experimental design and reliability estima-tion.

Mathematical ModelsIt is well known that mass transport in porous media

is governed by the following advection-dispersion equa-tion (1,2):

The distribution of average velocities can be com-puted according to Darcy's law. For instance, in thesaturated zone we have

Vji = - !i40 axj (3)

where Ku = components of hydraulic conductivity ten-sor (L/T); h = head (L); 0 = porosity (dimensionless).The distribution of head in Eq. (3) can be obtained bysolving the groundwater flow problem. The advection-dispersion equation [Eq. (1)] must be completed byboundary and initial conditions depending on the con-sidered problem. The initial condition is the concentra-tion distribution at t = 0:

C(x,O) = Co(x) (4)where CO (x) is a known function, point x belongs tothe flow region. The boundary conditions are generallydivided into three types:

a) Prescribed concentration boundary conditionsC(x,t) = g1 (x,t), x E rI (5)

b) Prescribed flux boundary conditions

Dij afclnj = 92(x,t),axi x e r2 (6)

c) Prescribed conditions and its normal derivative onthe boundary

ViC - Dij a - ni = g3(x,t), x E r3 (7)

In Eqs. (5)-(7), gl, g2, and g3 are known functions andF = F1 + F2 + F3 is the boundary of the flow region(Q).

a(Oc) a Dj ac - a (ViC) Mat ax1 axj -a(Ox (1)

i,j = 1,2,3where C = concentration of contaminants (M/L); Du =components of hydrodynamic dispersion coefficient ten-sor (L2/T); Vi = components of average velocity vectorof fluid in porous medium (L/T); M = source or sink (M/L2T); 0 = volumetric fraction of the fluid phase, in sat-urated zone; 0 = porosity; in unsaturated zone, 0 =water content; xi = space variables (L); t = time (T).The repeated subscripts indicate summation.For an isotropic porous medium, Dii may be ex-

pressed as:

Dij = aTSijV + (aL - aT) v + 6ijD (2)

where aL, aT = longitudinal and transverse dispersiv-ities of the isotropic medium, respectively (L); V =magnitude of the velocity (L/T); D* = molecular dif-fusivity of porous media (L/T); and bu = Kroneckerdelta, b& = O, when i + j; u = 1, when i = j.

Classical ModelsAnalytical solutions can be derived only for a few

classical models that are often used to simulate con-trolled experiments in the laboratory or in the field toobtain dispersion parameters. Another application ofanalytical solutions is testing and contrasting numericalsolutions. For this reason, two solutions are quotedhere.

Consider the movement of a tracer in a semi-infinitecolumn. The mathematical statement of the problem is

ac 2C acat D ax2 ax

(8)C(x,O) = 0C(O,t) = COC(00,t) = 0

The analytical solution of the classical model was givenby Fried and Combernous (46):

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APPLICATIONS OF NUMERICAL METHODS

C(X,t) =2 {erfc[ t] + exp [ ]D erfc[ } (9)

Another example is the dispersion in radial flow whena fully penetrating recharge well is located at the originwith a constant rate Q (the molecular diffusion is ne-glected). The mathematical statement of the problem is

ac a2c aca t ar2 ar

C(r,O) = 0,C(rw,t) = CO,C(oo,t) = 0,

r > rwt > Ot > O

where r, = radius of the injection well; the radial ve-locity V = Q/2-rBOr, where B = thickness of the aqui-fer; and CO = concentration of the recharge water. Theanalytical solution of problem (10) is given by Tang andBabu (47). A new formula that is simpler than the for-mer was presented recently by Hsieh (48).More analytical solutions can be found in Li (49), Bear

(2) and Sun (8). Some new results are given by VanGenuchten (50), Mironenko and Pachepsky (51), Lind-strom and Boersma (52), and Mornch (53) for 1-D dis-persion; by Al-Niami and Ruston (54) and Batu (55) forthe dispersion of stratified porous media; by Goltz andRoberts (56) for a 3-D model; and by Tang et al. (57),Sudicky and Frind (58), Chen (59,60), Lowell (61), andDillon (62) for transport in fractured porous media.

Field ModelsFor a field model, the groundwater flow equation

(GFE) and the advection-dispersion equation (ADE)must be solved individually or simultaneously. Severalcombinations of GFE and ADE are given below. Theseare often seen in practical problems.The model of3-D dispersion in a confined flow includes

a GFE

at ax haxand an ADE

a(oC) a [Dij ac 1 a (ViC) M3at ax, D, ax~J axi

(11)

(12)

where S. = specific storage coefficient (1/L); W3 (1/T)and M3 (M/L3T) = sink or source; 0 = porosity; i, j =1,2,3.The model of 2-D dispersion in a horizontal 2-D un-

confined aquifer includes a GFE

(13)

a(mc) = a [mDij ax - a (mvic) M2 (14)

where Sy = specific yield (dimensionless); m = satu-rated thickness of the aquifer (L); W2 (L/T) and M2 (M/L2 T) = sink or source; i,j = 1,2.The model of 2-D dispersion in a vertical saturated-

unsaturated zone inclues a GFE

( +ISs) a [K1J+ "[Ky+ 1 - W3 (15)

(10) and an ADE

at -I=a 8Dij a- ] a (OViC) - M3at aXL aXJJ aXi

where,

e = water content; v = pressure head (L);av

-=a ; in the saturated zone, I = 1;

in the unsaturated zone f3 = 0; i,j = 1,3

(16)

The more complex field models are also presented(63-73). Recently, the study of modeling the masstransport in fractured aquifers has become a hot topic(74-78).The GEF and ADE of a field model are coupled

through Darcy's law. In tracer cases (the density p andviscosity ,u of the fluid are constant); the two equationscan be solved individually. Otherwise (p and ,u dependupon the solute concentration) they must be solved si-multaneously with the state equations (1,79).The effects of ion exchange, chemical reactions, radio-

active decay, and adsorption, as well as pumping andrecharging of wastewater, are all included in the sourceor sink term of the ADE (2). For equilibrium transport,the linear adsorption isotherm F = kC is often used,where F is the solute concentration on the solid phaseand k is the slope of the isotherm. The ADE [Eq. (1)]then can be rewritten as

a(8c) a FD ac1 a Fovj c1at -axi [ Pd ax1J axi [Rd

where, (17)

Rd = 1 + k is called the retardation factor;

Os is the volumetric fraction of the solid phase.For nonequilibrium transport, the ADE should be

replaced by the coupled system.

a(OtF) + at =a [DijaI al(aViC)a t aat axi axjJ x

(18)aF

= k(C -aF)

SYa a = nKim W2

and an ADE

99

N-Z. SUN

The two above equations represent the mass transportand exchange in fluid phase and solid phase, respec-tively, where k and a are constants. [For review ofadsorption models, see (7,80-86).]

