**5. Conclusions**

In this paper, the anisotropic relative permeability of typical fluvial sandstone is studied using a self-developed anisotropic cubic core holder by unsteady-states relative permeability experiments. A new numerical simulator considering anisotropic relative permeability is established. The effect of anisotropic relative permeability in the flooding process is analyzed by the new simulator. An actual fluvial facies reservoir of Shengli Oilfield in China is selected as an example to validate the new simulator.


**Author Contributions:** Writing—original draft, C.L.; Writing—review & editing, S.W. and Q.Y. Data curation, C.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** I would like to thank Shuoliang Wang, Chunlei Yu and Qing You for their guidance in the process of writing and revising of this paper.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Nomenclature**



#### **Appendix A. The Oil Governing Equations**

Oil phase:

$$\nabla \cdot \left[ \frac{k k\_{\text{roamisatorpic}} \rho\_o}{\mu\_o} (\nabla p\_o - \gamma\_o \nabla D) \right] - q\_\vartheta = \frac{\partial}{\partial t} (\rho\_o \phi \mathbf{S}\_o) \tag{A1}$$

Water phase:

$$\nabla \cdot \left[ \frac{k k\_{rwunistropic} \rho\_w}{\mu\_w} (\nabla p\_w - \gamma\_w \nabla D) \right] - q\_w = \frac{\partial}{\partial t} (\rho\_w \phi \mathcal{S}\_w) \tag{A2}$$

where: *γ<sup>o</sup>* = *ρog*; *γ<sup>w</sup>* = *ρwg*, *Kroanisotropic* is the oil phase relative permeability, *K* is the absolute permeability tensor, *μo*, *po*, and *γ<sup>o</sup>* are the viscosity, pressure, and specific gravity of the oil phase, respectively, *Krwanisotropic* is the water phase relative permeability, *μw*, *pw*, and *γ<sup>w</sup>* are the viscosity, pressure, and specific gravity of the water phase, respectively, *D* is the depth, and *φ* is the porosity.

The results of the experiments in this paper showed that the relative permeability of oil and water is affected by the anisotropy of pore structure. In this paper, the *krlanisotropic* in the above formula is written into three relative permeability expressions that vary with different directions when dealing with the anisotropic relative permeability, namely *krox, kroy,* and *kroz*. After the replacement here, the traditional isotropic relative permeability is replaced by the anisotropic relative permeability.

Taking the oil phase as an example, the governing equation is expanded initially into a rectangular coordinate component as follows.

$$\begin{cases} \frac{\partial}{\partial x} \left[ \frac{\rho\_{\vartheta} \cdot k}{\mu\_{\vartheta}} \cdot k\_{\text{rox}} (\frac{\partial p\_{\vartheta}}{\partial x} - \gamma\_{\vartheta} \frac{\partial D}{\partial x}) \right] + \frac{\partial}{\partial y} \left[ \frac{\rho\_{\vartheta} \cdot k}{\mu\_{\vartheta}} \cdot k\_{\text{roy}} (\frac{\partial p\_{\vartheta}}{\partial y} - \gamma\_{\vartheta} \frac{\partial D}{\partial y}) \right] \\ + \frac{\partial}{\partial z} \left[ \frac{\rho\_{\vartheta} \cdot k}{\mu\_{\vartheta}} \cdot k\_{\text{roz}} (\frac{\partial p\_{\vartheta}}{\partial z} - \gamma\_{\vartheta} \frac{\partial D}{\partial z}) \right] + q\_{\vartheta} = \frac{\partial (\phi p\_{\vartheta} S\_{\vartheta})}{\partial t} \end{cases} \tag{A3}$$

For (*i,j,k,n +* 1) point, the subscript in the following expression is in an abbreviated form:

*ρ<sup>o</sup>* ·*k*·*kr*ox <sup>Δ</sup>*xi*·*μ<sup>o</sup>* [( *<sup>p</sup>n*+<sup>1</sup> *<sup>i</sup>*+<sup>1</sup> <sup>−</sup>*pn*+<sup>1</sup> *i* Δ*x i*+ 1 2 − *γoi*<sup>+</sup> <sup>1</sup> 2 *Di*+1−*Di* Δ*x i*+ 1 2 )+( *<sup>p</sup>n*+<sup>1</sup> *<sup>i</sup>*−<sup>1</sup> <sup>−</sup>*pn*+<sup>1</sup> *i* Δ*x <sup>i</sup>*<sup>−</sup> <sup>1</sup> 2 <sup>−</sup> *<sup>γ</sup>oi*<sup>−</sup> <sup>1</sup> 2 *Di*−1−*Di* Δ*x <sup>i</sup>*<sup>−</sup> <sup>1</sup> 2 )] +*ρ<sup>o</sup>* ·*k*·*kr*oy <sup>Δ</sup>*yj*·*μ<sup>o</sup>* [( *<sup>p</sup>n*+<sup>1</sup> *<sup>j</sup>*+<sup>1</sup> <sup>−</sup>*pn*+<sup>1</sup> *j* <sup>Δ</sup>*yj*<sup>+</sup> <sup>1</sup> 2 − *γoj*<sup>+</sup> <sup>1</sup> 2 *Dj*+1−*Dj* <sup>Δ</sup>*yj*<sup>+</sup> <sup>1</sup> 2 )+( *<sup>p</sup>n*+<sup>1</sup> *<sup>j</sup>*−<sup>1</sup> <sup>−</sup>*pn*+<sup>1</sup> *j* <sup>Δ</sup>*xj*<sup>−</sup> <sup>1</sup> 2 <sup>−</sup> *<sup>γ</sup>oj*<sup>−</sup> <sup>1</sup> 2 *Dj*−1−*Dj* <sup>Δ</sup>*xj*<sup>−</sup> <sup>1</sup> 2 )] +*ρ<sup>o</sup>* ·*k*·*kr*oz <sup>Δ</sup>*zk* ·*μ<sup>o</sup>* [( *<sup>p</sup>n*+<sup>1</sup> *<sup>k</sup>*+<sup>1</sup> <sup>−</sup>*pn*+<sup>1</sup> *k* Δ*z k*+ 1 2 − *γok*<sup>+</sup> <sup>1</sup> 2 *Dk*+1−*Dk* Δ*z k*+ 1 2 )+( *<sup>p</sup>n*+<sup>1</sup> *<sup>k</sup>*−<sup>1</sup> <sup>−</sup>*pn*+<sup>1</sup> *k* Δ*z <sup>k</sup>*<sup>−</sup> <sup>1</sup> 2 <sup>−</sup> *<sup>γ</sup>ok*<sup>−</sup> <sup>1</sup> 2 *Dk*−1−*Dk* Δ*z <sup>k</sup>*<sup>−</sup> <sup>1</sup> 2 )] +*qn*+<sup>1</sup> *<sup>o</sup>* = <sup>1</sup> <sup>Δ</sup>*<sup>t</sup>* [(*φρoSo*) *<sup>n</sup>*+<sup>1</sup> <sup>−</sup> (*φρoSo*) *n*] (A4)

Multiply both sides by *Vijk* = Δ*xi*Δ*yj*Δ*zk* and define the following conductivity:

⎧ ⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩ *TXoianisotropic*<sup>+</sup> <sup>1</sup> 2 = *Vijk* Δ*xi ρo* ·*k <sup>μ</sup><sup>o</sup>* ·*kr*ox Δ*x i*+ 1 2 = <sup>Δ</sup>*yj*Δ*zk* Δ*x i*+ 1 2 *ρ<sup>o</sup>* ·*k <sup>μ</sup><sup>o</sup>* · *kr*ox, *TXoianisotropic*<sup>−</sup> <sup>1</sup> 2 = *Vijk* Δ*xi ρo* ·*k <sup>μ</sup><sup>o</sup>* ·*kr*ox Δ*x <sup>i</sup>*<sup>−</sup> <sup>1</sup> 2 = <sup>Δ</sup>*yj*Δ*zk* Δ*x <sup>i</sup>*<sup>−</sup> <sup>1</sup> 2 *ρ<sup>o</sup>* ·*k <sup>μ</sup><sup>o</sup>* · *kr*ox *TYojanisotropic*<sup>+</sup> <sup>1</sup> 2 = *Vijk* Δ*yj ρo* ·*k <sup>μ</sup><sup>o</sup>* ·*kr*oy <sup>Δ</sup>*yj*<sup>+</sup> <sup>1</sup> 2 = <sup>Δ</sup>*xi*Δ*zk* <sup>Δ</sup>*yj*<sup>+</sup> <sup>1</sup> 2 *ρ<sup>o</sup>* ·*k <sup>μ</sup><sup>o</sup>* · *kr*oy, *TYojanisotropic*<sup>−</sup> <sup>1</sup> 2 = *Vijk* Δ*yj ρo* ·*k <sup>μ</sup><sup>o</sup>* ·*kr*oy <sup>Δ</sup>*yj*<sup>−</sup> <sup>1</sup> 2 = <sup>Δ</sup>*xi*Δ*zk* <sup>Δ</sup>*yj*<sup>−</sup> <sup>1</sup> 2 *ρ<sup>o</sup>* ·*k <sup>μ</sup><sup>o</sup>* · *kr*oy *TZokanisotropic*<sup>+</sup> <sup>1</sup> 2 = *Vijk* Δ*zk ρo* ·*k <sup>μ</sup><sup>o</sup>* ·*kr*oz Δ*z k*+ 1 2 = <sup>Δ</sup>*xi*Δ*yj* Δ*z k*+ 1 2 *ρ<sup>o</sup>* ·*k <sup>μ</sup><sup>o</sup>* · *kr*oz, *TZokanisotropic*<sup>−</sup> <sup>1</sup> 2 = *Vijk* Δ*zk ρo* ·*k <sup>μ</sup><sup>o</sup>* ·*kr*oz Δ*z <sup>i</sup>*<sup>−</sup> <sup>1</sup> 2 = <sup>Δ</sup>*xi*Δ*yj* Δ*z <sup>i</sup>*<sup>−</sup> <sup>1</sup> 2 *ρ<sup>o</sup>* ·*k <sup>μ</sup><sup>o</sup>* · *kr*oz (A5)

