Study on Variation Laws of Fluid Threshold Pressure Gradient in Low Permeable Reservoir

: A study on the seepage characteristics and laws of nano-micron pore throat in a low permeable reservoir matrix is of great signiﬁcance for promoting high e ﬃ cacy of low permeable reservoirs. Threshold pressure gradient (TPG) is an essential factor to reﬂect the seepage law. Here, variation laws of TPG and its inﬂuencing factors of low reservoir ﬂuid are analyzed systematically through physical simulation experiment. Throat diameter distribution of cores was measured by a mercury injection method, and it was found that with the decrease of pore throat median diameter, TPG increase appeared slowly ﬁrst and fast afterwards. The patterns of the TPG with permeability in water and oil were compared. Results showed that the TPG versus permeability gave power functions in a form and the TPG in oil was more than two times larger than that of water. Besides, TPG in two-phase ﬂow was investigated by the stabilization method. Tests revealed that the higher the oil saturation, the greater the TPG value, and the TPG in two-phase ﬂow is always higher than that of single-phase ﬂow under the same conditions, which function as the combined action of the capillary force. In addition, the e ﬀ ects of core length, ﬂuid type, and core wettability on the TPG were studied systematically, which has guiding signiﬁcance for the development of a low permeability reservoir.


Introduction
In recent years, studies on unconventional reservoirs such as low permeability and shale reservoirs have attracted extensive attention [1][2][3][4][5][6][7]. A low permeability reservoir has the characteristics of small pore throat, low permeability, prominent capillary phenomenon and complex structure, which leads to the great difference of its seepage mechanism and oil-water migration law from conventional reservoirs [6][7][8][9][10][11][12][13][14]. Main research results indicate that fluid flow in low permeability reservoirs has obvious characteristics of non-Darcy flow and no longer conforms to Darcy's law [15][16][17]. The characteristics of non-Darcy seepage flow are typically nonlinear, that is, the flow velocity is obviously lower than Darcy flow or the nonlinear section appears when the pressure gradient is small. Furthermore, there is a threshold pressure gradient (TPG), which is the intersection between the nonlinear part and the pressure gradient axis [18][19][20]. When the pressure gradient is lower than TPG, the fluid cannot flow. At this point, the relationship between the pressure gradient and flow velocity is not a simple linear relationship, but a complicated nonlinear relationship [21][22][23]. When the pressure gradient increases to a certain value, the relationship between the pressure gradient and the flow rate is close to a linear relationship [23][24][25][26][27][28].
Relevant studies show that it is necessary to take TPG into account in low permeability reservoir fluid flow. Pascal studied the effect of TPG on the fluid flow through porous media [29]. Yan Qinglai et al. summarized the experimental results of single-phase and oil-water two-phase flow in low-permeability reservoirs. It was proposed that oil and water are in non-Darcy seepage under low seepage velocity, and in quasi-linear seepage with the TPG at a higher seepage velocity [30][31][32][33][34]. Li Zhongfeng et al. discussed the influence of boundary layers on the seepage law in porous media. It is considered that the fluid in porous media is divided into two parts, the internal free phase and boundary layer fluid. The boundary layer fluid has a higher density than the internal free phase. They found that the thickness of the boundary layer decreases with the increase of pressure gradient and increases with the increase of fluid viscosity. The characteristics of non-Darcy flow are more serious while boundary layer thickness is bigger [17][18][19]35].
In addition, many investigations indicate that TPG has a significant impact on the determination of development scheme, productivity prediction of fractured wells, and distribution of remaining oil [15][16][17][18][19][34][35][36][37][38][39]. Pascal studied the effect of TPG on oil in non-steady flow by means of analysis, numerical, and integral methods [29]. Zeng analyzed the influence of fluid components and permeability on TPG. The results show that power function is more suitable to characterize the relationship between TPG and permeability [38]. Civan et al. established a seepage model for hydraulic fracturing reservoirs and considered the influence of TPG in fractures. With the presence of TPG, per-well production will continue to decline as permeability decreases. In fractured horizontal Wells, different fracture positions correspond to different TPG, making production prediction difficult.
However, while many experts have studied TPG in low permeability reservoirs, there are still few reports about the variation laws of TPG under different factors, especially for two-phase flow. A large number of research studies focus on numerical simulation and modeling, but the experimental research is not systematic enough. Therefore, the purpose of this work is to systematically elucidate the variation laws of TPG of different fluids in low permeable reservoirs by main experimental methods. We have explored the characteristics of pore structure by a throat diameter distribution curve measurement. To study the TPG in single-phase flow, variation patterns were compared. The effects of different factors on the TPG, including the pore throat median diameter, core length, fluid type and core wettability, were conducted. Moreover, the effect of oil-water flow ratio on the TPG in two-phase flow was investigated.

