Interaction between injection points during hydraulic fracturing

Authors


Corresponding author: K. M. D. Hals, Christian Michelsen Research, P.O. Box 6031, NO-5892 Bergen, Norway. (kjetil.hals@cmr.no)

Abstract

[1] We study the geomechanical stress interaction between two injection points during hydraulic fracturing (hydrofracking) and how this interaction in combination with disorder influences the fracturing process. To this end, we develop an effective continuum model of the hydrofracking of heterogeneous poroelastic media that captures the coupled dynamics of the fluid pressure and the fractured rock matrix and models both the tensile and shear failure of the rock. For injection points that are separated by less than a critical correlation length, our numerical simulations show that the fracturing process around each point is strongly correlated with the position of the neighboring point. The magnitude of the correlation length depends on the degree of heterogeneity of the rock and is on the order of 30–45 m for rocks with low permeabilities. In the strongly correlated regime, we predict a novel effective fracture force that attracts the fractures toward the neighboring injection point.

1. Introduction

[2] Hydrofracking is a technology that utilizes highly pressurized fluid to create fracture networks in rock layers with low permeabilities [Cosad, 1992; Brady et al., 1992]. A fracking fluid is injected into a cased wellbore, and the parts of the reservoir to be fractured are accessed by perforating the casing at the correct locations (Figure 1b). The injection well can be vertical or horizontal and several injection points may be active during the hydraulic stimulation of the system (Figures 1a, 1b). The highly pressurized fluid that flows into the reservoir increases the geomechanical stress around the injection points and causes the rock to fracture. Ideally, the hydrofracking creates long, distributed cracks that connect a large area of the reservoir to the well. If several injection points are active during the hydrofracking process, the final crack distribution strongly depends on the spacing of the points and the interaction between them during the injection process [Cosad, 1992; Brady et al., 1992]. A comprehensive understanding of this interaction and its implications for the fracturing process is therefore crucial to optimize the functionality of the reservoir.

Figure 1.

(a) Hydrofracking using two vertical wells. (b) Hydrofracking using a perforated horizontal well. (c) A simplified, discretized 2-D representation of the reservoirs shown in Figures 1a and 1b. (d) A heterogeneous system is modeled by distributing the material strengths of the discrete volume elements according to the Weibull distribution. The degree of disorder is tuned by varying the Weibull modulus,m.

[3] The complexity of the fracture networks and the multiple scales involved represent challenges in the mathematical and numerical modeling of both the fluid flow and the fracturing processes. The different modeling approaches of the complex fluid flow can be classified into three main categories: discrete fracture-matrix (DFM) models, effective continuum models (also known as equivalent continuum models), and multiple-continuum models.

[4] DFM models explicitly resolves the fractures, which is feasible in situations where only a limited number of fractures dominate the fluid flow. The approach requires gridding of the fractures, either with an equidimensional approach where the fractures have the same dimension as the matrix, or with a hybrid approach where the fractures are modeled as planes in three-dimensional domains and lines in two-dimensional domains. The DFM model requires a detailed knowledge of the geometric properties of the fracture network, and also requires unstructured grids to account for the given geometry, but provides an accurate description of the flow when this information is available.

[5] The effective continuum model (see, e.g., Bear et al. [1993]) represents the fractures and the rock matrix as a single effective continuum. The method has been extensively used to model fluid flow in fractured porous media because of its computational efficiency and the limited data that is required. However, in general, the method is only feasible when considering transport processes under steady state conditions. The advantage of the effective continuum model is that it is simple, and it also provides a sufficient model for the flow in the case where the equivalent characteristics of the fractured porous media may easily be estimated, e.g., in the case of fractures parallel to the fluid flow [Samardizioska and Popov, 2005].

