Mechanics of fluid-driven fracture growth in naturally fractured reservoirs with simple network geometries

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114,, doi: /2009jb006548, 2009 Mechanics of fluid-driven fracture growth in naturally fractured reservoirs with simple network geometries Xi Zhang, 1 Robert G. Jeffrey, 1 and Marc Thiercelin 2 Received 17 April 2009; revised 2 September 2009; accepted 23 September 2009; published 30 December [1] A numerical model has been developed for fluid-driven opening mode fracture growth in a naturally fractured formation. The rock formation contains discrete deformable fractures, which are initially closed but conductive because of their preexisting apertures. Fluid flow that develops along fractures depends on fracture geometry defined by preexisting aperture distribution, offsets along a fracture path, and intersections of two or more fractures. The model couples fluid flow, elastic deformation, and frictional sliding to obtain the solution, which depends on the competition between fractures for permeability enhancement. The fractures can be opened by fluid pressure that exceeds the normal stress acting on them and by interactions with intersecting closed fractures experiencing Coulombtype frictional slip. The Newtonian fluid is assumed to flow through the conductive fractures according to a lubrication equation that relates the cube of an equivalent hydraulic aperture to fracture conductivity. The rock material is assumed to be impermeable and elastic. This paper provides the governing equations for the multiple fracture systems and the solution methods used. Flow distribution and fracture growth in conductive fracture sets are simulated for a range of geometric arrangements and hydraulic properties. Numerical results show that elastic interaction between fracture branches plays a controlling role in fluid migration, although initial apertures can give rise to a preferential fluid flow direction during the early stage. In the presence of offsets, fracture segments subject to strong compression are difficult to open hydraulically, and their resulting smaller permeability can increase overall upstream fracture pressure and opening. The patterns of fluid flow become more complicated if fractures intersect each other. A portion of injected fluid is lost into closed empty fractures that cut across the main hydraulic fracture, and this delays the pressure increases required for fracture growth past the crosscutting fracture. The nonlinear fluid loss rate depends on the geometric complexities of the fracture sets and on the fluid viscosity. Sometimes fracture growth can be accelerated by the fast fluid transport along an intersected, relatively conductive natural fracture. Citation: Zhang, X., R. G. Jeffrey, and M. Thiercelin (2009), Mechanics of fluid-driven fracture growth in naturally fractured reservoirs with simple network geometries, J. Geophys. Res., 114,, doi: /2009jb Introduction 1 CSIRO Petroleum Resources, Ian Wark Lab, Clayton, Victoria, Australia. 2 Schlumberger RTC Unconventional Gas, Addison, Texas, USA. Copyright 2009 by the American Geophysical Union /09/2009JB006548$09.00 [2] The movement of hydrocarbons, water, and other fluids in naturally fractured reservoirs or conventional reservoirs with significant fracture permeability remains a topic of interest for academic and applied research. For example, fracture porosity is typically a small fraction of the total reservoir pore volume, but the presence of conductive fractures can result in significant and highly localized flow [Nelson, 1985; Warpinski and Teufel, 1987; Sanderson and Zhang, 1999; Berkowitz, 2002; Germanovich and Astakhov, 2004; Gale et al., 2007]. Researchers with diverse backgrounds have undertaken a broad range of theoretical, numerical, laboratory, and field investigations of both uncoupled and coupled flow in fractures. However, many details on how injection-forced fluids flow through the fractures in a rock mass and how the fracture paths are affected by full coupling of fluid flow and mechanical fracture opening and shearing deformation have not been completely analyzed. [3] During stimulation of naturally fractured reservoirs, the pressure required to create and extend a hydraulic fracture by fracturing fluid injected into a wellbore is strongly dependent on such a coupled flow and fracturing process. The resulting fracture and fluid flow pathways are often more complicated than assumed by fracture design and flow models owing to the dual nature of existing conductive fractures that can act as both primary flow pathways and barriers to fluid flow because of interactions. In one field case, because of channeling of gas and fluid in preexisting fractures, early fluid breakthrough was observed in water flooding operations of a fractured reservoir [Belayneh et al., 1of16

2 2006]. It is also commonly observed in field applications that stimulation treatments typically result in higher injection pressures compared to treatments carried out in unfractured formations [Warpinski and Teufel, 1987; Jeffrey et al., 1987; Daneshy, 2003]. Laboratory experiments have shown that some of the fluid injected was found to invade natural fractures close to the wellbore, depending on their permeability, regardless of the imposed stress conditions and fracture propagation direction [Beugelsdijk et al., 2000; de Pater and Beugelsdijk, 2005]. In highly naturally fractured rock, fractures mapped after mining show that a large part of the hydraulic fracture path is found to follow and widen preexisting fractures rather than to create new fractures [Warpinski and Teufel, 1987; Pollard and Aydin, 1988; van As and Jeffrey, 2002; Jeffrey et al., 2009b]. New fracture pathways are created to provide linkage between the otherwise disconnected natural fractures. Recent success in stimulation of natural gas production from the Barnett Shale in Texas demonstrates the practical implications of recognizing and exploiting the interaction of hydraulic fractures and natural fractures [Palmer et al., 2007]. Generally speaking, movement of fluid and associated pressure changes in the natural fracture network can result in disturbed fracture zones of enhanced conductivity connecting to a main propped hydraulic fracture, in contrast to a single conductive fracture produced by hydraulic fracturing of reservoirs containing few natural fractures. Preexisting conductive