Numerical methods for a model for compressible miscible displacement in porous media

Abstract

Abstract. A nonlinear parabolic system is derived to describe compressible miscible displacement in a porous medium. The system is consistent with the usual model for incompressible miscible displacement. Two finite element procedures are introduced to approximate the concentration of one of the fluids and the pressure of the mixture. The concentration is treated by a Galerkin method in both procedures, while the pressure is treated by either a Galerkin method or by a parabolic mixed finite element method. Optimal order estimates in I? and essentially optimal order estimates in Lx are derived for the errors in the approximate solutions for both methods.Introduction. We shall consider the single-phase, miscible displacement of one compressible fluid by another in a porous medium under the assumptions that no volume change results from the mixing of the components and that a pressure-density relation exists for each component in a form that is independent of the mixing. These equations of state will imply that the fluids are in the liquid state. Our model will represent a direct generalization of the model [3], [4], [7] that has been treated extensively for incompressible miscible displacement.The reservoir ti will be taken to be of unit thickness and will be identified with a bounded domain in R2. We shall omit gravitational terms for simplicity of exposition; no significant mathematical questions arise when the lower order terms are included.Let c¡ denote the (volumetric) concentration of the /th component of the fluid mixture, i= 1,...,«.We assume that a density pi can be assigned to the /'th component that depends solely on the pressure p; moreover, we shall take this equation of state in the form

Theory and Application of Characteristic Finite Difference Fractional Step Method of Capillary Force Enhanced Oil Production

“…This combination technique can also overcome numerical oscillation and dispersion, and can improve greatly the computational stability and accuracy. Douglas and other scholars presented a mathematical model of slight compression, numerical method and theoretical analysis for two-dimensional compressible displacement problem under periodic assumption and began a modern numerical model research (Douglas, Roberts, 1983;Yuan, 1992Yuan, , 1993Ewing, 1983). The authors dropped the period condition, gave a new modified characteristic finite difference algorithm and finite element algorithm, and derived optimal order error estimates in L 2 -norm (Yuan, 1994Axelsson, Gustafasson, 1979;Ewing, Lazarov, et al, 1994.…”

“…Note that numerical simulation is considered for the flow of interior region and the boundary condition affects the flow weakly, so the periodic assumption is reasonable and related contents can be found in the references (Douglas, 1983;Douglas, Russell, 1982;Douglas, 1981;Ewing, Russell, Wheeler, 1984;Douglas, Roberts, 1983). Then The boundary value conditions of no permeation case can be dropped.…”

A stabilized mixed discontinuous Galerkin method for the incompressible miscible displacement problem

“…The more popular approach in application so far has been based on the mixed formulation. In a previous work, Douglas and Roberts [1] presented a mixed finite element (MFE) method for the compressible miscible displacement problem. For the Darcy flow, Masud and Hughes [2] introduced a stabilized finite element formulation in which an appropriately weighted residual of the Darcy law is added to the standard mixed formulation.…”

Global weak solutions of a three-dimensional flow model for a multi-species mixture in porous media

“…The differential system describing the motion of a mixture consisting of n#1 chemical species is given in [6,7,14,16]. If we neglect the molecular diffusion, the dispersion and the gravitational terms, the differential system has the following form:…”

“…In [4], we have extended all these results for the n#1 multispecies mixture. For numerical approximations of miscible displacements, when convection essentially dominates diffusion, we refer to [7,9,15]. This paper is organized as follows.…”