Numerical MethodsThe previously mentioned field models are generally

intractable analytically because of irregular geometryand a nonhomogeneous structure of the flow region.Numerical methods are always required. In the past,the practical application of numerical solutions was al-most limited in 1-D and 2-D cases to avoid high expensesof 3-D models on both human and machine resources.However, dispersion of contaminants in groundwater isan inherently 3-D process. It is now recognized thateven a low-order transverse dispersion acting over along distance can substantially affect the configurationof a contaminant plume (14,87,88). Burnett and Frind(89) further examined the effects of dimension througha field-scale system that is simulated with a 3-D model,a 2-D vertical section model, and a 2-D vertically in-tegrated, horizontal plane model. In fact, there aremany instances of field problems where 2-D approxi-mations ofcontaminant plumes are not adequate. There-fore, numerous researchers have turned their attentionto 3-D numerical solutions in recent years (40,41,63,90-92). The major problem in investigating 3-D numericalmodels is knowing how to decrease the required inputdata, storage space, and computation time.

second derivatives in Eq. (20) by Green's theorem, onecan translate Eq. (20) into a system of ordinary differ-ential equations.

[A] C } ( { dt = (F) (21)

where the components of the matrices [A] and [B] are

A~J = f{ajvivi + aXi8Va)('i)

Aij~ ~ ij=

I

0 adOVcxi + ~~OVO0j

Bij = Oijd(fl)

(22)

(23)

when the boundary condition [Eq. (6)] is given on a partof boundary r2 then

Fi= JM30 dQ + 1g2.i dr(Q) (r2) (24)

The time derivative in Eq. (21) is usually replaced bysimple finite differencing, the equation then reduces tothe following finite element equations for each time stepAt

([A] +B

) (C}t+At = L{Ch + (Fh+AtAt At (25)

Galerkin Finite Element Method (FEM)In standard FEM, the unknown function C(x1,x2,x3,t)

of Eq. (1) is represented approximately by a trial func-tion

A

C(x1 ,x2,x3, t) = Ci(t)*,(x1 ,X2,X3)where, (19)

v i = 1,2, ... N is a set of basis functions

the repeated subscripts indicate summation. N is thenumber of all nodes where the concentrations are un-known. The equations of weighted residuals of ADE[Eq. (1)] are

Matrices [A] and [B] depend upon the division of theflow region and the selection of basis functions. Gupta(93) used the 3-D isoparameter elements and the 3-DGalerkin basis functions to obtain matrices A and Bwhere numerical integrations are needed.

In Sun et al. (40), the flow region is first divided intoa number of horizontal layers; these layers are thensubdivided into a number of triangular prisms so thatthe corresponding nodes in each layer are located alonga single vertical line. The six vertices of each elementare taken as nodes (Fig. 1). They are

P1 (x1,y1,zo) P2(xj,yj,zo)P4(xi,Yi,zl) P 5(xj;.yj,z 1)

P3(xk,Yk,zo) (26)P6(xk,Yk,z,)

In each element, the basis functions of the element's sixnodes are

A Al-

J atjjC - aX [ e a(Q)

+I-t (OVaC) + M3 } widV=Oax

(i = 1, 2, ...,N); (a,p = 1,2,3)

Oi(X,y,Z) = Z-j(x,y)W2(X,y,Z) = Z'+j(x,y)*3(x,y,Z) = Z-k(X,Y)

(20) where

z- z -z Z

AZ

X4(X,y,Z) = Z+Oi(X,y)*5(x,y,z) = z+Oj(x,y) (27)

*6(X4,yZ) = Z+Wk(X,y)

z . z0z+ = ZAAZ AZ = Z1 -Zo (28)

where {wi, i = 1, 2,... N}is a set ofweighting functions.In the Galerkin FEM, let {¢j} = {wi}. In eliminating the In this case, the input geometry information is almost

100

APPLICATIONS OF NUMERICAL METHODS

P6

P4

P,

P5

0,m)

(a)FIGURE 1. Three-dimensional triangular prism element.

equal to that of the 2-D models. The matrices [A] and[B] can be computed directly without any numericalintegration.

Mixed Finite Difference-Finite ElementMethod (FD-FE)To decrease the high expenses of 3-D models both on

human and machine resources, Sun (17) presented amixed FD-FE method for 3-D GFE that can be easilyextended to solving the 3-D ADE. A similar methodwas given by Babu and Pinder (90,94). The flow regionis divided into triangular prism elements as mentionedabove. All 3-D nodes can be numbered by a 2-D array(i,m) where m is the number of a layer that the nodelocates (m = 1,2, .. ,M) and i is the number of the nodein the layer (i = 1,2, . .. J,). Then a double division isintroduced to form an exclusive subdomain for each nodethat is a multiangular prism surrounding the node asshown in Figure 2. The local mass balance over theexclusive subdomain (di,m) of node (i,m) can be ex-pressed as the integral of ADE [Eq. (1)]. ApplyingGreen's theorem, it can be rewritten as

I {9_C + M3} dQ= f{Dap aC - VC nOdS

(di,m) (Si,m) (29)

where (Si,m) is the boundary of (di,m) and n is the normalvector of (Si,m). For the left-hand side of Eq. (29), oneuses the approximation.

-C~~dC(i.mj i-z{ {at + M3} dflJOim dt + Mi] i

(di,m) (30)

AZm

(b)

FIGURE 2. Exclusive subdomain of a node (i,m): (a) cross section; (b)profile.

where Pi is the area of the bottom or top of (di,m); AZmis its thickness. Let (Si,m) = (Si,m)i + (Si,m)2 + (Si,m)3,where (Si,m)i is the side of (di,m), (Si,m)2 and (Si,m)3 arethe top and bottom of (di,m), respectively. The integralover (Si,m) on the right-hand side of Eq. (29) then canbe computed in three parts. We have

[ ] dS = [ [ ] dLJ]*Zm = [A(i,i)C(i,m)(Si,m)l (Li) (31)

+ , A( ,j)C (j,m) ] *AZmj*i

where E represents the sum of all nodes that are neigh-.j- i

(k,m)

(

(i,m)

101

N-Z. SUN

boring nodes of node i on the same layer m. Aj,j and Acan be computed directly and will be given later in Eq.(42). Using finite difference approximation, we have

11(Sim)2

and

1f,(Si,m)3

where

dS = [ODZ ac- evzc I( ' Pi (32)

[LZZaz- (i,m+ -)

c~~~~

dS = OeDZc

a-vzc 1- Pi(33)

(i,m-i)

1lz (i,m+1)

k 1(i,m-1

C(i.m+1) - C(im)=2rAZm + AZm+1

2 C(i m) - C(i.m-1)AZm + AZm-1

12C l(i,m+-) = (C(i,m+1) + (m)2 (34)

C l(i,Mi-) = 2 (C(i,m) + C(i,m- ))2 '2

Let

(Cim = {C(1,m), C(2,m) .....C(l,m)l| dCl JdC(1.m) dC(2.mj dC(l.m)tdtfm l dt ' dt

.

