c Allerton Press, Inc., 2009. ISSN 0027-1330, Moscow University Mechanics Bulletin, 2009, Vol. 64, No. 6, pp. 135–142.  c V.R. Tagirova, 2009, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2009, Original Russian Text  Vol. 64, No. 6, pp. 33–41.

Hydraulic Fracture Crack Propagation Driven by a Non-Newtonian Fluid V. R. Tagirova Moscow State University, Faculty of Mechanics and Mathematics, Leninskie Gory, Moscow, 119899, Russia Received July 20, 2007; in final form, September 1, 2009 Abstract—The problem of hydraulic fracture crack propagation in a porous medium is considered. The fracture is driven by an incompressible viscous fluid with a power-law rheology of the pseudoplastic type. The fluid seepage is described by an equation generalizing the Darcy law in the hydraulic approximation. It is shown that the system of governing equations has a power-law self-similar solution, whereas, in the limiting cases of low and high fluid saturation in the porous medium, there are some families of power-law or exponential self-similar solutions. The complete self-similar solution is constructed. The effect of the nonlinear rheology of the fracturing fluid on the behavior of the solution is studied. The problem is solved analytically for an arbitrary boundary condition at the crack inlet when the viscous stresses in the non-Newtonian fluid are close to a constant.

DOI: 10.3103/S0027133009060016 INTRODUCTION This paper is devoted to the study of hydraulic fracture crack propagation in a porous medium under the assumption that the hydraulic fracturing fluid is non-Newtonian. It is also assumed that the crack width is much less than its length and height, the incompressible fluid flow is inertialess, the mass forces can be ignored, and the stress state in a cross section perpendicular to the direction of crack propagation is described in the framework of the plane section hypothesis. As a model of the non-Newtonian fluid, we adopt a fluid with a power-law rheology in which the viscous stress tensor and the strain rate tensor are related by a power law in such a way that the apparent viscosity coefficient decreases when the strain rate increases. The crack evolution is characterized by the crack opening width, the cross-section averaged fluid velocity in the crack, and the saturation depth of the hydraulic fracturing fluid in the medium. The system of governing equations has a self-similar solution when, at the crack inlet, a fluid flow rate is given as a quadratic function or a fluid pressure is given as a linear time function. When the fluid saturation is low or high, the problem has a one-parameter family of power-law and exponential self-similar solutions. As a parameter of this family, we consider the exponent in the power or exponential time-variation law for the fluid flow rate or the fluid pressure at the crack inlet. The exponential solutions become close to the power-law self-similar solutions when this parameter tends to infinity. The hydraulic fracture technology is one of the efficient ways of oil well recovery improvement in an oilbearing stratum. Cracks are developed under the pressure of a hydraulic fracturing viscous fluid and begin to propagate from the well to the stratum depth; this process increases the effective area of the surface through which the oil-bearing fluid is displaced. A number of theoretical works [1–10] are devoted to the problem of hydraulic fracture crack propagation. A crack-filling hydraulic fracturing fluid usually has a nonlinear rheology. The hydraulic fracture crack formation is considered in [5–7] in the framework of the model [3, 4] with a power-law rheology of the pseudoplastic type. The thixotropic fluid flow in a porous medium is discussed in [8]. In this paper we study the flow of a pseudoplastic fluid with a power-law rheology [1, 5–7, 11] in the framework of the model proposed in [1, 2]. Some families of self-similar solutions to this problem are considered in [9, 10] for a Newtonian fluid. Here we propose an extension of these families for the case of nonlinear viscous hydraulic fracturing fluids. The effect of the nonlinear rheology of the fluid on the behavior of the solution is studied. The problem is also solved in the limiting case when the viscous stresses in a non-Newtonian fluid are close to the constants for a simple shear flow. 135

