Introduction
Modern fracturing technology mainly focuses on volume fracturing, and its main characteristics are: staged fracturing of horizontal wells with low viscosity slickwater, large-scale, large displacement, which is different from traditional gel fracturing. The fractures formed by modern fracturing are no longer simple symmetrical fractures with two wings, but complex fracture network [1]. The complexity of fracture morphology and associated leak-off problem are the key factors affecting fracture closure calculation, geometric dimension and conductivity evaluation [2]. As a result, the techniques used in the traditional fracturing such as the G function pressure drop analysis and other technologies are no longer applicable to modern volume fracturing.
For the volume fracturing extension model, domestic and foreign scholars have carried out systematic research in many aspects [3]. With regard to fracturing fluid leak-off, Carter first proposed a one-dimensional leak-off model [4], which used three leak-off coefficients to characterize the effects of fracturing fluid viscosity, formation fluid compressibility and fracture wall building property. Later, scholars have made a series of amendments and improvements to the Carter leak-off model, such as dividing the wall building and fluid leak-off into three areas: filter cake, intrusion zone and reservoir [5], changing the one-dimensional leak-off into two-dimensional leak-off [6], and considering the viscosity change of fracturing fluid [7] and the leak-off of natural fractures [8]. Based on the theory of leak-off coefficient, scholars have established a series of interpretation models of pump-stopping pressure drop based on G function analysis [9⇓⇓⇓-13]. The classical G-function model was first proposed by Nolte in 1979 for two-dimensional simple fractures [9]. Later, Nolte associated the fracture geometry with the leak-off coefficient, and successively established PKN, KGD, Radial and other fracture pump-stopping pressure drop models [10-11]. In 2011, Mohamed [12] and others put forward an interpretation method for pump-stopping pressure drop of fracture network, proving that the log-log curve of Bourdet pressure drop derivative is consistent with the derivative of G function in the identification of fracture closure. In addition, it has more advantages in identification of flow pattern. In 2019, Liu et al. [13] proposed a series of pump- stopping pressure drop models under non-ideal leak-off conditions by simplifying the treatment method by substituting equivalent leak-off of primary and secondary fractures to constant and equal leak-off coefficients under different pressure stages.
Compared with the simple fractures formed by the traditional gel fracturing, the fracture closure of the complex fracture network formed by the volume fracturing is also different after pump stop. The field monitoring and interpretation results of pump-stopping pressure drop [14] found that the volume fracturing network has the characteristics of "late start of the closure period, long closure period". Specifically, the closure period can only be identified after 10-55 h of pump-stopping for single stage volume fracturing, and the closure period can be identified after 0.8-11.0 h of pump-stopping for simple fracture fracturing. Fracture closure can only be identified after 60-250 h of pump-stopping for volume fracturing, while it can be identified after 3-70 h of pump-stopping for traditional single stage fracturing [15]. The identification of the closure period is the basis for the interpretation of the pressure drop of the G function. The leak-off coefficient is calculated by identifying the linear flow or radial flow after closure, so as to ensure the accuracy of the leak-off calculation. However, in the case that the volume fracturing network is closed for a long period but not closed, the calculation result of the leak-off amount of the G-function model is often too high, and there is a large error with the actual situation [16]. This is also the main reason that the G function model is mostly used for small-scale test fracturing [17] to identify the fracture closure and calculate the closure stress, which cannot be used for the evaluation of the operation parameters of the main fracturing fractures.
In order to evaluate the main fracturing treatment effect when the fracture network is not completely closed, this paper proposes a pump-stopping pressure drop model under the volume fracturing mode. In this model, we consider the non-uniform transient leak-off of the primary and secondary fractures, obtain the log-log plot of the pump-stopping pressure drop through numerical simulation to determine the closure transient of the fracture network, and calculate the fracture network parameters by fitting the pressure drop history. It is expected to provide theoretical support for the evaluation of volume fracturing effect and understanding of fracturing network closure law.
