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A Semianalytical Solution for a Griffith Crack Nonuniformly Pressurized By Internal Fluid

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Abstract

Unintentional variations of fluid pressure within a hydraulic fracture will disturb the surrounding stress state, affect the stability of fracture propagation and complicate the fracture intersections during the process of hydraulic fracturing. However, little attention has been paid to the effect of nonuniform fluid pressure inside the hydraulic fracture. This paper presents a semianalytical solution for a Griffith crack nonuniformly pressurized by internal fluid in an impermeable elastic plane. The fluid pressure is hypothetically designated as a general polynomial with respect to the location of the fluid, such that the effect of an arbitrary form of fluid pressure distribution can be explored by polynomial fitting. The semianalytical solution is capable of being degenerated into constant pressure forms which confirms that the solution is an extension of classic constant pressure. In addition, the critical propagation conditions and stress distributions (e.g., σxx) under constant and nonuniform pressures are compared and discussed. The comparison results indicate that the effect of nonuniform fluid pressure accumulated by the number of polynomial terms increases the magnitude of stress (or displacement) but does not change its distribution. Subsequently, the semianalytical solution is validated by comparing the fracture intersection predicted by the semianalytical solution with the laboratory experimental observations and published predictions under constant fluid pressure. Good agreement with the experiments and sufficient advantages over previous predictions are observed. Finally, a sensitivity analysis of existing parameters in the semianalytical solution, including crack length, initial pressure, number of terms and number of subintervals, is conducted to evaluate their influence on surrounding stresses and critical propagation conditions, which further demonstrates the applicability and reliability of the presented semianalytical solution. The new solution enriches hydraulic fracturing theory by considering the nonuniform fluid pressure effect and provides important reference for fracture network design during hydraulic fracturing.

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Abbreviations

a :

Crack length

a C :

Critical fracture length at the beginning of fracture expansion under constant fluid pressure

a N :

Critical fracture length at the beginning of fracture expansion under nonuniform fluid pressure

b k :

Dimensionless coefficient variable of the assumed distribution of internal pressure

E :

Young’s modulus

\(E^{\prime}\) :

Plane-strain elastic modulus

K I :

Stress intensity factor (SIF)

K IC :

Mode I fracture toughness of rock (critical stress intensity factor)

\(K_{\text{I}}^{\text{C}}\) :

Stress intensity factor under the action of constant fluid pressure

\(K_{\text{I}}^{\text{N}}\) :

Stress intensity factor under the action of nonuniform fluid pressure

m :

Number of subintervals in the composite Simpson’s rule

n :

Number of terms of the polynomial that is assumed as the distribution of internal pressure

P(x):

Assumed net fluid pressure inside the crack

P 0 :

Constant term of the assumed distribution of internal pressure (initial fluid pressure excluding the effect of far-field stress)

P hf :

Critical fluid pressure required to drive the crack tip to move ahead

r c :

Critical radius of the plastic zone ahead of fracture tip

u x :

Displacement in the direction of x-axis

u y :

Displacement in the direction of y-axis

v :

Poisson’s ratio

μ :

Friction coefficient

β :

Intersection angle (≤ 90°) between a hydraulic fracture and a natural fracture

κ :

Dimensionless form of the critical intensity factor

σ H :

Maximum horizontal in situ stress

σ h :

Minimum horizontal in situ stress

σ xx :

Normal stress in the direction of x-axis induced by internal pressure

σ xy :

Tangential stress induced by internal pressure

σ yy :

Normal stress in the direction of y-axis induced by internal pressure

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Acknowledgements

This work was financially supported by the Research Fund for Talents of Guizhou University (Grant No. 201901), the Special Research Funds of Guizhou University (Grant No. 201903), the Project supported by graduate research and innovation foundation of Chongqing, China (Grant No. CYB19015) and the National Natural Science Foundation of China (Nos. 51374257, 50804060).

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Correspondence to Yu Zhao.

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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:

$$\left\{ \begin{aligned} \int\limits_{0}^{\infty } {\xi^{d} {\text{e}}^{ - \xi y} J_{v} (u\xi )\cos (\xi x){\text{d}}\xi } \hfill \\ \int\limits_{0}^{\infty } {\xi^{c} {\text{e}}^{ - \xi y} J_{v} (u\xi )\sin (\xi x){\text{d}}\xi } \hfill \\ \end{aligned} \right.(c = d = v, \, v + 1, \, v + 2,\; \ldots ),$$
(41)