Materials
Inorganic salts were proportionally prepared to simulate formation water; dehydrated crude oil and kerosene was prepared by 2:5 volume ratio to simulate formation oil, of which viscosity is 67 mPa·s. All inorganic salts were purchased from Sinopharm Chemical Reagent Co., Ltd., China, including NaCl, Na2SO4, NaHCO3, CaCl2, MgCl2·6H2O, and KCl. The dosage was referred to Table 1. The outcrop cores with three different Klinkenberg gas permeabilities were selected from the Changqing low permeability reservoir.

Mercury Intrusion Method
A mercury intrusion meter (AutoPore IV, Micromeritics Co., Ltd., China) was used to measure the core pore throat distribution curve. First, the sample cup was vacuumized. Second, the mercury was pressed into the porous sample, resulting in a change in the length of the mercury column in the capillary. Thus, the capacity was changed. After the changes in the quantity of electricity were transformed into mercury variation, the pore characteristics of the porous materials could be measured. Three kinds of cores (ϕ 2.5 cm × 10 cm) with different permeabilities in the low-permeability reservoir were screened out first. Their parameters, such as porosity Φ and permeability k were measured, respectively. Second, cores were cut into sections with length l of 0.5 cm, 1 cm, 2 cm and 5 cm, and the corresponding cores were vacuumized and saturated with water/oil according to the different requirements of displacement fluid. Third, each displacement fluid was packed into an intermediate container, which injects simulated formation water/oil into the core through a micro pump. The confining pressure was controlled by a hand pump. The steady inlet pressure p at different flow rate Q was recorded, and the curves of p/l versus Q were obtained. The above process was repeated three times to ensure the accuracy of the results. The TPG under the experimental condition was the vertical value of the curve extending over the intersection of the vertical axis.
The equipment used during single-phase TPG measurement included an Isco pump (Model 260D, Teledyne Isco, America), core holder, intermediate container, etc. The experimental equipment diagram is shown in Figure 1a.

Characteristics of Pore Structure in Low Permeable Reservoirs
The mercury injection method is an effective petrophysical method to investigate the pore structure of reservoirs, which take mercury as a non-wetting fluid and use the relationship between

Core Wettability Transformation
First, the water-wet cores were put into the oven at 75 • C. Second, a mixture of 20 wt% n-heptane solution and 80 wt% simulated oil was prepared. Cores were put into the mixture to be vacuumized and saturated. Then, cores and the mixed system were taken out and aged together at 75 • C for 4 days. Next, every core was displaced by n-heptane solution until the outlet liquid was colorless. All cores were put into the oven for 8 h at 110 • C. After the cores were taken out, the entire transformation process was at the end. Wettability of cores was measured by the goniometer (JC2000D2, Zhongchen Co., Ltd., China).

