Jim Douglas

Derivation of the double porosity model of single phase flow via homogenization theory

A general form of the double porosity model of single phase flow in a naturally fractured reservoir is derived from homogenization theory. The microscopic model consists of the usual equations describing Darcy flow in a reservoir, except that the porosity and permeability coefficients are highly discontinuous. Over the matrix domain, the coefficients are scaled by a parameter

Stabilized mixed methods for the Stokes problem

epsilon

A Survey of Numerical Methods for Parabolic Differential Equations

representing the size of the matrix blocks. This scaling preserves the physics of the flow in the matrix as

The modified method of characteristics with adjusted advection

epsilon

A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods

tends to zero. An effective macroscopic limit model is obtained that includes the usual Darcy equations in the matrix blocks and a similar equation for the fracture system that contains a term representing a source of fluid from the matrix. The convergence is shown by extracting weak limits in appropriate Hilbert spaces. A dilation operator is utilized to see the otherwise vanishing physics in the matrix blocks as

A dual-porosity model for waterflooding in naturally fractured reservoirs

epsilon

Three-phase immiscible displacement in heterogeneous petroleum reservoirs

tends to zero.

Immiscible displacement in vertically fractured reservoirs

SummaryThe solution of the Stokes problem is approximated by three stabilized mixed methods, one due to Hughes, Balestra, and Franca and the other two being variants of this procedure. In each case the bilinear form associated with the saddle-point problem of the standard mixed formulation is modified to become coercive over the finite element space. Error estimates are derived for each procedure.

A Limit Form of the Equations for Immiscible Displacement in a Fractured Reservoir

Publisher Summary The chapter introduces the theoretical view of finite difference methods for approximating the solutions of partial differential equations of parabolic type. A few preliminary definitions and facts about difference analogues of derivatives are first presented. The symbols “u” and “w” will be used to denote the solution of a differential equation and the solution of a difference equation, respectively. Numerical treatment of parabolic differential equations is done by considering the boundary value problem for the heat equation in one space variable. The chapter begins by deriving the backward difference equation and the Crank-Nicolson difference equation. The local error in the time direction is decreased by deriving the Crank-Nicolson difference equation. Crank-Nicolson equations can be applied to problems for which slope conditions are specified at a boundary, but the disadvantage of the Crank-Nicolson method is that greater smoothness is required of the solution of the differential equation to insure convergence. The chapter deals with unconditionally unstable difference equations and higher order correct difference equations, and presents a comparison of the calculation requirements.

Modelling of Compositional Flow in Naturally Fractured Reservoirs

Summary. The MMOC procedure for approximating the solutions of transport-dominated diffusion problems does not automatically preserve integral conservation laws, leading to (mass) balance errors in many kinds of flow problems. The variant, called the MMOCAA, discussed herein preserves the conservation law at a minor additional computational cost. It is shown that its solution, in either Galerkin or finite difference form, converges at the same rates as were proved earlier by Douglnas and Russell for the standard MMOC procedure.