[6] Multiple-continuum approaches are an extension of the effective continuum model, developed to handle the different characteristics of the flow in the fractures and the matrix. The fractures and the matrix are considered as different continuums over the same volume, interacting through mass transfer coefficients, and hence representing the fractured porous medium. The approaches include the double-porosity model [Warren and Root, 1963], the dual-permeability concept, the dual-continuum generalization [Pruess and Narasimhan, 1985], and the multicontinuum model [Wu and Pruess, 1988]. The method offers more information than the effective continuum approach on averaged properties of the flow and transport processes in the matrix and the fractures, without the need of detailed and complex information of the geometry of the problem.

[7] The failure of homogeneous materials is well understood. A fracture is created when the strain energy-release rate exceeds a critical value, and it propagates along the direction that maximizes the energy release [Knott, 1973]. The failure of heterogeneous materials, in contrast, depends on many microscopic mechanisms and is challenging to model [Hansen, 2005]. To understand the fracturing process of a heterogeneous material on the microscale, statistical models such as the fuse and the beam models have been particularly useful [Hansen, 2005; Herrmann and Roux, 1990]. In these models, the heterogeneity of the system is taken into account by defining a local, position-dependent material strength that is drawn from a statistical distribution. The fracturing process is therefore not entirely controlled by the energy-release rate, but also depends on the disorder of the system and the distribution of the local strengths. Depending on the degree of disorder, the statistical models have shown that heterogeneous systems exhibit fracturing regimes that are distinct from the single-crack development observed in homogeneous systems [Hansen, 2005].

[8] Several models for hydrofracking have been developed [Adachi et al., 2007], but a weakness of most of these models is that they do not account for the heterogeneities of the rock. However, some research based on extensions of the statistical beam model has been applied to the study of hydrofracking of heterogeneous systems [Tzschichholtz et al., 1994; Tzschichholtz and Herrmann, 1995; Tzschichholtz and Wangen, 1998; Flekkøy et al., 2002]. Recently, these studies have been further developed to provide a better description of the rock matrix. In the work of Wangen [2011] and Wang et al. [2009], the dynamics of the rock matrix are modeled using poroelastic theory, and the heterogeneity is treated by distributing the local material strength according to a probability distribution. Wangen [2011] applies an effective continuum approach to model the fluid flow, which captures the coupled dynamics of the fluid pressure and the fractured rock matrix.

[9] The creation of a fracture changes the geomechanical strain energy of the system. If the reservoir contains several fractures, the modification of the strain energy mediates an effective interaction between the cracks. The interactions between fractures have been studied theoretically in several works using both direct numerical simulations [Akulich and Zvyagin, 2008; Aghighi and Rahman, 2010] and statistical methods [Masihi and King, 2007; Shekhar and Gibson, 2011]. A geomechanical stress interaction also exists between two injection points during hydrofracking. However, a thorough investigation of this interaction and its consequences for the fracturing process is lacking. The main focus of the present paper is to investigate how this interaction in combination with disorder influences the hydrofracking process. To this end, we develop an effective continuum model of the hydrofracking of heterogeneous, poroelastic media. Our formalism is based on the previous work by Wangen [2011], which is extended to take into account the anisotropic fluid flow and the shear failure of the material. Our numerical simulations demonstrate, for the first time, how the fracturing process around each injection point is correlated with the neighboring point, and how this correlation depends on the degree of disorder in the system. We find that for two injection points that are separated by a distance that is smaller than a critical correlation length, the fracturing process around each injection point is strongly correlated with the position of the neighboring point. The critical correlation length at which this strongly correlated regime occurs depends on the degree of heterogeneity, with correlation lengths of ∼20 m for highly disordered systems and 45 m for weakly disordered systems. In the correlated regime, we predict a novel effective fracture forcethat attracts the fracture toward the neighboring injection point. Our results give important insights for optimizing the hydraulic stimulation of reservoirs. For well perforations separated by a distance that is less than the critical correlation length, the results imply a reduced effect of the stimulation because the fractures are attracted toward neighboring injection points. Knowing the correlation length of the system is therefore crucial for creating an effective and long-ranging fracture network.

