Rock Anelasticity, Pore Geometry and the Biot–Gardner Effect

Abstract

The anelastic properties of porous rocks depend on the pore characteristics, specifically, the pore aspect ratio and the pore fraction (related to the soft porosity). At high frequencies, there is no fluid pressure communication throughout the pore space and the rock becomes stiffer than at low frequencies, where the pore pressure is fully equilibrated. The models considered here include explicit pore geometry information in determining the poroelastic parameters. They are extensions of the EIAS (equivalent inclusion-average stress) and CPEM (cracks and pores effective medium) models to the whole frequency range, based on the Zener model. Knowing the degree of stiffness dispersion between the low- and high-frequency limits, we fit experimental data in the whole frequency range and obtain the average crack aspect ratio and soft porosity as a function of effective pressure. Then, we compute the dispersion and quality factor of the bulk, shear and Young moduli, and the P- and S-wave seismic velocities and quality factors as a function of frequency. However, when measuring axial or volumetric motions along a cylindrical sample, there is fluid flow at the ends of the sample in the experiments considered here. This generates dispersion and attenuation due to axial flow of the pore fluid, which does not occur for a plane wave in unbounded media. This phenomenon is called “drained/undrained transition" Pimienta et al. (J Geophys Res Solid Earth; https://doi.org/10.1002/2017JB014645, 2017). Actually, it is an axial version of the Biot–Gardner (BG) effect, and implies an “artificial" (mesoscopic) attenuation peak (and dispersion) due to the generation of slow (diffusion) Biot modes at the cylinder boundary, inducing a global flow at the scale of the sample. The classical BG effect is due to fluid flow along the radial direction, on the basis of open-pore conditions at the sides of the sample. In this case, the sides are sealed. To use the EIAS and CPEM models, the BG effect has to be removed to obtain the intrinsic Q of the rock. The models are applied here for measurements on sandstone. The axial BG effect is more evident if the intrinsic attenuation is weak or absent. An example is Lavoux limestone, which has a bimodal porosity distribution, with an equal proportion of intragranular microporosity and intergranular macroporosity (round pores). In this case, the attenuation and dispersion are related to the BG effect, since no squirt flow is detected due to the absence of cracks. We verified that the bulk and Young moduli obtained from the axial and hydrostatic oscillations are consistent with each other, and that the theoretical description of the axial BG effect shows some discrepancies with the data.

Appendices

Appendix A: EIAS (Equivalent Inclusion-Average Stress) Model

The so-called isolated pores or high-frequency moduli, predicted by Endres and Knight (1997; Eqs. (32) and (33)), are

$$\begin{aligned} \begin{array}{l} K_\mathrm{iso} = K_s + \displaystyle \frac{\phi (K_f-K_s) \gamma }{1 - \phi (1 - \gamma ) } , \\ \mu _\mathrm{iso} = \displaystyle \frac{\mu _s (1 - \phi )}{1 - \phi (1 - \chi )}, \end{array} \end{aligned}$$

(A.1)

where (Endres and Knight 1997; Eqs. (54) and (55)),

$$\begin{aligned} \begin{array}{l} \gamma = (1 - c) P_1 + c P_2 , \\ \chi = (1-c) Q_1 + c Q_2 \end{array} \end{aligned}$$

(A.2)

and (Berryman 1980a, b, 1995; Mavko et al. 2009, p. 187),

$$\begin{aligned} \begin{array}{l} P_1 = \displaystyle \frac{K_s + 4 \mu _s/3}{K_f + 4 \mu _s/3} , \\ \\ P_2 = \displaystyle \frac{K_s}{K_f + \pi a \beta } , \ \ \ \beta = \mu _s \cdot \displaystyle \frac{3 K_s + \mu _s}{3 K_s + 4 \mu _s} , \\ \\ Q_1 = 1 + \mu _s/ \zeta , \ \ \ \zeta = \displaystyle \frac{\mu _s}{6} \cdot \displaystyle \frac{9 K_s + 8 \mu _s}{K_s + 2 \mu _s} , \\ \\ Q_2 = \displaystyle \frac{1}{5} \left[ 1 + \displaystyle \frac{8 \mu _s}{\pi a (\mu _s + 2 \beta )} + 2 \cdot \displaystyle \frac{K_f +2 \mu _s/3}{K_f + \pi a \beta } \right] , \end{array} \end{aligned}$$

(A.3)

where $P_1$ and $Q_1$ correspond to spherical pores and $P_2$ and $Q_2$ are approximations for penny-shaped cracks. Actually, there are (more complex) exact expressions for P and Q, which hold for any aspect ratio of the oblate spheroidal pores, including spherical pores and thin cracks: $P = \frac{1}{3} T_{iijj}$ and $Q = \frac{1}{5} (T_{ijij} - P)$, where $T_{iijj}$ and $T_{ijij}$ are given in Appendix A of Berryman (1980b) or in page 189 of Mavko et al. (2009). The expressions $P_1$ and $Q_1$ for spherical pores are exact, while the approximations $P_2$ and $Q_2$ slightly deviate from the exact expressions at high aspect ratios of the cracks.

