Reinitiation or termination of fluid-driven fractures at frictional bedding interfaces

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi: /2007jb005327, 2008 Reinitiation or termination of fluid-driven fractures at frictional bedding interfaces Xi Zhang 1 and Robert G. Jeffrey 1 Received 14 August 2007; revised 17 April 2008; accepted 16 June 2008; published 28 August [1] A two-dimensional numerical model for coupled elastic deformation and fluid flow has been developed to examine reinitiation or termination of a vertical fluid-driven fracture in bedded rocks. The rocks on both sides of the bedding interface are assumed to be impermeable, and a Newtonian fluid, whose viscous dissipation cannot be neglected, is injected at a constant rate. Crack nucleation on the frictional interface is controlled by a tensile stress criterion. A fracture approaching the interface or terminating on it can generate a large bed-parallel tensile stress in the uncracked layer in excess of the local tensile strength to initiate a new fracture. The propagation of the nucleated fracture is assisted by interface sliding and pressurized fluid. The bedding interface can provide a conductive channel to connect the parent and the new fractures. Fluid partitioning among fracture branches depends on local stress states and triggers their competition to become the main fracture. The three types of fracture patterns generated numerically by the model are crosscutting through, terminating at, and offsetting at the interface. A large modulus or toughness contrast across the interface can lead to containment of the hydraulic fracture on the interface. For offsetting fractures, the presence of the bedding plane can reduce fracture permeability, and any associated high-stress barrier acts to slow or arrest further fluid-driven fracture growth. Propagating the new fracture results in high excess pressure, consistent with measured abnormal pressure in commercial fracture treatments. The predicted offset distances are on the order of centimeters, and their continued growth is perpendicular to the bedding contacts. Citation: Zhang, X., and R. G. Jeffrey (2008), Reinitiation or termination of fluid-driven fractures at frictional bedding interfaces, J. Geophys. Res., 113,, doi: /2007jb Introduction [2] Fluid-driven fractures or opening-mode hydraulic fractures, induced by the presence of elevated fluid pressure with respect to the in situ stresses, are commonly observed in geological formations at a wide range of scales. The permeable pathways provided by these tensile natural hydraulic fractures play an important role in the migration of water, hydrocarbons and ore forming fluids through the rock mass. The fracture deformation mechanisms, growth rates, and direction are still fundamental issues being addressed in the structural geology, hydrogeology and petroleum engineering literature. The coupled mechanical interaction of rock fracture and fluid flow is incompletely understood for fluid-driven fractures, especially when these fractures interact with other fractures or frictional interfaces [Taylor et al., 1999]. In particular, field and laboratory studies and numerical modeling continue to provide new insights and understanding about the nature and formation of fluid-driven fracture patterns occurring in layered rocks [e.g., Pollard, 1 Ian Wark Laboratory, CSIRO Petroleum Resources, Clayton, Victoria, Australia. Copyright 2008 by the American Geophysical Union /08/2007JB005327$ ; Anderson, 1981; Teufel and Clark, 1984; Warpinski and Teufel, 1987; Helgeson and Aydin, 1991; Baer, 1991; Gross, 1993; Renshaw and Pollard, 1995; Fischer et al., 1995; Bai et al., 2000; Cooke and Underwood, 2001]. [3] In the presence of a planar bedding plane, fracture branching often occurs when a fluid-driven fracture attempts to propagate vertically through a horizontally layered sequence. In addition to fracture branching, new fracture initiation on the interface can affect internal pressure evolution and overall conductivity. The asymmetric initiation of new fractures can generate shear displacement along fracture surfaces without any shear mode fracture surface characteristics. The commonly used assumption that the hydraulic fracture is completely planar is not often justified if fluid enters the bed contact after fracture propagation to the interface. Fluid pressure would drop once the fracture volume available to the fluid greatly increases, but the pressure would increase to overcome the stress barrier associated with the change in propagation direction. Evidence of abnormal treatment pressure has been found during hydraulic fracture stimulation of oil and gas wells [Medlin and Fitch, 1983] and branching of fractures has been postulated as a cause of elevated pressure. A complex deformation pattern arises from the process such as, fracture-interface coalescence, mode transition between tensile and shear fractures, fracture reinitiation, multiple fracture 1of16

2 interactions, fluid diversion and fluid loss into secondary fracture branches, interface frictional slippage, and so on. Accurate analyses of, and models for, prediction of likely fracture paths and whether and how fracture crossing occurs are thus valuable for commercial hydraulic fracture treatment design and can also assist in the understanding of the nature of fracture networks in layered or naturally fractured aquifers and reservoirs. [4] Because of its importance to the geological community and the petroleum industry, the effect of frictional bedding interfaces on fracture growth has been studied by a number of researchers. Earlier solutions for the response of a closed frictional interface to an approaching or intersecting fracture have been given by Weertman [1980], Keer and Chen [1981], Lam and Cleary [1984], and Dollar and Steif [1988]. Cooke and Underwood [2001] considered the effects of slip and opening along a weak bedding contact on stress variations. However, these previous studies have neglected the role of viscous flow by assuming a uniform pressure distribution in the fracture system. This assumption is valid if fracture growth is controlled by fracture toughness or if the fluid injection