[10] This paper is organized in the following manner: In section 2, we present the theory and governing equations of our hydrofracking model. The section concludes with a pseudo code of the algorithm. Section 3 describes the numerical solution strategy. The next two sections apply the formalism to the study of the interaction of two injection points during hydrofracking. Specifically, section 4 provides a description of the model system that we consider and section 5 presents our findings. We conclude and summarize our results in section 6.

2. Governing Equations

[11] In this section, we consider a poroelastic system and develop the mathematical description that captures the coupled dynamics of the fractured rock matrix and the fluid. At the end of the section, the tensile and shear failure criteria are defined.

2.1. Poroelastic Theory

[12] The dynamics of an elastic medium are described by the following fundamental law of elasticity theory [Landau et al., 1986]:

display math

Here ρ is the mass density, Fi represents the external forces, and math formula is the stress tensor arising from the internal stresses. The internal stresses are caused by the intermolecular forces that occur because of the relative displacement of the molecules under the deformation. The displacement of the material point located at the position r (prior to the deformation) is described by the displacement field u(r). In equation (1) (and in what follows), we apply the Einstein summation convention and sum over repeated indices. Usually, the only external force that appears is the gravitational force, math formula, where g is the gravitational acceleration. The equilibrium state of the system is determined by solving the stationary equation, i.e., equation (1) with math formula.

[13] In this study, we concentrate on a porous, elastic rock system with water-filled pores. In the linear response regime for an isotropic, poroelastic system, the stress tensor can be written phenomenologically in terms of the strain tensor math formula as [Bundschuh and Suárez Arriaga, 2010],

display math

where

display math

The tensor math formula is the Terzaghi effective stress tensor, which describes the stresses that act only on the rock matrix, while math formularepresents the stresses acting on the total fluid-rock system; math formula is the trace of the strain tensor, and math formula and math formula are elasticity coefficients, referred to as the Lamé and rigidity moduli, respectively. The parameter math formulais the effective stress coefficient (also known as the Biot-Willis parameter), which describes how the lithostatic pressure changes with the fluid pressurepf. A pressure math formula corresponds to a fluid pressure that is larger than the atmospheric pressure. For small deformations, the strain tensor can be expressed in terms of the displacement field u(r) as [Landau et al., 1986],

display math

We use the following sign conventions: math formula and math formula in tension and math formula and math formula in compression. The elastic energy stored in a strained system is [Landau et al., 1986],

display math

[14] Substituting the stress tensor in equation (2) into equation (1)and expressing the strain tensor in terms of the displacement field produces the equations of the poroelastic system. In this paper, we assume that the relaxation time of the elastic system is small compared with the timescale of the pressure evolution. We can therefore assume that for a given fluid-pressure profile, the elastic medium is always very close to the equilibrium state. Thus, in what follows, we will be concerned only with the stationary version ofequation (1).

[15] The above formalism models an isotropic system that does not contain any fractures. In the numerical implementation of the problem, a simple way to model a fracture is to set the elasticity parameters equal to zero (i.e., math formula) for the discrete volume elements containing a fracture (Figure 2). As shown by Wangen [2011], this method of modeling the fracture correctly produces the form of the stress field close to the fracture tip. A more correct description of the stress field close to the fracture tip requires a time-dependent grid with an extra fine mesh size near the fracture tip. Such a detailed description is beyond the scope of the present model in which the aim is to provide a qualitative description of the fracturing process.

Figure 2.

A system containing a fracture (blue area). The fracture volume is calculated numerically from the displacement field in the neighboring volume elements of the fracture. The displacement field is illustrated by the red arrows. The elasticity parameters of the elements containing parts of the fracture are equal to zero. The volume of each element is math formula.

