Poroelastic measurements resulting in complete data sets for granular and other anisotropic porous media

Heterogeneous pores

Another important type of heterogeneity that can occur in practice involves heterogeneity of the pore space. One obvious issue is whether the pores are all connected to each other, or whether there may be two (or more) distinct, but intertwining, pore systems. One well-known example of this situation is the double-porosity concept (Barenblatt and Zheltov, 1960; Gurevich et al., 2009; Berryman and Pride, 2002), in which one type of pore has high volume but low permeability, while the other has low volume (imagine a system of very flat cracks or fractures) and high permeability. I can also consider that some pores might be interior to some grains and not connected to any other pores (and might therefore also be empty of pore fluid), while other subsets of the grains have no inherent porosity of this type, and so are truly solid grains.

I will not try to deal with all these cases simultaneously, as even enumerating all the possibilities quickly becomes burdensome. I will limit myself instead to one of the more typical scenarios, considered for example by Brown and Korringa (1975) and by Rice and Cleary (1976) -- and also see the recent related work of Gurevich et al. (2009).

Heterogeneity of the pore space is most important when considering flow of fluid into and out of the boundaries of a porous sample. Then, the concept of increment of fluid content $\zeta$ comes into play, and special care is required. A straightforward definition of this dimensionless parameter (just as the strains $e_{11}$, $e_{22}$, $\ldots$, $e_{13}$ are all dimensionless) is given by:

$$\zeta \equiv \frac{\delta(\phi V) - \delta V_f}{V} \simeq \phi\left(\frac{\delta V_\phi}{V_\phi} - \frac{\delta V_f}{V_f}\right),$$ (10)

where $V$ is the overall volume of the initially fully fluid-saturated porous material at the first instant of consideration, $V_\phi = \phi V$ is the pore volume, with $\phi$ being the fluid-saturated porosity of the volume, and $V_f$ is the volume occupied by the pore-fluid, making $V_f = \phi V$ at the start. The $\delta$'s indicate small changes in the quantities following. For ``drained'' systems, there must be a reservoir of the same fluid just outside the volume $V$ that can either supply more fluid or absorb any excreted fluid as needed during the nonstationary phase of the poroelastic process; the amount of pore fluid can therefore either increase or decrease from the initial amount of pore fluid, and at the same time the pore volume can also be changing, but not necessarily at exactly the same rate as the pore fluid itself. The one exception to these statements is when the surface pores of the total volume $V$ are sealed, in which case the system is ``undrained'' and $\zeta \equiv 0$, identically. In these circumstances, it is still possible that $V_f$ and $V_\phi = \phi V$ are both changing, but because of the imposed undrained boundary conditions, they are necessarily changing at the same rate. The result is that, for an isotropic system, I have:

$$\zeta = \phi\left[\frac{\delta\sigma_c}{K_p} + \delta p_f\left(\frac{1}{K_p} - \frac{1}{K_R^\phi} + \frac{1}{K_f}\right)\right],$$ (11)

and where the various moduli in (11) are defined by the following relations [see Brown and Korringa (1975)]:

$$-\frac{\delta V_f}{V_f} = \frac{\delta p_f}{K_f},$$ (12)

$$-\frac{\delta V}{V} = \frac{\delta p_d}{K_R^d} + \frac{\delta p_f}{K_R^g},$$ (13)

and

$$-\frac{\delta V_\phi}{V_\phi} = \frac{\delta p_d}{K_p} + \frac{\delta p_f}{K_R^\phi}.$$ (14)

The changes in fluid pressure and differential pressure are respectively $\delta p_f$ and $\delta p_d \equiv \delta p_c - \delta p_f$, where $\delta p_c = -\delta \sigma_c$ is the uniform confining pressure, if the external confining pressure is uniform. If not, then this quantity is replaced in the definition of $\delta p_c$ by $-\delta \sigma_m$, which is the change in the mean confining pressure and where $\sigma_m \equiv (\sigma_{11} + \sigma_{22} + \sigma_{33})/3$ is the definition of the mean principal stress. Clearly, if the confining principal stress is uniform ( $\sigma_{11} = \sigma_{22} = \sigma_{33}$), then the mean stress equals this uniform confining stress. If not, then there can be additional shearing effects that need to taken into account, but these do not play any role in the changes of fluid content since this quantity is effectively a measure only of the total number of fluid particles contained in the pertinent pore volume.

It can also be shown using poroelastic reciprocity (and I will show this later as it very clearly develops in the following anisotropic analysis) that

$$\frac{\phi}{K_p} = \frac{\alpha_R}{K_R^d} = \frac{1}{K_R^d} - \frac{1}{K_R^g}.$$ (15)

So generalizing Gassmann's formula for undrained modulus gives:

$$\frac{1}{K_R^u} = \frac{1-\alpha_R B}{K_R^d},$$ (16)

where $\alpha_R = 1 - K_R^d/K_R^g$ and

$$B = \left(\frac{1}{K_R^d}-\frac{1}{K_R^g}\right)\left[\left(\frac... ...{K_R^g}\right) + \phi \left(\frac{1}{K_f}-\frac{1}{K_R^\phi}\right)\right]^{-1}$$ (17)

is the second coefficient of Skempton (1954). Combining these terms, I find that the most general form of the equation for the undrained bulk modulus in the isotropic case is:

$$\frac{1}{K_R^u} = \frac{1}{K_R^d} - \left(\frac{1}{K_R^d}-\frac{1... ...{K_R^g}\right) + \phi\left(\frac{1}{K_f}-\frac{1}{K_R^\phi}\right)\right]^{-1},$$ (18)

or, alternatively,

$$\frac{1}{K_R^u} = \frac{1}{K_R^d} - \frac{(\alpha_R/K_R^d)^2}{\alpha_R/K_R^d + \phi\left(\frac{1}{K_f}-\frac{1}{K_R^\phi}\right)},$$ (19)

which is the isotropic result of Brown and Korringa (1975), and should also be compared directly to (1). So, if the pore modulus and grain modulus are equal with $K_R^\phi = K_R^g$, then (19) reduces exactly to (1). Although this result is the same as that of Brown and Korringa (1975), I nevertheless write it differently to emphasize different features.

Poroelastic measurements resulting in complete data sets for granular and other anisotropic porous media