Fracture Propagation

Abstract

In previous theories regarding hydraulic fracture propagation, it is usually assumed the fluid pressure within the hydraulic fracture is constant in the calculation of the surrounding stress field. The actual fluid pressure during fracturing process, however, is often fluctuating, due to the disturbance of the irregular surface of rock fracture and the viscous flow of fracturing fluid.

1 Introduction

In previous theories regarding hydraulic fracture propagation, it is usually assumed the fluid pressure within the hydraulic fracture is constant in the calculation of the surrounding stress field. The actual fluid pressure during fracturing process, however, is often fluctuating, due to the disturbance of the irregular surface of rock fracture and the viscous flow of fracturing fluid. Considering the uneven distribution characteristics of fluid pressure inside a hydraulic fracture and investigating the propagation process of this hydraulic fracture is beneficial to reflecting the realistic dynamic propagation of hydraulic fractures and improving the theory of field hydraulic fracturing. Up to now, the specific form of fluid pressure inside the fracture remains unclear. The fluid pressure is not constant due to the flow and viscosity of the fluid inside the fracture. The fluid pressure can be any form, depending on the actual injection conditions (e.g., pulse hydraulic fracturing). Sneddon and Elliott [1] derived the theoretical solutions of the induced stress and displacement fields in response to the non-uniform pressure of internal fluid outside the crack, but merely obtained a form solution with an integral variable for the stress and displacement around the crack. The existence of the integral variable to a large extent complicates the state analysis near the crack and inhibits its potential application in investigating the fracturing process under non-uniform pressure.

Considering the nonuniform distribution of fluid pressure within the crack is beneficial to reflect the stress state around hydraulic fractures, refine the theory of hydraulic fracturing and provide support for the actual fracturing process. Though of apparent practical importance, the effect of nonuniform fluid pressure has not received due attention. In this chapter, the solution of Sneddon and Elliott [1] is reduced to a more simplified form and solved by means of the composite Simpson’s rule to facilitate a stability analysis in the hydraulic fracturing process. It is theoretically feasible because the validity of this semianalytical solution can be guaranteed within a reasonable margin of error if the number of subintervals is sufficiently small. The applicability of the semianalytical solution is validated by comparing the approximation results with those obtained from constant fluid pressure and laboratory experiments. The sensitivity of the initial pressure and crack length common in constant pressure and the perturbation of the number of subintervals and terms missed in constant pressure are also discussed to assess the solution’s reliability.

2 Mathematical Formulation

Assume a 2a-long Griffith crack nonuniformly pressured by internal fluid is propagating in an impermeable, homogeneous and linear elastic plane-strain medium (Fig. 7.1). As shown in Fig. 7.1, a rectangular coordinate system is established with the center of crack O as the origin and the crack extension direction as the y-axis. The fluid pressure within the crack is assumed to be $P^{\prime } (x)$, a general polynomial function of the crack length and the position coordinate x (x < a), which implies that the nonuniform fluid pressure can be replaced by a dimensionless ratio of x coordinates to half of the total crack length a.

Fig. 7.1

Model for a Griffith crack non-uniformly pressurized by internal fluid

In order to ensure the applicability and validity of this model, the following assumptions are made:

• (1) The rock medium is linear, elastic, homogeneous and impermeable.

• (2) The crack is propagating in the plane strain plane, ignoring the change in the height of the crack (perpendicular to the plane).

• (3) The fluid pressure in the hydraulic fracture is a function of the fracture length (a) and the transverse position (x), satisfying x < a.

• (4) The hydraulic fracture is filled with fluid, ignoring the lag length between the fracture tip and the fluid front.

Based on the stress superposition principle delineated in Fig. 7.2, the total stress acting on the crack inner face can be decomposed into far-field in-situ stress and induced stress. The far-field stress remains uniform and is generated by in-situ stress, while the induced stress originates from the effective pressure within the crack and is controlled by injection pressure. Thus, we may solely consider the induced stress to analyze the fluid effect of nonuniform pressure in consideration of the case in which the variable fluid pressures at different x locations will greatly disturb the stress and displacement field around the crack.

Fig. 7.2

Schematic of a Griffith crack stress decomposition (stress superposition principle)

2.1 Nonuniform Fluid Pressure Consideration

Due to variations in the fluid flow and viscosity inside the crack, the fluid pressure value cannot always remain constant. In contrast, the pressure may manifest as any form of function in terms of practical injection conditions. Taking this problem into consideration and using the superposition of far-field stress, the net fluid pressure acting on the crack is assumed to be a general polynomial.

$$P(x) = P^{\prime } (x) - \sigma_{h} = P_{0} \sum\limits_{k = 0}^{n} {b_{k} (\frac{x}{a})^{k} } \, (x \le a)$$

(7.1)

where $P^{\prime } (x)$ is the injection pressure, $\sigma_{h}$ represents the minimum horizontal in-situ stress, $P_{0}$ is defined as the initial fluid pressure (noting the fluid pressure mentioned here and after excluding the effect of far-field stress), k and n are both integers (≥0), and b_k is a series of dimensionless real coefficient variables that varies with k. With the fitting properties of this polynomial, any form of fluid pressure distribution can be approximately expressed if the pressure of certain points within the crack can be determined.

According to [2], the integral transformation solution of the induced stress and displacement field can be expressed as (compression is considered to be negative)

$$\left\{ \begin{aligned} \sigma_{xx} & = - \frac{2}{\pi }\int\limits_{0}^{\infty } {\xi^{2} A(\xi )(1 - \xi y)e^{ - \xi y} } \cos (\xi x)d\xi \\ \sigma_{yy} & = - \frac{2}{\pi }\int\limits_{0}^{\infty } {\xi^{2} A(\xi )(1 + \xi y)e^{ - \xi y} } \cos (\xi x)d\xi \\ \sigma_{xy} & = - \frac{2}{\pi }\int\limits_{0}^{\infty } {y\xi^{3} A(\xi )e^{ - \xi y} } \sin (\xi x)d\xi \\ \end{aligned} \right.$$

(7.2)

The components of the displacement vector are similarly found to be

$$\left\{ \begin{aligned} u_{x} & = - \frac{2(1 + \nu )}{{\pi E}}\int\limits_{0}^{\infty } {A(\xi )} \xi \left[ {(1 - 2\nu ) - \xi y} \right]e^{ - \xi y} \sin (\xi x){\text{d}}\xi \\ u_{y} & = \frac{2(1 + \nu )}{{\pi E}}\int\limits_{0}^{\infty } {A(\xi )} \xi \left[ {2(1 - \nu ) + \xi y} \right]e^{ - \xi y} \cos (\xi x){\text{d}}\xi \\ \end{aligned} \right.$$

(7.3)

As shown in Fig. 7.2, the following boundary conditions can be obtained (noting that only a quarter plane (x ≥ 0, y > 0) is studied in consideration of the symmetry of the Griffith crack):

$$\left\{ \begin{aligned} & x\,{\text{and}}\,y \to \infty :\sigma_{xx} = \sigma_{yy} = \sigma_{xy} = 0 \\ & y = 0,\quad {0 < }x < \infty {,}\quad \sigma_{xy} = 0 \\ & y = 0,\quad {0 < }x< a{,}\quad \sigma_{yy} = P(x) \\ & y = 0,\quad x > a,\quad u_{y} { = 0 } \\ \end{aligned} \right.$$

(7.4)

Combining Eqs. (7.2) ~ (7.4), we can obtain

$$\left\{ {\begin{array}{*{20}l} { - \frac{2}{\pi }\int\limits_{0}^{\infty } {\xi^{{2}} A(\xi )} \cos (\xi x){\text{d}}\xi = P(x),} \hfill & {0 < x< a} \hfill \\ {\int\limits_{0}^{\infty } {A(\xi )} \xi \cos (\xi x){\text{d}}\xi = 0,} \hfill & {x > a} \hfill \\ \end{array} } \right.$$

(7.5)

The form of Eq. (7.5) can be transformed into a dual integral equation,

$$\left\{ {\begin{array}{*{20}l} {\int\limits_{0}^{\infty } {\eta f(\eta )} J_{ - 1/2} (\eta \rho )d\eta = g(\rho ),\quad {0 < }\rho {< 1}} \hfill \\ {\int\limits_{0}^{\infty } {f(\eta )} J_{ - 1/2} (\eta \rho )d\eta = 0,\quad \quad \rho { > 1}} \hfill \\ \end{array} } \right.$$