2.2. Pressure Equation

[16] In the present paper, we consider a system in which the porous rock is incompressible and the fluid is slightly compressible. In this approximation, the continuity equation for the fluid mass becomes [Bear, 1972; Bundschuh and Suárez Arriaga, 2010],

display math

Here φ is the porosity, pf is the fluid pressure, and math formula is the mass density of the fluid. math formula is the permeability tensor, math formula is the fluid viscosity, and math formula represents the gravitational force. math formula is a source (sink) term that arises from the injection (extraction) of the fluid.

[17] In zones with fractures, a pressure gradient opens the fractures and consequently modifies the local permeability and porosity. The time variation of this process is of the same order as the pressure variation and is a nonnegligible contribution to the pressure evolution. Consequently, the pressure equation contains terms that couple the pressure dynamics to the fracture dynamics. In addition, the permeability becomes anisotropic with a much higher permeability along the fractures than across the fractures. As described by Wangen [2011], one method to include this coupling in a numerical implementation of the problem is to assign to each fractured volume element (of the discretized system) an effective porosity that depends on the opening of the fracture (Figure 2). The effective porosity model provides a suitable approximation as long as the typical length scale of the pressure variations is large compared with the opening of the fracture and the fluid flow is parallel to the fractures [Samardizioska and Popov, 2005]. The effective porosity of the volume element i is,

display math

where

display math

Here math formula is the volume of the fracture inside element i and math formula is the volume of element i. The fracture volume of each element is found by integrating the displacement field over the fracture surface (Figure 2),

display math

The effective porosity becomes equal to 1 if the fracture fills the entire volume element, and it reduces to the porosity of the rock if the fracture is closed or if element i does not contain a fracture.

[18] Let math formula denote the continuum limit of the effective porosity. Because of the time variation of the effective porosity, the continuity equation for the fluid mass becomes,

display math

where math formula is the compressibility factor under isothermal conditions.

[19] The fractures also modify the permeability, which becomes an anisotropic second-rank tensor. In our numerical implementation of the problem, we assign to each volume element a permeability tensor that depends on the fracture direction and the fracture opening. (In the numerical model, we allow the fracture to propagate only in thex- and they-directions for simplicity.) As an illustration, let us consider the situation shown inFigure 2. This 2-D system has an open fracture along they-axis, and the permeability is therefore larger along they-direction than along thex-axis. We model this effect by introducing the following tensor:

display math

where math formula is the permeability across the fracture, which is equal to the permeability of the rock, and math formula is the permeability along the fracture. The quantity math formula represents the permeability inside the fracture. In the parallel plate model for a single fracture, the fracture permeability is given by the cubic law math formula, where w is the fracture aperture. This yields a very large permeability that may cause numerical problems. As mentioned by Wangen [2011], a practical solution to this problem is to choose a permeability that is large enough to enact a minimal pressure drop along the fracture, but is small enough to avoid numerical instabilities. Equation (10) becomes math formula (I is the identity matrix) for elements that are not fractured or that contain a closed fracture.

2.3. Stress-Based Failure Criteria

[20] Porous materials fail under the action of a large fluid-pressure gradient and the failure can be induced by both shear and tensile forces. Several phenomenological failure criteria exist [Jaeger et al., 2007; Paterson and Wong, 2005; Herrmann and Roux, 1990]. To capture both tensile and shear failure, we adopt two distinct criteria.

[21] We model the tensile failure by locally defining (i.e., for each volume element i) a critical tensile stress, math formula. If one of the eigenvalues of the Terzaghi effective stress tensor (evaluated at element i) exceeds the critical stress, the volume element i fails. The compressive strength of the material can be included in a similar manner. However, during hydrofracking, the tensile failure is the dominant fracturing mechanism, and we therefore disregard compressive failure in our numerical simulation.

[22] The shear failure is modeled in 2-D by the Mohr-Coulomb criterion [Jaeger et al., 2007; Paterson and Wong., 2005],

display math

Here θ is the internal friction angle, which depends on the density, surface, and shape of the particles constituting the solid, and the cohesion, c, describes the minimal shear force that is required for fracturing when no normal stresses are present. Furthermore, math formula are the eigenvalues of the Terzaghi effective stress tensor evaluated for volume element i.