1. Pump-stopping pressure drop model
1.1. Assumptions and physical models
Assumptions: (1) The volume fracturing horizontal well is decomposed into multiple fracturing stage units. Each fracturing stage unit is composed of horizontal wellbore (W), main fractures with multiple clusters (F), secondary fractures (f) and rock matrix (m). Fluid exchange is carried out through the wellbore with the outside world. F, f and m are interconnected. (2) The main fractures are multiple clusters of vertical fractures symmetrically distributed on both sides of the wellbore with equal cluster spacing, and the properties of each cluster of main fractures in each fracturing stage are the same. The secondary fracture is perpendicular to the main fracture. The number and geometric parameters of primary and secondary fractures are preset [18]. (3) The fracturing fluid is a combination fluid of variable viscosity slick water, and the high viscosity and low viscosity slick water are characterized by viscosity and density. The viscosity of slick water before gel breaking is used to simulate the pumping process, and the viscosity of slick water after gel breaking is used to simulate the pump-stopping pressure drop, without considering the free flow interface of the combined fluid. (4) The leak-off of fracturing fluid after pump shutdown is considered as an isothermal flow process, which is related to the fluid property, fracture network size and conductivity, and does not consider the influence of crustal stress. (5) The fracture closure is considered as the coupling effect of fracturing fluid leak-off and fracture width stress sensitivity.
Based on the above assumptions, in the grid model of a single fracturing stage (Fig. 1 ), the flow and pressure continuity of the wellbore, main fracture, secondary fracture and matrix system are coupled through the contact surface. The three directions of model grid division are represented by x, y and z, the wellbore is the source sink term, and the bottom hole grid is located at the center of the main fracture. Multi-cluster main fractures are characterized by densified grid, and their dimensions are defined by fracture length, width and height. The conductivity of secondary fractures is characterized by fracture width and permeability. The matrix is evenly distributed around the main fracture and secondary fracture, and its reservoir capacity and flow capacity are characterized by porosity and permeability respectively. After the completion of fracturing pumping, the fracturing fluid in the fracture network system flows between the primary and secondary fractures, and further leaks to the matrix system. During this process, the fracture network continues to close.
Fig. 1. Grid model of fracturing stage unit and schematic diagram of coupled flow between fracture network and matrix after pump shutdown. |
1.2. Mathematical model
The mathematical model of pump-stopping pressure drop consists of three parts, including fracture network closure model, fracturing fluid leak-off model and fracture network material conservation model.
1.2.1. Fracture network closure model
After pump shutdown, the porosity and permeability in the primary and secondary fracture grids change with the pressure, which is shown as a pressure sensitive equation[19]:
where X={F, f}
The ratio between the number of secondary fractures and the area of fracture network is defined as the density of secondary fracture:
According to the fracture closure control equation [20], the change expressions of the fracture width and the average fracture width of the primary and secondary fractures during the pump-stopping pressure drop process are modified:
1.2.2. Fracturing fluid leak-off model
The leak-off coupling flow model of "primary fracture-secondary fracture-matrix" is established. During the pump-stopping, the leak-off of main fracture ($q_{F,m}$) and secondary fracture ($q_{f,m}$) changes in real time with their fracture width ($w_{F}$, $w_{f}$) and pressure difference (Δp) between inside and outside the fracture, and the expression is as follows:
The cross-flow between main and secondary fractures ($q_{F,f}$) is corrected by flow equation [21]:
1.2.3. Fracture network material conservation model
During the pump-stopping, the pumping volume of surface fracturing fluid is zero, and there is fracturing fluid flow between two adjacent underground double medium (F-f main and secondary fracture crossflow, F-m main fracture leak-off, f-m secondary fracture leak-off). The inflow and outflow of F, f, m offset each other, and the whole system accords with the material conservation.
The material conservation equation in the main fracture is:
The material conservation equation in secondary fracture is:
The material conservation equation in the matrix is:
Where the injection amount of the wellbore to the main fracture is shown as follows:
$q_{W,F}=\frac{2\pi h \rho w_{X}^{2}}{\Delta V\mu Bln\lgroup\frac{r_{e}}{r_{w}}+S\rgroup}(p_{w}-p_{F})$
The constraint equation of water saturation in each medium is:
where Y={F, f, m}
1.3. Initial and boundary conditions
The fracturing stage selected for simulation calculation should meet the closed outer boundary condition. The pumping process is constant displacement injection, and the relationship between the liquid inflow of each cluster and the pumping displacement is as follows:
At the initial time of fracturing fluid injection at the first stage, the porosity, permeability and pressure in each medium are consistent with the water saturation, which are all in the original formation state. At the initial time of subsequent fracturing fluid injection, the parameter values in each medium are calculated from the simulated pumping of fracturing fluid in the previous stage.
The initial conditions for fracturing fluid injection at the first stage are:
The initial conditions for subsequent fracturing fluid injection are:
The pressure field and saturation field obtained from the simulation of pumping process are used as the initial conditions for the pump-stopping pressure drop model:
1.4. Solution method
In this model, the stress sensitivity of porosity and permeability (Eq. (1), Eq. (2)) of the main and secondary fracture and the leak-off (Eq. (6)) of the main and secondary fracture are coupled through the fracture width formula (Eq. (4)) during the pump-stopping process. At the same time, by introducing wellbore injection rate equation (Eq. (11)) and crossflow rate equation (Eq. (7)) between main and secondary fractures into the material conservation equation, the continuous coupling of pressure and flow rate of wellbore-main and secondary fracture matrix system is realized.
Based on the preset initial and boundary conditions, the specific calculation includes the following steps (Fig. 2 ): (1) According to the geology and operation conditions of the fracturing stage, input the original pressure, porosity, permeability, number of main fracture clusters and other parameters of the reservoir, and preset the half length of the main fracture and the number of secondary fractures. (2) The porosity, permeability and stress sensitivity coefficients of the main and secondary fracture are set to simulate the pumping process equivalently, so that the simulated pumping pressure and pumping fluid volume are consistent with the actual treatment (simulation results of commercial software Meyer). (3) The pressure field, saturation field and main and secondary fracture width at the end of the pumping process are taken as the corresponding parameters at the initial time of pump-stopping, and the pressure drop process of pump shutdown is simulated continuously (the pump injection volume is zero). (4) Calculate the main and secondary fracture leak-off, crossflow, fracture width and bottom hole pressure in each step. (5) At the end of pump shutdown, the product ($K_{X}W_{a,X}$) of the average width of main and secondary fractures and their permeability is taken as the output of conductivity; The difference between the fracture width of the main and secondary fractures and the fracture width at the initial time of pump shutdown is output as the closed fracture width ($W_{b,X}-W_{a,X}$). The sum of the main and secondary fracture leak-off of each step at the pump-stopping stage is output as the network leak-off of the pump-stopping process.
Fig. 2. Algorithm flow chart. |
The solution algorithm adopts finite difference method and semi implicit method to linearize the above nonlinear equations, and then uses Gauss-Seidel iterative method to solve the equations iteratively. The calculation of each step in the model is only between adjacent grids, and the fluid migration in unit time is set to be less than or equal to the distance of a single grid. When the relative error is less than the minimum value (1×10−7), the calculation of this time step converges, and the calculated parameters of this time step are used as the initial values of the next step to continue iterative calculation until the set calculation time is reached.
2. Numerical simulation of pump-stopping pressure drop
2.1. Model description of fracturing stage
The simulated fracturing stage is established based on the geological and operation parameters of a typical fracturing horizontal well in shale reservoir [22⇓-24]. The original formation pressure is 36 MPa, the formation temperature is 120 °C, the length of the fracturing stage is 30 m. There are three clusters per stage, and the model scale (xyz) is 50 m×500 m×40 m. The reservoir matrix porosity is 8%, and the permeability is 0.010×10−3 μm2. The initial porosity and initial permeability of main and secondary fractures are set as 9% and 0.013×10−3 μm2. Combined fracturing fluid was selected, including low viscosity slick water and high viscosity slick water, with density of 1000 kg/m3 and 1100 kg/m3 respectively, viscosity of 5 mPa·s and 15 mPa·s respectively. The pumping procedure of fracturing fluid is as follows: low viscosity slick water with a fixed displacement of 10 m3/min was injected for 1.5 h, and the injected liquid volume was 900 m3, then high viscosity slick water with a fixed displacement of 10 m3/min was injected for 1.5 h, and the injected liquid volume was 900 m3.
The fracturing network parameters were simulated by the commercial software Meyer for fracture extension. The minimum horizontal principal stress was set to 62.1 MPa, the fracturing fluid leak-off during pumping was 1128.42 m3, and the converted fracturing fluid efficiency was 37.3%. It was simulated that the half length of the main fracture was 137.94 m, the fracture width of the main fracture and the secondary fracture were 6.74 mm and 3.30 mm respectively, and the density of the secondary fracture was 0.62 fractures/m2. The fracture network parameters were preset as the initial setting parameters of this model to conduct numerical simulation of fracturing pumping process, so as to ensure that the operation pressure (Fig. 3 ) and leak-off (1139.04 m3) in the pumping stage were consistent with Meyer's simulation results. The stress sensitivity coefficients of main and secondary fractures obtained by fitting were 0.10 MPa−1 and 0.07 MPa−1 respectively.
