Experimental Study on Connection Characteristics of Rough Fractures Induced by Multi-Stage Hydraulic Fracturing in Tight Reservoirs

Abstract

The well spacing for the development of tight reservoirs by multi-stage fracturing is continuously narrowed. Consequently, interwell interference during fracturing is more and more serious, accompanied by a host of issues in fracturing design and oil and gas production. However, the mechanism of interwell interference during fracturing is not explicit. The corresponding laws of the connectivity of rough fractures during fracturing, which plays a critical role in interwell interference, are not fully understood. In this study, on the basis of characterizing the roughness of fractures, a laboratory evaluation method for fracture connectivity was established. The connectivity characteristics of rough fractures and factors affecting the fracture connectivity are studied. The time and scale effects of fracture connectivity were discussed and their application in interwell interference was analyzed. The results show that the connectivity performance of rough fractures can be characterized by the time for pressure decay. The upstream pressure gradually decreases over time, and the decline rate is related to the fracture aperture, the fracture surface roughness, the contact area of the closed fractures, and liquid properties. Specifically, the decrease in fracture aperture and the increase in fluid viscosity leads to a significant reduction in fracture connectivity. While larger fracture surface roughness and contact area can make fracture connectivity better. The connectivity of the fracture system is one of the significant mechanisms causing interwell interference during fracturing. The connectivity of rough fractures formed during fracturing has remarkable scale and time effects. This study can effectively guide the fracturing design and the evaluation of the impact of fracture connectivity on production.

1. Introduction

The efficient development of tight reservoirs requires large-scale hydraulic fracturing to form complex fracture networks, thus increasing the contact area between the reservoirs and the fractures and creating high-permeability channels conducive to oil and gas flowing [1,2]. Although dense hydraulic fractures are formed, the recovery ratio of tight reservoirs is still relatively low. For example, the recovery ratio of shale reservoirs in the Midland Basin is only about 4% [3]. To further improve the producing degree of reservoirs, the distance between adjacent wells were continuously narrowed. As a result, the interference during fracturing is more and more severe, accompanied by a series of issues in oil and gas production [4,5,6,7].

Ajani et al. (2012) [8] studied the problem of interwell interference generated during large-scale fracturing of tight reservoirs. The impact of interwell interference on productivity was analyzed via data from 179 wells in the Arkoma Basin. Lawal et al. (2014) [9] used a rate transient analysis method to calculate the area of reconstruction fractures and studied the impact of interwell interference on production. They found that the area of reconstruction fractures in the parent well decreased by 60% after the interference and verified it by two cases of Haynesville and Marcellus. Tang et al. (2017) [10] also analyzed the impact of interwell interference during multi-stage fracturing on production with an EDFM method. The study showed that too large crossover area of the reconstruction volume between two wells would negatively affect production, while positive interference is formed if the crossover area is within a certain range. King et al. (2017) [11] studied the productivity loss caused by interwell interference in the Eagle Ford area. The study stated the reasons for the production loss, the locations where reservoir damage appears, corresponding prevention methods, and specific processes that can be adopted. Esquivel (2017) [12] studied the interwell interference during fracturing of the Haynesville shale and summarized the productivity of 65 disturbed wells. The data showed that 75% of the disturbed wells were positively affected, 16% were negatively affected, and the rest were not significantly affected. In summary, both the positive and negative effects of interwell interference on production were found, but the reasons for this phenomenon is unclear so far.