[23] As discussed by Wangen [2011], the critical stresses (i.e., the parameters math formula and math formula in our model) depend on the grid size because the stress field is singular at the fracture tip. Wangen [2011] solves this problem by scaling the critical stress with the square root of the grid size. An alternative procedure, is to use experimental data to fit the values for math formula and math formula, such that fracturing occurs for fluid pressures in the experimental range. In this paper, we apply the latter approach by tuning the mean values of math formula and math formula so that shear and tensile failure appear for borehole pressures that are typical for the present problem.

[24] The strength of a material usually follows a distribution that is well-described by the Weibull distribution (Figure 1d) [Herrmann and Roux, 1990],

display math

Here m is the Weibull modulus, math formula is the mean value of the critical stress, and math formula is the threshold stress below which no failure will occur (usually it is set as math formula). We incorporate the heterogeneity of the rock into our model by distributing the local strengths math formula and math formula according to the Weibull distribution. The Weibull modulus is a measure of the degree of disorder in the system. A system with a small Weibull modulus has a higher degree of disorder than a system with a larger modulus. This method of modeling the heterogeneity is analogous to what is done in microscopic fracturing models, such as the fuse and the beam models.

3. Numerical Solution Approach

[25] The model presented in section 2is an effective continuum model of a fractured poroelastic system. Our primary aim is not to provide a detailed description of the stress field close the fracture tip or the fluid flow inside the fractures. Instead, we seek a simplified description of the coupled fluid-rock system that captures the primary elements of the fracturing process to obtain a qualitative understanding of hydrofracking, for example, to sort out the mechanisms that dominate the process or to map out what types of fracturing regimes are produced by different material parameters.

[26] To solve the system of equations presented in section 2, we apply a sequential solution strategy using standard numerical discretization methods.

3.1. Discretization Methods

[27] The pressure equation, equation (9), is solved with a finite difference formulation of the complete pressure equation that captures the pressure dynamics in the fracture zones and in the homogeneous regions of the system. To isolate the effect of interactions between the two injection points, we disregard the gravitational force in the Darcy velocity, and for simplicity, we consider a 2-D rectangular region that is discretized with a lattice constant,a, as illustrated in Figure 1c. An explicit scheme is implemented using a forward Euler discretization in time for the temporal pressure derivative. The time derivative, math formula, is calculated from the last two time steps.

[28] The ordinary differential equations arising from the discretized pressure equation are solved using the Cash-Karp embedded Runge-Kutta method [Press et al., 1992], which is an adaptive algorithm that regulates the time step using an error estimate.

[29] The boundary condition for the pressure is, math formula. The Galerkin method with bilinear trial functions is used to solve the stationary stress equation, equation (1) (see Langtangen [2003]for more details). We use traction-free boundaries as boundary conditions in the present problem, i.e., math formula for all i, where math formula is the boundary of the system, and nj is its surface normal.

[30] Our simplified method of modeling the fractures results in a grid-dependent fracture aperture and effective porosity. In practice, we fix the grid size by claiming that the opening of the fracture is of the order 1 mm when the pressure inside the fracture is of the order 1 MPa. The fracture opening of the order 1 mm is estimated from the slip magnitudes reported from the Soultz test site [Evans et al., 2005a, 2005b], and lies in the intermediate range of the values reported when we related the fracture opening (w) to the slip magnitude (δ) by the relation math formula, where math formula is the dilation angle [Evans et al., 2005b].

3.2. The Fracture Event

[31] The present model assumes that the reservoir consists of hard rocks with a low porosity. Physically, this means that the relaxation time of the rock system is much less than the time variations of the pressure distribution. We can therefore assume that the fracture event happens instantaneously. This leads to a sudden fluid-pressure drop inside the fracture, while the pressure outside the fracture is not affected. The new fluid pressure inside the fracture is determined using the Newton iteration, based on the assumption that the mass of fluid inside the fracture is preserved during the fracture event. The iterative scheme is,

[32] 1. Adjust math formula.