natural fractures in a rock mass can cause a hydraulic fracture or other naturally generated veins, joints, or dikes that extend under a fluid pressure gradient to change direction, offset, and branch. [4] It has long been recognized that natural hydraulic fracturing is an important mechanism that explains the origin and formation of many joints and natural fractures [Clark, 1949; Hubbert and Willis, 1957; Secor, 1965; Berchenko et al., 1997]. The source of subsurface fluids for natural hydraulic fracture growth includes fluid exsolved from magma, fluid contained in rock pores, and mantle-derived water, magma, and carbon dioxide [Hickman et al., 1995]. [5] The importance of characterizing natural fracture sets lies in the fact that the flow of injected fluid through them depends on their size and orientation with respect to the in situ stress field. Pressurized fractures that are near one another will interact, changing their respective conductivities. For a given fluid source, the structural elements of a fracture network, such as offsets and intersections, can affect the formation and evolution of fracture branches and flow patterns. Some intersection points can act like a switch, with approaching fluid observed to slow and then turn into one or more branches [Glass et al., 2003; de Pater and Beugelsdijk, 2005]. Along offsets in the fracture channel, which are typically short in length and subject to higher compressive stress, the pressure-induced increase in fracture conductivity is limited [Zhang et al., 2007, 2008; Jeffrey et al., 2009a]. A pinch-swell opening distribution can develop along the fracture pathways that contain offsets and branches. The resulting high pressure required to force fluid flow through such fracture systems can sometimes lead to new fracture initiation from small flaws [Zhang and Jeffrey, 2008]. The distribution of apertures among multiple fractures can be affected, and flow rates into one region may become favored over others [Zhang and Sanderson, 1995]. Under other circumstances, there is a preferential pathway for fluid flow because of the coupling between fracture conductivity and pressure that exists when viscous dissipation associated with the fluid flow becomes important. [6] Many numerical studies on fluid flow through fracture networks have been conducted to characterize flow behavior in fractured geological media [e.g., Ferry, 1994; Berkowitz, 2002, and references therein]. Although most of these studies have focused on the assessment of fluid conductivity of joint networks [Witherspoon and Long, 1987; Brown and Bruhn, 1998], complete models should take into account the pressure-induced changes to fracture deformation and permeability in space and time. For example, the mechanics of fracture sets subjected to fluid pressure or remote loading has been numerically studied [Olson, 1993; Renshaw and Pollard, 1994; Taylor et al., 1999; Zhang and Jeffrey, 2008]. We have previously found that models without viscous fluid flow are poorly suited to capture features associated with coupled flow through fractured rocks, such as development of preferential flow patterns and nonlinear fluid leak-off [Zhang and Jeffrey, 2008]. Considerable effort has been devoted to the development of models that can account for the coupling of fracture deformation and fluid pressure, although some studies adopted a uniform pressure distribution in fractures, which is equivalent to using an inviscid fluid or assuming zero viscous dissipation. The use of the local cubic law for fluid flow in natural fractures is the standard approach for relating quantitatively the pressure change to fracture deformation [e.g., Brown, 1987; Zimmerman and Bodvarsson, 1996; Zhang and Jeffrey, 2008]. However, the modeling of a rock fracture as a pair of smooth, parallel plates is not adequate for the description of flow between rough fracture surfaces. In this sense, the cubic law hydraulic aperture should be understood as the equivalent parallel plate aperture that produces a conductivity equal to the real fracture with rough and nonparallel walls [Renshaw, 1995; Zimmerman and Bodvarsson, 1996]. Additionally, even for a closed fracture, the equivalent fluid conductivity is expected to change with the fluid pressure since contact deformation is a function of effective normal stress [Brown and Bruhn, 1998]. This pressure-induced dilatancy and the associated increase in conductivity are important in increasing flow through some segments of a hydraulic fracture path that may be subject to significant compressive contact stress. Also, any reduction in effective contact stress may result in fracture sliding, which can lead to local stress variations and slipinduced fracture dilation, which can in turn change the overall conductivity of fracture networks. [7] Apart from using an appropriate viscous fluid flow model, the study carried out here integrates some geometric features of deformable discrete fracture sets with stress analyses. As stated above, the fluid flux distributions are controlled not only by the physical characteristics of a single fracture but also by geometric relationships between it and neighboring fractures. When a propagating hydraulic fracture intersects natural fractures in the network, part of the total flux may then enter and follow these existing planes of conductivity. As a consequence of flux splitting at the intersections and development of preferential flow pathways, hydraulic fracture growth and flow patterns in fractured rocks can be radically different from those in unfractured rocks [e.g., Warpinski and Teufel, 1987; Berkowitz, 2002; Zhang and Jeffrey, 2008]. Basically, not all fractures play an equal 2of16

