Geofluids

Volume 2019, Article ID 2656747, 17 pages

https://doi.org/10.1155/2019/2656747

## Inversion for Geofluid Discrimination Based on Poroelasticity and AVO Inversion

^{1}State Key Laboratory of Petroleum Resources and Prospecting, CNPC Key Lab of Geophysical Exploration, China University of Petroleum, 102249, Changping, Beijing, China^{2}Research Institute of Petroleum Exploration and Development, CNPC, Xueyuan Road No. 20, Haidian, 10083 Beijing, China^{3}Oil & Gas Survey, CGS, Beijing 100083, China

Correspondence should be addressed to Hui Zhou; nc.ude.puc@uohziuh

Received 1 August 2019; Revised 21 October 2019; Accepted 1 November 2019; Published 26 November 2019

Academic Editor: Mohammad Sarmadivaleh

Copyright © 2019 Lingqian Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Geofluid discrimination plays an important role in reservoir characterization and prospect identification. Compared with other fluid indicators, the effective pore-fluid bulk modulus is more sensitive to the property of fluid contained in reservoirs. We combine the empirical relations with deterministic models to form a new kind of linearized relationship between the mixed fluid/rock term and the fluid modulus. On the one hand, the linearized relationship can decouple the fluid bulk modulus from the mixed fluid/rock term; on the other hand, the decoupled terms are more stable especially in low-porosity situations compared with previous approaches. In terms of the new linearized equation of the fluid modulus, we derive a novel linearized amplitude variation with offset (AVO) approximation to avoid the complicated nonlinear relationship between the fluid modulus and the reflectivity series. Convoluting this linearized AVO approximation with seismic wavelets, the forward modeling is constructed to combine the prestack seismic records with the fluid modulus. Meanwhile, we introduce the Bayesian inference with multivariable Cauchy prior to the fluid modulus inversion for a stable and high-resolution solution. Model examples demonstrate the accuracy of the proposed linearized AVO approximation compared with the exact Zoeppritz equation and Aki-Richards approximate equation. The synthetic and field data tests illustrate the accuracy and feasibility of the proposed fluid modulus inversion approach for geofluid discrimination.

#### 1. Introduction

Fluid discrimination plays an important role in seismic exploration and reservoir description. Qualitative interpretation [1–6] and direct quantitative estimation [7–11] for rock properties are two main methods to discriminate different pore-fluid types. However, the qualitative interpretation, such as the bright-spot, dim-spot, and flat-spot techniques, has difficulties in discriminating the geofluid in complicated lithologic reservoirs. Estimation of the rock-physical property directly from seismic records is a better way to differentiate the geofluid filled in reservoirs. The major challenge is the uncertainties in geofluid discrimination associated with two factors. First, the fluid indicators are very likely to provide ambiguous results for fluid identification due to the mixed effects of the fluid and rock porosity. Second, the instability and inaccuracy of the traditional elastic parameters, such as the P-wave velocity and S-wave velocity estimated by prestack seismic inversion, may magnify the uncertainty of the fluid indicators. A lot of efforts are taken to study various fluid indicators from prestack seismic records for reservoir prediction and fluid discrimination. Smith and Gidlow [12] initially incorporated the mudrock line of Castagna et al. [13] and the pseudo-Poisson’s ratio reflectivity to define the fluid factor in the form of the weighted difference between the P- and S-wave velocity reflectivities. Rutherford and Williams [2] determined gas sands based on the normal-incidence reflection coefficient and the contrast in Poisson’s ratio. Goodway et al. [14] converted the velocity measurements to Lame’s parameters to enhance the sensitivity to fluids. Quakenbush et al. [15] utilized the crossplot of acoustic impedance and shear impedance to investigate discrimination between different lithology and fluid types.

