Hydraulic Fracture Height Growth Limited by

Peer Reviewed

Hydraulic Fracture Height Growth Limited by Interfacial Leakoff Dimitry Chuprakov, Schlumberger

Abstract In this paper, we describe a model for the interfacial fracturing fluid leakoff during the contact of a hydraulic fracture with permeable horizontal interfaces such as interlayer bedding or lamination planes. The hydraulic conductivity of the planes is assumed to have different values near the junctions with the hydraulic fracture, where it is enhanced by slippage, and at far-field zone. We provide a solution giving the redistribution of the fluid volume in the system “fracture-interface”, the behavior of the net pressure and the fracture height growth. Dynamics of the model are illustrated through a parametric sensitivity analysis for the case of a hydraulic fracture crossing one permeable plane and propagating across a series of parallel bedding planes. The model could be a potential additional application for commercial hydraulic fracture simulators.

Introduction

Many geological discontinuities are filled by brittle minerals. Shear activation damages the low-permeability filling plug and greatly enhances hydraulic conductivity of such discontinuities [Olsson and Brown, 1993; Esaki et al., 1999; Samuelson et al., 2009]. Hydraulic fracture contact with the interfaces results in the fracturing fluid percolation into them, thus causing fluid leakoff from the hydraulic fracture [Athavale and Miskimins, 2008]. The fluid leakoff into the k-th interface is quantified by the advance of the percolated fluid front bf(k). The velocity of the latter depends, in particular, on the permeability of this interface κ(k). If the intrinsic part of the interface has low permeability, the maximum extent of fluid penetration bf(k) does not exceed the length of its shear activation bs(k). Figure 1 schematically depicts the shear activation and fluid infiltration into the interfaces that generally accompany the hydraulic fracture growth.

Problem statement

Consider an orthogonal junction of a vertical hydraulic fracture and a horizontal interface. The interface of finite thickness wint is filled with a permeable material. The intrinsic permeability of the filling material in intact interface parts is κi. Suppose a certain segment of the interface, -bs < x < bs, near the junction, is activated by shear displacement as a result of mechanical interaction with the hydraulic fracture. This results in the damage of the filling material within this segment and a change of its permeability to κs (Figure 2). In tight formations, κi can be negligibly small. This condition (κi = 0) can be used later to simplify the leakoff model. On the other hand, the activated part of the interface can be substantially more permeable than the intrinsic part due to the crushed grains of the filling material or shear dilation. Sliding activation of mineralized interfaces can be a dominant mechanism for the fracturing fluid leakoff in ultralow-permeability tight rocks. Let us assume that the fracturing fluid flow along a permeable interface is one-dimensional, steady and laminar. In these conditions, the flow can be described by Darcy’s law as given below:

q ( x ) = − wint

κ dp µ dx

(1)

where q(x) is the 2D rate of fluid percolation within the material of permeability κ, μ is the viscosity of the fluid and p(x) is the fluid pressure distribution along the interface (Figure 2). It is sometimes convenient to replace the product wintκ by the hydraulic conductivity of the interface c, typically measurable in the laboratory (notations cs and ci will be used hereafter, respectively). The total rate of the fracturing fluid leakoff from the hydraulic fracture into the particular interface at the junction point qL is doubled due to symmetrical fluid diversion into both sides of the interface:

qL = 2q ( 0 )

(2)

Due to the symmetry of the fluid percolation into both sides of the interface, in what follows we obtain the solution only for the positive OX direction (x > 0). Darcy’s law (1) establishes the relationship between the local flow rate q and associated fluid pressure decay dp/dx at every point of a permeable material infiltrated by fluid. We write this law first for the flow rate qs and pressure decay ps within the activated (sheared) part as

qs ( x ) = − www.petrodomain.com

cs dps ( x ) µ dx

x ≤ in ( b f , bs )

(3)

21

λ(0), KIIC(0), κ(0) λ(1),

(1),

KIIC

κ(1)

λ(2), KIIC(2), κ(2) λ(3), KIIC(3), κ(3)

σh(0), KIC(0) σh(1) , KIC(1) σh(2) , KIC(2)

λ(4), KIIC(4), κ(4) λ(5), KIIC(5), κ(5) λ(6), KIIC(6), κ(6) λ(7), KIIC(7), κ(7) λ(8),

(8),

KIIC

κ(8)

σh(8) , KIC(8) σh(9) , KIC(9)

Figure 1. Out-of-plane view of plane-strain hydraulic fracture growing vertically in a layered formation and interacting with horizontal interlayer interfaces. Each layer has different stress σh, Mode I fracture toughness KIC and other properties. Each interface has different coefficient of friction λ, Mode II fracture toughness KIIC, permeability κ, and other properties.

