Closed-Form Solution to the Poromechanics of Deep Arbitrary-Shaped Openings Subjected to Rock Mass Alteration

Dong, Xiangjian; Karrech, Ali; Basarir, Hakan; Elchalakani, Mohamed; Qi, Chongchong

doi:10.1061/(ASCE)GM.1943-5622.0001875

Technical Papers

Sep 21, 2020

Closed-Form Solution to the Poromechanics of Deep Arbitrary-Shaped Openings Subjected to Rock Mass Alteration

Authors: Xiangjian Dong https://orcid.org/0000-0002-0900-189X, Ali Karrech [email protected], Hakan Basarir https://orcid.org/0000-0003-4556-4953, Mohamed Elchalakani, and Chongchong QiAuthor Affiliations

Publication: International Journal of Geomechanics

Volume 20, Issue 12

https://doi.org/10.1061/(ASCE)GM.1943-5622.0001875

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Abstract

The aim of this paper is to propose a closed-form solution to the poromechanics problem of stress and pore pressure distribution around noncircular openings at great depth subjected to hydrostatic water pressure and far-field geostresses. The problem is solved by superposing the effects of fluid and solid skeleton obtained over simple circular geometries and generalizing the obtained expression to various geometries of mined ore bodies using complex variable functions and conformal mapping techniques. The principal stresses obtained analytically over the opening boundary and within the domain are compared with the results of finite-element analysis to verify the proposed approach. The comparison conducted for a representative noncircular opening indicates good agreement between the analytical and numerical methods. Hence, a parametric study is used to investigate in detail the stress variation under different opening dimensions, heterogeneous initial total stress conditions, and far-field pore pressure values. The proposed solution could be instrumental in the design of underground openings and deep mass alterations (that include local anthropogenic damage zones). It could also be used to provide reinforcement solutions where stresses can reach the mechanical stability thresholds.

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Acknowledgments

The first author would like to acknowledge financial support provided by the China Scholarship Council (CSC) under grant number 201606420056. The authors would like to thank the anonymous reviewers for their considerable effort in improving the paper.

Appendix. Detailed Derivation of the Coefficients a_n and b_n

Substituting Eqs. (9), (12), and (13) into (7), the equation is derived as

ψ_{1} (α) + \frac{ω (α)}{\bar{ω^{'} (α)}} \bar{ψ_{1}^{'} (α)} + \bar{χ_{1} (α)} = - 2 B ω (α) - B^{'} \bar{ω (α)}

(28)

where: α = an arbitrary point on the unit circle boundary in the ξ plane. It can be expressed as α = e^iθ, thus

\bar{α} = e^{- i θ} = 1 / α

.

Using Cauchy’s integral operator and integrating Eq. (28), the following equation is obtained:

\frac{1}{2 π i} \oint_{C} \frac{ψ_{1} (α)}{α - ξ} d α + \frac{1}{2 π i} \oint_{C} \frac{ω (α)}{\bar{ω^{'} (α)}} \frac{\bar{ψ_{1}^{'} (α)}}{α - ξ} d α + \frac{1}{2 π i} \oint_{C} \frac{\bar{χ_{1} (α)}}{α - ξ} d α = \frac{1}{2 π i} \oint_{C} \frac{f_{0} (α)}{α - ξ} d α

(29)

where the function f₀(α) can be expressed as

f_{0} (α) = - 2 B ω (α) - B^{'} \bar{ω (α)}

(30)

The ψ₁(α) in Eq. (29) is the boundary value of ψ₁(ξ), which is an analytic function exterior to the unit circle. Therefore, the first integral term on the left hand side of Eq. (29) by using Cauchy’s integral is obtained as:

\frac{1}{2 π i} \oint_{C} \frac{ψ_{1} (α)}{α - ξ} d α = - ψ_{1} (ξ)

(31)

The complex constant is omitted here and in what follows, since it does not contribute to the stresses. The right hand side (RHS) of Eq. (29) can be expressed as:

\frac{1}{2 π i} \oint_{C} \frac{f_{0} (α)}{α - ξ} d α = \frac{1}{2 π i} \oint_{C} \frac{- 2 B ω (α)}{α - ξ} d α + \frac{1}{2 π i} \oint_{C} \frac{- B^{'} \bar{ω (α)}}{α - ξ} d α

(32)

Expanding the first term on the right-hand side of Eq. (32) yields

\frac{1}{2 π i} \oint_{C} \frac{- 2 B ω (α)}{α - ξ} d α = - 2 B \frac{1}{2 π i} \oint_{C} \frac{c_{0} α + c_{1} α^{- 1} + \dots + c_{n} α^{- n}}{α - ξ} d α

(33)