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FORMULATION OF THE PROBLEM Let us consider the process of hydraulic fracture crack propagation in a porous medium in the framework of the model proposed in [1, 2, 9]. It is assumed that the crack width ω is much less than its length L and its constant height h (ω  h  L) and the plane section hypothesis is fulfilled, i.e., the stress states in two sections normal to the direction of crack propagation are independent. Near the crack, the rock is assumed to be elastic and porous in each cross section. The crack existing as a cut opens in the direction of the Ox axis. Near the lateral surface of the crack, the fluid seepage is locally one-dimensional along the Oy axis. The excess pressure of the fluid is measured from the rock pressure at a large distance of the crack. It is assumed that the hydraulic fracturing fluid is incompressible; the mass forces and the flow inertia are ignored. By virtue of the plane section hypothesis, we can assume [1, 2] that the relation between the excess pressure and the height-averaged width is linear (see [9]): P = bω,

b=

4μσ . π(1 − νσ )h

(1)

Here P (t, x) is the excess pressure of the fluid in the crack, ω(t, x) is the height-averaged width of the crack opening, νσ is Poisson’s ratio, and μσ is the material’s shear modulus. The mass conservation law takes the form ∂ω ∂ωu + + 2v = 0, (2) ∂t ∂x where u(t, x) is the width-averaged velocity of the fluid along the crack and v(t, x) is the seepage velocity of the fluid through the porous medium. The crack propagation proceeds under the pressure of the non-Newtonian fluid described by the relation τij = μa eij , where τij are the components of the viscous stress tensor, eij are the components of the strain rate tensor, and μa is the apparent viscosity coefficient dependent on the second invariant of the strain rate tensor. For the model of a fluid with the power-law rheology, we have [11]  β−1 I2 , I2 = eij eij , μa = M 2 where M is a constant expressed in units Pa cβ and β is a dimensionless constant characterizing the exponent in the rheological relation. For a Newtonian fluid, we have β = 1 and M = μ, where μ is the dynamic viscosity coefficient of the fluid. In oil industry applications, a hydraulic fracturing fluid is usually described as pseudoplastic without yield stress; in this case the apparent viscosity coefficient μa decreases when the shear rate increases, 0 < β < 1. The motion of an incompressible fluid is described in the projection onto the Ox axis in the hydraulic approximation by the equation [11]       ∂P   1/β (1+β)/β ∂P   ω , u = − sgn  ∂x  M ∂x where M  = φM and φ = (2β + 1)β 2β+1 /β β (for a Newtonian fluid, we have M  = 12μ). Based on the physical considerations, we assume that ∂P/∂x ≤ 0; hence, the average velocity of the fluid can be written as ω β+1 ∂P uβ = − . (3) M  ∂x The outflow of the fluid from the crack is considered in the framework of the locally one-dimensional approximation. In order to describe the fluid flow in the porous medium, we use the hydraulic approximation for the equation of motion for a non-Newtonian fluid through a narrow crack (this equation is averaged over the crack width). We also use the equation of motion for a non-Newtonian fluid in a porous medium as a set of thin small cracks of characteristic width d situated perpendicularly to the crack propagation axis. In other words, we assume that, in its dimensionless form, the outflow velocity v(t, x) satisfies the equation vβ = −

dβ+1 ∂Pr , M  ∂y

∂v = 0, ∂y

(4)

where Pr (t, x, y) is the excess pressure of the fluid in the porous medium. MOSCOW UNIVERSITY MECHANICS BULLETIN

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For the non-Newtonian fluid, we assume that the permeability coefficient κ of the medium is proportional to d2 /12 (if β = 1, then Eq. (4) can be reduced to the Darcy law). In this case the filtration law takes the form (12k)(β+1)/2 ∂Pr . (5) vβ = − M ∂y Initially, the surrounding stratum is saturated by an oil-bearing fluid whose viscosity is negligibly small compared to the viscosity of the hydraulic fracture fluid. We can ignore the drag of the less viscous fluid compared to the drag of the more viscous fluid. From the pressure continuity condition, then, we conclude that the excess pressure of the hydraulic fracturing fluid is equal to zero at the displacement boundary. The true relative velocity of the fluid in the porous medium is also defined as v(t, x) = ∂Γ/∂t, where 2β/(1+β) Γ(t, x) is the depth of saturation in the rock measured from the crack. Let k = κ/mr , where mr is the porosity of the medium; here k is expressed in the same units as for the permeability of the medium. Integrating (5) over the saturation region ω/2 ≤ y ≤ Γ + ω/2 and substituting the boundary conditions Pr (t, x, ω/2) = P (t, x) and Pr (t, x, Γ + ω/2) = 0, we obtain  v = (12k)(β+1)/(2β)

