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Use of thefiniteelement method in solving transient flow problems in aquifer systems P. A . Witherspoon, I. Javandel a n d S. P . N e u m a n University of California, Berkeley

A B S T R A C T : T h efiniteelement method is a new numerical approach for solving transient flow problems that is particularly well adapted to digital computers. In this method the partial differential equation together with the appropriate initial and boundary conditions are replaced by a corresponding variational problem. The continuum is replaced by afinitenumber of subrogions, and the variational principle is expressed as a summation of functionals over the entire network. The resulting functional is minimized by direct methods of the calculus of variations to yield the solution to the original boundary value problem. T o demonstrate the validity of this new technique, results obtained by thefiniteelement method are compared to analytical solutions from the literature. T o demonstrate the power of this approach, solutions for the more complex problem of transient flow in layered systems with crossflow are also presented. A n example of results for a two-layer aquifer is discussed. A second example of results for a three-layer system consisting of two aquifers enclosing an aquitard is also discussed. Thefiniteelement method is completely general with respect to geometry, boundary conditions and rock properties and provides a powerful new method of solvingfluidflowproblems in complex systems. R É S U M É : La méthode aux différencesfiniesest un nouveau procédé pour résoudre les problèmes de mouvements non permanents: elle est particulièrement bien adaptée pour les calculateurs digitaux. Dans cette méthode les équations aux dérivées partielles ainsi que les conditions initiales et aux limites sont remplacées par un problème de variation. Le continu est remplacé par un nombre fini de sous-régions et le principe du calcul des variations est exprimé par une s o m m e s'étendant à l'ensemble du réseau. Le résultat est minimisé par les méthodes directes du calcul des variations pour obtenir la solution ayant la limite d'origine du problème. Pour démontrer la validité de cette nouvelle technique, les résultats de cette méthode sont comparés à des solutions analytiques de la littérature. Pour montrer la puissance de ce procédé des solutions sont présentées pour le problème le plus complexe de mouvement non permanent dans des systèmes présentant différentes couches avec mouvement transversal. U n exemple de résultats pour une nappe à deux couches est discuté. U n deuxième exemple de résultats pour un système à trois couches consistant en deux couches perméables séparée par une couche de faible perméabilité est aussi discuté. La méthode aux différencesfiniesest complètement générale pour ce qui regarde la géométrie, les conditions aux limites, les propriétés des roches et elle fournit une nouvelle méthode puissante pour résoudre les problèmes du mouvement defluidedans des systèmes complexes.

INTRODUCTION Hydrologists have traditionally attacked problems of transient flow through porous media using the classical m e t h o d s of analytical mathematics. Theis (1935) a n d H a n t u s h (1957) are but t w o examples from a large n u m b e r that o n e could cite for the analytical approach. This approach, however, suffers because o n e usually has to adopt a simplified system that m a y not be sufficiently representative of the true situation in the field. Even w h e n the system has been simplified so that a solution is possible, one m a y still have difficulties. Recently, w e have been attempting ot obtain analytical solutions for transient flow in multiple-layered systems (Javandel a n d Witherspoon, 1968a) but the mathematical results are so complex that their evaluation has b e c o m e a major problem in

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itself. Moreover, w e arefindingthat the evaluation of our analytical expressions will often require far more computer time for the same degree of accuracy than would be required by numerical methods. Katz (1960) has also reported difficulties in his analytical approach to two-layered systems. Jacquard (1960) has presented analytical solutions for two-layered systems, which to our knowledge no one has been able to evaluate rigorously for all parts of his system. It appears to us that the application of analytical mathematics, at least in some cases, has s o m e serious limitations. Certainly, the complex systems that one encounters in thefieldcannot be handled analytically if one desires other than some kind of averaged result. W e have therefore embarked upon a program of developing numerical methods of solving transient flow problems in heterogeneous and anisotropic systems. O u r first approach w a s to use thefinitedifference methods that are well k n o w n in thefieldsof hydrology and petroleum. In thefinitedifference approach, heterogeneous bodies can be handled by dividing the system into a network of elements and writing afinitedifference equation for the flow into and out of each element. The solution of the resulting set of equations usually requires a high speed computer. This approach, however, also has its limitations and m a y require large amounts of computer time (Freeze and Witherspoon, 1968). In some particular cases, it has not been possible to obtain solutions for the more complex problems. For example, Vacher and Cabazat (1961), in using the alternating direction implicit method, which involves a more recentfinitedifference formulation, were unable to solve a problem of transient flow to a well in a bounded, two-layer, radial system w h e n the ratio of the outer to inner boundaries was 3,800. Using thefiniteelement method, w e were easily able to obtain the solution (Javandel and Witherspoon, 19686). This has led us to a second approach to our general problem of obtaining transient solutions for flow in complex systems. Thefiniteelement method was originally developed in the aircraft industry (Turner, et al., 1956) and is n o w receiving wide application in the field of structural machanics (Clough, 1965; Zienkiewicz, 1968). The method is similar to thefinitedifference approach in that the continuum is subdivided into a network of elements. The mathematics, h o w ever, is considerably different and requires an understanding of the calculus of variations and, in particular, the concept of the varitaional principle. Zienkiewicz and Cheung (1965) have described in detail h o w the method is applied to flow problems where a steady state solution is desired. Zienkiewicz et al. (1966) have used this approach in obtaining steady state solutions for fluid flow heterogeneous and anistotropic systems. Taylor and B r o w n (1967) have shown h o w to obtain solutions for a flow regime with a free surface. Gurtin (1964) has shown h o w variational principles can be developed for linear initial-value problems thus broadening this whole approach to include time dependence. Subsequently, Wilson and Nickell (1966) have shown h o w thefiniteelement method can be used to solve transient heat flow problems in complex systems. Javandel and Witherspoon (19686) have recently extended the application of thefiniteelement method to transientfluidflowproblems. W e believe it is n o w possible to attack m a n y situations that heretofore were either impractical, because of excessive computer time requirements, or simply impossible. W e shall briefly outline the mathematics of thefiniteelement method, show h o w results obtained with this method compare with previously published solutions, and then present some examples of the kind of results one can obtain for complex systems.

