Critical Porosity Model Research Articles

Decorrelated linearized seismic-petrophysics inversion

Seismic model and rock-physics models (RPMs) are of significance in petrophysics inversion to estimate petrophysical properties from seismic data. Nevertheless, the prediction of petrophysical properties is commonly restricted by two issues. One is that computationally intensive algorithms are required to invert petrophysical parameters coupled in most nonlinear RPMs. Another is that statistical error caused by correlation characterization between different petrophysical parameters can affect the accuracy of inverted results. To solve these two issues, a decorrelated linearized seismic-petrophysics inversion (DLSPI) based on a linearized seismic-petrophysics model is proposed. First, a novel linearized seismic-petrophysics model is constructed based on the Taylor first-order approximate RPMs including Nur's critical porosity model and Gassmann's equation. Next, a linearized seismic-petrophysics inversion (LSPI) is proposed under the linear Bayesian framework. This approach can estimate petrophysical parameters from seismic data directly and efficiently. Then, a decorrelation strategy based on principal component analysis (PCA) is introduced to the LSPI to form the DLSPI, which can avoid the effect of the statistical correlation between different petrophysical parameters and improve inversion accuracy. Finally, the proposed method is validated both on synthetic seismic data and real seismic data from a sandstone reservoir. The inverted results indicate the efficiency and stability of the proposed method.

AVA inversion for novel fluid indicator considering consolidation parameter

Abstract Geofluid identification from seismic data are crucial for understanding reservoir characteristics. However, fluid indicators based on elastic parameter combinations show strong ambiguity in terms of geofluid identification. Although the effective pore-fluid bulk modulus proves to be the superior fluid indicator in geofluid discrimination, it is limited to empirical models such as the critical porosity model. The consolidation parameter model can evaluate the consolidation and compaction of sediments and is widely used due to its better applicability. Therefore, a novel fluid indicator considering consolidation parameter is proposed and a boundary-constrained inversion strategy for geofluid identification is developed. First, the novel fluid indicator is defined based on the poroelasticity theory. Then, the linearized AVA (amplitude variation with angle) approximation equation related to the new fluid indicator is derived to directly link seismic data and properties of pore-filling materials. The model analysis verifies the accuracy of the derived approximation equation at moderate incident angles, which can be used for parameter prediction by pre-stack seismic inversion. Furthermore, the contribution of each attribute of the novel equation to the reflectivity is analyzed to validate the feasibility of new fluid indicator inversion. Finally, a boundary-constrained AVA pre-stack inversion method is presented to enhance the robustness of the inversion results of the model parameter. The reliability of the improved method is proved by accurate inversion results of synthetic seismic records. After determining the consolidation parameter using well-log data combined with BGT (Biot–Gassmann Theory), the field data further demonstrated that the proposed approach is accurate and effective.

Anisotropic Poroelasticity and AVAZ Inversion for in Situ Stress Estimate in Fractured Shale-Gas Reservoirs

Knowledge of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress is of great significance for hydraulic fracture stimulating of unconventional reservoirs. Quantitative estimation of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress, especially in the far field, is a major challenge. This research mainly focuses on developing a novel Bayesian AVAZ inversion approach to estimate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress of fractured shale-gas reservoirs with horizontal transversely isotropic (HTI) symmetry. Using the generalized Hooke’s law and Schoenberg’s linear slip model, we first deduce the anisotropic horizontal stress equation in an HTI medium formed by a single set of vertically aligned fractures embedded in an isotropic background rock. Based on the critical porosity model, we then obtain the saturated stiffness matrix with vertical effective stress-sensitive parameters and dry fracture weaknesses using the relationship between vertical effective stress and porosity. Combining the scattering function and the perturbed stiffness matrix, we deduce a linearized PP-wave reflection coefficient as a function of fluid bulk modulus, vertical effective stress-sensitive parameter, dry-rock P- and S-wave moduli, density, and fracture density. Finally, we propose a novel Bayesian amplitude variation with azimuth (AVAZ) inversion constrained by the Cauchy regularization and low-frequency regularization to estimate these model parameters, which are further used to calculate <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress. The analysis of synthetic data demonstrates that <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in situ</i> stress can be reasonably estimated even with moderate noise. A field data set test reveals that the inversion results agree well with the well log interpretation, and the proposed approach can generate meaningful results that are useful for seismic identification of potential fracturing barriers during fracturing stimulation of shale-gas reservoirs.