3 flow through the discrete fracture networks studied. In contrast to the previous studies, in which the preexisting fractures are assumed to be impermeable, here we specifically address the effect of initially closed but conductive fractures. The problem formulation and the governing equations are presented in section 2. Subsequently, we present simulations of 2-D networks containing multiple fracture branches, offsets, and intersections under various remote stress conditions to assess fluid pressure gradients, flux distributions, and resulting conductive fracture patterns. It should be emphasized that the study is within the framework of low-permeability fractured rocks such as the Barnett Shale, in which the porosity and the permeability of the rock matrix are assumed to play a negligible role in fluid flow. The authors acknowledge the limitation of the model for higherpermeability reservoirs. Figure 1. Evolution of natural fracture opening. The initial aperture w 0 along a closed preexisting natural fracture corresponds to its residual conductivity. It is equal to the effective aperture for the parallel plate model. With increasing fluid pressure, the hydraulic aperture w will be slightly changed due to microstructural change in the natural fracture, although the fracture is still closed and carries some contact stresses. In the end, the fracture would be opened mechanically as the fluid pressure exceeds the normal stress acting across the fracture. Variable w denotes the mechanical opening induced by pressurized fluids, and the hydraulic aperture is maintained unchanged after the contact is lost. role in conducting fluids in the subsurface. Detailed analysis of pressure and opening along hydraulic fractures that contain offsets and branches requires use of a coupled model to determine the subtle stress, deformation, and conductivity variations. The coupling between elastic deformation and fluid pressure is of primary importance for flux redistribution. We find that the interactions between fractures and flow pathways in a fracture network have not previously been fully analyzed using a coupled approach, especially in the presence of aperture differences, offsets, and intersections. [8] In this paper, the rigorous model developed by Zhang et al. [2007, 2008] and Zhang and Jeffrey [2008] is extended to the cases in which closed natural fractures, intersected by a hydraulic fracture, play an important role in continued fluid 2. Problem Formulation 2.1. Fluid Transport in Closed Natural Fractures [9] Conductive natural fractures have a strong effect on fluid flow in low-permeability rocks [Brown, 1987]. The initial conductivity of a closed natural fracture arises from the fact that its surfaces are rough and mismatched at fine scale. The surfaces can be propped apart by the contacting asperities, and the resulting space or aperture between the surfaces controls the fluid transport properties when the fluid pressure is below the fracture opening pressure. [10] For the closed natural fractures with varying aperture shown in Figure 1, we can define the so-called hydraulic aperture in terms of the actual conductivity [Tsang and Witherspoon, 1981; Brown et al., 1995]. It has been found in previous studies that the cubic law can still provide a firstorder approximation for fluid flow and pressure if we can find a suitable expression for the hydraulic aperture, either by direct testing or perhaps based on detailed computations using 2-D computational fluid dynamics models [Tsang and Witherspoon, 1981; Brown, 1987; Brown et al., 1995; Renshaw, 1995; Zimmerman and Bodvarsson, 1996]. In the calculations cited here, fluid turbulence was assumed to be unimportant when the Reynolds number was sufficiently small (e.g., Re 50 for gently sloping fracture walls [Brown et al., 1995]). For the flow tortuosity associated with contact areas, the hydraulic aperture is associated with the areal fraction of the fracture plane not occupied by the obstructions [Zimmerman and Bodvarsson, 1996]. In other words, the aperture is related to the fracture porosity, although the exact functional form is not unique. The cubic law has also been widely used in reservoir simulation, for example, by Warpinski and Teufel [1987] for closed joints. In addition, for injection-driven fluid flow, a natural fracture can be mechanically opened by fluid pressure at a point where no contact across the fracture faces exists. In this case, the effective hydraulic conductivity is equal to the sum of both hydraulic aperture and mechanical opening since the fracture opening augments the initial hydraulic aperture, as shown in Figure 1. It must be remembered that for smaller aperture relative to fracture roughness, this assumption may underestimate the role of deformation on fluid flow in fractures because of the inherent scale dependence [Brown, 1995]. The cases for extremely narrow fracture aperture are excluded in this model. 3of16

4 [11] The closed natural fracture with irregular asperities considered here is expected to exhibit several characteristic responses to injected fluid, namely, (1) the fracture is thin but long, (2) flow through fracture tortuosity can result from the development of a high fluid pressure gradient, and (3) significant flow in the fracture is channelized, being dominated by high-aperture channels [Brown et al., 1995]. In contrast to the slow seepage process that exists under small pressure gradients, fluid movement in this model is driven by externally generated pressure by an injection pump. The main fluid flow in closed fractures will still follow the conductive fracture path with least pressure loss. [12] Since the effective hydraulic aperture is small (<0.1 mm in most cases) and varies in a gentle slope along the actual natural fracture [Brown et al., 1995] (even though there are small roughness-induced changes in the hydraulic aperture along the natural fracture, the transition associated with this change occurs with a small slope so that the cubic law is still valid), the fracture conductivity is then given as follows [Tsang and Witherspoon, 1981]: k ¼ w2 12 ; ð1þ where k is the fracture permeability and w is the effective hydraulic aperture for the closed natural fracture. [13] Under normal compressive stress, asperity deformation in the elastic regime can be described by contact mechanics. When fluid pressure is increased, the decreasing effective contact stress produces an increase in the fracture aperture by partially recovered elastic deformation associated with the contacted asperities. Models for joint opening and closure have been developed by Walsh and Grosenbaugh [1979] and Brown and Scholz [1986] to relate contact stress changes to the mechanical aperture under normal loading conditions. Recent developments in this area have formulated the aperture change under oblique loading [Walsh, 2003; Zhu and Walsh, 2006]. Of course, shear sliding can also change the fracture conductivity, depending on the details of the sliding and how it reacts with surface roughness and asperity deformation [Olsson and Brown, 1993]. However, shearinduced dilatation is not taken into account here. Generally