Journal of Earth Science  2018, Vol. 29 Issue (6): 1335-1339 PDF     0
On the Fluid Dependence of Seismic Anisotropy: Beyond Biot-Gassmann
Leon Thomsen
University of Houston, Delta Geophysics, Houston, Texas 77042, USA
ABSTRACT: This work addresses the question of the fluid dependence of the non-dimensional parameters of seismic anisotropy. It extends the classic theory of the fluid-dependence of elasticity, and applies the approximation of weak seismic anisotropy. The analysis shows that reliance upon the classic theory leads to oversimplified conclusions. Extending the classic theory introduces new parameters (which must be experimentally determined) into the conclusions, making their application in the field context highly problematic.
KEY WORDS: Biot    Gassmann    incompressibility    fluids    fluid substitution    pore compressibility

0 INTRODUCTION

At seismic frequencies and below, the standard analysis of the fluid dependence of poro-elasticity is due originally to Biot (1941) and Gassmann (1951) ("B-G"). They showed that, for isotropic rocks, the shear modulus is independent of fluid content, and the incompressibility has a closed-form fluid dependence (given by Eq. (1), below). Gassmann (1951) also provided the fluid dependence of the anisotropic tensor components of poro-elastic compliance and of poro-elastic stiffness.

Subsequently, Brown and Korringa (1975) ("B & K") generalized this work to include a further parameter (often called the pore compressibility, κϕ). They showed that the B-G result was a special case, valid only if κϕ=κSol, the compressibility of the solid grains. Their proof that this condition fails for most rocks (except for rocks with uniform mineralogy) was shown by Thomsen(2017, 2010), to be more general than that, including even uniform mineralogy. B & K also gave the corresponding (post B-G) generalization for the fluid dependence of the anisotropic tensor components of poro-elastic compliance (Brown and Korringa, 1975). The present work extends that generalization to the poro-elastic stiffness tensor and to the non-dimensional parameters of anisotropy.

This B & K generalization introduces significant difficulty into the analysis of fluid dependence, since κϕ and its tensor analogs are properties of the rock (dependent on micro-geometry, pore pressure, etc.) and may not be found in a hand-book, as with the solid compressibility. Since seismic-band anisotropy may be caused by sub-seismic inhomogeneity (e.g., thin-layering or fracturing), the fluid dependence also varies on this sub-seismic scale (since the fluid pressure affects rock properties on the grain scale). This makes a fundamental difficulty in reaching general conclusions about the fluid dependence of anisotropy.

In the following sections, firstly the scalar equations are reviewed, to establish notation. Then the tensor equations are developed, in their general form. Then this formalism is applied to the case of weak polar anisotropy, wherein it simplifies substantially, but not as much as previously perceived.

1 SCALAR CASE

To establish notation, the well-known B-G prediction for the incompressibility difference is quoted here

 $\begin{array}{l} \;\;\;{\mathit{\boldsymbol{K}}_{fld}}\left({{\mathit{\boldsymbol{K}}_{Fld}}} \right) - {\mathit{\boldsymbol{K}}_{gas}}\\ = \frac{{{{({\mathit{\boldsymbol{K}}_{Sol}} - {\mathit{\boldsymbol{K}}_{gas}})}^2}}}{{\left[ {\phi {\mathit{\boldsymbol{K}}_{Sol}}\left({{\mathit{\boldsymbol{K}}_{Sol}}/{\mathit{\boldsymbol{K}}_{Fld}} - 1} \right) + ({\mathit{\boldsymbol{K}}_{Sol}} - {\mathit{\boldsymbol{K}}_{gas}})} \right]}} \end{array}$ (1)

where Kfld, Kgas, KFld, KSol are respectively the incompressibilities of: the rock saturated with a fluid, the rock saturated with highly compressible gas, the pore-filling fluid (brine, oil, gas or a combination), and the solid grains of the rock. Observe the capitalization in the subscripts: KFld is the incompressibility of the fluid which saturates the rock, which has incompressibility Kfld; and ϕ is porosity.

The B-G prediction above concerns only the difference KfldKgas, not either one of these separately. This result is valid only in the low frequency limit, and assumes that there are no chemical effects, and no static fluid pressure changes associated with 4D fluid substitution. It is widely applied within the exploration geophysics community, worldwide, especially in the 4D context.