[33] 2. Solve the stationary equation (1) with the new pressure profile.

[34] 3. Calculate the new fracture volume, math formula and

[35] 4. If math formula, stop the iteration process. If not, return to step 1.

[36] Here math formulais the equation of state of the fluid, which expresses the fluid-mass density as a function of the temperature and the pressure.

3.3. The Algorithm

[37] We end this theory section with a brief pseudo code of the numerical solution strategy:

[38] 1. Solve the pressure equation, equation (9), for the current time step.

[39] 2. Solve the stationary stress equation, equation (1), with the new pressure profile.

[40] 3. Check the fracture criteria for each volume element.

[41] 4. If no fracturing occurs:

[42] (a) Calculate the new fracture volumes. Update math formula, math formula, and math formula;

[43] (b) Calculate a new time step from the error estimate, and return to step 1.

[44] 5. If fracturing occurs,

[45] (a) Set math formula for the fractured volume elements.

[46] (b) Find the new fluid pressure and the volume of each fracture using the iterative scheme in section 3.2. Then update math formula, math formula, and math formula.

[47] (c) Calculate a new time step from the error estimate, and return to step 1. Use the updated pressure profile as the initial pressure for the next time step.

4. Model Specifications

[48] Next, we apply this fracturing model to the study of the geomechanical stress interactions between two injection points during the hydrofracking of a rock system with low permeability.

[49] The elasticity parameters of the system are math formula GPa and math formulaGPa. The mass density of the fluid-rock system is math formula kg m−3. The effective stress coefficient is math formula and the fluid compressibility is math formula math formula. The viscosity of the fluid, the porosity, and the permeability of the rock are math formula Pa s, math formula, and math formula mD, respectively. The viscosity and the compressibility of the fluid correspond to that of pressurized water (in the liquid phase) at a temperature of ∼413 K [Bundschuh and Suárez Arriaga, 2010]. These poroelastic parameters are collected from the technical data of the Los Humeros geothermal field [Bundschuh et al., 2010]. The permeability, math formula, inside the fractures is four orders of magnitude larger than the rock permeability.

[50] We represent the reservoir as a 2-D discretized system. To solve the pressure and stress equations, i.e.,equations (9) and (1), we use a quadratic grid, as illustrated in Figure 1c, with a grid size of math formula m. The dimensions of the system range from math formula m, depending on the separation of the injection points.

[51] The simulation starts with two injections points separated by a distance, L. Initially, the system contains no fractures. The elasticity parameters of the injection elements are equal to zero. The mean values for the cohesion, c, and the critical tensile stress, math formula, are 1 MPa, and the internal friction angle is 40°. These values result in tensile and shear failure for borehole pressures typically of the order of 1–8 MPa, which are in agreement with the critical pressure values found experimentally [Evans et al., 2005b]. We consider systems with a Weibull modulus of math formula (Figure 1d). The injection points are placed along the y-axis as illustrated inFigure 1.

[52] In the present paper, our main aim is to investigate how the stress interactions between two injection points in combination with disorder influences the hydrofracking process. To isolate this effect, we have disregarded the gravitational force in our numerical implementation.

5. Results and Discussion

[53] Equation (3) shows that geomechanical stresses arise from spatial variations of the displacement field. Large spatial variations result in large strain and stress fields. During hydrofracking, the stress field is largest close to the injection point and relaxes toward zero further away from the point (in the absence of gravity). Far from the injection point, the displacement field is equal to zero. The length scale over which the displacement field relaxes toward a constant vector field is referred to as the relaxation length. If two injection points are separated by a distance less than twice the relaxation length, the displacement field around each point is influenced by the presence of the neighboring point. This leads to effective stress interactions between the two points.

