Stochastic groundwater flow using boundary elements C. Z. Karakostas* and G. D. Manolis** ^Institute ofengineering Seismology and Earthquake Engineering, P. O. Box 53, Thessaloniki GR 55102, Greece **Department of Civil Engineering, Aristotle University, Thessaloniki GR 54006, Greece Abstract A boundary element solution for two dimensional, transient groundwater flow in confined aquifers with random diffusivity is presented. Firstly, fundamental solutions for the diffusion equation with a random coefficient are derived using the perturbation method.. Subsequently, perturbations are again used for building a boundary element solution for the covariance matrix of the response for general boundary-value problems. The entire methodology is defined in the Laplace transform domain, and an efficient inversion scheme is employed for capturing the transient response in terms of a mean solution plus a variance. A series of examples serves to illustrate the methodology. 1 Introduction In order to overcome difficulties associated with deterministic modeling of flow through heterogeneous porous formations, it has become quite common recently to view aquifer properties and flow variables as random variables/"^ Two main approaches have been followed so far : First, the geostatistical approach*, which is mainly concerned with the problem of statistical interpolation, i.e., how to predict best estimates and variances of variables measured on relatively few points in a spatial grid. Second, methodologies based on determination of the statistical structure of the flow given the statistical structure of material properties, geometry and boundary conditions of the problem. The latter approach is essentially based on stochastic differential
516 Boundary Elements equations for modeling groundwater flow and has attracted the attention of a number of researchers.^ The main difference between these two approaches is that geostatistical methods consider measured values as fixed and allow for random fluctuations at the remaining points of the grid, which means that conditional probabilities are used. This is in contrast to stochastic modeling of groundwater flow, where hydraulic properties fluctuate throughout the continuum according to their corresponding probability density functions. A variety of stochastic models have been introduced including the stochastic BEM for groundwater flow under random boundary conditions and recharge caused by stream stagefluctuations,tidal wave actions, random precipitation patterns, etc.^ The governing equation for steady-state groundwater flow in a rigid porous formation is replaced by a stochastic integral equation of the indirect type, whereby an unknown fictitious density source is used as an intermediate step. Routine numerical discretization gave response covariances which required double surface and volume integrals and resulted in good solution accuracy for two dimensional, confined aquifer flow. In this work, a direct boundary element formulation is developed for the mean vector plus covariance matrix solutions of two dimensional transient groundwater flow in confined aquifers with random diffusivity. The perturbation method is used in conjunction with the appropriate boundary integral equations to produce a compact boundary element scheme. The entire methodology is defined in the Laplace transform domain and an inverse transformation algorithm is employed for reconstructing the temporal response. Numerical examples for stochastic with mixed boundary conditions of arbitrary time variation serve to illustrate the proposed methodology. 2 Flow through a 2D confined aquifer Groundwater flow through a confined, compressible aquifer under 2D conditions is given as** (1) where h=p/y+z is the piezometric head (p, y and z respectively are pressure, specific weight and height) and c=t/s is the diffusion coefficient (S and T respectively are storativity and transmissivity). Finally, boundary conditions involve the piezometric head and the fluid flux q=9h/bn, where n is the direction normal to the surface. Application of Laplace's transform with respect to the time variable, which is defined as L(h)= h (x,s) = f (x,t) e^ dt (2)