However, as reviewed by Thomsen(2017, 2012), the experimental support for this theoretical result is very thin, because most of the experimental tests have violated its low-frequency assumption. Hence, the commonly observed discrepancies have usually been attributed to dispersive effects, not relevant at seismic frequencies. However, recent static compression effects (Hart and Wang, 2010) show discrepancies which cannot be explained away by dispersion, but rather require the generalization of B-G theory as first proposed by B & K, over 40 years ago.

The B & K result generalizing Eq. (1) is derived in terms of compressibility (κ=1/K) rather than incompressibility

 ${\mathit{\boldsymbol{\kappa }}_{gas}} - {\mathit{\boldsymbol{\kappa }}_{fld}} = \frac{{{{\left({{\mathit{\boldsymbol{\kappa }}_{gas}} - {\mathit{\boldsymbol{\kappa }}_M}} \right)}^2}}}{{\left[ {\left({{\mathit{\boldsymbol{\kappa }}_{Fld}} - {\mathit{\boldsymbol{\kappa }}_\phi }} \right)\phi + \left({{\mathit{\boldsymbol{\kappa }}_{gas}} - {\mathit{\boldsymbol{\kappa }}_M}} \right)} \right]}}$ (2)
 ${\mathit{\boldsymbol{\kappa }}_\phi } \equiv \frac{{ - 1}}{{{V_\phi }}}{\left({\frac{{\partial {V_\phi }}}{{\partial {p_{Fld}}}}} \right)_{{p_{dif}}}}$ (3a)
 ${\mathit{\boldsymbol{\kappa }}_M} \equiv \frac{{ - 1}}{V}{\left({\frac{{\partial V}}{{\partial {p_{Fld}}}}} \right)_{{p_{dif}}}}$ (3b)

Here V is the volume of an undrained mass element of rock, and Vϕ=ϕV is the volume of the pore space of that mass element. pFld is the fluid pressure in the pore space (assumed uniform at low frequencies); pdif=ppFld is the differential pressure (with p the average pressure on the exterior of the mass element). The B & K derivation of Eq. (2) utilizes only multi-variate calculus and the theorem of elastic reciprocity, applied to an undrained mass element of rock, with volume V (c.f., B & K Eq. (1)). Nowhere whether they consider a drained or an "unjacketed" experiment (the experimental measurement of κM (Eq. (3b)) appears to require an unjacketed procedure, but Thomsen (2017) shows how to measure it by compression of an undrained sample).

B & K showed that the two compressibility parameters (3a) and (3b) are related to each other and the compressibility κSol of the solid grains by

 ${\mathit{\boldsymbol{\kappa }}_M} = (1 - \phi){\mathit{\boldsymbol{\kappa }}_{Sol}} + \phi {\mathit{\boldsymbol{\kappa }}_\phi }$ (4)

The parameter κϕ is called the "pore compressibility". B & K do not indicate the mnemonic meaning of the subscript M on the parameter κM, but in view of Eq. (4) it may be called the "mean compressibility". Equation (4) may be used to eliminate κϕ from Eq. (2), in favor of κSol, leaving κM as the most convenient new B & K parameter.

B-G argued that

 ${\mathit{\boldsymbol{\kappa }}_M} = {\mathit{\boldsymbol{\kappa }}_\phi } = {\mathit{\boldsymbol{\kappa }}_{Sol}}$ (5)

in which case the B & K Eq. (2) reduces to the B-G Eq. (1). But B & K argued that this applies only to mono-mineralic rocks, and Thomsen(2017, 2010) argued that this restriction does not even apply in that special case (the error in logic of B-G, B & K, and many other analysts over the past 65 years, which leads to the conclusion (5), is the assumption that the fluid pressure and the differential pressure are independent variables, whereas in an undrained experiment, the two are coupled together by the boundary condition of no flow. So, this conclusion (5) by B-G in fact contains an implicit assumption about pore micro-geometry).

This extension of the B & K proof to mono-mineralic rocks seems to be a minor point, because almost all rocks are poly-mineralic. Further, the various minerals in a rock differ among themselves, (in their elasticity) much less than they differ from the fluid. And, it has not been clear how to experimentally determine κM on an undrained sample.