speaking, the stress- and deformation-induced mechanical aperture contributes to the variation of hydraulic-apertureassociated fluid flow along a closed, rough-walled fracture, as discussed by Witherspoon et al. [1980] and Renshaw [1995]. Renshaw [1995] has shown that the hydraulic aperture and the mechanical aperture are not linearly related to each other, especially for extremely narrow fractures in marble, whose mechanical aperture is less than 10 5 m [Witherspoon et al., 1980]. In this paper, the minimum mechanical aperture of a closed, rough-walled fracture is assumed to be larger than 10 5 m. For the sake of simplicity, a nonlinear spring response is used in our model for the hydraulic aperture increment associated with the changes in fluid pressure that cause mechanical aperture changes: dw=dp f ¼ cw; where p f is the fluid pressure and c is a small constant for characterizing the compliance of a natural fracture with ð2þ respect to pressure change (Pa 1 ) and is assumed to be less than 10 8 Pa 1 for most cases. [14] The governing equations for fluid flow in closed natural fractures are summarized as follows: [15] 1. Continuity of fluid mass can be þ _m ¼ 0; ð3þ where q m is the fluid mass flux and _m is the fluid mass change rate per unit volume of fracture channel, and [16] 2. Darcy s law relates fluid mass flux to pressure gradient q m ¼ r 0k ; where r 0 is the fluid density, s is a distance measured along the fracture length, and m is the fluid dynamic viscosity. Considering equation (1), it follows that the fluid volumetric flux q is expressed as [Tsang and Witherspoon, 1981] q ¼ f : [17] 3. When the fluid is assumed to be incompressible, a change in fluid mass may occur because of the elastic compressibility along the natural fracture in response to pressure variations (i.e., equation (2)). Similar to the case of a porous isotropic medium, the rate of change of fluid mass in a fracture is given by _m ¼ r 0 _w=w: Substituting equations (1), (2), (4), and (6) into equation (3), we w f where c = 1/(12cm). [18] Equation (7) provides the transport equation for fluid flow in closed fractures. It must be mentioned that the current hydraulic aperture w can be obtained by solving the evolution equation (equation (2)) at each time step, starting with an initially assigned hydraulic aperture w 0. The distribution of w 0 can be assigned according to information about the detailed microstructures in the fracture, if available. In this paper, w 0 is assumed to be constant and to vanish at the fracture tip. Substituting equation (2) into equation (7), we can obtain the governing equation in the same form as the lubrication equation (see below) if w is not allowed to change dramatically so as to avoid the formation of shock waves. As the fluid penetrates the natural fractures, the pressure can build up until it exceeds the level of the local confining normal stress arising from far-field stress and elastic deformation. The contact will be lost at these points where sufficient fluid pressure is developed and the natural fracture is opened mechanically. The coexistence of open and closed fluid-conducting fractures can be expected in naturally frac- ð4þ ð5þ ð6þ ð7þ 4of16

5 Figure 2. Geometrical configurations for multiple fracture systems and the global and local coordinate systems in an elastic homogeneous rock mass. Some fractures may propagate as a result of the fluid pressure. tured rock masses. In addition, it should also be remembered that for closed fractures, the fracture surfaces may slip relative to each other under shearing and an associated frictional stress is then generated that must be addressed by the elastic solutions discussed in section Coupled Flow and Deformation Model for Hydraulic Fractures [19] When the fracture is opened by the fluid pressure or the closed natural fracture slides, elastic deformation will give rise to significant stress changes in the rock mass. To account for elastic stress equilibrium for a fracture network, the governing equations that apply should be the same as those given in our previous studies [Zhang et al., 2007, 2008; Zhang and Jeffrey, 2008]. Since we neglect inertial effects, the equations of quasi-static elasticity provide a relationship between the normal w and shear v displacement discontinuities (DDs) of the fracture walls and any applied tractions, including the fluid pressure P f along the fracture. The formulation used is based on a superposition scheme of singular dislocation dipole solutions. The fracture branches are numbered on the basis of the tips in the fracture network, as shown in Figure 2; that is, there are N fractures if there are N tips. Summing up the contributions to the stresses from each fracture, the elasticity equilibrium equations for multiple fractures in the framework of a Cartesian coordinate system are Z s n ðx; tþ s 1 ðþ¼ x XN lr r¼1 0 Z t s ðx; tþ t 1 ðþ¼ x XN lr r¼1 0 ½G 11 ðx; sþws ðþþg 12 ðx; sþvs ðþšds ½G 21 ðx; sþws ðþþg 22 ðx; sþvs ðþšds; where x =(x, y); t is time; ds is an infinitesimal length increment along the fracture; l r is the fracture length with a subscript index r identifying the fracture branch number; s n ð8þ ð9þ is the normal stress, which is equal to P f within the fluid-filled open parts; and t s is the shear stress associated with frictional sliding along the interface. Variables s 1 and t 1 are the normal and shear stresses, respectively, along the fracture direction at location x caused by the far-field stresses. G ij are hypersingular Green s functions whose expressions are given by Zhang et al. [2005]. [20] During commercial hydraulic fracture treatments, the fluid flow velocity V at the fluid front is normally less than 1ms 1 since the hydraulic fracture grows quite slowly. For 2-D flow between parallel plates, the Reynolds number is defined as Re = rv w/m, where r is the fluid density and w is the average fracture aperture. In the extreme, taking water as the injected fluid and w < 0.1 mm near the crack tip or at the fluid front, we have Re =10 2, which is considerably less than the value associated with the onset of turbulent flow. Even for the worst cases, where the fracture channel has a steeply sloping roughness, the disagreement in flow velocity predicted by the lubrication theory and the full Navier-Stokes flow simulations is less than 25% at Re =10 2 [Brown et al., 1995]. However, in actual hydraulic fractures, the surface roughness