P M Γ



1/β ,

Γ

∂Γ ∂t

β =

(12k)(β+1)/2 P. M

(6)

From (1), (2), and (6), we come to the following complete system of equations describing the crack propagation in the porous medium under the action of the wedge flow of the non-Newtonian fluid: ωt + 2Γt + (ωu)x = 0, uβ = −

b 1+β ω ωx , M

 β b(12k)(β+1)/2 Γ Γt = ω. M

(7)

At the crack end x = L(t), we assume that its width ω(t, L), the saturation depth Γ(t, L), and the relative fluid flow rate u(L)ω(L) are equal to zero. At the crack inlet x = 0, we choose the volume fluid flow rate Q = huω or the fluid pressure P = bω. NONDIMENSIONALIZED EQUATIONS Let (b/M  )1/β = k  and (12k)(β+1)/2 = 12k  . From system (7), then, we get   (ω + 2Γ)t + K  ω 2+1/β (−ωx )1/β x = 0,

(8) Γ(Γt )β = (K  )β 12k  ω.   Here we have only the following two dimensional parameters: [K  ] = 1 t∗ (L∗ )1/β and [k  ] = (L∗ )β+1 , √  1/β where L∗ = 12k ∼ 10−6 m is the characteristic length scale. Let t∗ = φM/(bL∗ ) ∼ 10−2 s be the 2 characteristic time scale. In order to describe the above processes for x ∼ 10 m, we introduce the contraction factor ε = 10−8 . Using the change of variables √ √ ε ε ε ¯ t = ∗ t, ω ¯ = ∗ ω, x ¯ = ∗ x, L t L √ √ ∗ t ε ¯ = ε Γ, u ¯ = ε L, Γ ¯= u, L ∗ ∗ L L L∗ we nondimensionalize system (8) whose terms become of order unity. As a result, we obtain

 1/β  ¯ t¯ + ω ωx¯ ¯ 2+1/β −¯ = 0, (¯ ω + 2Γ) x ¯ (9)  1/β ¯ Γ ¯ t )β = ω u¯ = ω ¯ 1+1/β −¯ ωx¯ , Γ( ¯. u = K  ω 1+1/β (−ωx )1/β ,

 1/β ¯ we have ω ¯ = 0, and ω ωx¯ At the crack end x ¯ = L, ¯ = 0, Γ ¯ 2+1/β −¯ = 0. At the crack inlet, we choose ¯ = u¯ω the volume fluid flow rate Q ¯ or the fluid pressure P¯ = ω ¯. In order to find the self-similar solutions, the boundary condition is given in the form Q(t) =√Q0 Φ(t) or α in the form P (t) = P0 Φ(t), where the dimensionless time law has the power-law form Φ(t) = (t ε/t∗ ) or MOSCOW UNIVERSITY MECHANICS BULLETIN

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∗

the exponential form Φ(t) = eαt ε/t . Here α is the dimensionless parameter characterizing the variation of fluid injection into the crack as a time function. The proportionality coefficients Q0 and P0 characterize the intensity of fluid injection into the well. The dimensionless coefficients take the form ¯0 = Q

εt∗ Q0 , h(L∗ )2

√ t∗ ε P0 . P¯0 = 12μ

¯ = Q ¯ 0 t¯α or In terms of the dimensionless variables, the power-law boundary condition is given as Q ¯ ¯ α α t α t ¯=Q ¯ 0 e or P¯ = P¯0 e . P¯ = P¯0 t¯ , whereas the exponential boundary condition is given as Q In the plane (t, x), for each solution we can construct the line Γ = ω. Depending on the order of Γ/ω, the set of solutions can be divided according to the following three cases: the low saturation (Γ/ω  1), the high saturation (Γ/ω  1), and the finite saturation (Γ/ω ∼ 1). In the last case, system (9) is notchanged.  1/β In the first two limiting cases, the mass balance equation takes the form ω ¯ t¯ + ω ¯ 2+1/β −¯ = 0 or ωx¯ x ¯

   1/β ¯ t¯ + ω ¯ 2+1/β −¯ = 0. the form 2Γ ωx¯ x ¯

In our further discussion, we omit the bar over the dimensionless variables. From (9) it follows that, in each limiting case of saturation, there exists a one-parameter family of selfsimilar solutions of the power-law or exponential form. The family parameter α is given in the boundary condition at the crack inlet. In the case of finite saturation, we have only a power-law self-similar solution for each boundary condition at the crack inlet.

