Self-Adaptive Galerkin Methods For One-Dimensional, Two-Phase Immiscible Flow
Authors: Douglas Jim , Darlow Bruce L. , Wheeler Mary , Kendall Richard P. 
Published: December 2007

Abstract
The Galerkin methods for immiscible displacement are based on the weak formulation given by
(1)
for test functions v(x). The terms involving sigma g represent an artificial diffusivity that is required to enforce the entropy condition that selects the physical solution from the infinite set of solutions of physical solution from the infinite set of solutions of the conservation law St + f(S)x = 0. Two conceptionally different Galerkin methods have been studied. The first is based on the use of continuous, piecewise-polynomial functions for trial and test spaces. The piecewise-polynomial functions for trial and test spaces. The second method includes a penalty term that can be used to force the approximate solution to lie between C and C inclusively. This additional flexibility permits a better characterization of the solution of the permits a better characterization of the solution of the perturbed conservation law. In both methods the perturbed conservation law. In both methods the approximate location of the saturation front is determined by the method of characteristics. This permits a substantial reduction in the work required to advance the solution from one time step to the next by making possible self-adaptive refinement of the mesh in the neighborhood of the solution front. Finally, the convergence of the Galerkin approximations to the physical solution of the conservation law is physical solution of the conservation law is demonstrated experimentally.

Introduction
The use of finite element methods to solve immiscible flow equations is not novel. The pertinent literature, while not extensive, can be traced back to the late sixties. The authors of this literature have generally concluded that the immiscible flow equations can be efficiently solved by finite element methods, particularly when the solutions of these equations are sufficiently smooth. They have also noted that the finite element approach generally requires more work than conventional finite difference methods. We shall demonstrate that the use of a self-adaptive mesh, which is natural in the finite element setting, permits a substantial reduction in the work required to advance the solution from one time step to the next without sacrificing accuracy.
The principal problem to be confronted here, however, is the one created by vanishingly small capillarity. In this case the Buckley-Leverett equation is modeled by a hyperbolic conservation law. The solution of the boundary-initial value problem for such a hyperbolic partial differential equation need not be continuous. It is well known that discontinuous solutions are never unique. Spivak, et al introduced artificial dissipation to enforce the entropy condition and insure a unique solution. However, they did not relate this artificial dissipation to the level of accuracy of their Galerkin approximation. These will be explicitly, if not entirely rigorously, related in a subsequent section of this paper.
Equally important is the formulation of the Galerkin method itself. The procedure must be adapted to the problem at hand. In the present application it must be capable of producing approximations that reflect the abrupt changes in local smoothness that characterize the perturbed hyperbolic Buckley-Leverett equation.

THE BUCKLEY-LEVERETT EQUATION
In the absence of capillarity, the one-dimensional Buckley-Leverett equation for incompressible flow can be written as (in consistent units)
(2)
with boundary and initial conditions S(0,t) = 1, S(x,0) = 0, respectively. The symbols above are defined in the nomenclature immediately preceeding the references. Specific values of these quantities used in the figures to follow are also summarized there; they are essentially those used by Spivak, et al.