Further, the introduction of the parameter κM complicates the analysis of fluid dependence, since it is a property of the rock (dependent on micro-geometry, pore pressure, etc.) and may not be found in a handbook, as with the solid compressibility. For all these reasons, the generalization of B-G by B & K has usually been ignored, for the past 40 years, even as the importance of 4D seismic surveys has grown, as has the need to accurately understand the physical basis for the observed changes.

However, with the proof that Eq. (5) fails even for mono-mineralic rocks, a consequence is that any observed failure of this relation is due to the heterogeneity of the pore space, as well as of the mineralogy. Furthermore, Thomsen (2017) described a straightforward experimental procedure to determine κM on an undrained sample. And, this generalization of the B-G resolves a longstanding inconsistency with effective medium theory (Thomsen, 2017).

Recent compression data by Hart and Wang (2010) indicate that, for Berea sandstone, the differences between the three quantities in Eq. (5) are significant, beyond the experimental uncertainty. The H & W data show that κM differs from κSol by ~40% for this Berea sandstone, and κϕ differs from κSol by ~200% (Thomsen, 2017), consistent with H & W. The differences are attributable to the non-uniformity of the mineralogy was only a few percent. Although this is only a single case, it shows that the complications to fluid-dependency analysis, posed by Eq. (2), should be taken seriously. As discussed below, this applies especially to the fluid-dependency analysis of the anisotropy parameters.

Thomsen (2017) inverted Eq. (2), and incorporated Eq. (3) to eliminate κϕ, yielding the corresponding expression for the incompressibilities

 $\begin{array}{l} \;\;{\mathit{\boldsymbol{K}}_{fld}}({\mathit{\boldsymbol{\kappa }}_{Fld}}) - {\mathit{\boldsymbol{K}}_{gas}}\\ = \frac{{{{({\mathit{\boldsymbol{K}}_M} - {\mathit{\boldsymbol{K}}_{gas}})}^2}}}{{\phi \mathit{\boldsymbol{K}}_M^2(1/{\mathit{\boldsymbol{K}}_{Fld}} - 1/{\mathit{\boldsymbol{K}}_{Sol}}) + \mathit{\boldsymbol{K}}_M^2/{\mathit{\boldsymbol{K}}_{Sol}} - {\mathit{\boldsymbol{K}}_{gas}}}} \end{array}$ (6)

where KM=1/κM. It is easy to see that, with the B-G restriction (5), Eq. (6) reduces to Eq. (1). In the general case, Eq. (6) is the generalization of the B-G result (1) to include the B & K refinement, in analyses of isotropic wave propagation.

Equations (2) and (6) are valid for rocks which are both micro- and macro-anisotropic. For macro-anisotropic rocks, the scalar incompressibility (the anisotropic response of the rock to isotropic pressure) is given by

 $1/\mathit{\boldsymbol{K}} = \mathit{\boldsymbol{\kappa }} = {\mathit{\boldsymbol{S}}_{iikk}}$ (7)

where S is the (rank-four) poro-elastic compliance tensor, and the repeated indices indicate summation from 1 to 3.

2 TENSOR CASE

Gassmann (1951) presented (without derivation) the tensor generalization of Eq. (1); in the present notation his result is

 $\mathit{\boldsymbol{C}}_{ijkl}^{fld} - \mathit{\boldsymbol{C}}_{ijkl}^{gas} = \frac{{\left({{\mathit{\boldsymbol{K}}_{Sol}}{\mathit{\boldsymbol{\delta }}_{ij}} - \mathit{\boldsymbol{C}}_{ij}^{gas}/3} \right)\left({{\mathit{\boldsymbol{K}}_{Sol}}{\mathit{\boldsymbol{\delta }}_{kl}} - \mathit{\boldsymbol{C}}_{kl}^{gas}/3} \right)}}{{\phi \mathit{\boldsymbol{K}}_{Sol}^2\left({1/{\mathit{\boldsymbol{K}}_{Fld}} - 1/{\mathit{\boldsymbol{K}}_{Sol}}} \right) + {\mathit{\boldsymbol{K}}_{Sol}} - \mathit{\boldsymbol{K}}_{gas}^\sum }}$ (8)

where C is the (rank-four) poro-elastic stiffness tensor, and δij is the Kronecker delta tensor (the rank-two identity tensor). The other rank-two tensor in (8) is

