In the Tarim Basin, NW China, the Ordovician carbonate rock is rich in oil and gas resources, and it is the key area for increasing oil and gas reserves and production in the Tarim Basin. The exploration for the Ordovician carbonate rocks began in the 1990s. The exploration on the buried hill karst and interlayer karst was carried out around paleo-uplifts and paleo-slope. Many oil fields, including Lungu, Tazhong, Halahatang and Tahe ^[1,2,3,4] have been discovered successively, making a great contribution to the rapid increase in oil and gas reserves and production in the Tarim Basin. In recent years, the oil and gas exploration in the Ordovician carbonate rocks in paleo-uplifts and slopes has increased. PetroChina Company Limited (CNPC) and China Petroleum and Chemical Corporation (Sinopec) have expanded their exploration areas to A’man Transition Zone and great breakthroughs have been achieved. Since 2020, in the key exploration wells ManShen 1, ManShen 2, and ManShen 3 drilled by PetroChina Co Ltd. and exploration by Northwest Oilfield Company, Sinopec in A’man Transition Zone, great oil discovery has been made successively. Therefore, the fault-controlled fracture-cavity carbonate reservoirs controlled by intracratonic strike-slip faults have attracted widespread attention in the petroleum industry. The storage space of this type of oil reservoir is a fractured complex reservoir reformed by multi-stage fault activities, including fracture zone cavity, cave body, pore body and fractured body. During the multi-stage tectonic movement, deep-seated hydrocarbon source continued to supply hydrocarbons, and hydrocarbon migrated upwards along the strike-slip fault zone. The fractured complex reservoirs are hydrocarbon migration channels and hydrocarbon storage spaces. The drillings have confirmed that the height of the oil column is over 550 m, the daily oil production is more than 1.0×10³ t and 10×10⁸ t oil resources has been confirmed.

The fault-controlled carbonate reservoirs have considerable prospects for exploration and development. The research on fault identification, fault structure characteristics, stratigraphy, and sedimentation related to fault-controlled carbonate reservoirs are relatively abundant. The strike-slip faults are mainly manifested as "longitudinal layering, planar division, and segmentation along strike". The characteristics of "reservoir controlling, map controlling, and hydrocarbon controlling" are obvious. It has also been revealed that the fault-controlled reservoir controlled by Tabei-Tazhong intracratonic strike- slip fault system has the characteristics of small oil reservoirs and large oil fields, with huge exploration and development potential ^{[5,6,7,8,9,10]}. These results are significant for guiding the selection of favorable blocks, zones and high-efficiency wells. However, the research on the internal fluid flow patterns of special faulted complex reservoirs under deep burial conditions is still weak, and lack of effective techniques for enhancing oil recovery at the middle-late stages of development, which will restrict the long-term large-scale and high-efficiency development of this type of reservoir.

Based on the analysis of the geological characteristics of the ultra-deep fault-controlled fracture-cavity combination and the division of reservoir units, two physical models have been built in this study. In addition, water injection and oil displacement physical simulation tests have been carried out. Furthermore, the mechanisms of water injection and oil displacement under complex flow conditions in the fault-controlled fracture-cavity carbonate reservoirs have been studied. A flow mathematical model for the coexistence of seepage and free flow in fault-controlled carbonate reservoirs has been established, and pilot water injection experiments have been carried out on this basis to evaluate the effect of water injection in enhancing oil recovery. The field test results have confirmed that water injection gravity flooding can greatly increase the oil recovery rate, providing new ways for the efficient development of this type of reservoir.

The fault-controlled reservoirs are obviously different from the traditional weathering crust carbonate reservoirs. For example, in the Tarim Basin, in the Halahatang Oilfield, the fractured-cavity reservoirs in the denudation zone of the northern Halahatang are weathering crust karsts, distributed in continuous. While in the southern part of the Halahatang and Fuman Oilfield, the fractured-cavity reservoir in the Upper Ordovician is mainly distributed in strips along the fault zone (Fig. 1), with the reflection characteristics of "Long beads through strata" on the seismic profile (Fig. 2).