The main manifestation of interwell interference during fracturing is fracture hit. The fracture hit refers to a phenomenon that the fractures generated during fracturing are connected with the fractures in disturbed wells or cause the deformation of fractures in disturbed wells, resulting in the rapid increase in the pressure of the disturbed wells [13]. Jacobs et al. [14] found that the output of the sub-wells of the 300 shale horizontal wells in the Delaware Basin decreased by 30% compared with the parent wells. This mainly resulted from two factors. One is that the fractures in the sub-wells propagate in the direction of fractures of the parent wells in the low-pressure area, which leads to fracture connection. Another is that the inversion of the stress field during fracturing leads to the connection of fractures in the sub-wells and the parent wells. A set of coupling models of fluid flow and geomechanics were established by Sangnimnuan et al. [15] to characterize the changes in the stress field of complex reservoirs. On this basis, the mechanism of fracture hit was studied. The pressure drop caused by the pressure decay leads to the decrease in principal stress. During fracturing, an undesirable fracture network formed in an infilled well communicates with the fractures in an original well, resulting in a fracture hit. Consequently, the fractures in the infilled well asymmetrically propagate to connect with the original fracture networks. With the aid of numerical simulation methods, Morales et al. [16] proposed two mechanisms for mitigating the negative effects of fracture hit. One is to promote the propagation of transverse fractures to avoid excessive expansion of fractures in an infilled well toward the direction of fractures in a parent well. Another is to re-pressurize the pressure-released area of the parent well to avoid the expansion of new fractures in the infilled well toward the low-pressure area of the parent well. Using microfluidic technology and spontaneous imbibition experimental methods, Xu et al. [17] studied that the surface treatment agents can be added into the parent wells to relieve the impact of fracture hit on production, which can be regarded as a vital mechanism for improving oil recovery. It is easy to see that, in the research area of interwell interference, most scholars pay much attention to studying the interaction between newly formed fractures and original fractures during fracture propagation, rather than the connection degree of fractures after interacting with each other.

Interconnection of fractures is one of the important factors resulting in interwell interference [18,19]. A triple model involving fracturing fracture area, SRV area, and unreconstructed matrix area was established by Li Jiqing [20]. Based on the PEBI grid and finite volume method for numerical solution, the influence of factors such as connected permeability on the interference test results and the pressure field distribution characteristics were analyzed. Awada [21] argued that clarifying the law of interwell connectivity is the key to understanding the mechanism of interwell interference. The author proposed a method to identify interwell interference characteristics via on-site production data. This method can be used to reduce the negative impact of interwell interference and guide well completion. Haghshenas [22] proposed an analytical model to quantitatively analyze the connectivity between wells using the pressure data from monitoring wells. The governing equation involved hydraulic fractures and matrix systems into account, and the results obtained from the analytical model were consistent with that from the numerical simulations. Felisa [23] studied the flow of non-Newtonian fluids in fractures with variable fracture aperture. The result showed that the connectivity of fractures correlates closely with the fracture aperture and fluid rheology. Wang [24] designed and carried out nine sets of comparison experiments on water injection interference physical simulation according to the characteristics of the injection-production unit of fractured-vuggy reservoirs. The author analyzed the characteristics and influencing factors of water injection interference. Based on the seepage theory, an inversion model of water injection interference was established to study the influence of different factors on interference characteristics. The results indicated that the characteristics of the waterflooding interference are greatly affected by all of injection-production well spacing, permeability ratio, karst cave reserves, and the distribution of karst cave. Daneshy [25] argued that the interaction between wells significantly impacts short-term production. The connection of adjacent wells by fracturing fractures was affected by comprehensive factors such as stress environment, reservoir mechanical properties, completion type, the direction of fractures and the distance between them, well spacing, and perforation system. Although the connectivity of rough fractures plays a crucial role in interwell interference, the corresponding connectivity laws have not been fully mastered.

A single rough fracture is a basic unit consisting of fracture systems, and there already are research foundations for characterizing the rough fractures [26,27,28]. Fracture aperture, fracture surface roughness, etc., are critical parameters to characterize rough fractures. Generally, fracturing-related research assumes that the fractures are horizontal and straight. However, the rough fracture surface aggravates the complexity of fluid flow in the fractures and strengthens the interaction between liquids and rocks [29,30]. There are dominant channels for fluid flow in rough fractures in general. While the flow front of fluid in smooth fractures advances more uniformly, and the permeability is dozens of times higher than that in rough fractures [14,31]. Challenges still exist in studying the influence of the fracture surface roughness on the fracture connection, since there are no unified standards for the description of characteristics of rough fractures.