rate is very slow so that viscous dissipation in the fluid can be ignored. Such conditions may exist in some natural hydraulic fractures. In commercial stimulations and magma-driven dike propagation the fractures are driven by fluids whose viscous dissipation is typically larger than the energy consumed in overcoming fracture toughness. Analysis of such fractures must include the flow-induced pressure gradients and the local changes in fracture conductivity, which arise from the strong fluid-rock coupling. Few studies are available that consider the fracture-interface interaction by means of spatially and temporally coupled models for rock deformation and fluid flow [Barton et al., 1985]. Recently, studies dealing with the coupled problem have received much attention, e.g., the experimental work by Wu et al. [2004] and de Pater and Beugelsdijk [2005], and the numerical studies by Zhang et al. [2007, 2008], motivated by stimulation of low permeability and naturally fractured reservoirs. Compared to the earlier work, these recent efforts addressed the evolution through time of fluid pressure and fluxes, amount of opening and slip along fractures, and partially opened interfaces. [5] There are several methods to model nucleation of a fracture. In contrast to previous studies that treat the bedding contact as perfectly bonded and that select the location for fracture initiation as the location of the largest flaw [Baer, 1991; Helgeson and Aydin, 1991; McConaughy and Engelder, 2001], we employ a criterion based on a critical tensile stress. Using this criterion, fracture nucleation occurs where the local tensile stress exceeds this critical tensile stress value [Gross et al., 1995]. In particular, the critical stress is equal to the tensile strength of the rock [Renshaw and Pollard, 1995], which includes the effects of flaws such as fossils, minerals, grains and clasts on local concentrations of tensile stress [Pollard and Aydin, 1988]. Thereby, the use of the critical tensile stress criterion to select the initiation site for new fractures is consistent with microstructural considerations [Jaeger and Cook, 1979]. For lower tensile strength, the incipient fracture is expected to commence prior to the onset of frictional energy dissipation by slippage on the interface [Renshaw and Pollard, 1995]. But for higher tensile strength, a significant tensile stress concentration arises after interface slip and even after opening of the interface by a hydraulic fracture growing along it. The tip stress singularity is eliminated when the fracture terminates at the interface but a large tensile stress can still exist [Dollar and Steif, 1988]. The path- and history-dependent tensile stress variations give rise to different fracture paths ranging from straight penetrating fractures to offsetting ones having an offset on the bedding plane with respect to the parent fracture plane. These offset fractures are called composite fractures by Helgeson and Aydin [1991]. In addition, the growth of the newly created fracture to a significant size is ultimately dependent on fluid entering and pressurizing it. [6] In this article, a two-dimensional numerical fracture model is used to simulate the crossing process when a fluiddriven fracture intersects a frictional bedding interface. For this analysis of fluid-driven fracturing, poroelastic effects of the matrix rocks on fluid pressure are not accounted for since a much larger permeability exists along fractures and the crossing process normally lasts a short time. Also, inertial effects are not important for the quasi-static elastic deformation and fluid flow considered. The computational approach presented here is for fracture growth coupled with rock deformation and fluid flow in the fracture. We focus on the influence of critical stresses and the fluid dynamic viscosity, as well as modulus and toughness contrasts across the bedding interface, on fracture paths, distribution of fracture apertures and fluid loss into secondary fracture branches. In the following section, the formalism is given on which we base our two-dimensional fracture model. Following that, the numerical results are presented to interpret the observed features associated with commercial fracture treatments and fracture networks in layered rocks. 2. Problem Formulation [7] The system we are modeling is a bedded rock mass containing a single hydraulic fracture starting in one layer, as shown in Figure 1. The upward propagating fracture intersects a weak bedding contact between dissimilar linear-elastic half planes and may then deflect into the interface or may propagate through it. Although some researchers [Helgeson and Aydin, 1991; Fischer et al., 1995; McConaughy and Engelder, 2001] modeled the contact as a thin, strongly bonded interlayer, we considered it as a weak plane with a frictional strength. Cooke and Underwood [2001] have found that sliding and opening on the interface is important to fracture termination and the development of offsets which they call step-overs. The fracture geometry is assumed to be infinitely long in the out-of-plane z-direction so that plane-strain conditions apply to any cross section. This configuration might apply to the intersection with a bedding plane of a well contained vertical hydraulic fracture or of a long dike. Alternatively, if the leading edge of a large three-dimensional fracture has a very small curvature, its interaction with a bedding plane can be represented approximately by such a two-dimensional model. [8] Consider that, as shown in Figure 1, a new fracture is initiated at a distance d from the intersection site during the crossing. To avoid considering the existence of multiple new fractures, only the first initiated fracture is allowed to develop. Details of the initiation process and competition 2of16

