At seismic frequencies and below, the standard analysis of the fluid dependence of poroelasticity is due originally to Biot (1941) and Gassmann (1951) ("BG"). They showed that, for isotropic rocks, the shear modulus is independent of fluid content, and the incompressibility has a closedform fluid dependence (given by Eq. (1), below). Gassmann (1951) also provided the fluid dependence of the anisotropic tensor components of poroelastic compliance and of poroelastic stiffness.
Subsequently, Brown and Korringa (1975) ("B & K") generalized this work to include a further parameter (often called the pore compressibility, κ_{ϕ}). They showed that the BG result was a special case, valid only if κ_{ϕ}=κ_{Sol}, 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 BG) generalization for the fluid dependence of the anisotropic tensor components of poroelastic compliance (Brown and Korringa, 1975). The present work extends that generalization to the poroelastic stiffness tensor and to the nondimensional parameters of anisotropy.
This B & K generalization introduces significant difficulty into the analysis of fluid dependence, since κ_{ϕ} and its tensor analogs are properties of the rock (dependent on microgeometry, pore pressure, etc.) and may not be found in a handbook, as with the solid compressibility. Since seismicband anisotropy may be caused by subseismic inhomogeneity (e.g., thinlayering or fracturing), the fluid dependence also varies on this subseismic 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 CASETo establish notation, the wellknown BG 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 K_{fld}, K_{gas}, K_{Fld}, K_{Sol} are respectively the incompressibilities of: the rock saturated with a fluid, the rock saturated with highly compressible gas, the porefilling fluid (brine, oil, gas or a combination), and the solid grains of the rock. Observe the capitalization in the subscripts: K_{Fld} is the incompressibility of the fluid which saturates the rock, which has incompressibility K_{fld}; and ϕ is porosity.
The BG prediction above concerns only the difference K_{fld}–K_{gas}, 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 lowfrequency 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 BG 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 V_{ϕ}=ϕV is the volume of the pore space of that mass element. p_{Fld} is the fluid pressure in the pore space (assumed uniform at low frequencies); p_{dif}=p–p_{Fld} 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 multivariate 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 κ_{M} (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 κ_{Sol} of the solid grains by
$ {\mathit{\boldsymbol{\kappa }}_M} = (1  \phi){\mathit{\boldsymbol{\kappa }}_{Sol}} + \phi {\mathit{\boldsymbol{\kappa }}_\phi } $  (4) 
The parameter κ_{ϕ} is called the "pore compressibility". B & K do not indicate the mnemonic meaning of the subscript M on the parameter κ_{M}, but in view of Eq. (4) it may be called the "mean compressibility". Equation (4) may be used to eliminate κ_{ϕ} from Eq. (2), in favor of κ_{Sol}, leaving κ_{M} as the most convenient new B & K parameter.
BG 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 BG Eq. (1). But B & K argued that this applies only to monomineralic rocks, and Thomsen(2017, 2010) argued that this restriction does not even apply in that special case (the error in logic of BG, 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 BG in fact contains an implicit assumption about pore microgeometry).
This extension of the B & K proof to monomineralic rocks seems to be a minor point, because almost all rocks are polymineralic. 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 κ_{M} on an undrained sample.
Further, the introduction of the parameter κ_{M} complicates the analysis of fluid dependence, since it is a property of the rock (dependent on microgeometry, pore pressure, etc.) and may not be found in a handbook, as with the solid compressibility. For all these reasons, the generalization of BG 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 monomineralic 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 κ_{M} on an undrained sample. And, this generalization of the BG 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 κ_{M} differs from κ_{Sol} by ~40% for this Berea sandstone, and κ_{ϕ} differs from κ_{Sol} by ~200% (Thomsen, 2017), consistent with H & W. The differences are attributable to the nonuniformity of the mineralogy was only a few percent. Although this is only a single case, it shows that the complications to fluiddependency analysis, posed by Eq. (2), should be taken seriously. As discussed below, this applies especially to the fluiddependency analysis of the anisotropy parameters.
Thomsen (2017) inverted Eq. (2), and incorporated Eq. (3) to eliminate κ_{ϕ}, 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 K_{M}=1/κ_{M}. It is easy to see that, with the BG restriction (5), Eq. (6) reduces to Eq. (1). In the general case, Eq. (6) is the generalization of the BG 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 macroanisotropic. For macroanisotropic 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 (rankfour) poroelastic compliance tensor, and the repeated indices indicate summation from 1 to 3.
2 TENSOR CASEGassmann (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 (rankfour) poroelastic stiffness tensor, and δ_{ij} is the Kronecker delta tensor (the ranktwo identity tensor). The other ranktwo tensor in (8) is
$ \mathit{\boldsymbol{C}}_{ij}^{gas} \equiv \mathit{\boldsymbol{C}}_{ijkk}^{gas}/3 $  (9) 
Not to be confused with the ranktwo 6×6 matrix
$ \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
The stiffness tensor C is the rankfour 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 BG restriction (5), both in the denominator and in the appearance in the numerator of the scaled identity tensor K_{Sol}δ_{ij}.
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 S^{M} 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 η_{ij} is the average strain throughout the mass element, and
$ \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, S^{M} 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 ranktwo tensor
$ \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, C^{M} is a property of the rock (dependent on microgeometry, 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 BG 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 K_{Sol}δ_{ij} appears, whereas in (16) the corresponding tensor
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 BG 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 subseismic 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 fluidindependent for each subseismic element, it is also fluidindependent for the seismicscale ensemble.
The formulae for ε and δ which were found by Thomsen (2012) were too complicated to be useful, but surprisingly, the anelliptic parameter (ε–δ) was found to be approximately independent of fluid content. This made the analysis of (ε–δ) 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 BG restriction (5). Their formulae for ε and δ were simpler than those of Thomsen (2012), but their conclusion for (ε–δ) 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 C^{M} in the numerators.
The anelliptic parameter ε–δ 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 BG restriction (5), the first term in the numerator above,
$ {\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 ε–δ, like γ, has approximately null dependence on fluid content, and so field data may be analyzed without the complications of subseismic heterogeneity noted above.
However, in the general case (without the BG restriction (5)) the result does not simplify like this. Since
The generalization by B & K of the classic BG results for the fluiddependence of the poroelasticity 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" κ_{M}, and its tensor analogs) does not obey the BG 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 BG fluid dependence of poroelastic anisotropy, the simplification (previously found as a consequence of the BG analysis by Thomsen (2012) and by Collet and Gurevich (2013)), that the anisotropy parameter (ε–δ) 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 BG analysis, as expressed in the tensor C^{M} (c.f., Eq. (16)), which replaces a scaled identity tensor in the corresponding BG 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.
ACKNOWLEDGMENTSI 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/s1258301708069.
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