' dt |

Substituting Eq. (30)-(33) into Eq. (29), we obtain fol-lowing equations for all layers:

[A1lm{Clm + [A2]{Clm+1 + CA3]m{Cm-1dC (35)

+CBJm Xdtf m = (Fim (m = 1,2,..., M)

where [AiIm, [A2]m, [A3]m and [B]m are all

diagonal matrices

Eq. (35) can be solved by a layer-by-layer iteration pro-cess.

In Babu and Pinder (90), assume that

C(x,y,z,t) = Ci(z,t) *i(x,y)where Vi(x,y) are 2-D basis functions

(36)

Using the Galerkin process in the x-y space and thefinite difference approximation in the z space, they ob-tained Eq. (35) with nondiagonal matrices [Al]m, [A2]m,[A3]m and [B]m. In order to decrease the computationalexpense, these matrices are lump-diagonalized by sum-mation. An alternating direction algorithm is then usedto solve the equations.

Multiple Cell Balance Method (MCBM)The 2-D MCBM (13) is a variant of the linear FEM.

It combines finite-element approximation with the con-

cept of the local mass balance used in the integral finitedifference method (IFDM), presented by Narasimhan(12), Narasimhan and Witherspoon (42), and Rasmusonet al. (66). In IFDM, the flow region is divided intopolygon elements. The local mass balance of each ele-ment is represented by an integral form over the ele-ment and the derivatives arising in these integrals arereplaced by simple finite differencing. Consequently,the balance equation of each element becomes an al-gebraic equation that includes the unknown concentra-tions of the element and its neighboring elements.IFDM is easy to understand, but it is difficult to dealwith the tensor nature of dispersion coefficients andarbitrary orientation of flow velocities.

In MCBM, the flow region is divided into triangleelements, a twice division is then used to form the ex-clusive subdomain for each node as shown in Figure 3.The local mass balance of a node i can be representedby the integral of Eq. (14) over its exclusive subdomain(di):

D[co4-- VCc]ncdL = [ + M2] dD

(Li) (di)

Assume that in each element (Aijk) the unknown func-tion C can be represented as

C(x,y,z,t) = Ci(t) *g(x,y) + Cj(t) *j(x,y)

+ Ck(t) Xk(x,Y)(38)

where Ci, Cj and Ck are nodal values of unknown func-tion; Xi, (4j, and Xk are 2-D linear basis functions. Sub-stituting Eq. (38) into the mass balance Eq. (37) andcompleting integral calculations included in the equa-tion, it can be translated into an algebraic equation

e e eX(AiiCi + AijCj + AiA")(ei)

edcj e dC e dCk+ (Bii dt + B; dtS + Bik dj)=Fi(ei)

(39)

where E represents the sum of all elements that haveei

common node i. The general formulae of AF, Be (s =

i, j, k) are given by Sun and Liang (13). Ifm = constantand div V = 0, they reduce to

Ais = 4A [Dxxbibs + Dxy(bics + qbs) + Dyycics

+ 6 (Vxbs VVycs)(40)

102

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APPLICATIONS OF NUMERICAL METHODS

e A 22Bis=3 6

if i = s 3 36 (41)Fi = MziPi

where b8, cs are coefficients of basis functions; A is thearea of the element.

FIGURE 3. Elements of one-dimensional solute transport problem.Aij = O Aij,. .

(42)

Bii= L Bii Bij= XBel eij

where E represents the sum of all elements that haveei

common nodes i and j. Listing Eq. (39) for all nodeswhere concentrations are unknown and consideringgiven boundary conditions, the obtained equations canbe written as:

[A] (C) + [B] {d-} = (F) (43)

It is easy to verify that the matrix [A] in Eq. (43) isthe same as that of FEM, but in FEM

e A 1Bis=3- if i = s

e A 1is3 4, (44)

The MCBM has been extended to 3-D models (40,41).The division of the 3-D flow region is the same as thatmentioned before. Assume that in each triangular prismelement (Fig. 1) we have

6C (x,y,z,t) = , Cj(t)4I(x,y,z) (45)

1=1

are basis functions given in Eq. (27). Substituting Eq.(45) into the 3-D mass balance equation Eq. (29) allintegrals in the equation can be calculated explicitly.Consequently, we obtain a set of equations in as the 2-D case. A layer-by-layer iteration process can be usedto solve the algebraic system.The 1-D classical model [Eq.(8)], the radial dispersion

model [Eq. (10)], the advection-dispersion in a singlefracture with diffusion in a rock matrix (57) and theadvection-dispersion in a two-layered porous medium(54) were used to verify the proposed method (13,39,41).The results are satisfactory in comparison with analyt-ical solutions. For the classical model [Eq. (8)], the mesh

V -AxPeclet number is defined as Pe = . Let CO = 10,V = 1, Ax = 0.25, D = 10, then Pe = 0.25. A total of93 nodes and 30 triangular prism elements are used tosimulate this problem. Figure 3 adequately representsthe semi-infinite region in the given time interval 0 S

Table 1. Comparison between analytical and numericalsolutions of the 1-D classical model.

Concentrations5 Days 40 Days

Distance Analytical MCBM Analytical MCBM0.0 10.0000 10.0000 10.0000 10.00002.5 8.8970 8.8950 9.9290 9.92885.0 7.6157 7.6154 9.8405 9.83997.5 6.2495 6.2556 9.7326 9.7316

10.0 4.9014 4.9180 9.6037 9.602312.5 3.6645 3.6935 9.4524 9.450615.0 2.6061 2.6461 9.2776 9.275317.5 1.7600 1.8067 9.0785 9.075720.0 1.1269 1.1750 8.8548 8.851522.5 0.6833 0.7277 8.6063 8.602725.0 0.3920 0.4292 8.3336 8.329827.5 0.2125 0.2411 8.0378 8.033830.0 0.1088 0.1290 7.7202 7.716232.5 0.0526 0.0658 7.3827 7.379035.0 0.0240 0.0320 7.0278 7.024637.5 0.0103 0.0148 6.6582 6.655740.0 0.0042 0.0066 6.2770 6.275342.5 0.0016 0.0028 5.8874 5.886845.0 0.0006 0.0011 5.4930 5.493747.5 0.0002 0.0004 5.0972 5.099450.0 0.0001 0.0002 4.7038 4.7075

t - 40. Numerical results of MCBM and analytical so-lutions are listed in Table 1.The MCBM has been used to simulate field problems

in China. For instance, a simulation model of ground-water quality at Xian city, Shaanxi province, was cre-ated in 1985. The MCBM almost keeps the advantagesof the mixed FD-FE method on decreasing input dataand computational effort. In addition, it is easy to mod-ify in order to deal with field problems that have sharpfronts. A man-machine talking software package basedon 2-D and 3-D MCBM was presented by Wang et al.(95).Huyakorn et al. (96) gave a 3-D FEM for simulating

solute transport in multilayer systems. The division offlow regions is the same as that of the FD-FE method.The trial function [Eq. (36)] is used in x-y space and thetrial function

A

Cj(z,t) = Ci,m(t) om(Z) (46)is used in z space, where repeated subscripts indicatesummation as before. This method takes full advantageof the nature of flow and mass transport in multilayersystems of several aquifers and aquitards. Huyakorn et

103

eAii = Aii

N-Z. SUN

al. (73) presented a 3-D FEM for simulating saltwaterintrusion into aquifers. Recently, Burnett and Frind(92) designed a 3-D numerical model, the alternatingdirection Galerkin technique (ADG), that combines Gal-erkin finite elements with the alternating direction finitedifference technique. The 2-D pattern of the techniquewas given by Daus and Frind (97) and Frind and Ger-main (98). In the 3-D ADG, the advection-dispersionequation is split into three symmetric steps correspond-ing to directions x, y, and z, respectively; the final ma-trix equations can be solved in each direction by meansof the Thomas algorithm for tri-diagonal matrices. How-ever, the division of the flow region in the ADG methodmust be limited in a special way.