3 dipole solutions. The fracture branches are numbered based on the tips, that is, there are N fractures if there are N tips. Summing up the contributions to the stresses from each fracture, the elasticity equations for equilibrium fractures in the framework of a Cartesian coordinate system are Z s n ðx; tþ s 1 ðþ¼ x XN lr ½G 11 ðx; s; a; bþws ðþ r¼1 0 þ G 12 ðx; s; a; bþnðþšds s ð1þ Z t s ðx; tþ t 1 ðxþ ¼ XN lr ½G 21 ðx; s; a; bþwðsþ r¼1 0 þ G 22 ðx; s; a; bþnðþšds; s ð2þ Figure 1. Schematic of a hydraulic fracture interacting with and crossing a frictional bedding plane. The fracture geometry and in situ stresses are shown together with the Cartesian coordinate system used. The fracture path prior to intersection is drawn with solid lines, and the branched fracture path after intersection is drawn with dotted lines. among multiple new fractures is not considered here, but is left for future work. After the hydraulic fracture intersects an interface, it can split into two fracture branches, growing on the interface to the left and right of the intersection point, respectively. One of these two branches may well connect to the new fracture, leaving an offset on the interface to form a composite fracture, as illustrated in Figure 1. The fluid volume left in the interface fracture is treated as fluid loss in contrast to fluid available to open and extend the main fracture. [9] We model numerically the frictional interface as a long (compared with other lengths), initially closed fracture. The y-axis lies on the interface and the Cartesian coordinate system used is depicted in Figure 1. The origin of the coordinate system is at the intersection between the vertical hydraulic fracture and the bedding plane. The injection point is at a distance L away from the intersection point. It is assumed that the rock beds themselves are impermeable, isotropic and elastic. The Young s moduli, Poisson s ratios and fracture toughnesses for the two rock layers are, E i, n i, and K C i, where subscript i denotes the layer, i =1,2. A Newtonian fluid with a dynamic viscosity m is injected at a constant rate Q 0 into the vertical fracture. The vertical farfield stress (the least remote principal stress in most cases) is denoted by s 0 xx, and the layer-parallel far-field stresses in the lower and upper half planes are s 0(1) yy and s 0(2) yy, respectively. Fluid pressure and far-field stresses are taken as positive if in compression. [10] Since we neglect inertial effects, the equation of quasi-static elasticity provides the relationship between the normal, w, and shear, v, displacement discontinuities (DDs) of the fracture walls and the applied tractions including the fluid pressure P f inside the fracture. The formulation used is based upon a superposition scheme of singular dislocation where x =(x, y) and t is time; ds is the infinitesimal arc length along the fracture; l r is the fracture length with a subscript index r identifying the fracture branch number; s n is the normal stress which is equal to P f within the fluidfilled parts; and t s is the shear stress associated with frictional sliding along the interface; s 1 and t 1 are the normal and shear stresses along the fracture direction at location x caused by the far-field stresses. G ij are hypersingular Green s functions whose expressions can be found in Zhang et al. [2005]. These Green s functions are dependent on the Dundurs s parameters, defined by a ¼ m 2ðk 1 þ 1Þ m 1 ðk 2 þ 1Þ m 2 ðk 1 þ 1Þþm 1 ðk 2 þ 1Þ b ¼ m 2ðk 1 1Þ m 1 ðk 2 1Þ m 2 ðk 1 þ 1Þþm 1 ðk 2 þ 1Þ ; ð3þ where k i =3 4n I. The fracture index r will not be shown in the following equations for simplicity. [11] In order to maintain continuity of strain across the interface, s 0(1) yy and s 0(2) yy are related by [Rice and Sih, 1965] s 0ð2Þ yy ¼ # þ s 0ð1Þ yy n 2 # n 1 s 0 xx 1 n 2 1 n 1 ; ð4þ s 0ð1Þ yy where # = E 2 (1 n 2 )/E 1 /(1 n 1 ). [12] During sheet intrusions and commercial hydraulic fracture treatments, the fluid flow velocity V is less than 1 m/s for most cases. For two-dimensional flow between parallel plates, the Reynolds number is defined as Re = rvw/m in which r is the fluid density and w is the average fracture aperture. In the extreme, taking water as the injected fluid and w = 1 mm, we have Re = 10 3, less than for the onset of turbulent flow. The fluid flow in the fracture and the interface is thus governed by the lubrication equation [Batchelor, 1967; Witherspoon et w m 0 ; where m 0 =12m. Because we assume no fluid loss into the rock, the global mass balance leads to X Z l f wds ¼ Q 0 t; 0 ð6þ 3of16

4 where the flux is defined as q =(w 3 /m 0 )@P f /@s based on Poiseuille s law which is a good approximation because fracture width is at least one order of magnitude smaller than the length of the fluid branch. We recognized that Poiseuille s law cannot be applied to the sudden change in fracture width at the intersection. The details of the flow and stress fields at the intersection require numerical simulations of Navier-Stokes equations (e.g., Kosakowski and Berkowitz [1999], ignoring elastic deformation). This is outside the scope of this study. On the other hand, the flow velocity around the intersection is greatly reduced, and the corresponding Reynolds number can be less than unity. In this case, the pressure loss at the intersection is negligible. [13] The model includes a small hydraulic width w o h, which is assigned to the frictional interface to represent any preexisting conductivity before it is opened by the fracturing fluid. This hydraulic width does not result in any stress or displacement change of the elastic system, but contributes to the fracture permeability. Thereby, the fracture permeability along the interface is the sum of the permeability resulting from the mechanical opening and from this minimum aperture. w o h aids the process of fluid diversion into the two interface fracture branches. Once the fluid-filled interface portion is closed by the evolving stresses, its conductivity cannot decrease to be less than w o h. It has been found [Renshaw, 1995] that the Poiseuille s law is accurate in predicting the fluid flux through a roughwalled fracture as long as the appropriate average hydraulic fracture aperture is used. [14] At the injection point, the fluid flux is equal to the injection rate, that is, qð0; tþ ¼Q 0 : At the fracture tip, the opening and shearing DDs are zero, that is, ð8þ wðlþ ¼nðlÞ ¼0: ð9þ [15] The Coulomb frictional law is applied to the frictional interface and provides a proportional relation between the