3D fracture propagation simulation and pressure decline analysis research for I-shaped fracture of coalbed

After hydraulic fracturing, some treatments intended for production enhancement fail to yield predetermined effects. The main reason is the insufficient research about the fracture propagation mechanism. There is compelling evidence that I-shaped fracture, two horizontal fractures at the junction of coalbed and cover/bottom layer, and one vertical fracture in the coalbed have formed in part of the coalbed after hydraulic fracturing. Therefore, this paper aims at I-shaped fracture propagation simulation. A novel propagation model is derived on the basis of a threedimensional (3D) model, and the coupling conditions of vertical fracture and horizontal fractures are established based on the flow rate distribution and the bottom-hole pressure equality, respectively. Moreover, an associated PDA (pressure decline analysis of post-fracturing) model is established. Both models complement with each other and work together to guide fracturing treatment. Finally, a field case is studied to show that the proposed models can effectively investigate and simulate fracture initiation/propagation and pressure decline.


Introduction
Hydraulic fracturing (HF) can be defined as the process of fracture initiation and propagation. It is the process of squeezing liquid into the fracture through hydraulic action [1]. Today, HF is an effective means to improve the production of oil and gas wells in the field of petroleum industry [2]. As a result of HF in a coal seam, artificially induced fractures appear, and permeability increases. In the meantime, high pressure fluid can squeeze out the gas in the coal seam, which can release and absorb the gas around the well to increase the total gas volume [3]. However, not all measures can achieve the expected effect in the production after coal seam fracturing [4,5]. Research on new fracture propagation mechanisms and models will provide more methods and effective guidance for analyzing and solving this problem. Promote the output after fracturing treatment to meet the expectation.
On the basis of theoretical and field experimental work on fracture propagation mechanism, vertical fracture and horizontal fractures coexist in coal seam after fracturing [6]. In this kind of complex fracture, T-shaped and I-shaped fracture are special shapes [7,8]. The mutual influence of many factors leads to this phenomenon, such as the existence of the coalbed cleats at the interface of coalbed and cover/bottom layer [9], the large differences in the mechanical properties between coalbed and cover/bottom layer, tectonic stress of coal seams, coal blockage, interface effects of different lithologies [10,11], etc. The T-shaped and I-shaped fracture have been found in some realistic areas, and they were directly observed after HF [12]. Meanwhile, Guo et al. [13] and Tang et al. [14] have proved the existence of these kinds of fractures. Hence, correct understanding the propagation mechanism of these kinds of fracture is essential for fracturing effects.
Several models have been developed to describe the fracture propagation during HF before, such as 2D fracture models, P3D models, and fully three-dimensional (F3D) models. These models describe the propagation of vertical fracture or horizontal fracture and are also the basis for studying the propagation of I-shaped fracture. Before the 1980s, the study of fracture propagation used 2D fracture models, such as the PKN [15], KGD [16], and Penny [17] models. PKN is the basis of various complex models [18], which can still play an excellent effect on some specific problems [19]. The 2D model is a simple and fast alternative tool, but the effect is not very good because it can not simulate the propagation of fracture in horizontal and vertical directions at the same time [20]. Scholars proposed a variety of P3D fracture models and two F3D fracture models based on the past study. Developed from the 2D models, the 3D models assume that the height increases with fracturing time and varies along the propagating direction. P3D models have been widely used [21][22][23]. Guo et al. [9] and Dontsov et al. [24] have researched and developed the P3D fracture model. Zhang et al. [25] distinguished the material properties and in-situ stress of the conventional model, and proposed a new quasi three-dimensional model with its numerical solution. Zia and Lecampion [26] proposed an open-source simulator of P3D model on Python and gave a series of numerical examples. Aiming at the problem that it was difficult to couple hydraulic fracturing with proppant transportation, Skopintsev et al. [27] proposed an enhanced P3D model. Linkov and Markov [28] combined KGD with P3D to obtain an improved P3D model, which is more perfect in considering in-situ stress. Wen et al. [29] considered the rheology and filtration of fracturing fluid, improved the PKN model, and obtained a new P3D model. The even more complex F3D models can produce better results. Clifton et al. and Lam established the F3D models respectively and had developed them later [30,31]. The F3D models are more rigorous in mathematics, but the operation cost is much higher. For complex fractures, Karimi-Fard et al. [32] and Hossain et al. [33] established fracture simulation models for fracturing of shale gas and tight gas reservoir, respectively. Wu et al. [9] and Cheng et al. [8] established a 2D mathematical model of the T-shaped fracture of the coalbed and solved this model, but it is limited to the 2D plane-strain conditions, which is different from actual situation. In addition, the complex fracture models based on displacement discontinuity method (DDM) and embedded discrete fracture model (EDFM) can also address the fracture geometries [34][35][36]. However, these models are aimed at complex fracture networks, which are not suitable for the I-shaped fracture studied in this paper. Moreover, the conventional fracture models mentioned above can also be used as the basis for the study of I-shaped fracture, but they can not directly solve the problem of the propagation simulation of it.
It is well known that simulation of fracture propagation and simulation of PDA are two important tools for calculating the geometric sizes and mastering the dynamic propagation laws of fractures [37]. Both methods support and complement with each other, and work together to guide fracturing treatment. Therefore, the goal of this work is to establish propagation simulation model and PDA model based on the propagation mechanism of I-shaped fracture. The T-shaped fracture can be solved by the similar method. Guo et al. [38,39] and Zhao et al. [40] had proposed the P3D model of PDA, and obtained a series of research results. However, the PDA models which have existed are just for conventional fractures (vertical fracture or horizontal fracture). There has been no satisfactory model for the I-shaped fracture so far.
Therefore, this paper aims at the simulation of I-shaped fracture based on its propagation mechanism. A 3D fracture propagation simulator is developed based on the existing fracture propagation simulation model [38] and the propagation laws of I-shaped frac-Energies 2022, 15, 5811 3 of 20 ture [9]. The realistic coupling conditions is established in terms of the flow rate distribution and the bottom-hole pressure equality. A satisfactory numerical method of the P3D model is developed based on the iterative method. Similarly, the PDA model is obtained on the basis of convention methods and I-shaped fracture propagation laws [38,39].

I-Shaped Fracture Propagation Simulation Model
I-shaped fracture refers to the combined fracture in which the fracturing layer is vertical fracture and both the cover and bottom layer are horizontal fractures. According to the classical fracture propagation mechanism, the condition and process for the formation of type I-shaped fracture are as follows: (1) The hydraulic fracture starts from the coalbed. When the horizontal minimum principal stress of the fractured layer is less than the vertical stress, a vertical fracture is formed in the coalbed (Figure 1a). (2) Getting along with fracturing, the vertical fracture propagates longitudinally and break through the coal seam boundary. When the horizontal minimum principal stress of the cover and bottom layer is greater than the vertical stress, horizontal fractures are formed in the cover and bottom layer. Thus, an I-shaped fracture is produced ( Figure 1b). conventional fractures (vertical fracture or horizontal fracture). There has been no satisfactory model for the I-shaped fracture so far. Therefore, this paper aims at the simulation of I-shaped fracture based on its propagation mechanism. A 3D fracture propagation simulator is developed based on the existing fracture propagation simulation model [38] and the propagation laws of I-shaped fracture [9]. The realistic coupling conditions is established in terms of the flow rate distribution and the bottom-hole pressure equality. A satisfactory numerical method of the P3D model is developed based on the iterative method. Similarly, the PDA model is obtained on the basis of convention methods and I-shaped fracture propagation laws [38,39].

I-Shaped Fracture Propagation Simulation Model
I-shaped fracture refers to the combined fracture in which the fracturing layer is vertical fracture and both the cover and bottom layer are horizontal fractures. According to the classical fracture propagation mechanism, the condition and process for the formation of type I-shaped fracture are as follows: (1) The hydraulic fracture starts from the coalbed. When the horizontal minimum principal stress of the fractured layer is less than the vertical stress, a vertical fracture is formed in the coalbed (Figure 1a). shows the horizontal component of I-shaped fracture which formed firstly due to the minimum principal stress being the minimum horizontal principal stress in both cover and bottom layers. S2 and S3 are usually greater than S1, while there is little difference between S2 and S3.

Fracture Propagation Simulation Model of Vertical Components
The vertical fracture propagation is shown in Figure 2. The fracture height is calculated according to the fracture height equation, which is not necessarily a parabola in most cases.

Fracture Propagation Simulation Model of Vertical Components
The vertical fracture propagation is shown in Figure 2. The fracture height is calculated according to the fracture height equation, which is not necessarily a parabola in most cases.

Continuity Equation
Before the vertical fracture propagates to the cover layer/bottom layer, the continuity equation is [35] t t where A(x, t) is the cross-section area of fracture. Ct(x, t) is total fracturing fluid leak-off coefficient, m/min 0.5 . τ(x) is the needed time of fracturing fluid to reach the given point x at time t, s.
When the vertical fracture propagates to the cover and bottom layer, the height of the vertical fracture formed by the fracturing treatment will not change, and then the continuity equation of the vertical fracture is where qv is flow rate in vertical fracture, m 3 /s.

Pressure Drop Equations of the Vertical Component
According to the classic method of using pipe shape factor ) (n  in the pressure drop of fluid flowing in the parallel plate, pressure drop in 3D vertical fracture can be obtained:

Continuity Equation
Before the vertical fracture propagates to the cover layer/bottom layer, the continuity equation is [35] − is the needed time of fracturing fluid to reach the given point x at time t, s. When the vertical fracture propagates to the cover and bottom layer, the height of the vertical fracture formed by the fracturing treatment will not change, and then the continuity equation of the vertical fracture is where q v is flow rate in vertical fracture, m 3 /s.

Pressure Drop Equations of the Vertical Component
According to the classic method of using pipe shape factor Φ(n ) in the pressure drop of fluid flowing in the parallel plate, pressure drop in 3D vertical fracture can be obtained: where, p V (x, t) is pressure inside vertical fracture at the given time t and point x, MPa. q V (x, t) is flow rate in vertical fracture, m 3 /s. n is fracturing fluid flow behavior index, m/min 0.5 . K is fracturing fluid consistency index, mPa·s n . h(x, t) is vertical fracture height, m. d V (x, 0, t) is vertical fracture width at the center of cross section, m. t is fracturing time, s. The subscript "V" represents vertical fracture.

Fracture Width Equation of the Vertical Component
While fracture is propagating in coalbed, the net pressure in fracture (flowing pressure minus minimum horizontal principal stress) is where σ h1 is least horizontal principal stress in coalbed, MPa. At any height z, the width of the fracture width profile is [41] where ν 1 is Poisson's ratio of coalbed, dimensionless. E 1 is Young's modulus of coalbed, MPa.
When the vertical fracture propagates up and down to the cover/bottom layer, the net pressure (bottom-hole pressure minus the least horizontal principal stress) is where σ h2 is least horizontal principal stress of cover layer, MPa. σ h3 is least horizontal principal stress of bottom layer, MPa. h 1 (x, t) is the vertical fracture height in the bottom layer, m. h u (x, t) is the vertical fracture height in cover layer, m. H is the thickness of coalbed, m. According to the England and Green equation, the width of the fracture width profile at a specific height z is [38] If the net pressure of vertical fracture is p V (z) = f (z) + g(z), then the f (z) and g(z) are According to the theory of linear elastic fracture mechanics, when the vertical fracture is not propagated to the cover/bottom layer (h(x, t) < H), the stress intensity factors at ends up and down of fracture are mutually equal. Then, the fracture height is When the vertical fracture extends to the cover layer and the bottom layer, the horizontal fracture initiates at the interface between the coal seam and the cover/bottom layer. Then, the fracture height is

Pressure Drop Equations of the Horizontal Component
When starting to produce horizontal fractures, the pressure drop also appears in them. During HF, the shape of horizontal fracture is circular. We can obtain the equation of pressure drop in upper and lower horizontal fracture as follows: where p upH (r) is pressure inside upper horizontal fracture, MPa. p lowH (r) is pressure inside lower horizontal fracture, MPa. q upH (r) is flow rate in upper horizontal fracture, m 3 /s. q lowH (r) is flow rate in lower horizontal fracture, m 3 /s. d upH is the average fracture width of upper horizontal fracture, m. d lowH is the average fracture width of lower horizontal fracture, m. The subscript "H" represents horizontal fracture.

Fracture Width Equation of the Horizontal Component
When the vertical fracture propagates up and down to the cover/bottom layer, horizontal fractures will be generated. A part of horizontal fractures will generate in the cover/bottom layer, the other part generates in the coalbed. According to Sneddon equation, the maximum width of horizontal fracture at r = r w (r w is wellbore radius) is where σ(r) = p(r) − σ h , D Hmax is the maximum fracture width of horizontal fracture, m. r is radial coordinate of horizontal fracture, m. R is horizontal fracture radius at a given time, m. Thus, the maximum width of upper and lower horizontal fractures between the coal seam and cover/bottom layer are represented respectively as Therefore, where R up is upper horizontal fracture radius at a given time, m. R low is lower horizontal fracture radius, m.

Material Balance Equation
When the vertical fracture propagates to the cover/bottom layer, the flow rate was distributed to vertical fracture and generated horizontal fractures. Thus, the material balance equation is where q V is flow rate in vertical fracture, m 3 /s. Meanwhile, the continuity equation of the vertical fracture and two horizontal fractures are where C(r) is total fracturing fluid leak-off coefficient of horizontal fracture at x, m/s 0.5 .

Boundary Conditions
In the ends of vertical fracture, the fracture height and flow rate are zero, and the pressure is the minimum horizontal principal stress of coalbed. At the ends of horizontal fractures, the pressure is the minimum horizontal principal stress of cover/bottom layer, respectively. In the propagating process of horizontal fractures, we can suppose as follows: (1) The fracturing fluid is assumed to fully saturate the fractures; (2) Within the propagation radius, the width of horizontal fractures is in accordance with parabola distribution; (3) Outside the propagation radius, the width is zero. Thus, the boundary conditions are where L f is half-length of fracture, m. R up is radius of upper horizontal fracture, m. R low is radius of lower horizontal fracture, m.

Initial Conditions
At the initial time, all parameters of vertical fracture are zero. The initial horizontal fracture radius is regarded as the wellbore radius because the horizontal fractures are calculated from the wellbore axis. Thus, the initial conditions are where T is the time of horizontal fracture began to propagating, s.

Coupling Conditions
According to the fracture propagation mechanism of I-shaped fracture, the vertical fracture and horizontal fractures are not completely independent. The pressure of two horizontal fractures and vertical fracture are equal in the intersection point. The total displacement pumped into wells equals to the sum of two horizontal fractures flow rate and vertical fracture flow rate. Hence, the coupling conditions are: where Q is total pumping displacement, m 3 /s.

Numerical Method
The flow rate distribution of fracturing fluid is the key to solving the model. Once the flow rate ratio of horizontal fractures to vertical fracture is confirmed, the flow rate can be introduced into the horizontal and vertical fractures' simulation models, respectively. According to the solution of the respective models, the geometric sizes of horizontal and vertical fractures can be obtained. Figure 3 shows the flow chart of numerical method. The solving procedure of the above models involves five steps. Step 2: According to the flow rate distribution, the flow rates are substituted in vertical and horizontal fractures simulation models, respectively. At this time, according to solving steps of vertical and horizontal fracture simulation model, the bottom-hole pressures are calculated at the certain moment. The bottom-hole pressures of vertical fracture is pV(x = 0), the bottom-hole pressures of upper horizontal fracture is pH,up(rw), and the bottom-hole pressure of lower horizontal fracture is pH,low(rw); Step 3: Examine whether the bottom-hole pressures of two horizontal fractures are equal to the vertical fracture at the given moment. Generally, this is to determine whether the equation |pV(x = 0) − pH,up(rw)| 2 + | pV(x = 0) − pH,up(rw)| 2 < ξ (ξ is the maximum permissible error) is satisfied. If it is satisfied, the initial assumption of flow rate distribution is correct. Then, proceed to step 4. Otherwise, re-adjust the flow rate by the value of wellbore pressure and return to step 2; Step 4: Calculate the bottom-hole pressures of the vertical and horizontal fractures at the next moment. Determine whether the fracturing ends. If it ends, proceed to step 5. Otherwise, return to step 3; Step 5: The end.

PDA Model
The crustal stress of coalbed, cover layer, and bottom layer conforms to the distribution law of Figure 1. In this case, fracturing treatment will lead to the formation of I-shaped fracture, that is, the upper and lower horizontal fractures and a vertical fracture in the middle with symmetrical wings. In Section 2, the extension model of I-shaped fracture is established, and the following content is the establishment of PDA model of I-shaped fracture.
The end of fracturing?  Step 1: Assume that the flow rate distribution in the initial moment is q V : q H,up : q H,low = m 1 : m 2 : m 3 ; Step 2: According to the flow rate distribution, the flow rates are substituted in vertical and horizontal fractures simulation models, respectively. At this time, according to solving steps of vertical and horizontal fracture simulation model, the bottom-hole pressures are calculated at the certain moment. The bottom-hole pressures of vertical fracture is p V (x = 0), the bottom-hole pressures of upper horizontal fracture is p H,up (r w ), and the bottom-hole pressure of lower horizontal fracture is p H,low (r w ); Step 3: Examine whether the bottom-hole pressures of two horizontal fractures are equal to the vertical fracture at the given moment. Generally, this is to determine whether the equation |p V (x = 0) − p H,up (r w )| 2 + | p V (x = 0) − p H,up (r w )| 2 < ξ (ξ is the maximum permissible error) is satisfied. If it is satisfied, the initial assumption of flow rate distribution is correct. Then, proceed to step 4. Otherwise, re-adjust the flow rate by the value of wellbore pressure and return to step 2; Step 4: Calculate the bottom-hole pressures of the vertical and horizontal fractures at the next moment. Determine whether the fracturing ends. If it ends, proceed to step 5. Otherwise, return to step 3; Step 5: The end.

PDA Model
The crustal stress of coalbed, cover layer, and bottom layer conforms to the distribution law of Figure 1. In this case, fracturing treatment will lead to the formation of I-shaped fracture, that is, the upper and lower horizontal fractures and a vertical fracture in the middle with symmetrical wings. In Section 2, the extension model of I-shaped fracture is established, and the following content is the establishment of PDA model of I-shaped fracture.

PDA Model of Vertical Fracture
At the time of t 1 and t, during the closure of the vertical fracture, according to the volume balance principle of the fluid, we can obtain the following equation [41]: where p w,V (t) is bottom-hole pressure of vertical fracture at the given time t, MPa. p * V is pseudo pressure of vertical fracture, MPa. H w is vertical fracture height, m. β V is the ratio of average pressure to bottom-hole pressure in vertical fracture. C V is total fracturing fluid leak-off coefficient of vertical fracture, m/min 0.5 . t inj is the fracturing time, s.
For vertical fracture, the leak-off coefficient of post-fracturing fluid is The maximum width of vertical fracture at the time of pump shutdown is where p ISI is the instantaneous pressure of pump-stopping, MPa. According to injection volume balance principle, at the time of stopping the pump, the fracture half-length is: where Q inj,V is the flow rate in vertical fracture during fracturing, m 3 /s. The fracturing fluid efficiency is The time required for fracture closure is

PDA Model of Horizontal Fracture
According to the Carter filtration model and fracture propagation index, the of relationship between pressure fall-off and time is [42] p w,Hup (t 1 ) − p w,Hup (t) = p * Hup G(t 1 , t) p w,Hlow (t 1 ) − p w,Hlow (t) = p * Hlow G(t 1 , t) (43) where p w,Hup (t) is bottom-hole pressure of upper fracture, MPa. p w,Hlow (t) is bottom-hole pressure of lower fracture, MPa. p * Hup is pseudo pressure of upper horizontal fracture, MPa. In addition, p * Hlow is pseudo pressure of lower horizontal fracture, MPa. C Hup is total fracturing fluid leak-off coefficient of upper horizontal fracture, m/min 0.5 . C Hlow is total fracturing fluid leak-off coefficient of lower horizontal fracture, m/min 0.5 . β Hup is the ratio of average pressure to bottom-hole pressure in upper horizontal fracture, dimensionless. β Hlow is the ratio of average pressure and bottom-hole pressure in lower horizontal fracture, dimensionless. f p is the ratio of fracture leak-off area to fracture total area, dimensionless.
Based on the fracture leak-off coefficient which is fitted by the pseudo-pressure, some parameters of horizontal fracture are able to be obtained if the injection rate of fracture is known, such as leak-off coefficient of post-fracturing fluid, fracture radius and width, fracturing fluid efficiency and fracture closure time, etc.
The leak-off coefficient of post-fracturing fluid is The fracturing fluid efficiency is: where p c is fracture closure pressure, MPa. At the stop time of fracturing, the fracture radius is where Q inj,Hup is the flow rate of upper horizontal fracture during fracturing treatment, m 3 /s. At the stop time of fracturing, the average fracture width is Similarly, the lower horizontal fracture parameters can be obtained.

Coupling Conditions
The pressures of two horizontal fractures are equal to the vertical fracture value at the bottom of the well. Thus, the pseudo pressures of two horizontal fractures are equal to the vertical fracture value. The total pumping displacement is equal to two horizontal fractures flow plus vertical fracture flow. Then, the coupling conditions are By using the flow rate distribution principle and iterative method, we can obtain the leak-off coefficient of post-fracturing fluid, fracture length and width, fracturing fluid efficiency and fracture closure time of vertical fracture and the leak-off coefficient of postfracturing fluid, fracture radius and width, and the fracturing fluid efficiency of upper and lower horizontal fracture. In addition, the numerical calculation method of PDA model for I-shaped fracture is similar to that in Section 2.7.
In the process of solving different practical problems, given an initial flow allocation, the iterative process will automatically narrow the interval containing the solution according to the judgment criteria, so that, after a certain number of iterations, a solution that meets the preset accuracy will be obtained.

History Matching of Pseudo Pressure
In order to obtain pseudo pressure, m points were selected in the period during shut-in and prior to closure, and an objective function is built as where (p w (t i ) − p w (t j )) obs represents the observed pressure drop during time t i to t j , MPa. In addition, (p w (t i ) − p w (t j )) cal represents the calculated pressure drop during time t i to t j , MPa. By adjusting pseudo pressure of p * , the minimum error between observed pressure drop and calculated pressure drop can be obtained. Thus, this is an optimization model and can be expressed as minJ(p * ) (54) In this paper, the golden ration method was employed to solve the optimization model.

Results
The Hancheng area is a very important part in terms of the exploitation of CBM in China. Combining the I-shaped fracture propagation model with the PDA, a software on fracturing simulation has been applied in Hancheng area and achieved satisfactory effects.
Take well H-3-52 as an example. Table 1 shows the formation properties and the treatment parameters that will be input into the simulation model. The construction curve of well H-3-52 is as shown in Figure 4. The values of all parameters in Table 1 are interpreted according to conventional logging data, as shown in Figure 5. The construction curve of the well is shown in Figure 4. When fracturing treatment starts in the early stage, the fracture net pressure is less than the net overburden pressure. This phenomenon indicates that coalbed forms the vertical fracture. With the construction, the fracture net pressure began to exceed the net overburden pressure. This phenomenon indicates that formation forms the horizontal fractures. Therefore, the coalbed fracturing formed a complex fracture. Figure 5 shows the interpretation of rock mechanics parameters. In Figure 5, the difference between the minimum principal stress of cover/bottom layer and coalbed is 6 MPa. The minimum horizontal principal stress is the minimum stress in coal bed, and the overburden pressure is the minimum stress in cover/bottom layer. Thus, in Figure 4, the two labeled circle means that the horizontal fractures are formed.
indicates that formation forms the horizontal fractures. Therefore, the coalbed fracturin formed a complex fracture. Figure 5 shows the interpretation of rock mechanics parame ters. In Figure 5, the difference between the minimum principal stress of cover/bottom layer and coalbed is 6 MPa. The minimum horizontal principal stress is the minimum stress in coal bed, and the overburden pressure is the minimum stress in cover/bottom layer. Thus, in Figure 4, the two labeled circle means that the horizontal fractures ar formed.  Therefore, this paper simulates this fracturing treatment by the established models. Through continuous search and iteration, the ration of fractures flow rate for the propagation simulation model is: upper horizontal fracture: lower horizontal fracture: vertical fracture = 0.2376:0.2563:0.5061. We also simulate this treatment with a vertical fracture propagation model, a vertical fracture propagation model, and a T-shaped fracture propagation model. Figure 6 shows the bottom-hole pressure match using an I-shaped fracture propagation model, vertical fracture model, horizontal fracture model, and T-shaped fracture model. As seen in Figure 6, only the simulated pressure of I-shaped fracture model is consistent with the actual pressure on site (the error is 4.72%). The T-shaped fracture model deviates greatly from the actual situation in pressure simulation (the error is 16.59%). The pressure errors calculated by the conventional vertical and horizontal fracture models are completely distorted, and both of the trend and shape are different from the actual value.
Then, we set the above ration as the initial ration of PDA. Through continuous search and iteration, the ration of the fracture flow rate for the PDA model is: upper horizontal fracture: lower horizontal fracture: vertical fracture = 0.2503:0.2588:0.4909. In addition, we also analyzed the phase of pressure fall-off with T-shaped fracture PDA and convention PDA. Figure 7 shows the bottom-hole pressure match using I-shaped fracture PDA, Tshaped fracture PDA, and convention vertical and horizontal fracture PDA. As seen in Figure 7, an accurate pressure history match is obtained using a new model. However, the T-shaped fracture model and convention models cannot obtain the satisfactory pressure history match because of the inappropriate fracture shape.
The changes of fracture parameters during fracturing treatment are shown in Figure 8. In addition, the calculation of geometric sizes using a new model is shown in Table 2 history match because of the inappropriate fracture shape.
The changes of fracture parameters during fracturing treatment are shown in Figure  8. In addition, the calculation of geometric sizes using a new model is shown in Table 2. As seen from the table, the simulation results of the propagation model and PDA have a slight difference. Meanwhile, based on the PDA model, the fracturing fluid efficiency is 14.9%, the leak-off coefficient of post-fracturing fluid is 0.001206 m/min 0.5 , and the fracture closure time is 19.62 min.   The model proposed in this paper has good universality and is applicable to different geological conditions and rock types. Taking the construction data and logging data of well H-3-52 in the Hancheng area as an example, compared with the T-shaped fracture model and the conventional fracture models, the numerical simulation results of the I-shaped fracture model established in this paper are more accurate, and are in good agreement with the results of PDA model, which has been compared and analyzed in detail before. For the fracture model and PDA model proposed in this paper, it is one of the further research directions to adopt a more complex iterative algorithm as an alternative to the numerical simulation method. In general, the I-shaped fracture model and PDA model established in this paper can provide effective guidance for simulating fracture initiation/propagation and pressure decline.
shaped fracture model established in this paper are more accurate, and are in good agreement with the results of PDA model, which has been compared and analyzed in detail before.
For the fracture model and PDA model proposed in this paper, it is one of the further research directions to adopt a more complex iterative algorithm as an alternative to the numerical simulation method. In general, the I-shaped fracture model and PDA model established in this paper can provide effective guidance for simulating fracture initiation/propagation and pressure decline.   In order to study the variation law of I-shaped fracture geometry with crustal stress, under the condition that the stress state meets the formation conditions of I-shaped fracture, take two points respectively around the minimum principal stress of the cover and

Influence of Crustal Stress Variation on Fracture Geometry
In order to study the variation law of I-shaped fracture geometry with crustal stress, under the condition that the stress state meets the formation conditions of I-shaped fracture, take two points respectively around the minimum principal stress of the cover and bottom layer of well H-3-52 in 1 MPa steps, and calculate the variation of I-shaped fracture geometry. The specific results are shown in Tables 3 and 4.  In the above discussion, the horizontal minimum principal stress of the coal seam is less than the vertical stress, and the horizontal minimum principal stress of the cover and bottom layer is greater than the vertical stress. Therefore, these changes in these values can lead to the formation of I-shaped fracture.
It can be seen from Table 3 that, with crustal stress of cover layer increasing, the upper horizontal joint radius gradually decreases, and the half length of vertical fracture and the lower horizontal joint radius gradually increase. Similarly, with crustal stress of bottom layer increasing, the lower horizontal fracture radius gradually decreases, and the half length of vertical fracture and the upper horizontal fracture radius gradually increases. Compared with the horizontal fracture radius of cover and bottom layer, the change of crustal stress has a more significant impact on the half length of vertical fracture.

Influence of Fracturing Fluid on Fracture Geometry
The flow index is 1 and remains unchanged. First, the influence of fracturing fluid consistency coefficient K on fracture geometry is discussed. K gradually increases to 5 MPa·s n in steps of 1 MPa·s n . The simulated geometric results of I-shaped fracture are shown in Table 5. The higher the fracturing fluid consistency coefficient is, the smaller the fracture height will be, which will cause the fracture to propagate further. It can be seen from Table 5 that, with the increase of the fracturing fluid consistency coefficient, the radius of the horizontal fractures in the cover and bottom layers and the half length of the vertical fracture gradually increase. The simulation results are consistent with the actual situation. Table 6 shows the results of the change of I-shaped fracture geometry with the leak-off coefficient post-fracturing fluid. It can be seen from Table 6 that, with the increase of leakoff coefficient, the radius of horizontal fractures in the cover and bottom layers gradually decreases, while the half length of vertical fracture also gradually decreases. The greater the filtration is, the greater the limitation of I-shaped fracture propagation will be.

Conclusions
The I-shaped fracture model and PDA model proposed in this paper are effective tools to simulate the hydraulic fractures. The effectiveness of the methods is analyzed through examples, and the following conclusions can be summarized.
(1) The adequate research regarding the fracture propagation mechanism is essential for coal seam fracturing. The I-shaped fracture is often occurred in coal seam fracturing. This paper elaborates its propagation mechanism. We obtained the fracture geometries, fracturing fluid efficiency, etc. It is concluded that the established two models are effective tools for the simulation of fracture initiation/propagation and pressure decline of post-fracturing.