of an open fracture would be more gently sloping. So the fluid flow in the hydraulic fractures can be taken as laminar. Therefore, we assume that the flow is governed by the lubrication equation [Batchelor, 1967; Witherspoon et al., 1980] " ðw þ wþ ðw þ wþ f m 0 ; where m 0 =12m. It should be noted that the total hydraulic opening is the sum of fracture mechanical opening and hydraulic aperture. [21] Because no fluid loss occurs in the impermeable rock matrix, the sum of the fluid volume in all fractures, including both closed natural fractures and mechanically opened hydraulic ones, must equal the total injected fluid volume. Thus, assuming a constant injection rate, the global mass balance leads to X Z l f 0 ðw þ wþds ¼ Q 0 t; ð11þ where Q 0 is the injection rate and l f is the fluid-filled length of each fracture. [22] The model allows for a fluid lag, which is the part of the fracture between the fluid front l f and the fracture tip l,as shown in Figure 2 [Zhang and Jeffrey, 2008]. Inside the fluid lag, the pressure is zero because the rock matrix is assumed to be impermeable and pore pressure is not included in the model. The fluid front in the fractures, including that in closed natural fractures, can be found in terms of the flux q(l f ) and the opening w(l f ) at the fluid front in the form _l f ¼ ql f ; t =wlf ; t ; ð12þ where the fluid volumetric flux is defined by equation (5) on the basis of Poiseuille s law. Poiseuille s law is a good approximation because the fracture width is normally at least 1 order of magnitude smaller than the length of the fluid 5of16

6 branch, although it does not apply to the sudden change in fracture width at an intersection Fracture Slip Model [23] Frictional sliding occurs when the shear stress reaches the frictional shear strength of the natural fractures. Because frictional sliding is nonlinear, the increment of slip is found by iteratively solving a nonlocal elasticity fracture problem. Therefore, the total slip amount possesses a historical dependency on the stresses and displacement. The slip can cause the fracture to grow in a shearing mode and can give rise to the fracture opening of other fractures it intersects. Of course, slip along closed fractures can initiate or continue after the fracture surface is occupied by the injected fluid. In this paper, the local effective contact stress is defined as the difference between the elastic total stress and the fluid pressure. For a closed fluid-filled fracture portion, the effective stress is reduced as the pressure increases, resulting in a reduced shear strength and, potentially, fracture sliding. [24] The Coulomb frictional law, which is applied locally to the cohesionless but frictional fracture, provides a limit on the frictional stress t s in terms of the normal effective compressive stress s n p f : jt s j ls n p f ; ð13þ where l is the coefficient of friction and l(s n p f ) is the frictional strength. For the dry portion of the fracture, p f =0. [25] The contact behavior of fracture surfaces has three different modes. When the fracture is opened by the injected fluid pressure, the shear strength and the shear stress are zero. For closed surfaces, if the shear stress acting on them is less than the frictional strength l(s n p f ), the fracture surface is in sticking mode and the shear stress is transmitted across it elastically; otherwise, it is in sliding mode. Once full sliding at a specific point begins, the amount of slip cannot be determined by the local stress, but rather, through the elastic equilibrium in a nonlocal form. In addition, yielding frictional fractures can have two different slip directions on the sliding plane. The direction of frictional shear stress is reversed if the slip velocity changes sign. Strictly, the frictional stress t s and the slip velocity _v should meet the following condition for the contact sliding mode: t s _v 0: ð14þ Since the Coulomb frictional law only provides a limit on the frictional shear stress, the sliding status cannot be determined through the governing equations. To determine the correct contact modes, an initial guess is used, and a check is made to ensure that the guess is correct at each time increment. If not, the contact modes are updated on the basis of the calculated stresses and displacements. The new modes are then applied, and stresses and displacements are calculated again. This step is followed by another check and modification, until the contact frictional modes converge at each location. This solution strategy for determining contact modes has been implemented in our computation program [Zhang and Jeffrey, 2006] Auxiliary Conditions and Failure Criterion [26] To carry out an analysis of the responses during fluid injection into a fractured rock mass, the initial conductivity or hydraulic aperture has to be assigned to each fracture in the fracture network. Initially, all preexisting fractures are assumed to be evacuated, and the rock mass is assumed to be in static equilibrium with all tractions applied to it. At the injection point, the fluid flux is equal to the injection rate, i.e., qð0; tþ ¼ Q 0 : ð15þ At the fracture tip, the opening and shearing DDs are zero, i.e., wl ðþ¼vl ðþ¼0: ð16þ [27] When a hydraulic fracture intersects a natural fracture, the continuity of fluid flux must be met at the intersection. In addition, the induced pressure loss at the intersection is assumed to be negligible. Therefore, the fluid pressure should be the same for each fracture branch connected at the intersection [Zhang et al., 2007]. The coalescence of two intersecting fractures can be realized numerically through the algorithm given by Zhang et al. [2007]. Although the intersection-induced stress singularity variations have not been captured explicitly, the extra fine mesh near the intersection provides a rather good approximation of both stress and displacement when compared with finite element method solutions. [28] The magnitudes of mode I and II stress intensity factors at the fracture tip are determined by the displacement correlation method. In order to determine the propagation direction, the commonly used failure criterion is adopted, in which fracture propagation is along the direction of maximum tensile hoop stress [Erdogan and Sih, 1963]. Specifically, if both mode I and II stress intensity factors are known, the fracture propagation direction is determined by solving K I sin Q þ K II ð3 cos Q 1Þ ¼ 0; ð17þ where Q is the deflection angle from the current fracture line and K I and K II are mode I and II stress intensity factors, respectively. [29] The condition for quasi-static fracture growth is the same as used for dike propagation and fluid-driven or hydraulic fracture growth [Lister, 1990; Rubin, 1995; Detournay, 2004] and differs from that for subcritical crack growth in cases of long-term loading, as used by Olson [1993] and Savalli and Engelder [2005]. Additionally, in contrast to pure mode I crack growth, the mixed stress intensity factor along the propagation direction is required to meet the following condition at the onset of crack growth [Erdogan and Sih, 1963]: cos Q 2 where K c is the rock toughness. K I cos 2 Q K II sin Q ¼ K c ; ð18þ 6of16