Figure 2. The horizontal interface crossed by the vertical hydraulic fracture (top) and schematic distribution of the percolated fluid pressure along the interface (bottom).

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and for the fluid rate qi and pressure pi within the intact part of the interface

qi ( x ) = −

ci dpi ( x ) µ dx

bs < x < b f

(4)

where bf is the front of percolated fluid. Outside of the zone of penetrated fluid we assume in-situ pore pressure condition; i.e.

qi ( x ) = 0, p ( x ) = p p

x ≥ bf

(5)

The solution must include the position of the percolated fluid front bf and the pressure profile p(x) during the leakoff process.

Solution

From the fluid mass balance equation written for incompressible fluid within an interface with impermeable walls (except at the junction point) (6) where ϕ is porosity of the filling material or natural interface asperities, q = qs (x) for x ≤ bs and q = qi (x) for x > bs. It follows that if the width wint is constant (dwint/dt = 0), the flow rate q has uniform value along the interface, being only a function of time; i.e. (7) Taking into account (7) and boundary condition (5) at x = bf, the solution of (3)-(4) for the distribution of the percolated fluid pressure p(x) along the interface indicates a linear decay, as shown in Figure 3.

Figure 3. Profile of fluid pressure along the interface for the “in-slip” (top) and “out-of-slip” (bottom) regimes of percolation. The solution for the pressure profile is written separately for two regimes of fluid percolation into the interface: in-slip percolation, when the leaked fluid is totally contained within the slipped zone of the interface; i.e., bf ≤ bs, and out-of-slip percolation into the intact interface zone also; i.e., bf > bs. For the in-slip leakoff (Figure 3, top), we obtain the following linear pressure profile: (8) where pc = p(0) is the fluid pressure at “contact” with the hydraulic fracture; i.e., x = 0. For the out-of-slip leakoff (Figure 3, bottom), we obtain the following broken-line profile: (9) (10) where p1 = p(bs) is the fluid pressure at the slippage zone tip. In (8)-(10) we take into account that (11)

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.

where u is the lengthwise fluid velocity (upper dot stands for the differentiation with respect to time) equal to the velocity of the percolated fluid . propagation bf . Therefore, from (8)-(10) we obtain the following ordinary differential equations for the propagation of the fluid front bf (t) right after the contact (t > tc) for in-slip fluid penetration: (12) For out-of-slip penetration this is (13) where the fluid pressure at the slip zone tip ( p1 = p(bs) ) is found as (14) where κis = κi/κs, and H(x) is the Heaviside step function (0 for negative and 1 for positive arguments). The solution of (12)-(13) is found for both regimes of fluid penetration as follows: (15) (16) where tc is the time at the beginning of the fracture-interface contact, and Δpc (t´) = pc (t´) - pp is the differential fluid pressure at the interface. The evolution of the differential pressure with time therefore dictates the leakoff process in the given contacted interface.

Numerical evaluation

To include this interfacial leakoff model in a hydraulic fracture simulator, the numerical algorithm must be appropriately adapted. The volume of fracture fluid loss Vlo at every new time step t after the contact with a particular interface can be estimated using the derived percolation flow (15)-(16) as (17) It is therefore necessary to provide correct evaluation of the integral bf1 (15). Given the length of in-slip fluid percolation at the previous time step bf1 (t – Δt), the increment of bf1 at the current time step t is defined by the behavior of the differential pressure Δpc (i.e., contact pressure pc) during the last time step Δt, so that (18) We estimate the integral over the pressure using the values of pressure at the previous and current time steps, pc (t – Δt) and pc (t), respectively, and write (19) At the first time step of the fracture-interface contact, the percolation length can be estimated, assuming that bf1 (tc) = 0 and pc (tc) = pp, as (20) The viscosity of the injected fracturing fluid μ can sequentially change during the fracture treatment. If this change reaches the interfaces of interest, it changes the mobility of the fracturing fluid percolated at the interfaces. This effect is not accounted for in this model rigorously. However, provided that the fluid that penetrates into the interface does not mix, we suggest that the behavior will be defined only by the viscosity of the fluid that penetrates into the interface first.

