An accurate simulator of compressible flow in porous media with wells

An accurate and efficient simulator is developed for compressible flow porous media with wells. An Eulerian-Lagrangian localized adjoint method (ELLAM) is used to solve the transport partial differential equation for concentration, while a mixed finite element method (MFEM) is used to solve the pressure equation for the pressure and mass flow rate. The ELLAM-MFEM simulator symmetrizes the transport equation, eliminates nonphysical oscillation and/or excessive numerical dispersion in many large-scale simulators in industrial applications, and conserves mass. It minimizes the numerical difficulties occurring in standard methods caused by differentiation of the pressure and then multiplication by rough coefficients. Computational experiments show that the ELLAM-MFEM simulator can accurately simulate incompressible and compressible flows in porous media with wells, large mobility ratios, discontinuous permeabilities and porosities, and anisotropic dispersion in tensor form, even if very large time steps and spatial grids are used. 1 A MATHEMATICAL MODEL The objective of subsurface fluid flow modeling is to simulate complex fluid flow processes occurring in subsurface porous media sufficiently well to optimize the recovery of hydrocarbon or to accurately predict and effectively remediate the contamination in groundwater transport processes. In order to do this, one must build mathematical models to describe the essential phenomena and the fundamental laws, and design numerical methods to discretize these models and to represent the basic features as well as possible without introducing spurious nonphysical phenomena. 1.1 Conservation of mass Let p(x; t) and u(x; t) be the pressure and the Darcy velocity of a fluid mixture, and (x; t) be the concentration of an invading fluid or specified solute/solvent in the fluid mixture. The equation of mass conservation for the fluid mixture and Darcy’s law lead to the following coupled system of partial differential equations that describes fluid flow processes in a porous medium reservoir with injection and production wells (Bear 1979; Ewing 1984; Peaceman 1977) t( ) + r ( u) = q; u = K (rp grd); x 2 ; t 2 (0; tf ℄: (1) In many cases, the thickness of the medium is significantly smaller than its length and width. In this case, it is reasonable to average the medium properties vertically and to assume IR2 with a nonuniform local elevation. K(x) is the permeability tensor of the medium, ( ) is the concentration-dependent viscosity of the fluid mixture, which is determined by some mixing rule, for example, ( ) = o[(1 ) +M 1 4 ℄ 4 (2) where o is the viscosity of the resident fluid and M is the mobility ratio. is the density of the fluid mixture,