Techniques for Making Finite Elements Competitive in Modeling Three-Dimensional Flow and Transport
This paper presents the formulation of a three-dimensional finite element model designed to simulate ground-water flow and contaminant transport in complex multi-layered aquifer systems. The model combines the use of two-dimensional basis functions in the
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TECHNIQUES FOR MAKING FINITE ELEMENTS COMPETITIVE IN MODELING THREE-DIMENSIONAL FLOW AND TRANSPORT Peter S. Huyakorn, GeoTrans, Inc., Reston, Virginia, U.S.A.
INTRODUCTION This paper presents the formulation of a three-dimensional finite element model designed to simulate ground-water flow and contaminant transport in complex multi-layered aquifer systems. The model combines tne use of two-dimensional basis functions in the x-y plane and one-dimensional basis functions in the z-direction. In addition, coupling of adjacent aquifers and aquitards are performed in a highly efficient manner using one-dimensional analytical solutions. The formulation to be described in the following sections is very general and has several advantages which include great flexibility and highly efficient element matrix computational procedures and equation solution algorithms. PROBLEM STATEMENT A typical problem concerned is depicted in Figure 1. The problem is three-dimensional because of hydraulic connections between aquifer and aquitard units. In general, ground-water flow in the aquifers may be described by the following equation: __d__ (Kij
dX.
1
~) = dX.
J
s
s
ah at
- q
i,j
=
1,2,3
(1)
where Kij is the aquifer hydraulic conductivity tensor, h is the hydraulic head, S is the specific storage of the s aquifer, t is the time, and q is the volumetric flow rate via sources or sinks per unit volume of the porous medium.
J. P. Laible et al. (eds.), Finite Elements in Water Resources © Springer-Verlag Berlin Heidelberg 1984
188
/
____------------------- - - -'
L.ATERAL
~~'NFCOW ~
L.ANDFILl
o
Figure 1.
Typical three-dimensional groundwater and contaminant transport problems.
Slice 4
Slice 3
Z=ZT I
Slice 2
Slice 1
Figure 2.
Discretization of an aquifer unit.
189 Flow in the aquitards may be described approximately by
_a
aX 3
s's
(K'~)
aX 3
ah' at
(2)
where the prime is used to denote the aquitard properties. Equations (1) and (2) are coupled via the fluid leakage flux terms at the aquifer-aquitard interfaces where h = h'. Transport of a non-conservative species in the aquifer is governed by
a aX i
(Dij
~:)
Vi
j
~ ax i
~y.
(AC
+~) at
+q
(c-c*) (3)
where D.. is the hydrodynamic dispersion tensor, c is the solute ~~ncentration, V. is the Darcy velocity vector, ~ is the effective porosity,1\ is the decay coefficient, q is the volumetric fluid injection rate per unit volume, and c* is the solute concentration in the source fluid. Transport in the aquitards may be described by ( D'
aX 3
~
aX 3
~'
)
K' (
\'
c'
+~) at
(4)
Equations (3) and (4) are also coupled via solute flux terms at the aquifer-aquitard interfaces, where c = c'. FINITE ELEMENT APPROXIMATION To illustrate the development of the proposed finite element formulation, it is sufficient to deal with the transport equation. In three dimensional space (x,y,z), the governing equation for transport in an aquifer takes the form
~ ax
(Dxx l.£ ) ax V
de
+
d
ay V
(Dyy l.£) + ay dC
V
d
az dC
(Dzz
~zc a
)
z o ( 5) Y dy a
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