## 1 Introduction  
With the increasing demand for energy and the progressive exhaustion of shallow resources, the focus of resource extraction has gradually shifted to deep underground. However, deep underground mining faces complex geological environments and extreme engineering conditions, posing new challenges to traditional mining technologies [1; 2; 3; 4]. Hydraulic measures such as hydraulic slotting and hydraulic fracturing are commonly used in coal seam pressure relief and permeability enhancement. After these measures, the gas extraction is affected by the water in the coalbed. The gas-liquid two-phase flow in porous media is closely related to many geological engineering fields, including coalbed methane or oil extraction and CO\({}_{2}\) geological sequestration [5; 6; 7; 8; 9]. The key issues affecting the efficiency of coalbed methane extraction are the flow mechanisms in porous media and the instability of the flow interface in a two-phase flow [10]. Factors such as the heterogeneity of porous media, fluid viscosity, capillarity, and wettability are the main causes of the instability of the two-phase flow interface [11; 12]. Micromodels and numerical simulations offer a cost-effective and flexible means to investigate two-phase flow mechanisms at the pore scale, providing a direct reflection of varying flow characteristics. Therefore, micromodels and numerical simulation methods are used to study the mechanisms of water-oil, CO\({}_{2}\)-water, and other two-phase flowsat the pore scale. It is indicated that the mechanical imbalance causes the two-phase flow interface to be unstable, manifesting diverse advancing flow characteristics macroscopically [13]. The instability of this interface is mainly induced by the curved capillary forces and non-local viscous forces present between the fluids [14]. The slow displacement caused by capillary forces is primarily governed by the geometric shape of the pore structure, while the viscous forces generated by friction slow down the flow at the interface [15]. This competition of forces is largely influenced by factors such as pore geometry, wettability, gravity, surface roughness, and disorder [16; 17; 18]. Despite these influences, [PERSON] [19] introduced the classic phase diagram of immiscible displacement through microfluidic experiments, defining the invasion patterns as three modes: capillary fingering, viscous fingering, and stable displacement. [PERSON] [20] conducted extensive experiments using a two-dimensional homogeneous micromodel to study the effects of viscous and capillary forces on instability fingering and fluid saturation patterns. They plotted the observed displacement states onto a _logM-logCa_ stability phase diagram. By adjusting the capillary number (_Ca_) and viscosity ratio (_M_), three flow states--viscous fingering, capillary fingering, and stable displacement--could be identified. [PERSON] [21] studied non-isothermal water-oil displacement in homogeneous porous media through numerical simulations and found that reducing \(M\) exacerbated the water channeling effect in high-permeability layers, while lowering _Ca_ might lead to higher water sweep efficiency. [PERSON] [22] further extended the invasion pattern phase diagram proposed by [PERSON] on the _logM-logCa_ plane through water-oil displacement experiments in homogeneous porous media, quantifying the dynamic effects of capillary and viscous forces on the immiscible fluid-fluid displacement process. Heterogeneity affects fluid displacement by altering the state of forces present in the reservoir, while the wettability of the medium influences displacement by determining the microscopic distribution of fluid in the pore space. The wettability of the solid is represented by the contact angle on the wall surface, which is the static contact angle measured when the liquid is stable on a flat solid surface.