 $\mathit{\boldsymbol{C}}_{ij}^{gas} \equiv \mathit{\boldsymbol{C}}_{ijkk}^{gas}/3$ (9)

Not to be confused with the rank-two 6×6 matrix $\mathit{\boldsymbol{C}}_{\alpha \beta }^{gas}$. Latin indices here count from 1 to 3, whereas Greek indices count from 1 to 6. Also appearing in Eq. (8) is the quantity

 $\mathit{\boldsymbol{K}}_{gas}^\sum \equiv \mathit{\boldsymbol{C}}_{iikk}^{gas}/9 = \mathit{\boldsymbol{C}}_{ii}^{gas}/3 \ne {\mathit{\boldsymbol{K}}_{gas}}$ (10)

Although ${\mathit{\boldsymbol{K}}^\sum } \ne \mathit{\boldsymbol{K}}$ in general, these two are identical in the special case of isotropy, and are equal to first order in the anisotropy parameters {ε, δ, γ} defined by Thomsen (1986) in the special case of weak polar anisotropy (Thomsen, 2012).

The stiffness tensor C is the rank-four inverse of the compliance tensor S

 ${\mathit{\boldsymbol{C}}_{ijkl}}{\mathit{\boldsymbol{S}}_{klmn}} = {\mathit{\boldsymbol{I}}_{ijmn}} \equiv \left({{\mathit{\boldsymbol{\delta }}_{im}}{\mathit{\boldsymbol{\delta }}_{jn}} + {\mathit{\boldsymbol{\delta }}_{in}}{\mathit{\boldsymbol{\delta }}_{jm}}} \right)/2$ (11)

Of course, Gassmann's result (8) incorporates the B-G restriction (5), both in the denominator and in the appearance in the numerator of the scaled identity tensor KSolδij.

B & K presented the tensor generalization of their scalar compressibility Eq. (2); in the present notation their result is

 $\mathit{\boldsymbol{S}}_{ijkl}^{gas} - \mathit{\boldsymbol{S}}_{ijkl}^{fld} = \frac{{\left({\mathit{\boldsymbol{S}}_{ij}^M - \mathit{\boldsymbol{S}}_{ij}^{gas}} \right)\left({\mathit{\boldsymbol{S}}_{kl}^M - \mathit{\boldsymbol{S}}_{kl}^{gas}} \right)}}{{\left({{\kappa _{Fld}} - {\kappa _\phi }} \right)\phi + \left({{\kappa _{gas}} - {\kappa _M}} \right)}}$ (12)

where

 $\mathit{\boldsymbol{S}}_{ij}^{gas} \equiv \mathit{\boldsymbol{S}}_{ijkk}^{gas}$ (13)

B & K define the rank two tensor SM as

 $\mathit{\boldsymbol{S}}_{ij}^M \equiv - {\left({\frac{{\partial {\mathit{\boldsymbol{\eta }}_{ij}}}}{{\partial {p_{Fld}}}}} \right)_{\mathit{\boldsymbol{\sigma }}_{kl}^{dif}}}$ (14)

where ηij is the average strain throughout the mass element, and $\mathit{\boldsymbol{\sigma }}_{kl}^{dif} = {\mathit{\boldsymbol{\sigma }}_{kl}} + {p_{Fld}}{\mathit{\boldsymbol{\delta }}_{kl}}$ is the differential stress tensor. Note that the expression here for $\mathit{\boldsymbol{\sigma }}_{kl}^{dif}$ contains a plus sign, whereas the earlier expression for pdif contains a minus sign; this is a result of the standard sign convention relating stress to pressure. To make the notation more symmetrical, I introduce here a rank-four tensor $\mathit{\boldsymbol{S}}_{ijkl}^M$, such that its sum over the two stress-indices is the quantity in Eq. (14), $\mathit{\boldsymbol{S}}_{ijkk}^M \equiv \mathit{\boldsymbol{S}}_{ij}^M$. It is easy to show that the sum of tensor components

 $\mathit{\boldsymbol{S}}_{iikk}^M = \mathit{\boldsymbol{S}}_{ii}^M = {\mathit{\boldsymbol{\kappa }}_M} = 1/{\mathit{\boldsymbol{K}}_M}$ (15)

Note that the denominator in Eq. (12) is the same as that of B & K's scalar result, Eq. (2). In fact, the sum of Eq. (12), according to Eq. (7), reproduces Eq. (2) exactly. Of course, SM is a property of the rock (dependent on microgeometry, pressure, etc.) and does not appear in any handbook.

