**1. Introduction**

Old oilfields in eastern China have generally entered a dual-high development stage with high water cut and high recovery degree. The hydrodynamic adjustment of "liquid flow steering" has achieved good results in improving recovery. Fluid flow steering changes the flow direction of fluid and makes fluid flow paths more intricate. Fluid seepage laws thus become complicated. At present, relative permeability is isotropic in traditional numerical simulations, which cannot accurately calculate residual oil distribution and cannot describe the deep mechanisms of liquid flow steering.

Relative permeability (the ratio of effective permeability to absolute permeability) is a basic parameter to study the seepage law of multiphase fluid. There are many factors affecting relative permeability, among which the heterogeneous pore structure is of vital importance. It is found that relative permeability will change significantly due to microscopic local pore structure, fluid interaction, and rock–fluid interaction [1–5]. Therefore, the relative permeability must be different when the pore structure of the reservoir is different.

In the past several years, a number of works have been reported on the directional characteristics of pore structures. In fluvial sediments, due to long-term erosion and erosion by water flow, the particles that make up the rock skeleton of the reservoir are often in irregular ellipsoid shapes. In the process of deposition, the skeleton particles are arranged directionally with the transport medium, and the direction of the long axis is consistent

**Citation:** Li, C.; Wang, S.; You, Q.; Yu, C. A New Measurement of Anisotropic Relative Permeability and Its Application in Numerical Simulation. *Energies* **2021**, *14*, 4731. https://doi.org/10.3390/en14164731

Academic Editor: Marcin Kami ´nski

Received: 7 June 2021 Accepted: 2 August 2021 Published: 4 August 2021

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with the direction of the flow, while the direction of the short axis is perpendicular to the direction of the flow. The compaction in the diagenetic process also strengthens this directional arrangement, which makes the pore structure directional. Effected by the sedimentary environment and the inherent properties of the reservoir, including the original sedimentary bedding, vertical rhythm, and so on, the pore structure will be obviously different in the plane and vertical, and the fluid flow will have a dominant migration direction [5–11]. Recent theoretical and experimental studies have revealed that permeability is anisotropic [12–16]. Since it is the pore throats that govern the percolation threshold for porous media, permeability anisotropy is the behavior of the anisotropy of pore structure. In other words, the pore structure anisotropy has been widely observed.

Since Corey and Rathjens found that the relative permeability of bedded rocks is directional in 1956, research on anisotropic relative permeability has mainly focused on the following three aspects. Firstly, identification of core scales in the laboratory. It has been shown that relative permeability is related to the particle size of the rock composition of bedding structures (namely pore throat structure) and micro fractures. When driven along the direction of penetrating bedding, the change of rock properties leads to the change of capillary resistance so that the movement of fluid in this direction is blocked, and the oil in the core will be blocked, indicating that the relative permeability is directional. The displacement direction is different, the shape of phase permeability curve is different, the vertical displacement remaining oil is less, and the displacement efficiency is higher [17–24]. Secondly, discovery in numerical simulation. Some numerical simulation studies used pore network models to point out that the relative permeability parallel to the bedding direction is larger than that perpendicular to the bedding direction, and the residual saturation and relative permeability are sensitive to anisotropic correlation. The higher the absolute permeability value is, the greater the relative permeability value is, and the relative permeability is related to pore connectivity and pore inclination angles. The dendritic nature of gas-cluster topology, especially in the presence of other forces, such as gravity or strong viscous pressure gradients, clearly suggests that significant anisotropy may exist in relative permeability due to the balance of forces in the dissolved gas drive process [25–31]. Thirdly, microscopic visual seepage physical simulation. It is indicated that the change of injection-production angle will break the pressure balance of the original seepage field, promote the stripping and seepage of all kinds of remaining oil, and make the remaining oil migrate again, thus changing the microscopic remaining oil type and distribution law. The essence of this change is that the percolation law of the oil and water phases has changed due to different pore structures in different displacement directions [32,33]. Numerical models and physical experiments all prove that like porosity and permeability, relative permeability is a direction-dependent tensor, not a scalar.

Although the anisotropic characteristics of relative permeability were recognized earlier, the related studies focused on its embodiment in microscopic displacement experiments, or the demonstration of the characteristics by numerical simulation. In particular, few studies, to our knowledge, have considered two facts. Firstly, the test of anisotropic relative permeability. At present, most studies tested the anisotropic relative permeability curve by drilling cores in different directions. The testing device and the displacement direction is one-dimensional. However, the three-dimensional flow of fluid in the core is not equal to the simple superposition of three one-dimensional flows. The traditional relative permeability curve testing device cannot increase or change the displacement direction and cannot directly test the anisotropic relative permeability curve. It is quite difficult to test the relative permeability in three directions through one sample, However, the different test results of the same sample are the most comparable; Secondly, the application of anisotropic relative permeability in reservoir numerical simulation is very weak. There is a long way to go to test and apply the anisotropic relative permeability.

In this paper, firstly, the typical fluvial sandstone of the Shihezi fomation outcrop in Jiyang depression of Bohai Bay basin are used to measure the anisotropic relative permeability via a self-developed anisotropic cubic core holder; Secondly, we established a new simulator considering anisotropic relative permeability and compared the calculation results between the new simulator and traditional simulator. Finally, we apply the new simulator to the reservoir numerical simulation of Cheng 4 block in Shengli Oilfield (the reservoir is a typical fluvial sedimentary environment). The production data show that the numerical simulation method considering anisotropic relative permeability can describe the remaining oil distribution more accurately, especially in the situation that the reservoir has entered the development period of ultra-high water cut. In order to precisely apply anisotropic relative permeability to typical fluvial reservoir simulation or fractured reservoir, there is an urgent need for working on the upscaling of anisotropic relative permeability.
