Research Article | Open Access

Volume 2021 |Article ID 5559131 | https://doi.org/10.1155/2021/5559131

Shuangshuang Ren, Fei Shen, Shenglai Yang, Xiangyang Zhang, Hongwu Luo, Chaomin Feng, "Analysis and Calculation of Threshold Pressure Gradient Based on Capillary Bundle", Mathematical Problems in Engineering, vol. 2021, Article ID 5559131, 10 pages, 2021. https://doi.org/10.1155/2021/5559131

# Analysis and Calculation of Threshold Pressure Gradient Based on Capillary Bundle

Revised02 Apr 2021
Accepted06 May 2021
Published27 May 2021

#### 1. Introduction

During the development of low-permeability reservoirs, due to the existence of threshold pressure gradient, the injection-production pressure difference is large, which causes some low-permeability intervals being poorly developed. Therefore, studying the change rule of threshold pressure gradient as well as studying the relationship between the threshold pressure gradient and the reservoir physical properties has realistic guiding significance for the development of the reservoir development, well pattern deployment, and separated zone water injection, etc. .

The research is divided into the following parts. First, the experimental section is presented. Then, the results and discussion are put forward. Last, summary and conclusions are given.

Through the dynamic macro experiment and microcapillary bundle principle, the experiment can be divided into several sections for analysis, which can be more accurate.

The minimum start-up pressure gradient can not only guide the later development of the oilfield, but also enrich the theoretical study of non-Darcy low-velocity seepage.

##### 1.1. The Definition of Threshold Pressure Gradient

In a low-permeability reservoir, when the driving pressure gradient is small, the liquid cannot flow. Only when the driving pressure gradient reaches a certain value does the liquid start to flow. Now, the driving pressure gradient is called the threshold pressure gradient. The higher the threshold pressure gradient is, the more difficult it is for crude oil to flow and the poorer water absorption capacity in the oil interval. Therefore, the basic law of oil and water seepage in low-permeability reservoirs is different from that in high permeability reservoirs .

Predecessors have done some research on the mechanism of the threshold pressure gradient, mainly including assumptions: fluid boundary layer is abnormal in nature . The fluid in seepage is composed of body fluid and boundary fluid. The boundary fluid is plastic fluid, in which the flow and pressure difference follow the Buckingham–Reiner formula . From this, it can be seen that the fluid sees a nonlinear flow in the low-permeability reservoir, which is the comprehensive result of the fluid boundary layer anomaly and the behavior of the fluid in the porous medium as a plastic fluid.

#### 2. Experiment

##### 2.1. Conditions and Methods

The minimum threshold pressure gradient is generally measured using the capillary equilibrium method , but this method takes a long time. To measure the threshold pressure gradient is to use the steady-state method or the non-steady-state method to decide the pressure difference and flow in the seepage through the indoor physical simulation experiment. And then the mathematical model and the matching processing method are used to calculate the threshold pressure gradient. However, there are some disadvantages in the seepage law described of low-permeability cores.

In the experiment, because the flow rate and pressure gradient range need to be selected and determined as a straight line segment, the threshold pressure gradients calculated by the straight line segments corresponding to different experimental points are different. Besides, no experimental standard measures the threshold pressure gradient, and the results of measurements at different flow rates and pressure gradients are different [13, 14].

The main experimental equipment for this study is the high-precision microflow constant-speed pump and the pressure gauge. The flow rate of the water is controlled by a computer and the upstream pressure of the core is automatically collected. The experiment temperature is 20°C, and the core is the natural core from a low-permeability block in oilfield. The basic data are shown in Table 1. The experimental water is formulated according to the formation water composition in the study area, of which the salinity is 6813 mg/L, the density is 1.008 g/cm3, and the viscosity is 1.077 MPa s.

 Core number Diameter (cm) Length (cm) Gas logging permeability (10−3 μm2) Porosity (%) P11 2.516 5.242 1.97 13.43 P12 2.508 5.268 2.39 13.55 P13 2.516 5.594 3.45 13.47 P14 2.506 5.164 8.93 14.99 P15 2.504 6.242 9.96 14.43 P16 2.504 5.798 11.81 15.38 P17 2.512 5.20 14.96 15.27 P18 2.502 5.934 26.1 16.13 P19 2.504 6.046 28.91 17.86 P20 2.504 5.438 31.07 18.12
##### 2.2. Procedures

The experimental steps are as follows:(1)Dry core sample and weigh it, and then vacuum the core sample for 24 h. After that, keep the core sample saturated in formation water while keep vacuuming the core sample 12 h.(2)Keep the core sample stand for 24 h after saturated in formation water and then weigh the core sample.(3)Keep the constant-speed displacement at a certain speed while keeping the confining pressure higher than the flow pressure 2.5 MPa, and record the flow rate and pressure difference after the pressure is stabilized.(4)Increase flow rate and continue the constant-speed displacement procedure by using the increased flow rate and repeat step (3).(5)End the experiment and process the experimental data.