frictional strength t s, and the normal effective compressive stress, s n, that is, jt s j ¼ ls n ; ð10þ Figure 2. Schematic of the critical stress criterion for new fracture initiation. where l f is the fluid-filled length of each fracture. The formulation 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 1. Inside the fluid lag, the pressure vanishes because pore pressure is not included in the model. The fluid front can be found in terms of the flux l f ¼ ql f ; t =wlf ; t ; ð7þ where l is the coefficient of friction. [16] The contact behavior of fracture surfaces has three different modes. When the fracture is open, the shear stress is zero. For closed surfaces, if the shear stress acting is less than the frictional strength, the fracture surface is in sticking mode, and otherwise it is in sliding mode. In addition, the direction of shear stress can be reversed if the shear displacement increment changes sign. The frictional stress and the shear DD increment should meet the following condition during the contact sliding mode: t s Dn 0; ð11þ where Dv is the sliding increment. [17] In order to determine the correct contact mode for the interface, an initial guess is used and a check is then made to ensure that the guess is correct at each time increment. If not, the contact modes are updated based upon the calculated stresses and displacements. The new modes are then applied and stresses and displacements are recalculated. This is followed by another check and modification until the contact mode converges at each location. [18] Furthermore, we examine the maximum horizontal tensile stress along the top of the interface for fracture nucleation, as done by Cooke and Underwood [2001]. A new fracture is created when the layer-parallel tensile stress exceeds the tensile strength. As we know, two locations of the maximum stress may exist on either side of the fracture tip due to geometric symmetry. To simplify the problem for fracture reinitiation, only one new fracture is allowed to form, which is consistent with mapped fracture geometries in outcrops. The tensile strength s T is taken as constant, for the sake of simplicity. As shown in Figure 2, the inception of crack initiation occurs when the layer-parallel normal tensile stress adjacent to the interface, s yy, is equal to s T at the position y = d. [19] At the fracture tip, the magnitudes of the Mode I and II stress intensity factors are determined by the displacement correlation method. In order to determine the fracture 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 4of16

5 Table 1. Model Material Constants Used in the Calculations Material Property Granite Sandstone Siltstone Shale Young s modulus (GPa) Poisson ratio Fracture toughness (MPa m 0.5 ) are known, the fracture propagation direction is determined by solving the following equation: K I sin Q þ K II ð3 cos Q 1Þ ¼0; ð12þ where Q is the deflection angle from the current fracture line, and K I and K II are the Mode I and II stress intensity factors respectively. The condition for quasi-static fracture growth is the same as used for dike propagation and fluiddriven crack growth [Lister, 1990; Rubin, 1995; Detournay, 2004] in contrast to that for subcritical crack growth in cases of long duration loading used by Olson [1993] and Savalli and Engelder [2005]. Along the propagation direction, the mixed stress intensity factor should meet the following condition: cos Q 2 K I cos 2 Q K II sin Q ¼ Ki C ; ð13þ where K i C (i = 1, 2) is the rock toughness of the layer in which the fracture tip is located [Erdogan and Sih, 1963]. 3. Numerical Method [20] The method for finding a solution of the formulated problem is based on an implicit algorithm, as previously described by Zhang et al. [2005]. Since the 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-filled 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 interface is required. The interface is modeled as an initially closed long fracture that can carry frictional shear stress and compressive normal contact stress. The lubrication equation is solved for each time increment by means of the Finite Difference Method (FDM). [21] The numerical method uses a fixed size element length, Dh, for regular elements and crack increments and a small, but adjustable time step is used. The time step is adjusted to insure convergence of the coupled fluid and elastic solutions. When the propagation criterion is satisfied, a fracture increment equal to Dh, is added to the fracture tip along the determined propagation direction. To improve the accuracy for calculating the stress intensity factors, a mesh adaptive scheme is employed by using six small elements near the tip. These elements are one third in size of the standard sized 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. [22] Details of the numerical implementation of frictional contact mode selection and division of the fluid flux at the fracture-interface junctions, can be found in our previous article [Zhang and Jeffrey, 2006]. [23] The numerical treatment of fracture initiation is described as follows. The bed-parallel normal tensile stress at one end point of an element on the bedding interface is calculated. When this stress reaches the tensile strength, a new fracture 3Dh long and perpendicular to the interface is placed at this end point. In addition, the new fracture is assumed be closed at first. When loaded by interface frictional slip and/or by fluid pressure, this new fracture may propagate into the upper layer based on equations (12) and (13). Owing to the restriction of the element size, the position with maximum horizontal tensile stress cannot be identified to within less than one element length. However, with a sufficiently fine mesh, the fracture initiation site can be located with adequate resolution so that the time-dependent variations of fracture surface traction and opening, as well as fluid flux can be obtained. We note that the numerical algorithm used here for fracture initiation can be applied to other fracture initiation criteria. 