CONSTRUCTION OF SELF-SIMILAR SOLUTIONS The self-similar solutions are sought in the following power-law form: ξ = x/tm ,

u(t, x) = tr U (ξ),

Γ(t, x) = ts Y (ξ),

ω(t, x) = tn D(ξ).

(10)

If Γ/ω ∼ 1, such solutions exist only for n = r = s = 1 and m = 2, i.e., when the boundary condition at the crack inlet is Q(t) = t2 (then, α = 2) or P (t) = t (then, α = 1). The self-similar form of system (9) is nD − mξD + (DU ) + 2 U = −D β

1+β



D,



D Y

1/β = 0,

(11)

 β

Y (sY − mξY ) = D.

If Γ/ω  1 or Γ/ω  1, then the balance mass equation is modified; its new form is specified by the equation  1/β D = 0. (12) nD − mξD + (U D) = 0 or (U D) + 2 Y At the crack end ξ0 = L/tm , the following conditions are fulfilled: D(ξ0 ) = 0, Y (ξ0 ) = 0, and limξ→ξ0 D3 D = 0. At the crack tip ξ = 0, the boundary condition corresponds to D(0)U (0) = Q0 or D(0) = P0 , depending on the cases of the known volume fluid flow rate or the known fluid pressure, respectively. Here it is assumed that, at the crack tip, the boundary condition is given as Q(t) = Q0 tα or P (t) = P0 tα . The dependence n, m, r, and s on α and β is illustrated in Table 1. In the cases of low and finite saturation, near the crack end the asymptotic approximations for the crack width, the saturation depth, and the fluid velocity along the crack become as follows:  1/(2+β) , D(ξ) ≈ (mξ0 )β/(2+β) (2 + β)(ξ0 − ξ) (1+β)/(2+β)  (2 + β)(ξ0 − ξ) Y (ξ) ≈ , (1 + β)β/(1+β) (mξ0 )β/(2+β)

U (ξ) ≈ mξ0 .

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HYDRAULIC FRACTURE CRACK PROPAGATION Table 1 Parameters of the model

Boundary condition Φ

n

m Γ/ω  1,

Φ=t

r

s

α

Q

α(1 + β) + 1 2β + 3

α(β + 2) + 2β + 2 2β + 3

α(β + 2) − 1 2β + 3

α + 2β + 1 2β + 3

P

α

α(β + 2) + β β+1

α(β + 2) − 1 β+1

α+β β +1

Γ/ω  1,

Φ = tα

Q

α(1 + β)2 + 1 2β 2 + 4β + 3

3α(1 + β)3 + 1 + β 2β 2 + 4β + 3

α(β 2 + 2β + 2) + 1 2β 2 + 4β + 3

α(1 + β)2 + 2β 2 + 4β + 4 (1 + β)(2β 2 + 2β + 4

P

α

α(2 + β) + β (1 + β)2

α(2 + β)2 + 1 (1 + β)2

α+β 1+β

In the case of high saturation, the asymptotic approximation of the solution takes the form

mξ0 2β(1+β) (2β 2 + 3β + 2)β(2+β) β 2 +β+1 D(ξ) ≈ 2 +β−1 (ξ0 − ξ) 2 1+β 2 β (β + β + 1) (2β + 2β + 1) 

1/(2β 2 +3β+2)

2

2 2β (2β 2 + 3β + 2)2β +2β+1 (ξ0 − ξ)2β +2β+1 Y (ξ) ≈ (mξ0 )β(1+2β) (β 2 + β + 1)(2β 2 + 2β + 1)2β(β+1)



