Long‐term fracture conductivity in tight reservoirs

Reservoir stimulation using hydraulic fracturing technology is one of the most common stimulation measures to develop tight oil and gas reservoirs, and the conductivity of fractures after fracturing is the key to the stimulation effect of oil fields. In this paper, the laboratory simulation system of fracture conductivity is used to study the long‐term fracture conductivity under the displacement of formation water and fracturing fluid. The experimental results show that the fracture conductivity decreases with the increase of displacement time under the displacement of formation water and fracturing fluid, and the decrease of fracture conductivity under the displacement of fracturing fluid with time is greater than that of formation water displacement because of the plugging effect of fracturing fluid residue on pore channels. The use of formation water and fracturing fluid circulation displacement can delay the decline rate of conductivity with time. The conductivity increases with the increase of sand concentration, proppant particle size, and elastic modulus, and decreases with the increase of closure pressure. In addition, by studying the influence of elastic modulus on fracture width, it is found that the smaller the elastic modulus is, the more serious the embedding phenomenon of proppant is, and the smaller the fracture width is.


| INTRODUCTION
As the oil and gas exploration and mining technologies improve, the yield of unconventional hydrocarbon resources has grown in the past decade, 1 in which the yield of unconventional oil and gas all over the world in 2019 reached 1.984 × 109 tons. 2 As an important exploration and development field of unconventional oil and gas, tight oil and gas reservoirs are characterized by low reservoir permeability and poor conductivity of natural fractures.Therefore, it is necessary to realize economic production by applying some stimulation technologies such as hydraulic fracturing.In the actual production of unconventional oil and gas reservoirs, hydraulic fracturing is applied to generate as many fractures as possible in reservoirs to increase the contact area with the reservoirs. 3,4The fracture network formed by artificial fractures generated after hydraulic fracturing with natural fractures can improve the flow channel between reservoir fluids and the shaft to some extent and further increase the yield of oil and gas.The key to the success of hydraulic fracturing is whether the fracture conductivity can remain stable under the closure stress during production. 5The fracture conductivity is regarded as a key variable for estimating the productivity of fractured wells and maintaining long-term fracture conductivity enhances the productivity and recovery efficiency of oil and gas wells. 6s fracturing technology constantly develops, it is necessary to consider more factors when exploring fracture conductivity.1][12][13][14] In addition, the fracturing fluids significantly influence the fracture conductivity during the reconstruction of reservoirs with fracturing fluids.It is mainly shown as the damage of fracturing fluids to the reservoir permeability. 15,16hrough test research, Huitt and McGlothlin 17 found that the breaking and embedment of proppants are two main factors influencing fracture conductivity.The size of fractures is controlled according to different pressures and particle size and concentration of proppants.Elastic modulus is considered as the inherent characteristic of reservoir rock, which is also one of the important factors influencing fracture conductivity.Barton et al. suggested that relative to formations with a large elastic modulus, the fracture plane of soft formations is easier to be closed, thus resulting in a drastic decrease of the conductivity. 18,19y performing the evaluation test on the long-term fracture conductivity, Yang et al. found that the residues of guar gum fracturing fluids after gel breaking greatly impair fracture permeability owing to the difficulty in gel breaking, which leads to a great reduction of the fracture conductivity. 20uring hydraulic fracturing, successively injecting nanoparticles and conventional proppants with a larger particle size into the fractures is favorable for nanoproppants to penetrate into the fractures and support a longer fracture, thus improving the oil well productivity. 21The particle size of proppants shows a significant influence on the fracture conductivity while the breaking of proppant particles is affected by the closure stress and particle strength. 22By utilizing an integrated discrete element method combined with computational fluid dynamics to simulate the fluid flow in the proppant pack, Zhang et al. 23 evaluated the fracture conductivity after the fractures were closed.Guo et al. proposed a method for calculating the breaking rate of proppants and fracture conductivity under various test conditions by carrying out a series of laboratory tests and numerous numerical calculations. 24The influence of the flowback dynamics in multistage fractured well completion on the conductivity of a single fracture was surveyed on condition of considering various factors such as embedment of proppants, plastic deformation and creep of rock around fractures, and proppant breaking. 25As rock reacts with water, rock hardness reduces and proppants are embedded into soft formations.Furthermore, the fracture conductivity sharply decreases. 26Desouky et al. improved the rock strength through mineral alteration to maintain long-term fracture conductivity. 27The filling property of proppants is considered the main factor influencing the fracture permeability in fractures with sufficient proppants.After considering the influences of the proppant-pack permeability, proppant-pack porosity, and fracture width on the fracture permeability, Teng et al. proposed a novel model for the fracture permeability. 28Jin et al. studied the influence of different factors on fracture conductivity in the process of high-speed channel fracturing in tight sandstone reservoirs and analyzed each factor comprehensively.To study the influence of proppant embedment on tight sandstone fracture conductivity. 29Shen et al. evaluated the influence of proppant type, particle size, and sand concentration on the conductivity. 30fter conducting the hydraulic fracturing on the target zone, the fracture conductivity in reservoirs presents quite an important influence on the productivity of oil and gas wells. 31The fracture conductivity is affected by multiple factors.As for research on fracture conductivity, many researchers concentrate on analyzing the influence of a single factor while failing to comprehensively evaluate the fracture conductivity under the influences of various factors.The lack of digital representation of the effect of the fracture conductivity is due to insufficient related test research, and a mathematical evaluation model for the fracture conductivity under different influencing factors is absent.As a result, the popularization of the research result attained through the laboratory test to the field application is restricted.
Aiming at the above research conditions, the change of the fracture conductivity with the displacement time was further explored by applying a laboratory simulation test system for the fracture conductivity.The influences of various factors (e.g., proppant concentration, particle size of proppants, elastic modulus, closure pressure, and type of displacing fluids) on the long-term fracture conductivity were analyzed.In addition, a corresponding evaluation model was established.