#### 3. Results and Discussion

##### 3.1. The Comprehensive Analysis of Experimental Results
###### 3.1.1. The Conventional Method for Determining the Threshold Pressure Gradient

General capillary equilibrium method is used to measure the minimum threshold pressure gradient , but this method takes longer. Quasi threshold pressure gradient on indoor physical simulation experiment of measurement is using steady or unsteady seepage differential pressure and flow rate determination, using mathematical model and matching processing methods to solve the threshold pressure gradient. There are certain problems in the theory of obtaining the low permeability start-up pressure through linear slope regression based on the experimental data of flow and pressure because the data points selected for the experiment are different and the slope of the line obtained is different, so the start-up pressure values are different [13, 14].

What stands out in Figure 2 (the data are shown in Table 2) is the markedly nonlinear of the relationship between the driving pressure gradient and the flow rate. With the increase of the pressure gradient, most of the roars have been opened, and the relationship between core flow and pressure gradient tends to be a straight line.

 Flow Q (cm3/min) The driving pressure gradient dp/dx (MPa/m) 0.030 0.35859 0.030 0.36778 0.060 0.51490 0.061 0.53328 0.094 0.78154 0.102 0.80912 0.255 1.45274 0.257 1.43435 0.515 2.28025 0.506 2.24347 0.998 3.60427 1.029 3.53071 1.476 4.78117 1.526 4.68922 2.042 5.84774 2.058 5.86613 3.109 8.29349 3.167 8.31188 4.098 10.18757 4.060 10.16918 5.116 12.65171 5.976 14.26995
###### 3.1.2. Analysis of Threshold Pressure Gradient Based on Capillary Bundle Model

Because the permeability of the small radius of the capillary is small, the threshold pressure gradient is large, so when the pressure gradient increases, the tiny capillaries start, the active permeability; namely, the increase of the straight slope of amplitude (that is the active permeability of the model varies with the rate of change of pressure gradient) should be gradually reduced. If a certain period of straight line slope increase amplitude is bigger, it is illustrated that, at a certain pressure gradient, the larger is, the more is the started capillary tube with the same or similar radius. If the capillary radius of model is according to the normal distribution and is according to the normal distribution, should be first increased and then decreased. Both active permeability and threshold pressure gradient are in agreement, as shown in Figure 1.

At the same time, the analysis above figure shows that the relationship between the apparent permeability and the reciprocal of the pressure gradient is a stable straight line when the pressure gradient is large; i.e., the reciprocal of the pressure gradient is small. The starting permeability of the straight line segment can be obtained by fitting the linear equation. The starting permeability is and the quasistarting pressure gradient is .

###### 3.1.3. The Threshold Pressure Gradient of Cores with Various Permeabilities

Based on a capillary bundle model, the analysis of the experimental results can be fitted according to the neighboring data points, because two experimental data points may result in jumps because of experimental errors, if the pressure gradients included in the selected four data points are larger, and different levels of pore throats open more, then resulting is inaccurate. Therefore, three experimental data points are selected for one fitting. The linear fitting of the flow and pressure gradient of the first three points (two measurements for each point) can be obtained. Thus, from the analysis, it can be seen that the whole low-permeability seepage flow is nonlinear, but it is divided into some stages. The first stage is the strong nonlinear stage, the intermediate stage is quasilinear stage, and then the last stage is the linear stage.

The method can be used to obtain the threshold permeability and the threshold pressure gradient value fitted by the neighboring points, and then the relationship between the threshold permeability and the threshold pressure gradient value and the flow velocity can be made. Through , the experimental flow value is divided by the cross-sectional area of the core to convert the flow velocity, and the relationship between the starting permeability, the starting pressure gradient, and the flow velocity is obtained. The results are shown in Figures 3 and 4.