4. Numerical Results [24] For the calculations presented below, a typical value of the coefficient of friction for the interface (l = 0.5) is used for all cases, although whether the fracture crosses the interface or not is sensitive to this value [Renshaw and Pollard, 1995]. The injection point is at L = 1 m from the interface and the injection rate is Q 0 = m 2 /s if not otherwise specified. The minimum hydraulic width is taken as w h 0 = 0.01 mm which corresponds to a permeability of 100 md. In addition, the initial length of the new fracture is 3 cm, much less than L and other geometric lengths. If the assumed length is too large, the new fracture will first grow in a nonequilibrium way, and meanwhile the fluid pressure is reduced because of the rapid development of new fracture permeability. An equilibrium state is then reached rapidly and subsequent crack propagation is stable. Therefore the assumed initial fracture length will not significantly influence the overall results. The choice of element size has been tested to provide accurate numerical results. [25] To evaluate the possibilities of fracture reinitiation, two types of rock layers are considered. One type consists of a granite and a sandstone layer and the other type is a siltstone and a shale layer. The elastic proprieties of these rock pairs are summarized in Table 1. The fracture crossing interaction also depends on the fluid properties such as fluid dynamic viscosity and injection rate. The dynamic viscosity of fluids varies with fluid composition, temperature and pressure and can range from less than Pas for water to 10 4 Pas for basalt or higher for other molten rock types. In particular, we do not consider toughness dominated fracture regimes (uniform pressure) in this article. The viscosity and the injection rate not only determine the timescale of the fracture process, since the fluid provides the driving force, but also affect the mechanical interaction and the final fracture pattern because of the strong coupling between the fluid and elastic parts of the problem, as 5of16

6 Figure 3. Variations of layer-parallel normal stresses near the intersection with time prior to and post-fracture-interface intersection at very large critical stress values for granitesandstone layers. demonstrated in our previous articles [Zhang et al., 2007, 2008] Granite-Sandstone Layers [26] In this case, the fracture is initially located in the lower granite layer and it propagates toward the upper 0 sandstone layer. The vertical confining stress is s xx = 8 MPa, applied across the interface, corresponding to a depth of burial of about 320 m. The horizontal in situ stress s 0(1) yy is 7 MPa, and s 0(2) yy is 4.1 MPa based upon equation (4). A relatively larger viscosity is chosen (m = 0.1Pas). The critical stress is varied from 0.5 to 50 MPa, covering a range of possibilities from fracture reinitiation with an offset, to fracture penetration through the bedding interface at the intersecting point, and to fracture arrest. For example, an arrested fracture occurs for the case of s T = 50 MPa. An arrested bedding plane fracture, in this sense, means that fracture growth and fluid flow are contained along the interface. [27] To explore the reasons for fracture reinitiation and termination, we tracked the tensile layer-parallel normal stress along the top of the interface. Figure 3 shows the distributions of this normal stress along the interface at several times for the case of s T = 50 MPa. Owing to geometric symmetry, only half of the interface (y 0) is considered. The time t starts when the constant rate injection begins. Note that the normal compressive stress is defined as positive. There are two stress peaks on either side of the existing (parent) fluid-driven fracture except for the case at t = s when the tip is on the interface. The dual-lobed stress distributions along the interface suggest the possibility for nucleation of two small fractures at one time. However, only one new fracture on the right-hand side is allowed and considered here. For the case without crack nucleation (s T = 50 MPa), the greatest peak tensile stress (around 26 MPa) occurs at t = s when the fracture tip is at the interface and the pressure is increased to a level sufficient to open the interface. This largest normal tensile stress arises after the loss of stress singularity as the tip intersects the frictional interface [Dollar and Steif, 1988]. In contrast to pure elastic solutions with fixed uniform internal pressure, the maximum tensile stress keeps increasing for the fluid pressure increases. As the fracture grows along the interface, the dual-lobed normal stress profiles reemerge and a zone with compressive stress develops at the central part of the interface fracture as shown by the curves for t = s and s. Note that the magnitudes of the tensile stress peaks attained decrease slowly as the fracture grows along the interface. [28] Since the largest normal tensile stress calculated is around 26 MPa for the given material and geometric parameters, the fracture can only continue propagating along the interface and containment of fracture growth at the bedding plane is predicted if the critical stress for the creation of a new fracture is higher than 26 MPa. On the basis of the results shown in Figure 3, the single stress peak at the intersection point passes through a range of values. If the critical stress is within this range, a new fracture will be initiated at the intersecting point, resulting in fracture growth straight through the interface. In addition, the nonsingular and decreasing stress peaks associated with growth of the fracture on the interface imply that a new fracture will not be initiated unless we allow the critical stress to decrease at some points away from the intersection point. On the other hand, if s T is small enough, there is a possibility to induce a new fracture at an offset to the intersecting point before the fracture touches the interface because of the duallobed stress profiles shown in Figure 3 for times less than s. As expected, the remote horizontal stress acting parallel to the interface and the vertical stress acting across the interface are important because they significantly affect the magnitude of the layer-parallel tensile stress which produces the new crack nucleation [Pollard, 1973]. [29] Figure 4 shows the variations of sliding displacement discontinuity (DD) and frictional stress on the interface at t = 0.184, s, as the hydraulic fracture grows along the interface. The interface experiences a large slip transferred from the opening of the parent hydraulic fracture. This transfer of opening to shear displacement is a typical feature Figure 4. Distributions of shear displacement and frictional stress along the interface flank for the case of a large critical stress for granite-sandstone layers. 6of16