The effective moduli, when complete fluid pressure communication occurs (low frequencies), are (Endres and Knight 1997; Eqs. (34) and (35)),

$$\begin{aligned} \begin{array}{l} K_\mathrm{com} = K_s + \displaystyle \frac{\phi K_s (K_f-K_s) \gamma _0}{(1 - \phi ) (K_s-K_f) + [K_f +\phi (K_s-K_f)] \gamma _0} , \\ \mu _\mathrm{com} = \displaystyle \frac{\mu _s (1 - \phi )}{1 - \phi (1 - \chi _0)}, \end{array} \end{aligned}$$

(A.4)

where

$$\begin{aligned} \begin{array}{l} \gamma _0 = (1 - c) P_{01} + c P_{02} , \\ \chi _0 = (1-c) Q_{01} + c Q_{02} , \end{array} \end{aligned}$$

(A.5)

where

$$\begin{aligned} \begin{array}{l} P_{01} = P_1 (K_f=0) = 1 + \displaystyle \frac{3 K_s}{4 \mu _s}, \\ \\ P_{02} = P_2 (K_f=0) = \displaystyle \frac{K_s}{\pi a \beta }, \\ \\ Q_{01} = Q_1 (K_f=0) = Q_1 , \\ \\ Q_{02} = Q_2 (K_f=0) = \displaystyle \frac{1}{5} \left[ 1 + \displaystyle \frac{4 \mu _s}{\pi a} \cdot \displaystyle \frac{\mu _s + 8 \beta }{3 \beta (\mu _s + 2 \beta )} \right] . \end{array} \end{aligned}$$

(A.6)

The moduli $K_\mathrm{com}$ and $\mu _\mathrm{com}$ [Eqs. (34) and (35) in Endres and Knight (1997)], with $K_f = 0$, are the dry-rock moduli to be used in Gassmann equations, i.e.,

$$\begin{aligned} K_{m 0}= \frac{K_s (1 - \phi ) }{1 + \phi (\gamma _0 - 1) }, \ \ \ \mu _{m 0}= \frac{\mu _s (1 - \phi )}{1 + \phi (\chi _0 -1 )}, \end{aligned}$$

(A.7)

since those moduli with $K_f \ne 0$ [equations (A.4)], are identical to the Gassmann moduli [Eqs. (52) and (53) in Endres and Knight (1997)]. Values of a and c can be obtained by fitting the relaxed and unrelaxed moduli.

The EIAS model has no restrictions on the crack density, since it considers the interactions between cracks. Endres and Knight (1997) also developed a dilute approximation given by their Eqs. (48)–(51), which shows a better agreement with the CPEM model (see next section) than the EIAS model.

For this model, we evaluate the crack density based on the stiffness moduli as in O’Connell and Budiansky (1974) and Budiansky and O’Connell (1976), which is summarized in page 187 of Mavko et al. (2009), identifying $\nu$ with the uncracked Poisson ratio (obtained from $K_u$ and $\mu _u$), and $\nu _\mathrm{SC}^*$ with $\nu _\mathrm{iso}$ (the same for the bulk modulus). The uncracked wet-rock moduli $K_u$ and $\mu _u$ are obtained from $K_\mathrm{iso}$ and $\mu _\mathrm{iso}$ by setting the crack fraction c = 0.

