Simultaneous multiple fracturing is a key technology to facilitate the production of shale oil/gas. When multiple hydraulic fractures propagate simultaneously, there is an interaction effect among these propagating hydraulic fractures, known as the stressshadow effect, which has a significant impact on the fracture geometry. Understanding and controlling the propagation of simultaneous multiple hydraulic fractures and the interaction effects between multiple fractures are critical to optimizing oil/gas production. In this paper, the FDEM simulator and a fluid simulator are linked, named FDEMFluid, to handle hydromechanicalfracture coupling problems and investigate the simultaneous multiple hydraulic fracturing mechanism. The fractures propagation and the deformation of solid phase are solved by FDEM; meanwhile the fluid flow in the fractures is modeled using the principle of parallelplate flow model. Several tests are carried out to validate the application of FDEMFluid in hydraulic fracturing simulation. Then, this FDEMFluid is used to investigate simultaneous multiple fractures treatment. Fractures repel each other when multiple fractures propagate from a single horizontal well, while the nearby fractures in different horizontal wells attract each other when multiple fractures propagate from multiple parallel horizontal wells. The in situ stress also has a significant impact on the fracture geometry.
In the past two decades, a great number of researchers have been attracted to improving hydraulic fracturing techniques, which provide large surface area contact with formations and facilitate the production of oil and gas, as evident from the experience in North America [
Multiple hydraulic fractures treatment is a complex process, as it needs to consider not only the hydraulic fracturing process but also fracture interaction between multiple fractures. The hydraulic fracturing process is a typical hydromechanicalfracture (HMF) coupling problem, in which the following three basic processes require discussion [
With the development of computers, numerical techniques have become reliable and convenient tools for studying hydraulic fracturing processes. The finite element method (FEM) [
Although FDEM has been welldeveloped and applied by many researchers to study the process of hydraulic fracturing, very little literature can be found about the study of stressshadow effect with FDEM. In this paper, FDEM coupled with a fluid algorithm, referred to as FDEMFluid, is developed to handle simultaneous multiple hydraulic fractures propagation. The organization of this paper is as follows. In Section
As shown in Figure
Discretized intact body with 3node triangular elements and 4node joint elements.
The movement of discrete bodies and the nodal coordinates are updated by the following governing equation:
The schematic diagram of the internal resisting forces, the external loads, and the contact forces (the contact force happens when two blocks contacted).
A highly efficient No Binary Search (NBS) algorithm [
A distributed contact force is generated according to the shape and size of the overlap between the contactor and target elements. As the contactor penetrates an area
Contact force due to an infinitesimal overlap around points
Failure of rock material is simulated by the combined single and smeared crack model [
(a) Softening part of stressstain at the FPZ zone; (b) representation in FDEM by using triangular elements and joint elements.
Based on the local stress and deformation field, fractures initiate and propagate according to three different failure modes, which are often denoted as Mode I (opening failure), Mode II (sliding failure), and Mixed Mode III. The constitutive behavior of the joint element is illustrated in detail in the following.
Constitutive behavior of the joint element. (a) Mode I. (b) Mode II. (c) Mixed Mode III.
Upon undergoing critical slip,
The values of the residual opening and slip depend on Mode I and Mode II fracture energy release rates,
As shown in Figure
Fluid flow in the hydraulic fracture network and the influence domain of node
The hydraulic fracturing process is complicated, especially in regard to HMF coupling. Therefore, the following hypotheses are introduced:
A cubic law based on parallelplate flow model [
Hydraulic elements connected with the hydraulic boundary are defined as the hydraulic fracture network, which is the only flow path for the fluid flow. Taking hydraulic nodes
The flow rate from
Considering the fluid flow saturation, the actual flow rate from
The total flow rate of node
Note that there is a convergence criterion herein; that is, the hydraulic time step should be less than a critical value:
The hydromechanicalfracture (HMF) coupling process is illustrated in Figure
Interaction in HMF coupling process.
This FDEMFluid consists of the following two parts: the solid solver, which calculates the deformation of rock mass and fracture propagation, and the fluid solver, which calculates the fluid diffusion process and fluid pressure in fracture networks. The schematic diagram of the coupling between the two solvers is presented in Figure
Interaction between the solid solver and fluid solver in FDEMFluid.
A typical time step in FDEMFluid is illustrated in Figure
Flowchart of the calculation process in the FDEMFluid method.
As shown in Figure
Fluid pressure acting on element boundaries.
The normal force on each node is perpendicular to the edge and can be calculated as
The shear force on each node is opposite to the fluid flow and can be calculated as
The fracture aperture can be obtained directly through the displacement of the broken joint element surfaces. Considering the HM coupling, the fracture aperture is assumed to be a linear distribution along the fracture surface, as shown in Figure
Two numerical tests for the present FDEMFluid are carried out, and the results are compared with previous analytical and experimental results.
This test investigates the fluid flow in a single fracture by FDEMFluid. The mesh sensitivity of fluid flow in FDEMFluid is also discussed. The geometry and boundary conditions of fluid flowing through a single fracture are shown in Figure
Rock sample with a single fracture.
Sample geometry
Three sets of mesh with mesh sizes of 0.1 m, 0.05 m, and 0.02 m
The hydraulic pressure distribution with time and location is investigated. The analytical solution of this example is as follows [
Three sets of meshes, as shown in Figure
The hydraulic pressure distributions at different time with the three sets of meshes are shown in Figure
Hydraulic pressure distribution at different time with three sets of mesh size (Pa).
A comparison between the analytical and numerical solution is presented in Figure
Comparison between analytical and numerical solutions of hydraulic pressure distribution with three sets of meshes.
The mesh sensitivity of fluid flow is shown in Figure
Error between numerical solution and analytical solution under three sets of mesh sizes at
Error analysis at
Error analysis at
An experiment designed by Jiao et al. [
The geometry and cross section of the sample are shown in Figure
A cubic specimen with preset fractures.
Sample geometry
Sample cross section
Model mesh
Two models are taken in to account: in model I, the confining stress is ignored; in model II, the compressive confining stresses
Parameters of validation 2.
Parameters  Value 