7 Figure 5. Fracture trajectories for different critical stress levels for granite-sandstone layers. for fracture-interface interactions and is accompanied by the development of high internal fluid pressure in the parent fracture, as discussed by Zhang et al. [2007, 2008]. Without using a coupled model, the stress and slip evolution cannot be captured and some ad hoc assumptions used to replace such detailed calculations may lead to unreasonable results. The frictional shear stress on the interface may generate a layer-parallel normal tensile stress along the top of the interface (the intact side of the interface), but the normal compressive stress acting along the interface does not contribute to this tensile stress. On the basis of the Cerruti s solution for a point force F tangential to the interface at the location (0, s), the induced layer-parallel stress at the position (0, y) is given by [Kachanov et al., 2003] s yy ¼ 2 p F y s : ð14þ If the frictional shear stress has a distribution t s (y) within two zones [ d, c] and [c, d], the resultant layer-parallel normal stress can be calculated as s yy ¼ 4y p Z d c t s ðsþ y 2 s 2 ds: ð15þ [30] As shown in Figure 4, the frictional stress tends to increase rapidly near the ends of the open portion of the interface fracture to an approximately constant value, which we designate by B that is determined by the Coulomb frictional law. On the basis of the numerical results, the shear stress distribution can be approximated by an exponential function: h i t s ðsþ ¼B 1 e ks c ð Þ ; ð16þ where B and k are constants. The value of k can be adjusted to fit the gradient of the shear stress at the front of the opening interface portion. Upon substituting equation (16) into equation (15), we can obtain an expression for s yy (c)at the opening front. The detailed derivation is given in Appendix A. It is obvious from equation (A6) that the singular terms are eliminated and the finite layer-parallel normal tensile stress should be continuous along the interface, as shown in Figure 3. It can be seen that using other equations to fit the shear stress distribution will result in the same conclusion. [31] There are three different fracture trajectories for various values of s T, as shown in Figure 5. For s T =0.6 and 5 MPa, a composite fracture is formed with an offset distance of 3 and 1 cm, respectively. As anticipated, the offset size increases with decreasing tensile strength. Similar composite fractures have been observed in laboratory experiments [Renshaw and Pollard, 1995] and in the field [Helgeson and Aydin, 1991]. If s T is between 5 and 26 MPa, the nucleation site is at the intersection point or is so close to it that our model cannot detect any difference. In Figure 5, the offsetting fractures initiated within a few centimeters of the intersecting point tend to propagate perpendicular to the frictional contact soon after they are nucleated. This is consistent with the outcrop fractures observations by Cooke and Underwood [2001]. It must be remembered that due to the nature of our computer program, other possible fracture patterns such as multiple fracture patterns are not addressed here. Multiple hydraulic fracture networks are considered by Olsen (1993) using uniformly pressurized fracture models for fracture growth driven by zero viscosity fluids. In contrast, our model considers fracture growth driven by fluid sources that are so strong that the pressure distribution in the fracture is nonuniform and viscous dissipation must be included. Such a model tends to produce fractures with localized flow and channel structures, rather than a branched fracture network. The use of one approach or the other depends primarily on the source and time frame of the fluid that propagates the fracture. If the fluid of small viscosity enters the fracture at a very slow rate so that no significant pressure gradients are established in the fracture, the uniform pressure models are appropriate. [32] While the main fracture which includes the offset on the interface is opened and extended by the pressurized fluid, secondary branches may extend for some distance along the interface. For some cases, the fluid storage in the left interface fracture is not negligible, for example, in the case of s T = 0.6 MPa. Figure 6 shows the fracture aperture distribution at a specific time. The loss of fluid from the main fracture path leading to the tip in the upper layer will slow the fracture growth by reducing fluid volume available to extend the main fracture. The secondary branches also interact mechanically with the main branch. A long and wide left interface fracture helps open and establish the flow channel through the offset region. It is found that the fracture width along the offset is much smaller than other fracture segments in the upper and lower layers. The narrow channel results in a large pressure drop as fluid is forced through it and may lead to proppant bridging or plugging [Daneshy, 2003]. [33] Figure 7 provides the evolution of fluid pressures at three positions (P 0 f at the borehole, P 1 f at the intersecting point and P 2 f at the nucleating point) using the same parameters as in Figure 6. The borehole pressure does not decrease monotonically as would be the case for a planar 7of16