, 1/(2β 2 +3β+2) ,

1/(2β 2 +3β+2) 2+β β(1+β) 2 mξ 0 U (ξ) ≈ (β 2 + β + 1)(2β 2 + 3β + 2)(β +2β +β−2)/β (ξ0 − ξ)β(1+β) . (2β 2 + 2β + 1)β 2 +β−1 Now we consider the general case of finite saturation when the volume fluid flow rate Q(t) = Q0 t2 is given at the crack tip (see (11)). For this case the solutions are illustrated in Figs. 1–3. When the conditions at the crack end are fulfilled, the numerically obtained values of ξ0 are equal to 0.2066, 0.2108, and 0.2156 for β = 1, 1/2, and 1/3, where ξ0 is the self-similar coordinate of the crack end. In the case of (12), we have ξ0 = 0.4265, 0.4516, and 0.4639 for the same values of β. The self-similar coordinate ξ0 specifies the crack length L = ξ0 tm . The dimensionless value of this length increases with decreasing β. The dependence of the dimensionless self-similar solutions on β are also represented in these figures. For the fluids with a nonlinear rheology (β < 1), the fluid velocity U along the crack is larger, whereas the crack width D is smaller than those for the case of a Newtonian fluid (β = 1) if the boundary condition is the same (Figs. 1 and 3). The behavior of the saturation depth Y is ambiguous with in- Fig. 1. Dependence of the crack self-similar width creasing β (at least, Y increases at the crack tip, see Fig. 2). D on the self-similar coordinate ξ for t = 1 and In the crack the dimensionless velocity of the non- Q(t) = t2 and for the following values of β: (1) 1, (2) 1/2, and (3) 1/3. Newtonian fluid becomes larger than that of the Newtonian fluid. For the above condition at the crack inlet, in other words, the use of the non-Newtonian fluid may cause an increase in the crack propagation velocity compared to the case of the Newtonian fluid. This conclusion is valid when the inlet pressure is given as a linear time function. Remark. The parameter β is implicitly used in the nondimensionalization process, since the time scale t∗ depends on β. The effect of β on the dimensional solution is specified by the parameters α, t, Q0 , or P0 . The general dependence of the solution on these parameters can be studied by analyzing the asymptotic behavior of the solution near the crack end when the boundary condition is formally satisfied at the crack tip. It can 

3

2

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be shown that the effect of β on the behavior of the dimensional self-similar solutions for the crack width and the fluid velocity is in good agreement with that of the dimensionless solutions for α = 2 and Q0 = 1. When the fluid pressure P is equal to t, the dimensional fluid velocity and the crack length decrease with increasing β.

Fig. 2. Dependence of the self-similar saturation depth Y on the self-similar coordinate ξ for t = 1 and Q(t) = t2 and for the following values of β: (1) 1, (2) 1/2, and (3) 1/3.

Fig. 3. Dependence of the fluid self-similar velocity U on the self-similar coordinate ξ for t = 1 and Q(t) = t2 and for the following values of β: (1) 1, (2) 1/2, and (3) 1/3.

EXPONENTIAL SELF-SIMILAR SOLUTIONS In the cases of low and high saturation, problem (9) has only families of power-law and exponential self-similar solutions with the parameter α. The exponential solutions are also called the limit ones to the self-similar solutions [12, 13]. Now we show that it is possible to obtain a family of exponential self-similar solutions from a family of power-law self-similar solutions with the aid of the following passage to the limit: α → ∞. Let us consider the family of power-law solutions of the form (10) for n, m, r, and s taken from Table 1. Let x = xe be a shift and t = 1 + me te /m be the extension in the direction of t, where me is a constant. These transformations preserve the form of the equations. When α → ∞ and β is fixed, we have m → ∞; then, xe xe x ξ = m =  m/(me te ) me te → ξe = eme te , t 1 + me te /m where te , xe , and me are fixed. For example, now we consider the Newtonian fluid (β = 1) and the case of low saturation when the volume fluid flow rate is given at the crack inlet. Since lim n/m = 2/3 (see Table 1), the following passage α→∞ to the limit is valid when α → ∞ and te and D are fixed: nme te /m

ω = tn D (1 + me te /m)m/(me te ) D → ωe = e2me te /3 D.