Two-Phase TPG Measurement
First, three cores with length of 5 cm and permeability of 0.98 mD, 8.16 mD, and 29 mD, respectively, were vacuumized and saturated with simulated water. Then, while the total flow was constant, oil and water were pumped into the core according to a certain flow ratio. The steady inlet pressure was recorded. While the ratio was constant, the previous step was repeated after reducing the total flow. The oil-water flow ratio was changed and the oil-water two-phase TPG under different oil-water flow ratios was measured. A different two-phase TPG can be measured after changing the oil-water flow ratio. The experimental equipment diagram is shown in Figure 1b

Characteristics of Pore Structure in Low Permeable Reservoirs
The mercury injection method is an effective petrophysical method to investigate the pore structure of reservoirs, which take mercury as a non-wetting fluid and use the relationship between cumulative immersion volume and inlet pressure to get the pore diameter corresponding to the different pressure [40]. Figure S1 in the Supporting Information shows the pore throat distribution curves obtained by the rate-controlled mercury injection experiment of three outcrop cores [Supplementary material Figure S1]. It can be seen that for every core, there is a pore throat median diameter which corresponds to an immersion volume. Besides, the pore throat median diameter increases along with gas permeability. The results demonstrate that the filtration capacity of reservoir fluid can be characterized not only by permeability, but also by pore throat median diameter. Table 2 gives the pore throat data of three outcrop cores. When the pore throat curvatures of different cores are close, the permeability of the core is proportional to the mean value of the pore throat median diameter. For the core with higher tortuosity, its pore throat median diameter is higher than those with similar permeability.

The Variation Pattern of the TPG in Single-Phase Flow
The most commonly used corrected Darcy's law is as follows: Energies 2020, 13, 3704 where v-flow rate, m/s; K-permeability, 10 −3 µm 2 ; µ-viscosity, mPa·s; λ-threshold pressure gradient, MPa/m; ∇p-pressure gradient, MPa/m [41]. The above formula is based on the complicated distribution of pore throat sizes in low permeability reservoirs. In a low permeability reservoir, the fluid mainly passes through a large throat when the pressure gradient is small. This is because the viscous force in the large throat is relatively small, so the fluid can easily overcome it to flow. Meanwhile, the flow space of fluids changes into a small throat when the pressure gradient increases slowly, on account of high viscous force. Therefore, the maximum TPG, which is denoted as λ max , is determined by the smallest pore throat. Similarly, the minimum TPG, which is denoted as λ min , is determined by the largest pore throat. As shown in Figure S2 in the Supporting Information [Supplementary material Figure S2], the relationship between seepage velocity and pressure gradient in a low permeability reservoir can be divided into three stages: no flow section, nonlinear section and quasi-linear section. If the value of λ min is zero, it means that the fluid which flows through the largest pore throat does not need the TPG. If the value of λ min is bigger than zero, it means that there is the TPG for fluid flowing through the largest pore throat [42]. The figure shows the locations of λ max , λ min , and λ pesudo on the curve of flow rate versus pressure gradient. In the following sections, the pseudo TPG is regarded as the TPG in this study.
The variation of TPG with permeability for the fluids of water and oil was compared in Figure 2. Although the properties of the two fluids vary a lot, the TPG variations follow the same pattern. By fitting the curves of these two fluids, the following functions were obtained.
Energies 2020, 13, x FOR PEER REVIEW 6 of 18 From Equation (7), the TPG versus permeability gives a straight line with the slope value of −1 in the semi logarithmic coordinate system, which indicates the reverse proportionality between TPG and permeability.
All analysis above indicates that the TPG versus permeability gives power functions in low permeable reservoirs, which can be summarized in the following form [44]:

The Effect of Pore Throat Median Diameter on the TPG
To clarify the laws between the TPG and the pore throat median diameter, two fluids were injected into cores with different permeabilities. Figure 3a,b show the relationship between TPG and the pore throat median diameter of water and oil in cores with different lengths. Whether the fluid is water or oil, the TPG decreases gradually along with the increase in the pore throat median diameter, which means that there is an inverse relationship between flowing space and the TPG. This is related to the boundary layer thickness [30][31][32]. For single-phase liquid flow, there would be interaction between the solid and liquid, which causes the boundary layer on the pore wall. The existence of the boundary layer is the main cause of TPG. Especially for low-permeability reservoirs, which has a relatively thick boundary layer due to the small pores, this cannot be ignored. The pore throat median While λ is higher than p, the commonly used corrected Darcy's law formula is as follows: From the definition of TPG, the mathematical equation can be described as [43]: By combining Equations (5) and (6) and taking logarithms on both sides of the equal sign, the equation can be written in the following implicit form: From Equation (7), the TPG versus permeability gives a straight line with the slope value of −1 in the semi logarithmic coordinate system, which indicates the reverse proportionality between TPG and permeability.
All analysis above indicates that the TPG versus permeability gives power functions in low permeable reservoirs, which can be summarized in the following form [44]:

