Time-Dependent Shape Factors for Interporosity Flow in Naturally Fractured Gas-Condensate Reservoirs

Abstract

A rigorous interporosity flow equation incorporating a time-dependent shape factor is derived and validated for improved dual-porosity modeling of naturally fractured gas-condensate reservoirs. The equation expresses theinterporosity molar rate in terms of the pseudo-pressure gradient in thematrix, fracture surface area, matrix permeability, and a variable matrix-blockshape factor. This approach can accommodate the flow directed from matrix tofractures in a simulator that represents the permeability of the interconnected fractures by a tensor inside each grid-block. This feature distinguishes thismodel from the popular sugar-cube approach to modeling naturally fracturedreservoirs. Compositional simulation is performed to verify the flow equation using thetime-dependent shape factor. Numerical experiments with various matrix-blocksizes indicate that the shape factor varies with time but converges to values derived by Lim and Aziz (1995). The average matrix-pressure location in the matrix block shifts from near the fracture face to the block center as thefluid flows from the matrix into the fracture. This phenomenon indicates that neglecting the time dependency of the shape factor can introduce significant errors in numerical simulation of naturally fractured reservoirs. The time dependency can contribute significantly to fluid production once the pressure front moves through the reservoir along the highly permeable fracture network.This phenomenon is not considered in present commercial simulators that usepseudo-steady state factors. The model equations are expressed in dimensionless form so they can bereadily integrated into current simulators. The dimensionless time used in the proposed method includes pseudo-functions that capture multiphase effects ofthe gas-condensate systems. Applications to single- and multiphase black-oils ystems are also discussed. Introduction Numerical simulation of naturally fractured reservoirs has received significant attention and its application has increased in recent years withthe advent of highly efficient computers. Much of the research on naturally fractured reservoir modeling has focused on accurately representing thematrix-fracture fluid transfers. Various mechanisms, including gravity andcapillary effects,1–3 reinfiltration and capillary continuity of the matrix blocks,4–7 and cocurrent/counter current imbibition phenomena,8–10 have been extensively investigated. However, in spiteof the great level of current model sophistication, the highly anisotropic andheterogeneous nature of a fractured formation makes fractured reservoir modeling a challenging task, frequently with uncertain results inforecasting. Typically, numerical simulation of naturally fractured reservoirs assumesthere are two continua, matrix and fractures, within each grid-block. The flow equations are written for each system with a matrix/fracture transfer function to relate the loss or gain of matrix fluids to or from the fracture. For single-phase fluid flowing through an interconnected fracture system, thefollowing governing equation applies: Equation 1 where the fluid transfer rate per unit volume of rock, q, is commonly calculated as a function of the pressure difference between the matrix and fracture systems, matrix flow capacity and matrix geometry considered through aconstant shape factor.

Matrix–fracture transfer shape factor for modeling flow of a compressible fluid in dual-porosity media

“…These include single-phase models, which are used in well test analysis, and a multiphase approach, which is applied in multiphase flow simulation in fractured media [27]. In this section, a brief review of the relevant literature dealing with the single-phase shape factor for slightly compressible fluids is presented.…”

Derivation of the Kazemi–Gilman–Elsharkawy Generalized Dual Porosity Shape Factor

“…Otherwise, the time depending Block-geometry function must be considered and a function cannot be named factor. Nevertheless, some authors (Penuela et al 2002;Rangel-German and Kovscek 2003) published such functions calling them time-dependent shape factors. The supposition that the exchange term for all possible physical situations will depend on the geometry of the discontinuous element (the matrix) of the model was postulated from the beginning, at first time by Warren and Root (1963).…”

“…Starting from a multi phase, immiscible but compressible fluid formulation the matrix-fracture fluid exchange will be derived using the Control volume finite difference discretization scheme on irregularly shaped matrix blocks formed by Voronoi polyhedra. Some authors (Penuela et al 2002 andRangel-German andKovscek 2003) refer to Kazemi's (1976) shape factor, which can be easily derived from the generalized Kazemi et al (1992) shape factor, as a single-phase solution. Nevertheless, the presented derivation is based on a two-phase formulation to ensure consistency with the two-phase (water and oil) imbibition experiments of Mattax and Kyte (1962).…”

Three-dimensional and two-phase numerical simulation of fractured dry gas reservoirs

“…An experimental method was developed which relative permeability around well in steady-state, high-pressure and highvelocity conditions was determined (Mott 2000). Effect of the capillary number and the non-Darcy flow on phase's relative permeability in gas condensate reservoir were investigated (Penuela et al 2002). Equations of gas injection into gas condensate reservoir were solved, and MMP was determined (Seto et al 2003).…”