The second-order difference operator is defined as follows:

$$\begin{cases} \Delta\_{\mathbf{x}} T X\_{\text{oamistorpci}} \Delta\_{\mathbf{x}} P = T X\_{\text{omisstorpci} + \frac{1}{2}} (p\_{i+1} - p\_i) + T X\_{\text{omisstorpci} - \frac{1}{2}} (p\_{i-1} - p\_i) \\\ \Delta\_{\mathbf{y}} T Y\_{\text{omisstorpci}} \Delta\_{\mathbf{y}} P = T Y\_{\text{omisstorpci} + \frac{1}{2}} (p\_{j+1} - p\_j) + T Y\_{\text{omisstorpci} - \frac{1}{2}} (p\_{j-1} - p\_j) \\\ \Delta\_{\mathbf{z}} T Z\_{\text{omisstorpci}} \Delta\_{\mathbf{z}} P = T Z\_{\text{omisstorpci} + \frac{1}{2}} (p\_{k+1} - p\_k) + T Z\_{\text{omisstorpci} - \frac{1}{2}} (p\_{k-1} - p\_k) \end{cases} \tag{A6}$$

The oil phase governing equation is discretized as a sample in this section, which is shown as follows:

<sup>Δ</sup>*xTXoanisotropic*Δ*xPn*+<sup>1</sup> <sup>+</sup> <sup>Δ</sup>*yTYoanisotropic*Δ*yPn*+<sup>1</sup> <sup>+</sup> <sup>Δ</sup>*zTZoanisotropic*Δ*zPn*+<sup>1</sup> <sup>−</sup> <sup>Δ</sup>*xTXoanisotropicγog*Δ*xD* <sup>−</sup>Δ*yTYoanisotropicγog*Δ*yD* <sup>−</sup> <sup>Δ</sup>*zTZo anisotropicγog*Δ*zD* <sup>+</sup> *<sup>q</sup>n*+<sup>1</sup> <sup>o</sup> *Vijk* <sup>=</sup> *Vijk* <sup>Δ</sup>*<sup>t</sup>* [(*φρoSo*) *<sup>n</sup>*+<sup>1</sup> <sup>−</sup> (*φρoSo*) *<sup>n</sup>*] (A7)

Then, the formula above can be further simplified as follows:

$$
\Delta T\_{\mathcal{O}\_{\text{anisotropic}}} \Delta P^{n+1} - \Delta T\_{\mathcal{O}\_{\text{anisotropic}}} \Delta D + q\_o^{n+1} V\_{ijk} = \frac{V\_{ijk}}{\Delta t} [ (\phi \rho\_o \mathcal{S}\_{\mathcal{O}})^{n+1} - (\phi \rho\_o \mathcal{S}\_{\mathcal{O}})^n ] \tag{A8}
$$

The governing equation of the water phase is expressed with the same format. So far, the anisotropic relative permeability is introduced into the traditional numerical simulation method by dealing with the relative permeability in the traditional oil and water phase governing equation.