| Test apparatuses
The laboratory simulation experiment system for fracture conductivity is used for the studies, which consists of a displacement system, a loading system, a vacuum pumping system, and an information acquisition system.The equipment flowsheet of the experiment system is shown in Figure 1.
The test on the fracture conductivity was conducted according to Darcy's law.At first, the width W of rock plates, the length L of the gripper along the flow direction of fluids, the density ρ L of displacing fluids, the density ρ w of distilled water, the fluid viscosity μ, and pressure P were measured by using measuring tools; the fracture width ω of rock plates during the test was recorded.Afterward, the fracture permeability under the closure pressure was calculated by substituting the data into Equation (1) 32   ( ) Fracture conductivity equals the product between the fracture width and the fracture permeability under the closure pressure 33 and therefore the fracture conductivity D s can be attained according to Equation (2) where D s , K, Q, W, L, ω, ρ L , ρ w , μ, and P denote fracture conductivity (μm 2 cm), permeability (μm 2 ), the rate of flow (cm 3 s −1 ), the width of rock plates (cm), the length of the gripper along the flow direction of fluids (cm), the fracture width of rock plates (cm), the density of displacing fluids (g cm −3 ), the density of distilled water (g cm −3 ), fluid viscosity (mPa s), and pressure (kPa), respectively.

| Test materials
As for rock plates, the elliptic rock plates based on the American Petroleum Institute standard used in the Flowchart of fracture conductivity evaluation equipment.1-evacuation at high-pressure end; 2-evacuation at lowpressure end; 3-feeding flow at high-pressure end; 4-feeding flow at low-pressure end; 5-upstream of the gripper; 6-vaccum pumping; 7-downstream of the gripper; 8-cylinder valve; 9-return pressure; 10-evacuation of oil pressure.
fracture conductivity test were prepared according to the composition control method by adjusting the proportion of kaolin, quartz sand, cement, and water.Moreover, ceramsites were used as the proppants during the tests.The elastic modulus of the rock plate we used in the paper is obtained according to the equation E σ ε = / based on the measurement of the core sample stress and strain, where E is the elastic modulus, σ is the stress, and ε is the strain.Parameters of partial rock plates and core samples, as shown in Table 1. Figure 2 shows the rock plates and the steel plate used in the test.
To explore the fracture conductivity under the displacement effect of either formation water or fracturing fluids, the formation water and fracturing fluids after gel breaking were separately used as seepage fluids to test the fracture conductivity.Table 2 shows the dosages of various ions in NaHCO 3 formation water used for the tests.
The percentage of each solute in the fracturing fluid is, respectively, 0.55% of guar gum + 0.2% of gel breaker (ammonium persulfate) + 0.6% of cross-linking agent.After gel-breaking, the supernatant was taken and its apparent viscosity (T = 95°C) was measured in the range of 0.7-0.8MPa s.According to industry standards such as General Technical Specifications of Fracturing Fluid and The Evaluation Measurement for Properties of Waterbased Fracturing Fluid, the fracturing fluid prepared using the proportions was completely gel-broken.