The Effect of Pore Throat Median Diameter on the TPG
To clarify the laws between the TPG and the pore throat median diameter, two fluids were injected into cores with different permeabilities. Figure 3a,b show the relationship between TPG and the pore throat median diameter of water and oil in cores with different lengths. Whether the fluid is water or oil, the TPG decreases gradually along with the increase in the pore throat median diameter, which means that there is an inverse relationship between flowing space and the TPG. This is related to the boundary layer thickness [30][31][32]. For single-phase liquid flow, there would be interaction between the solid and liquid, which causes the boundary layer on the pore wall. The existence of the boundary layer is the main cause of TPG. Especially for low-permeability reservoirs, which has a relatively thick boundary layer due to the small pores, this cannot be ignored. The pore throat median diameter is the critical parameter reflecting the size of the pore throat. While the pore throat median diameter is lower, that is, the throat is fine, the proportion of the boundary layer becomes larger. The effective pore size for fluid to flow through is smaller, so the TPG value is higher [33,34].
Besides, the TPG of water and oil both decrease with the increase of the core length under the same median diameter of the pore throat ( Figure 3). Comparing two diagrams together, the TPG of oil is higher than that of water under the same condition, indicating that the properties of the fluids such as viscosity, affect the TPG. In addition, it can be seen from the diagrams that with the increase of the pore throat median diameter, the TPG downward trend of the two fluids is different. The slope in Figure 3a becomes smaller while the slope in Figure 3b becomes bigger along with the increase of the pore throat median diameter. This is mainly because viscosity is another important factor. With the increase of the pore throat median diameter, more fluid flows through the channel. Viscosity plays a more important role in determining the value of TPG, so the trend of oil with high viscosity is different from that of water with low viscosity [28,29].  Besides, the TPG of water and oil both decrease with the increase of the core length under the same median diameter of the pore throat ( Figure 3). Comparing two diagrams together, the TPG of oil is higher than that of water under the same condition, indicating that the properties of the fluids such as viscosity, affect the TPG. In addition, it can be seen from the diagrams that with the increase of the pore throat median diameter, the TPG downward trend of the two fluids is different. The slope in Figure 3a becomes smaller while the slope in Figure 3b becomes bigger along with the increase of the pore throat median diameter. This is mainly because viscosity is another important factor. With the increase of the pore throat median diameter, more fluid flows through the channel. Viscosity

The Effect of Core Length on the TPG
In order to investigate the effect of core length on the TPG, two fluids were injected into cores with different lengths of 0.5 cm, 1 cm, 2 cm, and 5 cm. While fluids flowed through these core sections, their TPG were measured. The relationship between TPG and core length is shown in Figure 3c,d.
From Figure 3c,d, with the increase of core length, the TPG of water and oil decreases under the same permeability. The flow of fluid in different lengths of rock is analogous to the flow in a long tube and short tube. From the knowledge of fluid mechanics, the total head loss of the short tube includes local head loss and frictional head loss, while the total head loss of the long tube is similar to frictional Energies 2020, 13, 3704 9 of 17 head loss. As shown in Figure 4, frictional head loss is caused by viscous resistance while flowing. Local head loss includes the interaction of fluid clusters in the local range and the vortex produced in the fluid. With the decrease of the core length, the local head loss cannot be ignored. Therefore, as the total head loss increases, the TPG of fluid increases.
In order to investigate the effect of core length on the TPG, two fluids were injected into cores with different lengths of 0.5 cm, 1 cm, 2 cm, and 5 cm. While fluids flowed through these core sections, their TPG were measured. The relationship between TPG and core length is shown in Figure 3c,d.
From Figure 3c,d, with the increase of core length, the TPG of water and oil decreases under the same permeability. The flow of fluid in different lengths of rock is analogous to the flow in a long tube and short tube. From the knowledge of fluid mechanics, the total head loss of the short tube includes local head loss and frictional head loss, while the total head loss of the long tube is similar to frictional head loss. As shown in Figure 4, frictional head loss is caused by viscous resistance while flowing. Local head loss includes the interaction of fluid clusters in the local range and the vortex produced in the fluid. With the decrease of the core length, the local head loss cannot be ignored. Therefore, as the total head loss increases, the TPG of fluid increases.