The B & K generalization of Gassmann's Eq. (8), including the B & K refinement, is

 $\begin{array}{l} \;\;\;\mathit{\boldsymbol{C}}_{ijkl}^{fld} - \mathit{\boldsymbol{C}}_{ijkl}^{gas}\\ = \frac{{\left({\mathit{\boldsymbol{C}}_{ij}^M - \mathit{\boldsymbol{C}}_{ij}^{gas}/3} \right)\left({\mathit{\boldsymbol{C}}_{kl}^M - \mathit{\boldsymbol{C}}_{kl}^{gas}/3} \right)}}{{\phi \mathit{\boldsymbol{K}}_M^2\left({1/{\mathit{\boldsymbol{K}}_{Fld}} - 1/{\mathit{\boldsymbol{K}}_{Sol}}} \right) + \mathit{\boldsymbol{K}}_M^2/{\mathit{\boldsymbol{K}}_{Sol}} - \mathit{\boldsymbol{K}}_{gas}^\sum }}\\ \, \equiv \frac{{\left({\mathit{\boldsymbol{C}}_{ij}^M - \mathit{\boldsymbol{C}}_{ij}^{gas}/3} \right)\left({\mathit{\boldsymbol{C}}_{kl}^M - \mathit{\boldsymbol{C}}_{kl}^{gas}/3} \right)}}{{\mathit{\boldsymbol{\tilde K}}}} \end{array}$ (16)

where the last form introduces notation for the denominator which will simplify equations below. The rank-two tensor $\mathit{\boldsymbol{C}}_{ij}^M$ is, from Eq. (11), given by the matrix inverse of $\mathit{\boldsymbol{S}}_{ij}^M$. From Eq. (11)

 $\mathit{\boldsymbol{C}}_{kl}^M\mathit{\boldsymbol{S}}_{kl}^M = \frac{1}{3}\mathit{\boldsymbol{C}}_{iikl}^M\mathit{\boldsymbol{S}}_{klmm}^M = \frac{1}{3}{\mathit{\boldsymbol{I}}_{iimm}} \equiv \frac{{{\mathit{\boldsymbol{\delta }}_{im}}{\mathit{\boldsymbol{\delta }}_{im}} + {\mathit{\boldsymbol{\delta }}_{im}}{\mathit{\boldsymbol{\delta }}_{im}}}}{6} = 1$ (16a)

Of course, CM is a property of the rock (dependent on micro-geometry, pressure, etc.) and does not appear in any handbook.

It is clear that the denominator of Eq. (16) reduces to that of the Gassmann result Eq. (8) with the B-G restriction, Eq. (5), but differs from this in the general case. However, the major difference between Eqs. (8) and (16) lies in the numerator, where in (8) the scaled identity tensor KSolδij appears, whereas in (16) the corresponding tensor $\mathit{\boldsymbol{C}}_{ij}^M$ is full, in general (6 independent components). Of course, both expressions refer to a mass element; the application to wave propagation assumes homogeneity of the mass element on the scale of the wavelength used.

3 ANISOTROPY PARAMETERS

While Eqs. (11)–(16) present a formal solution to the problem of the fluid dependence of anisotropic rocks, they do not lead to much physical insight. To address this issue, Thomsen (2012) found the fluid dependence, in the case of weak polar anisotropy of the dimensionless anisotropy parameters {ε, δ, γ}, starting from Eq. (12), but incorporating the B-G restriction (5).