(7.6)

where

$$\begin{aligned} & x = \rho a, \, \eta = a\xi ,\cos (\xi x) = (\frac{\pi \xi x}{2})^{\frac{1}{2}} J_{ - 1/2} (\xi x), \\ & g(\rho ) = aP_{0} \sqrt {\frac{\pi a}{2}} \sum\limits_{k = 0}^{n} {b_{k} } (\rho )^{k} \rho^{ - 1/2} ,A(\xi )\xi^{3/2} = f(\xi ) = f(\eta ) \\ \end{aligned}$$

(7.7)

The general form of a dual integral equation can be expressed as

$$\left\{ \begin{gathered} \int\limits_{0}^{\infty } {\eta^{\alpha } f(\eta )} J_{v} (\eta \rho ){\text{d}}\eta = g(\rho ),\quad 0 < \rho< 1 \hfill \\ \int\limits_{0}^{\infty } {f(\eta )} J_{v} (\eta \rho ){\text{d}}\eta = 0,\quad \rho > 1 \hfill \\ \end{gathered} \right.$$

(7.8)

whose solution is given by [5] in the form of

$$f(\eta ) = \frac{{(2\eta )^{1 - \alpha /2} }}{\Gamma (\alpha /2)}\int\limits_{0}^{1} {u^{1 + \alpha /2} } J_{v + \alpha /2} (u\eta ){\text{d}}u\int\limits_{0}^{1} g (\rho u)\rho^{v + 1} (1 - \rho^{2} )^{\alpha /2 - 1} {\text{d}}\rho$$

(7.9)

where α and v are the power of a power function (η^α) and the order of a Bessel function (J_v) both in Eqs. (7.8) and (7.9).

It is easy to find that Eq. (7.8) can be simplified to Eq. (7.6) in the case of α = 1, $v = - \frac{1}{2}$. After substituting the determined α = 1 and $v = - \frac{1}{2}$ into Eq. (7.9), the expression of f(η) becomes

$$f(\eta ) = \sqrt {\frac{2\eta }{\pi }} \int\limits_{0}^{1} {u^{\frac{3}{2}} } J_{0} (u\eta ){\text{d}}u\int\limits_{0}^{1} {g(\rho u)\rho^{\frac{1}{2}} } (1 - \rho^{2} )^{{ - \frac{1}{2}}} {\text{d}}\rho$$

(7.10)

If the function of g(ρ) in Eq. (7.7) is substituted into Eq. (7.10), we have

$$f(\eta ) = aP_{0} \sqrt {a\eta } \sum\limits_{k = 0}^{n} {b_{k} } \int\limits_{0}^{1} {u^{k + 1} } J_{0} (u\eta ){\text{d}}u\int\limits_{0}^{1} {\rho^{k} (1 - \rho^{2} )^{{ - \frac{1}{2}}} } {\text{d}}\rho$$

(7.11)

According to the Tables of Integrals [3], the definite integrals of power function and algebraic function can be found in the form of

$$\int\limits_{0}^{a} {x^{\alpha } (a^{c} - x^{c} )^{\beta } } {\text{d}}x = \frac{{\Gamma (\frac{\alpha + 1}{c})\Gamma (\beta + 1)}}{{n\Gamma (\frac{\alpha + 1}{c} + \beta + 1)}}a^{\alpha + c\beta + 1} \,$$

(7.12)

where c is a nonzero positive integer, and a, α and β are nonzero real numbers.

For the function of $\int_{0}^{1} {\rho^{k} (1 - \rho^{2} )^{{ - \frac{1}{2}}} } {\text{d}}\rho$ in Eq. (7.11), it is easy to find that a = 1, x = ρ, c = 2, α = k and $\beta = - \frac{1}{2}$. Thus, Eq. (7.11) has a more simplified form of

$$f(\eta ) = f(\xi ) = aP_{0} \frac{{\sqrt {a\xi \pi } }}{2}\sum\limits_{k = 0}^{n} {b_{k} } \frac{{\Gamma (\frac{k}{2} + \frac{1}{2})}}{{\Gamma (\frac{k}{2} + 1)}}\int\limits_{0}^{1} {u^{k + 1} } J_{0} (u\xi ){\text{d}}u$$

(7.13)

Substituting Eq. (7.13) into Eq. (7.7), we can obtain

$$A(\xi ) = \xi^{{ - \frac{3}{2}}} f(\xi ) = aP_{0} \frac{{\sqrt {a\pi } }}{2\xi }\sum\limits_{k = 0}^{n} {b_{k} } \frac{{\Gamma (\frac{k}{2} + \frac{1}{2})}}{{\Gamma (\frac{k}{2} + 1)}}\int\limits_{0}^{1} {u^{k + 1} } J_{0} (u\xi ){\text{d}}u$$

(7.14)

Combining Eq. (7.2) and Eq. (7.14), the induced stress and displacement fields under nonuniform fluid pressure become

$$\left\{ \begin{aligned} \sigma _{{xx}} & = - aP_{0} \frac{{\sqrt a }}{{\sqrt \pi }}\sum\limits_{{k = 0}}^{n} {b_{k} } \frac{{\Gamma (\frac{k}{2} + \frac{1}{2})}}{{\Gamma (\frac{k}{2} + 1)}}\int\limits_{0}^{1} {u^{{k + 1}} } \\ & \quad \int\limits_{0}^{\infty } {\xi J_{0} (u\xi )(1 - \xi y)e^{{ - \xi y}} \cos (\xi x){\text{d}}\xi {\text{d}}u} \\ \sigma _{{yy}} & = - aP_{0} \sqrt {\frac{a}{\pi }} \sum\limits_{{k = 0}}^{n} {b_{k} } \frac{{\Gamma (\frac{k}{2} + \frac{1}{2})}}{{\Gamma (\frac{k}{2} + 1)}}\int\limits_{0}^{1} {u^{{k + 1}} } \\ & \quad \int\limits_{0}^{\infty } {\xi J_{0} (u\xi )(1 + \xi y)e^{{ - \xi y}} \cos (\xi x){\text{d}}\xi {\text{d}}u} \\ \sigma _{{xy}} & = - aP_{0} \frac{{\sqrt a }}{{\sqrt \pi }}\sum\limits_{{k = 0}}^{n} {b_{k} } \frac{{\Gamma (\frac{k}{2} + \frac{1}{2})}}{{\Gamma (\frac{k}{2} + 1)}}\int\limits_{0}^{1} {u^{{k + 1}} } {\text{d}}u \\ & \quad \int\limits_{0}^{\infty } {y\xi ^{2} e^{{ - \xi y}} J_{0} (u\xi )\sin (\xi x){\text{d}}\xi } \\ u_{x} & = - \frac{{aP_{0} (1 + \nu )}}{E}\sqrt {\frac{a}{\pi }} \sum\limits_{{k = 0}}^{n} {b_{k} } \frac{{\Gamma (\frac{k}{2} + \frac{1}{2})}}{{\Gamma (\frac{k}{2} + 1)}}\int\limits_{0}^{1} {u^{{k + 1}} } \\ & \quad \int\limits_{0}^{\infty } {J_{0} (u\xi )\left[ {(1 - 2\nu ) - \xi y} \right]e^{{ - \xi y}} \sin (\xi x){\text{d}}\xi {\text{d}}u} \\ u_{y} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\leftarrow}$}}{ = } & \frac{{aP_{0} (1 + \nu )}}{E}\sqrt {\frac{a}{\pi }} \sum\limits_{{k = 0}}^{n} {b_{k} } \frac{{\Gamma (\frac{k}{2} + \frac{1}{2})}}{{\Gamma (\frac{k}{2} + 1)}}\int\limits_{0}^{1} {u^{{k + 1}} } \\ & \quad \int\limits_{0}^{\infty } {J_{0} (u\xi )\left[ {2(1 - \nu ) + \xi y} \right]e^{{ - \xi y}} \cos (\xi x){\text{d}}\xi {\text{d}}u} \\ \end{aligned} \right.$$

(7.15)

The pure analytical solution of Eq. (7.15) can be derived under the premise that the integrals about u and ξ are solved and integrated in the proper order. It is easy to see that the integrals about ξ in Eq. (7.15) (hereafter called ξ-integrals) primarily consist of Bessel, trigonometric, exponential and power functions, which can be unified into two general forms: $\int_{0}^{\infty } {\xi^{c} e^{ - \xi y} J_{v} (u\xi )\sin (\xi x){\text{d}}\xi }$ (c = 0, 1, 2; v = 0) and $\int_{0}^{\infty } {\xi^{d} e^{ - \xi y} J_{v} (u\xi )\cos (\xi x){\text{d}}\xi }$ (d = 0, 1, 2; v = 0). These ξ-integrals can all be deduced based on the recursion formula of the Bessel function and integration by parts (see Appendix 1 for details).

