A Semi-Implicit Formulation for Compositional Simulation of Fractured Reservoirs
Authors: Arana-Ortiz Victor H. , Rodriguez Fernando 
Published: December 2007

Abstract
In this paper we present a Semi-Implicit formulation for the numerical simulation of compositional flow in dual-porosity, naturally fractured reservoirs. Newton's method is applied on the non-linear difference equations that result from treating compositions explicitly in the flow terms of the fracture equations, one iteration behind. Matrix equations are treated in a fully-implicit fashion.
The linearized system of equations. 2x(2nc+6) per gridblock, obtained from the Semi-Implicit formulation possess a peculiar matrix structure that allows its reduction to three equations in three unknowns per gridblock, regardless of the number of hydrocarbon components, nc, thus providing the optimal solution of the matrix problem. The size of the reduced system is similar to that obtained in the case of the multiphase flow of beta type fluids in non fractured reservoirs.
The Semi-Implicit formulation possess an intermediate level of implicitness between the IMPECS and Fully-Implicit methods, and provides a convenient approach for the simulation of compositional flow in naturally fractured reservoirs when IMPECS method does not provide stable solutions and the Fully-Implicit method results too costly.
Applications to the simulation of a well test of a gas-condensate well and the natural depletion and gas cycling of a volatile oil reservoir are presented.

Introduction
Naturally fractured reservoirs contribute worldwide in a large extent to the total production of oil and gas. Predicting the behavior of compositional naturally fractured reservoirs has represented a challenge for the reservoir engineer in the last years. Many improvements have been developed since the pioneering works of Kazemi et al. and Thomas et al. on the simulation of naturally fractured reservoirs and since the works of Kazemi et al. and Fussell and Fussell on the simulation of compositional reservoirs.
Regarding the fractured reservoir problem, emphasis has been placed on the study and modelling of the various processes that occur in the matrix to fracture transfer of fluids. Gravity segregation and reinfiltration of oil in the gas zone have called the attention of several authors.
On the compositional reservoir flow problem, emphasis has been placed on looking for optimal ways to solve the non-linear sets of flow difference equations. In this regard, Branco and Rodriguez presented a semi-implicit approach to solve the compositional flow problem.
The compositional and the naturally fractured reservoir flow problems are by themselves unique, in the sense that they produce large matrix problems when solved in a fully-implicit manner. In the compositional problem, the computer requirements increase with the number of components considered in the hydrocarbon mixture.
A paper that addresses the numerical simulation of compositional flow in dual-porosity naturally fractured reservoirs, based on a fully-implicit formulation, was presented by Coats, in 1989.
Seeking for alternative less expensive formulations, in this paper we present a semi-implicit method for compositional fractured reservoir simulation, which is an extension of Branco and Rodriguez paper.

The Mathematical Model
The equations that govern the multiphase compositional flow in fractured assuming a dual-porosity model are given elsewhere. They account for isothermal flow of nc components and water in both fractures and matrix-blocks. The nc components form a two phase hydrocarbon mixture, oil and gas, in a continued state of equilibrium at all points in the reservoir. The water phase and the nc components are considered mutually insoluble. Under these assumptions, the flow equations, written in finite difference form, are the following: P. 345