[54] A consequence of the geomechanical stress interaction is that the strain energy increases in the area between the two points. Figures 3a and 3b show the elastic energy density, math formula, stored in the elements close to the two injection points when the borehole pressure is 5 MPa. In Figure 3a, the injection points are separated by 15 m. In this case, there is significant stress interaction between the two points, which causes the elastic energy density to be largest in the region between the points. The rock here is under particularly strong tensile stress. In Figure 3b, the points are separated by 65 m. For such a large separation, the interactions between the points becomes negligible, and the elastic energy density is equally distributed around each of the two injection points.

Figure 3.

(a) Elastic energy density of a system containing two injection points separated by a distance of 15 m. The borehole pressure is 5 MPa and the injection rate is 10−4 m3 s−1. (b) Elastic energy density of a system containing two injection points separated by a distance of 65 m. The borehole pressure is 5 MPa and the injection rate is 10−4 m3 s−1. Note that the rotational symmetry of the system is broken because the injection element is quadratic, i.e., the rotational symmetry around the injection point is reduced to quadratic symmetry. This is why the strain energy in Figure 3b is not isotropically distributed around each injection point.

[55] In homogeneous materials under tensile stress, a fracture propagates along the direction that causes the largest strain energy-release rate. During hydrofracking, we therefore expect the stress interaction between two injection points to mediate an effective force on the fractures that are created close to one of the points. This effective fracture force is expected to drive the fractures toward the neighboring point because this leads to the highest release rate of elastic energy. In heterogeneous materials, this effective force is accompanied by disorder effects. Whether the fracturing process is disorder-driven or effective force-driven, i.e., as the dominant fracturing mechanism, depends on the degree of disorder.

[56] Let us, at this point, briefly discuss the effect of the gravitational force. The gravitational force yields a large compressive stress of the order −25 MPa per distance of 1000 m beneath the surface, which is approximately one order of magnitude larger than the tensile stresses produced by the injection of fluid. The effect of the gravitational force is to align the fractures along g. Thus, there are two distinct forces acting on the fractures: one caused by the gravitational field that drives the fractures in the vertical direction, and one arising from the geomechanical stress interaction that drives the fractures toward neighboring injection points. The effect of the gravitational field is well known. In contrast, a thorough investigation of how the stress interactions between the injection points in combination with disorder influences the hydrofracking process is lacking.

[57] To investigate how the stress interactions influence the fracturing process of heterogeneous systems, we calculate the ensemble-averaged propagation direction, math formula. The vector, math formula, is a unit vector that denotes the initial propagation direction for a fracture created from one of the two injection points. By definition, math formula points toward the neighboring point. The value of nis a function of the microstate of the system. In other words, its value depends on the distribution of the local material strengths and the heterogeneity of the system. To map the disorder dependency of the fracturing process, we ensemble-average,n, by averaging over several microstates. A nonvanishing math formula implies that a fracture has a larger probability of propagating along this direction than in the other directions. In this case, the fracturing process is strongly influenced by the stress interaction and the location of the neighboring injection point. In contrast, a vanishing math formula implies an isotropic distribution of n, and the governing fracturing mechanism is the disorder of the system. Figure 4a shows math formula as a function of the separation length, L, of the two injection points. The injection points are placed along the y-axis. math formula is not shown because it is equal to zero for the systems we consider. As expected, math formula decreases as the separation becomes larger. This is because of a weaker interaction between the two injection points, which results in a weaker effective fracture force and a reduced probability for the fracture to propagate toward the neighboring point. How quickly math formula decays depends primarily on the degree of disorder. For a Weibull modulus of 30, there exists an effective attraction toward the neighboring point for a separation <45 m. We refer to this separation as the critical correlation length. If the Weibull modulus is 5, which corresponds to a highly disordered system, the critical correlation length decreases to ∼30 m. For separations less than the critical correlation length, the fracturing process around each injection point is strongly correlated with the position of the neighboring point, and the fracturing process is governed by the effective fracture force. For separations larger than the critical correlation length, the disorder effects dominate the fracturing process.