Advection-Dominated TransportProblemsThe numerical methods mentioned previously are all

successful for dispersion-dominated transport prob-lems. For instance, when the Peclet number Pe = 0.5in the classical model [Eq. (8)], the Galerkin FEM, thelumped mass FEM, and the MCBM all gave numericalsolutions that nearly coincide with the analytical solu-tion of the model (Fig. 4). However, for advection-dom-inated problems, they suffer from two kinds of numer-ical difficulties: overshoot and numerical dispersion. Theformer is characterized by oscillations around the com-puted concentration profile, the latter, by the smearingof the concentration front (Fig. 5, where Pe = 100).Gray and Pinder (99) assumed that the spatial dis-

tribution of concentration can be expressed as a Fourierseries with components of different wavelengths. Thebehavior of a numerical scheme is then evaluated interms of its ability to propagate each component of theseries with the correct speed and amplitude. They dis-cussed several FD and FE schemes for the classical

10

8

6

4

- ANALYTICAL SOLUTION-*- NUMERICAL SOLUTION

t = 40Pe = 0.5

10

8

6

4

2

- ~ANALYTICAL SOLUTION-0-- LMFEM-- FEM-0O- MCB (A = 0)t = 50Pe = 100

10 20 30 40 50 60 70 80

FIGURE 5. Comparison between analytical and numerical solutions:Pe = 100, t = 50.

model [Eq. (8)]. When advection dominates dispersion,the solutions generated from both FEM and FDM arecharacterized by overshoot, although the oscillationsare less pronounced for the finite element solution. Grayand Pinder (99) also pointed out that overshoot andnumerical dispersion are closely related in FD and FEschemes. Generally, one can select a numerical schemethat decreases the magnitude of one problem at theexpense of increasing the effect of the other.

Various improved numerical methods have been pre-sented in recent years to overcome these numerical dif-ficulties, such as upstream weighting methods (UWM)(100-103), moving point methods (MPM) (104-108), andother alternative methods (109-111). Perhaps, the mostsimple and efficient approach is the mesh refinementtechnique that improves numerical solutions by decreas-ing the local Peclet number. Some adaptive forms ofthis technique were given by Shepherd and Gallagher(111) and Yeh (112). However, when Peclet numbertends to infinity, the mesh refinement technique maynot be economically feasible. A perfect numericalmethod for advection-dominated transport should elim-inate overshoot and simultaneously improve the accu-racy of numerical solutions with only a little increase ofcomputational effort. Beside that, it should deal withcomplex 3-D field problems.

Upstream Weighting Method (UWM)If one uses the FDM to solve 1-D advection-dispersion

equations, the first and second derivatives, with respectto the space variable x at a grid i, are usually replacedby central difference approximations as follows:

10 20 30 40 50 60 70 80 90 100 110

FIGURE 4. Comparison between analytical and numerical solutions:Pe = 0.5, t = 40.

ac I Ci+1 - Ci-iax ji 2Ax

a2Cax2

Ci+1 - 2Ci + Ci-1(Ax)2

(47)

(48)

104

APPLICATIONS OF NUMERICAL METHODS

For the classical model [Eq. (8)], serious oscillation ofthe FD solution is observed when the Peclet number islarge. If Eq. (47) is replaced by upstream differencing

ac Ci- Ci-1iax j AXac Ci+1 - Ciax j AX

if V > 0

if V < 0

(49a)

(49b)

are used as basis functions in the isoparametric element,- 1 - e - 1, then the weighting functions are givenby

wi (t) = +j (4) - aF(k) W2(V) = +2(t - aF(t) (53)

where a is a parameter to be determined and

F(E) =3

O1- 00( + t) (54)

then the oscillation of the numerical solution is elimi-nated at the expense of increasing the numerical dis-persion significantly as shown by Gray and Pinder (99).In order to obtain an optimal compromise solution, theweighting average of Eq. (47) and Eq. (49) is used:

acax

C-C 01+1 -C1Aa + (1- a) 2lx

if V > 0

acax

Ci+1 - Cj Ci+1 - Cj=

+(1AX 2&x

if V < 0

where 0.5 < a. < 1 is the upstream weighting factor.Substituting Eqs. (48) and (50) into Eq. (8), we obtaina numerical scheme that is called the first-orderUWFDM. However, it is possible to present high-orderupstream FDM, for example, Leonard (113) presenteda third-order scheme. Its accuracy is higher then thatof first order UWFDM, but at the expense of increasedcomputational effort, because the coefficient matrix offinite difference equations in this scheme is no longertri-diagonal.

In 2-D models, an upstream weighting factor a isintroduced into FD equations by changing Ci = 1/2 (C1+ Cj) to

Cij = aCj + (1- a)Cj (51)where CU is the concentration of water flowing betweennodes i and j; Ci and Cj are the concentrations at thenodes. If fluid is moving from node i to node j, then 0.5< a < 1. Generally, let a = 0.75. Eq. (51) is also usedin IFDM [Rasmuson et al. (66)] to damp the oscillationsof numerical solutions with a = 0.65.Note that the tensor nature of the dispersion coeffi-

cient is not considered in the UWFDM. Therefore, it isdifficult to deal with field problems. In the late 1970s,the idea of upstream weighting was extended to FEM.The terms "upwind FEM," "asymmetric weightingfunction FEM" and "Petrov-Galerkin FEM" are oftenused in the literature. In Petrov-Galerkin FEM, the setof weighting functions {wi} included in Eq. (20) differsfrom the set of basis functions. For 1-D ADE, if thepiecewise linear functions:

has been called the modifying function.It is interesting to note that for 1-D ADE with con-

stant coefficients, the discretization equations ofUWFEM using Eqs. (52) and (53) are exactly the sameas that of the first-order UWFDM. The parameter a inEq. (54) corresponds to the upstream weighting factora in Eq. (50). Because the weighting functions [(Eq.(53)] are quadratic functions, the UWFEM needs morecomputational effort then that of the linear FEM. Someauthors suggested that the upstream weighting func-tions should only be used for the advection term of Eq.(20). For the other terms of the equation, the basisfunctions are still used as weighting functions (114).