Parametric sensitivity analysis

Power-law analysis of fluid percolation into an interface Let us assume that the growing hydraulic fracture reaches the permeable interface at t = tc. After the contact, let the differential pressure Δpc grow according to the following polynomial of power m: (21) where tc is the time of first contact. The corresponding law of fluid percolation within the interface according to (15)-(16) is found as (22) 24

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(23) where βf = bf/bs is the fluid-filled-zone extent relative to the slip-zone extent, which designate in-slip and out-of-slip percolation regimes, when βf ≤ 1 and βf > 1, respectively; βf0 =

The presented solution is valid for any m > -1. The fluid percolation length βf is growing with time, with

the rate dependent on the pressure buildup rate m. The rate is also proportional to the increment of differential fluid pressure Δpc (tc) and the permeability of the interface κI and inversely proportional to the viscosity of fracturing fluid μ, as typical in Darcy flow.

5

5

m=-0.5 m=0 m=0.5 m=1

4

Dimensionless percolation length βf

Relative differential pressure buildup ∆pc (t)/∆pc (tc)

Both decreasing and increasing pressure curves for different powers m are plotted in Figure 4 along with the associated propagation of the fluid front with time. Figure 4 (left) shows the differential pressure normalized by the initial value at contact (t = tc). For any variant of the pressure behavior, decrease or increase, the evolution of the fluid is always progressive (Figure 4, right). For the constant fluid pressure at the junction (m = 0), the fluid front monotonically propagates following Carter’s leakoff law ~(t – tc)1⁄2. The faster the pressure builds up, the faster the fluid penetrates. When the pressure is growing in the fracture linearly with time (m = 1), the fluid front also propagates linearly with time ~(t – tc). One can also see that the contrast of permeabilities within the interface κis creates significant changes in the out-of-slip percolation. If the filling material is impermeable (κis = 0), the propagation stops once the fluid reaches the interface. The higher the permeability of the intact interface, the faster the fluid propagates (Figure 5).

3

2

1

0

1

2

3

4

5

m=-0.5 m=0 m=0.5 m=1

4

3

2

1

0

1

2

3

4

5

Dimensionless time of leak-off t/tc Dimensionless time of leak-off t/tc Figure 4. Relative differential pressure buildup within the fracture ∆pc (t)/∆pc (tc) (left) and associated propagation of the fluid penetration front βf (t) (right) for βf0 = 1 and four different powers m within a uniformly permeable interface (κis = 1).

Dimensionless percolation length βf

3

m=1, κis=0

2.5 2

m=1, κis=0.1 m=1, κis=10 m=-0.5, κis=0

1.5

m=-0.5, κis=0.1 m=-0.5, κis=10

1 .5 0

1

1.5

2

2.5

Dimensionless time of leak-off t/tc Figure 5. Growth of the relative length of fluid percolation βf (t) along the interface with various contrasts of permeability in the activated (slipped) zone and intact (non-slipped) zones κis = κi/κs. The red line denotes the faster rate of pressure buildup with m = 1, while the black line denotes slower rate with m = -0.5.

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The presented analysis is simple and illustrative to understand the dynamics of the interfacial fluid leak-off in various pressure decline/growth regimes. However, to understand the realistic pressure behavior within the fracture after contact with a permeable interface, it must be coupled with the fluid injection into the fracture. Fluid-coupled modeling of fracture contacting an interface Consider a vertical plane-strain fracture extending at a constant injection rate and growing symmetrically upward and downward in a homogeneous rock (Figure 6). Let a permeable interface be placed at some distance y = hc from the injection point y = 0. Once the height of the fracture reaches h = hc, the fluid begins to percolate into the interface. At time t = tc, the fracture may stop or continue growing with given leakoff, as shown in Figure 6.

