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The conjugate-gradient method can solve efficiently and accurately finite-difference approximations to the ground-water flow equation. An aquifer-simulation model using the conjugate-gradient method was applied to a problem of ground-water flow in an alluvial aquifer at the Rocky Mountain Arsenal, Denver, Colorado. For this application, the accuracy and efficiency of the conjugate-gradient method compared favorably with other available methods for steady-state flow. However, its efficiency relative to other available methods depends on the nature of the specific problem. The main advantage of the conjugate-gradient method is that it does not require the use of iteration parameters, thereby eliminating this partly subjective procedure. INTRODUCTION Aquifer-simulation models commonly use finite-difference methods to solve the partial differential equation that describes ground-water flow. Several numerical procedures are available and commonly used to solve the system of finite-difference equations generated by approximating the flow equation. Trescott and others (1976) include the option of using LSOR (line successive overrelaxation), ADIP (alternating direction implicit procedure), or SIP (strongly implicit procedure) in their well-documented program for aquifer simulation. However, the efficiency of these numerical methods is sensitive to the choice of the iteration (or acceleration) parameters. The optimum values for these parameters depend on the problem and are not estimated easily. The method of conjugate gradients offers an alternative procedure for solving iteratively large systems of simultaneous linear equations (Beckman, 1960). This method requires no iteration parameters, and it is relatively easy to implement. The purposes of this report are to: (1) Describe the derivation and application of the method, including the use of matrix scaling to increase the efficiency of the solution; (2) demonstrate the application of the method to a complex field problem; and (3) compare the results with other available methods. GROUND-WATER FLOW EQUATION The partial differential equation describing ground-water flow in a confined aquifer in two dimensions may be written as: _JL CT QL\ + -A CT QL\ = G QL 3x CTxx 8x} + 8y C yy 8y} b 3t where x and y = the coordinate axes, L; 2 T and T = the components of transmissivity tensor, L /T; xx 77 h = the hydraulic head, L; s = the storage coefficient (dimensionless); t = the time, T; and W(x,y,t) « the volume flux per unit area (positive for outflow and negative for inflow), L/T. Equation 1 assumes that the prinicipal components of the transmissivity tensor are alined with the coordinate axes x and y. A standard approach to solving equation 1 is to approximate h at a discrete set of points arranged in a rectangular grid. The continuous derivatives then are replaced by finite-difference approximations. Suppose we approximate h at N evenly spaced points (or nodes) where N=WT?. The grid spacing is given by Ax = v -v t-*A A * A i Ay = y .-y . J J ft J si \ (2a) (2b) ordered from left to right and bottom to top. Letting h..(t) = h(x.,y.,t), ' we can replace equation 1 by

Application of the conjugate-gradient method to ground-water models