Figure 3 shows the quadratic function between the threshold permeability and the threshold pressure gradient with the flow rate. According to Figure 3, threshold permeability and threshold pressure gradient values at different pressure gradients (or flow rates) can be interpolated. According to the first three points in Figure 3, the minimum threshold pressure gradient can be obtained by extrapolation. The data listed in Table 3 include the minimum threshold pressure gradient of each core, the threshold permeability and the threshold pressure gradient within the flow range of 0.1–0.6 mL/min, the threshold pressure gradient and threshold permeability obtained from the line segment without the influence of velocity sensitive effect, and the threshold pressure gradient.

 Core number Permeability, 10−3 μm2 (md) Minimum threshold pressure gradient (MPa/m) 0.1–0.6 mL/min, threshold pressure (MPa/m) 0.1–0.6 mL/min, start permeability (md) Threshold pressure gradient before the velocity sensitive (MPa/m) Start permeability before the velocity sensitive (md) P11 1.97 0.3205 3.504 0.440 22.246 0.420 P12 2.39 0.4209 1.711 0.561 19.572 0.697 P13 3.45 0.3383 2.559 0.546 18.515 1.154 P14 8.93 0.0626 0.576 0.166 — — P15 9.96 0.0495 1.138 0.197 3.843 4.216 P16 11.81 0.0161 0.450 0.092 — — P17 14.96 0.0521 0.324 0.185 1.816 6.661 P18 26.1 0.0299 0.213 0.073 — — P19 28.91 0.0198 0.290 0.062 — — P20 31.07 0.0276 0.484 0.160 1.179 16.582

In Figure 4(a), the threshold pressure gradient value is the minimum threshold pressure gradient fitted according to the matching pressure gradient of 0.03 mL/min, 0.06 mL/min, and 0.12 mL/min. In Figure 4(b), the threshold pressure gradient value is the value when the flow rate is zero according to the threshold pressure gradient and flow rate curve showed.

The minimum threshold pressure gradient value does not decrease monotonically as the permeability increases, but rather fluctuates. The relationship between threshold pressure gradient and permeability in the range of 0.1–0.6 mL/min can be fitted aswhere is the threshold pressure gradient, ; is the permeability, ; and is the correlation coefficient.

The relationship between the threshold pressure gradient and the permeability obtained from the line slice without the influence of velocity sensitive effect can be fitted aswhere is the threshold pressure gradient, ; is the permeability, ; and is the correlation coefficient.

It can be seen that the threshold pressure gradient has a better power function with the permeability. When permeability of core is more than 4 md, the threshold pressure gradient is small; when the permeability of core is less than 4 md, the threshold pressure gradient increases rapidly. The minimum threshold pressure gradient decreases as the permeability increases. The power function can better fit the variation of the data points. In Figure 4(b), the data points are smaller than those in Figure 4(a), and they can show the true minimum threshold pressure gradient better.

##### 3.2. The Application of the Threshold Pressure Gradient in Oil Field Radial Flow

The condition of the threshold pressure gradient obtained in laboratory is one dimension, but the flow is radial flow in oilfield. From the foregoing analysis, it can be seen that the threshold pressure gradient changes with the flow velocity, while, in the radial flow, the velocity at different distances is different, so the threshold pressure gradient is different.

From the foregoing analysis, the relationship between the threshold pressure gradient and velocity can be fit as a quadratic function. The relationship between the start-up pressure gradient and the flow velocity is further analyzed and matched with the actual reservoir water injection, by the formula ; the change of flow velocity with the radius under different injection flow rate can be obtained. And its expression is shown as follows:where is the threshold pressure gradient, ; , , and are the coefficients; is the flow rate, ; is the radius, ; and is the thickness, .

According to equation (6), the threshold pressure gradient can be obtained at different distances from the center of the well. Integrating the radius can get the threshold pressure at different distances from the well center, as shown in the following formula:where is the threshold pressure, ; is the reservoir radius, m; is the wellbore radius, m; is the flow rate, ; and is the thickness, m.

Through analyzing the flow rates relations at different cores permeability, it shows that the permeability is approximate to linear with the flow rate (or velocity) before 0.002 cm/s, and after this flow rate, the slope of the permeability in linear relationship with the flow rate decreases. Therefore, the seepage equation can be divided into two sections before and after 0.002 cm/s (1.728 m/d); that is, the relationship between core pressure gradient and velocity can be divided into low-speed and high-speed sections for quadratic function fitting.

Fitting the seepage equations (5) and (6) to the form of equation (8) yieldswhere is the driving pressure gradient, ; , , and are the coefficients; and is the velocity, .

The seepage equation coefficients of cores with different gas logging permeability are listed in Table 4.