Appendix B: CPEM (Cracks and Pores Effective Medium) Model

Adelinet et al. (2011) proposed alternative equations to the EIAS model, based on the non-interactive crack approximation, i.e., valid for low crack density. The high-frequency wet-rock moduli are obtained from

$$\begin{aligned} \frac{K_s}{K_\mathrm{iso}}=\, & {} 1 + \phi (1-c ) \frac{3 (1-\nu _s)}{2 (1 - 2 \nu _s)} \left( \frac{\delta _p}{1+\delta _p}\right) \nonumber \\&+ \rho _c \frac{16 (1-\nu _s^2)}{9 (1 - 2 \nu _s)} \left( \frac{\delta _c}{1+\delta _c} \right) \end{aligned}$$

(B.1)

and

$$\begin{aligned} \frac{\mu _s}{\mu _\mathrm{iso}}= \,& {} 1 + \phi (1-c) \frac{15 (1-\nu _s)}{ 7 - 5 \nu _s} + \rho _c \left[ \frac{16 (1-\nu _s)}{15 (1 - 0.5 \nu _s)}\right. \nonumber \\&\left. + \frac{32 (1-\nu _s)}{45} \left( \frac{\delta _c}{1+\delta _c} \right) \right] , \end{aligned}$$

(B.2)

where

$$\begin{aligned} \rho _c = \frac{3 \phi c}{4 \pi a} \end{aligned}$$

(B.3)

is the crack density,

$$\begin{aligned} \delta _p = \frac{2 Y_s}{9 (1 - \nu _s)} \left( \frac{1}{K_f}-\frac{1}{K_s} \right) , \ \ \ \delta _c = \frac{\pi Y_s a}{4 (1 - \nu _s^2)} \left( \frac{1}{K_f}-\frac{1}{K_s} \right) , \end{aligned}$$

(B.4)

and

$$\begin{aligned} Y_s = \frac{9 K_s \mu _s}{3 K_s + \mu _s} \ \ \ \mathrm{and} \ \ \ \nu _s = \frac{3 K_s-2 \mu _s}{2 (3 K_s + \mu _s)} \end{aligned}$$

(B.5)

are the mineral Young modulus and Poisson ratio, respectively.

The high-frequency dry-rock moduli can be obtained from equations (B.1) and (B.2) by taking $\delta _p \rightarrow \infty$ and $\delta _c \rightarrow \infty$, so that $\delta _p/ (1+ \delta _p) = 1$ and $\delta _c/ (1+ \delta _c) = 1$. The low-frequency wet-rock moduli $K_\mathrm{com}$ and $\mu _\mathrm{com}$ are given by Gassmann equations (1), where $K_m$ and $\mu _m$ are the high-frequency dry-rock moduli previously obtained.

Appendix C: The Radial Biot–Gardner Effect

White (1986) [Eq. (3)] reports the complex Young modulus related to the Biot–Gardner effect,

$$\begin{aligned} Y = 4 \mu \cdot \frac{U-\Theta ^*}{V-\Theta ^*} , \ \ \ \Theta = \frac{2 J_1 (x)}{x J_0 (x)} , \ \ \ x = \mathrm{i}^{5/2} r_0 \sqrt{\omega q} , \end{aligned}$$

(C.1)

where $J_n$ are Bessel functions, $r_0$ is the radius of the cylinder,

$$\begin{aligned} U= \,& {} \displaystyle \frac{(3W/4) (D + 4/3)}{W-D} , \ \ \ V = \displaystyle \frac{(W+1/3) (D + 4/3)}{W-D} , \nonumber \\ W= & {} \displaystyle \frac{K_G}{\mu } , \ \ \ D = \displaystyle \frac{K_m}{\mu } , \end{aligned}$$

(C.2)

$$\begin{aligned} q= \,& {} \frac{b H}{R P - Q^2} , \end{aligned}$$

(C.3)

$$\begin{aligned} H=\, & {} P + R + 2 Q = K_G + 4 \mu /3, \ \ \ K_G = K_m + \alpha ^2 M,\nonumber \\ P= & {} K_m + (\alpha - \phi )^2 M + \displaystyle \frac{4}{3} \mu , \ \ \ Q = \phi M (\alpha - \phi ) , \ \ \ R = \phi ^2 M , \nonumber \\ M= & {} \displaystyle \frac{ K_s }{1 - \phi - K_m / K_s + \phi K_s / K_f } , \ \ \ \alpha = 1 - \displaystyle \frac{K_m}{K_s} , \nonumber \\ b= & {} \displaystyle \frac{\eta \phi ^2}{\kappa } , \end{aligned}$$

(C.4)

$\eta$ is the fluid viscosity and $\kappa$ is the permeability [see Eqs. (2.20) and (3.3) in Gardner (1982), and Eqs. (7.16)–(7.18) in Carcione (2014)]. The factor $\mathrm{i}^{5/2}$, instead of $\mathrm{i}^{3/2}$, in the argument of the Bessel functions is due to the fact that we use here the opposite sign convention for the Fourier transform ($\omega \rightarrow - \omega$).