The lithology of the Ordovician Yijianfang Formation- Yingshan Formation in Fuman Oilfield is mainly bright crystal sandy debris limestone, bright crystal sandy gravel debris limestone, bright crystal oolitic limestone, bright - micritic bioclastic limestone, micritic algae sandy debris algal mass limestone, tray-type biocrust and micritic limestone. The carbonate matrix has the characteristics of low porosity and low permeability, with limited contribution to the physical properties of the reservoir. The reservoir space mainly depends on the fractured-cavity bodies related to the fault reformation. The permeability of the reservoir space obtained by well test interpretation ranges from (0.077-10 607.420)×10^-3 μm². The wells with the fractured-cavity reservoirs permeability greater than 500×10^-3 μm² accounted for 57.1%.

The fractured-cavity reservoirs were formed mainly due to multi-stage fault activities, characterized by the complex of fault and fracture-cavity, composed of fault cavity, cave body, hole body and fracture body ^[11]. The primary matrix pores are underdeveloped, and the reservoir is of strong heterogeneity. The carbonate reservoirs in Fuman Oilfield are divided into three types on the basis of the dynamic and static data, namely cave type, hole type and fracture-hole type ^[12]. At present, 79 % of oil wells drilled into cave type reservoirs in Fuman Oilfield.

According to the spatial forms of the reservoir and the strength of natural energy, oil reservoir units can be divided into three main types, namely the strong natural energy connected type, the weak natural energy connected type, and the weak natural energy isolation type. The test production characteristics of different types of oil reservoir units are obviously different. The strong natural energy connected reservoir is good in continuity, with strong bottom water. The oil pressure and production are stable before water breakthrough, and the water cut increases sharply after water breakthrough. The oil pressure and production decrease rapidly. The flowing production period is long, and the recovery degree of dynamic reserves is relatively high. The weak natural energy connected reservoirs is relatively poor in continuity, without bottom water, and the early production and oil pressure of production wells decrease slowly. Due to weak natural energy, the flowing production period is short, and the recovery degree of dynamic reserves is low. Weak natural energy isolated reservoirs are characterized by poor in continuity, constant in volume, no bottom water, rapid reduction of oil pressure and production at the early stage of production wells, rapid energy loss and rapid stop of flowing production, the shortest flowing production period, and the lowest recovery of dynamic reserves (Fig. 3). The main production parameters related to three types of reservoir units are shown in Table 1.

The strong natural energy connected reservoir unit is continuous in reservoir space. The bottom natural water body is connected directly to the upper oil body vertically. In this type of reservoir unit, the recovery factor can be improved by bottom water flooding through using natural water energy reasonably. The most effective development strategy for improving oil recovery is to optimize the reasonable working system at different driving stages, to ensure production with reasonable production capacity, and extend the water free oil production period. At present, a set of development technology policies with reasonable productivity have been formed through classification of water energy and driving stages, optimization of reasonable production capacity at different stages. For the weak natural energy connected reservoir unit, the reservoir is poor in continuity, being connected but not smooth. The bottom natural water body and the upper oil body is not connected vertically. The large amount of remaining oil distributed at the bottom of the well is the main target for potential exploration after the elastic drive. The water injection gravity flooding technology can be used. At the early stage, water injection is used to supplement the formation energy, and the residual oil in the dominant channel is replaced by gravity flooding. At the late stage, the residual oil in the blocked area is replaced by high pressure water injection. In the isolated reservoir unit with weak natural energy, the reservoirs are isolated, not connected or poorly connected, with no water connection. The remaining oil at the bottom of the well and the unutilized reserves at the far end of the wellbore after elastic driving are the main targets for potential exploration. Similarly, the water injection gravity flooding technology is used. The high-pressure water injection is used to open the separation barrier to connect the fracture and cavity at the far end of the wellbore, and then the oil is gradually replaced by the gravity differentiation of injected water to improve the recovery efficiency.