Extensions of UWFEM with linear basis and quad-ratic weighting functions to 2-D or 3-D cases arestraightforward (115). The UWFEM using triangularelements was presented by Huyakorn (116). Theweighting functions are

wl(x,y) = Oi(x,y) + F(x,y)(55)

= i,j,k

where 4l are the linear basis functions of the triangularelement; F1 are the modifying functions given by

F = -3akOiOj + 3ajOiOk

Fj = -3aiVjik + 3akjOi (56)

Fk = -3ajOkOi + 3aiOkWjwhere ai, aj, Ok are upstream weighting factors (Fig.6). In Eq. (56) the upstream weights are introduced

k

(52) FIGURE 6. Triangular element with upstream weighting coefficients.

105

i i

1 1O (t) - O - 4) .2(0 = (1- t)2 2

N-Z. SUN

along the sides of elements that are not the flow ditions. Consequently, the effect of so-called "crossvdispersion" may decrease the accuracy of numericalutions. In order to lessen the crosswind dispersiomodified form named the streamline upwind/PetGalerkin (SU/PG) method was presented (102,117,1Generally, the accuracy of SU/PG method is higher tthe PG method.

If the set of weighting functions is required t(orthogonal to the set of basis functions, the corresping method is named as the orthogonal upstrweighting (OUW) finite element scheme. Yeh (noted that the OUW scheme provides an alternativ4large-scale problems because it gives a convergent psuccessive overrelaxation (SOR) computation for allclet numbers.Although UWFEMs have been extended to 2-D

even 3-D cases, it is necessary to remark that tpractical applications are limited because of comptional complexity.Sun and Yeh (39) presented a new upstream wei

ing method for 2-D models based on the MCBM ntioned above. In this method, the center of eachangular element is divided into three subtriangelements by linking the center and vertices of thement. In Figure 7, m is an invented node. The contration at the invented node is defined as the weiglaverage of the three vertices of the element:

Cm = wiCi + wjCj + WkCk

Wi + Wj + Wk = 1

where concentration values at points i,j,k and mnoted by Ci,Cj,Ck and Cm, respectively; wi,wj,wkupstream weighting coefficients corresponding to n(i,j,k in the element. They are dependent on the]Peclet number of the element. In each subelement,unknown function can be represented by Eq. (38).example, in the subelement Aijm, we have

k

FIGURE 7. Triangular element and its three subelements.

irec-windLI so-)n, a;rov-f18).than

o beond-eam'103)e for)ointIPe-

and

C(x,y,t) = OkiCi + 4kjCj + fkmCm(58)

(x,y) E Aijkwhere 4qd, Rj, and k4)km are linear functions of the sub-element. Substituting Eq. (57) into Eq. (58), we obtain

C(x,y,t) = * kiCi + *kjCj + 0 WCk(59)

(x,y) e Aijkwhere

* ki = Oki + WiOkm

*kj = Okj + WjOkm (60)

kk = Xkk + WkO+neir are new basis functions with upstream weighting coef-'uta- ficients. The mass balance equation can then be derived

for each original node as before, except the unknownLght- function has different representations in different sub-nen- elements.tri- Sun and Yeh (39) tested the proposed UWMCBM by

rular several numerical examples including a field problem ofele- contaminant transport in a stream-aquifer system. Be-icen- cause upstream weighting coefficients in the method arehted introduced into basis functions directly, the representa-

tion of unknown functions in each element remains lin-ear. The coefficient matrix of discretization equationscan be computed without numerical quadrature. Addi-

(57) tionally, the method is specially available for the solu-tion of field problems. In fact, the method has been usedto predict groundwater pollution at several aquifers inae China.

are The 3-D form of UWMCBM was given by Sun et al.Odes (40). Several numerical examples were given by Wanglocal et al. (41). A multilayer transport problem explains thatthe the elimination of oscillations of numerical solutions isFor also important for field problems.

However, all UWMs previously mentioned have twodisadvantages: overshoot is controlled, but at the ex-pense of increasing numerical dispersion, and the up-stream weighting coefficients arising in each UWM haveto be designated artificially.

Moving Point Methods (MPM)All UWMs can be seen as Eulerian methods. The

advantage of Eulerian methods is the use of a fixed grid,but they are generally not well suited for the handlingof sharp concentration fronts. In contrast, Lagrangianmethods use either a deforming grid or a fixed grid indeforming coordinates. For example, O'Neill (123) pro-posed a moving, deforming coordinate system that dealswith steep concentration fronts with no additional nodes

\ and time steps. However, for lack of a fixed grid, it isdifficult at handling fixed sources, fixed boundary con-ditions, and nonhomogeneous structures of media. Inaddition, it requires more computational effort in gen-

106

APPLICATIONS OF NUMERICAL METHODS

erating coordinate systems, especially for 3-D fieldproblems. Mixed Eulerian-Lagrangian methods at-tempt to eliminate such difficulties by combining thesimplicity of a fixed Eulerian grid with the computa-tional power of the Lagrangian approach (106). A com-plete review of these methods was given by Farmer(124).The method of characteristics combining with FDM

(MOC/FDM) was presented and successfully applied tofield problems by several researchers (125-127). Con-sidering the 2-D ADE [Eq. (14)] that can be combinedwith the GFE [Eq. (13)] to give

at =1 a mDijac -Viaac+F (61)

where

F- [W2 + SYat at (62)me

Eq. (61) possesses a characteristic defined bydx,dt = Vi, (i = 1,2) (63)

Along this characteristic, Eq. (61) reduces to

dC=

1 a mDj]+F (64)

dt = maag a +F

The unknown concentration is then divided into twoparts: the advection concentration and the dispersionconcentration. The former is determined by Eq. (63),which can be solved numerically by tracking the move-ment of a set ofmoving points or particles (Lagrangian).The latter is determined by Eq. (64), which can besolved by FDM. Note that there is no advection termin Eq. (64) and the problem of numerical dispersion canbe avoided.

In MOC, an interpolation process is always requiredto generate mesh concentrations from the concentra-tions of moving points and vice versa. Generally, theinterpolation process decreases the accuracy of numer-ical solutions significantly. Another disadvantage ofMOC is that the sources (or sink) and boundary con-ditions must be specially treated to generate or elimi-nate moving points that make the MOC cumbersome toprogram and execute. The reports on field applicationsof the MOC are still limited in 2-D problems.