The theory predicts a relaxation frequency of

$$\begin{aligned} f_{BG} \propto \frac{\kappa K_m}{\eta } \end{aligned}$$

(C.5)

(Pimienta et al. 2017).

Appendix D: The Axial Biot–Gardner Effect

Pimienta et al. (2016a) obtained the Skempton coefficient and bulk modulus in the case that the rock sample satisfies open (“drained") and semi-open boundary conditions at the ends, contrary to the Biot–Gardner theory, which holds for open conditions at the sides of the cylindrical sample (Gardner 1962; White 1986; Dunn 1987).

The Young modulus is

$$\begin{aligned} Y (\omega ) = \frac{9 K (\omega ) \mu }{3 K (\omega ) + \mu }, \end{aligned}$$

(D.1)

where $\mu$ is assumed to be a real quantity and, according to Eqs. (9) and (11) in Pimienta et al. (2016a), the bulk modulus is

$$\begin{aligned} K (\omega ) = \frac{L K_m}{B} \left[ L ( B^{-1} - \alpha ) + \alpha \int _0^L f(\omega , z) dz \right] ^{-1}, \end{aligned}$$

(D.2)

where

$$\begin{aligned} f (\omega , z) = \frac{\sinh [ k (L-z)] + \sinh (k z)}{\sinh (k L)} , \ \ \ k = (1 + \mathrm{i}) \sqrt{\frac{\omega }{2 D}} . \end{aligned}$$

(D.3)

The various quantities are as follow: z is the axial spatial variable, L is the length of the sample,

$$\begin{aligned} B = \frac{\alpha M}{K_G} , \end{aligned}$$

(D.4)

is the Skempton coefficient.

$$\begin{aligned} M = \displaystyle \frac{ K_s }{1 - \phi - K_m / K_s + \phi K_s / K_f } , \ \ \ \alpha = 1 - \displaystyle \frac{K_m}{K_s} , \end{aligned}$$

(D.5)

are the fluid modulus and Biot coefficient, respectively,

$$\begin{aligned} K_G = K_m + \alpha ^2 M, \end{aligned}$$

(D.6)

is the Gassmann bulk modulus,

$$\begin{aligned} D = \frac{\kappa }{S \eta } \end{aligned}$$

(D.7)

is the hydraulic diffusivity,

$$\begin{aligned} S= \frac{\alpha }{B K_m} \end{aligned}$$

(D.8)

is the storage coefficient, $\eta$ is the fluid viscosity and $\kappa$ is the permeability. Performing the integration in Equation (D.2), we obtain

$$\begin{aligned} K (\omega ) = \frac{L K_m}{B} \left[ L ( B^{-1} - \alpha ) + \frac{2 \alpha [ \cosh (kL) -1 ]}{k \sinh (k L)} \right] ^{-1}. \end{aligned}$$

(D.9)

In real experiments, the “drained" condition is difficult to achieve, Pimienta et al. (2016a) combined the purely drained and undrained conditions, and obtained the more realistic “experimentally undrained" condition. An approximate solution for the pressure $p_f$ is given by their Eq. (15) (see also Sun et al. 2019), which holds for equal dead volumes on both ends of the sample. Since the local volumetric strain is $\epsilon _v = (1/K_m) (P - \alpha p_f)$, where P is the applied source pressure, it can be shown that we obtain Eq. (D.2) with

$$\begin{aligned} f (\omega , z) = \frac{\cosh [ k (L/2-z)]}{ b \sinh (k L/2) + \cosh (k L/2)} , \ \ \ \ b = \frac{2 \pi r_0^2 S}{k S_V} , \end{aligned}$$

(D.10)

where $S_V = V_d/K_f$ is the storage capacity of the total dead volume, with $V_d$ the dead volume and $r_0$ the radius of the rock sample [see Borgomano et al. (2017) for values of the dead volume and other properties (below their Eq. (28))].

Performing the integration of equation (D10) as above [see equation (D2)], we obtain

$$\begin{aligned} K (\omega ) = \frac{L K_m}{B} \left[ L ( B^{-1} - \alpha ) + \frac{(2 \alpha /k)}{ b + \coth (k L/2)} \right] ^{-1}. \end{aligned}$$

(D.11)

If $b \rightarrow 0$ (e.g., infinite dead volume), equations (D.9) and (D.11) coincide.

Sun et al. (2019) developed the theory in three dimensions, basically showing that in this case the peak frequency is lower than that predicted by the theory of Pimienta et al. (2016a), which indicates that the fluid flow needs more time to equilibrate the pore pressure.