2.1.1. Physical model design and building

Fault-controlled fracture-cavity carbonate reservoirs are often trans-layer. Exploration practices have shown that the vertical extension space is large, up to 100 m, or even more than 1000 m. The fracture-cavity space is generally characterized by a “V” shape, with a wide top and a narrow bottom. Within the reservoir, the cave-type storage space is dominated, which are connected to each other through fracture zones of different levels, and the caves and fracture zones form relatively independent fracture-cavity units. Drilling and well test data show that the physical properties of the reservoir in the fracture-cavity unit are good, and it is very easy for the fluid to flow and mass exchange vertically in the cave or along the high-angle fracture zone.

According to the reservoir characteristics and fluid flow rules of fault-controlled fracture-cavity carbonate reservoirs, the organic glass etching method is used to design and make two experimental models for isolated units and weak natural energy connected units. The oil for experiment is made of silicone oil and kerosene, and dyed red with Sudan red dye, with a density of 0.8281 g/cm³ and a viscosity of 20.64 mPa•s. The experimental water is prepared according to the actual formation water component of the reservoir, and dyed with methylene blue dye. The viscosity is 1.00 mPa•s.

2.1.2. Development mechanisms of water injection in isolated unit

For the isolated unit, the gravity differentiation formed by the density difference between injected water and oil is used to replace the oil to the upper part of the fracture-cavity body for production. Fig. 4 shows the oil-water distribution at different stages of production when conducting the physical simulation of water injection in the isolated unit. Each water injection cycle in the isolated unit is divided into three phases, including water injection, simmering and oil production. The oil displacement mechanism is: (1) At the initial stage, oil production relies on natural energy. When the formation pressure drops to unable to maintain normal production, water injection is conducted to provide energy and restore formation pressure. (2) Under the action of gravity differentiation, the injected water will replace oil continuously during the simmering process, forming the secondary bottom water. As time goes by, the cone formed by the injected water will gradually become smoother, and the injected water will move towards the surrounding cracks. When the pressure is basically stable, the remaining oil will be re-enriched in the upper part of the reservoir. (3) The remaining oil in the upper part will be produced under the action of bottom hole pressure. In the isolated unit, each water injection cycle including water injection, simmering and oil production. After multiple cycles of water injection, the oil recovery rate can be improved gradually.

2.1.3. Development of water injection in weak natural energy connected unit

For weak natural energy connected units, the remaining oil in the fault-controlled fracture-cavity reservoir space can be mobilized effectively through water injection, showing the complex flow characteristics of lateral displacement and vertical gravity displacement (Fig. 5). Based on the spatial distribution of fractured complex reservoirs, remaining oil distribution and well-storage connectivity, an irregular volumetric injection-production well pattern can be established to ensure the optimal control of remaining oil reserves and the degree of three-dimensional production. The experimental results show that: (1) The injected water flows mainly in vertical direction, and the gravity displacement formed by the difference in oil-water density has been fully utilized during the flowing process, and the lower part of the reservoir space is occupied preferentially for vertical oil displacement. (2) The reservoir controlled by fault gravity displacement is obvious, and the well pattern of "shallow injection and deep production" also shows the characteristics of a wide swept range and an overall uplift of the oil-water interface. The water injection effect is good. (3) The normal pressure water injection cannot effectively drive the fracture-cavity area poorly connected with reservoir, and the effect of water injection gravity flooding can be improved by high pressure water injection and fracturing.

Fracture-cavity carbonate reservoirs have diverse storage and seepage spaces, including pores, fractures, and caves, with large differences in spatial scale, spanning from micrometers to meters, leading to the existence of both seepage and free flow in large fractures and caves in this type of reservoir. The traditional seepage theory based on Darcy’s equation is no longer fully applicable. In this regard, Yao et al. ^[13] proposed the model of Discrete Fracture Vug Network (DFVN), in which matrix rock blocks and cave filling parts are seepage areas, which are characterized by Darcy equation; unfilled caves and large fractures which regarded as a free flow area are characterized by the Navier-Stokes equation. The two areas are coupled using extended two-phase Beavers-Joseph-Saffman boundary conditions ^[14,15]. Taking into account the rapid gravity differentiation, Cui et al. assumed that the gravity differentiation in the free flow of the cave was completed instantaneously, simplified the above model and proposed an effective oil-water two-phase numerical simulation technical model ^[16,17]. Recently, Liu et al. ^[18] proposed a new embedded discrete fracture-cavity model based on this model to describe the oil-water two-phase flow.

