2.2. P-Wave High-Order AVO Approximations for the Effective Pore-Fluid Bulk Modulus
An effective pore-fluid bulk modulus has shown a clear advantage as a fluid indicator for identifying pore-fluid. AVO inversion is a useful technology for extracting an effective pore-fluid bulk modulus from seismic data directly, though a high-order approximation for inversion is nonexistent. Here, a new estimation is deduced based on the Zoeppritz equations and Biot–Gassmann theory. The Zoeppritz equations are the foundation of AVO methodology [
10]. Based on the relationships among seismic amplitudes, this equation can be expressed as (Keys, 1989) [
11]
where
represents
, with
as the incident angle;
and
are P- and S-wave reflection coefficients, respectively; and
and
are P- and S-wave transmission coefficients, respectively.
For the incident P-wave case,
,
,
, and
are defined as (Kim and Innanen, 2013) [
12]
where
and
denote the velocity ratios of P- and S-wave in dry rock and saturated rock, respectively.
,
and
are calculated by
where
and
represent the Gassmann fluid factor and shear modulus, respectively, which can be calculated by (Russell et al., 2011) [
5]
where
is density, and
and
are the upper and lower layers of the interface, respectively.
According to the Gassmann equation, the Gassmann fluid factor can be derived as (Han and Batzle, 2003) [
13]
where
is the effective pore-fluid bulk modulus, and
where
,
, and
stand for the porosity of porous rock, the bulk modulus of solid grain, and the bulk modulus of dry porous rock, respectively.
Based on the critical porosity model, Nur et al. (1998) provide the following equations [
14]:
where
,
, and
denote critical porosity, the shear modulus of solid grain, and the shear modulus of dry porous rock, respectively.
For the same reservoir, we assume that critical porosity is constant. Replacing
with
, and substituting Equations (5) and (7a,b) into Equation (2a–d), we have
By substituting Equation (8a–d) into (1) and applying the Taylor expansion to
,
and
, the P-wave reflection coefficient is deduced as follows:
where,
are the different order terms of P-wave reflection coefficients, involving the effective pore-fluid bulk modulus, porosity, and density. The
ith-order approximation is written as
. Using the method of numerical analysis, we study the effect of the expansion order number on the accuracy of the deduced approximation. For small incident angle and parameter differences across the interface, the relative errors of different order approximations are all less than 1%. With an increase of the incident angle and parameter differences across the interface, the accuracy of the first- and second-order approximations decreases heavily. The relative error of the third-order approximation increases but is still less than 1%. When the expansion order number is greater than 3, the accuracy of the deduced approximation increases slightly, but the calculation efficiency decreases heavily. We considered both accuracy and efficiency, and decided to use the third-order approximation, which is expressed as
Using the sand model in
Table 3 (Yin and Zhang, 2014) [
7], we compare the accuracy of these approximations. P-wave reflection coefficients are computed through the Zoeppritz equations [
10], Aki–Richards approximation (Aki and Richards, 1980) [
15], and different order approximations, respectively. As shown in
Figure 2a, the curves of the Aki–Richards approximation and different order approximations all are close to the Zoeppritz curve. Small elastic parameter perturbations across reflectors are caused only by pore-fluid (Model 1). In
Figure 2b,c, note that the AVO curves computed from the Aki–Richards approximation and different order approximations are close to the Zoeppritz curve for incident angles less than 30°. However, for larger incident angles the curve of the deduced third-order approximation is the closest to the Zoeppritz curve. This is because the elastic parameter perturbations across reflectors are caused by only porosity or both pore-fluid and porosity (Models 2 and 3).
Next, we analyze the information contained in reflections and the stability of inversion for the first-order approximation. If the second- and higher-order terms of Equation (9) are neglected, it becomes a linear approximation. The P-wave reflection coefficient is written as
In this equation, the effective pore-fluid bulk modulus, porosity, and density have different proportions of information in their reflection coefficients. As a result, the stability of inversion for these three elastic parameters is different. Equation (11) can be expressed as
where the first term on the right of Equation (12) is the mapping operator between elastic parameters and reflection coefficients.
Nicolao et al. (1993) proposed that the eigenvalues of this mapping operator matrix can be used to measure the information of elastic parameters [
16]. The eigenvalues are proportional to the information content. The direction cosine of the eigenvector of the mapping operator matrix can reflect the difficulty of inverting these elastic parameters. The greater the absolute value of the direction cosine is, the easier the inversion is. Moreover, when the direction cosine equals zero, the inversion is unviable. By using the first model in
Table 3, the eigenvalues corresponding to the effective pore-fluid bulk modulus, porosity, and density are shown in
Figure 3a. When the incident angle is smaller than 30°, the energy of the first eigenvalue (blue) is larger than that of the second eigenvalue (red) and the third eigenvalue (black). Consequently, we only need to analyze the eigenvector of the first eigenvalue to get a different degree in inverting these elastic parameters. As displayed in
Figure 3b, the direction cosine of the effective pore-fluid bulk modulus, porosity, and density is far away from zero. This means that these three elastic parameters can be inverted by Equation (11).