REVIEW OF MATHEMATICAL FORMULATION In thefiniteelement method, the general approach is to replace the appropriate partial differential equation together with its initial and boundary conditions by a corresponding

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Use of thefiniteelement method in solving transientflowproblems in aquifer systems

variational problem. The method is not restricted to linear boundary value problems, and Ciarlet et al. (1967) have shown h o w the variational principle can be used to solve nonlinear boundary value problems. However, for our purposes here, w e shall only consider h o w the usual diffusivity equation in terms of hydraulic head: V2/z = ^ K at

(1)

can be treated using thefiniteelement method. According to the analysis of Gurtin (1964), equation (1) isfirstcombined with the initial condition to obtain an integro-differential equation. For flow to a well in a radial system, one usually has an initial condition: h(r,z,0)

= ho

(2)

and after combining equations (1) and (2), one has: K*V2h

= Ss(h-h0)

(3)

where * stands for convolution. For the purposes of simplification, w e shall only deal with heterogeneous systems in the analysis that follows, but if one is also faced with two-dimensional anisotropy, as might be the case for radial flow, then equation (3) must be written: V-((Kl*Vh) = Ss{h-h0)

(4)

where [K] represents the appropriate permeability matrix. Gurtin has shown that the expression: Í2W =

[_Ssh*h + K*Vh*T7h-2Ssh0*h](r,

z, t)dV

(5)

represents the variational principle for equation (3). In order to include the effect of flow boundary conditions, one must alter (5) to: Í2W =

[Ssh*h + K*Vh*Vh-2Ssh0*h](r,

-2

K* — *hdA

z, t)dV

(6)

on where dh/dn represents the hydraulic gradient normal to the boundary under consideration. Having obtained the appropriate variational principle, the general procedure is to minimize the functional Û (h) with respect to the generalized coordinates (values of h at the nodal points of the network), and in so doing, one obtains the solution to equation (1). W e shall follow the Rayleigh-Ritz method of minimizing Q{h). It should be kept in mind that the minimizing procedure can either be applied to the system as a whole, if that is possible, or individually to any number of sub-divisions that m a k e up the whole. In attacking problems that involve heterogeneous systems, one usually divides the continuum into a network of elements such that the flow properties within each element remain constant. For an axi-symmetric problem, w e choose a ring-shaped element with a triangular cross-section, as shown infigure1. If w e can obtain the values of head at each corner (h¡, h¡, hk) from the minimization procedure and extend such a treatment to a 689

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network of elements that encompasses the continuum under consideration, w e have the solution to our problem.

F I G U R E 1. Triangular cross-section of axi-symmetric element

W efirstexpress hydraulic head at any point within the element (or on the boundaries) as some function of the coordinates. T h e simplest procedure is to assume the linear relationship: h{r, z, t) = a + br + cz

(7)

where a, b, and c are constants that are determined by the fact that equation (7) must reduce to the values h¡, h¡ and hk at points i, j and k, respectively (Argyris, 1960). O n e can use higher ordered expressions instead of equation (7) if desired, but the crux of the matter will hinge upon whether the elements have been made small enough so that a linear relationship is appropriate over all values of time. It can be shown that when one has a network of elements, the hydraulic head at any instant of time in the mth element can be expressed in matrix form as: hm{r,z,t) =

(Nm(r,z)>{h(t)}

(8)

where {/¡(i)} is a column matrix of all hydraulic heads at the entire nodal points of the network and (Nm(r, z)) is a row matrix composed of finite values for the nodal points of the mth element and zeroes at all other points. If one carries out the operations indicated by equation (6) and expresses the results in matrix form (Javandel and Witherspoon, 19686), one will obtain: Q(h) = {/i}r[i)] * {h} + {h}T* \_AA~] * {h}

(9) T

T

-2{h} U)-]*{h0}-2{h} *{Q}

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Use of thefiniteelement method in soloing transientflowproblems in aquifer systems

where: M

SSmT