| Experimental
The change of the fracture conductivity under the influences of different factors was explored by using the simulation test system for the fracture conductivity.During the experiments, the flow velocity was set as 5 cm 3 • min −1 and the experiment time was designed as 60 or 250 h.The changes in the fracture conductivity at different proppant concentrations, particle sizes of proppants, elastic moduli, and closure pressures over time under the displacement effect of formation water and fracturing fluids were separately investigated.Moreover, rock plates and steel plates were used as the test materials, using ceramsites with a particle size of 20/40 mesh as proppants, with a proppant concentration of 10 kg m −2 and a closure pressure of 40 MPa.Each material was displaced for 5 h with the formation water at first and then for another 5 h with fracturing fluids, in turn.On this basis, the change of the fracture conductivity with time was analyzed.
The fracture conductivity after displacement for 60 h was used as the test datum.The influencing laws of various factors (including different proppant concentrations, particle sizes of proppants, and elastic moduli) on the fracture conductivity under the closure pressures of 20, 40, and 60 MPa were successively investigated at first; afterward, the influencing laws of different closure pressures on the fracture conductivity at the proppant concentrations of 5, 10, and 15 kg m −2 were explored.The single-factor mathematical models for the fracture conductivity were established on the basis of the above test data.To explore the change of the fracture width over time, three test materials (i.e., a steel plate (200,000 MPa), a rock plate 1 (7284 MPa), and a rock plate 2 (3696 MPa)) were applied to perform the test at the proppant concentration of 10 kg m −2 and the closure pressure of 40 MPa.It can be seen from Figure 3 that the fracture conductivity declines with the increasing displacement time under the displacement effect of either formation water or fracturing fluids.At first, under the displacement effect of formation water, the fracture conductivity slowly drops with the growing displacement time within 130 h after the test starts; however, the fracture conductivity accelerates to reduce in the following 70 h.The value maintains at a low level after the test is carried out for 200 h.This is because, under the action of immersion of formation water, the damage of the rock plate increases with the increase of time, resulting in the decrease of rock plate strength.At the same time, the change of fracture conductivity is affected by the compaction and embedding of proppant, 10 and the permeability and fracture width are further reduced, resulting in the reduction of fracture conductivity.With the increase of the embedded degree of proppant, when the proppant strength is exceeded, it will be broken, and the flow channel will gradually shrink and be filled with debris, resulting in the decrease of conductivity. 9In addition, the displacement fluid will carry some small sand particles in the process of flow, and these sand particles will eventually redeposit at the downstream location with the increase of time, forming a sand plug and causing the decrease of fracture conductivity.
Compared with the results under the displacement effect of fracturing fluids, the fracture conductivity rapidly decreases within the first 80 h of the displacement; afterward, the fracture conductivity increasingly insignificantly declines with the growing displacement time, and finally, it tends to stabilize.It is attributed to two aspects: on the one hand, under the displacement effect of the fracturing fluids, the residues of fracturing fluids are constantly accumulated in the fractures to block the flow channel of fluids, as a result, the fracture conductivity rapidly reduces in the early test stage; on the other hand, due to the large pores in the filling layer at the beginning of the experiment, the relative displacement between proppant particles will occur under the action of the closure pressure, resulting in a rapid decline in fracture width and a decrease in fracture conductivity. 34,35The influence of the residues of fracturing fluids accumulated in the fractures on the reduction amplitude of the fracture conductivity becomes increasingly insignificant with the increasing displacement time.36-38Thus, the reduction rate of the fracture conductivity constantly reduces, and finally, the reduction amplitude approximates to zero.
From the difference in fracture conductivity under the displacement of fracturing fluid and formation water, it can be seen that the type of fluid has a significant impact on the long-term conductivity of the fracture.Therefore, during the production process, if the type of fluid flowing into the bottom of the well through the fracture is different (oil, gas, water), there will also be differences in the long-term conductivity of the fracture.This is mainly because, first, the interaction laws between different types of fluid and the rock on the fracture surface are different, which can lead to different softening properties of the fracture surface under the long-term action of the fluid, and the degree of support agent embedded in the fracture surface during the production process also varies.Second, due to the differences in flow rate and viscosity of different types of fluids during the production process, the carrying capacity of sand particles during the flow process is also different, the migration and deposition of sand particles can cause blockage of cracks and affect their conductivity.

