Theory and verification of residual flow procedure for 3-D free surface seepage

Abstract

The Residual Flow Procedure (RFP) in conjunction with the finite difference method was first proposed and used for solution of free surface seepage problems in river banks t . Subsequently, it was developed and used with the finite element method for two-dimensional steady and transient free surface flow problems 2,3. In contrast to the variable domain or mesh (VM) procedures 4-6, Fig. l(a), in which the finite element mesh for the saturated domain below the free surface is modified through iterative procedures, the RFP involves fixed domain or invariant mesh (IM), Fig. l(b), for the entire domain containing both saturated and partially saturated zones separated by the free surface. In the RFP, the governing equations are first assumed to be valid for the entire domain, which is treated as saturated. During the iterative solutions, separation of the saturated and unsaturated zones with different permeabilities leads to the residual or correction load vector. This procedure thus is different from other invariant mesh procedures such as that proposed by Neumann v, and variational inequality methods 9 t2. It may be mentioned, however, that Westbrook t3 has shown that the algorithms resulting from the RFP and the variational inequality method proposed by Alt 14 are essentially similar. The methods used by Cathie and Dungar is, Bathe and Khosgoftaar TM and Lacy and Prevost Iv are considered similar to the RFP. Some of the attributes and advantages of the RFP are: (1) it involves invariant mesh which avoids the need of mesh modifications, (2) it permits use of arbitrary shapes of (isoparametric) elements, whereas the inequality methods may often be found difficult to implement for such shapes, (3) it can accommodate variable material properties whereas in the VM schemes they can involve considerable difficulties, (4) it can incorporate partially saturated zones, and (5) it is found to be relatively economical; e.g., in comparison with the inequality methods ~ s.~9, which is shown subsequently in this paper. The major new contributions in this paper are: