1 Introduction

Multi-stage hydraulic fracturing technology in a horizontal well has become one of the most important technology in the development of tight oil resources. Because of the super low permeability of the tight rock matrix, the fast production rate decline can be observed under oil production for a while. Under this condition, the refracturing method should be performed on pre-stimulated tight oil wells without more costs on drilling new wells (Goulty 2003; Mortazavi and Atapour 2018). Recompletion is one important and common way to increase the fracture stimulated area during the refracturing treatment. To more effectively stimulated the horizontal wells, the temporary plugging method in the horizontal well was proposed for fracture blunt at the fracture tip or perforation entrance. This method can divert fracturing fluid and proppant into the understimulated zones along the wellbore with different temporary plugging materials (Yi et al. 2019; Minghui et al. 2022). The application of the suitable diverting agent is beneficial for the entire perforation accepting fracturing fluid and making sure uniform fracture propagation along the horizontal well, thus more stimulated zones can be obtained. As for the refracturing scenario on the horizontal well, the addition of new perforations in the unstimulated zones is necessary. However, the selection of recompletion position in the horizontal well is a challenge because the continuous dynamic depletion induced dynamic in-situ stress change generally suffers from hydrocarbon depletion and fracture open.

In fact, both reservoir depletion and fluid injection into a reservoir will change the original in-situ stress condition. Some authors tend to understand the effect of hydrocarbon depletion effects in in-situ stress alternation with lab experiments. They found the variation in in-situ stress is not only affected by the fluid extraction but also by the reservoir lithology (Kuzmina and Rosneft 2009; Liu and Harpalani 2014; Saurabh and Harpalani 2018). The dynamic in-situ stress change is a hot topic in petroleum engineering because the incorrect updated in-situ stress information can induce serious engineering issues, such as fault activation, wellbore instability, stimulation, and enhanced oil recovery failure. Pouria et al. (2016) investigated the wellbore stability problem during the reservoir life, and variation of pore pressure near the wellbore was used to study the optimum well trajectory. Guo and Wu (2017) applied the finite element method to characterize the stress update by history matching production data while both the parent well operation and reservoir properties affect the updated stress field. Furthermore, the non-uniform hydraulic fracture can be found. Gao et al. (2019) captured the hydraulic fracture reorientation considering the pre-existing well production. They demonstrated both the breakdown pressure and fracture propagation pressure can increase near the injection well. Rezaei et al. (2019) investigated the fracture propagation path for a horizontal well and they indicated that the local stress distribution can result in an asymmetric fracture path for a child well. In addition, they also analyzed the parameters for the infill fracture path, and fracture interaction and unstable fracture propagation can occur. Pei et al. (2020) discussed the optimum drilling window for shale reservoirs with the infill drilling techniques. They considered the effects of natural fracture on pressure depletion and thought before 0.8 years or after 5 years can assure a good fracture path for a child well. Compared to a vertical well, the change of pressure depletion on the horizontal well is more complex because of previous stimulated area difference for each cluster along the horizontal well in previous hydraulic fracturing, thus the non-uniform pore pressure and stress profile near the wellbore exist, thus affecting the recompletion scenario design. To consider the poroelastic effect from the depleted fracture in the hydraulic fracture propagation from recompletion during the refracturing treatment, the integration of coupled fluid flow and the geomechanics model is necessary. In general, the fully coupled governing equations of fluid flow and rock deformation can yield the highest computational accuracy but heavy computational load (Rick 2003; Jie et al. 2019). However, the application of the fully coupling method in the dynamic stress analysis and following fracture propagation is rarely studied.

Various numerical methods are employed to handle the complexities of hydraulic fracturing quandaries, among which include the Boundary Element Method (BEM) (Kan and Jon 2016), Finite Element Method (FEM) (Deng et al. 2018), Discrete Element Method (DEM) (Chen et al. 2018), and the Extended Finite Element Method (XFEM) (Gao et al. 2019a, b). Of these, the XFEM stands out for its proficiency in tackling issues characterized by localized features and the modeling of diverse discontinuities. Wang et al. (2018) established the model using the cohesive zone model (CZM) based on the XFEM to simulate the applicability of temporary plugging and diverting technique under various reservoir conditions. Liao et al. (2022) applied XFEM to establish a five-spot well pattern model, optimizeing refracturing orientation under non-uniform pore pressure conditions. From previous research, although some studies have demonstrated that hydraulic fracture propagation will be affected during infill wells and refracturing work, few studies have considered the effect of depletion on hydraulic fracture propagation.

In this reserach, a fully coupled method was applied to map the dynamic changes in in-situ stress resulting from fluid extraction using the finite element method. After that, the XFEM method (extended finite element method) was used to simulate the hydraulic fracture propagation from recompletion in the horizontal well during the refracturing. When multiple fractures are stimulated simultaneously from the recompletion during the refracturing, curved fractures can be commonly seen and the deflected fracture generally diverts towards the primary stimulated area with low pore pressure. Finally, the different treatment parameters on the recompletion efficiency concerning fracture geometry were investigated. The results learned in the study can be a guide for optimal development of the recompletion scenario on the horizontal well in refracturing.

2 Governing equations for dynamic in-situ stress modeling

The governing equations for dynamic in-situ stress modeling should integrate the geomechanical model and fluid flow model.

2.1 Geomechanical model

The formulation of the geomechanical problem takes into account the equilibrium equation, stress–strain displacement equation, and constitutive equation.

The geomechanical equilibrium is expressed as:

$$\nabla \sigma + f = 0$$
(1)

According to the effective stress law (Warpinski and Teufel 1993), the total stress can be written with the effective stress:

$$\sigma_{ij}^{\prime } = \sigma_{ij} - \alpha \delta_{ij} p$$
(2)

where σ is the total stress tensor, α is Biot’s effective stress, p is pore pressure, σeff is effective stress, and I is a second order identity tensor. The calculation of the Biot parameter can be obtained by:

$$\alpha = \frac{{K_{d} }}{{K_{s} }}$$
(3)

where Ks is the and Kd is the effective volume modulus. The bulk modulus of the rock, K, and the bulk modulus of the rock matrix material Km.

The relationship between the strain and displacement tensor can be expressed by:

$$\varepsilon_{ij} = \frac{1}{2}\left( {u_{i,j} + u_{j,i} } \right)$$
(4)

The stress–strain displacement equation follows the generalized Hook law, which can be written as:

$$\sigma = D_{e} \varepsilon$$
(5)

where σ, De, and ε are the second order stress tensor, the fourth order elastic stiffness tensor, and the second order strain tensor respectively.

2.2 Fluid-flow model

The fluid flow model can be described with the two-phase-flow equation, which is derived by use of a mass-balance equation combined with an equation of state and Darcy’s law. The Darcy law was used to characterize the flow in the porous media, which can be written as:

$$v_{w} = - \frac{a\mu }{{n_{w} \rho_{w} g}}k\left( {\frac{{\partial p_{w} }}{\partial x} - \rho_{w} g} \right)$$
(6)

where a is a constant, μ is the fluid viscosity, and k is the formation permeability parameter. pw is the fluid pressure, ρw is the fluid density, and g is the gravitational acceleration.

The fluid continuity equation for fluid flow can be represented by:

$$\int_{V} {\frac{1}{J}\frac{d}{dt}(J\rho_{w} (n_{w} + n_{t} ))} dV + \int_{s} {\rho_{w} n_{w} nv_{w} } dS = 0$$
(7)
$$n_{w} = \frac{{dV_{w} }}{dV}$$
(8)
$$n_{t} = \frac{{dV_{t} }}{dV}$$
(9)
$$J = \frac{dV}{{dV_{0} }}$$
(10)

where vw is fluid velocity. V0 is the initial fluid volume, V is the total element volume, Vt is the bound fluid volume, and Vw is the free fluid volume.

2.3 Coupling parameters

According to the effective stress principle, the fluid flow and rock deformation within the porous medium are affected by a pair of coupled variables. The water injection or fluid extraction will cause rock skeleton deformation. In the injection and extraction process, stress changes belong to the mutual coupling of rock deformation and reservoir physical properties, i.e., physical parameters such as porosity and permeability also change in reservoir deformation, and physical properties change and in turn affect rock deformation. When the stress-sensitive reservoir is depleted or waterflooded, porosity and permeability variations should have been connected to the stress field and deformation. Assuming that elastic deformation of shale cannot cause damage, plastic deformation and damage occur simultaneously (Pierre and de Gennaro 2007; Rodriguez 2011; Gao and Gray 2017). After yielding failure of rock, the internal pore and fracture in rock gradually tend to be deformed, thus the porosity and permeability of rock increase obviously. According to the equation of Kozeny-Carman, the permeability and porosity evolution of shale can be defined as Shanpo et al. (2019):

$$\left\{ \begin{gathered} k = k_{0} \left[ {\left( {\frac{1}{{n_{0} }}} \right)\left( {1 + \varepsilon_{v} } \right)^{2/3} - \left( {\frac{{1 - n_{0} }}{{n_{0} }}} \right)\left( {1 + \varepsilon_{v} } \right)^{ - 1/3} } \right]^{3} \hfill \\ \phi = 1 - \frac{{1 - \phi_{0} }}{{1 + \varepsilon_{v} }} \hfill \\ \end{gathered} \right\}$$
(11)

where k0 is the initial permeability of shale, Φ0 is the initial porosity, and εv is the volumetric strain.

Therefore, the governing equation used in the fully coupled scheme during hydraulic fracturing is given by Eq. (12):

$$(\alpha - \phi_{0} )\frac{\partial p}{{\partial t}} - \frac{k}{\mu }\nabla^{2} p = - \alpha \frac{{\partial \varepsilon_{v} }}{\partial t}$$
(12)

where k is the absolute permeability, µ is the fluid viscosity and p is the pore pressure

$$G\nabla^{2} u + \frac{G}{1 - 2\nu }\nabla \nabla \cdot u = \alpha \nabla p$$
(13)

where G is the shear modulus.

Critical parameters (pore pressure, and displacement) were selected as the variables between two simulators. Then coupling parameters are exchanged at each time step until reaching the convergence. The convergence equation can be represented as:

$$\left\{ \begin{gathered} \Delta p^{(t + 1)} \hfill \\ \Delta u^{(t + 1)} \hfill \\ \end{gathered} \right\} = \left\{ \begin{gathered} p^{(t + 1)} \hfill \\ u^{(t + 1)} \hfill \\ \end{gathered} \right\} - \left\{ \begin{gathered} p^{t} \hfill \\ u^{t} \hfill \\ \end{gathered} \right\}$$
(14)

3 XFEM fracture propagation modeling

Considering the arbitrary fracture propagation path, the XFEM was presented (Goulty 2003; XiaoLong et al. 2016; Escobar et al. 2019; Fang et al. 2022). To model initial fracture, enrichment functions are added to the finite element approximation. The discontinuity of fracture is characterized by the enrichment functions associated with the additional degree of freedom. The expression for the displacement vector function u has the following form:

$$u = \sum\limits_{I = 1}^{N} {N_{I} } \left( x \right)\left[ {u_{I} + H\left( x \right)a_{I} + \sum\limits_{\alpha = 1}^{4} {F_{\alpha } \left( x \right)b_{I}^{\alpha } } } \right]$$
(15)

where \(N_{I} (x)\) are the usual nodal shale functions, \(u_{I}\) is a continuous part of the solution to displacement vector, which relates to continuous part of the finite element solution, \(a_{I}\) is the nodal enriched degree of freedom vector, \(b_{I}^{\alpha }\) refers to nodal extended enriched of freedom degree vector, and \(F_{\alpha } (x)\) is an elastic asymptotic function, which can be only used at the fracture tip, \(H(x)\) represents associated discontinuity jump function across the fracture surfaces.

The discontinuous jump function \(H(x)\) has the following form:

$$H\left( x \right) = \left\{ \begin{gathered} 1\quad \,\,\,\,\,if\quad (x - x^{ * } ) \cdot n \ge 0 \hfill \\ - 1\quad otherwise \hfill \\ \end{gathered} \right.$$
(16)

where n is the unit outward normal to the fracture at x*, x* is the point on the fracture nearest to x, and x is a sample (Gauss) point.

The equation of elastic asymptotic function \(F_{\alpha } (x)\) can be expressed as Warpinski and Teufel (1993):

$$F_{\alpha } \left( x \right) = \left[ {\sqrt r \sin \frac{\theta }{2},\sqrt r \cos \frac{\theta }{2},\sqrt r \sin \theta \sin \frac{\theta }{2},\sqrt r \sin \theta \cos \frac{\theta }{2}} \right]$$
(17)

where r and θ represent as a local polar coordinate system with the crack tip as its origin and θ= 0 as a tangent to the fracture.

The fracture propagation requirements for rock deformation and fluid flow coupling, as well as the laws of conservation of momentum, mass, and energy, should all be computed simultaneously. the stress can be expressed by Biot’s effective stress law. Within the current arrangement, the virtual work principle is defined as follows:

$$\int {(\overline{\sigma } - p_{w} I)} :\delta \varepsilon dV = \int {_{S} t \cdot } \delta vdS + \int {_{V} f \cdot } \delta vdV$$
(18)

where ′ and ′ are effective stress and virtual rate of deformation, respectively. I is the unit matrix, and t and f are surface traction per unit area and body force per unit volume, respectively. The nodal variables in this equation are displacements, which are discretized using a Lagrangian formulation.

The LEFM mechanism served as the foundation for the fracture initiation and propagation criteria. The traction–separation relationship can be used to represent the rock deformation, which can be displayed in Fig. 1. When one of the stress components reaches the maximum strength of the rock material in that direction, which can be represented by a quadratic law, the damage is presumed to begin. where \(t_{s}^{0}\) and \(t_{n}^{0}\) stand for the normal and shear strengths of an undamaged cohesive element, respectively; the symbol stands for the Macaulay bracket; and tn and ts stand for the normal and shear stress components, respectively. The definitions of the normal and shear stress elements are as follows:

$$\left\{ {\frac{{t_{n}^{{}} }}{{t_{{_{n} }}^{0} }}} \right\}^{2} + \left\{ {\frac{{t_{s}^{{}} }}{{t_{{_{s} }}^{0} }}} \right\}^{2} + \left\{ {\frac{{t_{t}^{{}} }}{{t_{{_{t} }}^{0} }}} \right\}^{2} = 1$$
(19)
Fig. 1
figure 1

Traction–separation relationship for fracture initiation and propagation

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The normal and shear stress components can be defined as:

$$t_{n} = \left\{ \begin{gathered} \left( {1 - D} \right)T_{n} ,T_{n} \ge 0 \hfill \\ T_{n} ,T_{n} < 0 \hfill \\ \end{gathered} \right.$$
(20)
$$t_{s} = \left( {1 - D} \right)T_{s}$$
(21)
$$t_{t} = \left( {1 - D} \right)T_{t}$$
(22)

Tn and Ts are the normal and shear stress components, respectively, anticipated by the elastic separation-traction behavior for current displacement without damage. D is a scalar damage variable that is used to describe the entire fracturing process.

The relative proportions of normal and shear deformation are determined by damage evolution, which is based on the mix mode energy theory. The fracture energy is equal to the area under the traction separation curve. In this study, the BenzeggaghKenane (BK) criterion is employed to calculate the damage progression and fracture propagation. The criterion is formatted as follows:

$$G_{n}^{C} + \left( {G_{s}^{C} - G_{n}^{C} } \right)\left\{ {\frac{{G_{s} }}{{G_{T} }}} \right\}^{\eta } = G^{C}$$
(23)

\(G^{C}\) is the calculated total critical energy release rate due to mix mode failure; \(G_{n}^{C}\) and \(G_{s}^{C}\) are the critical energy release rates due to Mode I and Mode II fractures, respectively. The energy release rate of Mode I and Mode II fractures, GT and Gs, is a constant that varies depending on the material.

The fluid flow in the hydraulic fracture can be described with the cubic law. The equation to calculate the fracture permeability for a viscous fluid flow with Newtonian rheology can be expressed as:

$$q_{f} = - \frac{{w^{3} }}{12\mu }\nabla p_{f}$$
(24)

where \(\nabla p_{f}\) is the pressure gradient along the cohesive element, μ is the fluid viscosity and w is the fracture opening width.

In order to explain the typical flow of hydraulic fracturing, the Carter leak-off model is also presented. The fully coupled poroelastic analysis is used in the proposed model to determine the leak-off of the fracturing fluid through the fracture faces into the surrounding medium. The regular flow in the fracture is written as follows:

$$\begin{gathered} q_{t} = c_{t} \left( {p_{i} - p_{t} } \right) \hfill \\ q_{b} = c_{b} \left( {p_{i} - p_{b} } \right) \hfill \\ \end{gathered}$$
(25)

where qt and qb are the typical flow flux cross into the cohesive element’s top and bottom surfaces, respectively. pt and pb are the pore pressures on the top and bottom surfaces, respectively, while pi is the fluid pressure within the cohesive element gap.

According to the fluid mass balance, a part of that injected fluid at a certain period fills the fracture and the rest will be lost to the rock matrix as shown in the following equation.

$$\frac{\partial d}{{\partial t}} + \nabla \cdot q + (q_{t} + q_{b} ) = q_{inj} \delta (x,y)$$
(26)

Then the control equation of the rock matrix involves coupling fluid flow and rock deformation as follows.

$$\sigma_{ij} - \sigma_{ij}^{0} = \frac{E}{1 + v}\left( { + \frac{v}{1 - 2v}\varepsilon_{kk} \delta_{ij} } \right) - \alpha \left( {P_{w} - P_{w}^{0} \delta_{ij} } \right)$$
(27)

4 Model construction

In a conventional coupling algorithm, the fluid flow and geomechanics equations require different discretization methods, thus inconsistent grid/element between the two models might occur. To offset these limitations, this fully coupled approach that solves simultaneously for fluid-flow variables and displacement variables in one platform was presented. Modeling and generation of gridding are operated in Software Abaqus, and fluid–solid coupling for fluid depletion from hydraulic fracture of the horizontal well is simulated by its inherent Module Soil. The fully coupled algorithm between fracture simulators and fluid production simulators can increase the computational efficiency and accuracy of the fluid depletion. Propagation of fracture is simulated by XFEM-based CZM (Cohesive Zone Model) in Abaqus and the application of the XFEM method can capture the arbitrary fracture path from recompletion during the refracturing. The “ABCD” domain in Fig. 2 (1) shows the study area for multi-stage horizontal fracturing simulation in the fracturing model (three initial fracture clusters in one stage and two fractures from the recompletion). A fully saturated porous domain was discretized with CPE4RP elements in Abaqus software. In this case, multiple fracture propagation started simultaneously with multiple injection ports from recompletion. The total injection rate was set to 0.2m3/min. The fracturing fluid was slick water with a viscosity of 3 mPa·s. Considering the middle hydraulic fractures suffered from serious stress interaction due to neighboring close hydraulic fractures, thus the fracture length of the middle fracture is the shortest. The child fracture refers to the old hydrualic fracture while the parent fractures represent the new hydrualic fractures in refracturing (Table 1).

Fig. 2
figure 2

Study area for hydraulic fracture propagation from recompletion during the refracturing

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Table 1 Model input parameter in fracturing and simulation models
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Figure 3 shows the comparison of in-situ stress field before and after reservoir production. At the far end, the direction of the minimum horizontal principal stress remains unchanged (such as Fig. 3(a)), but it can be seen in Fig. 3(b) that the direction of the minimum principal stress around the old crack changes significantly, which is caused by the production of the parent fractures. Figure 4 shows the spatial distributions of the in-situ stress after the production. The stress steering Angle at the parent fractures location is 90 degrees, and both sides gradually decrease.

Fig. 3
figure 3

Comparison of in-situ stress field before and after reservoir production. The red line in the panel is the direction of the minimum horizontal principal stress

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Fig. 4
figure 4

Spatial distributions of the in-situ stress after the production

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5 Numerical simulation

5.1 Influence of production time on fracture propagation during the recompletion

The pressure depletion in the reservoir continues with time, thus the updated in-situ stress should be time-dependent. Figure 4 shows the production time effect on the hydraulic fracture propagation path from recompletion in the refracturing of the horizontal well. The depletion reaction of the reservoir not only impacts the in-situ stress change but also induces the in-situ stress orientation, thus affecting the following recompletion (Fig. 5(a-d)). Pressure depletion is a common production method that, as the name implies, involves depressurizing the rock. Since two outer initial fracture has large fracture production capability, thus the change of in-situ stress around two outer initial fracture is large. With the increase in production time, a more curved hydraulic fracture path during the refracturing can be observed. The comparison of hydrualic fracture length was shown in Fig. 6(a). It illustrates that this stress depletion response can affect the fracture stimulation effectiveness for recompletion, the optimum production time is about 600 days, and the new fracture length is the largest (29 m). When the production time is 200 days, the fracture length is 25 m, which is the smallest. The new fracture width comparison at the injection port is shown in Fig. 6(b). The results show that the width increases with the injection time, and the maximum width is 0.02 m during 600 days of production. However, after 800 days of production, the width is 0.015 m, which is the smallest fracture width in this group of models.

Fig. 5
figure 5

Hydraulic fracture propagation path from recompletion during the refracturing with different production times. Different colors in the panels correspond to diffeent pore pressure. a 200 d, b 400 d, c 600 d, d 800 d

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Fig. 6
figure 6

The new fracture length and width for hydraulic fracture propagation from recompletion during the refracturing with different production times. The different color symbols in the panel correspond to different production times. a New fracture length, b New fracture width

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5.2 Influence of the horizontal stress contrast on fracture propagation during the recompletion

Horizontal stress difference directly affects the hydraulic fracture propagation direction, which could be an important factor for final fracture morphology. Figure 7(a-d) shows the influence of horizontal stress contrast on the hydraulic fracture propagation path. Ensure other parameters remain unchanged, the horizontal stress difference is increased by reducing the minimum horizontal principal stress. The horizontal stress contrast is 3 MPa, 5 MPa, 7 MPa, and 9 MPa. The results showed that the hydraulic fracture deviated from the original maximum horizontal principal stress direction after a certain distance from the perforation positions, and more curved fracture path can be seen at the end of fracture tip. In addition, the smaller of the horizontal stress contrast, the stronger the ability of induced stress fields caused by oil production activities, and the easier it is for new hydraulic fractures to extend from the recompletion during the refracturing. Figure 8(a) shows the new fracture length for hydraulic fracture propagation from recompletion during the refracturing on the horizontal well. Because of the increase in the horizontal stress contrast, the fracture length increases. When the stress difference is 9 MPa, the fracture length is 32.16 m, which is the largest. When the stress difference is 3 MPa, the length is the smallest, only 25.02 m. The comparison of fracture widths is shown in Fig. 8(b). In contrast to the fracture length, the fracture width decreases with the increase of stress difference. When the stress difference is 3 MPa, the fracture width is 0.025 m, which is the largest. When the stress difference is 9 MPa, the seam width is the smallest, only 0.018 m.

Fig. 7
figure 7

Hydraulic fracture propagation path from recompletion during the refracturing with different horizontal stress contrasts. Different colors in the panels correspond to diffeent pore pressure. a 3 MPa, b 5 MPa, c 7 MPa, d 9 MPa

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Fig. 8
figure 8

The new fracture length and width for hydraulic fracture propagation from recompletion during the refracturing with different horizontal stress contrasts. The different color symbols in the panel correspond to different horizontal stress contrasts. a New fracture length, b New fracture width

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5.3 Influence of initial completion spacing on fracture propagation during the recompletion

Perforation spacing is the key parameter to control the induced stress field between the multiple fractures, and the smaller the perforation spacing, the more intense stress shadowing effect can be generated. Large initial perforation spacing can increase the recompletion spacing in the refracturing. Figure 9 shows the hydraulic fracture propagation from recompletion during the refracturing with different initial perforation clusters in the refracturing of the horizontal well. It can be seen that the increase of initial cluster spacing also increases the cluster spacing during the recompletion, and the degree of stress interference seen in fractures will be reduced, thus the relative straight fracture can be seen in Fig. 9(d). Figure 10(a) shows the new fracture length for hydraulic fracture propagation from recompletion during the refracturing with different initial completion spacings. The large fracture length (35.07 m) can be observed for the case with the initial completion spacing of 41 m. With the fracturing spacing decreasing, the fracture propagation from the recompletion also can be restricted, when the initial completion space is 20 m, the fracture length is 25.16 m. The comparison of fracture width was shown in Fig. 10(b), when the initial completion space is 20 m, the fracture width is 0.023 m and the initial completion space is 41 m, and the fracture width is 0.019 m. In contrast to the hydraulic fracture length, the width of the hydraulic fracture gradually decreases as the initial completion space increases, resulting from lower production-induced in-situ stress change.

Fig. 9
figure 9

Hydraulic fracture propagation path from recompletion during the refracturing with different initial completion spacings. Different colors in the panels correspond to diffeent pore pressure. a 20 m, b 27 m, c 34 m, d 41 m

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Fig. 10
figure 10

The new fracture length and width for hydraulic fracture propagation from recompletion during the refracturing with different initial completion spacings. The different color symbols in the panel correspond to different initial completion spacings. a New fracture length, b New fracture width

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5.4 Influence of initial fracture length on fracture propagation during the recompletion

The initial fracture length also refers to the depletion stimulation effectiveness Fig. 11 shows the hydraulic fracture propagation path from recompletion during the refracturing with different initial fracture lengths. It shows the large fracture length can affect a larger in-situ stress change area, thus more curved fracture path can be seen from the recompletion during the refracturing. Figure 12(a) shows the new fracture length for hydraulic fracture propagation from recompletion during the refracturing with different initial fracture lengths. It can be seen that the increase of initial fracture length can increase the length of the new fracture from recompletion during the refracturing. When the initial fracture length is 6 m, the new fracture length is 25.16 m. When the initial fracture length is 18 m, the new fracture length is 39.12 m. The longer the initial fracture length, the larger the area of low pore pressure formed by depletion production, and the more easily the new fracture formed by re-fracturing spreads. Figure 12(b) shows the fracture width for hydraulic fracture propagation from recompletion during the refracturing with different initial fracture lengths. As shown in Figure, when the initial fracture length is 6 m, the fracture width is 0.01486 m and the initial fracture length is 18 m, the fracture width is 0.01224 m. The fracture width increased by about 21% compared to the case of the initial fracture length of 18 m.

Fig. 11
figure 11

Hydraulic fracture propagation path from recompletion during the refracturing with different initial fracture lengths. Different colors in the panels correspond to diffeent pore pressure. a 6 m, b 10 m, c 14 m, d 18 m

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Fig. 12
figure 12

The new fracture length and width for hydraulic fracture propagation from recompletion during the refracturing with different initial fracture lengths. The different color symbols in the panel correspond to different initial fracture lengths. a New fracture length, b New fracture width

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5.5 Influence of varying recompletion perforation cluster spacings on fracture propagation during the recompletion

Considering the later heterogeneity of horizontal well, varying perforation clusters can be an alternative method in recompletion during the refracturing. Figure 13 shows the hydraulic fracture propagation path from recompletion during the refracturing with different varying perforation cluster spacings of 8 m, 16 m, 24 m, and 32 m. It was shown that with the close of the initial depleted hydraulic fracture, then the fracture tends to reorient towards the depleted hydraulic fracture. Figure 13(a) and (b) show that hydraulic fractures tend to be affected by two outer fractures from the initial completion. Figure 13(c) shows the effect of depletion effect by initial completion decrease thus a straight hydraulic fracture during the refracturing can be seen. Figure 13(d) displays that hydraulic fracturing from the recompletion was strongly affected by the middle fractures from the initial completion. Because of the close spacing between recompletion and the middle initial completion, the reorientation of hydraulic fractures towards the middle initial completion can be observed. Figure 14(a) shows the new fracture length for hydraulic fracture propagation from recompletion during the refracturing with varying perforation cluster spacings. It found that enough high spacing can assure enough long fracture length from the recompletion, the new fracture length for the case with the spacing of 16 m and 24 m are the largest, which are higher than the other two cases that with the close instance to the hydraulic fracture from initial completion. Figure 14(b) shows the fracture width for hydraulic fracture propagation from recompletion during the refracturing with different varying perforation cluster spacings. As shown in this figure, when the space is 8 m, the fracture width is optimal while lower stress shadowing effect from depletion can be observed. As for this given case, a strong stress shadowing effect can be found for new fractures from the initial middle fracture rather than the initial two side fractures. As can be seen from the figure, when the distance between the new fracture and the initial fracture on both sides is larger, the new fracture will expand better and the fracture length will be longer. For example, as shown in Fig. 13(b) and c, the joint length of the new fracture is 32.40 m and 32.11 m respectively. The lengths of the new fracture in Fig. 13(a) and d are 27.71 m and 24.33 m, respectively. However, the fracture width is opposite to the length rule, and the fracture width is smaller when the spacing between new fracture and initial fractures on both sides is larger. When the fracture spacing is 8 m, the fracture width is the best, the fracture width is 0.024 m, and the dissipative stress shadow effect is small. In this case, the new fracture produced from the initial intermediate fracture rather than the initial bilateral fracture have a strong stress shadow effect.

Fig. 13
figure 13

Hydraulic fracture propagation path from recompletion during the refracturing with varying perforation cluster spacings. Different colors in the panels correspond to diffeent pore pressure. a 8 m, b 16 m, c 24 m, d 32 m

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Fig. 14
figure 14

The new fracture length and width for hydraulic fracture propagation from recompletion during the refracturing with varying perforation cluster spacings. The different color symbols in the panel correspond to different perforation cluster spacings during the refracturing process. a New fracture length, b New fracture width

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6 Conclusions

In this work, the fully coupled method on dynamic in-situ stress condition reflects the depletion-induced stress can change the original in-situ stress and affects the following new simultaneous fracture clusters from recompletion during the refracturing treatment. Furthermore, a methodology on hydraulic fracture propagation considering dynamic depletion stress from the parent hydraulic fracturing is developed for recompletion during the refracturing. Some conclusions can be summarized as follows:

  1. (1)

    The stress interference on new child fractures decrease and a strongly curved fracture path was observed on the fracture tip while the fracture direction generally follows the primary depletion zone. The production time affects the optimal refracturing time window and it is possible to contact larger parts of the reservoir by perforating new clusters at the optimal time of refracturing.

  2. (2)

    The dynamic in-situ stress condition has a close relationship with initial parent fracture parameters and refracturing scenarios. The smaller the old completion spacing and new recompletion perforation cluster spacings can induce the more distinct stress shadowing effect due to rapid oil depletion. The larger of the old parent hydraulic fracture length can induce a more distinct stress change.

  3. (3)

    The original horizontal stress contrast also affect the child fracture path from recompletion. A high horizontal stress contrast is beneficial for larger fracture extend areas. However, the increased horizontal stress can also increase the closure stress and induce small fracture width in the fracturing treatment.

  4. (4)

    The non-uniform reservoir pressure and stress field can be created along the horizontal well, thus affecting the following child hydraulic fracture path. The different fluid distribution for different new fractures with varying fracture widths can result in screening out and promote non-uniform fracture propagation, affecting the temporary plugging agents injection. Reducing the number of perforations near the old perforation position can decrease the possibility of curved fracture.