As c, d and v all equal 0, the solution of ξ-integrals can be obtained referring to [3],

$$\begin{gathered} \int\limits_{0}^{\infty } {e^{ - \xi y} J_{0} (u\xi )\cos (\xi x){\text{d}}\xi } = R^{ - 1} \cos \frac{1}{2}\varphi \hfill \\ \int\limits_{0}^{\infty } {e^{ - \xi y} J_{0} (u\xi )\sin (\xi x){\text{d}}\xi } = - R^{ - 1} \sin \frac{1}{2}\varphi \hfill \\ \end{gathered}$$

(7.16)

where R and φ, hereafter, are both functions of the location coordinates and the integral variable u and can be expressed as $R^{4} = (y^{2} + u^{2} - x^{2} )^{2} + 4x^{2} y^{2}$ and $\varphi = \arg (y^{2} + u^{2} - x^{2} - 2ixy)$, respectively.

In the case of c = d = 1 and v = 0, according to [3], the following solution of ξ-integrals can be provided:

$$\left\{ {\begin{array}{*{20}l} {\int\limits_{0}^{\infty } {\xi e^{ - \xi y} J_{0} (u\xi )\cos (\xi x)d\xi = } R^{ - 3} (y\cos \frac{3}{2}\varphi - x\sin \frac{3}{2}\varphi )} \hfill \\ {\int\limits_{0}^{\infty } {\xi e^{ - \xi y} J_{0} (u\xi )\sin (\xi x)d\xi } = - R^{ - 3} (x\cos \frac{3}{2}\varphi + y\sin \frac{3}{2}\varphi )} \hfill \\ \end{array} } \right.(y > 0)$$

(7.17)

When both c and d equal 2 and v = 0, ξ-integrals become

$$\left\{ \begin{aligned} & \int\limits_{0}^{\infty } {\xi ^{2} e^{{ - \xi y}} J_{0} (u\xi )\cos (\xi x){\text{d}}\xi = \frac{1}{u}\left[ {3uR^{{ - 5}} ((y^{2} - x^{2} )\cos \frac{5}{2}\varphi } \right.} \\ & \quad - \left. {2xy\sin \frac{5}{2}\varphi ) - uR^{{ - 3}} \cos \frac{3}{2}\varphi } \right] \\ & \int\limits_{0}^{\infty } {\xi ^{2} e^{{ - \xi y}} J_{0} (u\xi )\sin (\xi x){\text{d}}\xi = \frac{1}{u}\left[ {3uR^{{ - 5}} ((x^{2} - y^{2} )\sin \frac{5}{2}\varphi } \right.} \\ & \quad - \left. {2xy\cos \frac{5}{2}\varphi + uR^{{ - 3}} \sin \frac{3}{2}\varphi } \right] \\ \end{aligned} \right.$$

(7.18)

Substituting Eqs. (7.16) ~ (7.18) into Eq. (7.15) and changing the order of integration allow one to specify the relation between the stress (or displacement) vector and the variable u in terms of a single integration. Simultaneously, the variable ξ in Eq. (7.15) can be eliminated by separating the variables and solving the definite integration on ξ. The new form of Eq. (7.1) is shown as follows:

$$\left\{ \begin{aligned} \sigma _{{xx}} & = - aP_{0} \frac{{\sqrt a }}{{\sqrt \pi }}\sum\limits_{{k = 0}}^{n} {b_{k} } \frac{{\Gamma (\frac{k}{2} + \frac{1}{2})}}{{\Gamma (\frac{k}{2} + 1)}}\int\limits_{0}^{1} {u^{{k + 1}} } [R^{{ - 3}} (y\cos \frac{3}{2}\varphi - x\sin \frac{3}{2}\varphi ) \\ & \quad - (3R^{{ - 5}} ((y^{3} - x^{2} y)\cos \frac{5}{2}\varphi - 2xy\sin \frac{5}{2}\varphi ) - yR^{{ - 3}} \cos \frac{3}{2}\varphi )]{\text{d}}u \\ \sigma _{{yy}} & = - aP_{0} \frac{{\sqrt a }}{{\sqrt \pi }}\sum\limits_{{k = 0}}^{n} {b_{k} } \frac{{\Gamma (\frac{k}{2} + \frac{1}{2})}}{{\Gamma (\frac{k}{2} + 1)}}\int\limits_{0}^{1} {u^{{k + 1}} } du[R^{{ - 3}} (y\cos \frac{3}{2}\varphi - x\sin \frac{3}{2}\varphi ) \\ & \quad + (3R^{{ - 5}} ((y^{3} - x^{2} y)\cos \frac{5}{2}\varphi - 2xy\sin \frac{5}{2}\varphi ) - yR^{{ - 3}} \cos \frac{3}{2}\varphi )] \\ \sigma _{{xy}} & = - aP_{0} \sqrt {\frac{a}{\pi }} \sum\limits_{{k = 0}}^{n} {b_{k} } \frac{{\Gamma (\frac{k}{2} + \frac{1}{2})}}{{\Gamma (\frac{k}{2} + 1)}}\int\limits_{0}^{1} {u^{{k + 1}} \left[ {15R^{{ - 7}} ((x^{3} y - 3xy^{3} )\cos \frac{7}{2}\varphi } \right.} \\ & \quad - \left. {(y^{4} - 3x^{2} y^{2} )\sin \frac{7}{2}\varphi ) + 9R^{{ - 5}} (y^{2} \sin \frac{5}{2}\varphi + xy\cos \frac{5}{2}\varphi )} \right]{\text{d}}u \\ u_{x} & = aP_{0} \frac{{\sqrt a (1 + v)}}{{E\sqrt \pi }}\sum\limits_{{k = 0}}^{n} {b_{k} } \frac{{\Gamma (\frac{k}{2} + \frac{1}{2})}}{{\Gamma (\frac{k}{2} + 1)}}\int\limits_{0}^{1} {u^{{k + 1}} } [(1 - 2\nu )R^{{ - 1}} \sin ( - \frac{1}{2}\varphi ){\text{d}}u \\ & \quad - y(R^{{ - 3}} (x\cos \frac{3}{2}\varphi + y\sin \frac{3}{2}\varphi ))]{\text{d}}u \\ u_{y} & = aP_{0} \frac{{\sqrt a (1 + v)}}{{E\sqrt \pi }}\sum\limits_{{k = 0}}^{n} {b_{k} } \frac{{\Gamma (\frac{k}{2} + \frac{1}{2})}}{{\Gamma (\frac{k}{2} + 1)}}\int\limits_{0}^{1} {u^{{k + 1}} } [2(1 - \nu )R^{{ - 1}} \cos (\frac{1}{2}\varphi ) \\ & \quad + R^{{ - 3}} (y^{2} \cos \frac{3}{2}\varphi - xy\sin \frac{3}{2}\varphi )]{\text{d}}u \\ \end{aligned} \right.$$

(7.19)

Since R and φ are implicit functions about the variable u, it becomes difficult to solve the integrals about u at the interval of [0, 1]. To solve this nested integral, a numerical approximation is suggested for calculating these integrals by discretizing the integral interval of [0, 1], which can facilitate to approach the real solution as long as the computational accuracy is high enough.

2.2 Semianalytical Solution

The composite Simpson’s rule is an approximated method for numerical integration, which can reduce errors by quadratic interpolation. Suppose that the interval of u, [0, 1] is split up into m subintervals of equal length, with m as an even number. Then, the composite Simpson’s rule is given by

$$\int\limits_{0}^{1} {f(u){\text{d}}u} = \frac{1}{6m}\left[ {f(0) + 4\sum\limits_{i = 0}^{m - 1} {f(u_{i + 1/2} ) + 2\sum\limits_{i = 1}^{m - 1} {f(u_{i} ) + f(1)} } } \right]$$

(7.20)

Based on Eq. (7.20), the variable u can also be removed from the integrand in Eq. (7.19). Consequently, semianalytical and seminumerical solutions (called semianalytical solutions for short) of the stress and displacement field subjected to nonuniform fluid pressure can be completely derived.

To evaluate the applicability of the solution, let us consider a computational area for –5 ≤ x ≤ 5 and -5 ≤ y ≤ 5. Notably, the values of stress for the negative x- and y- axes are sketched by symmetric transformation considering the limitation of the positive coordinate position in Eq. (7.4). Adopting a set of rock and fluid properties [4], $E^{\prime } = 30$ GPa, v = 0.25, and P₀ = 7 MPa, the total length of the Griffith crack is assumed to be l = 2a = 2 m, the real variable b_k remains constant at 1, and the number of term n is taken as 100 for the sake of reducing the computational load and facilitating the analysis. Figure 7.3 shows the regularity of the stress distribution of a Griffith crack induced by nonuniform fluid pressure. It is clear that the value of stress increasingly approaches infinity near the crack tip, which illustrates the existence of a tip-stress singularity. Another obvious feature is that the contour marked in red dotted boxes in Fig. 7.3a, b are caused by the precision of subinterval division as well as the error (y = 0), which violates the limitation of Eq. (7.17). To ensure the continuity of stress contours, the values of contours at y = 0 are assumed to equal those at y = 1/m. This assumption can be valid as the number of terms m is great enough.

Fig. 7.3

Stress contours of σ_xx (a), σ_yy (b) and σ_xy (c) for a Griffith crack subjected to non-uniform fluid pressure (m = 100)

2.3 Propagation Conditions Under Nonuniform Fluid Pressure

Considering nonuniform net fluid pressure in terms of a polynomial (Eq. 7.1) to simulate the arbitrary pressure distribution inside the crack, the assumed net fluid pressure at the inlet, crack tip and other places can be expressed piecewise as

$$P(x) = \left\{ {\begin{array}{*{20}l} {P_{0} b_{0} {,}} \hfill & {x = 0} \hfill \\ {P_{0} \sum\limits_{k = 0}^{n} {b_{k} (\frac{x}{a})^{k} , \, } } \hfill & {0 < x < a} \hfill \\ {P_{0} \sum\limits_{k = 0}^{n} {b_{k} ,} } \hfill & {x = a} \hfill \\ \end{array} } \right.$$

(7.21)

For the existence of stress singularity as x equals a, the near-crack-tip behavior holds the key to fracture propagation conditions according to the linear fracture mechanics (LFEM). In LFEM, the stress intensity factor (SIF) is usually adopted to describe the stress state of the crack tip, which can be expressed in terms of the Bueckner–Rice weight function [6]

$$K_{I} = \frac{2\sqrt a }{{\sqrt \pi }}\int\limits_{0}^{a} {P(x)\frac{f(x/a)}{{\sqrt {a^{2} - x^{2} } }}{\text{d}}x}$$

(7.22)

where f is a ‘configurational’ function modified by Nilson and Proffer (1984) and $f(x/a) \simeq 1$, in line with [7].

Integrating Eq. (7.22) by parts and after some simplification, we can obtain,

$$\frac{{K_{I} }}{{\sqrt {\pi a} }} = \int\limits_{0}^{a} {F(\frac{x}{a})\frac{{{\text{d}}P(x)}}{{{\text{d}}x}}{\text{d}}x} + P_{0} b_{0}$$

(7.23)

where

$$F(\frac{x}{a}){ = }\frac{2}{\pi }\int\limits_{x/a}^{1} {\frac{f(x/a)}{{\sqrt {1 - (x/a)^{2} } }}} d(\frac{x}{a})$$

(7.24)

(The closed form for function F(ξ) is given in Appendix 2).

In the case of a Griffith crack, [7] provided a more simplified form of Eq. (7.24)

$$F(\frac{x}{a}) = 1 - \frac{2}{\pi }\arcsin \frac{x}{a}$$

(7.25)

Substituting $f(x/a) \simeq 1$ and Eq. (7.25) into Eq. (7.23), we can finally obtain

$$\frac{{K_{I} }}{{\sqrt {\pi a} }} = \int\limits_{0}^{a} {(1 - \frac{2}{\pi }\arcsin \frac{x}{a})\frac{{{\text{d}}P(x)}}{{{\text{d}}x}}{\text{d}}x} + P_{0} b_{0}$$

(7.26)

If Eq. (7.21) in the case of x→a is substituted into Eq. (7.26), the stress intensity under nonuniform fluid pressure becomes

$$K_{I} = P_{0} \sqrt {\pi a} \sum\limits_{k = 0}^{n} {\frac{{kb_{k} }}{{a^{k} }}} \int\limits_{0}^{a} {(1 - \frac{2}{\pi }\arcsin \frac{x}{a})x^{k - 1} {\text{d}}x} + \sqrt {\pi a} P_{0} b_{0}$$

(7.27)

Integrating Eq. (7.27) by parts and consulting the Tables of Integrals [3], we can get

$$K_{I} = P_{0} \sqrt a \sum\limits_{k = 0}^{n} {b_{k} } \frac{{\Gamma (\frac{k}{2} + \frac{1}{2})}}{{\Gamma (\frac{k}{2} + 1)}} + \sqrt {\pi a} P_{0} b_{0}$$

(7.28)

The stress intensity near the crack tip shows noticeable dependence on the number of polynomial terms, the crack length and the initial fluid pressure.

Based on LFEM theory, a Griffith crack starts to initiate and propagate in its favored direction when the SIF reaches its critical value, i.e., rock fracture toughness $K_{I} = K_{IC}^{N}$. Consequently, Eq. (7.28) can be expressed as

$$K_{IC}^{N} = P_{0}^{\prime } \sqrt {a^{\prime } } \sum\limits_{k = 0}^{{n^{\prime } }} {b_{k} } \frac{{\Gamma (\frac{k}{2} + \frac{1}{2})}}{{\Gamma (\frac{k}{2} + 1)}} + \sqrt {\pi a^{\prime } } P_{0}^{\prime } b_{0}$$

(7.29)

3 Validation of the Semianalytical Solution

In the case of constant fluid pressure when the first item of Eq. (7.1) is considered for n = 0, b₀ = 1, [1] deduced the form of

$$A(\xi ) = \xi^{{ - \frac{3}{2}}} f(\xi ) = \frac{{\pi aP_{0} }}{2}\xi^{ - 2} J_{1} (a\xi )$$

(7.30)

To verify the validity of the semianalytical solution under nonuniform fluid pressure, we compare the derived stress and critical intensity factor with those under classic constant fluid pressure. All of the related parameters are selected the same as those assumed in Sect. 7.2.2.

3.1 Degradation from Nonuniform Pressure to Constant Pressure

In reality, the nonuniform pressure herein considered can be expressed in the form of constant pressure when the number of terms equals 0, as stated in Eq. (7.21). Nevertheless, whether it is feasible for the semianalytical solution to be degraded into the analytical solution of constant fluid pressure while simultaneously enabling reasonable accuracy still needs further verification. Consequently, we make a comparison between the two forms of the solutions by changing the number of subintervals m. The comparison aims to evaluate the approximation degree of the semianalytical solutions to the analytical solutions when the nonuniform pressure is degraded to be constant, by which the validity of the semianalytical solution is further validated.

Due to the symmetrical stress distribution of σ_xx, the tensile and compressive stresses on the positive y-axis are separately considered to analyze the difference, as shown in Fig. 7.4. Focusing on the same position on the line of x = 5 m, the semianalytical solution of stress under constant fluid pressure gradually approximates the real analytical solution with the increase in the number of subintervals m. It seems that the discrepancy of the two solutions for both tensile stress and compressive stress can be roughly eliminated when m exceeds 30. This elimination makes it possible to supplant the analytical solution of constant pressure with the approximated semianalytical solution for unified analysis if the number of subintervals can ensure adequate calculation accuracy. In other words, the semianalytical solution is suitable for analyzing the stress induced by constant fluid pressure under the conditions in which a small reasonable error is accepted. This conclusion to some degree verifies the applicability of the semianalytical solution.

Fig. 7.4

Degradation from non-uniform pressure to constant pressure on the line of x = 5 m (P₀ = 7 MPa): a σ_xx > 0 (tension); b σ_xx < 0 (compression)

3.2 Stress Distribution

The approximate stress field induced by nonuniform fluid pressure has been analyzed in Sect. 7.2.2. Correspondingly, the stress distribution under constant fluid pressure is depicted in Fig. 7.5. Comparing Figs. 7.3 and 7.5, the induced stresses generated by constant fluid pressure and assumed nonuniform fluid pressure show similar stress distribution laws in contour shape. However, the magnitude of the stresses under the nonuniform fluid pressure is much greater than that induced by the constant fluid pressure amid the same conditions. Note that this result is concluded under the assumptions of b_k = 1 and n = 100 for the convenience of calculation. In fact, the difference will change as n and b_k vary. The influence of n will be further discussed in Sect. 4.3.

Fig. 7.5

Stress contours of σ_xx (a), σ_yy (b) and σ_xy (c) for a Griffith crack subjected to constant fluid pressure

We may take the stress at x = a or y = 0.1 as a separate example to evaluate the varying regularity of the normal and tangential stresses. On the line of x = a shown in Fig. 7.6a–c, with the increase of the y coordinate, the stress variation influenced by constant pressure is nearly the same as that under nonuniform pressure. It is also of interest to find that as y tends to zero, regardless of different fluid pressure effects, σ_xx experiences the tension–compression stress transformation twice (y/a =  ± 2.1, y =  ± 0.08), σ_yy experiences it once (y/a =  ± 0.8), but σ_xy does not at all. These similarities partly illustrate the consistency of the stress distribution between constant fluid pressure and nonuniform fluid pressure.

Fig. 7.6

Comparison of σ_xx (a), σ_yy (b), σ_xy (c) and similarity ratio (d) induced by non-uniform and constant pressure as x = a (Unit: MPa)

A similarity ratio is introduced as the ratio of the stress components (σ_xx, σ_yy and σ_xy) under nonuniform fluid pressure to that under constant fluid pressure. As shown in Fig. 7.6d, the normal stresses (σ_xx and σ_yy) separately induced by constant pressure (n = 0, b₀ = 1) and nonuniform pressures (n = 100) present an obvious similarity, with average similarity ratios of 3500.57 and 3353.89 (neglecting the singular abrupt points), while the variation of the similarity ratio of tangential stress σ_xy mainly behaves as a fluctuation of a concave function. Moreover, two abrupt points on the curve of normal stresses are found symmetrically distributed on both sides of the crack tip (y = 0) in Fig. 7.6d. The emergence of these abrupt points marks the transition of tension–compression stress. For example, when a point moves from afar to y = 0, the normal stress σ_xx of this point changes from tension to compression (T → C) at y/a = 2.1, then returns to tension (C → T) at y/a = 0.08 and finally increases abruptly towards infinity at the crack tip. However, for the tangential stress σ_xy, the similarity ratio remains symmetrically and uniformly distributed around the crack tip and progressively approaches infinity with the increase of the y coordinate. There is no abrupt point on the similarity curve of σ_xy, and correspondingly, there is no tension–compression transition, which precisely illustrates the correlation between the stress state and singularity ratio.

A similar variation of the induced stresses is found in Fig. 7.7 as y equals 0.1. The main stress discrepancy between Figs. 7.6 and 7.7 lies in the position change of the tension–compression transition point. Less influenced by tip singularity compared with x = a, which covers the singular point of y = 0 into the computational domain, the values of stress remain finite, and the tension–compression transitions all appear at the nearest point to the crack tip, x/a =  ± 1. Moreover, the tangential stress (σ_xy) starts to initially experience the tension–compression transition at x/a =  ± 1. These results differ from those in Fig. 7.6 and can be explained by the absence of stress singularity on the line of y = 0.1.

Fig. 7.7

Comparison of σ_xx (a), σ_yy (b), σ_xy (c) and similarity ratio (d) induced by non-uniform and constant pressure as y = 0.1 (Unit: MPa)

As evident from the two examples of x = a and y = 0.1, nonuniform fluid pressure in the form of a polynomial (n = 100, b_k = 1) can produce a linear increasing effect on the normal stress, whereas a nonlinear effect is apparent on the tangential stress. This distinction implies that the stress state has a great correlation with the singularity ratio under the action of different fluid pressures and that the stress state can also be a reflection of stress singularity to a certain extent.

3.3 Critical Propagation Condition

The crack influenced by constant fluid pressure starts to move ahead as the magnitude of fluid pressure exceeds the critical value P_hf. According to [8] and [9], the SIF can be calculated using

$$K_{IC}^{C} = \sqrt {\pi a} (P_{hf} - \sigma_{h} )$$

(7.31)

Together with Eq. (7.29) which gives the expression of the critical SIF (fracture toughness) under nonuniform fluid pressure, the relationships and discrepancies of critical propagation conditions under different fluid pressures can be found by changing the parameters of P0, a and KIC as follows:

• (i) Regarding the same fracture geometry, rock medium and stress environment, the physical and mechanical properties of rock and in-situ stress are identical, which means that the critical propagation conditions of the crack for both constant and nonuniform pressures are the same. Thus, we can obtain

  $$P_{0}^{\prime } \sqrt a \sum\limits_{k = 0}^{100} {b_{k} } \frac{{\Gamma (\frac{k}{2} + \frac{1}{2})}}{{\Gamma (\frac{k}{2} + 1)}} + \sqrt {\pi a} P_{0}^{\prime } b_{0} = (P_{hf} - \sigma_{h} )\sqrt {\pi a}$$

  (7.32)

Assuming the coefficient variable and term number of the designated polynomial to be the same as before (bk = 1, n = 100), Eq. (7.32) can be simplified to

$$P_{hf} - \sigma_{h} = 15.401P_{0}^{\prime }$$

(7.33)

This expression exhibits a linear relationship between critical constant fluid pressure and critical nonuniform fluid pressure in an identical rock medium and stress environment. Additionally, the net fluid pressure under the effect of constant pressure $\left( {P_{hf} - \sigma_{h} } \right)$ is much greater than that of nonuniform fluid pressure at the injection point $\left( {P_{0}^{\prime } } \right)$, which indicates that greater pump pressure is required to promote crack extension under constant fluid pressure than nonuniform fluid pressure.

• (ii) In a two-dimensional horizontal plane, the magnitude of the in-situ stress is a direct reflection of the fracture geometry. Based on Eqs. (7.28) and (7.31), the relation of fracture length under different fluid pressures can be expressed by (Note that only a is changed for different fluid pressures and bk = 1, n = 100)

  $$P_{0}^{\prime } \sqrt {a^{N} } \sum\limits_{k = 0}^{100} {b_{k} } \frac{{\Gamma (\frac{k}{2} + \frac{1}{2})}}{{\Gamma (\frac{k}{2} + 1)}} + \sqrt {\pi a^{N} } P_{0}^{\prime } b_{0} = P_{0}^{\prime } \sqrt {\pi a^{C} }$$

  (7.34)

where aN and aC represent the critical fracture length at the beginning of fracture propagation under nonuniform and constant fluid pressure, respectively. After simplification, we obtain

$$269.112a^{N} = a^{C}$$

(7.35)

From Eq. (7.35), it is easy to conclude that the critical fracture length, which is needed to maintain the continuous propagation of a hydraulic fracture under constant pressure, is much greater than that under nonuniform fluid pressure.

• (iii) For the same initial net fluid pressure and fracture length, the SIF under different forms of critical fluid pressure can be expressed as

  $$\left\{ \begin{aligned} K_{{_{I} }}^{C} & = \sqrt {a\pi } (P_{hf} - \sigma_{h} ) = 1.772\sqrt a P_{0} \\ K_{I}^{N} & = 29.069\sqrt a P_{0} \\ \end{aligned} \right.$$

  (7.36)

where $K_{I}^{N}$ and $K_{I}^{C}$ are the SIFs under critical nonuniform and constant fluid pressures, respectively.

Crack propagation occurs when the SIF reaches the critical value, rock fracture toughness KIC, i.e.,

$$K_{{\text{I}}} = K_{IC}$$

(7.37)

Considering the inequation of $K_{I}^{N} > K_{I}^{C}$, the SIF under nonuniform fluid pressure is more likely to reach or exceed the fracture toughness of rock formations than that under constant pressure.

Generally, this simplified nonuniform fluid pressure (bk = 1, n = 100) yields a higher SIF and requires lower pump pressure as well as a smaller fracture length to drive a crack forward than constant fluid pressure. Thus, it is reasonable to conclude that nonuniform fluid pressure is more likely to promote crack extension than constant pressure under this assumed form of fluid pressure (bk = 1, n = 100). This conclusion can be verified by field and experimental observations. For example, pulse hydraulic fracturing, which causes fluctuating variations in fluid pressure inside the hydraulic fracture, usually requires lower breakdown pressure and results in a more complicated fracture network than low-speed fluid fracturing operations [11, 12]. The fluid pressure acting on the crack’s inner face is hard to remain constant in consideration of the pressure difference of fluid flow even during constant rate fracturing treatment [10]. Therefore, a nonuniform fluid pressure distribution is more realistic and applicable to reflect the crack propagation state in the field.

4 Parametric Sensitivity Analysis

It is notable that the semianalytical solution deduced herein is implemented for the analysis of the stress and displacement around a Griffith crack, different from the work conducted by Garagash and Detournay [13], which focused on the crack itself and the internal fluid pressure. The reliability and accuracy of this solution are determined by a series of parameters, such as the number of subintervals m, the crack length a, the initial fluid pressure P₀ and the number of terms n.

4.1 Reliability Analysis of the Numerical Solution (Perturbation of the Number of Subintervals m)

As mentioned above, the calculation accuracy of the derived semianalytical solution depends on the number of subintervals m in composite Simpson’s rules. In theory, the greater the value of m is, the more accurate the solution becomes. However, a large m will conversely aggravate the computational load. In this regard, we tried to analyze the sensitivity of m to evaluate an appropriate value that can both ensure accuracy and reduce computational cost.

The function of Eq. (7.19) to be integrated can be uniformly expressed in the form of      

$$P(k)\int_{0}^{1} {u^{k + 1} F(u)} {\text{d}}u.$$

(7.38)

Thus, we may take Eq. (7.19) as an example (here we choose σ_xx) to elaborate the solution reliability resulted by m.

The difference of σ_xx under adjacent m must be below a certain small numerical tolerance, i.e., convergence is considered to be achieved if the following conditions are fulfilled:

$$\left| {\sigma_{xx}^{{m_{i + 1} }} - \sigma_{xx}^{{m_{i} }} } \right| < \varepsilon$$

(7.39)

where ε is the small numerical tolerance convergence, which can be ≤ 0.1% according to [14].

When m increases from 100 to 8000 in Table 7.1, 6 significant figures (shown in bold) in the value of σ_xx are observed at (a, 5), while only 4 significant figures are found at (a, 0.1). This difference means that the convergence rate of the solution decreases at a closer distance to the crack tip, which can be ascribed to the influence of the tip-stress singularity. In other words, the results of this semianalytical solution can be reliable outside the zone of tip-stress singularity and are suitable for the calculation of the induced fields. However, the specific range of this singular zone still needs further study.

Table 7.1 Value of σ_xx for different numbers of subintervals m

4.2 Sensitivity Analysis of the Initial Fluid Pressure P₀ and Crack Length a

Both P₀ and a are assumed to be initially constant in the calculation process of nonuniform fluid pressure. By means of the control variable method, we can analyze the sensitivity of the two parameters. Specifically, 3 groups of crack lengths (a = 1, 5 and 10 m) are initially considered, and P₀ was successively set to 7, 14, 21 and 28 MPa.

Figure 7.8a, b shows the distribution of σ_xx on the lines x = a and y = 0.1 under different crack lengths. Obviously, the larger crack length value indicates the greater magnitude of σ_xx for the same position. The varying crack length will not change the tension and compression stress range divided by x coordinates, which implies that an increase in crack length will result in a change in the stress value. To analyze the stress value variations at the same position, we extract three points on the line of x = a (y = 5, 2.5 and 0.1 m). Another two cases of a = 15 m and a = 20 m are considered for better regularities of the curves. As shown in Fig. 7.8c, the values of σ_xx at the selected locations all exhibit increasing trends when the crack length changes from 1 to 20 m. So, the crack length has a nonlinear on the stress variation.

Fig. 7.8

Distribution of σ_xx on the lines of x = a (a) and y = 0.1 m (b) as well as the variation of stress value (c) under different crack lengths a

In addition, the distribution of σ_xx induced by nonuniform fluid pressure under different initial fluid pressures is depicted in Fig. 7.9. A similar increase of σ_xx is also found with the increase of P₀ in Fig. 7.9a, b. However, as shown in Fig. 7.9c, the stress value pressure exhibits linear growth with the increase of initial fluid P₀. As P₀ is changed from 7 MPa to 14, 21 and 28 MPa, the value of σ_xx at the point of (a, -0.1) will be accordingly expanded to 2, 3 and 4 times as much as its initial value (P₀ = 7 MPa). Similar outcomes can also be found at other positions on the line x = a, which means that a linear increment in the initial fluid pressure P₀ will lead to the same magnitude of stress increasing at different locations.

Fig. 7.9

Distribution of σ_xx on the lines of x = a (a) and y = 0.1 m (b) as well as the variation of stress value (c) under different initial fluid pressures P₀

Based on Eq. (7.29), the critical intensity factor $K_{IC}^{N}$ for a crack to propagate under nonuniform fluid pressure is a function of P₀ and a. If the number of terms n and the coefficient variable b_k are considered to remain invariable as before (n = 100, b_k = 1), the sensitivity of P₀ and a can be separately evaluated in Fig. 7.10. Similar to the variation of the stress value, the initial fluid pressure P₀ is linearly related to the critical intensity factor, while the crack length poses a nonlinear effect of a power function with a power of 1/2 on the critical propagation condition.

Fig. 7.10

Curves of critical stress intensity factor versus the crack length a and the initial fluid pressure P₀

In summary, the varying initial fluid pressure P₀ shows a linear correlation with the stress value and the critical intensity factor. Meanwhile, with the increase of the crack length a, the magnitude of the induced stress near a Griffith crack and the critical intensity factor present a nonlinear increase, and the increase rate also increases.

4.3 Perturbation Analysis of the Number of Terms n

In terms of fluid with power-law (non-Newtonian) rheology, the number of terms n in Eq. (7.1) represents the distribution form of fluid pressure inside the Griffith crack, which may fluctuate the surrounding stress state of certain locations. To explore the perturbation effect of different n, a series of numbers of terms (n = 0, 1, 2, 3, 4, 5) were considered when analyzing the variation of σ_xx on the line x = 5 m (a = 1 m). It is also important to mention that the number of subintervals m is chosen to be 1000 in consideration of the calculation accuracy.

As Fig. 7.11 shows, an equal increase in the number of terms n results in an incremental increase in stress value according to the peak line of σ_xx (the rate of increase becomes greater and greater). The demarcation between the compressive and tensile stress zones remains stable despite the varying number of terms n. It can be inferred that fluid with power-law rheology may merely influence the magnitude of the stress value instead of the form of stress distribution around the crack. Additionally, as a point at line x = 5 m moves from y = −5 m to y = 0, the stress in the x-axial direction (σ_xx) decreases gradually and then increases until to the peak value. Symmetrical law is also observed on the positive y-axis. It seems that the closer a point is to y = 0, the greater the σ_xx becomes, which exactly reveals the effect of stress singularity near the crack tip.