Several modified forms of the MOC have been pre-sented. One of them is the single-step reverse particletracking technique that is used to replace the forwardmoving particle tracking procedure of the MOC(105,128,129). A single-step reverse point of a node isthe point where a particle leaving it at the time tk willreach the node location exactly at the time tk+, . In thefinite element version of the modified MOC, the con-centration of a reverse point at tk is determined by thefinite element interpolation, and it is used as the ad-vection concentration ofthe node at tk+ 1. The dispersion

concentration ofthe node can be obtained by the solutionof Eq. (64) with FEM. For second-order and third-orderversions of the method, see Farmer (124).Neuman (106) presented an adaptive Eulerian-La-

grangian FEM where the forward moving particletracking technique is used around each sharp front toimprove the accuracy ofnumerical solutions. Away fromsuch fronts the single-step reverse particle trackingtechnique is used to save the computational effort. Thenumber of moving particles is adaptive. When the frontsharpens, moving particles are inserted; when the frontflattens, they are eliminated. The adaptive scheme max-imizes computational efficiency.A similar method that is named the hybrid moving-

point method (HMPM) was presented by Farmer andNorman (107). Farmer (124) noted that the finite ele-ment version of the modified MOC is a good method forsmooth problems (fronts spread over three or moremesh lengths), and the HMPM a good method for prob-lems with sharp fronts.A variation of MPM is the random-walk method

(RWM). It is based on the concept that dispersion inporous media is a random process. In this method, ad-vection transport is simulated by a large number ofmoving particles like the MRM but the particle move-ment takes place in continuous space. Dispersion trans-port is simulated by adding a random-walk process toeach particle in each time-step, instead of the solutionof the dispersion equation. Consequently, it is not nec-essary to calculate the mesh concentration and updatethe particle concentration in every time-step. The con-centration distribution needs to be calculated only whenit is of interest. The main drawback of the RWM is thatthe computed concentrations are strongly dependent onsample size. If the number of moving particles is notadequate, a coarse solution may be observed. However,the RWM is conceptually simple and relatively easy toprogram. It has been used to solve 2-D field problemswith many thousands of moving particles. A completereport with a computer program was given by Prickettet al. (108).

Other Alternative MethodsThere are some alternative methods for the solution

of advection-dominated transport problems. The finiteanalytic method (FAM) was presented by Chen and Li(130) and Hwang et al. (109). The basic idea of the FAMis to invoke the local analytic solution of the governingequation in the numerical solution of the problem. Inthis method, the flow region is divided into small rec-tangular elements. For each element, the boundary con-ditions prescribed on the four sides are approximatedby either a second-degree polynomial or by combininga linear and an exponential function. The local analyticsolution is then expressed in an algebraic form relatingan interior nodal value to its neighboring nodal values.The system of algebraic equations derived from localsolutions is then solved to provide the solution of thetotal problem. Because the discretization equation as-

107

N-Z. SUN

sociated with each node is derived from the analyticsolution technique, the numerical difficulties caused bythe truncation error can be avoided. Hwang et al. (109)noted that the coefficients in interior nodal expressionhave upstream shift that make the behavior of numericalsolutions of the FAM be like that of UWMs. For thecase of a high Peclet number, the FAM also cannotproduce sharp fronts as predicted by analytical solu-tions. An advantage of the FAM is that it does notrequire artificially designated upstream parameters asin UWMs. However, the FAM has not met much ap-proval as yet.When the boundary element method (BEM) is used

to solve the contaminant transport problem, the gov-erning equation, e.g., 2-D ADE [Eq. (14)] must first betranslated into the following form:

a2Cax2 +

a2C FaY2F

where

F = aC a (VxC) + a (VYC) + M2 (66)at ax a

The translation can be done by assuming that the co-ordinate system is defined in the principal axes of thedispersion tensor. Molecular diffusion can be ignored,and dimensionless variables are used in Eqs. (65) and(66). Using Green's second identity, we have

XC(rit) = { aG- Gac dF + GFd (67)

in which: (F) is the boundary of the flow region (Qi);function G = In (r-ri) where r-ri is the distance from anyfield point r = (x,y) in (fQ) to a base point ri = (x,,yi);X = 2ni, -Fr, 0, if (xi,yi) in (fQ), on a smooth part of (F),or a kink of (F), respectively, where 0 is the internalangle at the kink. The integrals on the right-hand sideof Eq. (67) can be calculated approximately by linearinterpolation along (F) and the finite element interpo-lation in (fQ), respectively, then discretization equationsare obtained. Unfortunately, the coefficient matrix ofthe system of linear equations is fully populated. Fordetails of the method, see Taigbenu and Liggett (110).Numerical examples show that numerical solutions ofthe BEM display small oscillations and large numericaldispersion when the Peclet number is large.

Recently, Sun and Liang (45) presented a new nu-merical method that is named as advection controlMCBM for the solution of 2-D ADEs. The fundamentalidea of the method differs from general UWMs and theMOC in such a way that a given concentration termdepending upon advection is added to the right-handside of multiple cell balance equations to control thecharacter of numerical solutions. The flow region is di-vided into triangular elements. The vertices i,j,k andthe center m of each triangular element Aijk are con-sidered as real and invented nodes, respectively (Fig.

7). The concentration Cm of the invented node m isdecomposed into two parts: the regular part Cm,r andthe advection control part Cm,a. Let

Cm = WCm,r + (1- W)Cm,a (68)

where 0 < w < 1 is an advection contr9l coefficient thatcan be determined by an adaptive procedure. Cm,r(Ci + Cj + Ck)13, Ci, Cj, and Ck are the unknown con-centrations of real nodes i,j, and k, respectively; Cm,.is the pure advection concentration ofthe invented nodem that can be obtained by a modified single-step reverseparticle tracking technique. By constructing the MCBequations for all nodes and eliminating the unknownconcentrations of invented nodes with Eq. (68) one ob-tains a set of discretization equations that include allunknown concentrations of real nodes on its left-handside and all pure advection concentrations of inventednodes on its right-hand side. The character of the nu-merical solution ofthe equation is controlled by w. Whenw = 1, the method reduces to the general MCB methodthat is accurate enough for dispersion-dominated cases.When w = 0, the numerical solution is usually deter-mined by pure advection. Several classical problems anda field problem are solved in order to verify and dem-onstrate the usefulness of the proposed method. Themethod is very simple both in concept and computation.Also, it is easy to extend to 3-D cases.A finite-volume method with high-resolution upwind

schemes is introduced by Putti et al. (131) for the so-lution of the ADE. Due to an efficient interpolation tech-nique that is employed, the results of this method showgood agreement with analytical solutions for a full rangeof cell Peclet numbers.

Inverse ProblemsAt the initial stage of constructing a model for a field

problem, often the average velocity and dispersion coef-ficient included in the ADE are unknown. These param-eters are difficult to measure directly from the physicalpoint of view and have to be determined from historicalobservations. The inverse problem of advection-disper-sion simulation (ADIP) is determining unknown param-eters imbedded in the simulation model by using a setof observations of concentration and head distributionsin time and space. Sometimes, the unknown parametersalso include the strength of sink (or source) and bound-ary conditions.

