This paper compares the efficiency of seventeen different methods for the solution of the simultaneous nonlinear finite difference approximating equations for groundwater flow in a watertable aquifer in three dimensions. Solution methods include those of Newton, Picard, and generalized linear methods. The Newton and Picard iteration methods were implemented by making use of several different linear methods including successive overrelaxation, the strongly implicit procedure, and eight different preconditioned conjugate gradient methods. The best methods were found to be those using Picard iteration implemented with the preconditioned conjugate gradient method.

A numerical code is developed for the solution of the three‐dimensional steady state groundwater flow equation in which groundwater density varies with spatial position and is treated as known spatially dependent parameter. The integrated finite difference grid elements of the numerical code are rectangular when viewed along the vertical direction, but their top and bottom surfaces parallel or are coincident with the interfaces between the geologic strata within the solution region. Conditions that are sufficient for the accuracy of the code are derived. Approximate solutions for pressure head, as determined from the code, are compared for accuracy with several exact analytic test solutions.

This paper compares the efficiency of the strongly implicit procedure (SIP) and the incomplete Cholesky‐conjugate gradient method (ICCG) applied to the solution of the finite difference approximating equations for groundwater flow. Results for five isotropic two‐dimensional test problems are presented. Three are linear confined aquifer problems, and two are nonlinear water table aquifer problems; in all but one of the test problems the aquifer was considered to be nonhomogeneous. Both SIP and ICCG as applied to water table aquifer problems make use of iteration parameters. These were varied for each of the test problems to reduce the amount of computational work needed to find a solution. ICCG was usually substantially more efficient than SIP when applied to the confined aquifer test problems. For the water table aquifer test problems, SIP and ICCG performed equally well.

Two geostatistical approaches for the estimation of hydraulic conductivity and hydraulic head from hydraulic conductivity and hydraulic head measurements are developed for two‐dimensional steady flow with sinks. For both approaches the field of the logarithm of hydraulic conductivity (log‐conductivity) is represented as a random field with mean θ1+θ2x+θ3y where x and y denote Cartesian coordinates, variance σ2, and covariance σ2, exp (−αR), σ2, exp (−αR2), or, σ2ΔRr where α, Δ, and r are constants and R is separation distance. The first approach uses linearization of the discretized flow equations to allow the construction of the joint covariance matrix of the hydraulic head and log‐conductivity measurements as functions of the parameters θ, θ2, θ3, σ2, and α or Δ. It then uses maximum likelihood estimation to obtain these parameters and also a parameter associated with log‐conductivity measurement error. Having found values for the parameters, it then uses kriging to form predictors for log‐conductivity and hydraulic head from measured values of hydraulic conductivity and hydraulic head. The second approach uses kriging to form a parameter‐dependent predictor for log‐conductivity from measured hydraulic conductivity, and then uses this predicted log‐conductivity placed into the discretized flow equations to compute hydraulic head. The parameters are determined by the minimization of the sum of the squares of the difference between the measured and computed hydraulic heads. A third approach simply allows the hydraulic conductivity field to be a step function with a different value for hydraulic conductivity assigned to each of several chosen regions in the two‐dimensional aquifer. The assigned hydraulic conductivities are determined by the minimization of the sum of the squares of the difference between the measured and computed hydraulic heads. The three approaches are tested for hydraulic head prediction accuracy on two generated test problems, one of which is statistically generated, and also on a field problem. The third approach, despite its simplicity, performs as well or better than the other approaches.

The Galerkin procedure when it is applied to the equation for horizontal two-dimensional flow of groundwater in a nonhomogeneous isotropic aquifer generates approximating equations of the following form: Rc • G[do/dt] • • = O, where R and G are square matrices, o and • are column matrices, and t is time. This matrix equation is decoupled and solved for the unknown column matrix c(t). In the case of a confined aquifer that approaches a steady state solution, R, G, and • are constant. An analytic solution to the matrix equation for c(t) is given for this case. In the case of a water table aquifer that approaches a steady state solutio.n, R and • are explicitly dependent on c(t), and G is constant. For this case, c(t --•) is found in a simple iterative manner, and an iterative procedure is given to approximate c(t). These methods are compared with the approximate numerical Crank-Nicholson procedure by applying both to a particular problem for which the unknown column matrix c(t) has 49 elements. The Crank-Nicholson procedure is found usually to require less computation time to evaluate c(t) for the confined aquifer case but to give errors for drawdown averaging approximately 10%. The Crank-Nicholson procedure is found to take considerably more computation time to evaluate c(t --o•) for both the confined and the water table cases but to take considerably less time to evaluate c(t) for the water table case. 1097 to find c(t') and c'(t') were approximately 40 sec and 300 sec, respectively. P•EFERENCES Douglas, J., Jr., and T. Dupont, Galerkin methods for parabolic equations, SIAM J. Numer. Anal., 7 (4), 575-626, 1970. Finlayson, B. A., The Method o/ Weighted Residuals and Variational Principles, pp. 16, 29, 35-36, Academic, New York, 1972. Perlis, S., Theory o/ Matrices, pp. 140-142, Addison-Wesley, Reading, Mass., 1958. Pinder, G. F., and E. O. Frind, Application of Galerkin's procedure to aquifer analysis, Water