Figure 6. Hydraulic fracture propagating upward and downward in plane-strain geometry (vertical cross-section). There are three distinct stages: (left) precontact with growing fracture without leakoff, (middle) early contact with nongrowing fracture with leakoff and (right) late contact with growing fracture with leakoff. We will suppose that prior to a direct contact with an interface at t = tc, the hydraulic fracture propagates without any elastic or hydraulic interaction. The remotely placed permeable interface is not mechanically activated due to the approaching fracture and thus it does not change the stress state around it. Before the contact, the injected fluid is totally contained within the fracture, as the medium is assumed impermeable. Right after the contact with the interface (t = tc), the fluid flows within the interface and causes a loss of fluid stored in the hydraulic fracture. The fracture continues to grow once fluid loss is compensated for by the injected volume at a later time t = tr > tc. We provide a detailed analysis of the mechanics of fracture propagation affected by the presence of a hydraulically conductive interface on the path of its height growth. In the present analysis for simplicity we will assume 2D plane-strain geometry of fracture propagation in height. Consequently, it will be intrinsically assumed that the injection pressure, flow rate and elastic fracture opening follow 2D fracture propagation behavior rather than 3D with lateral growth. Governing equations The description of the hydraulic fracture height growth in the inviscid fluid limit is given by the following solution of the elasticity equation for the opening w of a pressurized crack with the half-height h [Kachanov et al., 2003; Janssen et al., 2004]: (24) where y is the vertical coordinate, p´ = p – σh is the net fluid pressure, p is the fluid pressure uniformly distributed along the crack, σh is the minimum horizontal in-situ principal stress applied normal to the crack, E´ = E/(1 – ν2) is the modified plane strain Young’s modulus of the rock. We will assume that the plane strain Young modulus E´ and the net fluid pressure p´ do not change along the vertical direction. Integrating this equation gives the elasticity relationship between the 2D volume of injected fluid V (equal to the entire 2D volume of fracture cross section) and the net pressure p´ for a given fracture height h (25) The fluid mass balance equation specifies that the fluid constantly injected into the fracture at the rate Q is distributed between the storage volume of an elastically open hydraulic fracture V and the leaked-off volume Vlo at every time t (t = 0 corresponds to the beginning of the fracture growth from an infinitesimally small size): (26) where the leaked-off volume obeys (17) for the case of a single contacted permeable interface. One can rewrite (15) into a differential equation for bf1 and arrive at the following equation for the leakoff volume in the particular case of a homogeneously permeable interface with permeability κ: (27) where σh´ = σh – pp is the effective minimum horizontal in-situ principal stress. We note that given the different shear-induced permeability κs along the interface in the zone –bs < x < bs (Figure 2), the law of the in-slip percolation is identical to the above: 26

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(28) where Vlo (tc) = 0. For the percolation extending out of the slip zone at t = tcs, the leaked-off volume with (16) obeys the following modified differential equation: (29) where Vlo (tcs)=2wint bs. The growth of the fracture can progress only provided that the stress intensity factor at the tips KI reaches the critical value KIC for the given type of rock. For a plane-strain uniformly pressurized crack we have [Kachanov et al., 2003; Janssen et al., 2004] (30) and, therefore, the fracture with half-height does not grow if (31) Otherwise, the fracture height is growing (32) Normalized equations We normalize all dimensional quantities by their values at the beginning of contact with the interface, i.e. (33) (34) (35) (36) where we introduce the following scales (37) (38) (39) With these dimensionless quantities, we rewrite the equations as follows: Elasticity equation: (40) Tip growth condition, provided that

: (41)

Fluid volume balance: (42) Leakoff dynamics:

(43) (44)

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(45)

and index k denotes s and i, respectively, Σ´h = σh´/p´c and vlos = 2ϕwint bs/Vc.

where

The given set of equations with the initial conditions for the fracture allows one to solve for the problem of the fracture height growth in the presence of a permeable discontinuity. Propagating fracture without leakoff Consider first the precontact fracture propagation at t < 1. Since the leakoff is absent (vlo = 0) (46) If the fracture of a given initial size h = h0 is pressurized by fluid, initially, the growth condition may not be satisfied, i.e.,

, and h(t) = h0 over

some time t < tbr. During this nonfracturing stage, the fluid pressure increases linearly with the time of injection t until the breakdown pressure is achieved at

as (47)

After the breakdown, the fracture will be steadily growing in height proportionally to the injecting volume: (48) The net pressure will begin declining with the fracture tip propagation and the continued fluid injection as (49) The interdependent variation of fluid injection, fracture height and associated pressure decline are plotted in Figure 7.

