Analytic solution of spatially discretized groundwater flow equations

Abstract

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 [ dc/dt ] + f = 0, where R and G are square matrices, c and f 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 f 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 solution, R and f 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 =∞) for both the confined and the water table cases but to take considerably less time to evaluate c ( t ) for the water table case.