Fig. 3. Comparison of pumping simulation results with software Meyer simulation results. |
2.2. Simulation results of pump-stopping pressure drop
Based on the pressure and saturation fields at the end of pumping, the viscosity of fracturing fluid after pump shutdown was set as 5 mPa·s after gel breaking to simulate the bottom hole pressure drop and fracture closure during the period of 1 d after pump shutdown. The transient leak-off simulation results of main and secondary fractures within 1 d of pump-stopping are shown in Fig. 4 . At the initial time of pump shutdown, the crossflow phenomenon between fractures is obvious, and then the crossflow speed drops rapidly. After the end of Stage I, the crossflow is no longer dominant. With the extension of pump-stopping time, the fracturing fluid in the main and secondary fractures will further leak to the matrix, showing a non-uniform decline trend. The leak-off rate of the main fractures is slow, and the leak-off rate of the secondary fractures is fast. The fracture network as a whole shows the leak-off characteristics of fast first (Stage II), slow later (Stage III), and close to zero (Stage IV). After pump shutdown for one day, the total leak-off is 221.46 m3, including 51.62 m3 for the main fracture, 169.84 m3 for the secondary fracture, and 28.58 m3 for the main fracture to the secondary fracture. The average width of the main fracture is closed with 1.56 mm, and the average width of the secondary fracture is closed with 1.28 mm.
Fig. 4. Simulation results of crossflow velocity between main and secondary fractures, leak-off velocity of main fracture, leak-off velocity of secondary fracture. |
Fig. 5. Bottom hole pressure drop and derivative log-log plot simulation results. |
2.3. Verification of pump-stopping pressure drop model
In order to verify the accuracy of the calculation results of pump-stopping pressure drop model in this paper, the simulated bottom hole pressure drop was analyzed with a G function curve. The closure stress explained by the classical tangent analysis method of the G function is 62.1 MPa, and the corresponding closure time is 278.5 min (0.19 d), which is consistent with the result of pc=62.1 MPa explained by the start time tc=0.19 d of the model closure control stage (IV) in this paper (Fig. 5 ). However, as the fracture network has not been completely closed, the fracturing fluid efficiency estimated by G function is only 10.6%, which is far less than 37.3% at the end of pumping, which means that 71.6% of the fluid in the fracture at the end of pumping has been leaked to the formation within 1 d after pump shutdown, which is contrary to the view that shale reservoir has low leak-off after fracturing at home and abroad [30]. This is also consistent with the description in the Reference [14] that "using the G function analysis will overestimate the leak-off when the fracture is still open". With the model in this paper, we obtained through calculation that the fluid loss within one day of pump shutdown is 221.46 m3. After conversion, 32.9% of fracturing fluid in the fracture is lost in the formation (23.3% in the main fracture and 76.7% in the secondary fracture). The results are more in line with the actual situation.
2.4. Sensitivity analysis of pump-stopping pressure drop characteristic curve
In order to study the relationship between the pump- stopping pressure drop dynamics and the fracture network parameters, sensitivity simulation and analysis were carried out. Fig. 6 and Fig. 7 respectively show the simulation results of the characteristic curve of the pump-stopping pressure drop under the condition of different network size and conductivity. The comparison with the results of the above basic model shows that (Table 1 ): the scale of the main and secondary fractures affects the derivative shape of the flow stage and closure control stage in the fractures. When the scale of the main fractures is large and the scale of the secondary fractures is small, the pressure drop derivative presents a "steep upward" shape, and the 0.5 MPa/d and 0.25 MPa/d slope segments cannot be identified. When the main fracture is small in scale and the secondary fracture is large in scale, the derivative curve takes on a "flat" shape as a whole, with the 0.5 MPa/d slope segment obviously ahead of time, and a long transition between the 0.5 MPa/d slope segment and the 0.25 MPa/d slope segment. The conductivity of the main and secondary fractures mainly affects the early flow stages I and II of pressure drop. When the conductivity of the fracture network is weak, the pressure drop derivative will also show a "steep upward" shape, but hardly affects the shape of stages III and IV. The conductivity of fracture network is positively correlated with leak-off volume and fracture closure. The secondary fracture size is positively correlated with leak-off volume and closure of the secondary fracture, but negatively correlated with closure of the main fracture.