Figure 7. Dependency of normalized fracture height h/hc, and relative net pressure within the fracture p/p ´ ´0 on the relative fluid volume v/v0. This fracture propagation behavior will be stopped by the presence of a permeable interface, when the fracture tip comes in contact with it. Mechanics of post-contact leakoff and propagation Consider now the post-contact stage of the fracture propagation at t > 1. At this stage, the total injected fluid is partitioned between the hydraulic fracture and the interface where the fluid percolates. Let us take the particular case of uniformly permeable interface, i.e., κis = 1, for simplicity. The set of equations we solve is as follows: Elasticity equation:

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(50)


Condition of tip growth, provided that for the given h,

: (51)

Fluid volume balance:

(52)

Dynamics of leakoff into an interface: (53) (54) where Κ = Κs = Κi. Rewriting the leakoff equation (53) in terms of the fracture storage volume rate at the early time of fracture contact, i.e., t – 1 > 2Κ (1+Σ´h) . We note that the steep pressure drop at the moment of contact with a conductive interface can hardly be visible in most of hydraulic fracturing operations. One reason is that the duration of the early contact stage can be short. Another reason is that this modeling neglects viscous fluid friction within a hydraulic fracture that may smooth this effect away. Propagation through a series of permeable planes Practically, a hydraulic fracture goes through many permeable planes. This enhances the total fluid leakoff from the fracture into the formation and creates an intermittent growth. If the number of interfaces is large, the impact of the interfaces on the height growth can be significant even if the hydraulic conductivity of each plane is small.

Figure 9. Hydraulic fracture propagating across a series of parallel permeable planes. Contact with every subsequent plane triggers new fracturing fluid leakoff into this plane. (j)

Consider the propagation of a hydraulic fracture across a series of parallel permeable planes located at hc positions (j = 1,N), respectively, as shown in Figure 9. For description we can use the same system of equations (50)-(54), and normalize the physical variables by their values reached at contact (1) with the first interface, i.e., hc = hc . In the current case, we assume that fluid leakoff takes place at multiple crossed interfaces simultaneously, so that N (j) vlo (t) = ∑j=1 vlo (t). The dimensionless equations we are solving are as follows: Elasticity equation: (57) Tip growth condition, provided that for the preexisting fracture height h,

: (58)

Fluid volume balance: (59) Leakoff dynamics: (60) (61)

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(62) (j)

(j)

where vlo is the volume of fluid leakoff into the j-th plane normalized on fracture volume at contact with the first interface, vlos is the maximum fluid volume to fill a slip zone of the interface and κ(j)is = Κ(j)i /Κ(j)s, t(j)c is the normalized time of the hydraulic fracture contact with the j-th plane (tc(1)= 1). (j)

The leakoff equations (60)-(61) can be solved, but may cause computational instability at the beginning of contact (t = tc ) because of very small values in the denominator. For the sake of computational robustness, we rewrite these equations in terms of the square of the leakoff volume as follows: (63) (64) where (j)

(j)

(j)

(1)

The solution of these equations for 10 uniformly permeable interfaces with Κs = 1, κis = 1 ( j = 1,10) located equidistantly at hc = jhc is shown in Figure 10.

Figure 10. The injected, fracture and leaked-off fluid volumes (upper), net pressure (middle) and hydraulic fracture half-height during the whole cycle of fluid injection into the fracture. One can see the effect of the dominant leakoff fraction due to numerous permeable planes. The pressure drops are significant only for the few first interactions with the planes, whereas at a later stage, the oscillations of the net pressure become less prominent. In contrast, the delay of fracture height growth becomes longer for more remote interfaces. This prolongation of the early contact stage (where fracture growth stops) is associated with the increase of the total fluid leakoff into all crossed interfaces, and, therefore, the decrease of the fracture volume pumping. The velocity of vertical fracture tip growth is also decreasing with the number of crossed permeable planes (Figure 10). The next numerical study demonstrates the comparison of hydraulic fracture height growth for the various hydraulic conductivities of the interfaces (Figure 11). In Figure 11 one can see the fastest growth of the fracture in a fully impermeable rock (red line), and the slowest growth for interfaces that have the maximum hydraulic conductivities (black line). The planes with permeability limited within the slip zone bs and totally impermeable outside it (blue line) produce less fluid waste than infinitely permeable planes with less conductivity (green line). Of course, this comparison depends on the extent of the slip zone with enhanced permeability, which can progress during the fluid percolation. This investigation requires a separate study.