Fig. 7.11

Distribution of σ_xx on the line x = 5 m under different the number of terms n

The continuing crack propagation (K_I ≥ K_IC) or arrest (K_I < K_IC) partly depends on the changing number of terms n. For the convenience of calculation, b_k is assumed to be the same as that in Sect. 7.2.2. According to Eq. (7.29), we can derive a dimensionless form of the critical intensity factor κ:

$$\kappa = \frac{{K_{IC}^{N} }}{{P_{0} \sqrt {\pi a} }} = \frac{1}{\sqrt \pi }\sum\limits_{k = 0}^{{n^{\prime } }} {\frac{{\Gamma (\frac{k}{2} + \frac{1}{2})}}{{\Gamma (\frac{k}{2} + 1)}}} + 1$$

(7.40)

Figure 7.12 illustrates the relationship between the dimensionless critical intensity factor and the number of terms n. As n increases from 0 to 300, the dimensionless critical intensity factor exhibits an increasing trend, while its increase rate decreases gradually from 0.636 and tends to be constant at 0.051. Therefore, the accumulated number of terms of the polynomial form of fluid pressure ultimately possesses a linear effect on the critical propagation as n exceeds 200. The crack propagation behavior is related to the compound effect of injection fluid pressure and in-situ stress. Assuming b_k equals 1 is too simplified to reflect realistic and practical hydraulic fracturing. Nonetheless, analyzing the variation of n can still help evaluate the perturbation of the fluid pressure form on the crack propagation conditions.

Fig. 7.12

Curves of dimensionless critical SIF κ versus the number of terms n

A mechanical model of a Griffith crack nonuniformly pressurized by interval fluid in a linear elastic and homogeneous plane is formulated. The initial state involving stress and displacement around the crack and the critical propagation condition of the crack are studied in detail. Based on Fourier transformation and the composite Simpson’s rule, a semianalytical solution for a crack under nonuniform fluid pressure is obtained as a function of two numbers: the number of subintervals (m) and the number of terms (n).

Appendices

Appendix 1. ξ-Integrals Function

A pair of general form functions composed of Bessel, trigonometric, exponential and power functions (ξ-integrals) can be expressed as

$$\left\{ {\begin{array}{*{20}l} {\int\limits_{0}^{\infty } {\xi^{d} e^{ - \xi y} J_{v} (u\xi )\cos (\xi x){\text{d}}\xi } } \hfill \\ {\int\limits_{0}^{\infty } {\xi^{c} e^{ - \xi y} J_{v} (u\xi )\sin (\xi x){\text{d}}\xi } } \hfill \\ \end{array} } \right.\quad (c = d = v,v + 1,v + 2, \ldots )$$

(7.41)

where c, d and v are integers ranging from zero to infinity.

Gradshteyn and Ryzhik [3] also provided the following solution of ξ-integrals when c and d both equal v + 1:

$$\begin{aligned} & \left\{ \begin{aligned} \int\limits_{0}^{\infty } {\xi ^{{v + 1}} e^{{ - \xi y}} J_{v} (u\xi )\cos (\xi x)d\xi } & = \frac{{2(2u)^{v} }}{{\sqrt \pi }}\Gamma (v + \frac{3}{2})R^{{ - 2v - 3}} [y\cos (v + \frac{3}{2})\varphi \\ & \quad - x\sin (v + \frac{3}{2})\varphi ] \\ \int\limits_{0}^{\infty } {\xi ^{{v + 1}} e^{{ - \xi y}} J_{v} (u\xi )\sin (\xi x)d\xi } & = - \frac{{2(2u)^{v} }}{{\sqrt \pi }}\Gamma (v + \frac{3}{2})R^{{ - 2v - 3}} [x\cos (v + \frac{3}{2})\varphi \\ & \quad + y\sin (v + \frac{3}{2})\varphi ] \\ \end{aligned} \right.{\text{ }} \\ & \quad (y > 0,v > - \frac{3}{2}) \\ \end{aligned}$$

(7.42)

A recursion formula of the Bessel function is given by

$$(x^{v} J_{v} (ax))^{\prime } = a \cdot x^{v} J_{v - 1} (ax)$$

(7.43)

Making v in Eq. (7.43) equal to w + 1 (w is an integer greater than or equal to 0) and substituting this equation into the ξ-integrals, we obtain

$$\begin{aligned} \int\limits_{0}^{\infty } {\xi^{w + 1} e^{ - \xi y} J_{w} (u\xi )\cos (\xi x){\text{d}}\xi } & = \frac{1}{u}\int\limits_{0}^{\infty } {(\xi^{w + 1} J_{w + 1} (u\xi ))^{\prime } e^{ - \xi y} \cos (\xi x){\text{d}}\xi } \\ \int\limits_{0}^{\infty } {\xi^{w + 1} e^{ - \xi y} J_{w} (u\xi )\sin (\xi x){\text{d}}\xi } & = \frac{1}{u}\int\limits_{0}^{\infty } {(\xi^{w + 1} J_{w + 1} (u\xi ))^{\prime } e^{ - \xi y} \sin (\xi x){\text{d}}\xi } \\ \end{aligned}$$

(7.44)

Using integration by parts, Eq. (7.44) can be expressed as

$$\begin{aligned} \int\limits_{0}^{\infty } {\xi ^{{w + 1}} e^{{ - \xi y}} J_{w} (u\xi )\cos (\xi x){\text{d}}\xi } & = \frac{1}{u}[\xi ^{{w + 1}} J_{{w + 1}} (u\xi )e^{{ - \xi y}} \cos (\xi x)\left| {_{0}^{\infty } } \right. \\ & \quad - \int\limits_{0}^{\infty } {\xi ^{{w + 1}} J_{{w + 1}} (u\xi )(e^{{ - \xi y}} \cos (\xi x))^{\prime } {\text{d}}\xi } ] \\ \int\limits_{0}^{\infty } {\xi ^{{w + 1}} e^{{ - \xi y}} J_{w} (u\xi )\sin (\xi x){\text{d}}\xi } & = \frac{1}{u}[\xi ^{{w + 1}} J_{{w + 1}} (u\xi )e^{{ - \xi y}} \sin (\xi x)\left| {_{0}^{\infty } } \right. \\ & \quad - \int\limits_{0}^{\infty } {\xi ^{{w + 1}} J_{{w + 1}} (u\xi )(e^{{ - \xi y}} \sin (\xi x))^{\prime } {\text{d}}\xi } ] \\ \end{aligned}$$

(7.45)

After simplification, Eq. (7.45) becomes

$$\begin{aligned} \int\limits_{0}^{\infty } {\xi^{w + 1} e^{ - \xi y} J_{w} (u\xi )\cos (\xi x)d\xi } & = \frac{x}{u}\int\limits_{0}^{\infty } {\xi^{w + 1} J_{w + 1} (u\xi )e^{ - \xi y} \sin (\xi x)d\xi } \\ & \quad + \frac{y}{u}\int\limits_{0}^{\infty } {\xi^{w + 1} J_{w + 1} (u\xi )e^{ - \xi y} \cos (\xi x)d\xi } \\ & = \frac{x}{u}Y + \frac{y}{u}X = M_{1} \\ \int\limits_{0}^{\infty } {\xi^{w + 1} e^{ - \xi y} J_{w} (u\xi )\sin (\xi x)d\xi } & = - \frac{x}{u}\int\limits_{0}^{\infty } {\xi^{w + 1} J_{w + 1} (u\xi )e^{ - \xi y} \cos (\xi x)d\xi } \\ & \quad + \frac{y}{u}\int\limits_{0}^{\infty } {\xi^{w + 1} J_{w + 1} (u\xi )e^{ - \xi y} \sin (\xi x)d\xi } \\ & \quad - \frac{x}{u}X + \frac{y}{u}Y = M_{2} \\ \end{aligned}$$

(7.46)