Fig. 6. Pressure drop and its derivative curves of pump-stopping under different network sizes. |
Fig. 7. Pressure drop and its derivative curves during pump-stopping under different conductivities of fracture network. |
Table 1. Comparison results of sensitivity analysis of characteristic curves |
| Type | Main fracture half-length/ m | Secondary fracture density/ (fractures·m−2) | Main fracture conductivity/ (μm2·cm) | Viscosity of gel breaking liquid/ (mPa∙s) | Leak-off after pump shutdown for 1 d/m3 | Main fracture width closed/ mm | Secondary fracture width closed/mm |
|---|---|---|---|---|---|---|---|
| Basic model | 137.94 | 0.62 | 8.84 | 5 | 221.46 | 1.56 | 1.28 |
| Large main fracture and small secondary fracture | 150.00 | 0.48 | 8.84 | 5 | 214.38 | 1.85 | 1.22 |
| Small main fracture, large secondary fracture | 120.00 | 0.81 | 8.84 | 5 | 313.06 | 1.41 | 1.66 |
| Strong fracture network conductivity | 137.94 | 0.62 | 15.63 | 5 | 282.12 | 2.64 | 1.50 |
| Weak fracture network conductivity | 137.94 | 0.62 | 4.22 | 5 | 198.31 | 1.26 | 1.15 |
3. Application cases
The pump-stopping pressure drop model was used to fit the pump-stopping pressure drop of a typical volume fracturing horizontal well with multiple clusters (Well J14) in Jimusar sag, Junggar Basin. The fracture network parameters and closed amount of 39 fracturing stages were obtained by continuously adjusting the parameters for inversion.
The vertical depth of the horizontal stage of Well J14 is 3800 m, the formation pressure is 56.46 MPa, the porosity is 17.7%, and the permeability is 0.048×10−3 μm2. Fracturing was divided into 39 stages, each stage was perforated with 6-8 clusters, with the average cluster spacing of 5.8 m and the average stage length of 45.3 m. The combined fracturing pumping process of low viscosity slick water and high viscosity slick water was adopted. The fracturing fluid consumption was about 70 000 m3, and the ratio of low viscosity slick water to high viscosity slick water was 1:1.
After field treatment, a long period of continuous pressure drop was recorded. The pressure drop and pressure drop derivative curves of each fracturing stage were fitted using the pump-stopping pressure drop model in this paper. The fitting effect between the simulated data and the actual data is good. At the same time, it can be seen that the pressure drop derivative curves of the actual data are in the form of rising when the pump stops, which is consistent with the understanding obtained previously. This shows that the pump-stopping pressure drop model in this paper can effectively reflect the closure characteristics and dynamics of the fracture network (Fig. 8 ). The fracture network of each fracturing stage has not been completely closed in the monitoring period, and the average time to enter the fracture network closure stage is 0.17 d. The fitting and interpretation results of operation pressure drop are shown in Fig. 9 and Fig. 10 . The specific data are: the average half-length of the main fracture is 123.47 m, and the average conductivity is 8.73 μm2•cm, the average density of secondary fractures is 0.62 fractures/m2, and the average closed width of main and secondary fractures is 2.02 mm and 1.11 mm.
Fig. 8. Effect of pressure drop history matching at stages 6, 21 and 39 of Well J14. |
Fig. 9. Interpretation results of fracturing stage in Well J14. |
Fig. 10. Comparison between simulation results and microseismic monitoring data in Well J14. |
By comparing the interpretation results of each fracturing stage of horizontal well J14 with the interpretation results of microseismic monitoring (Fig. 10 ), the average half-length of fractures interpreted by microseismic method for the target well is 181.07 m, and the average half-length of main fractures interpreted by this model is 123.47 m, which is 68.1% of the microseismic monitoring results. This is because the existence of noise type invalid event points affected the accuracy of microseismic interpretation fracture parameters [31]. From the difference rule of fracture size, the change trend of fracture half- length in each fracturing stage explained by the model in this study is consistent with the microseismic monitoring results, indicating that the interpreted fracture size can reflect the non-uniform stimulation effect of each fracturing stage of the actual horizontal well.
4. Conclusions
At the initial time of pump-stoppping during volume fracturing of horizontal wells, the crossflow between main and secondary fractures is obvious, and then the leak-off is dominated. The leak-off of main and secondary fractures shows a non-uniform decreasing trend. The leak-off rate of main fractures is slow, while that of secondary fractures is fast. The overall fracture network shows a leak-off law of "fast first, then slow, and close to zero".