As foreseen by B & K, γ was found to be independent of fluid content. This is an important result, not fully appreciated by B & K, who considered only a macroscopically homogeneous mass element (like an unfractured shale). But seismic wavelengths include significant inhomogeneity; if this inhomogeneity has a preferred orientation (as with thin layers or oriented fractures), it leads to seismic anisotropy. But, since the fluid effects modify elastic properties on the grain scale, they affect each different sub-seismic element differently, so it is not feasible to reach general conclusions which apply on the seismic scale. An exception to this conclusion is evident for γ: since γ is fluid-independent for each sub-seismic element, it is also fluid-independent for the seismic-scale ensemble.

The formulae for ε and δ which were found by Thomsen (2012) were too complicated to be useful, but surprisingly, the anelliptic parameter (εδ) was found to be approximately independent of fluid content. This made the analysis of (εδ) feasible despite the complications of subseismic layering (just as with γ). However, this conclusion was inconsistent with numerical results for a model of aligned cracks based on effective medium theory, e.g., as reported by Tsvankin and Grechka (2011).

Subsequently, Collet and Gurevich (2013) addressed the same problem, starting from Gassmann's result (8), also containing the B-G restriction (5). Their formulae for ε and δ were simpler than those of Thomsen (2012), but their conclusion for (εδ) was similar.

With the use of Eq. (16), this same problem may be addressed without the restriction (5). Of course, the previous conclusions regarding γ are unaffected by this generalization. From Eq. (16) and the definition of ε from Thomsen (1986)

 $\begin{array}{l} \mathit{\boldsymbol{C}}_{1111}^{fld} - \mathit{\boldsymbol{C}}_{3333}^{fld} = \mathit{\boldsymbol{C}}_{1111}^{gas} + \frac{{{{\left({\mathit{\boldsymbol{C}}_{11}^M - \mathit{\boldsymbol{C}}_{11}^{gas}/3} \right)}^2}}}{{\mathit{\boldsymbol{\tilde K}}}} - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathit{\boldsymbol{C}}_{3333}^{gas} - \frac{{{{\left({\mathit{\boldsymbol{C}}_{33}^M - \mathit{\boldsymbol{C}}_{33}^{gas}/3} \right)}^2}}}{{\mathit{\boldsymbol{\tilde K}}}}\\ 2{\mathit{\boldsymbol{\varepsilon }}^{fld}}\mathit{\boldsymbol{C}}_{3333}^{fld} = 2{\mathit{\boldsymbol{\varepsilon }}^{gas}}\mathit{\boldsymbol{C}}_{3333}^{gas} + \\ \quad \quad \quad \quad \, \, \frac{{{{\left({\mathit{\boldsymbol{C}}_{11}^M - \mathit{\boldsymbol{C}}_{11}^{gas}/3} \right)}^2} - {{\left({\mathit{\boldsymbol{C}}_{33}^M - \mathit{\boldsymbol{C}}_{33}^{gas}/3} \right)}^2}}}{{\mathit{\boldsymbol{\tilde K}}}} \end{array}$ (17)

whence

 ${\mathit{\boldsymbol{\varepsilon }}^{fld}} = \frac{{\mathit{\boldsymbol{C}}_{3333}^{gas}}}{{\mathit{\boldsymbol{C}}_{3333}^{\mathit{\boldsymbol{f}}ld}}}{\mathit{\boldsymbol{\varepsilon }}^{gas}} + \frac{{{{\left({\mathit{\boldsymbol{C}}_{11}^M - \mathit{\boldsymbol{C}}_{11}^{gas}/3} \right)}^2} - {{\left({\mathit{\boldsymbol{C}}_{33}^M - \mathit{\boldsymbol{C}}_{33}^{gas}/3} \right)}^2}}}{{2\mathit{\boldsymbol{\tilde KC}}_{3333}^{\mathit{\boldsymbol{f}}ld}}}$ (18)

Similarly, using the weak form of δ (Thomsen, 1986)