Note that the first integrand (c₋₁α + c₀) is a boundary value of an analytic function (c₋₁ξ + c₀) interior to the unit circle, whereas the other terms are analytical functions exterior to the unit circle. The denominator (α − ξ) is also an analytical function interior to the unit circle, given that |ξ| > 1. Therefore, the integral of the first term is zero and the rest can be obtained using the Cauchy-Goursat theory:

\begin{aligned} \frac{1}{2 π i} \oint_{C} \frac{- 2 B ω (α)}{α - ξ} d α & = - 2 B \frac{1}{2 π i} \oint_{C} \frac{c_{1} α^{- 1} + \dots + c_{n} α^{- n}}{α - ξ} d α \\ = 2 B (\frac{c_{1}}{ξ} + \frac{c_{2}}{ξ^{2}} + \dots + \frac{c_{n}}{ξ^{n}}) \end{aligned}

(34)

Similarly, the second term on the right-hand side of Eq. (32) gives

\begin{aligned} \frac{1}{2 π i} \oint_{C} \frac{- B^{'} \bar{ω (α)}}{α - ξ} d α & = - B^{'} \frac{1}{2 π i} \oint_{C} \frac{c_{- 1} α^{- 1} + c_{0} + c_{1} α + \dots + c_{n} α^{n}}{α - ξ} d α \\ = - B^{'} \frac{1}{2 π i} \oint_{C} \frac{c_{- 1} α^{- 1} + c_{0}}{α - ξ} d α - B^{'} \frac{1}{2 π i} \oint_{C} \frac{c_{1} α + \dots + c_{n} α^{n}}{α - ξ} d α \\ = - B^{'} \frac{1}{2 π i} \oint_{C} \frac{c_{- 1} α^{- 1} + c_{0}}{α - ξ} d α \\ = B^{'} \frac{c_{- 1}}{ξ} \end{aligned}

(35)

Combining Eqs. (34) and (35), the right-hand side of Eq. (29) can be expressed as

\frac{1}{2 π i} \oint_{C} \frac{f_{0} (α)}{α - ξ} d α = 2 B (\frac{c_{1}}{ξ} + \frac{c_{2}}{ξ^{2}} + \dots + \frac{c_{n}}{ξ^{n}}) + B^{'} \frac{c_{- 1}}{ξ}

(36)

The analytic function of ψ₁(ξ) is determined by

- ψ_{1} (ξ) + \frac{1}{2 π i} \oint_{C} \frac{ω (α)}{\bar{ω^{'} (α)}} \frac{\bar{ψ_{1}^{'} (α)}}{α - ξ} d α = 2 B (\frac{c_{1}}{ξ} + \frac{c_{2}}{ξ^{2}} + \dots + \frac{c_{n}}{ξ^{n}}) + B^{'} \frac{c_{- 1}}{ξ}

(37)

By conjugating Eq. (28) and following a similar procedure, the analytic function of χ₁(ξ) can be rewritten as

- χ_{1} (ξ) + \frac{1}{2 π i} \oint_{C} \frac{\bar{ω (α)}}{ω^{'} (α)} \frac{ψ_{1}^{'} (α)}{α - ξ} d α = 2 B \frac{c_{- 1}}{ξ} + B^{'} (\frac{c_{1}}{ξ} + \frac{c_{2}}{ξ^{2}} + \dots + \frac{c_{n}}{ξ^{n}})

(38)

The second term in Eq. (37) can be expressed as

\begin{aligned} \frac{ω (α)}{\bar{ω^{'} (α)}} & = \frac{c_{- 1} α + c_{0} + c_{1} α^{- 1} + c_{2} α^{- 2} + \dots + c_{n} α^{- n}}{c_{- 1} - c_{1} α^{2} - 2 c_{2} α^{3} - \dots - n c_{n} α^{n + 1}} \\ = \frac{1}{α^{n}} \frac{c_{- 1} α^{n + 1} + c_{0} α^{n} + c_{1} α^{n - 1} + c_{2} α^{n - 2} + \dots + c_{n}}{c_{- 1} - c_{1} α^{2} - 2 c_{2} α^{3} - \dots - n c_{n} α^{n + 1}} \end{aligned}

(39)

and

\begin{aligned} \bar{ψ_{1}^{'} (α)} & = - \bar{a_{1}} α^{2} - 2 \bar{a_{2}} α^{3} - 3 \bar{a_{3}} α^{4} - \dots - n \bar{a_{n}} α^{n + 1} - \dots \\ = α^{2} (- \bar{a_{1}} - 2 \bar{a_{2}} α - 3 \bar{a_{3}} α^{2} - \dots - n \bar{a_{n}} α^{n - 1} - \dots) \end{aligned}

(40)

As shown previously, the ratio

ω (α) / \bar{ω^{'} (α)}

has (n) order pole at the origin. Therefore, it can be written in the following form:

\frac{ω (α)}{\bar{ω^{'} (α)}} = \frac{d_{n}}{α^{n}} + \frac{d_{n - 1}}{α^{n - 1}} + \dots + \frac{d_{1}}{α} + \sum_{j = 0}^{\infty} d_{j}^{'} α^{j}

(41)

The product of

ω (α) / \bar{ω^{'} (α)}

and

\bar{ψ_{1}^{'} (α)}

has (n − 2) order pole at the origin and it takes the form

\frac{ω (α)}{\bar{ω^{'} (α)}} \bar{ψ_{1}^{'} (α)} = \frac{k_{n - 2}}{α^{n - 2}} + \frac{k_{n - 3}}{α^{n - 3}} + \dots + \frac{k_{1}}{α} + \sum_{j = 0}^{\infty} k_{j}^{'} α^{j}