| Particle size of proppants
The changing trend of the fracture conductivity with time under the displacement effect of formation water and fracturing fluids at different particle sizes of proppants is shown in Figure 4.It can be seen from the figure that the fracture conductivity rapidly declines at first and then slowly reduces with the growth of the displacement time.Under the influence of the residues of fracturing fluids, the fracture conductivities at the particle sizes of proppants of 20/40 and 30/60 mesh separately drop by 70.61% and 80.60% within 15 h under the displacement effect of fracturing fluids; afterward, the reduction amplitude of the fracture conductivity rapidly reduces and approximates to zero.However, within the first 60 h of the displacement of formation water, the fracture conductivity stably drops and separately reduces by 67.03% and 77.75% at the particle sizes of proppants of 20/40 and 30/50 mesh.It can be seen that different types of displacing fluids significantly influence the fracture conductivity and the residues in fracturing fluids accelerate the reduction, and increase the reduction amplitude, of the fracture conductivity.
As shown in the figure, the fracture conductivity equals 20.8 and 35.1 × μm 2 cm at the particle size of proppants of 30/50 and 20/40 mesh under the proppant concentration of 10 kg m −2 after 60 h of displacement using formation water.It can be found that the fracture conductivity increases by 68.7% using 20/40-mesh proppants.After the displacement of fracturing fluids, the fracture conductivities are 13.28 and 17.69 μm 2 cm at particle sizes of 30/50 and 20/40 mesh, respectively, with the latter showing an increase of 33.2%.It indicates that the fracture conductivity grows with the increasing particle size of proppants.The reason is that the proppants with a large particle size support a larger flow channel and there is a low resistance when fluids flow, which is favorable for fluids to go through.The fracture conductivity is lower after 60 h under the displacement effect of fracturing fluids compared with that under the displacement effect of formation water.

| Elastic modulus
Figure 5 shows the changing trend of the fracture conductivity with time under the displacement effect of formation water and fracturing fluids at different elastic moduli.After displacement using formation water for 60 h, the fracture conductivities at the elastic moduli of formations of 3696 and 7284 MPa are 27.6 and 35.0 μm 2 cm, respectively, with the latter growing by 26.8%.After 60 h of displacement using fracturing fluids, the fracture conductivities at the elastic moduli of formations of 3696 and 7284 MPa are 11.5 and 18.3 μm 2 cm, with the latter showing an increase of 37.16%.It can be seen that the fracture conductivity increases with the growth of the elastic modulus.The reason is that the larger the elastic modulus of formations is, the more difficult the embedment of proppants into formations under the influence of the closure pressure and the lower the embedment degree of the proppants into rock plates.
Under the displacement effect of formation water, the fracture conductivity in the early test stage is 104.while it drops to 27.6 μm 2 cm after 60 h at the elastic modulus of formations of 3696 MPa, showing a reduction amplitude of 73.6%; at the elastic modulus of formations of 7284 MPa, the fracture conductivity in the early test stage equals 150.2 μm 2 cm and it reduces to 35.0 μm 2 cm after 60 h, with the reduction amplitude of 76.7%.Under the displacement effect of fracturing fluids, the reduction amplitudes of the fracture conductivity at the elastic moduli of formations of 3696 and 7284 MPa are 90.5% and 87.9%, respectively.It implies that the fracture conductivity declines with the increasing displacement time and its reduction amplitude under the displacement effect of fracturing fluids is larger than that under the displacement effect of formation water.
Under the displacement effect of either the formation water or fracturing fluids, the fracture conductivity rapidly reduces within 15 h and then its reduction rate constantly decreases and eventually approximates to zero.In the early stage, the contact between the particles in the proppant packing layer is loose and the pores are large.Therefore, the pores between the particles are reduced under the action of the closing pressure, which results in the reduction of fracture conductivity.With the increase of displacement time, the rock plates deliver decreasing strength after being in contact with the displacing fluids.Therefore, the proppants are easy to be embedded into rock plates under the closure pressure, which leads to the reduction of the fracture conductivity.In addition, the deposition of free sand in fractures and the accumulation of residues in fracturing fluids cause the blocking of the pore channel and thereby the reduction of the fracture conductivity.However, the influence of the factors on the fracture conductivity is limited.After being in contact with the displacing fluids, rock plates present a finite reduction amount of strength.The influences of the residues of fracturing fluids constantly accumulated in fractures and small sand particles on the reduction amplitude of the fracture conductivity are gradually weakened over time.Therefore, the reduction amplitude of the fracture conductivity gradually reduces and finally approximates to zero after the test proceeds for 15 h.