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

Gradshteyn and Ryzhik (2007) also provided the following solution of ξ-integrals when c and d both equal v + 1:

$$\left\{ \begin{aligned} \int\limits_{0}^{\infty } {\xi^{v + 1} {\text{e}}^{ - \xi y} J_{v} (u\xi )\cos (\xi x){\text{d}}\xi = } \frac{{2(2u)^{v} }}{\sqrt \pi }\varGamma (v + \frac{3}{2})R^{ - 2v - 3} \left[ {y\cos \left( {v + \frac{3}{2}} \right)\varphi - x\sin \left( {v + \frac{3}{2}} \right)\varphi } \right] \hfill \\ \int\limits_{0}^{\infty } {\xi^{v + 1} {\text{e}}^{ - \xi y} J_{v} (u\xi )\sin (\xi x){\text{d}}\xi } = - \frac{{2(2u)^{v} }}{\sqrt \pi }\varGamma (v + \frac{3}{2})R^{ - 2v - 3} \left[ {x\cos \left( {v + \frac{3}{2}} \right)\varphi + y\sin \left( {v + \frac{3}{2}} \right)\varphi } \right] \hfill \\ \end{aligned} \right. \, \left( {y > 0,v > - \frac{3}{2}} \right).$$
(42)

A recursion formula of the Bessel function is given by:

$$(x^{v} J_{v} (ax))' = a \cdot x^{v} J_{v - 1} (ax).$$
(43)

Making v in Eq. (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:

$$\begin{aligned} \int\limits_{0}^{\infty } {\xi^{w + 1} {\text{e}}^{ - \xi y} J_{w} (u\xi )\cos (\xi x){\text{d}}\xi } = \frac{1}{u}\int\limits_{0}^{\infty } {(\xi^{w + 1} J_{w + 1} (u\xi ))'{\text{e}}^{ - \xi y} \cos (\xi x){\text{d}}\xi } , \hfill \\ \int\limits_{0}^{\infty } {\xi^{w + 1} {\text{e}}^{ - \xi y} J_{w} (u\xi )\sin (\xi x){\text{d}}\xi } = \frac{1}{u}\int\limits_{0}^{\infty } {(\xi^{w + 1} J_{w + 1} (u\xi ))'{\text{e}}^{ - \xi y} \sin (\xi x){\text{d}}\xi } . \hfill \\ \end{aligned}$$
(44)

Using integration by parts, Eq. (44) can be expressed as:

$$\begin{aligned} \int\limits_{0}^{\infty } {\xi^{w + 1} {\text{e}}^{ - \xi y} J_{w} (u\xi )\cos (\xi x){\text{d}}\xi } = \frac{1}{u}\left[ {\xi^{w + 1} J_{w + 1} (u\xi ){\text{e}}^{ - \xi y} \cos (\xi x)\left| {_{0}^{\infty } } \right. - \int\limits_{0}^{\infty } {\xi^{w + 1} J_{w + 1} (u\xi )({\text{e}}^{ - \xi y} \cos (\xi x))'{\text{d}}\xi } } \right], \hfill \\ \int\limits_{0}^{\infty } {\xi^{w + 1} {\text{e}}^{ - \xi y} J_{w} (u\xi )\sin (\xi x){\text{d}}\xi } = \frac{1}{u}\left[ {\xi^{w + 1} J_{w + 1} (u\xi ){\text{e}}^{ - \xi y} \sin (\xi x)\left| {_{0}^{\infty } } \right. - \int\limits_{0}^{\infty } {\xi^{w + 1} J_{w + 1} (u\xi )({\text{e}}^{ - \xi y} \sin (\xi x))'{\text{d}}\xi } } \right]. \hfill \\ \end{aligned}$$
(45)

After simplification, Eq. (45) becomes

$$\begin{aligned} \int\limits_{0}^{\infty } {\xi^{w + 1} {\text{e}}^{ - \xi y} J_{w} (u\xi )\cos (\xi x){\text{d}}\xi } &{ = }\frac{x}{u}\int\limits_{0}^{\infty } {\xi^{w + 1} J_{w + 1} (u\xi ){\text{e}}^{ - \xi y} \sin (\xi x){\text{d}}\xi + \frac{y}{u}} \int\limits_{0}^{\infty } {\xi^{w + 1} J_{w + 1} (u\xi ){\text{e}}^{ - \xi y} \cos (\xi x){\text{d}}\xi } \hfill \\& { = }\frac{x}{u}Y + \frac{y}{u}X{ = }M_{1} , \hfill \\ \int\limits_{0}^{\infty } {\xi^{w + 1} {\text{e}}^{ - \xi y} J_{w} (u\xi )\sin (\xi x){\text{d}}\xi }& { = } - \frac{x}{u}\int\limits_{0}^{\infty } {\xi^{w + 1} J_{w + 1} (u\xi ){\text{e}}^{ - \xi y} \cos (\xi x){\text{d}}\xi } + \frac{y}{u}\int\limits_{0}^{\infty } {\xi^{w + 1} J_{w + 1} (u\xi ){\text{e}}^{ - \xi y} \sin (\xi x){\text{d}}\xi } \hfill \\ &{ = } - \frac{x}{u}X + \frac{y}{u}Y{ = }M_{2} , \hfill \\ \end{aligned}$$
(46)