Boundary Elements 517 for the direct transform and L-'(h) = h(x,t) = f(x,s)e*'ds (3) 2711 'b-ioo for the inverse transform (s is the transform parameter), to the governing equation of groundwater flow yields V^ h(x,s)- -h(x,s) = 0 (4) c for zero initial conditions. The above formulation can be recast as a boundary integral equation in the Laplace transformed space as" a(xo) h (xo) = JJ { G(x,Xo) q (%)_ F(x,x«) h (x) } ds(x) (5) where G is the fundamental solution (Green's function) for eq.(l) under a unit point load 8(x-Xo) and radiation boundary conditions, while XD and x are respectively known as source and receiver. Furthermore, F=BG/9n, while where KO is the modified Bessel function of second kind and zero order, while radial distance r = I x-xj. If both x, Xo are defined at the surface of the problem, then jump term a(xp) is equal to 0.5 for a smooth surface.. Routine numerical discretization of the above boundary integral equation using quadratic surface elements yields The above system of algebraic equations can now be used for solving well posed boundary value problems. The final step is to perform an inverse Laplace transform so as to recover the unknown piezometric heads h and fluid fluxes q from their transformed values h and q, respectively. This is achieved numerically through integration over the complex plane as dictated by eq.(3) using Durbin's algorithm^. It requires that the function of interest be sampled at Sm = b + im (2%/T) (8)
518 Boundary Elements points, where 5 < bt < 10, m = 0,1,2,...,M-1, M > 50 in multiples of 2 and 5 and T is the total time interval of interest. The inverted function is then obtained at M equidistant points U= m At = m T/M. 3 Fundamental solutions for random diffusivity In order to simulate some of the complications arising from groundwater flow through geological media, it is necessary to resort to a stochastic representation of the diffusion coefficient as perturbation of the form c(y) = c, + 6Ci(y) (9) about mean value c«with fluctuation ci being a zero-mean function of random parameter y and with known variance o^ «1.0. A similar Taylor series expansion about the mean is used in conjunction with fundamental solution G, i.e. G(r,e) = G(r,6=0) 4- e " / L=o +... (10) de In view of eq.(6), we have that G(r,6=0) = Go(r) = K«(u) (1 la) and - I s=o = Gi(r,y) = KI(U) u = Ci(y) Gj (u) (1 Ib) where argument u*= sr^/co and KI is the modified Bessel function of second kind, first order. The same expansion is used in conjunction with the second fundamental solution corresponding tofluidflux,i.e. where F(r,e) = Fo(r)4-GFi(r,y) (12) Fo(r) = -Ki(u) v - dn (13a) {2Ki(u) v - v" r K:(u)} -= Ci(y) F^ (r ) (13b)
Boundary Elements 519 with u defined as before and v^ = s/c<,. Statistical moments for the aforementioned expansions are derived through introduction of the expectation operator to give a mean value (for the first fundamental solution) mg(r) = <G(r,6)> = Go(R) (14) and a covariance defined about the mean as where u^= sr^/co and similarly for U2. Furthermore, a<? is the variance of the diffusion coefficient, for which any distribution can be assumed. As far as the variance of the fundamental solution is concerned, we have varg(r) = covg(r,=r2) = ^ G / ( r ) = Oc' K^(u) u' / (4c/) ( 1 6) Exactly the same procedure yields the statistical measures varf for the second fundamental solution. rrif, covf and 4 BEM formulation for stochastic groundwater flow Assuming the dependent variables of the problem to be functions of random parameter y, the boundary integral statement given by eq.(5) is rewritten as a(xo) h (xo) = jj { G(r,y) q (x,y) - F(r,y) h (x,y) } ds(x) (17) By perturbing both piezometric head and flux in the manner employed for the fundamental solutions G and F, substituting all these expansions in eq.(17) and sorting out powers of 8 we obtain the following zeroth and first order solutions a(xo) h, (xo) = jj { G,(r) q,(x) - F,(r) h, (x) } ds(x) (18a) a(xo) h\ (x,,y) = Jj { Go(r) q^ (x,y) - F«(r) h^ (x,y) } ds(x) { Gi(r,y) q,(x) - Fi(r,y) h,(x) } ds(x) (18b)
