Pore pressure coefficient B for saturated geomaterials with compressible, isotropic and uniform solid and fluid phases

Abstract

This paper derives an expression for pore pressure coefficient $B$ in terms of a geomaterial’s mesoscopic properties and jacketed and unjacketed macroscopic compressibilities. The mesoscopic properties include phase compressibilities and pressure distribution ratios. The pressure distribution ratios represent the geomaterial’s meso-heterogeneity, while the difference between the unjacketed bulk compressibility and the solid grain compressibility represents macroscopic effects, such as solid grain rearrangement. The difference in phase compressibilities is shown to be the principal determinant of $B$, with the effects of meso-heterogeneity and macroscopic change relatively negligible. The unjacketed pore compressibility appears to be quite sensitive to the effects of both meso-heterogeneity and macroscopic change. Although $B$ can be derived reliably from a measured value of unjacketed pore compressibility, unjacketed pore compressibility derived from a measured value of $B$ is considered unreliable. The expression for $B$ includes five elastic coefficients. The necessary and sufficient condition for constant porosity in an unjacketed test is proposed. The published expressions for $B$, the coefficient of fluid content and porosity change are recovered by introducing appropriate assumptions with regard to meso-heterogeneity and phase compressibilities.

Appendices

Appendix 1: The reciprocity relation

Consider a material that exhibits reversibility in all processes. Its response to any change in its state is independent of the path between the start and end states. Internal energy potentials exist for such material, and the internal energy associated with any given state is a scalar measure of that state. If the applied pressure and pore pressure are identified as the independent variables that define the element’s state, then the element’s volumetric strain and seepage can be expressed as partial derivatives of the element’s complementary internal energy potential, $U^{ *} \left( {p, \omega } \right)$:

$$e = - \frac{\delta V}{V} = \left( {\frac{{\partial U^{*} }}{\partial p}} \right)_{\omega }$$

(66)

$$\xi = \frac{{\left( {\delta V_{\text{p}} } \right)^{s} }}{V} = \left( {\frac{{\partial U^{*} }}{\partial \omega }} \right)_{p}$$

(67)

where compressive volumetric strain and inflow seepage are positive-valued.

The volumetric strain and seepage rates follow from differentiation of Eqs (66) and (67):

$$\delta e = \left( {\frac{{\partial^{2} U^{*} }}{{\partial p^{2} }}} \right)_{\omega } \delta p + \left( {\frac{{\partial^{2} U^{*} }}{\partial p\partial \omega }} \right)_{p} \delta \omega$$

(68)

$$\delta \xi = \left( {\frac{{\partial^{2} U^{*} }}{\partial \omega \partial p}} \right)_{\omega } \delta p + \left( {\frac{{\partial^{2} U^{*} }}{{\partial \omega^{2} }}} \right)_{p} \delta \omega$$

(69)

If $U^{ *}$ is $C^{2}$-continuous in its independent variables, $p, \omega$, Schwarz’s theorem states that its cross-derivatives are identical:

$$\left( {\frac{{\partial^{2} U^{*} }}{\partial \omega \partial p}} \right)_{\omega } = \left( {\frac{{\partial^{2} U^{*} }}{\partial p\partial \omega }} \right)_{p}$$

(70)

The dependent variable rates are defined as

$$\delta e = \left( {\frac{\partial e}{\partial p}} \right)_{\omega } \delta p + \left( {\frac{\partial e}{\partial \omega }} \right)_{p} \delta \omega$$

(71)

$$\delta \xi = \left( {\frac{\partial \xi }{\partial p}} \right)_{\omega } \delta p + \left( {\frac{\partial \xi }{\partial \omega }} \right)_{p} \delta \omega$$

(72)

Comparing Eqs. (68) and (69) to (71) and (72), yields

$$\left( {\frac{\partial e}{\partial \omega }} \right)_{p} = \left( {\frac{{\partial^{2} U^{*} }}{\partial p\partial \omega }} \right)_{p}$$

(73)

$$\left( {\frac{\partial \xi }{\partial p}} \right)_{\omega } = \left( {\frac{{\partial^{2} U^{*} }}{\partial \omega \partial p}} \right)_{\omega }$$

(74)

From Eqs. (70), (73) and (74),

$$\left( {\frac{\partial \xi }{\partial p}} \right)_{\omega } = \left( {\frac{\partial e}{\partial \omega }} \right)_{p}$$

(75)

or

$$\left( {\frac{{\partial V_{\text{p}} }}{\partial p}} \right)_{\omega }^{s} = - \left( {\frac{\partial V}{\partial \omega }} \right)_{p}$$

(76)

The volumetric change of the fluid phase in full-drained condition is entirely due to seepage. Therefore,

$$\left( {\frac{{\partial V_{\text{p}} }}{\partial p}} \right)_{\omega } = - \left( {\frac{\partial V}{\partial \omega }} \right)_{p}$$

(77)

Appendix 2: The comparative notations

See Table 4.

Table 4 Comparison between notations used in different references