Numerical Simulation of Water Injection in Fractured Media Using the Discrete-Fracture Model and the Galerkin Method
- Mohammad Karimi-Fard (Reservoir Engineering Research Inst.) | Abbas Firoozabadi (Reservoir Engineering Research Inst.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- April 2003
- Document Type
- Journal Paper
- 117 - 126
- 2003. Society of Petroleum Engineers
- 4.1.2 Separation and Treating, 4.1.5 Processing Equipment, 5.4.1 Waterflooding, 5.5 Reservoir Simulation, 5.3.2 Multiphase Flow, 5.3.1 Flow in Porous Media, 6.5.2 Water use, produced water discharge and disposal
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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 used mainly for sugar-cube representation of fractured media. In such a representation, the transfer function between the fracture and the matrix block can be calculated readily for water-wet media. For a mixed-wet system, the evaluation of the transfer function becomes complicated because of the effect of gravity.
In this work, we use a discrete-fracture model in which the fractures are discretized as 1D 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.
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 models1-3 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 disconnected 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 owing to the effect of gravity.
The single-porosity model provides the accuracy, but it is not practical because of the 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 1D entities. The heterogeneity is accounted accurately, and there is no need for the transfer function; it also can 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 et al.5 were among the early authors to use 1D 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 with 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.
For two-phase flow with capillary pressure, very limited work can be found in the literature using the discrete-fracture model. The work of Bourbiaux et al.6 is based on finite-volume discretization; they use the so-called joint-element technique to represent the fractures. The same approach is used by Granet et al.7 for single-phase flow. The work of Kim and Deo8,9 is based on the finite-element method and the use of the superposition principle to couple the two media. Their approach is similar in principle to the work of Noorishad and Mehran4 and Baca et al.5 Kim and Deo employ a fully implicit formulation and solve the set of nonlinear differential equations using the inexact Newton method. The numerical results presented by Kim and Deo show that the agreement between the discrete-fracture model from the finite-element method and the single-porosity results from the conventional approach are sometimes not close.
The main advantage of the finite-element method in reservoir simulation is the possibility to discretize a geometrically complex reservoir with an optimal use of mesh points. Mesh refinement around a well can be limited to the well zone, which may not be practical in the standard finite-difference method. In a recent work,10 multiblock gridding is used in the finite-difference discretization to represent a complex geometry. This approach, although effective, may cause the loss of the simplicity and the efficiency of the standard finite-difference method.
The standard finite-element approach is more complicated than the finite-difference method and numerically less efficient. These disadvantages can be overcome by using a simplified finiteelement approach. Dalen11 presents a simplified approach for 2D flow in homogeneous media.
In this work, we use a Galerkin variational method with the finite-element discretization. The variational method has been used successfully in single-phase flow12-14 and two-phase flow.15-19 Our approach is similar to the work of Kim and Deo.9 In our work, we use the implicit pressure/explicit saturation method (IMPES), whereas Kim and Deo used a fully implicit approach. We demonstrate excellent agreement between the discrete-fracture model and fine-grid simulation. An extensive set of examples, including a 25×25-m cross section with a set of discrete fractures, is studied for water injection in water-wet and mixed-wet systems and for the effect of fracture capillary pressure. Currently, the cross-section example cannot be modeled in a finite-difference simulator.
In the following, we first present the mathematical formulation of the problem, followed by the numerical approach to solve the system of equations that defines the problem. In the Results section, we first present the effect of gridding on numerical dispersion and then validate the discrete-fracture model. We also include the numerical results from water injection in complex fractured media of different dimensions and wettability.
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