2.2.1. Mathematical model of fluid flow in seepage area

The mathematical model of the seepage area includes the continuity equation, the motion equation and the auxiliary equation, etc. ^[18], the continuity equation is

The motion equation is

The auxiliary equation is

2.2.2. Mathematical model of fluid flow in free flow region

The unfilled cave area (free flow area) adopts the instantaneous gravity differentiation model. The cave satisfies the mass conservation equation of the oil and water phases and the water is located in the lower part of the cave, that is, it occupies the lower area grid in the numerical calculation. The fluid exchange between the cave and the seepage area grid is characterized by conductivity, as follows ^[18]

Since the unfilled cave can be regarded as an equipotential body with infinite conductivity, it can be regarded as the permeability of the cave tends to infinity. Therefore, the conductivity value of the unfilled cave can be approximated by the unilateral conductivity of the matrix or fracture grid in the seepage area. For two-phase flow, according to the fluid fraction of the fluid exchange between the grid and the surrounding seepage area, the phase flow fraction is defined as follows ^[18]

In Eq. (6), the value of ${{\tilde{K}}_{\text{r}\beta }}$needs to be taken according to the upstream wind. When the upstream grid is the seepage area, the value is the relative permeability of the matrix rock block or fracture, that is${{\tilde{K}}_{\text{r}\beta }}={{K}_{\text{r}\beta }}$. When the upper grid is the cave free flow area, the calculation is carried out according to Formula (5). The saturation definition of the cave grid (Fig. 6, where the volume of a similar hemisphere below the light blue line is V_u and the volume of a similar hemisphere below the green line is V_d) is as follows

The recovery of formation pressure shows wave rise under the action of rapid gravity differentiation of oil and water during the process of water injection and oil replacement (Fig. 7). The mathematical equation of cavity flow can be used to describe quantitatively the regularity, and the continuous prediction of formation pressure can be realized. Then the key parameters such as soaking time, injection volume in each stage and reasonable adjustment time can be optimized in real time.

The superior connecting channel between the fracture-cavity bodies is the key factor which affects the effectiveness of water injection. The connectivity of fault- controlled fractures and caves is controlled by multiple factors, such as its own state (such as cave distribution, fault-fracture occurrence, etc.) and external environment^[19,20,21] (such as present ground stress field, cave internal pressure, etc.). The occurrence and activity of fractures and fractures are important factors for controlling the connectivity of fractures and caves from the perspective of geomechanics.

For a fault (fracture), its activity mainly depends on the normal stress σ_ne and the shear stress τ on the fault (fracture) surface. Each fault (fracture) surface in a critical slip state satisfies the following relationship ^[22]

In the Eq. (8), μ (defined as the ratio of normal to shear stress) is not only a key parameter affecting the sliding of the fracture surface, but also an important index reflecting the fracture permeability and fluid flow. The normal stress σ_ne and shear stress τ can calculate through the relationship between the fault (fracture) surface and the in-situ stress field ^[22,23].

In the formula, n_i,j are directional cosines, and the stress tensor at a certain point on the fracture surface calculated by this parameter is defined as M

According to the above method, based on the quantitative prediction of the three-dimensional distribution of natural fractures, the relationship between the in-situ stress tensor and the occurrence of fractures can be used to clarify the activity (value of μ), development location and occurrence information of fractures. It can be seen that the ratio of the normal stress to shear stress on the fault (fracture) surface is an important geological parameter that controls the permeability of the fault (fracture) zone, and is a positive indicator reflecting the connectivity of the fracture-cavity body. The larger the value the better the connectivity ^[24]. In addition, changes in reservoir pressure will also cause changes in fault (fracture) activity, which in turn affects the connectivity of fracture-cavity bodies. For this reason, the critical injection pressure is defined to describe the activity of the fault (fracture) surface

Critical injection pressure indicates the required reservoir pressure when the fault (fracture) is active, and is an inverse indicator reflecting the fault (fracture) activity. The smaller the value, the stronger the fault (fracture) activity.