The essence of interwell interference is the communication of the fracture networks. As shown in Figure 1, there is no interwell interference between well-I and well-II in the case of larger well spacing, while interwell interference between well-III and well-IV appears through the connected area in the case of smaller well spacing. In general, the connected area is the connection of multiple fractures, and a single rough fracture is the basis for studying the connection of fracture networks. Based on characterizing fracture roughness, this paper studied the connectivity characteristics of rough fractures and the factors influencing the same with laboratory experiments. An indoor evaluation method for the connectivity of fractures was put forward and the time and scale effects of fracture connectivity were discussed. This study intends to provide effective guidance on the fracturing design and the evaluation of the impact of fracture connectivity on production.

2. Experimental Sections

2.1. Experiment Materials

In this study, samples of shale oil reservoirs in the Lucaogou Formation of the Junggar Basin were taken with a core depth of about 2850 m. Samples of tight oil reservoirs in the Yanchang Formation of the Ordos Basin were taken with a core depth of about 1875 m. Samples of shale gas reservoirs in the Longmaxi Formation of the Sichuan Basin were taken with a core depth of about 2520 m. Samples of volcanic gas reservoirs in the Yingcheng Formation of the Songliao Basin were taken with a core depth of about 2470 m. As shown in

Table 1. Sampling number and the formation information.

Number	Basin	Formation	Lithology	Depth/m
L	the Junggar Basin	the Lucaogou Formation	Shale	2850
Y	the Ordos Basin	the Yanchang Formation	Tight sandstone	1875
M	the Sichuan Basin	the Longmaxi Formation	Shale	2520
C	the Songliao Basin	the Yingcheng Formation	tight volcanic rock	2470

, the samples described above are respectively numbered as L, Y, M, and C. The actual samples that are cylinders with a sample diameter of 2.5 cm, 3.8 cm, and 5.0 cm are shown in Figure 2.

The liquid system includes distilled water and guar gum liquids with mass fractions of 0.2% and 0.5%, and the viscosity of the latter is 1 mPa·s, 10 mPa·s, and 30 mPa·s. The distribution of whole-rock minerals and clay minerals of the samples is shown in

Table 2. Whole-rock minerals and clay minerals contents.

Number	Mineral Content/%						Clay Mineral Content/%
	Quartz	Feldspar	Calcite	Dolomite	Iron Ore	Clay Mineral	Illite	Montmorillonite	Illite/Smectite Mixed Layer	Chlorite	Kaolinite
L	23.7	56.5	0.0	13.1	0.0	6.7	21.0	0.0	0.0	55.0	24.0
Y	70.5	9.2	1.6	3.9	0.0	14.8	33.0	0.0	43.0	20.0	4.0
M	40.3	8.8	7.5	6.5	0	36.9	15.9	4.3	62.3	8.7	8.8
C	36.8	48.8	6.0	0.0	0.0	8.4	14.0	0.0	44.0	42.0	0.0

. Sample L mainly consists of quartz and feldspar and contains a certain amount of dolomite and fewer clay minerals. Sample Y largely consists of quartz and contains some amount of feldspar, calcite, and dolomite, as well as about 15% of clay minerals. Quartz represents 40% of sample M, and clay minerals account for 37%. Sample C essentially comprises quartz and feldspar and contains 8% of clay minerals. On the whole, sample M has the highest clay mineral content, sample L and sample C have the lowest clay mineral content, and sample Y has a medium clay mineral content. The higher the content of clay minerals is, the more intense the interaction between the fracturing fluid and the rock is after the fracturing fluid enters the reservoirs. It causes a series of changes in physical and chemical properties such as rock surface mechanical properties, flow channels of fluids, and even microscopic pores.