(42)

Hence, the Cauchy integral is solved

\frac{1}{2 π i} \oint_{C} \frac{ω (α)}{\bar{ω^{'} (α)}} \frac{\bar{ψ_{1}^{'} (α)}}{α - ξ} d α = - \frac{k_{n - 2}}{ξ^{n - 2}} - \frac{k_{n - 3}}{ξ^{n - 3}} - \dots - \frac{k_{1}}{ξ}

(43)

The coefficients d_n and d_j′ are known for a given conformal mapping function whereas k_n and

\bar{a_{n}}

are unknown. Combining Eqs. (39), (41), and (42), the coefficient k_n is expressed as

{\begin{cases} k_{0}^{'} = - d_{2} \bar{a_{1}} - 2 d_{3} \bar{a_{2}} - \dots - (n - 1) d_{n} \bar{a_{n - 1}} \\ k_{1} = - d_{3} \bar{a_{1}} - 2 d_{4} \bar{a_{2}} - \dots - (n - 2) d_{n} \bar{a_{n - 2}} \\ \dots \\ k_{n - 2} = - d_{n} \bar{a_{1}} \end{cases}

(44)

Another set of equations is then required to obtain these coefficients. Substituting Eq. (43) into (37) yields

- \frac{a_{1}}{ξ} - \frac{a_{2}}{ξ^{2}} - \dots - \frac{a_{n}}{ξ^{n}} - \frac{k_{n - 2}}{ξ^{n - 2}} - \frac{k_{n - 3}}{ξ^{n - 3}} - \dots - \frac{k_{1}}{ξ} = 2 B (\frac{c_{1}}{ξ} + \frac{c_{2}}{ξ^{2}} + \dots + \frac{c_{n}}{ξ^{n}}) + B^{'} \frac{c_{- 1}}{ξ}

(45)

therefore, k_n can also be expressed as

{\begin{cases} k_{1} = - a_{1} - 2 B c_{1} - B^{'} c_{- 1} \\ k_{2} = - a_{2} - 2 B c_{2} \\ \dots \\ k_{n - 2} = - a_{n - 2} - 2 B c_{n - 2} \\ a_{n} = - 2 B c_{n} \end{cases}

(46)

where c_n, B, and B′ are known. Combining Eqs. (44) and (46) the coefficient a_n is obtained. Therefore, the analytic function ψ₁(ξ) is derived and the series number (n) is related to the conformal mapping terms.

Using the same procedure, the second term in Eq. (38) is written as

\frac{\bar{ω (α)}}{ω^{'} (α)} ψ_{1}^{'} (α) = \bar{k_{n - 2}} α^{n - 2} + \bar{k_{n - 3}} α^{n - 3} + \dots + \bar{k_{1}} α + \bar{k_{0}^{'}} + \sum_{j = 1}^{\infty} \frac{\bar{k_{j}^{'}}}{α^{j}}

(47)

To derive Eq. (47), the following equality is introduced:

g (α) = g_{*} (α) + g_{0} (α)

(48)

where

\begin{aligned} g (α) & = \frac{\bar{ω (α)}}{ω^{'} (α)} ψ_{1}^{'} (α) \\ g_{*} (α) & = \bar{k_{n - 2}} α^{n - 2} + \bar{k_{n - 3}} α^{n - 3} + \dots + \bar{k_{1}} α + \bar{k_{0}^{'}} \\ g_{0} (α) & = \sum_{j = 1}^{\infty} \frac{\bar{k_{j}^{'}}}{α^{j}} \end{aligned}

(49)

Therefore, the Cauchy integer of Eq. (48) reads

\begin{aligned} \frac{1}{2 π i} \oint_{C} \frac{\bar{ω (α)}}{ω^{'} (α)} \frac{ψ_{1}^{'} (α)}{α - ξ} d α & = \frac{1}{2 π i} \oint_{C} \frac{g (α)}{α - ξ} d α \\ = \frac{1}{2 π i} \oint_{C} \frac{g_{*} (α)}{α - ξ} d α + \frac{1}{2 π i} \oint_{C} \frac{g_{0} (α)}{α - ξ} d α \\ = - g_{0} (ξ) \\ = g_{*} (ξ) - g (ξ) \end{aligned}

(50)

Substituting Eq. (50) into (38), the coefficient b_n of analytic function χ₁(ξ) in Eq. (15) is obtained.

References

Ahmed, S., T. M. Müller, M. Madadi, and V. Calo. 2019. “Drained pore modulus and biot coefficient from pore-scale digital rock simulations.” Int. J. Rock Mech. Min. Sci. 114: 62–70. https://doi.org/10.1016/j.ijrmms.2018.12.019.

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Appendix. Detailed Derivation of the Coefficients an and bn

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Appendix. Detailed Derivation of the Coefficients a_n and b_n