In Eq. (7.46), two binary linear equations of $\int_{0}^{\infty } {\xi^{w + 1} e^{ - \xi y} J_{{w{ + }1}} (u\xi )\cos (\xi x){\text{d}}\xi }$ (X) and $\int_{0}^{\infty } {\xi^{w + 1} e^{ - \xi y} J_{{w{ + }1}} (u\xi )\sin (\xi x){\text{d}}\xi }$ (Y) can be solved using M₁ and M₂ derived by Eq. (7.42). Notably, these deductions, held as variables of c, d, v and w, are all integers ranging from zero to infinity. Thus, it is appropriate to replace variable w with v in X and Y for the sake of consistent expression with Eq. (7.45). Thus, the unknown functions X and Y are separately expressed as

$$\left\{ \begin{aligned} X & = \int\limits_{0}^{\infty } {\xi^{v + 1} J_{v + 1} (u\xi )e^{ - \xi y} \cos (\xi x){\text{d}}\xi } = \frac{{(2u)^{v + 1} }}{\sqrt \pi }\Gamma (v + \frac{3}{2})R^{ - 2v - 3} \cos (v + \frac{3}{2})\varphi \\ Y & = \int\limits_{0}^{\infty } {\xi^{v + 1} J_{v + 1} (u\xi )e^{ - \xi y} \sin (\xi x){\text{d}}\xi } = - \frac{{(2u)^{v + 1} }}{\sqrt \pi }\Gamma (v + \frac{3}{2})R^{ - 2v - 3} \sin (v + \frac{3}{2})\varphi \\ \end{aligned} \right.$$

(7.47)

In the case of c = d = v+2, using the recurrence relation of Eq. (7.43), we can obtain

$$(x^{v + 2} J_{v + 1} (ax))^{\prime } = a \cdot x^{v + 2} J_{v} (ax) + x^{v + 1} J_{v + 1} (ax)$$

(7.48)

Then, this relation reduces to

$$x^{v + 2} J_{v} (ax) = \frac{1}{a} \cdot [(x^{v + 2} J_{v + 1} (ax))^{\prime } - x^{v + 1} J_{v + 1} (ax)]$$

(7.49)

Substituting Eq. (7.49) into ξ-integrals, we can obtain

$$\begin{aligned} \int\limits_{0}^{\infty } {\xi ^{{v + 2}} e^{{ - \xi y}} J_{v} (u\xi )\cos (\xi x){\text{d}}\xi } & = \frac{1}{u}\int\limits_{0}^{\infty } {[(\xi ^{{v + 2}} J_{{v + 1}} (u\xi ))^{\prime } } \\ & \quad - x^{{v + 1}} J_{{v + 1}} (ux)]e^{{ - \xi y}} \cos (\xi x){\text{d}}\xi \\ \int\limits_{0}^{\infty } {\xi ^{{v + 2}} e^{{ - \xi y}} J_{v} (u\xi )\sin (\xi x){\text{d}}\xi } & = \frac{1}{u}\int\limits_{0}^{\infty } {[(\xi ^{{v + 2}} J_{{v + 1}} (u\xi ))^{\prime } } \\ & \quad - x^{{v + 1}} J_{{v + 1}} (ux)]e^{{ - \xi y}} \sin (\xi x){\text{d}}\xi \\ \end{aligned}$$

(7.50)

Based on integration by parts, Eq. (7.50) can be further simplified to

$$\begin{aligned} \int\limits_{0}^{\infty } {\xi^{v + 2} e^{ - \xi y} J_{v} (u\xi )\cos (\xi x){\text{d}}\xi } & = \frac{x}{u}\int\limits_{0}^{\infty } {\xi^{v + 2} J_{v + 1} (u\xi )e^{ - \xi y} \sin (\xi x){\text{d}}\xi } \\ & \quad + \frac{y}{u}\int\limits_{0}^{\infty } {\xi^{v + 2} J_{v + 1} (u\xi )e^{ - \xi y} \cos (\xi x){\text{d}}\xi } \\ & \quad - \frac{1}{u}\int\limits_{0}^{\infty } {x^{v + 1} J_{v + 1} (ux)} e^{ - \xi y} \cos (\xi x){\text{d}}\xi \\ \int\limits_{0}^{\infty } {\xi^{v + 2} e^{ - \xi y} J_{v} (u\xi )\sin (\xi x){\text{d}}\xi } & = - \frac{x}{u}\int\limits_{0}^{\infty } {\xi^{v + 2} J_{v + 1} (u\xi )e^{ - \xi y} \cos (\xi x){\text{d}}\xi } \\ & \quad + \frac{y}{u}\int\limits_{0}^{\infty } {\xi^{v + 2} J_{v + 1} (u\xi )e^{ - \xi y} \sin (\xi x){\text{d}}\xi } \\ & \quad - \frac{1}{u}\int\limits_{0}^{\infty } {x^{v + 1} J_{v + 1} (ux)} e^{ - \xi y} \sin (\xi x){\text{d}}\xi \\ \end{aligned}$$

(7.51)

When v in Eq. (7.42) equals v + 1, it is easy to derive

$$\begin{aligned} \int\limits_{0}^{\infty } {\xi ^{{v + 2}} e^{{ - \xi y}} J_{{v + 1}} (u\xi )\cos (\xi x){\text{d}}\xi } & = \frac{{2(2u)^{{v + 1}} }}{{\sqrt \pi }}\Gamma (v + \frac{5}{2})R^{{ - 2v - 5}} [y\cos (v + \frac{5}{2})\varphi \\ & \quad - x\sin (v + \frac{5}{2})\varphi ] \\ \int\limits_{0}^{\infty } {\xi ^{{v + 2}} e^{{ - \xi y}} J_{{v + 1}} (u\xi )\sin (\xi x){\text{d}}\xi } & = - \frac{{2(2u)^{{v + 1}} }}{{\sqrt \pi }}\Gamma (v + \frac{5}{2})R^{{ - 2v - 5}} [x\cos (v + \frac{5}{2})\varphi \\ & \quad + y\sin (v + \frac{5}{2})\varphi ] \\ \end{aligned}$$

(7.52)

Substituting Eqs. (7.47) and (7.52) into Eq. (7.51), ξ-integrals in the case of c = d = v + 2 are solved:

$$\left\{ {\begin{array}{*{20}l} {\int\limits_{0}^{\infty } {\xi ^{{v + 2}} e^{{ - \xi y}} J_{v} (u\xi )\cos (\xi x)} {\text{d}}\xi = \frac{1}{u}\left[ {\begin{array}{*{20}l} {\frac{{2(2u)^{{v + 1}} }}{{\sqrt \pi }}\Gamma (v + \frac{5}{2})R^{{ - 2v - 5}} ((y^{2} - x^{2} )} \hfill \\ {\cos (v + \frac{5}{2})\varphi - 2xy\sin (v + \frac{5}{2})\varphi )} \hfill \\ { - \frac{{(2a)^{{v + 1}} }}{{\sqrt \pi }}\Gamma (v + \frac{3}{2})R^{{ - 2v - 3}} \cos (v + \frac{3}{2})\varphi } \hfill \\ \end{array} } \right]} \hfill \\ {\int\limits_{0}^{\infty } {\xi ^{{v + 2}} e^{{ - \xi y}} J_{v} (u\xi )\sin (\xi x)} {\text{d}}\xi = \frac{1}{u}\left[ {\begin{array}{*{20}l} {\frac{{2(2u)^{{v + 1}} }}{{\sqrt \pi }}\Gamma (v + \frac{5}{2})R^{{ - 2v - 5}} ((x^{2} - y^{2} )} \hfill \\ {\sin (v + \frac{5}{2})\varphi - 2xy\cos (v + \frac{5}{2})\varphi )} \hfill \\ { + \frac{{(2a)^{{v + 1}} }}{{\sqrt \pi }}\Gamma (v + \frac{3}{2})R^{{ - 2v - 3}} \sin (v + \frac{3}{2})\varphi } \hfill \\ \end{array} } \right]} \hfill \\ \end{array} } \right.$$

(7.53)

Appendix 2. Closed—Form of F(ξ)

The function F(ξ) in Eq. (7.24) can be expressed as

$$\begin{aligned} F\left( \xi \right) & = \frac{2}{\pi }\int\limits_{\xi }^{1} {\frac{f(\xi )}{{\sqrt {1 - \xi^{2} } }}} d\xi \\ & = F_{0} - \frac{2}{\pi }(\chi \arcsin \xi - (\chi - 1)\frac{{\xi^{\alpha + 1} }}{\alpha + 1}{}_{2}F_{1} (\frac{\alpha + 1}{2},\frac{1}{2};\frac{\alpha + 3}{2};\xi^{2} )) \\ \end{aligned}$$

(7.54)

where ₂F₁ is a hypergeometric function.