| Closure pressure
The changing trend of the fracture conductivity with time under the displacement effect of formation water and fracturing fluids at different closure pressures is shown in Figure 6.It can be seen from Figure 6 that the fracture conductivity rapidly reduces at first and then slowly declines over time.Under the closure pressure of 40 MPa, the fracture conductivity drops from the initial 80.47 to 19.88 μm 2 cm by an amplitude of 75.3% after 60 h under the displacement effect of formation water; under the closure pressure of 55 MPa, the fracture conductivity decreases from 75.19 to 16.09 μm 2 cm, with the reduction amplitude of 78.6%.After 60 h of displacement by virtue of fracturing fluids, the fracture conductivity reduces from 148.44 to 11.26 μm 2 cm under the closure pressure of 40 MPa, with the reduction amplitude of 92.4%; it declines from 109.49 to 11.03 μm 2 cm by 89.9% under the closure pressure of 55 MPa.It can be seen that the higher the closure pressure is, the lower the fracture conductivity.With the growth of the closure pressure, the fracture conductivity will gradually decrease affected by the dual effects of proppant compaction and embedding.Relative to the results under the displacement effect of formation water, the fracture conductivity under the displacement effect of fracturing fluids more significantly decreases due to the blocking effect of residues in fracturing fluids.
The fracture conductivity under the displacement effect of fracturing fluids significantly reduces within the first 30 h.However, its reduction rate slightly declines in the following 30 h because the influence of the residues in fracturing fluids constantly accumulated in fractures on the reduction amplitude of the fracture conductivity is constantly weakened.Compared with the result under the displacement effect of fracturing fluids, the fracture conductivity under the displacement effect of formation water stably and marginally drops within the first 60 h.

| Displacement mode
The alternative displacement with the aid of formation water and fracturing fluids has not been researched in previous tests on the fracture conductivity.Therefore, it The change of the fracture conductivity over time at different closure pressures (under the proppants concentration of 10 kg m −2 particle size of proppants of 30/50 mesh, and elastic modulus of 3696 MPa).
was explored in the study to analyze whether the alternative displacement of displacing fluids is favorable for improving the fracture conductivity or decelerating the attenuation rate of the fracture conductivity.
Ceramsites with a particle size of 20/40 mesh were used as the proppants during the test.The cyclic displacement (5 h with formation water at first and then another 5 h with fracturing fluids) was performed at the flow rate of 5 cm 3 min −1 , proppant concentration of 10 kg m −2 and the closure pressure of 40 MPa.To improve the comparability of tests, the changes in the fracture conductivity under two working conditions (steel and rock plates) with time were investigated.The specific results are shown in Figure 7.
As shown in Figure 7, the fracture conductivities under rock and steel plates gradually decrease and finally stabilize with the growing time.During the cyclically alternative displacement, the fracture conductivity insignificantly declines within 5 h of displacement using formation water, which fluctuates within a small range; within 5 h of displacement by virtue of fracturing fluids, the fracture conductivity sharply and significantly decreases due to the influence of residues in fracturing fluids.The fracture conductivity basically stabilizes under the displacement effect of formation water while it sharply declines under that of fracturing fluids.This is because the residues in fracturing fluids block the pore throat and further increase the flow resistance of fluids after displacement of fracturing fluids.After performing the cyclic the residues in fracturing fluids are increasingly accumulated and therefore effective pores increasingly reduce.Furthermore, the formation water only seepages outwards along the damaged effective pores when performing displacement by using formation water again.Thus, after a long displacement period, the accumulated residues in fracturing fluids have greatly blocked the pore throat under either rock plates or steel plates.In this case, the cyclic displacement of formation water and fracturing fluids fails to effectively improve the fracture conductivity.
As shown in Figure 5, the initial fracture conductivity under the displacement effect of fracturing fluids is 150.77μm 2 cm and it declines to 17.08 μm 2 cm after 50 h at the elastic modulus of 7284 MPa.It can be seen from Figure 7 that the fracture conductivity drops from the initial 151.06 to 21.04 μm 2 cm after the alternative displacement of formation water and fracturing fluids.It indicates that the reduction amount of the fracture conductivity after the alternative displacement for 50 h is lower than that under the displacement effect of fracturing fluids on the same test conditions.Through analysis, it can be attained that the cyclic displacement of formation water and fracturing fluids can decelerate the attenuation rate of the fracture conductivity with time while failing to effectively improve the fracture conductivity.