The pump-stopping pressure drop of volume fracturing with horizontal well can be divided into four main control stages, namely, "inter-fracture crossflow", "fracture network leak-off", "fracture network closure " and "residual leak-off", which respectively correspond to the 1.00 MPa/d slope segment, 0.50 MPa/d and 0.25 MPa/d slope segments, 1.00 MPa/d slope segment and negative slope segment of the log-log plot of Bourdet pressure drop derivative. The network conductivity has weaker influence on pump-stopping pressure drop curve shape compared with the network size. The conductivity of fracture network is positively correlated with leak-off volume and fracture width closed. The secondary fracture size is positively correlated with leak-off volume and width closed of the secondary fracture, but negatively correlated with width closed of the main fracture.
According to the field treatment data, the model in this study can effectively reflect the closure characteristics of the fracture network. The interpretation results are reliable and can reflect the non-uniform stimulation effect of each fracturing stage of the actual horizontal well.
Nomenclature
B—fracturing fluid volume coefficient, m3/m3;
cn—number of clusters, dimensionless;
dX—stress sensitivity coefficient of fracture porosity, MPa−1;
D—longitudinal migration distance of fluid, m;
g—acceleration of gravity, taking values 9.8 m/s2;
h—fracture height, m;
i—fracture grid serial number;
j—fracture cluster serial number;
KY—fluid flow space medium permeability, m2;
K0—initial permeability, m2;
L—main fracture half-length, m;
M—curve slope, MPa/d;
nw,X—number of grids contained in the fracture;
nf—fracture density, fractures/m2;
num—number of secondary fractures;
p0—original formation pressure, Pa;
pY—fluid flow space medium pressure, Pa;
pj,Y—fluid flow space medium pressure at initial time of subsequent fracturing fluid pumping, Pa;
pb,Y—fluid flow space medium pressure at the end of fracturing pump injection, Pa;
pw—bottom hole flowing pressure, Pa;
pc—closure stress, Pa;
pisip—instantaneous pump shutdown pressure, Pa;
Δp—pressure difference, Pa;
Δpc—pressure difference corresponding to closure stress, Pa;
qT—pumping displacement, m3/s;
qj—liquid inflow of jth cluster fracture, m3/s;
qW,F—fracturing fluid injection volume, kg/(m3·s);
qF,f—crossflow between main and secondary fractures, kg/(m3·s);
qF,m, qf,m—the leak-off of main and secondary fractures to matrix, kg/(m3·s);
re—reservoir equivalent radius, m;
rw—wellbore radius, m;
s—skin coefficient, dimensionless;
Sw,Y—water saturation of fluid flow space medium, %;
So,Y—fluid flow space medium oil saturation, %;
S0—initial water saturation, %;
Sw,j,Y—water saturation of fluid flowing space medium at initial time of subsequent fracturing fluid pumping, %;
Sw,b,Y—water saturation of fluid flow space medium at the end of fracturing pump injection, %;
t—time, s;
t0, tj—initial time of first and subsequent fracturing fluid injection, s;
tb—initial pump shutdown time, s;
tc—start time of the closure control stage (closed time), d;
ΔV—grid size, m3;
wX—fracture width, m;
wa,X—average fracture width, m;
wb,X—fracture width at pump-stopping, m;
x, y, z—rectangular coordinate system, m;
xmin, ymin—minimum grid coordinates of the fracture network extending in x and y directions, m;
xmax, ymax—the maximum grid coordinates of the fracture network extending in the x and y directions (the difference between ymax and ymin is the length of the main fracture), m;
$\alpha_{F,f}$—crossflow coefficient between main fracture and secondary fractures, m−2;
$\alpha_{F,m}$,$\alpha_{f,m}$—crossflow coefficient between main fracture and matrix, secondary fracture and matrix, m−2;
μ—fracturing fluid viscosity, mPa·s;
ρ—fracturing fluid density, kg/m3;
τ—tortuosity, dimensionless;
ϕY—fluid flow space medium porosity, %;
ϕ0—initial porosity, %.
Subscript:
f—secondary fracture;
F—main fracture;
m—matrix;
max—maximum value;
min—minimum value;
X—fracture type, values of F, f;
Y—fluid flow space medium type, values of F, f, m.