7 smaller elements near the tip. These elements are one third the size of the standard DD elements. When the fracture grows, the mesh is adjusted so that only six of these fine elements are ever used. The traction and opening displacements must be mapped to the new mesh after each growth step. In addition, at the fracture tip, a singular element (square root shape function) is used. [32] Details of the numerical implementation of the frictional contact mode selection and the division of the fluid flux at fracture intersections can be found in our previous paper [Zhang and Jeffrey, 2006]. At intersections, we enforce the conditions that the fluid pressure is equal for all intersecting branches and that the fluxes into and out of the intersection are the same. Figure 3. Variations of influxes for three fractures with different initial apertures, which are connected at the borehole. The apertures for fractures 1, 2, and 3 are 0.01, 0.05, and 0.1 mm, respectively. The angle between fractures 2 and 3is Description of Numerical Methods [30] The numerical method for finding solutions to the previously formulated problem is based on an implicit algorithm, as previously described by Zhang et al. [2005]. Since the fracture interaction involves crack nucleation, growth, and coalescence, the numerical model was validated for complex crack geometries through comparisons with available results for both dry- and fluid-pressurized fractures [Zhang and Jeffrey, 2006; Zhang et al., 2007, 2008]. For the elasticity equations, we use a numerical method that employs constant strength DD elements [Crouch and Starfield, 1990]. Discretization along both the hydraulic fractures and the natural fracture is required. The natural fractures are modeled as initially closed with assigned hydraulic apertures, and they can carry frictional shear stress and compressive normal contact stress. More importantly, they can turn into opening mode hydraulic fractures if the pressure increases sufficiently. Also, an initial hydraulic aperture is assigned to each node so that its distribution along each fracture is smooth and without rapid changes. The hydraulic aperture in the model is typically set to zero at the fracture tip. Both the lubrication equation for hydraulic fractures (equation (9)) and the transport equation for fluid pressure in closed conductive fractures (equation (6)) are solved for each time increment by means of the finite difference method (FDM). Also, for closed fractures, the evolution of hydraulic aperture based on equation (2) is incorporated into the FDM solution method. The finalsolution must simultaneously satisfy equations (8) (14) and boundary conditions (15) (18). [31] The numerical method uses a fixed size element length D for regular elements and crack increments and a small time step that is adjusted on the basis of fluid front advancement. The time step is also adjusted to ensure convergence of the coupled fluid and elastic solutions. When the propagation criterion is satisfied, the fracture tip is extended along the selected propagation direction by an increment D. To improve the accuracy of calculating the stress intensity factors, a mesh adaptive scheme is employed by using six 3. Numerical Results 3.1. In Situ Stresses [33] We start by modeling fluid injection into a set of three natural fractures connected at a borehole, as shown in the inset of Figure 3. The time t is recorded from the onset of fluid injection. These three preexisting fractures are of different initial effective hydraulic apertures, and their ability to accept fluid is accounted for in the calculation. The vertical and horizontal remote stresses are 10 and 15 MPa, respectively. The two fractures perpendicular to the least principal stress have a lower conductivity than the inclined one, which has a hydraulic aperture w 0 = 0.1 mm. The left and right horizontal fractures have effective hydraulic apertures of 0.05 mm and 0.01 mm, respectively. The inclined fracture is at an angle a = 60 with respect to the left horizontal fracture. The material and fluid properties used are as listed in Table 1, if not specified otherwise. In response to the increasing pressure, the fluid penetrates into the three fractures simultaneously. Figure 3 shows the evolution of the inflow rates. The inclined fracture with the largest initial conductivity accepts more fluid at the beginning than the other two fractures. However, it is in a less favorable direction to be opened by the pressure because a larger confining stress acts across it. With continued fluid injection into the system, the fluid pressure at the borehole keeps increasing. The fluid pressure eventually reaches a level that will open up the two horizontal fractures because of the lower compressive stress acting across them. The mechanically opened hydraulic fractures then become much more conductive as their fracture opening increases further with fracture growth (movement of the fracture opening). The direct result of this enhancement of fracture conductivity is the rapid increase in fluid influx for these two fractures. This varying trend is reflected in Figure 3 as the influxes into them take most of the injected fluid eventually. On the other hand, the left horizontal fracture, with Table 1. Material Constants Used in the Numerical Calculations Material Property Value for Granite E (GPa) 55 n 0.22 K c (MPa m 1/2 ) 1.0 l 0.8 c (Pa 1 ) 10 8 Q 0 (m 2 s 1 ) m (Pa s) of16