Time step, 

Injection pressure 
0.8 
Rock parameters  
Bulk density, 
1100 
Young’s modulus, 
4.0 
Poisson’s ratio, 
0.3 
Joint element parameters  
Tensile strength, 
0.3 
Cohesion, 
1.0 
Friction angle, 
32 
Energy release rate, 
1.0, 10.0 
Fluid parameters  
Bulk modulus of fluid, 
2.2 
Viscosity, 
0.001 
Initial value of aperture, 

The fracture geometry evolution and fluid pressure distribution are shown in Figures
(a) Fracture geometry and hydraulic pressure distribution of model I; (b) displacement magnitude contours.
(a) Fracture geometry and hydraulic pressure distribution of model II; (b) displacement magnitude contours.
Comparing the two fracture geometries, the fractures propagate strictly along the direction of preset fractures in model I. However, in model II, the fractures obviously turn toward
The numerical result of model I agree well with the experimental result conducted by Jiao et al. [
Experimental result of model I test [
Experimental result of Chen et al. [
Top view
Sectional view
These two examples validate the ability of FDEMFluid to model hydraulic fracture correctly with and without considering the influence of in situ stress.
Simultaneous multiple hydraulic fracturing in combination with horizontal drilling has been the key technology in the exploitation of unconventional gas/oil. This technology can significantly increase the volume and surface area of the flow path, so as to improve the productivity and recovery efficiency. For simultaneous multiplefracture propagation, fracture interaction (i.e., the stressshadow effect) plays an important role in fracture geometry. In this section, two examples are studied to investigate fracture interactions during simultaneous fracturing processes.
In this section, a single horizontal wellbore with two parallel initial fractures is simulated and compared to the results of Wu et al. [
Parameters for simultaneous multifracture propagation from a horizontal well.
Parameters  Value 