8 Figure 6. layers. Opening profile along both fracture and interface at s T = 0.6 MPa for granite-sandstone fracture. There is a small increase in P 0 f prior to the establishment of the flow channel across the offset region. The existence of such an offset along the composite fractures not only changes the fracture propagation direction, but also increases the treatment or injection pressure. It should be noted that the treatment pressure for fracture propagation in softer sandstone is much lower than that shown in Figure 7. The relatively large fluid pressure in the parent hydraulic fracture also generates additional compressive stress across the interface and this makes fluid-driven fracture growth on the interface more difficult, as discussed by Savalli and Engelder [2005]. From Figure 7, we see that the time taken for the fluid front to pass through the offset is more than 0.15 s, during which the fluid pressure tends to be uniform along the parent fracture since the difference between P 0 f and P 1 f decreases. After the fluid front reaches the new fracture, the fluid pressure starts to decrease at t = 0.25 s. This time-dependent variation of fluid pressure is an important feature of hydraulic fracture crossing frictional interfaces in the presence of significant viscous dissipation. The geometry of the composite fracture as found by fracture mapping is always expected to produce elevated treatment pressure [Medlin and Fitch, 1983]. P 2 f increases with time to provide a driving force for fracture growth in the upper layer. In addition, Figure 7 also gives results for the timedependent variations of P 1 f for the penetrating and arrested fracture cases. For penetrating fractures, there is a sharp reduction in pressure as the hydraulic fracture enters the softer upper layer and thereafter the pressure begins to decrease slowly as expected for a single planar fluid-driven fracture. The maximum value of P 1 f for the arrested fracture is less than that for the composite fracture because the opening compliance of the composite fracture is smaller Siltstone-Shale Layers [34] Elastic constants for siltstone and shale listed in Table 1, were selected to be similar to those chosen by Helgeson and Aydin [1991] and McConaughy and Engelder [2001]. The interface is located underground at around 200 m deep. A vertical fracture in the siltstone is subjected to a remote horizontal compressive stress (3 MPa) and a vertical one (5 MPa). The injected fluid is water with a dynamic viscosity Pas. Figure 8 gives the distribution of fracture width at t =0.2sfors T = 0.7 MPa. In contrast to the large viscosity cases, the fluid with low viscosity can Figure 7. Time dependence of fluid pressures at the borehole P 0 f, the intersection point P 1 f, and the new fracture mouth P 2 f in the case of s T = 0.6 MPa and comparisons with other cases for granite-sandstone layers. 8of16

9 Figure 8. Fracture opening profile at s T = 0.6 MPa for siltstone-shale layers. readily pass through the narrow channel along the bedding interface to enter the new fracture. The fracture aperture is smaller, compared to Figure 6, especially along the offset region. The magnitude of the new fracture aperture is close to that of the parent fracture. It should be noted that the same scale is used in these two figures for fracture opening. Three fracture pathways are generated as shown in Figure 9. The offset size is on the order of a few centimeters and the new fractures are perpendicular to the bedding contact. In addition, the fracture nucleation site moves closer to the intersecting point with increasing critical stress. These features associated with initiation of new fractures and fracture patterns exist independent of the choices of rock and fluid properties. [35] The fluid pressure in the fracture is shown in Figure 10 for the siltstone-shale interface case using s T = 0.7 MPa. The time for the low-viscosity water to penetrate through the offset region is less than 0.01 s. The fluid pressure at the intersecting point increases rapidly to display a slowly increasing trend and approaches the injection pressure at point 0 because the lower viscosity of the fluid results in less viscous dissipation in the parent fracture. It should be noted that across the offset, there still is a significant pressure loss because the channel width is small. The small value of P f 2 in Figure 10 reflects the large compressive stress and the lower opening compliance of the offset portion. Use of a uniform Figure 9. Fracture pathways for different critical stress levels for siltstone-shale layers. Figure 10. Time-dependent variations of fluid pressures at the borehole P 0 f, the intersection point P 1 f, and the new fracture mouth P 2 f in the case of s T = 0.7 MPa for siltstoneshale layers. 9of16

10 Figure 11. Fracture paths for different modulus ratios for granite-sandstone layers. pressure approach would not be accurate in this case where viscosity is small but injection rate is reasonably large. The restricted width at the offset results in significant viscous dissipation there. Furthermore, because pressure and width are coupled, the overall fracture pressure, width, and growth rate are affected Effects of Contrasting Elastic Moduli [36] The nucleation and growth of a new fracture depends on the horizontal far-field stress in the upper layer, which is, in our model, a function of the material constants according to equation (4). To examine the effects of contrasting elastic moduli on fracture crossing, we study fracture responses in the granite and sandstone layers by changing the Young s modulus of the upper sandstone layer. The tensile strength for fracture nucleation is 0.5 MPa, and the fluid dynamic viscosity is taken as 0.01 Pas. All other material and fluid properties and the in situ stresses in the lower granite layer remain the same as those used in Section 4.1. The Young s modulus of the intact rock is given values of 8, 20, 40 and 80 GPa (E 2 /E 1 = 0.2, 0.5, 1.0, 2.0) in the calculations and the bed-parallel far-field stress for the upper layer is accordingly assigned values of 2.6, 4.1, 6.6 and 11.5 MPa in compression. Figure 11 displays the fracture paths calculated for these different modulus ratios except for E 2 /E 1 = 2.0 which is discussed below to illustrate the effect of large bed-parallel far-field stresses on new fracture growth. From Figure 11, the nucleation site for new fracture becomes closer to the intersection point as the modulus ratio E 2 /E 1 is reduced for the cases considered. Generally speaking, a very