 Gas logging permeability × 10−3 μm2 Low flow, <1.728 m/d High flow, >1.728 m/d a b c a2 b2 c2 1.97 4401173 41495.94 0.270135 270974.2 29381.52 11.54866 2.39 965176.1 23992.24 0.550795 299825.1 19745.3 6.815821 3.45 2483235 20298.96 0.447727 106402.5 11171.91 11.77325 8.93 690991.8 3742.243 0.12773 224498.1 3066.88 0.328465 9.96 2120510 6538.008 0.078849 229873.3 3800.372 0.873352 11.81 639844.8 6462.558 0.055449 205292.8 5856.899 0.220347 14.96 546029.8 2904.804 0.125703 23705.57 1948.148 0.811546 26.1 133718.5 1105.043 0.065861 0 775.07 0.2131 28.91 416471.6 1405.897 0.035546 0 680.41 0.2896 31.07 822100.9 2226.526 0.131521 6839.687 757.4748 1.009064 Fitting result 4E + 06k−0.724 79472k−1.176 0.6633k−0.722 4E + 6k−1.034 68163x−1.309 21.622x−1.317

By fitting the equation, the relationship between the coefficient of the quadratic term of the equation and the permeability can be obtained, and the percolation equation at any permeability can be obtained.

The statistics of water injection data in the block: the results (Table 5) show that the injection intensity Q/h is mainly between .

 r (m) Q/h (m3/d/m) 0.3 1 2 3 4.5 7.3 0.1 0.478 1.592 3.185 4.777 7.166 11.624 0.5 0.096 0.318 0.637 0.955 1.433 2.325 1 0.048 0.159 0.318 0.478 0.717 1.162 2 0.024 0.080 0.159 0.239 0.358 0.581 5 0.010 0.032 0.064 0.096 0.143 0.232 10 0.005 0.016 0.032 0.048 0.072 0.116 50 0.001 0.003 0.006 0.010 0.014 0.023 100 0.000 0.002 0.003 0.005 0.007 0.012 150 0.000 0.001 0.002 0.003 0.005 0.008

Figure 5 shows that the flow rate of injection intensity Q/h at varies with the distance from the well center. According to the relationship between velocity and distance from the center of well, the relationship between starting pressure gradient and starting pressure and distance from the center of well can be obtained. As shown in Figure 6, the blue curve is the injection intensity , the red curve is the injection intensity , and the middle small figure is the local enlargement within 0.1∼5 m. The relationship between pressure gradient and pressure drop and distance from well center can be obtained.

As the method similar to equations (3) to (8), at the injection intensity of and conditions, take the threshold pressure values at the distance of 150 m away from the well center, which is obtained from cores with different permeability, and plot it. The result is shown in Figure 6.

With an injection rate of , the pressure drop at a certain distance can be obtained by integrating the distance by driving the pressure gradient. The pressure drop values at a distance of 150 m from the center of the well with different permeability are plotted in Figure 6. The blue line is the direct calculation result of the pressure drop values calculated from the core driven pressure gradient and flow rate. The red line is the pressure drop values which are calculated from the value of the quadratic coefficient of the fitting percolation equation at different permeability. It can be seen from Figure 7 that the blue direct calculation value curve is very close to the power function curve fitted by the red fitting calculation value.

It is known from the following equation:where is the pressure, ; is the seepage pressure, ; is the threshold pressure, ; is the viscosity, Pas; is the total cross-sectional area, ; is the threshold permeability, ; is the threshold pressure gradient, ; and is the flow rate, .

The total pressure drop is divided into two parts, which is the pressure drop caused by seepage and the threshold pressure . The small proportion of the threshold pressure to the total pressure drop indicates that the pressure drop used to overcome the threshold pressure is small which is advantageous. According to the fitting formulas , in when the injecting intensity is , we can getwhere is the threshold pressure, ; is the pressure, ; and is the permeability, .

From equation (10), it can be seen that as the permeability increases, the threshold pressure occupies a smaller proportion of the total pressure drop. The ratio of starting pressure to total pressure drop is about 0.5, and the higher the permeability is, the lower the ratio is.

#### A. The Relationship of the Starting Permeability and the Quasistarting Pressure Gradient

Because the largest radius capillary of the boundary layer thickness is small, the threshold pressure gradient is low. If the pressure gradient is tiny, then the fluid in the largest radius capillary firstly begins to flow. Therefore, the minimum threshold pressure gradient matches the maximum capillary radius. There is a certain threshold value in the pore radius, and the flow cannot start until the threshold value is reached, that is to say, the pore radius is related to the threshold pressure gradient. With the increase of pressure gradient, other capillary is in seepage flow.