| Single-factor models
The influences of different proppant concentrations, particle sizes of proppants, elastic moduli, and closure pressures on the fracture conductivity under the displacement effect of the formation water and fracturing fluids within the first 60 h were experimentally explored.
Figure 8 displays the change of the fracture conductivity under the displacement effect of formation water and fracturing fluids with the proppant concentration.According to the test data in the figure, the fracture conductivity increases with the increase of proppant concentration by analyzing the above change of the fracture conductivity with the proppant concentration.Moreover, the quantitative relationships under the displacement effect of formation water and fracturing fluids are separately shown in Equations ( 3) and ( 4) where C, D s , and R 2 denote proppant concentration (kg m −2 ), fracture conductivity (μm 2 cm), and correlation coefficient, respectively.The R 2 values of Equations ( 3) and ( 4) are both close to 1, indicating that the fitting curve is highly consistent with the experimental points.
As shown in Figure 8 and Equations ( 3) and ( 4), the fracture conductivity increases with the growing proppant concentration.The fracture conductivities under The change of the fracture conductivity with time.
different closure pressures show a great difference.The reason is that the breaking, compaction, and embedment degree of proppants in fractures are associated with the closure pressure.The higher the pressure is, the higher the breaking rate of proppants, and the higher the compaction and embedment degree in rock plates, the weaker the fracture conductivity.
Figure 9 shows the influence of the particle size of proppants under the displacement effect of formation water and fracturing fluids on the fracture conductivity.It can be seen from the figure that the fracture conductivity increases with the increase of the particle size of proppants.The quantitative relationships under the displacement effect of formation water and fracturing fluids are separately displayed in Equations ( 5) and ( 6) where d 50 , D s and R 2 denote particle size (mm) of proppants and fracture conductivity (μm 2 cm) and correlation coefficient, respectively.The R 2 values of Equations ( 5) and ( 6) are both greater than 0.9, indicating that the fitting curve can well represent the experimental data points, and its fitting quality is very high.
According to Figure 9 and Equations ( 5) and ( 6), the fracture conductivity increases with the growing particle size of proppants.The larger the particle size of proppants is, the larger the space supported in the fractures, which makes it easy for fluids to go through.The proppants are broken and embedded into rock plates under high closure pressure.As a result, the higher the closure pressure is, the weaker the fracture conductivity.
The influence of the elastic modulus under the displacement effect of formation water and fracturing fluids on the fracture conductivity is shown in Figure 10.It can be seen from the figure that the fracture conductivity increases with the increase of the elastic modulus.The quantitative relationships under the displacement effect of formation water and fracturing fluids are separately displayed in Equations ( 7) and ( 8) where E, D s , and R 2 denote elastic modulus (MPa), fracture conductivity (μm 2 cm), and correlation coefficient, respectively.
F I R E 8 The influence of the proppant concentration on the fracture conductivity (under the particle size of proppants of 20/40 mesh and elastic modulus of 3696 MPa).
G U R E 9 The influence of the particle size of proppants on the fracture conductivity (under the proppants concentration of 15 kg m −2 and elastic modulus of 7284 MPa).

