The theory of the dynamic bulk modulus, K(ω), of a porous rock, whose saturation occurs in patches of 100% saturation each of two different fluids, is developed within the context of the quasi-static Biot theory. The theory describes the crossover from the Biot–Gassmann–Woods result at low frequencies to the Biot–Gassmann–Hill result at high. Exact results for the approach to the low and the high frequency limits are derived. A simple closed-form analytic model based on these exact results, as well as on the properties of K(ω) extended to the complex ω-plane, is presented. Comparison against the exact solution in simple geometries for the case of a gas and water saturated rock demonstrates that the analytic theory is extremely accurate over the entire frequency range. Aside from the usual parameters of the Biot theory, the model has two geometrical parameters, one of which is the specific surface area, S/V, of the patches. In the special case that one of the fluids is a gas, the second parameter is a different, but also simple, measure of the patch size of the stiff fluid. The theory, in conjunction with relevant experiments, would allow one to deduce information about the sizes and shapes of the patches or, conversely, to make an accurate sonic-to-seismic conversion if the size and saturation values are approximately known.

Theory of Frequency Dependent Acoustics in Patchy-Saturated Porous Media

The fluid term in the Biot-Gassmann equation plays an important role in reservoir fluid discrimination. The density term imbedded in the fluid term, however, is difficult to estimate because it is less sensitive to seismic amplitude variations. We combined poroelasticity theory, amplitude variation with offset (AVO) inversion, and identification of P- and S-wave moduli to present a stable and physically meaningful method to estimate the fluid term, with no need for density information from prestack seismic data. We used poroelasticity theory to express the fluid term as a function of P- and S-wave moduli. The use of Pand S-wave moduli made the derivation physically meaningful and natural. Then we derived an AVO approximation in terms of these moduli, which can then be directly inverted from seismic data. Furthermore, this practical and robust AVO-inversion technique was developed in a Bayesian framework. The objective was to obtain the maximum a posteriori solution for the P-wave modulus, S-wave modulus, and density. Gaussian and Cauchy distributions were used for the likelihood and a priori probability distributions, respectively. The introduction of a low-frequency constraint and statistical probability information to the objective function rendered the inversion more stable and less sensitive to the initial model. Tests on synthetic data showed that all the parameters can be estimated well when no noise is present and the estimated P- and S-wave moduli were still reasonable with moderate noise and rather smooth initial model parameters. A test on a real data set showed that the estimated fluid term was in good agreement with the results of drilling.

AVO inversion and poroelasticity with P- and S-wave moduli

Resolving thin layers and clearly delineating layer boundaries in inverted seismic sections are very important goals for exploration and production. Many seismic inversion methods based on a least-squares optimization approach with Tikhonov-type regularization can lead to unfocused transitions between adjacent layers. A basis pursuit inversion (BPI) algorithm based on the L1 norm optimization method can, however, resolve sharp boundaries between layers. We have formulated a BPI algorithm for amplitude-versus-angle inversion and investigated its potential to improve contrasts between layers. Like the BPI for poststack case, the sparse layer constraint, rather than the sparse spike constraint, is used to construct the model space as a wedge dictionary. All the elements of the dictionary are bed reflectivities, which include solutions consisting of thin beds as well. With this dictionary, we use an L1 norm optimization framework to derive three reflectivities, namely, Rp, Rs, and Rρ. Although BPI doe...

A prestack basis pursuit seismic inversion

Seismic fluid identification works as an effective approach to characterize the fluid feature and distribution of the reservoir underground with seismic data. Rock physics which builds bridge between the elastic parameters and reservoir parameters sets the foundation of seismic fluid identification, which is also a hot topic on the study of quantitative characterization of oil/gas reservoirs. Study on seismic fluid identification driven by rock physics has proved to be rewarding in recognizing the fluid feature and distributed regularity of the oil/gas reservoirs. This paper summarizes the key scientific problems immersed in seismic fluid identification, and emphatically reviews the main progress of seismic fluid identification driven by rock physics domestic and overseas, as well as discusses the opportunities, challenges and future research direction related to seismic fluid identification. Theoretical study and practical application indicate that we should incorporate rock physics, numerical simulation, seismic data processing and seismic inversion together to enhance the precision of seismic fluid identification.