soft upper layer (E 2 /E 1 is much less than 1) is more likely to result in a straight crossing fracture. Because the softer layer causes an increase in the stress intensity factor of the parent fracture before the tip reaches the interface, the excess pressure driving the parent fracture is somewhat lower than it would otherwise be and the stress on the interface generated by the approaching fracture is reduced in magnitude The interface region with high tensile stress concentration is therefore smaller and more localized ahead of the crack tip. For the case that considers a higher Young s modulus of the upper layer, the stress intensity factor at the tip of the parent fracture is reduced. The pressure in the fracture must then be increased to generate crack growth. This, in turn, produces a larger tensile stress concentration zone ahead of the crack tip. The critical stress criterion hence is more likely to be met before the fracture tip reaches the interface, resulting in a composite fracture. However, it must be noted that increasing the elastic modulus of the upper layer gives a higher compressive remote in situ stress by equation (4). This higher compressive stress will force the nucleation site to be closer to the intersecting point. The fracture reinitiation relies on the competition between the extent of stress concentration caused by the parent fracture and the magnitude of the remote layer-parallel stress, although Figure 11 shows the tendency for the daughter fracture to deviate from a planar path with increasing elastic modulus of the upper layer. Wherever the nucleation site is, the new fracture eventually propagates in a direction normal to the bedding contact as dictated by the far-field stress. [37] Figure 12 shows the normal stress profile along the fracture branches at one instant for the case of E 2 /E 1 = 2.0. In this figure, the bars above the interface represent fluid pressure and the bars below the interface represent contact stress. The location where the bars switch from above to below the interface coincides with the position for fluid front. For this case, the higher remote in situ stress causes the new fracture to be nucleated at a small distance from the intersection point. The new fracture is propagated easily by large slip along the interface. However, the slip induced growth is soon stopped by the high bed-parallel confining stress. On the other hand, since s 0 xx is less than s 0(2) yy, growth of an interface fracture to the left of the intersection requires a lower pressure level than for extending the new fracture. As shown in Figure 12, the interface fracture branch continues growing under a pressure that is too low to propagate the new fracture and this interface fracture becomes the main hydraulic fracture extended by the injected fluid. [38] Figure 13 shows the variations of fluid pressure at the injection point with time for various values of E 2 /E 1. The injection pressure does not vary in a consistent manner. We recall that changes in modulus ratio result in different stresses in each layer and the fracture then develops with a different offset and overall geometry. At E 2 /E 1 = 0.2 for fracture growth through the interface, the pressure at the injection point becomes nearly constant at a higher value because the small parent fracture aperture leads to a high pressure gradient. For the fractures that cross with an offset, the opening compliance of the offset increases as the elastic modulus of the upper layer is reduced. As a result, the injection pressure for these cases decreases to a lower level than for the E 2 /E 1 = 0.2 case. However, the decreasing trend is limited by the pressure required to cause the fluid to open and extend the new fracture. At E 2 /E 1 = 2.0, the stiff upper layer not only provides a high resistance to fracture growth, but decreases the opening compliance of a fracture on the interface. The injection pressure has to increase until the fracture branch can be opened and extended by the fluid. At E 2 /E 1 = 0.5, the pressure peak at about 0.15 s is associated with the establishment of fluid filled fracture across the offset. Prior to this the new fracture is extended some distance by interface slip induced from opening of the 10 of 16

11 Figure 12. layers. Normal stress profile along the fracture and the interface at E 2 /E 1 = 2.0 for granite-sandstone parent fracture, which for this case it is not far away. The growth of the new fracture results in a higher compressive stress that acts across the offset region. However, the compressive stress induced on the interface by this process diminishes for E 2 /E 1 = 1.0 because the offset size is larger and the new fracture is not extended significantly by the opening of the parent fracture Effects of Contrasting Fracture Toughness [39] In addition to modulus contrasts, we now investigate the effects of the toughness contrast on fracture crossing in the siltstone and shale layer combination. The toughness of siltstone is fixed as 0.8 MPa.m 0.5, but that of the shale is assumed to be 0.3, 1.0, 3.0 and 6.0 MPa m 0.5. The extremely large toughness is assigned to the softer shale perhaps justified by plastic deformation, while it is still considered as elastic material for the sake of simplicity. In addition, a medium value of fluid viscosity (m = 0.01 Pas) and a small critical stress (s T = 0.5 MPa) are used. The offset size which is independent of the fracture toughness of the upper layer is 3 cm based on our calculations. [40] Figure 14 shows the fracture trajectories for various values of fracture toughness of the upper layer. It should be noted that the new flaw does not extend and develop into a fracture for K 2 C = 6 MPa m 0.5. In the case of K 2 C = 3 MPa m 0.5, the new fracture path has a larger inclined angle to the original fracture plane compared to those paths for smaller values of K 2 C shown in Figure 14. This change in fracture Figure 13. Time dependence of fluid pressure at the borehole for different modulus ratios in the case of granitesandstone layers. Figure 14. Fracture patterns for different values of the upper-layer rock toughness in the case of siltstone-shale layers. 11 of 16