F G U R E 10
The of the elastic modulus on the fracture conductivity (under the proppants concentration of 10 kg m −2 and particle size of proppants of 20/40 mesh).
The R 2 values of Equations ( 7) and ( 8) are both close to 1, indicating that the quality of the fitted curve is very high, and the obtained fitting equations are reliable.
According to Figure 10 and Equations ( 7) and ( 8), it can be found that the fracture conductivity rises with the increasing elastic modulus, and the higher the closure pressure is, the weaker the fracture conductivity under the displacement effect of either formation water or fracturing fluids.As the fracture conductivity is greatly influenced by the type of displacing fluids, the fracture conductivities under the displacement effect of different types of displacing fluids deliver a great difference.
During the test on the influence of the closure pressure on the fracture conductivity, curves shown in Figure 11 are drawn according to the test data.According to the change of the fracture conductivity under the displacement effect of formation water and fracturing fluids with the closure pressure, it can be seen that the fracture conductivity decreases with the increasing of closure pressure.The quantitative relationships under the displacement effect of formation water and fracturing fluids are displayed in Equations ( 9) and ( 10) where P c , D s , and R 2 denote closure pressure (MPa), fracture conductivity (μm 2 cm), and correlation coefficient, respectively.The R 2 values of Equations ( 9) and ( 10) are both greater than 0.9, indicating that the fitting quality of the fitted curve is very high and can well reflect the experimental laws.
From Figure 11 and Equations and (10), the influence of the closure pressure under the displacement effect of formation water on the fracture conductivity is basically consistent with that under the displacement effect of fracturing fluids.That is, the fracture conductivity first rapidly and then slowly exponentially reduces with the growing closure pressure.The proppant concentration is closely related to the fracture width, and the higher the proppant concentration is, the larger the fracture width. 31Therefore, the fracture conductivities at different proppant concentrations greatly differ, that is, the higher the proppant concentration is, the stronger the fracture conductivity.Due to the influence of residues in fracturing fluids, the fracture conductivity under the displacement effect of formation water is larger than that under the displacement effect of fracturing fluids at the same concentration.

| The change of the fracture width over time at different elastic moduli
The elastic modulus of the steel plate, rock plate 1, and rock plate 2 used in this test were 200,000, 7284, and 3696 MPa, respectively.As shown in Figure 12, the fracture width under the steel plate is basically constant and the proppants are basically not broken and embedded during the test for 60 h; within 30 h, the fracture width in rock plate 1 slightly decreases while it rapidly drops to about 0.4 cm within a short time.The fracture width in rock plate 1 maintains at about the value until the test is ended.It implies that the proppants are slightly embedded therein; the fracture width in rock plate 2 constantly declines during the test: the fracture width significantly reduces within the first 36 h and then it fluctuates at about a low value.In this case, the embedment of proppants into the rock plate is significant.
The rock strength slightly decreases under the influence of displacing fluids in the whole test process.The fracture width in rock plate 1 drops from the initial value of 0.419 to 0.40 cm, showing a reduction amplitude of 4.5%.The fracture width under rock plate 2 reduces from 0.408 to 0.375 cm, showing a reduction amplitude of 8.09%.It can be found that the fracture width rises with the increasing elastic modulus of formations under the same conditions and the reduction amplitude of the fracture width over time gradually declines.