Based on comprehensive consideration of the above two parameters, guided by geomechanical theory, combined with practical experience in oilfield production, Zhang et al. proposed a calculation model of fault (fracture) activity index suitable for high geostress, high pore pressure and complex tectonic background ^[24,25]

where

G₁ is the normalized value of fault (fracture) normal shear stress ratio, G₂ is the normalized value of fault (fracture) critical injection pressure; W₁, W₂ are the weights of normal shear stress ratio and critical injection pressure geological attributes respectively. W₂ comprehensively considers regional current in-situ stress field, reservoir pressure, rock mechanical properties and fault (fracture) geometric occurrence and other factors, which can directly reflect the potential mechanics of the fault (fracture) zone under the control of multiple factors such as the activity behavior under current in-situ stress field. It’s suitable for quantitative analysis of the connectivity between different fracture-cavity bodies. The larger the value, the stronger the potential activity of the fault (fracture).

On the basis of previous understanding of development mechanism, cave connected reservoirs with weak natural energy and isolated units with weak natural energy are preferentially selected for water injection gravity flooding development. According to the fracture-cavity space connection structure and the well storage space configuration relationship, it can be divided into three types: single well water injection huff and puff for oil replacement, unit water injection to drive oil, and single well high-pressure water injection. The Fuman Oilfield ultra-deep fault-controlled fracture-cavity reservoir was tested in 2014 for water injection development. By the end of 2020, the cumulative water injection was 180.0× 10⁴ m³, and the cumulative increased oil was 61.6×10⁴ t.

The reservoir type is a cave reservoir, and the oil production testing was conducted in weak-energy and isolated reservoir unit, which is mainly developed by single-well water injection. Taking Well Yueman 20C as an example, the well drilled 11 m into the Yijianfang Formation, and was completed ahead of schedule due to drilling fluid loss. The controlled reserve is of 52.00×10⁴ t in single well. It was put into production on May 10, 2017. The flowing production stage was 636 days. The oil production was 3.94×10⁴ t. Since May 29, 2019, a total of 9 rounds of water injection huffing and puffing have been carried out. A total of 6.80×10⁴ m³ of water was injected, and the increased oil was 2.69×10⁴ t. The oil recovery rate was increased by 5.17% through single well water injection huffing and puffing.

The spatial distribution of fault controlled fractured-cavity body is complex, and there are partial separated zones in some reservoir units ^[26]. Due to the limited penetrated volume of poor connected fractured-cavity reservoir in single well, the initial production of single well was high, and the production decreased quickly, resulting in low recovery degree. The separated zone was opened by high pressure water injection to break through the separated zones. High-pressure injected water is a high-energy carrier. Under the action of high pressure, the injected water entered into the separation barrier in the isolated fracture-cavity body to form a seepage channel, thereby achieving effective capacity expansion ^[27,28]. The seismic study of the fracture-cavity body in Well Yueman 25 shows that there are two fracture-cavity systems in this well (Fig. 8a). The in-situ stress analysis shows that the two fracture-cavity systems are located in the dominant fracture development direction, and are 25 m apart. The results of the geomechanical analysis on the connectivity of the fault controlled fractured-cavity body show that if the water injection pressure in the strike-slip fault zone of the well reaches 24 to 35 MPa, the internal separation barrier of fracture-cavity bodies with a thickness of more than 200 m can be broken vertically. The connection of the upper and lower part of the reservoir can be realized to achieve effective expansion.