The length of the experimental samples is mainly 5 cm. The average porosity of samples L, Y, M and C are respectively 7%, 14%, 4%, and 9%, and the average permeability of which are 0.01 mD, 0.015 mD, 0.002 mD, and 0.006 mD. The fundamental parameters of samples and the liquid type used correspondingly are as shown in

Table 3. Fundamental parameters of samples and liquid types.

Serial Number	Diameter/cm	Length/cm	Permeability/mD	Porosity/%	Liquid Types
L-1	2.5	5.0	0.0012	7.2	Distilled water
L-2	2.5	5.0	0.0018	6.8	Distilled water/Slick water
Y-1	2.5	5.0	0.011	13.1	distilled water
Y-2	2.5	10.0	0.025	12.7	Distilled water/Slick water
Y-3	2.5	5.0	0.017	15.3	Distilled water
Y-4	3.8	5.0	0.016	13.9	Distilled water
Y-5	5.0	5.0	0.022	14.5	Distilled water
M-1	2.5	5.0	0.0011	3.2	Distilled water
M-2	2.5	5.0	0.0018	5.1	Distilled water
M-3	2.5	5.0	0.0032	4.9	Distilled water
M-4	3.8	5.0	0.0025	5.7	Distilled water
M-5	5.0	5.0	0.0021	4.1	Distilled water
C-1	2.5	5.0	0.0072	10.2	Distilled water
C-2	2.5	5.0	0.0081	9.7	Distilled water
C-3	2.5	5.0	0.0055	8.6	Distilled water

.

2.2. Experiment Method

Driven by the pressure pump, the fluid flows from an upstream container through the fractures with certain fracture aperture into a downstream container. The schematic diagram of the experimental device for fracture connectivity is as shown in Figure 3. Both ends of the core holder were equipped with a pressure sensor (PA-33X) with a measuring range of 0–100 MPa. The output signals were collected by the RS485 port, which has a control accuracy of 0.05% at room temperature to achieve automatic data collection in real-time. The physical picture of the laser microscope (VK-X250K) is as shown in Figure 4. The laser microscope has a resolution of 5 nm for height measurement and a resolution of 10 nm for width measurement. The operating range of the stage on the horizontal plane is 100 mm × 100 mm, which can meet the test requirements for rock surface topography at the micron or centimeter level.

Before the experiment, the physical parameters of samples such as core diameter, length, porosity, and permeability were recorded. In addition, the fracture surface roughness was quantitatively characterized to facilitate the analysis of influencing factors. Then, the flow rate of the stable injection stage was obtained to calculate the fracture aperture and permeability. Finally, the characteristics of pressure drop were arrived at with the upstream constant volume and the downstream constant pressure. The law of fracture connectivity was thus obtained after the analysis, and the fracture connectivity characteristics were evaluated by the half-life time. The above experimental methods include the following steps:

(1) A core sample was loaded into the core holder with the confining pressure increased to a set value and the downstream pressure adjusted to a set value. (2) Pressure sensors began to work, and the pumping system was initiated to pump liquids into the core fracture with a set pressure. The flow rate at the time when the flow reaches stable was recorded to calculate the fracture aperture and permeability. (3) The upstream liquid pumping valve was closed, and as the upstream pressure gradually decays when it through the fracture, the change in the decayed pressure over time was recorded. (4) The above processes were repeated with the change in factors such as the roughness of fractures, fracture aperture, liquid properties, and the scale of the sample to research the fracture connectivity under different factors.

3. Results

3.1. The Roughness of Fracture Surface

The roughness of the fracture surface in this study mainly refers to the roughness of the partial fracture surface. Figure 5 shows the three-dimensional view of the roughness of the partial fracture surface. Parameters S_a and S_z were employed to quantitatively characterize the roughness of fracture surface. Specifically, S_a represents the arithmetic mean height which is the arithmetic mean of the absolute value of height from the mean surface, and S_z represents the maximum height which is the absolute value of the maximum height from the mean surface.