Figure 4.

(a) The ensemble average, math formula, as a function of the separation of two injection points. math formula is studied for three systems with different degrees of disorder, with Weibull moduli, m, of 5, 15, 30. The injection rate is 2 × 10−4 m3 s−1. The figure illustrates how strongly the fracturing process around each point is influenced by the neighboring point and how this correlation depends on the separation between the points and the degree of disorder in the system. (b) The ensemble average, math formula, for the injection rates math formula m3 s−1. The separation between the two injection points is 15 m and the Weibull modulus is 30. In both plots, the lines are guides to the eye and the ensemble average is obtained by averaging over 100 statistical disorder realizations of the system.

[58] Figure 4b shows math formula as a function of the injection rate for a system in which the Weibull modulus is 30, and the two injection points are separated by 15 m. A decreasing injection rate results in a larger math formulavalue. This is because of a stronger stress interaction between the two points. The stronger interaction arises because of a slower pressure build-up at the injection points, which leads to a longer time interval before the fracturing appears compared with a system with a higher injection rate. The pressure therefore diffuses a larger distance into the rock medium before the fracturing appears, which results in a larger relaxation length of the displacement field. The larger relaxation length yields a larger stress interaction between the points.

[59] The effective fracture force enhances the chance to create a connecting fracture network between two injection points. Figure 5a shows how many times a number of fracturing processes creates a connecting fracture between the two injection points as a function of the separation for systems with different degrees of disorder. Two examples of a connecting fracture are shown in Figures 5b and 5c for systems with Weibull moduli of 5 and 30, respectively. In the typical time evolution of such a fracture, the fracture first connects the two points before a new fracture arm is created from each point. In addition, for a system with a high Weibull modulus (Figure 5c), the fracture is most likely to show the same character as a fracture in a homogeneous system, i.e., a straight line fracture, while the fractures in highly disordered systems (Figure 5c) propagate more randomly. The ability to create a connecting fracture network is crucial for creating a geothermal system, such as an Enhanced Geothermal System (EGS), as well as in the exploitation of unconventional hydrocarbon resources. In the case of hydrofracking, the effect of the stress interaction is important. During hydrofracking, the perforation zones ideally are completely uncorrelated to create long and deep cracks that connect a large area of the reservoir to the well. If the zones are spaced by a distance of less than the critical correlation length, the stress interactions lead to a reduced hydraulic stimulation of the system. We therefore believe that our results can be used in the optimization of the hydrofracking process.

Figure 5.

(a) The histogram shows how many times the fracturing process creates a connecting fracture between the two injection elements. For each length, L, and Weibull modulus, m, 10 fracturing events are simulated. The figure illustrates how the effective fracture force enhances the connectivity between the two injection points. (b) The fluid pressure along a connecting fracture between two injection points in a system with a Weibull modulus of 5. (c) The fluid pressure along a connecting fracture between two injection points in a system with a Weibull modulus of 30. In Figures 5b and 5c, the injection points are separated by 25 m.

6. Conclusions

[60] In this paper, we developed a model of hydrofracking and applied the formalism to the study of how geomechanical stress interactions between two injection points influence the fracturing process. We found that when the separations between the two injection points are less than a critical correlation length, the fracturing process around each injection point is strongly correlated with the position of the neighboring point. The magnitude of the critical correlation length depends on the degree of heterogeneity of the rock. For weakly disordered systems, the correlation length can be as large as 45 m, and for highly disordered rock systems, it reduces to ∼30 m. In the strongly correlated regime, there exists an effective fracture force that drives the fractures toward the neighboring injection point. An important observation in this work is that the fracture force reduces the effectiveness of the hydraulic stimulation if the injection points are separated by a distance less than the critical correlation length.

Acknowledgment

[61] We are grateful to Ø. Pettersen at Uni CIPR for stimulating discussions and comments on the manuscript.

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