In Well Yueman 25, flowing production lasted for 200 days, with cumulative oil production of 1.22×10⁴ t. After the oil pressure dropped to zero, water injection began. During the first two rounds of water injection, the cumulative water injection volume was 20 763 m³. The water injection at this stage mainly made up for the underground deficit caused by production. The oil well pressure did not increased obviously. The injection and production were basically balanced. There was no effective connection between near well fracture cave system 1 and far well fracture cave system 2 (Fig. 8b). During the third round of water injection, the production well pressure increased significantly. The water injection volume was 17611 m³ in the third round, and the cumulative water injection volume was 38 374 m³. When the water injection pressure reached 5.5 MPa, a short plateau area occurred in the pressure curve. It was predicted that the fracture and cave separation barrier around the well began to break gradually. With the increase in the cumulative water injection, the water injection pressure increased rapidly. During the fourth round of water injection, when the cumulative water injection volume reached 50977 m³, the pressure dropped slightly, indicating that a small amount of water had entered the fracture-cavity system 2. The fracture-cavity system 2 was not affected by production, and the pressure was maintained at a high level. Therefore, the amount of entered water was limited. The pressure rose rapidly, the highest pressure before the end of the water injection reached 21 MPa.

Before the first three rounds of high-pressure water injection in the Well Yueman 25 taking effect, with the increase of water injection rounds, oil replacement rate (ratio of the ground volume of the produced crude oil to the injected water volume) by the water injection huff and puff was 1.63, 0.64, 0.45, and the water injection effect became poor gradually. After the high-pressure water injection, the new fracture-cavity system was communicated, the peripheral area started to supply fluid. During the 4^th to 6^th rounds of water injection huffs and puffs, the oil replacement rate increased gradually to 0.63, 0.65, 0.73, and the water injection effect was improved as a whole. Up to August 10, 2021, 8 rounds of water injection had been carried out in Well Yueman 25. The cumulative water injection volume was 106 747 m³, and the cumulative increased oil was 61 611 t. The effect of replacing oil by water injection is good.

Take the Fuyuan 210 fault zone in Fuman Oilfield as an example, which currently has 11 oil wells in production. Statically, we focus on analyzing the geomechanical activity of the fault zone, judging the connectivity of the fault, and designing the interference well test plan according to the static study result. Finally, combined with the dynamic characteristics of production, the Fuyuan 210 fault zone is divided into four connected units from north to south (Fig. 9). FY210-H7 in the north is an isolated unit, and the fracture geomechanical activity around the well was very weak; FY210H, FY210-H1, FY210-H3, FY210-H4, and FY210-H6 are the same connected unit; FY210-H10 and FY210-H12 are the same connected unit; FY210-H14 in the southern part is an isolated unit.

Through the single-well energy evaluation, it is clear that the N_pr of the single well in the Fuyuan 210 fault zone is generally lower than 2.5, and the D_pr is generally higher than 2.0. The fault zone as a whole belongs to the weak energy connection type. After the oil well was put into production, the oil pressure and production decreased rapidly, and the annual decline rate was over 25%. The oil wells in the Fuyuan 210 fault zone relied on natural energy to produce, and it was difficult to stabilize production. After an average single well was put into production for 404 days, the productivity decreased from the initial 1246 t/d to 304 t/d. Before the implementation of unit water injection, 5 wells had stopped producing. The remaining 6 wells were close to stopping production. The average single well production degree was only 6% after stopping production.

In Unit FY210-H3, unit water flooding field test is carrying out currently. Water injection is conducted in wells FY210-H1 and FY210-H4 in this unit, being shallow unit water injection. Two wells have been drilled, and 81.5 m and 85.0 m reservoirs have been penetrated, respectively (Fig. 10). After only 15 days of water injection (June 30, 2021), Well FY210-H3, drilled in the deep part of the reservoir (240 m reservoir was penetrated) was effective, and the oil pressure rose steadily, while the response in the Well FY210H in the shallow part (73.85 m reservoir was penetrated) was not obvious. On August 10, 2021, after the daily water injection rate of Well FY210-H1 increased, the oil pressure of deep Well FY210-H3 increased significantly higher than that in the shallow Well FY210H. Practice has proved that water injection in fault-controlled reservoirs is mainly affected by gravity, and water flows downwards preferentially into the lower part, the deep well will take effect first. At present, the accumulative water injection volume in this unit is 7.4×10⁴ m³, and the accumulative increased oil in effective well is 2.6×10⁴ t.