The Effect of Fluid Type on the TPG
To study the effect of fluid type on TPG, two fluids were injected into cores with a different permeability and length, separately. The TPG of oil is higher than that of water in different core sections under the same permeability, as shown in Figure 5.

The Effect of Fluid Type on the TPG
To study the effect of fluid type on TPG, two fluids were injected into cores with a different permeability and length, separately. The TPG of oil is higher than that of water in different core sections under the same permeability, as shown in Figure 5.
From Figure 3c,d, with the increase of core length, the TPG of water and oil decreases under the same permeability. The flow of fluid in different lengths of rock is analogous to the flow in a long tube and short tube. From the knowledge of fluid mechanics, the total head loss of the short tube includes local head loss and frictional head loss, while the total head loss of the long tube is similar to frictional head loss. As shown in Figure 4, frictional head loss is caused by viscous resistance while flowing. Local head loss includes the interaction of fluid clusters in the local range and the vortex produced in the fluid. With the decrease of the core length, the local head loss cannot be ignored. Therefore, as the total head loss increases, the TPG of fluid increases.

The Effect of Fluid Type on the TPG
To study the effect of fluid type on TPG, two fluids were injected into cores with a different permeability and length, separately. The TPG of oil is higher than that of water in different core sections under the same permeability, as shown in Figure 5. The results show that the boundary layer fluid is one of the main factors affecting the TPG of fluids. It can be known from the boundary layer theory that because of the interfacial interaction between solid and liquid, polar molecules in oil adsorb the mineral particles while flowing, forming a boundary fluid on the particle surface, which is not easy to flow. The ultimate shear stress and viscosity of the liquid in the layer should be much higher than those of the bulk fluid. Because of the presence of the oil boundary layer, the cross-sectional area for fluid seepage is smaller than the crosssectional area of channel ( Figure 6). The results show that the boundary layer fluid is one of the main factors affecting the TPG of fluids. It can be known from the boundary layer theory that because of the interfacial interaction between solid and liquid, polar molecules in oil adsorb the mineral particles while flowing, forming a boundary fluid on the particle surface, which is not easy to flow. The ultimate shear stress and viscosity of the liquid in the layer should be much higher than those of the bulk fluid. Because of the presence of the oil boundary layer, the cross-sectional area for fluid seepage is smaller than the cross-sectional area of channel ( Figure 6). Oil has a higher viscosity than water. Heavy components of oil are more likely to be adsorbed by the wall of porous media. While flowing through the same rock, the thickness of boundary layers in oil is greater than that of water, and the space for fluid seepage is smaller. From the Laplace Equation at arbitrary interface: where is the capillary force (mN), is the interfacial tension (mN/m), and 1 and 2 are the radius of curvature of the liquid film formed between two phases (m). Decrease of the pore radius makes an increase in the capillary force, resulting in greater flowing resistance for fluid. Thus, the TPG is higher.