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Figure 11. The fracture height growth for various permeable properties of the interfaces: (black line) uniformly permeable interfaces with (j) (j) (j) (j) Κs = Κi = 1; (blue line) impermeable interfaces with shear-enhanced permeability (Κs = 1, Κi = 0), (green line) uniformly permeable interfaces with (j) (j) Κs = Κi = 0.1; and (red line) impermeable interfaces.

Conclusions

The analytical model for the fluid leakoff from a hydraulic fracture into permeable horizontal discontinuities such as interlayer bedding or lamination planes shows that the discrete leakoff into the discontinuities produces noticeable effect on the vertical fracture growth. Sufficiently conductive interfaces temporarily terminate the fracture tip growth at the early contact time. After the crossing of the interfaces, the fracture propagation velocity decreases due to injected fluid losses. Fracture height growth in a formation with series of hydraulically conductive interfaces is reduced proportionally to the number of interfaces and their conductivity. The partial enhancement of the interface conductivity in the near-junction slip zone may not produce large effect on the fluid loss, as infinitely permeable interfaces do. Depending on the permeability of the in-fill material, the leakoff can be a dominant mechanism for containment of hydraulic fracture height growth.

Nomenclature

bf bf(k) . bf bf1 bs bs(k) cs ci E E´ H(∙) h h0 hc (k) hc j KI KIC (k) KIC (k) KIIC k m N p p´ pc´ pc pp 32

= length of the fracturing fluid percolation along an interface nearby a contact with hydraulic fracture, m = length of the fracturing fluid percolation along the k-th interface nearby a contact with hydraulic fracture, m = velocity of fracturing fluid front percolation within an interface, m/s = length of the “in-slip” fracturing fluid percolation along the interface, m = length of the shear activated zone along an interface nearby a contact with hydraulic fracture, m = length of the shear activated zone along the k-th interface nearby a contact with hydraulic fracture, m = hydraulic conductivity of the interface within an activated (damaged) zone, Darcy*m ≈ 10-12 m3 = hydraulic conductivity of the interface within an intact (undamaged) zone, Darcy*m ≈ 10-12 m3 = Young modulus of the rock layer, Pa = modified Young modulus for plane strain, Pa = stepwise function of Heaviside = fracture height, m = initial fracture height, m = critical hydraulic fracture height, when its tip reaches an interface, m = critical hydraulic fracture height, when its tip reaches the k-th interface, m = index of the interface = Mode I stress intensity factor at the fracture tip, Pa*m1/2 = Mode I fracture toughness in the rock, Pa*m1/2 = Mode I fracture toughness within the k-th layer, Pa*m1/2 = Mode II fracture toughness along the k-th interface, Pa*m1/2 = index of interface/layer numeration, ascending from the top to the bottom = arbitrary power of the contact pressure increase or decrease with time = total number of interfaces = fluid pressure within a hydraulic fracture, Pa = net pressure within a hydraulic fracture, Pa = scaling net pressure within a hydraulic fracture, Pa = fluid pressure at the contact point of the hydraulic fracture and interface, Pa = pore fluid pressure at the interface far away from the contact point, Pa

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ps pi p1 Δpc Q q qL qs qi t tc

= fluid pressure within a shear activated part of the interface, Pa = fluid pressure within an intact part of the interface, Pa = fluid pressure at the boundary of the slippage zone, Pa = difference between the contact and pore fluid pressures within an interface, Pa = 2D rate of constant injection into a hydraulic fracture, m2/s = 2D rate of fluid percolation within one wing of the interface, m2/s = total 2D fluid percolation rate within from the hydraulic fracture into the interface, m2/s = 2D fluid percolation rate within a shear activated part of the interface, m2/s = 2D fluid percolation rate within an intact part of the interface, m2/s = time of fluid injection into a hydraulic fracture, s = time of the contact between a hydraulic fracture and an interface, s

tc(j) t tbr Δt V Vc Vlo v vlo vlos

= time of the contact between a hydraulic fracture and the j-th interface, s = normalized time of fluid injection into a hydraulic fracture = normalized time of fluid injection into a hydraulic fracture, when the initial fracture starts to grow = time step, s = total 2D fracturing fluid volume injected into a hydraulic fracture, m2 = scaling 2D fluid volume, m2 = total 2D fluid leakoff volume along one interface, m2 = normalized total 2D fracturing fluid volume injected into a hydraulic fracture = normalized total 2D fluid leakoff volume along one interface = normalized total 2D fluid leakoff volume along one interface when the slip zone boundary is reached by the percolated fluid

vlo(j)

