Numerical Simulation of Water Injection in 2D Fractured Media Using Discrete-Fracture Model

Abstract

Numerical simulation of water injection in discrete fractured media with capillary pressure is a challenge. Dual-porosity models in view of their strength and simplicity can be mainly used for sugar-cube representation of fractured media. In such a representation, the transfer function between the fracture and the matrix block can be readily calculated for water-wet media. For a mixed-wet system, the evaluation of the transfer function becomes complicated due to the effect of gravity. In this work, we use a discrete-fracture model in which the fractures are discretized as one dimensional entities to account for fracture thickness by an integral form of the flow equations. This simple step greatly improves the numerical solution. Then the discrete-fracture model is implemented using a Galerkin finite element method. The robustness and the accuracy of the approach are shown through several examples. First we consider a single fracture in a rock matrix and compare the results of the discrete-fracture model with a single-porosity model. Then, we use the discrete-fracture model in more complex configurations. Numerical simulations are carried out in water-wet media as well as in mixed-wet media to study the effect of matrix and fracture capillary pressures. Introduction Numerical simulation of oil recovery from fractured petroleum reservoirs remains a challenge. The heterogeneity of the porous media and the connectivity of the fractures have a significant effect on two-phase flow with capillary pressure and gravity effects. Dual-porosity models1,2,3 have been used to simulate two-phase flow with connected fractures; the sugar-cube model configuration has been studied in such a model. This approach, although very efficient, suffers from some important limitations. One limitation is that the method cannot be applied to discon-nected fractured media and cannot represent the heterogeneity of such a system. Another shortcoming is the complexity in the evaluation of the transfer function between the matrix and the fractures. In fact, in mixed-wet fractured media, a dual-porosity model may lose accuracy due to the effect of gravity. The single-porosity model provides the accuracy, but it is not practical due to very large number of grids. A large number of grids is required because of two different length scales (matrix size and fracture thickness). A geometrical simplification of the single-porosity model can make it applicable to larger configurations. The simplified model is called the discrete-fracture model. In this model, the fractures are discretized as one dimensional entities. The heterogeneity is accounted accurately and there is no need for the transfer function; it can also be applied to both water-wet and mixed-wet media. The discrete-fracture model was first introduced for single-phase flow. Noorishad and Mehran4 and Baca, Arnett and Langford5 were among the early authors to use one dimensional entities to represent fractures. These authors used finite element formulation to simulate 2D single-phase flow through fractured porous media. Noorishad and Mehran4 solved the transient transport equation in fractured porous media using an upstream finite element method to avoid oscillation for convective-dominated flow. Baca et al.5 considered a 2D single-phase flow with heat and solute transport. The two media (matrix and fractures) are coupled using the superposition principle.

Control‐volume method for numerical simulation of two‐phase immiscible flow in two‐ and three‐dimensional discrete‐fractured media

“…[ Deo, 1999, 2000;Karimi-Fard and Firoozabadi, 2003], implying that x m = x f = x, which is true only for some specific cases, as we will see later. Equation (16) lacks a relationship between the matrix and fracture variables based on physical grounds.…”

“…It also reduces computational time by orders of magnitude in 2-D and 3-D as pointed out earlier. Despite all these advantages, there is no loss of accuracy when the results from the discrete fracture model are compared with the full n-dimensional fracture flow [Karimi-Fard and Firoozabadi, 2003]. The discrete-fracture model has been employed by several authors to develop codes for multiphase flow in fractured porous media Deo, 1999, 2000;Bastian et al, 2000;Karimi-Fard and Firoozabadi, 2003;Geiger et al, 2003].…”

“…[13] The FU-FEM has been recently employed for 2-D numerical simulation of two-phase immiscible flow in the context of discrete-fractured media Deo, 1999, 2000;Karimi-Fard and Firoozabadi, 2003]. However, Kim and Deo [1999] reported the failure of the method when extended to 3-D problems.…”

“…These authors did not use the concept of crossflow equilibrium, and therefore the computational speed for fractured media may be very low. They also neglected capillary pressure, despite its significance in immiscible fluid flow in fractured porous media [Terez and Firoozabadi, 1999;Karimi-Fard and Firoozabadi, 2003].…”

“…In section 2 we present a mathematical formulation for twophase incompressible flow in fractured media, clearly establishing physically based relations between matrix and fracture variables. This formulation has not been presented in the previous related works [ Deo, 1999, 2000;Bastian et al, 2000;Karimi-Fard and Firoozabadi, 2003;Geiger et al, 2003]. In section 3 we provide an efficient procedure to numerically solve the two-phase flow equations in fractured media using the discrete-fracture model within the framework of the CV method.…”