8 Figure 4. Schematic of fluid injection into a parallelogram fracture system with high-conductivity segments. The lowconductivity segments are assumed to be impermeable. The lengths of two adjacent segments are c and d, and their angle is denoted by b. The injection point is located at the middle of one longer fracture segment or step, which is at angle a with respect to the y axis. For simplicity, there are five steps for this particular fracture geometry, which is the geometry used for the results shown in Figures 5 and 6. w 0 = 0.05 mm, is more conductive than the right one, with w 0 = 0.01 mm, during the simulation time, although the influx of the right one still follows an increasing trend. Therefore, fracture opening and sliding are coupled through stress equilibrium, and the deformation-induced flow redistributions depend on both the initial hydraulic aperture and the fracture geometric arrangement with respect to remote stress fields Fracture Offsets [34] As another example, we consider a fracture network with some high-conductivity segments regularly aligned in the steplike manner shown in Figure 4. Such offset fracture paths have been found to form as hydraulic fractures grow through naturally fractured rocks, typically with a small or zero angle a [Daneshy et al., 2004; Jeffrey et al., 2009a]. Such fractures may also form as a hydraulic fracture alternately follows bedding planes and steps through connecting natural fractures that provide high-permeability pathways in the rock. Normally, the fracture spacing in one direction is much larger than the other, as observed in outcrops and at hydraulic fracture mine through sites [Jeffrey et al., 2009b]. All the short segments, called offsets, are assumed to have equal length for modeling purposes, as shown in Figure 4. Likewise, the longer segments, or steps, are of equal length. The step is assigned a length of c = 4 m and is oriented at an angle a with respect to the y axis. The offset connecting these steps has a length of d and is oriented at an angle b with respect to the step orientation. In the calculations, all steps and offsets are of the same initial conductivity (i.e., the hydraulic aperture w 0 = 0.05 mm), and the remote stresses are 15 and 10 MPa horizontally and vertically, respectively. A discontinuous, segmented fracture geometry in the absence of offsets has also been considered by other researchers [e.g., Segall and Pollard, 1980]. In contrast to their work, our analysis includes viscous fluid flow in a fully coupled model. As shown in Figure 4, a borehole into which the fluid is injected is located at the center of the middle step. Of course, the geometric parameters, such as angles, lengths, and spacing, determine the fracture network shape. [35] Figure 5 shows the effects of fracture geometry on the injection pressure under the in situ stress field shown in Figure 4. In Figure 5a, a =0, and in Figure 5b, b =60 for all cases. An increase in either a or b makes fluid penetration along the offset more difficult because the compressive stress acting on the offset portion is increased when a or b increases. An increase in a also increases the compressive stress acting on the steps. The direct result of the increased difficulty in fracture growth along the preexisting channel is an increase of injection pressure, as shown in Figure 5. In general, the longer the offset is, the higher the injection pressure peak for those cases studied here. The injection pressure overall displays a decreasing trend with time, but it fluctuates as the fracture front grows through the steps and offsets, as shown in Figure 5a. This response can be explained by looking at the overall response of the fracture as described by Jeffrey et al. [2009a], where higher pressure and wider fracture opening are generated along the segment Figure 5. Time-dependent variations of borehole pressure for different offset sizes at w 0 = 0.05 mm: variations of angles (a) b and (b) a. 8of16

9 Figure 6. Fracture mechanical opening profiles at the onset of fracture propagation: (a) a =0,b = 60, d = 0.4 m and (b) a = 30, b = 60, d = 0.4 m. upstream of the offset. This situation results in slower fracture growth. In addition, the oscillating pressure variation shown in Figure 5 reflects the difficulty for the fracture opening front to penetrate through successive offsets along the defined path. The slower advance of the opening front increases the wavelength of the pressure variations while the resistance to growth through the offsets increases their amplitude. These responses differ from each other because of the coupling between pressure and opening. [36] For some cases, an extremely high pressure is required to open the offsets, and the flow through them is solely dependent on the narrow preexisting hydraulic aperture. Therefore, the fluid flux through these offsets is much less than the injection rate and the pressure gradient across the offset rapidly increases. The higher pressure and fracture width on the upstream side of such a restriction will be maintained until the offset is opened sufficiently to allow the fluid to pass through it readily. However, a much higher internal pressure implies that the fracture might well choose alternative paths to bypass or avoid the offsets through a process of new fracture initiation at preexisting secondary flaws instead of following the natural fractures. It must be mentioned that some cases studied here are chosen for the purpose of demonstration of offset-induced pressure enhancement and, especially if 3-D effects are considered, they would not be physically possible. The results provided here for a 2-D geometry are expected to produce the upper bound pressure and opening responses because the fluid must flow through the predefined offset rather than around it, as might well happen in a 3-D case. [37] Fracture opening profiles are displayed in Figure 6 for two stepped fracture geometries: a =0, b =60, d = 0.4 m (Figure 6a), and a =30, b =60, d = 0.4 m (Figure 6b). The offset fracture path is shown by a dotted line and the bars, drawn perpendicular to this path, denote the opening amount based on the scale shown in Figure 6. Figures 6a and 6b correspond to the time when the fracture has grown to the end of the natural fracture and fracture propagation from the ends is about to occur. It is clear that the fracture opening along the offsets is restricted. If the restriction at the offset is severe, fluid flow will primarily depend on the small values of hydraulic aperture w. The restricted width at offsets, when experienced during a commercial fracture treatment, can result in proppant