Premium
A Finite Difference Method for Unsteady Flow in Variably Saturated Porous Media: Application to a Single Pumping Well
Author(s) -
Cooley Richard L.
Publication year - 1971
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/wr007i006p01607
Subject(s) - mathematics , finite difference method , nonlinear system , finite difference , groundwater flow equation , aquifer , partial differential equation , mathematical analysis , boundary value problem , flow (mathematics) , iterative method , matrix (chemical analysis) , porous medium , rotational symmetry , mechanics , geometry , geotechnical engineering , groundwater flow , porosity , geology , mathematical optimization , physics , materials science , groundwater , quantum mechanics , composite material
Finite difference equations were derived by using the divergence theorem to convert the nonlinear partial differential equation (which approximately describes liquid flow in a variably saturated, elastic porous medium) to an integral equation, and then to integrate around individual mesh volume elements. Original nonlinearity of the differential equation was preserved by keeping saturations and relative conductivities current with hydraulic heads during the iterative matrix solution method. The problem of axisymmetric flow to a water well that penetrates one or more elastic rock units, the upper one of which is unconfmed, provides a convenient vehicle for analysis of the procedural and theoretical study of unconfined and semiconfined flow. Of the three methods tried to solve the matrix equation that resulted from the finite difference equations (which included a form of the direct alternating direction implicit method, the iterative alternating direction implicit method, and the line successive overrelaxation method), the line successive overrelaxation method was the fastest and was selected for use in a general computer program. A comparison with analytical solutions that use Boulton's convolution integral as a velocity boundary condition at the water table for a single aquifer and an aquifer‐aquitard system demonstrates close correspondence of the numerical and analytical solutions, even for a case where the water table is lowered appreciably.