520 Boundary Elements As expected, the first equation is the deterministic solution for mean values h\ and q^, while the solution for the fluctuating terms hj and q^ follows the convolution-like pattern of the second equation. A numerical discretization of eqs.(18) yields the following systems of algebraic equations (19a) [Go] {q,} + [G,] {q,} = [Fo] {h,} + [F,] {k} (19b) Assume, for simplicity, that the piezometric head is prescribed and that the correspondingfluidfluxesare unknown Then {<U = [Go]-'[Fo]{hJ = [A]{nJ (20a) { q, (Y)} = [Go] -' [Fo] { h, (Y)}+ [Go] -' ([F,] - [G,] [Go] "' [F.]) {hj = [A]{h,(Y)} + [B(Y)]{hJ (20b) Introducing the expectation operator leaves eq.(20) unchanged, while eq.(20b) yields the covariance matrix of { q } as = [A] < {h,(y)} {h,(y>r> [Af + [A] < { {ho} {ho}^[b(y)f > (21) As special case we identify deterministic boundary conditions, i.e. {hi(y)}={0}. Then, eq.(21) simplifies considerably and yields [covj = < [B(Y)J {ho} {ho}lb(y)f > (22) Taking into account the structure of the fundamental solution, eq.(22) can be re-written as [covj =<[B]ci(y){ho} {ho}^ci(y)[bf > = c^<[b] {ho} {hor[b]^> (23) Thus, the covariance matrix of the response [ q (y)] is proportional to the variance Ci(y) of the diffusion coefficient. Both mean value { q^ } and covariance [covj are defined in the Laplace transform domain, which implies that the inverse Laplace transformation discussed must be utilized. Furthermore, mean values and covariances can be obtained for any mixed boundary value problem by introducing appropriate partitioning in eqs.(20).
Boundary Elements 521 5 Numerical examples Consider the simple example of Fig.(l), whereby a confined aquifer of dimensions 1=2 m and d=l m has prescribed piezometric heads of hi= -1 f(t) and \\2= 1 f(t) (in meters, with f(t) a dimensionless function of time) along the left and right vertical boundaries, respectively, while there is a no-flow condition (q=0) along the two horizontal boundaries. As far as the aquifer mean properties are concerned, we have T= 0.01 mvsec and S= 0.0005, hence Co = T/S= 200 nf/sec and K = T/d= 0.01 m/sec. We further assume that the boundary conditions are deterministic, while the variance of the diffusion coefficient is simply a constant, i.e. o/ = 0.4 nwsec^. The confined aquifer surfaces are discretized using quadratic, three-noded elements and the basic mesh consists of 12 elements and 24 nodes. 5.1 Mean solution At first, the quasi-static solution is recovered by using the H(t-O) time variation in conjunction with a very large observation time, namely T=400000 sec. It is compared against the constant flow solution in Fig.(2a) and excellent accuracy is observed for 50 steps in the inverse Laplace transformation algorithm. Next, Fig.(2b) plots the results to the same problem but for a total time duration of T=0.25 sec, where the transient character of the long-term solution is clearly manifested. More examples are found in [13]. 5.2 Response Covariance The covariance matrix of the response is evaluated here. Specifically, due to the fact that the inverse Laplace transformation is computationally intensive, only the diagonal terms (autocovariances) are inverted to the time domain for the nodes specified. Figure 3 plots the variance o^ of the piezometric head at 1=1.67 m on the bottom side (node 6) as a function of time for a duration of T=0.05 sec. Concurrently plotted is the variance G<? of the fluid flux at the middle of the left vertical side (node 22). We first note that the randomness in the medium affects the flux much more than the piezometric head. Furthermore, a variance of order 10"* in the diffusion coefficient translates into a variance of the order of 10" for the flux and 10" for the piezometric head. Finally, the piezometric head and flux variances, at nodes 6 and 22 respectively, for the rectangular aquifer with boundary conditions that vary as f(t)=t^ are plotted in Fig.4. It is observed that the time variation of the variance is similar to the time variation of the corresponding mean solution.