 $\begin{array}{l} \mathit{\boldsymbol{C}}_{1133}^{\mathit{\boldsymbol{f}}ld} - \mathit{\boldsymbol{C}}_{3333}^{\mathit{\boldsymbol{f}}ld} + 2\mathit{\boldsymbol{C}}_{1313}^{\mathit{\boldsymbol{f}}ld} = \mathit{\boldsymbol{C}}_{1133}^{gas} - \mathit{\boldsymbol{C}}_{3333}^{gas} + 2\mathit{\boldsymbol{C}}_{1313}^{gas}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{\left({\mathit{\boldsymbol{C}}_{11}^M - \mathit{\boldsymbol{C}}_{11}^{gas}/3} \right)\left({\mathit{\boldsymbol{C}}_{33}^M - \mathit{\boldsymbol{C}}_{33}^{gas}/3} \right)}}{{\mathit{\boldsymbol{\tilde K}}}} - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{{{\left({\mathit{\boldsymbol{C}}_{33}^M - \mathit{\boldsymbol{C}}_{33}^{gas}/3} \right)}^2}}}{{\mathit{\boldsymbol{\tilde K}}}}\\ \mathit{\boldsymbol{C}}_{3333}^{fld}{\mathit{\boldsymbol{\delta }}^{fld}} \approx \mathit{\boldsymbol{C}}_{3333}^{gas}{\mathit{\boldsymbol{\delta }}^{gas}} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{\left({\mathit{\boldsymbol{C}}_{11}^M - \mathit{\boldsymbol{C}}_{11}^{gas}/3} \right)\left({\mathit{\boldsymbol{C}}_{33}^M - \mathit{\boldsymbol{C}}_{33}^{gas}/3} \right) - {{\left({\mathit{\boldsymbol{C}}_{33}^M - \mathit{\boldsymbol{C}}_{33}^{gas}/3} \right)}^2}}}{{\mathit{\boldsymbol{\tilde K}}}} \end{array}$ (19)

whence

 $\begin{array}{l} {\mathit{\boldsymbol{\delta }}^{fld}} \approx \frac{{\mathit{\boldsymbol{C}}_{3333}^{gas}}}{{\mathit{\boldsymbol{C}}_{3333}^{fld}}}{\mathit{\boldsymbol{\delta }}^{gas}} + \\ \;\;\;\;\;\;\;\;\left({\mathit{\boldsymbol{C}}_{33}^M - \mathit{\boldsymbol{C}}_{33}^{gas}/3} \right)\frac{{\left({\mathit{\boldsymbol{C}}_{11}^M - \mathit{\boldsymbol{C}}_{11}^{gas}/3} \right) - \left({\mathit{\boldsymbol{C}}_{33}^M - \mathit{\boldsymbol{C}}_{33}^{gas}/3} \right)}}{{\mathit{\boldsymbol{\tilde KC}}_{3333}^{fld}}} \end{array}$ (20)

These results Eqs. (18), (20) generalize the corresponding results of Thomsen (2012) and Collet and Gurevich (2013), avoiding Eq. (5), modifying the denominators of the fluid correction terms (c.f., Eq. (16)), and including the full tensor CM in the numerators.

The anelliptic parameter εδ has fluid dependence, from Eqs. (18)–(20)

 $\begin{array}{l} {\mathit{\boldsymbol{\varepsilon }}^{\mathit{\boldsymbol{f}}ld}} - {\mathit{\boldsymbol{\delta }}^{\mathit{\boldsymbol{f}}ld}} = \frac{{\mathit{\boldsymbol{C}}_{3333}^{gas}}}{{\mathit{\boldsymbol{C}}_{3333}^{\mathit{\boldsymbol{f}}ld}}}({\mathit{\boldsymbol{\varepsilon }}^{gas}} - {\mathit{\boldsymbol{\delta }}^{gas}}) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{{{\left[ {\left({\mathit{\boldsymbol{C}}_{11}^M - \mathit{\boldsymbol{C}}_{33}^M} \right) - \left({\mathit{\boldsymbol{C}}_{11}^{gas} - \mathit{\boldsymbol{C}}_{33}^{gas}} \right)/3} \right]}^2}}}{{2\mathit{\boldsymbol{\tilde KC}}_{3333}^{fld}}} \end{array}$ (21)

With the B-G restriction (5), the first term in the numerator above, $\left({\mathit{\boldsymbol{C}}_{11}^M - \mathit{\boldsymbol{C}}_{33}^M} \right) \to \left({{\mathit{\boldsymbol{K}}_{Sol}} - {\mathit{\boldsymbol{K}}_{Sol}}} \right) = 0$, as reported by Gassmann (1951), c.f., Eq. (8). The remaining term in the numerator, ${\left({\mathit{\boldsymbol{C}}_{11}^{gas} - \mathit{\boldsymbol{C}}_{33}^{gas}} \right)^2}$, is proportional to the (gas-saturated) anisotropy parameters squared, hence is negligible to this approximation. This leaves