Pressure effects in Nur's critical porosity model estimates for carbonate rocks

The forecasting of fluid saturation changes within the reservoirs is one of the main issues of rock physics, which is often done using Biot-Gassmann theory. However, this theory demands the knowledge of the dry rock elastic moduli. Nur's (critical porosity) model is one of the theories to evaluate those dry rock properties from porosity and the mineral elastic properties of rock constituents. Therefore, that model is appropriate for estimating the dry rock bulk and shear moduli in well log analysis, especially in situations where there are no sonic logs available. In this paper, we analyzed four carbonate rock datasets reported in the literature, which consisted of measurements of porosity, mineral content and elastic velocities. Our approach focused on investigating how the misfits between the model and the experimental data varied according to porosity and pressure, through best-fitting regressions. Those observations allowed the setting of a modification on the Nur's model that can account for the pressure influence in porosity. This modified Nur's model significantly improved the accuracy of the results, exhibiting average relative errors 50% smaller than the original model estimates. The performance of that model varied for the different carbonate textures and pointed out that the siliciclastic content influences the predictability error of that modified critical porosity model. Rocks exhibiting the highest siliciclastic content showed an error reduction trend as pressure increases, while the other samples with minor siliciclastic content did not show significative accuracy variation.

Seismic AVO statistical inversion incorporating poroelasticity

Seismic amplitude variation with offset (AVO) inversion is an important approach for quantitative prediction of rock elasticity, lithology and fluid properties. With Biot–Gassmann’s poroelasticity, an improved statistical AVO inversion approach is proposed. To distinguish the influence of rock porosity and pore fluid modulus on AVO reflection coefficients, the AVO equation of reflection coefficients parameterized by porosity, rock-matrix moduli, density and fluid modulus is initially derived from Gassmann equation and critical porosity model. From the analysis of the influences of model parameters on the proposed AVO equation, rock porosity has the greatest influences, followed by rock-matrix moduli and density, and fluid modulus has the least influences among these model parameters. Furthermore, a statistical AVO stepwise inversion method is implemented to the simultaneous estimation of rock porosity, rock-matrix modulus, density and fluid modulus. Besides, the Laplace probability model and differential evolution, Markov chain Monte Carlo algorithm is utilized for the stochastic simulation within Bayesian framework. Models and field data examples demonstrate that the simultaneous optimizations of multiple Markov chains can achieve the efficient simulation of the posterior probability density distribution of model parameters, which is helpful for the uncertainty analysis of the inversion and sets a theoretical fundament for reservoir characterization and fluid discrimination.

Rock physics modelling of porous rocks with multiple pore types: a multiple‐porosity variable critical porosity model

ABSTRACTA critical porosity model establishes the empirical relationship between a grain matrix and a dry rock by the concept of critical porosity. The model is simple and practical and widely used. But the critical porosity in the model is a fixed value that cannot relate to pore structure. The aim of this paper is to establish the theoretical relationship between critical porosity and pore structure by combining Kuster–Toksöz theory with the critical porosity model. Different from the traditional critical porosity model, critical porosity is not an empirical value but varied with pore shape and the ratio of bulk modulus versus shear modulus of the grain matrix. The substitution of the theoretical relationship into Kuster–Toksöz theory will generate the formulae for the bulk and shear moduli of multiple‐porosity dry rocks, which is named the multiple‐porosity variable critical porosity model. The new model has been used to predict elastic moduli for sandstone and carbonate rock. We compare the modelling results for P‐ and S‐wave velocities and elastic moduli with the experimental data. The comparison shows that the new model can be used to describe the elastic properties for the rocks with multiple pore types.