The Effect of Core Wettability on the TPG
To investigate the effect of core wettability on the TPG, all the water-wet cores used above were transformed into oil-wet cores by immersing them with a mixture of 20 wt% n-heptane solution and 80 wt% simulated oil. Wettability can be determined by measuring the contact angle of the oil/chip/water slice system [45][46][47]. Taking the 1.03 mD core as an example, Figure 7 shows the results of the contact angles of the oil/water/water-wet core slice system and the oil/water/oil-wet core slice system. As shown in Figure 7a, the contact angle of oil on the surface of the water-wet core slice was 122°. While in Figure 7b, the contact angle of oil on the oil-wet core slice decreased to 28.8°. By comparison, the addition of a mixture of n-heptane solution and simulated oil can obviously alter the wettability of the core from water-wet to oil-wet.  Oil has a higher viscosity than water. Heavy components of oil are more likely to be adsorbed by the wall of porous media. While flowing through the same rock, the thickness of boundary layers in oil is greater than that of water, and the space for fluid seepage is smaller. From the Laplace Equation at arbitrary interface: where P C is the capillary force (mN), σ is the interfacial tension (mN/m), and R 1 and R 2 are the radius of curvature of the liquid film formed between two phases (m). Decrease of the pore radius makes an increase in the capillary force, resulting in greater flowing resistance for fluid. Thus, the TPG is higher.

The Effect of Core Wettability on the TPG
To investigate the effect of core wettability on the TPG, all the water-wet cores used above were transformed into oil-wet cores by immersing them with a mixture of 20 wt% n-heptane solution and 80 wt% simulated oil. Wettability can be determined by measuring the contact angle of the oil/chip/water slice system [45][46][47]. Taking the 1.03 mD core as an example, Figure 7 shows the results of the contact angles of the oil/water/water-wet core slice system and the oil/water/oil-wet core slice system. As shown in Figure 7a, the contact angle of oil on the surface of the water-wet core slice was 122 • . While in Figure 7b, the contact angle of oil on the oil-wet core slice decreased to 28.8 • . By comparison, the addition of a mixture of n-heptane solution and simulated oil can obviously alter the wettability of the core from water-wet to oil-wet. Oil has a higher viscosity than water. Heavy components of oil are more likely to be adsorbed by the wall of porous media. While flowing through the same rock, the thickness of boundary layers in oil is greater than that of water, and the space for fluid seepage is smaller. From the Laplace Equation at arbitrary interface: where is the capillary force (mN), is the interfacial tension (mN/m), and 1 and 2 are the radius of curvature of the liquid film formed between two phases (m). Decrease of the pore radius makes an increase in the capillary force, resulting in greater flowing resistance for fluid. Thus, the TPG is higher.