| CONCLUSIONS
1.By analyzing the change of the fracture conductivity with time, it can be found that the long-term fracture conductivity rapidly declines at first and then slowly reduces with the growth of the displacement time.It is attributed to two aspects: on the one hand, in the early stage, the contact between the particles in the proppant packing layer is loose and the pores are large, so that the pores between the particles are reduced under the action of the closing pressure, which resulting in the reduction of fracture conductivity; on the other hand, the displacing fluids seepage into rock plates over time and therefore the proppants are easy to be embedded into the rock plates due to the decreasing strength of the rock plates, and the displacing fluids carry some loose small sand particles to fractures and pores in the flow process and these sand particles deposit downstream of the fractures with time to block the flow channel of fluids.Therefore, when predicting the productivity of fractured wells, it is necessary to fully consider the dynamic changes in fracture conductivity during the long-term production process.2. The type of displacing fluids is the main factor influencing the fracture conductivity.Compared with the formation water, the fracturing fluids significantly reduce the fracture conductivity when used as displacing fluids under the influence of residues therein.When designing fracturing fluids, the compatibility between the fracturing fluid and the formation should be enhanced to minimize the impact of the fracturing fluid on the fractures and formation as much as possible.3. The residues of fracturing fluids in fractures are increasingly accumulated and therefore effective pores gradually reduce during cyclic displacement of formation water and fracturing fluids.The fracture conductivity will decrease from 150 to 17 μm 2 cm after 50 h of fracturing fluid flooding, and the fracture conductivity will decrease to 21 μm 2 cm after 50 h of alternate formation water and fracturing fluid flooding.Furthermore, the formation water only seepages outwards along the damaged effective pores when performing the displacement by using formation water again.Thus, the cyclic displacement of formation water and fracturing fluids can decelerate the attenuation rate of the fracture conductivity with time while failing to improve the fracture conductivity.4. The long-term fracture conductivity increases with the increasing of proppant concentration, particle size of proppants, and elastic modulus of the formations, but decreases with the increasing of closure pressure.When designing hydraulic fracturing, reasonable process parameters should be selected based on the oilfield production requirements and construction technology. 5.When the elastic modulus of the experimental material reaches 200,000 MPa, the crack width remains at 0.456 cm with time, and when the elastic modulus is 7284 MPa, the crack width decreases rapidly from 0.410 to 0.4 cm in a short time after 30 h.When the elastic modulus is 3696 MPa, the fracture width gradually decreases from 0.408 to 0.375 cm.The fracture width decreases with time and formation of elastic modulus.When predicting the production capacity of fractured wells, the impact of formation elastic characteristics on long-term conductivity should be fully considered.

T A B L E 1 35 F
Parameters of partial rock plates and core sample.I G U R E 2 Picture of experimental rock and steel plates.

F
I G U R E 3 The change of the fracture conductivity over time at different proppant concentrations (under the closure pressure of 55 MPa, particle size of proppants of 40/70 mesh, and elastic modulus of 7284 MPa).

4 μm 2 cmF R 4
Figure5shows the changing trend of the fracture conductivity with time under the displacement effect of formation water and fracturing fluids at different elastic moduli.After displacement using formation water for 60 h, the fracture conductivities at the elastic moduli of formations of 3696 and 7284 MPa are 27.6 and 35.0 μm 2 cm, respectively, with the latter growing by 26.8%.After 60 h of displacement using fracturing fluids, the fracture conductivities at the elastic moduli of formations of 3696 and 7284 MPa are 11.5 and 18.3 μm 2 cm, with the latter showing an increase of 37.16%.It can be seen that the fracture conductivity increases with the growth of the elastic modulus.The reason is that the larger the elastic modulus of formations is, the more difficult the embedment of proppants into formations under the influence of the closure pressure and the lower the embedment degree of the proppants into rock plates.Under the displacement effect of formation water, the fracture conductivity in the early test stage is 104.4 μm 2 cm

F
G U R E 11 The of the closure pressure on the fracture conductivity (under the particle size of proppants of 40/70 mesh and elastic modulus of 3696 MPa).F I G U R E 12 The change of the fracture width with time (under the proppants concentration of 10 kg m −2 and closure pressure of 40 MPa).
The amount of ions per liter of formation water.Figure3shows the changing trend of the fracture conductivity with time under the displacement effect of formation water and fracturing fluids at different proppant concentrations.It can be seen from the figure that the larger the proppant concentration is, the higher the fracture conductivity.Under the displacement effect of formation water, the fracture conductivity drops by 90.9%, 89.7%, and 87.8% within 250 h at the proppant concentrations of 5, 10, and 15 kg m −2 , respectively; under the displacement effect of fracturing fluids, the fracture conductivity separately decreases by 96.6%, 92.7%, and 89.2% within 250 h at the same proppant concentrations.It can be seen that the fracture conductivity more significantly reduces under the displacement effect of fracturing fluids at a certain proppant concentration.This is because residues in fracturing fluids are accumulated with time to block the fractures and pores.
T A B L E 2