(13)

From Table 1 it follows that limα→∞ s/m = 1/3, limα→∞ r/m = 1, and limα→∞ α/m = 5/3. Similarly to (13), from (10) and the above boundary condition we get the following relations when α → ∞ and te , Y , U , and Q are fixed: Γ → Γe = eme te /3 Y,

u → ue = eme te U,

Q → Qe = e5me te /3 Q.

Let αe = 5me /3, ne = 2αe /5, se = αe /5, and re = 3αe /5. Then, the functions ξe = xe /eme te , ωe = ene te D, Γe = ese te Y , and ue = ere te U can be considered as a family of exponential self-similar solutions to system (9) with the boundary condition Qe = Q0 eαe te at the crack tip. MOSCOW UNIVERSITY MECHANICS BULLETIN

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Table 2 contains the values of ne , me , se , and re for the case of the non-Newtonian fluid (0 < β < 1) when the inlet volume fluid flow rate and the inlet fluid pressure are given and the saturation modes are low and high. Table 2 Boundary condition Φ

Parameters of the model n

m Γ/ω  1,

Φ∼e

r

s

αt

Q

α(1 + β) 2β + 3

α(β + 2) 2β + 3

α(β + 2) 2β + 3

α 2β + 3

P

α

α(β + 2) β+1

α(β + 2) β+1

α β+1

Γ/ω  1,

Φ ∼ eαt

Q

α(1 + β)2 2β 2 + 4β + 3

3α(1 + β)3 2β 2 + 4β + 3

α(β 2 + 2β + 2) 2β 2 + 4β + 3

α(1 + β)2 (1 + β)(2β 2 + 2β + 4)

P

α

α(2 + β) (1 + β)2

α(2 + β)2 (1 + β)2

α 1+β

THE CASE OF CONSTANT VISCOUS STRESSES Let us consider system (7) with the boundary conditions when the viscous stresses τxy in the fluid are constant (τxy = M ) under a simple shear. Let the apparent viscosity μa be equal to M 2/I2 (eij ) = M/exy . In this case, β = 0. Near β = 0, then, an approximate solution to this system can be sought in the form ω(t, x) = ω0 (t, x) + βω1 (t, x),

Γ(t, x) = Γ0 (t, x) + βΓ1 (t, x),

u(t, x) = u0 (t, x) + βu1 (t, x).

(14)

Using the representation of an exponential function in the form aβ ≈ 1 + β ln a, where β ≈ 0, we substitute (14) into (9). Equating the terms at the zeroth and first powers of β (the terms of higher order are ignored), from the third equation of (9) we get Γ0 = ω 0 ,

Γ1 + Γ0 ln Γ0,t = ω1 .

(15)

At the zeroth approximation, thus, the crack width and the saturation depth are equal. From the second equation of (9), we obtain    u0 (ω1 ω0 )x + ln = 0. ω0 = 2(L(t) − x), ω0

(16)

From the first equation of (9), it follows that ˙ u0 = 3L,

(ω1 + 2Γ1 )t + (ω1 u0 + u1 ω0 )x = 0.

Using (15) and (16), we conclude that     u0 ω0 ω1 = +1 , 2 ln 4 ω0 Substituting (18) into (17), we get

Γ1 =

ω0 4

    9ω0 +1 . 2 ln u0

(17)

(18)

  ¨ 2 Lω0 ω0 − u1 = 2L˙ ln . L˙ 6L˙

If we know the inlet volume fluid flow rate Q(t) or the inlet fluid pressure P (t), then we can find the crack length L(t): Q(t) dL3/2 = √ . Q) Q(t) = u0 (t, 0)ω0 (t, 0), dt 2 2 2 P (t) . P ) P (t) = ω0 (t, 0), L(t) = 2