The Effect of Core Wettability on the TPG
To investigate the effect of core wettability on the TPG, all the water-wet cores used above were transformed into oil-wet cores by immersing them with a mixture of 20 wt% n-heptane solution and 80 wt% simulated oil. Wettability can be determined by measuring the contact angle of the oil/chip/water slice system [45][46][47]. Taking the 1.03 mD core as an example, Figure 7 shows the results of the contact angles of the oil/water/water-wet core slice system and the oil/water/oil-wet core slice system. As shown in Figure 7a, the contact angle of oil on the surface of the water-wet core slice was 122°. While in Figure 7b, the contact angle of oil on the oil-wet core slice decreased to 28.8°. By comparison, the addition of a mixture of n-heptane solution and simulated oil can obviously alter the wettability of the core from water-wet to oil-wet.  As shown in Figure 8a,b, the regulations of pore throat median diameter, length, and fluid type are still applicable in oil-wet cores. The TPG decreases with the increase of the pore throat median diameter. When the core length is longer, the TPG of the fluid becomes lower. Besides, under the same condition, the TPG of oil is higher than that of water. Compared to the initial water-wet cores (Figure 3c,d), the TPG of oil is reduced and the TPG of water is increased conversely. Take a core with k = 1.03 mD as an example, TPG of oil in a 1cm core has fallen from 30 MPa/m to 21.8 MPa/m, dropped by about 27%, while the TPG of water rose from 11.5 MPa/m to 13.9 MPa/m, up 20.8%.
Energies 2020, 13, x FOR PEER REVIEW 13 of 18 As shown in Figure 8a,b, the regulations of pore throat median diameter, length, and fluid type are still applicable in oil-wet cores. The TPG decreases with the increase of the pore throat median diameter. When the core length is longer, the TPG of the fluid becomes lower. Besides, under the same condition, the TPG of oil is higher than that of water. Compared to the initial water-wet cores (Figure 3c,d), the TPG of oil is reduced and the TPG of water is increased conversely. Take a core with k = 1.03 mD as an example, TPG of oil in a 1cm core has fallen from 30 MPa/m to 21.8 MPa/m, dropped by about 27%, while the TPG of water rose from 11.5 MPa/m to 13.9 MPa/m, up 20.8%. The TPG of water and oil in water-wet cores and oil-wet cores is compared in Figure 8c,d, respectively. The TPG of water in oil-wet cores is higher than that in water-wet cores ( Figure 8c). As the wettability of the core was transformed from water-wet into oil-wet, the thickness of the boundary layer increased. The effective seepage channel became smaller, so the TPG was higher. Figure 8d shows that the TPG of oil in oil-wet cores is lower than that in water-wet cores. It was easier for oil to spread on the surface of oil-wet cores than water-wet cores. Thus, it was more conductive for oil to flow through oil-wet cores, so the TPG was lower. Specifically, the water film on the surface of mineral particles in water-wet cores is thick and difficult to break, and the capillary force generated The TPG of water and oil in water-wet cores and oil-wet cores is compared in Figure 8c,d, respectively. The TPG of water in oil-wet cores is higher than that in water-wet cores ( Figure 8c). As the wettability of the core was transformed from water-wet into oil-wet, the thickness of the boundary layer increased. The effective seepage channel became smaller, so the TPG was higher. Figure 8d shows that the TPG of oil in oil-wet cores is lower than that in water-wet cores. It was easier for oil to spread on the surface of oil-wet cores than water-wet cores. Thus, it was more conductive for oil to flow through oil-wet cores, so the TPG was lower. Specifically, the water film on the surface of mineral particles in water-wet cores is thick and difficult to break, and the capillary force generated by it has an obvious blocking effect on oil migration. The oil needs a higher driver pressure to overcome the resistance to achieve a stable continuous flow state, so the TPG becomes bigger. Instead, the water film on the oil-wet core surface is easily destroyed, which leads to the oil-phase being attached to the pore surface and spread out. Therefore, the capillary force acts as the power at this time; the TPG of oil-wet core is small [48][49][50].

The TPG in Two-Phase Flow
The TPG of oil-water two-phase is the pressure gradient that should be overcome when oil and water begin to flow simultaneously at any water saturation. The oil-water two-phase TPG consists of two parts: one part is the viscous resistance between fluid and pore surface, which is the same as the TPG of single-phase flow, and the other part is the capillary resistance produced by interactions between two fluids. In the process of oil-water two-phase seepage, oil and water flow through the occupied pore space, respectively. They militate with the pore wall and produce their respective boundary layer fluid and the TPG. The flow of one phase necessarily drives the other phase, so the TPG of the two phases is equal [51]. There is no experimental standard for two-phase TPG measuring. A large number of experiments indicate, so long as the oil-water flow ratio is certain, the saturation of fluids in cores will keep constant. Hence, the stabilization method was adopted in the experiment [52].
In order to compare the TPG difference between single-phase flow and two-phase flow, oil and water were injected into three kinds of permeability water-wet cores with various flow ratios. As shown in Figure 9, the TPG value of the two-phase is always higher than the single-phase (ratio 1:0 and 0:1) at the same permeability. And in the process of two-phase flow injection, TPG increased sharply with the oil saturation. TPG reached a maximum value of 26 MPa/m when the oil-water flow ratio was 3:1, which was 2.4~4.3 times that of the oil/water injection alone. The presence of two phases results in the mutual interference between solid-liquid interaction and interface interaction in the porous media, so that the phase permeability of each phase changed, the seepage behavior varied along with it, and the TPG changed greatly as well.
Energies 2020, 13, x FOR PEER REVIEW 15 of 18 by it has an obvious blocking effect on oil migration. The oil needs a higher driver pressure to overcome the resistance to achieve a stable continuous flow state, so the TPG becomes bigger. Instead, the water film on the oil-wet core surface is easily destroyed, which leads to the oil-phase being attached to the pore surface and spread out. Therefore, the capillary force acts as the power at this time; the TPG of oil-wet core is small [48][49][50].