In Eq. (46), two binary linear equations of \(\int\nolimits_{0}^{\infty } {\xi^{w + 1} {\text{e}}^{ - \xi y} J_{{w{ + }1}} (u\xi )\cos (\xi x){\text{d}}\xi }\) (X) and \(\int\nolimits_{0}^{\infty } {\xi^{w + 1} {\text{e}}^{ - \xi y} J_{{w{ + }1}} (u\xi )\sin (\xi x){\text{d}}\xi }\) (Y) can be solved using M1 and M2 derived by Eq. (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. (45). Thus, the unknown functions X and Y are separately expressed as:

$$\left\{ \begin{aligned} X = \int\limits_{0}^{\infty } {\xi^{v + 1} J_{v + 1} (u\xi ){\text{e}}^{ - \xi y} \cos (\xi x){\text{d}}\xi } = \frac{{(2u)^{v + 1} }}{\sqrt \pi }\varGamma \left( {v + \frac{3}{2}} \right)R^{ - 2v - 3} \cos \left( {v + \frac{3}{2}} \right)\varphi \hfill \\ Y = \int\limits_{0}^{\infty } {\xi^{v + 1} J_{v + 1} (u\xi ){\text{e}}^{ - \xi y} \sin (\xi x){\text{d}}\xi } = - \frac{{(2u)^{v + 1} }}{\sqrt \pi }\varGamma \left( {v + \frac{3}{2}} \right)R^{ - 2v - 3} \sin \left( {v + \frac{3}{2}} \right)\varphi \hfill \\ \end{aligned} \right..$$
(47)

In the case of c = d=v + 2, using the recurrence relation of Eq. (43), we can obtain:

$$(x^{{v{ + }2}} J_{{v{ + }1}} (ax))' = a \cdot x^{v + 2} J_{v} (ax){ + }x^{{v{ + 1}}} J_{{v{ + }1}} (ax).$$
(48)

Then, this relation reduces to:

$$x^{v + 2} J_{v} (ax) = \frac{1}{a} \cdot \left[ {(x^{v + 2} J_{v + 1} (ax))' - x^{v + 1} J_{v + 1} (ax)} \right].$$
(49)

Substituting Eq. (49) into ξ-integrals, we can obtain:

$$\begin{aligned} \int\limits_{0}^{\infty } {\xi^{v + 2} {\text{e}}^{ - \xi y} J_{v} (u\xi )\cos (\xi x){\text{d}}\xi } = \frac{1}{u}\int\limits_{0}^{\infty } {\left[ {(\xi^{v + 2} J_{v + 1} (u\xi ))' - x^{v + 1} J_{v + 1} (ux)} \right]{\text{e}}^{ - \xi y} \cos (\xi x){\text{d}}\xi } , \hfill \\ \int\limits_{0}^{\infty } {\xi^{v + 2} {\text{e}}^{ - \xi y} J_{v} (u\xi )\sin (\xi x){\text{d}}\xi } = \frac{1}{u}\int\limits_{0}^{\infty } {\left[ {(\xi^{v + 2} J_{v + 1} (u\xi ))' - x^{v + 1} J_{v + 1} (ux)} \right]{\text{e}}^{ - \xi y} \sin (\xi x){\text{d}}\xi } . \hfill \\ \end{aligned}$$
(50)

Based on integration by parts, Eq. (50) can be further simplified to:

$$\begin{aligned} \int\limits_{0}^{\infty } {\xi^{v + 2} {\text{e}}^{ - \xi y} J_{v} (u\xi )\cos (\xi x){\text{d}}\xi } = \frac{x}{u}\int\limits_{0}^{\infty } {\xi^{v + 2} J_{v + 1} (u\xi ){\text{e}}^{ - \xi y} \sin (\xi x){\text{d}}\xi } \hfill \\ \, + \frac{y}{u}\int\limits_{0}^{\infty } {\xi^{v + 2} J_{v + 1} (u\xi ){\text{e}}^{ - \xi y} \cos (\xi x){\text{d}}\xi } - \frac{1}{u}\int\limits_{0}^{\infty } {x^{v + 1} J_{v + 1} (ux)} {\text{e}}^{ - \xi y} \cos (\xi x){\text{d}}\xi , \hfill \\ \int\limits_{0}^{\infty } {\xi^{v + 2} {\text{e}}^{ - \xi y} J_{v} (u\xi )\sin (\xi x){\text{d}}\xi } = - \frac{x}{u}\int\limits_{0}^{\infty } {\xi^{v + 2} J_{v + 1} (u\xi ){\text{e}}^{ - \xi y} \cos (\xi x){\text{d}}\xi } \hfill \\ \, + \frac{y}{u}\int\limits_{0}^{\infty } {\xi^{v + 2} J_{v + 1} (u\xi ){\text{e}}^{ - \xi y} \sin (\xi x){\text{d}}\xi } - \frac{1}{u}\int\limits_{0}^{\infty } {x^{v + 1} J_{v + 1} (ux)} {\text{e}}^{ - \xi y} \sin (\xi x){\text{d}}\xi . \hfill \\ \end{aligned}$$
(51)

