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Mechanical Mechanism of Rock with Fluids under Elastic Deformation
DOI: 10.12677/AG.2018.88141, PDF, HTML, XML, 下载: 1,050  浏览: 5,242  科研立项经费支持

Abstract: The effective stress theorem reveals that the mechanical mechanism of confining pressure is shared by the porous skeleton and pore fluid in porous elastic materials, but Terzahi’s effective stress theorem can not explain the validity of the theory by the relationship between strain and stress. Biot and Willis’ effective stress theorem describes the relationship between strain and stress of saturated porous rocks when pore pressure remains constant, but can not explain the mechanical mechanism of pore pressure increasing with confining pressure. In this paper, the establishment condition and validity of Biot and Willis’ effective stress theorems are deduced theoretically. According to Gassmann’s equation, the mechanical mechanism of pore pressure increase caused by the increase of confining pressure of saturated rock is described, which has important application value in petroleum exploration and exploitation.

1. 引言

Terzahi (1923)的有效应力定理揭示了孔隙压力在多孔弹性土材料中能引起土材料明显的体积增大，在数量级上类似于围压引起的体积变化，与围压的方向相反，并且土壤在应力条件下的应变和破坏取决于压力差(有效应力) [1]

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Biot和Willis (1957)根据实验室测量，在保持多孔材料孔隙压力不变的情况下，建立了样品应变与应力的关系，提出了基于各向同性材料的有效应力定理 [2]

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Nur和Byerlee (1971) [4] 根据Terzahi (1923) [1] 和Biot (1941)的研究结果 [6] ，给出了一个基于统计学各向同性多孔集料的应力和孔隙压力的应变的张量通式

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2. 多孔岩石可压缩性定义

Figure 1. Confining pressure Pc and pore pressure Pp acting on a porous body [8]

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${C}_{bp}=\frac{1}{{V}_{b}}{\left[\frac{\partial {V}_{b}}{\partial {P}_{p}}\right]}_{{P}_{c}}$ (5)

${C}_{pc}=\frac{-1}{{V}_{p}}{\left[\frac{\partial {V}_{p}}{\partial {P}_{c}}\right]}_{{P}_{p}}$ (6)

${C}_{pp}=\frac{1}{{V}_{p}}{\left[\frac{\partial {V}_{p}}{\partial {P}_{p}}\right]}_{{P}_{c}}$ (7)

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${C}_{m}=\frac{-1}{{V}_{b}}{\left[\frac{\partial {V}_{b}}{\partial {P}_{c}}\right]}_{{P}_{d}=const}$ (12)

${C}_{\varphi }=\frac{-1}{{V}_{p}}{\left[\frac{\partial {V}_{p}}{\partial {P}_{c}}\right]}_{d{P}_{p}=d{P}_{c}}$ (13)

3. 有效应力定理的力学机制

$\frac{1}{{K}_{dry}}=\frac{1}{{K}_{ma}}+\frac{\varphi }{{K}_{\varphi }}$ (14)

$\frac{{P}_{c}}{{K}_{dry}}=\frac{{P}_{e}}{{K}_{dry}}+\frac{\varphi {P}_{p}}{{K}_{\varphi }}$ (15)

Figure 2. The diagrammatic sketch of three-dimensional stress of the saturated rock

${\epsilon }_{b}=\frac{\Delta {V}_{b}}{{V}_{b}}=f\left({P}_{c},{P}_{p}\right)$ (16)

$d{\epsilon }_{b}=-{C}_{bc}d{P}_{c}+{C}_{bp}d{P}_{p}$ (17)

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$-{C}_{bc}{P}_{e}=-{C}_{bc}{P}_{c}+\left({C}_{bc}-{C}_{m}\right){P}_{p}$ (19)

$\frac{{P}_{e}}{{K}_{dry}}=\frac{{P}_{c}}{{K}_{dry}}-\left(\frac{1}{{K}_{dry}}-\frac{1}{{K}_{ma}}\right){P}_{p}$ (20)

$-{C}_{u}{P}_{c}=-{C}_{bc}{P}_{c}+\left({C}_{bc}-{C}_{m}\right){P}_{p}$ (21)

$\frac{{P}_{p}}{{P}_{c}}=\frac{\frac{1}{{K}_{dry}}-\frac{1}{{K}_{sat}}}{\frac{1}{{K}_{dry}}-\frac{1}{{K}_{ma}}}$ (22)

4. Gassmann方程隐含的力学机制

$-{C}_{u}d{P}_{c}=-{C}_{bc}d{P}_{c}+\left({C}_{bc}-{C}_{m}\right)d{P}_{p}$ (23)

$\frac{d{P}_{p}}{d{P}_{c}}=\frac{\frac{1}{{K}_{dry}}-\frac{1}{{K}_{sat}}}{\frac{1}{{K}_{dry}}-\frac{1}{{K}_{ma}}}$ (24)