The TPG in Two-Phase Flow
The TPG of oil-water two-phase is the pressure gradient that should be overcome when oil and water begin to flow simultaneously at any water saturation. The oil-water two-phase TPG consists of two parts: one part is the viscous resistance between fluid and pore surface, which is the same as the TPG of single-phase flow, and the other part is the capillary resistance produced by interactions between two fluids. In the process of oil-water two-phase seepage, oil and water flow through the occupied pore space, respectively. They militate with the pore wall and produce their respective boundary layer fluid and the TPG. The flow of one phase necessarily drives the other phase, so the TPG of the two phases is equal [51]. There is no experimental standard for two-phase TPG measuring. A large number of experiments indicate, so long as the oil-water flow ratio is certain, the saturation of fluids in cores will keep constant. Hence, the stabilization method was adopted in the experiment [52].
In order to compare the TPG difference between single-phase flow and two-phase flow, oil and water were injected into three kinds of permeability water-wet cores with various flow ratios. As shown in Figure 9, the TPG value of the two-phase is always higher than the single-phase (ratio 1:0 and 0:1) at the same permeability. And in the process of two-phase flow injection, TPG increased sharply with the oil saturation. TPG reached a maximum value of 26 MPa/m when the oil-water flow ratio was 3:1, which was 2.4~4.3 times that of the oil/water injection alone. The presence of two phases results in the mutual interference between solid-liquid interaction and interface interaction in the porous media, so that the phase permeability of each phase changed, the seepage behavior varied along with it, and the TPG changed greatly as well. The TPG of two-phase flow is affected not only by the viscous resistance caused by the boundary layer, but also by the capillary resistance between two-phase fluids. As the oil saturation becomes higher, the viscosity of the fluid is bigger. More heavy components adsorb on the wall of the porous The TPG of two-phase flow is affected not only by the viscous resistance caused by the boundary layer, but also by the capillary resistance between two-phase fluids. As the oil saturation becomes higher, the viscosity of the fluid is bigger. More heavy components adsorb on the wall of the porous media, and boundary layers thicken, so the TPG is smaller. Under the condition of high oil saturation, oil flow will be cut into an oil column by the first kind of capillary force. When truncated oil column flows, additional resistance is brought by meniscus deformation, which is called the second kind of capillary force. Besides, the Jamin effect happens when non-wetting liquid beads flow through the throat and raise the flow resistance. In a word, the combined action of the capillary force makes the TPG of two-phase flow higher than that of single-phase flow [53].

Conclusions
The variation law of TPG in single-phase flow and two-phase flow in a low-permeability reservoir was studied systematically. Based on this study, the following conclusions can be drawn.
(1) Core pore throat distribution curves were obtained by the mercury injection method. Median diameter of the pore throat is an important index of reservoir filtration capacity and increases along with gas permeability. With the decrease of the throat median diameter, TPG increased progressively; the rate was rather low at the initial stage, and then faster.
(2) TPG variation follows the same pattern in water and oil. In the semi logarithmic coordinate, the relationship between TPG and permeability is a straight line with the slope of −1. It shows that the two are inversely proportional.
(3) The value of TPG is inversely proportional to the core length. With shorter core length, flow is not only affected by frictional head loss, but also by local head loss. In addition, the value of TPG in oil is about two times higher than that in water under the same conditions, which depend on the fluid properties.
(4) By mixing with a mixture of 20 wt% n-heptane solution and 80 wt% crude oil, cores can be transformed from water-wet into oil-wet. The TPG of water in oil-wet cores is higher than that in water-wet cores, while that of oil is opposite. The main reason is the water film adsorbed on the pore surface of the oil-wet core is easily damaged, which causes the oil phase to adhere and diffuse, resulting in easier flow in the porous media.
(5) When two-phase seepage exists, the capillary phenomenon is more prominent. The combined effect of three kinds of capillary force makes the TPG of two-phase flow higher than that of single-phase flow. Furthermore, in two-phase flow, TPG increases with the proportion of oil.