= normalized total 2D fluid leakoff volume along the j-th interface

(j)

vlos

= normalized total 2D fluid leakoff volume along the j-th interface when the slip zone boundary is reached by the percolated fluid within this j-th interface

v~lo(j) w wint x z

= dimensionless parameter = mechanical opening of the hydraulic fracture, m = interface thickness, m = horizontal coordinate along the interface, m = vertical coordinate perpendicular to interfaces, m

βf βf0 βf1

= ratio of the percolated fluid length bf to the slip zone length bs = dimensionless parameter = ratio of the “in-slip” percolated fluid length bf to the slip zone length bs

κ(k) Κ Κs Κi

= hydraulic permeability of the k-th interface, Darcy≈10-12 m2 = normalized permeability of the uniformly permeable interface = normalized permeability of the interface within an activated (damaged) zone = normalized permeability of the interface within an intact (undamaged) zone

Κs (j)

= normalized permeability of the j-th interface within an activated (damaged) zone

Κi (j) κs κi κis

= normalized permeability of the j-th interface within an intact (undamaged) zone = hydraulic permeability of the interface within an activated (damaged) zone, Darcy≈10-12 m2 = hydraulic permeability of the interface within an intact (undamaged) zone, Darcy≈10-12 m2 = ratio of hydraulic permeabilities outside and inside of the activation zone

κis (j)

= ratio of hydraulic permeabilities outside and inside of the activation zone of the j-th interface

(k)

λ μ ν

= coefficient of friction between the edges of the k-th interface = fracturing fluid viscosity, Pa*s = Poison coefficient of the rock layer

Πbr´ Σ´h

= normalized net pressure within a hydraulic fracture, when the initial fracture starts to grow = ratio of the effective minimum stress and net pressure at the fracture-interface contact

σh´

= minimum effective horizontal stress acting normally to the hydraulic fracture (along OX), Pa

σh ϕ

= minimum horizontal stress acting normally to the hydraulic fracture (along OX) within the k-th layer, Pa = porosity of the filling material inside an interface

Π´

Σ´h(j) σh (k)

= normalized net pressure within a hydraulic fracture

= ratio of the effective minimum stress and net pressure at the contact of a hydraulic fracture with j-th interface = minimum horizontal stress acting normally to the hydraulic fracture (along OX), Pa

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Acknowledgements

Author is grateful to Dr. Romain Prioul for careful revision of the paper, Dr. Xiaowei Weng, Dr. Jeffrey Burghardt, and Dr. Wenyue Xu for the fruitful discussion and useful remarks about this study that helped to improve it. I also thank Schlumberger for allowing us to publish this work.

References

Athavale, A. S. and J. L. Miskimins (2008) “Laboratory hydraulic fracturing tests on small homogeneous and laminated blocks”, paper presented at the 42nd U.S. Rock Mechanics Symposium (USRMS), 08-067, American Rock Mechanics Association, San Francisco, CA. Esaki, T., S. Du, Y. Mitani, K. Ikusada and L. Jing (1999) “Development of a shear-flow test apparatus and determination of coupled properties for a single rock joint”, Int J Rock Mech Min, 36(5), pp. 641-650. Janssen, M., J. Zuidema, and R. J. H. Wanhill (2004) “Fracture Mechanics”, Spon Press, Taylor & Francis Group. Kachanov, M., B. Shafiro and I. Tsukrov (2003) “Handbook of Elasticity Solutions”, 324 pp., Kluwer Academic Publishers, Dordrecht. Olsson, W. A. and S. R. Brown (1993) “Hydromechanical response of a fracture undergoing compression and shear”, Int J of Rock Mech and Min, Geomechanics Abstracts, 30(7), pp. 845-851. Samuelson, J., D. Elsworth and C. Marone (2009) “Shear-induced dilatancy of fluid-saturated faults: Experiment and theory”, J. Geophys. Res., 114(B12), p. B12404.

Biography Dimitry Chuprakov is a research scientist at Schlumberger Moscow Research, Moscow, Russia. He was also working at Schlumberger-Doll Research center, Boston, USA from 2010 to 2015. He joined Schlumberger in 2004. His current research interests include modeling of the hydraulic fracture propagation, fracture interaction, rock failure, poroelastic and other geomechanical problems. He holds a PhD degree in Physics and Mathematics from Moscow State University.

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Volume 2 - Number 4