For different radius of capillary tube threshold, with the linear fitting of the velocity and pressure gradient, the active permeability is formed:where is the total flow of capillary bundle model; A is total cross-sectional area of bundle of capillary tubes model; is the active permeability of the bundle of capillary tubes model; and is the threshold pressure gradient of the bundle of capillary tubes model.

Combining equations (A.1) and (A.2), equations (A.3) and A.4) are obtained:

Types (A.3) and (A.4) show that the active permeability is the permeability weighted superposition that has been started in the threshold of the different radius capillary. With the increase of pressure gradient, small capillary is opened; the threshold pressure gradient and the active permeability gradually increase. The change of the active permeability with pressure gradient is as follows:

#### Data Availability

The experiment data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

1. Y. Huang, Flow Mechanics for Low Permeability Reservoir, Petroleum Industry Press, Beijing, China, 1998.
2. C. Lv, J. Wang, and Z. Sun, “An experimental study on starting pressure gradient of fluids flow in low permeability sandstone porous media,” Petroleum Exploration and Development, vol. 29, no. 2, pp. 86–89, 2002. View at: Google Scholar
3. Q. Yan, Q. He, W. Ligang et al., “An experimental study on one phase fluid flow in low permeability reservoir,” Journal of Xi’an Shiyou University, vol. 5, no. 2, pp. 1–6, 1990. View at: Google Scholar
4. L. K. Thomas, D. L. Katz, and M. R. Tek, “Threshold pressure phenomena in porous media,” Society of Petroleum Engineers Journal, vol. 8, no. 2, 1968. View at: Publisher Site | Google Scholar
5. H. Han, L. Cheng, M. Zhang et al., “Physical simulation and numerical simulation of ultra-low permeability reservoir in consideration of starting pressure gradient,” Journal of the University of Petroleum, vol. 28, no. 6, pp. 49–53, 2004. View at: Google Scholar
6. V. A. Baikov, A. Y. Davletbaev, and D. S. Ivaschenko, “Non-darcy flow numerical simulation and pressure/rate transient analysis for ultra-low permeable reservoirs,” in Proceedings of the SPE Russian Oil and Gas Exploration & Production Technical Conference and Exhibition, Moscow, Russia, October 2014. View at: Publisher Site | Google Scholar
7. J. Liu, X. Zhao, X. Liao et al., “New method to confirm the starting pressure gradient for low permeability reservoir,” Science Technology and Engineering, vol. 12, no. 32, pp. 8518–8520, 2012. View at: Google Scholar
8. A. Li, M. Liu, S. Zhang et al., “Experimental study on the percolation characteristic of extra low-permeability reservoir,” Journal of Xi’an Shiyou University, vol. 23, no. 2, pp. 35–39, 2008. View at: Google Scholar
9. J. Xu, L. Cheng, Y. Zhou et al., “A new method for calculating kickoff pressure gradient,” Petroleum Exploration and Development, vol. 34, no. 5, pp. 594–597, 2007. View at: Google Scholar
10. L. Sun, Wufan, W. Zhao et al., “The study and application of reservoir start-up pressure,” Fault Block Oil & Gas Field, vol. 5, no. 5, pp. 30–33, 1998. View at: Google Scholar
11. Q. Wang, H. Tang, D. Lv et al., “An experimental study on threshold pressure gradient in low permeability reservoir,” Petroleum Geology and Recovery Efficiency, vol. 18, no. 1, pp. 97–100, 2011. View at: Google Scholar
12. M. Kutiĺek, “Non-darcian flow of water in soils—laminar region: a review,” Developments in Oil Science, vol. 2, pp. 327–340, 1972. View at: Publisher Site | Google Scholar
13. H. Fei, L. Cheng, C. Li et al., “Study on threshold pressure gradient in ultra-low permeability reservoir,” Journal of Southwest Petroleum Institute, vol. 28, no. 6, pp. 29–32, 2006. View at: Google Scholar
14. J. Yan, Y. Qi, and X. Liu, “Analysis on starting pressure gradient of oil phase at irreducible water in low permeability oil reservoirs,” Fault Block Oil & Gas Field, vol. 21, no. 3, pp. 344–347, 2014. View at: Google Scholar
15. W. Xiong, Q. Lei, X. Liu et al., “Pseudo threshold pressure gradient to flow for low permeability reservoirs,” Petroleum Exploration and Development, vol. 36, no. 2, pp. 232–236, 2009. View at: Publisher Site | Google Scholar