In terms of the poroelasticity theory presented by Biot [16, 17] and Gassmann [18], Russell et al. [19] generalized the formulation of Goodway et al. [14] by defining the fluid indicator as the weighted difference between the P-wave impedance and the S-wave impedance. Zong et al. [20] defined the P- and S-wave moduli as a new fluid indicator and introduced them into the prestack seismic inversion. Russell et al. [21] and Zong et al. [22] derived a new AVO approximate equation with different approaches to estimate the fluid/porosity term directly. It may be that these geofluid indicators can estimate the fluid type effectively to a certain extent, but they are ambiguous in complicated situations due to the integrated response of the geofluid and the petrophysical properties. In theory, fluid modulus is the most sensitive parameter to discriminate the geofluid contained in reservoirs. Yin and Zhang [23] extracted the fluid modulus as the fluid factor based on the empirical critical porosity and estimated the fluid modulus directly from the prestack seismic data. Further, a rock-physical empirical formula [24, 25] and AVO theory were combined to generate a new linearized AVO approximate equation to estimate the fluid modulus directly [26]. However, one of the inversion terms containing the porosity in the denominator means that the inversion results will be unstable in low-porosity situations. Meanwhile, the hypothesis and approximation will magnify the instability and inaccuracy. In this paper, we combine the deterministic model proposed by Han and Batzle [27] with the velocity-porosity empirical relations to derive the linearized relationship between the fluid-saturation effect term in Gassmann’s equation and the fluid modulus. The stability of the inversion results can be improved and some approximations can be omitted for more accurate results.

It is important to estimate the rock-physical properties for reservoir prediction and fluid discrimination. Empirical relations and deterministic models can provide the relationship between the petrophysical parameters and the elastic parameters, such as P-wave velocity (), S-wave velocity (), and density (). Compared with the poststack seismic inversion, prestack seismic inversion can provide more elastic property information taking advantage of the AVO phenomenon [28]. However, the prestack inversion also suffers difficulties stemming from noise contamination, band limitation, and nonuniqueness like other inversion problems [29]. Therefore, it is important to improve the stability and accuracy of prestack seismic inversion results with different methods. AVO inversion techniques can be classified into two categories: the stochastic inversion approaches and the deterministic inversion approaches; each has its own merits and demerits. The former category is carried out with forward modeling time by time in terms of the prior knowledge derived from well logs and multivariate geostatistical modeling [30–35]. The major limitations of its widespread application are the expensive computation and the tough choice and evaluation of inversion results. In contrast, the deterministic inversion approach can provide a determinate solution with high computational efficiency [36–38]. The Bayesian approach, as a popular method for inversion, takes advantage of available prior information of inversion parameters and the assumed distribution of noise in observed data as constraints to stabilize the inversion results [39–41]. The main concern in this approach is choosing an appropriate prior probability distribution for the inversion parameters [42]. The solution of conventional Gaussian probability distribution lacks sparsity and hence cannot improve resolution. In this paper, we utilize the Bayesian linearized methods proposed by Alemie and Sacchi [43] to obtain stable and accurate inversion results. The objective function is formulated with the pore-fluid bulk modulus and other elastic parameters based on the Bayesian framework with the assumption that the likelihood model has a Gaussian probability distribution and the prior model has a Cauchy distribution for high-resolution inversion results.

In this paper, the geofluid-discrimination approach is conducted by incorporating poroelasticity and AVO inversion. Firstly, we decouple the fluid modulus from the mixed fluid/rock term in Gassmann’s equation. The deterministic models provided by Han and Batzle [27] are combined with the linearized empirical velocity-porosity relations to derive the fluid modulus. Compared with the decoupled approaches in Zong et al. [26], the decoupled terms in this paper are more stable, especially in low-porosity situations. Furthermore, we derive a new AVO approximate equation to relate the reflectivity series with the fluid modulus. The new AVO approximate equation allows us to estimate the fluid modulus of the reservoir in a more direct and stable manner than previous formulations. We also test the accuracy of the AVO approximate equation compared with the exact Zoeppritz equation and the Aki-Richards approximate equation. Eventually, we invert the fluid modulus with the linearized Bayesian methods for a more stable and accurate solution. We conduct synthetic and field data case studies to illustrate the feasibility and accuracy of this method in geofluid discrimination.