The stability of inversion is very important. Therefore, the method proposed by Yin et al. (2018) is used to analyze the stability of Equation (11). Yin et al. (2018) deduced a three-term linear approximation including the effective pore-fluid bulk modulus [
8], an approximation of which can be displayed as
where
,
is the fit coefficient of
and
according to field data.
The logarithm of the condition number of weighting coefficients can measure the stability of the inversion equation. The smaller the logarithm is, the more stable the inversion is. We compute the values of Equations (11) and (13), the results of which are shown in
Figure 4. With the increase of incident angle, both of these two conditional numbers decrease. However, the conditional number of Equation (13) is larger than that of Equation (11). This indicates that the stability of inverting the effective pore-fluid bulk modulus, porosity, and density increases with an increase in the incident angle, and that the stability of Equation (11) is better than that of Equation (13).
The weighting coefficients before elastic parameters in Equation (11) include
.
is the ratio of P- to S-wave velocities in saturated rock and can be calculated by logging data, while
is the ratio of P- to S-wave velocities in dry rock. Russell et al. (2003, 2006, 2011) [
5,
17,
18] estimated the value of
through theoretical analysis and practical calculations, and proposed a possible range value for
.
Here, we analyze the effect of
on the weighting coefficients of the effective pore-fluid bulk modulus, porosity, and density. According to Russell et al. (2011) [
5], we assume that
is 4 and
is 2.33, 2.50, and 3.00, respectively. As shown in
Figure 5, the weighting coefficient curves of these three elastic parameters are computed with an incident angle from 0° to 45°. With the variation of
, the pore-fluid bulk modulus (
Figure 5a) and porosity (
Figure 5b) curves change, while the density curve (
Figure 5c) is unaffected. Therefore,
influences only the effective pore-fluid bulk modulus and porosity, while the density is unchanged.
In this paper, we use the third-order approximation to establish a nonlinear AVO inversion method. This approximation includes the first-, second-, and third-order terms:
2.3. Effective Pore-Fluid Bulk Modulus Nonlinear AVO Inversion Method
AVO inversion includes linear and nonlinear approaches. By adopting the third-order approximation, the AVO inversion method is strictly nonlinear. Because the efficiency of the nonlinear inversion method is pretty low in practice, series reversion is applied. Series reversion is an effective approach to transforming a nonlinear relationship into a linear relationship (Frank and Tat, 2001) [
19]. For example, for the expression
the solution is
By using series reversion to solve Equation (15),
can be expanded in series as
where
is the
ith-order item of
.
Substituting Equation (17) into (15), we can express
as
By adopting Equation (18) into (17),
can be calculated as
In this way, the P-wave reflection coefficient, the effective pore-fluid bulk modulus, porosity, and density can be expanded in series. There are some tests for the effect of the expansion item number on accuracy and efficiency. When these parameters are expanded into three items, this method has advantages in both accuracy and efficiency. These parameters can be expressed as
Combining Equation (20a–d) with (10a–d), we obtain
Additionally, using the approach of Zhang and Weglein (2009) [
20], we can calculate
,
, and
through
in four steps. First, according to Equation (21c),
,
, and
are inverted by the Bayesian theory. Second,
,
, and
are inverted via the least square method based on Equation (21b). Third,
,
, and
are inverted using the last step from Equation (21a). Finally,
,
, and
are computed using Equation (20a–d).
In the first three steps, a linear inversion AVO method is functional. The AVO inversion is based on the convolution model, which is written as
where,
,
,
, and
are the seismic data, seismic wavelet, elastic parameters, and mapping operator between
and
, respectively. Because
is usually an irreversible matrix, its conjugate matrix is commonly adopted to solve Equation (22). In general, the objective function of AVO inversion takes the form of an L2 norm, which is expressed as
AVO inversion is an ill-conditioned and ill-posed problem. The single point estimation method is commonly used to obtain optimal solutions for the objective function. However, it is unable to evaluate solutions. Thus, this paper introduces the Bayesian inversion theory. The mathematical expression of the Bayesian theory can be expressed as
where
denotes the posterior probability function,
is the likelihood function,
represents the prior distribution function (on which the results of this theory mainly depend), and
is marginal distribution, which is always a constant. It is assumed that the prior information complies with multivariate Cauchy distribution, which is expressed as
where
is the number of samples, and
is a
scale matrix.
The likelihood function obeys multivariate Gaussian distribution as
where
denotes the noise covariance matrix.
The posterior probability distribution of parameters to be solved can be indicated as
In terms of Equations (23) and (27), the objective function becomes
By taking the derivative of Equation (28) with respect to , is given.