$\frac{d{P}_{p}}{d{P}_{c}}=B=\frac{1}{1+{K}_{\varphi }\left(\frac{1}{{K}_{f}}-\frac{1}{{K}_{ma}}\right)}=\frac{1}{1+\varphi \left(\frac{1}{{K}_{f}}-\frac{1}{{K}_{ma}}\right){\left(\frac{1}{{K}_{dry}}-\frac{1}{{K}_{ma}}\right)}^{-1}}$ (25)

$\frac{1}{{K}_{dry}}-\frac{1}{{K}_{sat}}=\frac{{\left(\frac{1}{{K}_{dry}}-\frac{1}{{K}_{ma}}\right)}^{2}}{\varphi \left(\frac{1}{{K}_{f}}-\frac{1}{{K}_{ma}}\right)+\left(\frac{1}{{K}_{dry}}-\frac{1}{{K}_{ma}}\right)}$ (26)

${K}_{sat}=\frac{{K}_{dry}}{1-\left(1-\frac{{K}_{dry}}{{K}_{ma}}\right)\left(\frac{d{P}_{p}}{d{P}_{c}}\right)}$ (27)

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Brown和Korringa (1975)放宽了Gassmann方程的各向同性的内在假设，并且推导出广义的Gassmann方程 [17]

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Brown和Korringa (1975)描述了岩石在不排水的状态下，Skempton系数B值(Skempton, 1954)的表达式为 [17]

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5. 结论

Terzahi (1923)的有效应力定理 [1] 描述了地表饱和土壤的孔隙流体与土壤骨架(土壤骨架相当于排水状态下的多孔土壤)分担围压的力学机制，但是其应变与应力关系是模糊的，难以从土壤的三维力学模型解释其应变与应力的关系。Biot和Willis (1957) [2] 提出的有效应力定理能从理论上解释岩石围压增加的过程中，孔隙流体压力保持不变时岩石的应变与应力的关系，并且Biot和Willis (1957)提出的有效应力定理能解释Terzahi (1923)的有效应力定理成立的条件，对于地表的饱和土壤而言，土壤骨架的体积模量Kdry要远小于土壤矿物颗粒的体积模量Kma (Kdry/Kma ≈ 0)，因此有效应力系数n ≈ 1，Biot和Willis (1957)的有效应力定理式(2)可简化为Terzahi(1923)的有效应力定理式(1)。

Biot和Willis (1957)提出的有效应力定理 [2] 描述了饱和多孔岩石在孔隙压力保持不变的情况下岩石应变与应力的关系，饱和岩石在围压Pc不断增加的过程中，要保持孔隙流体压力Pp不变，岩石孔隙流体必须与外界连通，排出部分流体才能保证孔隙流体压力不变(释放多余的压力)。从理论的严密性方面来讲，Biot和Willis (1957)的有效应力定理 [2] 是不足以描述孔隙不连通(非渗透)或者孔隙与外界不连通的饱和岩石的力学机制的，如果岩石孔隙不连通，意味着孔隙流体与外界不连通，当围压Pc不断增加时，孔隙内的部分流体无法排出，多余孔隙压力无法得到释放，随着围压Pc不断增加，孔隙流体压力Pp不断增大。

Gassmann方程 [3] 描述了饱和岩石在不排水的状态下(孔隙流体质量不变)，围压增量dPc所引起孔隙压力增量dPp的这一力学机制，在数学上是孔隙压力Pp对围压Pc的导数。事实上，对于饱和岩石，孔隙流体与外界不连通时(孔隙流体质量不变)，在围压Pc不断增加的过程中，孔隙压力Pp逐渐增大，岩石骨架(相当于排水状态下的岩石)和孔隙流体如何分担围压的问题并未得到完全地解决，Gassmann方程只是确定了孔隙压力增量dPp与围压增量dPc的关系dPp/dPc = B，而不是孔隙压力Pp与围压Pc的关系。

Nur和Byerlee (1957) [5] 对Biot和Willis (1957) [2] 提出的有效应力定理进行了理论推导，假定有一个具有任意形状、孔隙互相连通的固体材料各向同性集料，承受一个围压Pc和一个均匀的孔隙压力Pp，压缩方向为正方向。该应力状态可从概念上分两步完成：第一步，施加孔隙压力Pp和一个与之相等的围压，即；第二步，施加剩余的围压，对孔隙压力不做任何进一步改变。如果现在假设集料在Pc的压力范围内呈线性弹性，那么可以将这两步导致的应变场进行叠加。

Figure A1. A homogeneous aggregate with pores (The lines S represent the surfaces of the pores that are subject to a pore pressure that is equal to the confining pressure)

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