Figure 6 is a quantitative characterization parameter diagram of the roughness of the partial fracture surface. The arithmetic mean height of each sample is as shown in Figure 6a and the maximum height of each sample is as shown in Figure 6b. There is little difference in the roughness of the partial fracture surface between sample L of the Lucaogou Formation and sample Y of the Yanchang Formation. The partial fracture surface roughness of sample M of the Longmaxi Formation is the smallest. The partial fracture surface roughness of sample C of the Yingcheng Formation is the largest.

3.2. The Characteristics of Pressure Evolution

The pressure gradually decreases over time (Figure 7). The injection pressure applied to all the experimental samples is 3 MPa. While the confining pressures applied to samples are different. To samples L-1 and L-2, the confining pressure applied is 4–6 MPa. The confining pressure for samples Y-1 and Y-2 is 4–6 MPa and 8–10 MPa, for sample Y-3 is 4–6 MPa and 15 MPa, for sample Y-4 is 4–6 MPa, and for sample Y-5 is 18 MPa, 20 MPa, and 22 MPa. The confining pressure applied to sample M-1 is 5–10 MPa and 12 MPa, to sample M-2 is 4–6 MPa and 8 MPa, and to samples M-3, M-4 and M-5 are 4–6 MPa. As to sample C-1, the confining pressure applied is 4–9 MPa. Comparative experiments on the pressure drop laws under high confining pressure and low confining pressure were carried out with samples Y-1, Y-2, Y-3, Y-5, M-1, M-2, and C-1. The speed and the time required for pressure drop are different. The fracture with low confining pressure has a relatively large aperture, rapid pressure drop, and good connectivity. With the increase in confining pressure, the fracture aperture decreases, and the pressure drops with a slow speed and gradually tends to a linear drop mode. It leads to the deterioration of fracture connectivity. The faster the pressure drops, the better the fracture connectivity is.

3.3. The Factors Influencing Fracture Connection

The fracture aperture is affected by in-situ stress conditions. The roughness of the fracture surface is an inherent property of rock. The contact area is closely related to the closure of the fracture. The properties of a fracturing fluid also affect pressure transmission. Therefore, a single factor analysis of fracture aperture, the roughness of fracture surface, contact area, and the properties of the fracturing fluid is carried out to arrive at the law of their influence on fracture connectivity.

3.3.1. Fracture Aperture

Samples L-1 and Y-3 were used as examples to analyze the influence of fracture aperture on connectivity. As shown in Figure 8, as the fracture aperture decreases, the time required for pressure drop increases, and fracture connectivity decreases. When the fracture aperture of sample L-1 is 27 μm, 23 μm, and 20 μm, as the fracture aperture decreases, the time it takes for the pressure to drop from 3 MPa to 1.5 MPa increases significantly. When the fracture aperture is 27 μm, the time for the pressure drop is about 80 min. When the fracture aperture is 23 μm, the time needed is about 380 min. When the fracture aperture is 20 μm, it takes more than 800 min. The decrease in the fracture aperture leads to the increase in time needed for the pressure drop time in a multiplied manner, indicating that the fracture connectivity decreases sharply with the reduction in the fracture aperture. As to sample Y-3, the fracture aperture of which is 32 μm, 29 μm, 27 μm, and 24 μm. When the fracture aperture is 32 μm, 29 μm, and 27 μm, the time required for the pressure drop increases slowly. When the fracture aperture reduces to 24 μm, the time for pressure drop increases from 30 min to more than 120 min. The fracture aperture decreases under high closure stress, resulting in a significant decrease in fracture connectivity. Therefore, the fracture connectivity decreases with the reduction in the fracture aperture, and particularly, and the main performance on site is that the connectivity of fractures decreases with the increase in closure stress. Meanwhile, affected by the morphology of the fracture surface, although the fracture aperture is similar, there is an obvious difference in fracture connectivity.