#### 2. AVO Approximation Equation with Fluid Modulus

The prestack seismic response is directly influenced by the P-wave velocity, S-wave velocity, density, and incident angle based on the exact Zoeppritz equation. The P-wave velocity and S-wave velocity can be expressed as where , , and are the bulk modulus, shear modulus, and density of the fluid-saturated rock, respectively. From equation (1), we see that the fluid effects are mainly contained in the density and the bulk modulus of the fluid-saturated rock. Han and Batzle [27] have demonstrated that the bulk modulus of a fluid-saturated rock is more sensitive to the fluid-saturation effect than the P-wave velocity. The bulk modulus of a fluid-saturated rock containing the fluid effects can be expressed as following from Gassmann’s equation, where , , , and are the bulk modulus of the dry rock, the saturated-rock frame, mineral grain, and geofluid, respectively, is the porosity of the rock, is the increment resulting from the fluid saturation, and and are saturated and dry rock shear moduli. These equations indicate that the fluid in pores will affect bulk modulus but not shear modulus. Russell et al. [21] and Zong et al. [22] linearized the poroelasticity to estimate the fluid properties directly from prestack seismic records. The linearized AVO approximation with fluid factor can be expressed as where is the average of incident and transmission angles and and are the ratio of P-wave velocity to S-wave velocity of the saturated rock and dry rock, respectively. From equation (3), we see that the fluid factor has the mixed effects of the fluid component and the rock matrix component. Therefore, the fluid factor cannot reflect the fluid property directly.

In this paper, we develop a novel P-wave reflectivity equation in terms of fluid modulus. The P-wave velocity, S-wave velocity, and density are expressed as a linearized function of the porosity for dry, clean sands [44]:

The bulk modulus of the dry rock frame is expressed in terms of the P-wave velocity, S-wave velocity, and density as

Substituting equation (6) into equation (7), we obtain a function of to express the bulk modulus. Since the modulus is the product of the density and the square of velocity, we obtain an equation that is cubic in terms of porosity. In terms of [27], the bulk modulus is written as

In low-porosity situations, equation (8) is further simplified as where is an empirical parameter. Based on the derivation of equation (9), is determined by the statistical parameters in equation (6). Further, Han and Batzle [27] illustrated that ranges from 1.45 to slightly more than 2.0 for consolidated rocks at high differential-pressure conditions with experience, and is associated with the lithology. In this paper, the lithology of the target stratum is sandstone; we set as 2 in terms of the experience results provided by Han and Batzle [27]. Substituting equation (9) into the linearized equation of the fluid modulus and the mixed pore/fluid term proposed by Han and Batzle [27], the new linearized relationship of the fluid modulus and the mixed pore/fluid term is written as

In practice, the rock physics model parameter should be tested against the well log data and calibrated in the work area.

Compared with equation (10), the expression in equation (11) is more stable even in low-porosity situations. For simplification, we rewrite as relating to the lithology and porosity, then we express equation (11) as

We incorporate equations (5) and (12) to derive a new AVO approximation to estimate the fluid modulus directly. From equation (12), the fluid/pore term reflectivity can be divided into the fluid modulus reflectivity and the porosity-related reflectivity:

Substituting equation (13) into equation (5) and merging the shear modulus with , we define the combined new parameter as a shear modulus related term without physical meaning. The combination of these two parameters leads to the simplification of the new P-wave reflectivity and the decoupling of the fluid modulus from the mixed fluid/pore term. The new shear modulus related term of the rock matrix can be written as

The form of the new shear modulus related term is the same as the relationship between the fluid modulus and the mixed fluid/pore term shown in equation (12). It should be noted that the new variable has no physical meaning which is proposed here only for the simplification of the expression. Substituting equation (13) into equation (5), we write the new AVO approximation as

If is equal to , which means that there is no fluid in the reservoir, the coefficient of the fluid modulus goes to zero, and the coefficient of the second term tends to be the opposite of that of the third term. In equation (15), the ratio of the P-wave velocity to the S-wave velocity in dry rock can be estimated from the laboratory measurements and real calculation [19]. The range of and the range of their equivalent elastic constant ratios are shown in Table 1 provided by Russell et al. [21]. We see from Table 1 that ranges from 4 to 1.333. However, there is no physical meaning when equals 2 or 1.333; it means that the Poisson’s ratio is 0 or -1, respectively. Assigning 2.333 to is appropriate for clean unconsolidated sandstones in most cases. However, it is also necessary to obtain the value of from laboratory measurements due to its dependency on the reservoir type.