bridging. Although the opening distribution undergoes a large change around the offset, the overall fracture opening profile is similar to that for a single fracture without offsets subjected to the higher net pressure. A consideration to simplify the interactions between stepping segments in the absence of offsets has been taken by Germanovich and Astakhov [2004]. However, the situations here seem more complicated. Indeed, the offsets nearest to the fracture center are opened sufficiently so that they no longer act as a significant restriction to fluid flows, as discussed by Jeffrey et al. [2009a]. It is also noted that the highpressure upstream of the offsets for the slant fracture (a = 30 ) can result in a very large fracture volume, as shown in Figure 6b, and thus generate much slower fracture growth (i.e., the slower advance of the fracture opening front) Fluid Loss [38] In a naturally fractured rock, fluid can invade conductive natural fractures that are intersected by a newly created opening mode hydraulic fracture. The fluid loss into these conductive natural fractures may significantly affect the main hydraulic fracture growth. In general, when designing fracture treatments in naturally fractured reservoirs, this process must be taken into account [Warpinski and Teufel, 1987; Jeffrey et al., 1987; Thiercelin and Makkhyu, 2007]. To quantify the volume for fluid loss during injection, we consider a special fracture geometry, as shown in Figure 7, in Figure 7. Schematic of the intersection of a preexisting kinked natural fracture by a propagating hydraulic fracture, used for calculating the fluid loss into a natural fracture. Subsequently, the shorter branch of the kinked fracture along the horizontal direction propagates straight ahead, while some fluid continues to invade the infinite inclined natural fracture branch. The angle between the longer branch and the y axis is denoted by a. 9of16

10 Figure 8. Variations of fluid pressure at the intersection with time, as well as the position for the fluid front in the infinite vertical branch. The narrow vertical branch conductivity is fixed. The fluid penetration into the vertical branch is similar to the process of fluid filling into a rigid empty pipe under constant injection pressure. which a fluid-driven fracture propagates toward and intersects a preexisting branched fracture. This branched natural fracture is arranged so that the fluid-driven fracture is allowed to continue propagating in its initial direction after intersection. The length of the angled conductive fracture branch is long enough so that it can be assumed to be infinite. The resulting geometry in Figure 7 allows consideration of the coupled process of fluid leakage into a natural fracture from a propagating hydraulic fracture. The time-dependent flux for fluid leakage is thus a function of both the geometric parameter defined by the angle a, the initial hydraulic aperture w 0, and the far-field stresses. The fluid loss problem studied here is sometimes referred to as a nonlinear leak-off process because its rate is dependent on how pressure develops in the natural fracture and the associated change of the natural fracture s conductivity. Such a coupled leak-off process is an important feature of hydraulic fracture growth in naturally fractured rocks. For the cases considered here, a viscosity m = Pa s, representative of water, is used, and the remote stresses used are given in Figure 7. The injection point is located at a distance of 1 m from the intersection point, and the initial length of the shorter branch of the branched fracture is 6 cm, as shown in Figure 7. The fracture geometry and the fluid properties are chosen so that the natural fracture is not hydraulically opened during the fluid injection process, although the hydraulic aperture does increase with increasing pressure as explained above. [39] To verify the numerical accuracy, a special case is first considered where the natural fracture is so rigid that its hydraulic aperture is nearly unchanged during fluid invasion. At the intersection, the fluid pressure increases rapidly to a constant value, as shown in Figure 8. Therefore, the fluid movement into the natural fracture, representing fluid loss from the main fracture, is equivalent to the solution for the case where the fluid penetrates an infinite evacuated channel at a constant injection pressure. There is an analytic solution for the fluid front movement in this case. The formula for fluid front length from the intersection is given as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðþ¼ t 2P f kt=m: ð19þ In particular, the value of injection pressure can be extracted from Figure 8. Upon substituting the injection pressure into equation (19), we can compare the analytic solution (dotted line) to the numerical solution (dashed line), as shown in Figure 8. It is evident that the numerical results fit very well with the analytical solutions. [40] For the problem illustrated in Figure 7, the fracture will continue to grow past the natural fracture in the horizontal direction. This implies that most of the injected fluid will eventually be stored in the main horizontal hydraulic fracture with a large conductivity instead of being lost into the natural fracture. For the cases considered here, we find that after a sufficiently long time, the fluxes Q 1 and Q 2 shown in Figure 9 will both eventually approach half of the injection rate. However, the natural fracture can act as a fluid loss point to the main fracture, and this will impact fluid movement and fracture growth past the junction point, especially at the early stage. Some of these early time responses of flux changes are highlighted in Figure 9 for a case where a natural fracture is perpendicular to the main fracture. After the hydraulic fracture intersects the natural fracture, there is a drop in Q 1 since mechanical interactions between the natural fracture and the hydraulic fracture can slightly increase the compressive stress on that side of the borehole. The flux Q 4 either increases or decreases as the natural fracture is filled, depending on interactions between this branch and other hydraulic fracture branches. The time integration of the curve for Q 4 yields the fluid volume lost into the infinite evacuated natural fracture. From the variations with time of the normalized fluxes for each fluid branch, the effect of the initial aperture 10 of 16