Support-Operators Method in the Identification of Permeability Tensor Orientation

Abstract

The dependence of rock permeability on direction, or permeability anisotropy, is confirmed by numerous field examples. Therefore, the ability to carry out a numerical simulation of an anisotropic reservoir is very important. The support-operators method provides a conservative discretization scheme, allowing one to solve nonisotropic problems on a grid of practically arbitrary structure. Moreover, a discretization designed with the supportoperators method provides a natural and convenient way of deriving and solving the adjoint system for evaluation of the gradient and second-order differential in inverse problems. After a theoretical introduction into the support-operators method, we consider an illustrative parameter-identification problem. More specifically, we evaluate the orientation angle of a nonisotropic permeability tensor in a horizontal reservoir. We assume that the principal permeabilities near a cored or otherwise logged well are already known. To accomplish this task, we need pressure measurements in monitoring wells. We consider both rectangular and curvilinear grids. In either case, the orientation angle has been recovered with a high accuracy.

A Mimetic Finite-Volume-Discretization Operator for Reservoir Simulation

“…In the petroleum engineering literature, the first implementation of the support-operators method is demonstrated for the identification of permeability tensor orientation. 38 In our approach for solving multiphase multicomponent reservoir simulation equations, flow and transport mechanisms are separately accounted for in terms of a parabolic pressure and a set of nearly hyperbolic mass balance equations. In the case of geologically complex reservoir models, coefficients of the pressure equation often exhibit anisotropic and highly discontinuous characteristics.…”

A Second-Order Mimetic Approach for Tracer Flow in Oil Reservoirs

“…A comparison of equations (5) and (6) shows that a mimetic grid has the same characteristics of a block center grid at inner nodes [12] . However, it has additional nodes at the boundaries to evaluate gradient and the main dependent variable.…”

“…It could be said that mimetic methods are an improved version of finite differences methods that provides an alternative choice to finite element methods in the case of fluid flow problems. Second order mimetic schemes have been developed since mimetic method's earlier years, a full account of them can be found in the literature [1,2,3] and their only application to petroleum engineering [5] , up to year 2006. However, these second order versions have the limitation that their discretizations at the boundary produce lower orders of convergence than in the interior points.…”