Time step, 

Injection rate, 
0.106 
Stress anisotropy, 
0.9 
Rock parameters  
Young’s modulus, 
30 
Poisson’s ratio, 
0.35 
Joint element parameters  
Tensile strength, 
0.3 
Cohesion, 
1.0 
Friction angle, 
32 
Energy release rate, 
1.0, 10.0 
Fluid parameters  
Viscosity, 
0.001 
Bulk modulus of fluid, 
2.2 
A single horizontal wellbore with two parallel preset fractures.
This model is simulated under isotropic and anisotropic in situ stress conditions. The differential stress for the anisotropic stress is 0.9 MPa, with the maximum horizontal stress oriented in the
Figures
Propagation of two initially parallel fractures driven by hydraulic pressure: isotropic in situ stress field.
Step 4000
Step 8000
Step 12000
Step 16000
Propagation of two initially parallel fractures driven by hydraulic pressure: anisotropic in situ stress field.
Step 4000
Step 8000
Step 12000
Step 16000
As shown in Figures
As shown in Figure
Influence domain of fractures under isotropic and anisotropic in situ stress conditions.
Influence domain of fractures in the
Influence domain of fractures in the
The reason fractures turn is because of the mechanical fracture interaction (i.e., the stressshadow effect), which exerts additional stress on the surrounding rock, resulting in a local change in direction of the maximum horizontal stress and a deviation of the fracture path.
As shown in Figures
Maximum principal stress contours under an isotropic in situ stress field (positive denotes tension).
Step 4000
Step 8000
Step 12000
Step 16000
Maximum principal stress contours under an anisotropic in situ stress field (positive denotes tension).
Step 4000
Step 8000
Step 12000
Step 16000
As shown in Figure
Comparison of propagation paths by FDEMFluid and Wu et al. [
Under isotropic in situ stress field condition
Under anisotropic in situ stress field condition
In this example, simultaneous propagation of two sets of initial fractures from two horizontal wells is investigated. As shown in Figure
Two horizontal wells with two sets of parallel preset fractures.
The evolution of the preset fracture geometry and the distribution of fluid pressure at different time steps are shown in Figure
Evolution of fracture geometries and the distribution of fluid pressure.
Step 4000
Step 8000
Step 12000
Step 16000
Figure
Counter of (a) maximum principle stress and (b) displacement magnitude.
Direction of maximum principal stress at steps 4000 and 8000.
The interaction (i.e., the stressshadow effect) during simultaneous multiplefracture propagation plays an important role in fracture geometry, which can significantly affect the productivity and recovery efficiency. Therefore, understanding and controlling the interaction effects between multiple fractures are critical. In this study, a hydraulic fracturing simulator, named FDEMFluid, based on FDEM and the parallelplate flow model is presented to capture the development of complex hydraulic fracturing propagation. This FDEMFluid is verified by comparing it with existing analytical and experimental results, and good agreements are observed. Then, the multifracture treatment in the hydraulic fracturing process is investigated with FDEMFluid. The main conclusions from the study include the following:
When multiple fractures are orthogonal to the horizontal well and propagate from a single horizontal well simultaneously, fractures repel each other as a result of the stressshadow effect, resulting in a complex fracture geometry. Furthermore, the anisotropic in situ stress, in which the maximum horizontal stress is orthogonal to the horizontal well, has a great influence on the curvature and length of the fracture geometry because of the competition between the stressshadow effect and the anisotropic stress.
When multiple fractures are orthogonal to the horizontal wells and propagate from multiple parallel horizontal wells simultaneously, the nearby fractures in different horizontal wells attract each other as a result of the stressshadow effect, which alters the local in situ stresses and the direction of the maximum principle stress. The fracture tips in these fractures tend to reorient themselves to be perpendicular to the main fractures. This leads to the development of a more complex fracture geometry and prevent fractures from penetrating deeper into the opposite wells.
The authors declare that there are no conflicts of interest related to this paper.
This research is supported by the National Basic Research Program of China (973 Program) (Grant no. 2014CB046900) and the National Natural Science Foundation of China (Grant no. 41602296).