*2.2. Experimental Process*


#### **3. Methodology**

#### *3.1. Characterization of Anisotropic Relative Permeability*

The test process is continuous. After measuring the x direction, the oil is saturated in the x direction, and then the direction is changed to do the water flooding process, so as to ensure that the core porous medium and the oil and water conditions are the same in each test. The main goal of this research is to observe the difference in residual oil saturation after water flooding in different directions. Therefore, minor changes in the irreducible water saturation are ignored.

During the experimental process and numerical simulation research process, we realized that the relative permeability of the XY, XZ, and YZ directions are also important to this experiment. Different inlet and outlet combinations could help to test the relative permeability of the XY, XZ, and YZ directions. For example, if we use the X direction as the injection end, we can use the Y direction or the Z direction as the production end. However, the existing JBN calculation method cannot be used to solve the relative permeability curve of such a combined inject-product method. The automatic history matching method or other methods need to be proposed to solve the relative permeability curve.

In this experiment, the JBN method was used to solve the relative permeability curve. The JBN method is based on the Buckley–Leverett one-dimensional two-phase water flooding front advancement theory, ignoring capillary pressure and gravity, assuming that the two-phase immiscible fluid is incompressible and the oil–water saturation in any cross section of the rock sample is uniform. In the process of water flooding, only when the displacement front breaks through the end, can the relative permeability of the oil and water phase be calculated. For the entire core, the displacement process has a breakthrough time, but for the end face, the flow is pure oil phase before the breakthrough. The saturation in the end face has not changed. After the breakthrough, the process of the end face water saturation from the irreducible water saturation to the maximum water saturation is complete. JBN projects the seepage law of the entire core to the end face. It studies the law of water saturation and oil–water seepage at the end face. The relative permeability calculated by JBN represents the end face, not the entire core.

There is a marked difference between the relative permeability curves derived from the x direction, y direction, and vertical displacements. Different residual oil saturations were obtained for three directions displacements, the residual oil saturation in X direction is 0.56, the residual oil saturation in Y direction is 0.61, the residual oil saturation in Z direction is 0.82. The displacement was more efficient in the vertical than other directions. The greater the angle between the displacement direction and the bedding strike, the lower the residual oil saturation is, the higher the displacement efficiency is, and the relative permeability curve tends towards a rightward shift. The displacement efficiency in the three directions of x, y, and z are 0.664, 0.721, and 0.84, respectively.

#### *3.2. Numerical Simulation with Anisotropic Relative Permeability*

According to the experimental results, JBN (calculation of relative permeability from displacement experiments proposed by Johnson, E.F., Bossler, D.P. and Naumann, V.O.) method was used to obtain the relative permeability curves in three directions. The anisotropic relative permeability is applied to the traditional numerical simulator and the traditional numerical simulator is modified.

In this paper, based on the oil–water two-phase black oil model, the isotropic relative permeability in the water phase governing equation and the oil phase governing equation is replaced by the anisotropic relative permeability, and the new oil phase and water phase motion equations are solved by the finite difference method. There are many numerical simulation methods, such as the finite difference method, finite element method, etc. Different solving methods have great influence on the results of fracture–matrix seepage, different mesh sizes, and poor mesh quality. The stochastic perturbation-based finite element approach proposed by Kaminski, M. can even treat uncertainty in inflow or wall boundary conditions, in parameters of the equations, in profile shape, etc. [34]. The model in this paper is relatively simple, so we chose the traditional finite difference method to solve the problem. The assumptions of the numerical simulation model established in this paper are as follows:


The oil governing equations used in this study are presented in Appendix A, where the derivation process is discussed.

The fully implicit method is selected to ensure accuracy during the calculation. The heterogeneity of the reservoir is usually strong, and the fluid properties are also complex and diverse. In addition, the time span of historical production data is large, and as a result, the stability and robustness of implicit pressure, explicit saturation (IMPES) implementation are not enough to meet the requirements of reservoir numerical simulation [35]. The fully implicit method is very stable and can solve the problems of non-convergence and computational instability in a heterogeneous model.

#### **4. Numerical Simulation Calculating Results and Discussion**

In this study, according to the experimental results, we set up two kinds of numerical simulation models: homogeneous permeability model and heterogeneous permeability model. The numerical simulation calculating results obtained from the new simulator considering anisotropic relative permeability is compared with that from a traditional numerical simulator (isotropic relative permeability). The basic parameters of the numerical simulation model are as shown in Table 1, and the relative permeability curve used in the numerical simulation model is shown in Figure 3.


**Table 1.** Simulation model parameters.

**Figure 3.** Relative permeability of three directions.