Fault-controlled fracture-cavity carbonate reservoir units can be divided mainly into three types, namely the strong natural energy connected type, the weak natural energy connected type, and the weak natural energy isolated type. Faults are the main controlling factor for the formation of different types of reservoirs, and determined that the reservoirs are mainly developed along the longitudinal direction of the fault on a large scale, forming a complex fracture-cavity bodies composed of fractures, cavities and caves, which is favorable for the vertical flow of injected water.

The fault (fracture) activity index of fault-controlled fractures and cave bodies can be used to characterize effectively the reservoir connectivity and predict the effective direction of water injection. The flow mathematical model of fault-controlled carbonate reservoirs under the coexistence of seepage and free flow states can be used to describe quantitatively the regularity of fluid flow in the fracture-cavity body.

The injected water in the fault-controlled fracture- cavity body is weakly affected by the capillary force of the lithology body, and the oil-water movement is mainly controlled by gravity. It has been confirmed by pilot tests that the three methods, namely the single well water injection huff and puff method, high-pressure water injection method, and the water injection gravity flooding are effective methods in improving the oil recovery efficiency.

During the process of the physical simulation of water flooding and the research on the mechanisms of unit water injection development, we got support from senior engineer WANG Qi and engineer ZHANG Qi of Oilfield Development Research Institute, PetroChina Exploration and Development Research Institute. The numerical simulation technology has been guided by associate professor HUANG Chaoqin, Dr. LIU Lijun and other researchers of Petroleum Engineering Research Institute, China University of Petroleum (East China). The authors appreciate their help.

A—unit contact area, m²;

d—distance between the center points of the two grids, m;

D—reservoir depth (downward is positive), m;

D_pr—formation pressure drop value of 1% of the recovered geological reserves, MPa;

F_in—fracture activity index, dimensionless;

g—acceleration of gravity, m/s²;

G₁—normalized value of fracture shear normal stress ratio, dimensionless;

G₂—normalized value of fracture critical injection pressure, dimensionless;

i, j—direction identification number;

K—absolute permeability, m²;

K—absolute permeability tensor, m²;

K_r—relative permeability, dimensionless;

${{\tilde{K}}_{\text{r}}}$—phase flow fraction, dimensionless;

l—the unit distance vector between the center points of the two grids, dimensionless;

M—stress tensor at a point on the crack surface, Pa;

n—unit normal vector, dimensionless;

N_pr—flexible output ratio, dimensionless;

p—pressure, Pa;

p_cow—capillary pressure, Pa;

p_in—critical injection pressure, Pa;

p_p—formation pore pressure, Pa;

q—fluid source and sink, s^-1;

S—fluid phase saturation,%;

t—time, s;

T—conductivity, m²•m;

v—fluid phase percolation velocity, m/s;

V—volume, m³;

W₁—the weight of the ratio of normal shear stress, dimensionless;

W₂—weight of critical injection pressure, dimensionless;

γ—the angle between the normal of the crack surface and the minimum principal stress, (°);

λ—the maximum principal stress and the median principal stress constitute the angle between the in-plane crack strike projection and the maximum principal stress, (°);

μ—ratio of normal stress and shear stress, dimensionless;

μ_β—fluid phase viscosity, Pa•s;

ρ—fluid phase density, kg/m³;

σ—stress, MPa;

σ_ne—effective normal stress, MPa;

τ—shear stress, MPa;

ϕ—porosity, %;

Δp—pressure difference on both sides of gravel layer, Pa;

c—cave;

d—lower part;

f—crack;

m—matrix rock block;

min, max—maximum and minimum value;

o—oil phase;

u—upper part;

w—water phase;

β—fluid phase, the value is o, w.