Literature published in this area is limited becausemost concepts and methods used in the inverse problemof groundwater flow (GFIP) can also be used in theADIP. Another reason is that the ADIP is more difficultto solve than the GFIP and new results are hard to find.In the ADIP, we have to deal with two problems: thenonlinear problem arising from the dispersion coeffi-cient (depending on the velocity of groundwater flow)and the problem of differentiating the effects of advec-tion transport and dispersion transport. These problemsincrease the uncertainty of the ADIP. Therefore, in-

108

APPLICATIONS OF NUMERICAL METHODS

vestigating the ADIP should include two fields: ex-tending methods used in GFIP to the ADIP, and de-veloping new methods for the ADIP.Note that the ADIP is generally coupled with the

GFIP (32,132-135). The observed concentrations de-pend on the porous velocity and the latter depends onthe unknown porosity and hydraulic conductivity thathave to be identified by solving the GFIP.

Parameter IdentificationIn order to solve the ADIP numerically, the unknown

parameter must be first replaced by a vettor with finitedimension N. This can be done by either using the zon-ation method or the finite element interpolation method(22, 136). As the GFIP, the ADIP also can be translatedinto an optimization problem, depending upon what kindof performance criterion is used.When the equation error criterion is used, the dis-

cretization equations derived from any numericalmethod are rewritten as

[A] {p) = (b) + (F) (69)in which: [A]LXN = coefficient matrix and b = L-di-mensional column vector with [A] and b dependent onthe observed concentration distribution; p = N-dimen-sional unknown parameter vector; E = residual columnvector with components ei, (i = 1,2, ... ,L); and L isthe number of observations. Numerical methods basedon minimizing the residual E are classified as direct

L

methods (15). If minE lEil is taken as the minimizationi= 1

criterion, then the ADIP is translated into a linear pro-gramming problem. If min TE iS taken as the minimi-zation criterion, then the unknown parameter vectorcan be estimated as

(p) = ([A]T[A])-1 [A]T{b) (70)

Direct methods require an interpolation process to ob-tain the observed concentration at all nodes; the obser-vation noise and the interpolation error cannot beavoided. Unfortunately, direct methods are generallyunstable in the presence of noise.Another approach is the use of output least square

error criterion. The objective function of minimizationis the weighted sum of squared differences betweensimulated and observed concentrations:

min E = (C* - C(p))T [W (C* - C(p)) (71)

in which: C* = observed concentration vector withnoise; C(p) = nodal output, when p is used as the nodalparameter; [W] = diagonal matrix of weights, its ele-ment wii may be taken as 1/Ci2 (p) (29) where Ci (p) isthe nodal output corresponding to the ith observation.The problem [Eq. (71)] can be solved by the Gauss-

Newton method (30):

{Pr+1) = (Prl + ([Jr]T[W][Jr])-1[Jr]T[W]{C -C(Pr) (72)

where Pr, Pr + 1 are values of the unknown parametervector at the rth and (r + 1) th iterations, respectively,and

[Jr] = [ apU] P=Pr (73)

is the Jacobian called sensitivity matrix. Its elementsare called sensitivity coefficients. Eq. (71) can be solvedby any nonlinear programming method as well. For ex-ample, Umari et al. (26) used a quasi-linearization tech-nique, Strecker and Chu (137) used the quadratic pro-gramming method, and Wagner and Gorelick (29) useda quasi-Newtonian algorithm. In fact, all methods men-tioned in Yeh (25) for the GFIP can be extended to theADIP directly. A variety of statistical methods can bedone in the same way (16,18,138).The identification of the sink or source term is a little

easier than that of dispersion parameters. For example,Gorelick (139) used the linear programming method toidentify sources of groundwater pollution.However, the difficulty of inverse problems cannot

be moved by the application of numerical methods. Itasserts itself either by the ill-condition of the coefficientmatrix or by the existence of many local minimizers inthe optimization problem.The identifiability of unknown parameters is defined

initially as the one-to-one property of mapping from thespace of system outputs to the space of parameters(140). However, the uniqueness of such mapping is ex-tremely difficult to establish and often nonexistent. An-other definition of identifiability was given by Chavent(141). The parameter is said to be output least squareidentifiable if and only if a unique solution of the optim-ization problem Eq. (71) exists and the solution dependscontinuously on observations. It is also very difficult toachieve in field problems.Yeh and Sun (21) presented a concept of the extended

identifiability. The parameter p is said to be 8-identi-fiable if the solutions of Eq. (71) must be equivalentwhen they are used in the prediction model (a small,prescribed error is allowed). For field problems, the 8-identifiability generally can be achieved by only pointmeasurements.Chavent (142) reviewed the concepts of identifiability

of distributed parameters systems. He summarized andcompared five definitions of identifiability.

Recently, Sun and Yeh (135) presented a new conceptof identifiability for coupled inverse problems that isnamed management equivalence identifiability (MEI).An identified parameter is said to be management equiv-alence indentifiable, if it can satisfy the accuracy re-quirement of a given management model. Using MEI,the whole procedure ofgroundwater modeling, from theexperimental design, parameter identification to man-agement decision can be considered systemically.

Sensitivity AnalysisIn order to solve the optimization problem [Eq. (71)],

we have to calculate sensitivity coefficients aCi(p)/apj in

109

N-Z. SUN

Eq. (73) or gradient components aE(p)/hpj, j = 1,2, ...,N. There are three approaches that can be selected: theinfluence coefficient method (differencing method), thesensitivity equation method, and the variational method(25). If the number of components of the parameter tobe identified is greater than the number of observations(N > L) the variational method would be advantageous.The adjoint sensitivity method presented by Carter(143) and Chavent (144) is easy to extend to advection-dispersion equations. If the unknown parameter p is thelongitudinal dispersivity aL, we have

OA= J fVq(x,y,t-¶) DLVC(X,Y,¶) dtdQ (74)

where (fi) is the exclusive subdomain of node i; V is thegradient operator; and C(x,y,t) is the solution of the2-D ADE with additional conditions. q' is the time de-rivative of q(x,y,t) which is the solution of the followingadjoint problem:

x[Dap -] + a (Vaq) + i= (75)with final and boundary conditions:

q(x,y,T) = 0; (x,y) e (Q); T is the final time (76a)

q(x,y,t) I, = 0 (76b)

Dap axc .n r2 = 0 (76c)

In Eq. (75), Gi = 1/Pi, if (x,y) E (fi); otherwise Gi =0, where Pi is the area of the subdomain (Qi),

F 2vX vxvy

DL = 2 (77)VXVY Vy

If the unknown parameter p is the transverse dis-persivity alT it is only required to change DL in Eq.(77) to

- 2 1vx -vXvy

DT=[ 2 (78)

L-VXVY VyJHowever, a general form of adjoint sensitivity analysiswas given (145-147). In Schmidtke et al. (148), a generalperformance

tE = J Jf(p,C)dadt (79)

0 (a)was considered. If f(p,C) = W[C* - C(p)],2 it reducesto the least square error criterion. The continuum aswell as finite element discretization representations ofaElapj can be found in (148).