3.3.2. Fracture Roughness

According to the results of the fracture roughness characteristics, the roughness of the partial fracture surface of samples C-1 and Y-3 is larger, followed by that of sample M-3, and the roughness of samples L-1 and L-2 is relatively small. The influence of roughness on fracture connectivity is as shown in Figure 9. With an injection pressure of 3 MPa and a confining pressure of 5 MPa, the pressure drop of samples C-1 and Y-3 is significantly faster than that of other samples. Relatively, the pressure drop rate of sample M-3 is at a middle level, and the pressure of samples L-1 and L-2 decreases with a significantly slower drop rate. Specifically, the pressure of samples C-1 and Y-3 decreases from 3 MPa to 1.5 MPa within 20 min. While for sample M-3, it takes about 120 min, and that of samples L-1 and L-2 is close to 300 min. The fracture connectivity of samples C-1 and Y-3 is the best, followed by M-3, and the fracture connectivity of samples L-1 and L-2 is relatively poor. The higher the roughness is, the better the fracture connectivity is. The contribution of the fracture surface roughness of tight volcanic rock and tight sandstone to fracture connectivity is better than that of the Longmaxi Formation shale and significantly better than that of the Lucaogou Formation shale. Higher fracture surface roughness and larger shear slip distance lead to better fracture connectivity.

3.3.3. The Contact Area

The Yanchang Formation tight sandstones and the Longmaxi Formation shales with a length of 50 mm, and diameters of 25 mm, 38 mm, and 50 mm were used to analyze the influence of contact area on fracture connectivity. As the contact area increases and the time required for the pressure drop decreases, the fracture connectivity is improved. As shown in Figure 10, Y-# represents the Yanchang Formation tight sandstones, and M-# represents the Longmaxi Formation shales, with an injection pressure of 3 MPa and a confining pressure of 5 MPa, it can be seen from Y-# that the pressure of the tight sandstones of the Yanchang Formation with diameters of 50 mm and 38 mm drops rapidly and the pressure drops from 3 MPa to 1.5 MPa within less than 20 min, while the sample with a diameter of 25 mm needs 140 min. It can be seen from M-# that, for shale with a diameter of 50 mm, the pressure drops from 3 MPa to 1.5 MPa needs 20 min. For the same situation, shale with a diameter of 38 mm needs 130 min, and shale with a diameter of 50 mm needs 160 min. During fracturing, the larger the areas connected in the fractures are, the greater the impact on pressure communication is. Therefore, the closed contact of fractures should be properly controlled during fracturing design to avoid the large-area continuous disturbance of interference fractures to reduce the negative impact of interference.

3.3.4. The Property of Fracturing Fluid

The experiments on the influence of viscosity on fracture connectivity were carried out with samples Y-1 and C-1 with liquids having a viscosity of 1 mPa·s, 10 mPa·s, and 30 mPa·s. As shown in Figure 11, as the viscosity increases, the time for the pressure decay increases sharply, indicating the rapid deterioration of fracture connectivity. For sample Y-1, when the viscosity is 1 mPa·s and 10 mPa·s, the pressure decay time is within 500 min. When the viscosity increases to 30 mPa·s, the pressure decay time exceeds 2000 min. Similarly, the pressure of the sample C-1 decays rapidly when using a fracturing fluid with lower viscosity (1 mPa·s and 10 mPa·s), and decays very slowly when high-viscosity fracturing fluid (30 mPa·s) is used. The experimental results show that the properties of fracture connectivity can be characterized by the pressure decay time when considering the properties of liquid. The viscous force of the liquid increases significantly due to the increase in the viscosity of a fracturing fluid. The significant increase in friction between the liquid and the fracture surface is also the reason that leads to the above phenomenon. The movement of fluids in the fractures is restricted, significantly reducing the possibility of interconnection between fracturing fractures. Slick water fracturing fluids with lower viscosity and friction resistance are mostly used for large-scale multi-stage fracturing, which leads to a greater impact on fracture connectivity. It is suggested that the influence of the viscosity of fracturing fluids on fracture connectivity should be fully considered when designing the fracturing fluid used in layers prone to disruption, to reduce the impact of the disturbance during fracturing on the pressure of adjacent wells.