Research on seismic fluid identification driven by rock physics

Geofluid discrimination plays an important role in the fields of hydrogeology, geothermics, and exploration geophysics. A geofluid discrimination approach incorporating linearized poroelasticity theory and pre-stack seismic reflection inversion with Bayesian inference is proposed in this study to identify the types of geofluid underground. Upon the review of the development of different geofluid indicators, the fluid modulus is defined as the geofluid indicator mainly affected by the fluid contained in reservoirs. A novel linearized P-wave reflectivity equation coupling the fluid modulus is derived to avoid the complicated nonlinear relationship between the fluid modulus and seismic data. Model examples illustrate the accuracy of the proposed linearized P-wave reflectivity equation comparing to the exact P-wave reflectivity equation even at moderate incident angle, which satisfies the requirements of the parameter estimations with P-wave pre-stack seismic data. Convoluting this linearized P-wave reflectivity equation with seismic wavelets as the forward solver, a pragmatic pre-stack Bayesian seismic inversion method is presented to estimate the fluid modulus directly. Cauchy and Gaussian probability distributions are utilized for prior information of the model parameters and the likelihood function, respectively, to enhance the inversion resolution. The preconditioned conjugate gradient method is coupled in the optimization of the objective function to weaken the strong degree of correlation among the four model parameters and enhance the stability of those parameter estimations simultaneously. The synthetic examples demonstrate the feasibility and stability of the proposed novel seismic coefficient equation and inversion approach. The real data set illustrates the efficiency and success of the proposed approach in differentiating the geofluid filled reservoirs.

Geofluid Discrimination Incorporating Poroelasticity and Seismic Reflection Inversion

Young’s modulus and Poisson’s ratio are related to quantitative reservoir properties such as porosity, rock strength, mineral and total organic carbon content, and they can be used to infer preferential drilling locations or sweet spots. Conventionally, they are computed and estimated with a rock physics law in terms of P-wave, S-wave impedances/velocities, and density which may be directly inverted with prestack seismic data. However, the density term imbedded in Young’s modulus is difficult to estimate because it is less sensitive to seismicamplitude variations, and the indirect way can create more uncertainty for the estimation of Young’s modulus and Poisson’s ratio. This study combines the elastic impedance equation in terms of Young’s modulus and Poisson’s ratio and elastic impedance variation with incident angle inversion to produce as table and direct way to estimate the Young’s modulus and Poisson’s ratio, with no need for density information from prestack seismic data. We initially derive a novel elastic impedance equation in terms of Young’s modulus and Poisson’s ratio. And then, to enhance the estimation stability, we develop the elastic impedance varying with incident angle inversion with damping singular value decomposition (EVA-DSVD) method to estimate the Young’s modulus and Poisson’ sr atio. This method is implemented in a two-step inversion: Elastic impedance inversion and parameter estimation. The introduction of a model constraint and DSVD algorithm in parameter estimation renders the EVA-DSVD inversion more stable. Tests on synthetic data show that the Young’s modulus and Poisson’s ratio are still estimated reasonable with moderate noise. A test on a real data set shows that the estimated results are in good agreement with the results of well interpretation.

Elastic impedance parameterization and inversion with Young's modulus and Poisson's ratio

Elastic impedance (EI) and amplitude variation with offset or angle (AVO/AVA) inversion are two cardinal methods to estimate elastic parameters underground with reflection seismic data Conventional EI inversion as a kind of pre-stack and post-stack joint inversion method has been widely applied in the industry because of its high efficiency and high stability of wavelet extraction; however, the robustness of extracting elastic parameters in conventional EI inversion is still controversial The robustness of three-term AVO inversion has improved a lot; however, it is still challenging to extract reasonable space variant wavelets for each offset or incident angle In this paper, a robust three-parameter estimation method, named elastic impedance variation with angle (EVA) inversion, is proposed in the Bayesian framework, which can estimate elastic parameters directly from EI This method supposes that the parameters to be inverted are Cauchy distributed and it is implemented based on a normalized EI equation in a logarithmic domain which can reduce the nonlinearity of inversion Application of a covariance matrix to decorrelate the parameters and constraint of well log curves introduced in an objective function enhances the robustness of EVA inversion A model test shows that the proposed EVA inversion method enables one to estimate reasonable elastic parameters with extremely smooth initial models and moderate Gaussian noise A real data example shows that the inverted P-wave velocity, S-wave velocity and density are identical to well log interpretation results, which shows the validity of the proposed method

http://iopscience.iop.org/article/10.1088/1742-2132/9/3/247/pdf

Guochen Wu

Papers

AVO inversion and poroelasticity with P- and S-wave moduli

Research on seismic fluid identification driven by rock physics

Geofluid Discrimination Incorporating Poroelasticity and Seismic Reflection Inversion

Elastic impedance parameterization and inversion with Young's modulus and Poisson's ratio

Elastic impedance variation with angle inversion for elastic parameters