Simulation of Naturally Fractured Reservoirs With Semi-Implicit Source Terms
Authors: Rossen R.H. 
Published: December 2007

Abstract
Conventional reservoir simulation techniques prove to be inadequate when applied directly to the prove to be inadequate when applied directly to the study of fractured reservoir systems. Such systems are characterized by extremes in porosity, permeability, and saturation. The vast bulk of the permeability, and saturation. The vast bulk of the reservoir volume is occupied by relatively low-permeability, disjoint matrix blocks of various sizes surrounded by a small volume of high-permeability, interconnected fracture space.
Our approach to this complex problem bas been to treat the matrix blocks as source and sink terms in an otherwise conventional simulation that models only the fracture system. The source/sink terms are functions of matrix rock and fluid properties with fracture saturation and pressure defining the boundary conditions. These functions are derived either by history-matching simulations or independently by laboratory experiments or single matrix-block simulation.
This basic concept of a source/sink treatment is not unique to this work. However, the numerical formulation and implementation of these terms in the fracture simulation offers significant advantages over existing modeling procedures. The fundamental advantage of our approach is that these source terms are handled semi-implicitly in both the pressure and saturation calculations involved in pressure and saturation calculations involved in the fracture simulation. This avoids instability problems that are inherent in a sequential problems that are inherent in a sequential fracture-matrix solution and links more closely the behavior of the matrix and the fracture. Special techniques are developed for modeling the effects of fluid contact movement within a large simulation grid block and for treating receding gas-oil and water-oil contacts. Hysteresis effects are included in matrix blocks that begin to imbibe oil after drainage has begun.

Introduction
Conventional reservoir simulation techniques are not capable of adequately modeling large, naturally fractured reservoir systems. Extreme discontinuities in porosity, permeability, and saturation exist throughout the reservoir. Most of the fluids are found in very low-permeability, disjoint matrix blocks of various sizes, while most of the fluid mobility is in a small volume of high-permeability, interconnected fracture space.
The distribution of fluids within the fracture is usually governed primarily by gravity segregation while the behavior of the individual matrix blocks depends on pressure, fluid environment, and matrix fluid saturation. Fluids can move readily throughout the reservoir in the fracture space, but fluids that reside in matrix rock must enter the fracture to move any great distance. The behavior of individual matrix blocks in response to various drive mechanisms has been studied experimentally by Crawford and Yazdil and has been simulated in two dimensions by Kleppe and Morse and Yamamoto et al. Other investigators have studied the single-phase pressure behavior of fractured reservoirs and its effect on pressure buildup curves. Kazemi investigated single-phase flow in a radial reservoir dominated by horizontal fractures. Closman simultaneously solved equations for flow between matrix and fracture and for flow along the fracture planes for a radial aquifer to develop relationships similar to those developed by van Everdingen and Hursts for water influx.
Simulation of an entire reservoir system with multiple phases further complicates the problem and makes additional simulator modifications necessary. Asfari and Witherspoon have developed a modeling approach for reservoirs with a regular pattern of noncommunicating vertical fractures by pattern of noncommunicating vertical fractures by assigning constant pressures along each fracture. Several investigators have applied finiteelement techniques to the fracture-matrix flow problem. problem. Our approach to this complex problem has been to model the flow in the fracture system and to treat fluid transfer to and from the matrix much as injection and production are modeled in conventional simulators. Transfer of fluid into the fracture will be represented by a "source" term and transfer from the fracture to the matrix will be represented by a "sink" (or negative source) term.
SPEJ
P. 201