4. Discussion

4.1. The Effect of Scale and Time

In this study, experiments were carried out to research the law of fracture connection. From Section 3.3 the factors influencing fracture connection, it can be seen that the fracture connectivity is affected by the width of the fracture contact area. The fracture connectivity improves with the increase in the contact area and the number of connected channels. A fracture hit phenomenon appears to cause a long-term impact on production if larger areas of connected fractures are formed during fracturing. On the contrary, if connected fractures are formed with relatively small areas during fracturing, although a certain degree of fracture hit phenomenon appears, the influence of the fracture connectivity on the production gradually becomes smaller. This is related to the scale of the connected fractures, which shows as a typical scale effect.

The connectivity of some fractures with similar fracture scales gradually decreases over time, and the fractures may even gradually disappear. In this case, the connectivity of fractures shows a typical time effect. Zhou et al. [32,33] argued that the water absorption in tight reservoirs results in the weakening of the mechanical strength of the fracture surface. With the increase in time, the degree of weakening aggravates, making the conductivity of fractures decrease and the connectivity of fractures deteriorate. Using the comprehensive technology of microseismic and radioactive tracers and numerical simulation, Manchanda [7] concluded that with the leakage of the fracturing fluids, the opened and unsupported fractures gradually close over time and the fracture connectivity decreases. As a result, the impact of interwell interference on later production is reduced. It can be seen from Table 2 that the brittle minerals in the Lucaogou Formation samples are mainly feldspar, and the clay minerals of which are mostly chlorite. For samples from the Longmaxi Formation, the brittle minerals are mainly quartz, and the clay minerals are essentially illite. For samples from the Yanchang Formation, the brittle minerals largely consist of quartz, and the clay minerals are mainly illite/smectite mixed layers. For samples from the Yingcheng Formation, the brittle minerals are mainly quartz and feldspar, and the clay minerals are essentially montmorillonite. After entering reservoirs, the fracturing fluid is easy to interact with the fracture surface of samples with high content of swelling clay minerals. Consequently, fracture surface damages appear over time and the fracture connectivity is affected to some extent. With relatively strong time dependence, the connectivity of fractures shows a strong time effect.

4.2. Application of Interwell-Fracturing Interference

During fracturing, the form of fracture connection in the minefield is complex, and four probable situations are as shown in Figure 12a–d. Figure 12a shows the direct connection due to the excessive extension of the fractures. Figure 12b shows the connection of two branch fractures. Figure 12c shows the connection of two opposing fractures through the matrix. Figure 12d shows two parallel fractures connected through the matrix. The pressure in the fracture system needs to pass through the connected fractures or matrix, so the total time of interwell interference is equal to the sum of the time for the pressure to pass through both the fractures and the matrix, as shown in the Formula (1).

$$T_{total}{{}_{}}_{interference}=T_{pressure}{{}_{}}_{transmission}{{}_{}}_{in}{{}_{}}_{fracture}+T_{pressure}{{}_{}}_{transmission}{{}_{}}_{in}{{}_{}}_{matrix}$$

(1)

According to the upstream constant volume and downstream constant pressure equations as well as boundary conditions, the calculation formula of permeability can be derived [32]. The equation can be deformed to obtain:

$$\Delta t=\frac{\mu L{C}_1}{AK}\cdot \ln \frac{\Delta P_t}{\Delta P_{t+\Delta t}}$$

(2)

$$C_1=\frac{d\left({\rho }_w{V}_1\right)}{{\rho }_{w}d{p}_1}=\frac{d{V}_1}{d{p}_1}$$