When v in Eq. (42) equals v + 1, it is easy to derive:

$$\begin{aligned} \int\limits_{0}^{\infty } {\xi^{v + 2} {\text{e}}^{ - \xi y} J_{v + 1} (u\xi )\cos (\xi x){\text{d}}\xi = } \frac{{2(2u)^{v + 1} }}{\sqrt \pi }\varGamma (v + \frac{5}{2})R^{ - 2v - 5} \left[ {y\cos \left( {v + \frac{5}{2}} \right)\varphi - x\sin \left( {v + \frac{5}{2}} \right)\varphi } \right], \hfill \\ \int\limits_{0}^{\infty } {\xi^{v + 2} {\text{e}}^{ - \xi y} J_{v + 1} (u\xi )\sin (\xi x){\text{d}}\xi } = - \frac{{2(2u)^{v + 1} }}{\sqrt \pi }\varGamma (v + \frac{5}{2})R^{ - 2v - 5} \left[ {x\cos \left( {v + \frac{5}{2}} \right)\varphi + y\sin \left( {v + \frac{5}{2}} \right)\varphi } \right]. \hfill \\ \end{aligned}$$
(52)

Substituting Eqs. (47) and (52) into Eq. (51), ξ-integrals in the case of c = d=v + 2 are solved:

$$\left\{ \begin{aligned} \int\limits_{0}^{\infty } {\xi^{v + 2} {\text{e}}^{ - \xi y} J_{v} (u\xi )\cos (\xi x){\text{d}}\xi = \frac{1}{u}\left[ \begin{aligned} \frac{{2(2u)^{v + 1} }}{\sqrt \pi }\varGamma \left( {v + \frac{5}{2}} \right)R^{ - 2v - 5} \left( {(y^{2} - x^{2} )\cos (v + \frac{5}{2})\varphi - } \right. \hfill \\ \left. {2xy\sin \left( {v + \frac{5}{2}} \right)\varphi } \right) - \frac{{(2a)^{v + 1} }}{\sqrt \pi }\varGamma (v + \frac{3}{2})R^{ - 2v - 3} \cos \left( {v + \frac{3}{2}} \right)\varphi \hfill \\ \end{aligned} \right]} \hfill \\ \int\limits_{0}^{\infty } {\xi^{v + 2} {\text{e}}^{ - \xi y} J_{v} (u\xi )\sin (\xi x){\text{d}}\xi = \frac{1}{u}\left[ \begin{aligned} \frac{{2(2u)^{v + 1} }}{\sqrt \pi }\varGamma \left( {v + \frac{5}{2}} \right)R^{ - 2v - 5} \left( {(x^{2} - y^{2} )\sin \left( {v + \frac{5}{2}} \right)\varphi - } \right. \hfill \\ \left. {2xy\cos \left( {v + \frac{5}{2}} \right)\varphi } \right) + \frac{{(2a)^{v + 1} }}{\sqrt \pi }\varGamma \left( {v + \frac{3}{2}} \right)R^{ - 2v - 3} \sin \left( {v + \frac{3}{2}} \right)\varphi \hfill \\ \end{aligned} \right]} \hfill \\ \end{aligned} \right..$$
(53)

Appendix 2: Closed Form of F(ξ)

The function F(ξ) in Eq. (24) can be expressed as:

$$\begin{aligned} F\left( \xi \right) &= \frac{2}{\pi }\int\limits_{\xi }^{1} {\frac{f(\xi )}{{\sqrt {1 - \xi^{2} } }}} {\text{d}}\xi \hfill \\ \, &= F_{0} - \frac{2}{\pi }\left( {\chi \arcsin \xi - (\chi - 1)\frac{{\xi^{\alpha + 1} }}{\alpha + 1}{}_{2}F_{1} \left( {\frac{\alpha + 1}{2},\frac{1}{2};\frac{\alpha + 3}{2};\xi^{2} } \right)} \right), \hfill \\ \end{aligned}$$
(54)

where 2F1 is a hypergeometric function.

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Zhang, Y., Zhao, Y., Yang, H. et al. A Semianalytical Solution for a Griffith Crack Nonuniformly Pressurized By Internal Fluid. Rock Mech Rock Eng 53, 2439–2460 (2020). https://doi.org/10.1007/s00603-020-02052-z

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