522 Boundary Elements 6 Conclusions The introduction of stochasticity in the description of a geological medium offers an attractive alternative, due to the fact that randomness is able to approximate various complications associated with a naturally occurring medium. This paper presented a method of analysis based on boundary integral equation formulations for random groundwater flow in two dimensional aquifers under transient conditions. Acknowledgment We greatfully aknowledge support through grand PENED 1483/94 of the Greek General Secretariat for Research and Technology. References 1. Delhomme, J.P., Spatial variability and uncertainty in groundwater flow parameters : a geostatistical approach, Water Resources Res., 1979, 15 (2), 269-280 2. Dagan, G. Stochastic modeling of groundwater flow by unconditional and conditional probabilities, 1. Conditional simulation of the direct problem, ^^^ewwrc^^&, 1982, 18 (4), 813-33. 3. Smith, L & Freeze, R.A. Stochastic analysis of steady - state groundwater flow in a bounded domain : One dimensional simulations, Water Resources #&?., 1979, 15(3)521-528. 4. Tang, D.H. & Finder, G.F. Simulation of groundwater flow mass transport under uncertainty, Advan. Water Res., 1977, 1 (1), 25-30. 5. Gutjahr, A L & Gelhar, L.W. Stochastic models for subsurface flow : Infinite versus finite domain and stationarity, Water Resources Res., 1981, 17 (2), 337-350 6. Dagan, G., Analysis of flow through heterogeneous random aquifers by the method of embedding matrix 1. Steady flow, Water Resources Res., 1981, 17(1), 107-121. 7. Tompson, A.F.B. & Gelhar, L.W. Numerical simulation of solute transport in three-dimensional, randomly heterogeneous porous media, Water /^owrc&s7?e&, 1990, 26 (10), 2541-2562. 8. Cawlfield, J.D. & Sitar, N., First order reliability analysis of groundwater flow, Probabilistic Methods in Civil Eng., ed. P.D. Spanos, pp. 144-147, ASCE Publication, New York, 1988. 9. Lafe, O.E. & Cheng, A.H.D. A perturbation boundary element code for steady-state groundwater flow in heterogeneous aquifers, Water Resour. 7&%., 1987,23(6), 1079-84. 10. Lafe, O.E. & Cheng, A.H.D. A stochastic boundary element method for groundwater flow with random boundary conditions and recharge, Betech
Boundary Elements 523 '89 Conference, Windsor, Canada, Computational Mech. Publications, Southampton, 1989. 11. Liggett, J.A. & Liu, P.L.F. The Boundary Integral Equation Method for Porous Media Flow, Allen & Unwin, London, 1983. 12. Durbin, F, Numerical inversion of Laplace transforms : an efficient improvement to Dubner and Abate method, Computer J., 1974, 17, 371-76. 13. Manolis, G.D. & Karak6stas, C.Z. Diffusion of pollutants in randomly structured soil through groundwaterflow, (in Greek), Final report to Greek General Secretariat for Research and Technology, Thessaloniki, 1995. y = h. V b q = 0 v c h = h A D /v%/yyy/c/x/%^/yy%x/yyyy/v%/c/yx/yy/'////// q = 0 Figure 1 : Confined aquifer : configuration. 100000.00 200000.00 300000.00 400000.00 0.00 Time (sec) (a) 0.10 0.15 Time (sec) (b) Figure 2 : Mean (a) long - term solution for T= 400000 sec and (b) short term solution for T= 0.25 sec.
524 Boundary Elements 0.01 0.02 0.03 0.04 Time (sec) (a) (b) Figure 3 : Variance response for rectangular aquifer, f(t)= H(t-O) and T=0.05 sec for (a) piezometric head and (b) flux. «4.00E-6-0.00 1.00 2.00 3.00 Time (sec) (a) 4.00 5.00 0.00 1.00 2.00 3.00 Time (sec) (b) Figure 4 : Variance response for rectangular aquifer, f(t)= (a) piezometric head and (b) flux. and T=5.0 sec for