 ${\mathit{\boldsymbol{\varepsilon }}^{fld}} - \mathit{\boldsymbol{\delta }}_{wk}^{fld} \approx \frac{{\mathit{\boldsymbol{C}}_{3333}^{gas}}}{{\mathit{\boldsymbol{C}}_{3333}^{\mathit{\boldsymbol{f}}ld}}}({\mathit{\boldsymbol{\varepsilon }}^{gas}} - \mathit{\boldsymbol{\delta }}_{wk}^{gas}) \approx ({\mathit{\boldsymbol{\varepsilon }}^{gas}} - \mathit{\boldsymbol{\delta }}_{wk}^{gas})$ (22)

which is the result of Thomsen (2012) and of Collet and Gurevich (2013). In this case, the anelliptic parameter εδ, like γ, has approximately null dependence on fluid content, and so field data may be analyzed without the complications of sub-seismic heterogeneity noted above.

However, in the general case (without the B-G restriction (5)) the result does not simplify like this. Since $\mathit{\boldsymbol{C}}_{ij}^M > \mathit{\boldsymbol{C}}_{ij}^{gas}$ it is plausible that $\left({\mathit{\boldsymbol{C}}_{11}^M - \mathit{\boldsymbol{C}}_{33}^M} \right)$ could be much larger than $\left({\mathit{\boldsymbol{C}}_{11}^{gas} - \mathit{\boldsymbol{C}}_{33}^{gas}} \right)$, and not negligible as in the previous argument. In particular, if the anisotropy is caused by cracks aligned in the 3-plane, it is intuitively clear that $\mathit{\boldsymbol{C}}_{11}^M - \mathit{\boldsymbol{C}}_{33}^M > 0$. Its magnitude depends upon the microgeometry of the cracks, and must be determined by suitable experimentation, or (as in the case of idealized modeling) by computation, cf., Tsvankin and Grechka (2011).

4 CONCLUSIONS

The generalization by B & K of the classic B-G results for the fluid-dependence of the poro-elasticity of rocks should be taken more seriously by the geophysical community. The experiments of Hart and Wang (2010) demonstrate, for Berea sandstone, that the additional parameter introduced by B & K (most conveniently taken as the "mean compressibility" κM, and its tensor analogs) does not obey the B-G restriction (5). Thomsen (2017) provides theoretical support for this experimental conclusion. Of course, much more experimental work needs to be done, in order to gain more confidence in this conclusion.

Applying the B & K generalization of the B-G fluid dependence of poro-elastic anisotropy, the simplification (previously found as a consequence of the B-G analysis by Thomsen (2012) and by Collet and Gurevich (2013)), that the anisotropy parameter (εδ) is approximately independent of fluid content, is not valid (c.f., Eq. (21). This new complexity arises directly from the B & K generalization of the B-G analysis, as expressed in the tensor CM (c.f., Eq. (16)), which replaces a scaled identity tensor in the corresponding B-G result (8). This theoretical result appears to be consistent with a numerical result for an idealized model by Tsvankin and Grechka (2011).

This analysis of course assumes inhomogeneity on the grain/pore scale, but that, when averaged over many grains and pores, that average is uniform within the seismic wavelength. So, it is suitable for application to homogeneous shale formations, for example. But, if the formation varies locally (on the subseismic scale) with a preferred orientation (e.g., with layers), then the seismic wave propagates anisotropically (Bckus, 1962). In this case, as discussed by Thomsen (2017), the present analysis must be applied separately in each local element (e.g., each layer). Obviously, this makes the application in the field context even more problematic. This conclusion is discouraging, but it may avoid a false conclusion which ignores this complexity.

ACKNOWLEDGMENTS

I appreciate many tough discussions with J. Berryman (LBNL), and B. Gurevich (Curtin), and a tough review by I. Tsvankin (CSM) of a different manuscript. The final publication is available at Springer via https://doi.org/10.1007/s12583-017-0806-9.

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