Improved Calculations for Viscous and Gravity Displacement in Matrix Blocks in Dual-Porosity Simulators (includes associated papers 17851, 17921, 18017, 18018, 18939, 19038, 19361 and 20174 )
Authors: Gilman J.R. , Kazemi H. 
Published: December 2007

Summary
This paper presents two innovations to improve the mathematical representation of viscous and gravity displacement from matrix blocks in dual-porosity simulators. The first improvement accounts for viscous displacement and convective mass transport in matrix blocks caused by potential gradients in the fracture network. Computation time is not significantly increased compared with current dual-porosity simulators. The second improvement provides two computationally efficient methods to refine simulation of gravity displacement in the matrix. Examples show that the improvements are highly desirable in many situations.

Introduction
Simulation of dual-porosity reservoirs has received much attention over the last decade.1–8 The most widely accepted and practical approach has been to treat fractures as the continuum and matrix as discontinuous source or sink terms embedded in the fracture network. In a finite-difference approximation of reservoir flow equations, all rnatrix blocks in a given grid cell are assumed to behave identically. The matrix has high porosity and low permeability; the fractures have very high permeability and low PV. Viscous displacement in matrix blocks caused by pressure (potential) gradient in the fracture network is generally neglected. This paper describes a practical way to correct this problem, and discusses when this correction is important.
In the state-of-the-art dual-porosity simulators, capillary/gravity interaction between fractures and matrix blocks is not rigorously accounted for. Saidi,2 Litvak,3 and Sonier et al.4 simulated gravity effects by segregating all phases in each computational grid and in each timestep. This was done by simple material balance without accounting for the time-dependent nature of gravity segregation as affected by capillary pressure and the vertical transmissibility of the matrix. In this paper we present an example that shows that these approaches can lead to optimistic oil recoveries.
Gilman5 and Pruess and Narasimhan6 have reported a grid refinement scheme to calculate heat transfer and fluid movement in the matrix blocks of a fractured reservoir more accurately. In this paper, we show that this approach of matrix-block grid refinement improves the calculation of gravity displacement.
We also compare dual-porosity and dual-permeability concepts reported by Hill and Thomas.7 The latter concept allows both matrix-to-matrix and fracture-to-fracture flow between gridblocks. Dual-permeability models require much more computing time than the dual-porosity models in which the matrix is discontinuous. In this paper, we used the dual-permeability concept to develop a computationally efficient algorithm to account accurately for gravity effects both in the fracture and the matrix.
We also show to what extent the more relaxed assumptions used in current dual-porosity simulators and the new ideas presented in this paper are compatible with results from fine-grid simulations that more accurately describe the fracture and matrix network.

Flow Equations in Dual-Porosity Simulators
The basic flow equations are represented in the same manner as those used by Gilman and Kazemi.8 For simplicity, we consider only the equations for immiscible flow of insoluble phases. The inclusion of solubility of one phase into another phase does not affect our presentation. The finite-difference form for the fracture isEquation 1
and for the matrix,Equation 2
where tamaf is the rate of fluid transfer between matrix and fracture in a gridblock and is given byEquation 3
The subscript a represents water, oil, and gas phases. Fracture transmissibility, Taf, in the x direction is given byEquation 4
Similar equations are defined for the y and z directions. Upstream mobilities are used in the equations.
The equations with our new modifications will be presented later. For the simulations presented in this paper, single-point upstream relative permeability, without any modifications, is used for finite-matrix flow in Eq. 3 so that the dual-porosity models can be compared with fine-grid single-porosity models that use separate nodes for the fracture and matrix.