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The functions ω0 , Γ0 , u0 , and L constitute a dimensionless solution to system (7) for β = 0. These functions are specified by the boundary condition of the form Q(t) or P (t), which is an advantage of this solution. CONCLUSION In the framework of the hydraulic approximation and the plane section hypothesis and for the model under consideration, we propose a system of equations to describe the hydraulic fracture crack propagation in a porous medium when the driven fluid with a power-law rheology of the pseudoplastic type is nonNewtonian. In the limiting cases of fluid saturation through a stratum, this system allows the existence of only power-law and exponential self-similar solutions. If the fluid saturation is finite, then the complete self-similar solution exists when the inlet volume fluid flow rate is given as a quadratic function or the inlet fluid pressure is given as a linear time function. An exact self-similar solution is constructed when the volume fluid flow rate is given at the crack inlet. It is shown that the use of a non-Newtonian hydraulic fracturing fluid may increase the velocity of hydraulic fracture crack propagation. It is shown analytically that the family of exponential solutions is a limit one with respect to the family of power-law solutions when the family parameter tends to infinity. When the components of the viscous stress tensor in the non-Newtonian fluid are constant for simple shear flows, the complete solution to the problem can be obtained for arbitrary injection conditions. This allows one to construct and study approximate solutions for pseudoplastic fluids with a small exponent in the rheological law. ACKNOWLEDGMENTS The author is grateful to N.N. Smirnov and A.N. Golubyatnikov for useful discussions and helpful remarks. This work was supported by the Russian Foundation for Basic Research (project nos. 06–08–00009 and 09–08-00265). REFERENCES 1. T. K. Perkins and L. R. Kern, “Widths of Hydraulic Fractures,” J. Pet. Technol. 13 (9), 937–949 (1961). 2. R. Nordgren, “Propagation of Vertical Hydraulic Fractures,” Soc. Pet. Eng. J. 12 (4), 306–314 (1972). 3. Yu. P. Zheltov and S.A. Khristianovich, “On Hydraulic Fracturing of an Oil-Bearing Stratum,” Izv. Akad. Nauk SSSR, Otdel Tekh. Nauk, No. 5, 3–41 (1955). 4. J. Geertsma and F. De Klerk, “A Rapid Method of Predicting Width and Extent of Hydraulic Induced Fractures,” J. Pet. Technol. 21 (12), 1571–1581 (1969). 5. J. I. Adachi and E. Detournay, “Self-Similar Solution of a Plane-Strain Fracture Driven by a Power-Law Fluid,” Int. J. Numer. Anal. Methods Geomech. 26 (6), 579–604 (2002). 6. D. I. Garagash, “Evolution of a Plane-Strain Fracture Driven by a Power-Law Fluid,” in Electronic Proc. 16th ASCE Eng. Mech. Conf., July 16–18, 2003 (University of Washington, Seattle, 2003), pp. 219–234. 7. D. I. Garagash, “Transient Solution for a Plane-Strain Fracture Driven by a Shear-Thinning, Power-Law Fluid,” Int. J. Numer. Anal. Methods Geomech. 30 (14), 1439–1475 (2007). 8. D. Pritchard and A. Pearson, “Viscous Fingering of a Thixotropic Fluid in a Porous Medium or a Narrow Fracture,” J. Non-Newtonian Fluid Mech. 135 (2), 117–127 (2006). 9. O. E. Ivashnev and N. N. Smirnov, “Formation of Hydraulic Fracture Cracks in a Porous Medium,” Vestn. Mosk. Univ. Ser. 1: Mat. Mekh., No. 6, 28–36 (2003) [Moscow Univ. Mech. Bull. 58 (6), 12–20 (2003)]. 10. N. N. Smirnov and V. R. Tagirova, “Self-Similar Solutions of the Problem of Formation of a Hydraulic Fracture in a Porous Medium,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 1, 70–82 (2007) [Fluid Dynamics 42 (1), 60–70 (2007)]. 11. M. J. Economides and K. G. Nolte, Reservoir Simulation (Wiley, New York, 2000). 12. L. I. Sedov, Similarity and Dimensional Methods in Mechanics (Nauka, Moscow, 1981; CRC Press, Boca Raton, 1993). 13. G. I. Barenblatt, Similarity, Self-Similarity, and Intermediate Asymptotics (Gidrometeoizdat, Leningrad, 1978; Consultants Bureau, New York, 1979).

Translated by O. Arushanyan

MOSCOW UNIVERSITY MECHANICS BULLETIN

Vol. 64

No. 6

2009