Analytic solution of spatially discretized groundwater flow equations

摘要

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+Gdc/dt +f= 0, whereRandGare square matrices,candfare column matrices, andtis time. This matrix equation is decoupled and solved for the unknown column matrixc(t). In the case of a confined aquifer that approaches a steady state solution,R,G, andfare constant. An analytic solution to the matrix equation forc(t) is given for this case. In the case of a water table aquifer that approaches a steady state solution,Randfare explicitly dependent onc(t), andGis constant. For this case,c(t= ∞) is found in a simple iterative manner, and an iterative procedure is given to approximatec(t). These methods are compared with the approximate numerical Crank‐Nicholson procedure by applying both to a particular problem for which the unknown column matrixc(t) has 49 elements. The Crank‐Nicholson procedure is found usually to require less computation time to evaluatec(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 evaluatec(t=∞) for both the confined and the water table cases but to take considerably less time to evaluatec(t) for the water