(3)

$$k_f={10}^3\frac{w_f^2}{12}$$

(4)

$$w_f=\sqrt[3]{30\pi d\left(k_{effective}-k_m\right)}$$

(5)

where Δt is the time interval of the data point pair, min; μ is the dynamic viscosity of water, mPa s; L is the height of the sample, cm; C₁ is the water capacity of the upstream pressure vessel, m³/Pa; A is the cross-sectional area of the sample, cm²; K is the permeability of the porous medium, mD; ΔP_t, ΔP_t+Δt is the pressure difference between the two control points on the differential pressure curve, MPa; k_f is the permeability of propped fracture, mD; w_f is thefracture width of propped fracture, μm; and k_m is thematrix permeability, mD.

The connectivity of a single rough fracture is characterized by half-decayed pressure. According to the above research, it can be seen that the factors affecting the pressure drop include the fracture aperture, the roughness of the fracture surface, the contact area of the fracture, liquid properties, and the permeability of the fracture, etc. The half-life time is calculated according to Formula (2) and the results are as shown in

Table 4. The fracture connectivity characterized by the half-life time.

Number	Flow Rate/(mL/min)	Injection Pressure/MPa	Outlet Pressure/MPa	Liquid Type	Half-Life Time/min
L-1	0.0035	3	1	distilled water	230
L-2	0.04	3	1	distilled water	220
Y-3	0.5	3	1	distilled water	15
M-3	0.08	3	1	distilled water	110
C-1	0.3	3	1	distilled water	20

. For the five groups of samples, L-1, L-2, Y-3, M-3, and C-1, the calculated half-life time is respectively 230 min, 220 min, 15 min, 110 min, and 20 min, which is consistent with the half-life time obtained from the experiment shown in Figure 9. In terms of fracture connectivity, samples Y-3 and C-1 are the best, followed by sample Y-3, and the connectivity of samples L-1 and L-2 is the worst. The difference in connectivity can be quantitatively described with the half-life time to provide foundations for the analysis of pressure interwell interference during fracturing.

4.3. The Limitations of This Study

Fracture permeability can be calculated, and matrix permeability can be assumed. However, the range of both the confining pressure and the injection pressure applied during the experiment is limited by the instruments used. Thus the experiments were carried out under a relatively low-pressure level, making it difficult to simulate the real confining pressure and the injection pressure of the original formation. In the next step, the performance of the instrument will be continuously improved to meet the experimental requirements of the real formation temperature and pressure system.

5. Conclusions

In this study, on the basis of characterizing the roughness of fractures, a laboratory evaluation method for fracture connectivity was established. The connectivity characteristics and factors influencing same were studied. The time and scale effects of fracture connectivity were discussed and their application in interwell interference was analyzed. The conclusions are as follows:

(1)

As an important mechanism of interwell interference, fracture connectivity has a strong scale effect and time effect. If larger areas of fractures are formed during fracturing, it will lead to a longer-term impact on production, which is shown as a strong scale effect. After entering the reservoirs, the fracturing fluid is easy to interact with the fracture surface of samples with a relatively high content of swelling clay minerals. With the increase in time, the fracture surface is damaged and the fracture connectivity decreases, which shows a strong time effect;

(2)

Fracture connectivity is related to the fracture aperture, the roughness of the fracture surface, the fracture closure contact area, and the viscosity of a fracturing fluid. The decrease in fracture aperture and the increase in the fluid viscosity leads to a significant reduction in fracture connectivity, while higher fracture surface roughness and larger contact area can make fracture connectivity better;

(3)

The connectivity of rough fractures is characterized by half-decayed pressure and a laboratory evaluation method for fracture connectivity is established. The fracture connectivity evaluated based on the calculated half-life time is consistent with the experimental results. The total time of the interwell interference can be calculated with the fracture interference time and the matrix interference time. It is equal to the sum of the time for the pressure to pass through both the fracture and the matrix.