In Sun and Yeh (134), the adjoint problem is derivedfor general coupled problems through introducing a setof adjoint operation rules. An example of parameteridentification for coupled groundwater flow and masstransport is given.

Experiment DesignAn experiment design X is a decision that includes

the following components: a) locations of observationwells (space distribution of observations; b) observationtimes (time distribution of observations); c) locations ofinjection and pumping wells (locations of stimulation);and d) pumping or recharge rate and the concentrationof injection water (strength of stimulation).The method of sensitivity analysis can help engineers

to design field tracer experiments. Mercer and Faust(149) presented an example where a two-well tracer testwas designed to determine the well spacing, the injec-tion flow rate, and the injection period.The reliability of parameter estimate depends upon

the quantity and quality of observations. If the esti-mated parameter p is located in a neighborhood nearthe true parameter p* in the parameter space, then wehave the theoretical covariance matrix:

COV(p) - a2([JJT[W][JJ)1 (80)

where a2 is the common variance of the random errorin observations. Obviously, the maximization of the de-termination of ([J]T[J]) will both improve the conditionnumber of Eq. (72) and decrease the uncertainty of theestimated parameter. Therefore, the objective

max A = det ([J]T[J]) (81)is often used as a criterion of the optimal experimentdesign (30). The senstivity coefficients that are elementsof [J] in Eq. (81) depend upon the unknown true pa-rameter; Eq. (80) is correct only if p is a good approx-imation of p*. It is very difficult to make an optimalexperiment design by using Eq. (81) for a field problem.Yeh and Sun (21) presented a new concept named the

sufficiency of observations. The observations of an ex-perimental design are said to be sufficient if any twoestimations Pi and P2 that are non-equivalent for theobjective of model prediction are 5-identifiable whenthese observations are used in the identification pro-cedure. In other words, these observations are ade-quate for determining a weak, unique solution of theinverse problem. The sufficiency of design X can bejudged by solving a nonlinear programming problem.Obviously, this method is very useful for practicallydesigning a field experiment.

Recently, this concept has been extended to deal withmanagement problems (135). The question of whetherthe observations are sufficient for the MEI can be an-swered by solving a constrained optimization problem.Hsu and Yeh (150) presented a design criterion that

minimizes experimental cost while meeting the relia-bility of model prediction. They formulated the optimalexperimental design in terms of a nonlinear mixed-in-

110

APPLICATIONS OF NUMERICAL METHODS

teger programming problem and solved it through aheuristic approach.

Reliability EstimationAfter getting a parameter from an identification pro-

cess, we should estimate the reliability of the identifiedparameter. The covariance estimation [Eq. (80)] maynot be available because the true parameter is unknown.In general cases, Monte Carlo simulation can be usedto evaluate the reliability without any additional as-sumption. In the Monte Carlo method, each trial is adifferent realization of observations used to solve theinverse problem. The output set of Monte Carlo runs isthen used to estimate the statistical properties of theidentified parameter. Several hundreds or thousands ofrealizations may be necessary to arrive at a stable es-timate of statistical results. If the kth realization ofparameter estimate is Pk, then the expected value andvariance of the parameter are

1KE(p) X, Pk (82)k=1

andK

Var (p) = [Pk - E(p)]2 (83)k=1

respectively, where K is the number of realizations.Once the expected value and the variance of the.un-

known parameter are obtained, it is necessary to furtherestimate the uncertainty of the output (concentration)of the constructed model. For this reason, we need toestimate the uncertainty of the solution of an ADE fromthe uncertainty of its coefficients. Tang and Pinder (151,152), introduced a numerical scheme based on a per-turbation expansion technique and its extension to solvethe problem. From numerical examples, they concludedthat the greatest uncertainty of the solution occurs atthe concentration front, and the transport equation ap-pears to dampen the uncertainty associated with thephysical parameters: velocity and dispersivity.

In fact, the ADE governing the contaminant trans-port in groundwater should be seen as a stochastic par-tial differential equation because the uncertainty is in-cluded in its coefficients, initial and boundaryconditions, and sink or source terms. If the model isused in a prediction or a decision-making process, quan-tifying the uncertainty of the model may play an im-portant role. For a complex problem, it is convenientto use the Monte Carlo method to evaluate the uncer-tainty. Nelson et al. (153) presented an analysis se-

quence that begins with field data and results in theconfidence limits of the model output. A field model isused to explain the technique. When the covariance ma-trix of parameter estimation error is provided as a partof the model calibration, then the velocity field and fluidflow paths with the associated travel times are gener-ated for each realization. Also, the confidence limits of

the model output are obtained by the Monte Carlo sim-ulation.Note that the reliability of a model prediction is al-

ready included in the 8-identifiability (21,135). Providedobservations are sufficient for the 8-identifiability; theidentified parameter must be equivalent with the trueone in the sense of satisfying prediction and manage-ment requirements.To obtain the uncertainty of using an estimated pa-

rameter, another approach is the use of stochasticmodels (20,154-156), where the covariance of the esti-mated parameters can be obtained directly by the useof conditioning or unconditioning geostatistical meth-ods.

SummaryIn this paper, we focus our attention on numerical

methods for solving 3-D problems, advection-dominatedproblems, and inverse problems. The solution of theseproblems is very important, not only in theoretical in-vestigations, but also in practical applications.

In order to describe the shape of contaminant plumescorrectly, we need 3-D and multilayer models. The mainproblem in applying 3-D models is decreasing the com-putational expenses both on human and machine re-sources. Thus the Galerkin FEM with special elementsand basis functions, the mixed FD-FE method, and theMCBM are introduced. The MCBM keeps the local massconservative. It is simple in concept and easy to use forfield problems. A layer-by-layer iteration process is ef-ficient for solving the discretization equations. A soft-ware package with the functions of figure input andfigure output was also presented.The UWM, modified MOC, BEM, FAM, and ACM

were introduced and compared for solving advection-dominated transport problems. The advantages and dis-advantages of each method were shown. The adaptiveEulerian-Lagrangian method seems to be a good ap-proach for getting a high-quality solution. For fieldproblems, a kind of modified MCBM orACM is a simple,efficient method to obtain a satisfactory solution withoutoscillations.Most methods used for ADIP are those that directly

extend the one for GFIP. Several optimization tech-niques based on different criteria were introduced. Inthe solution of ADIP, the main difficulty was its ill-posedness. We do not know how many observations areadequate for getting a unique solution of the inverseproblem. However, a weak identifiability, i.e., the 8-identifiability, can be determined with limited obser-vations. The calculation of sensitivity coefficients wasnecessary, both in the numerical methods of the ADIPand in the experimental design. The adjoint sensitivityanalysis was discussed for the ADE and general coupledprobems. In order to construct a practical model, a sys-tematic procedure from the data collection to the un-certainty estimates of the built model is required, inwhich the Monte Carlo method plays an important role.

ill

112 N-Z. SUN

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