An efficient rock-physics workflow for modeling and inversion in anisotropic organic-shales

In this work we present a workflow for modeling and inversion of physical parameters using ultrasonic wave velocities measured on dry and saturated samples of Bakken, Bazhenov shales and Niobrara marls. The forward problem is based on the well known theory of Ciz and Shapiro (2007) combined with the anisotropic generalization of two empirical elastic models for the rock matrix: the critical porosity model Nur et al. (1998) and Krief et al. model (1990). Moreover, taking into account the high variability of the physical properties of kerogen and clay minerals, and their influence on the mechanical properties of these rocks, the procedure also involves the numerical inversion of such properties, using an efficient numerical routine. The results found for the effective elastic properties and density of clay minerals and kerogen are reasonable taking into account those published in the reference literature. In all cases the quality of fit between real and synthetic data is analysed by means of an adequate error measure. The stability of the predicted elastic coefficients is verified using strain energy restrictions. The calibrated models are also used to quantify the influence of the effective porosity on the computations and for a fluid substitution problem. Finally, we found that even for weak anisotropy the generalized Krief's model requires at least three empirical parameters. We conclude that our novel anisotropic critical porosity model is a simpler and more efficient choice for rock-physics applications.

Tight carbonate reservoir characterization based on the modified rock physics model

Because tight carbonate reservoirs have outstanding potential of reserves and economic benefits, it is significant to recognize their typical geophysical responses using rock physics models. Nevertheless, conventional models are not satisfactory to interpret multi-scale or in-situ observations, since their computational generalization. To enhance the predictive and computational power, this paper derives an upscale rock physics model which includes both modeling and inverting schemes. Its main idea is to study and utilize inherent statistical natures in naturally occurring tight carbonate rocks. Through extending the typical critical porosity model (CPM), our modeling scheme relates pore system to wave propagation and fluid flow responses in dolomite-calcite mixtures. Thus, it can recognize the effect of pore structure changes on velocity and permeability heterogeneity. On the other hand, in terms of deterministic and probabilistic methods, pore systems can be characterized within different scopes of observations. It also allows a model based inverting scheme for data interpretation and correction, according to the statistical comparability between lab measurements and well logs. Therefore, if mineralogy contents and noisy sonic logging curves are available, one can simultaneously obtain the corrected porosity, permeability and velocity curves. In site data tests, predictive power of this model is well demonstrated by the high data consistency.

Bayesian linearized petrophysical AVO inversion

Seismic reservoir characterization aims to provide a 3D model of rock and fluid properties based on measured seismic data. Petrophysical properties, such as porosity, mineral volumes, and water saturation, are related to elastic properties, such as velocity and impedance, through a rock-physics model. Elastic attributes can be obtained from seismic data through seismic modeling. Estimation of the properties of interest is an inverse problem; however, if the forward model is nonlinear, computationally demanding inversion algorithms should be adopted. We have developed a linearized forward model, based on a convolutional model and a new amplitude variation with offset approximation that combined Gray’s linearization of the reflectivity coefficients with Gassmann’s equation and Nur’s critical porosity model. Physical relations between the saturated elastic moduli and the matrix elastic moduli, fluid bulk modulus, and porosity are almost linear, and the model linearization can be obtained by computing the first-order Taylor series approximation. The inversion method for the estimation of the reservoir properties of interest is then developed in the Bayesian framework. If we assume that the distributions of the prior model and error term are Gaussian, then the explicit analytical solution of the posterior distribution of rock and fluid properties can be analytically derived. Our method has first been validated on synthetic seismic data and then applied to a 2D seismic section extracted from a real data set acquired in the Norwegian Sea.

Rock critical porosity inversion and S-wave velocity prediction

A critical porosity model is often used to calculate the dry frame elastic modulus by the rock critical porosity value which is affected by many factors. In practice it is hard for us to obtain an accurate critical porosity value and we can generally take only an empirical critical porosity value which often causes errors. In this paper, we propose a method to obtain the rock critical porosity value by inverting P-wave velocity and applying it to predict S-wave velocity. The applications of experiment and log data both show that the critical porosity inversion method can reduce the uncertainty resulting from using an empirical value in the past and provide the accurate critical porosity value for predicting S-wave velocity which significantly improves the prediction accuracy.

Estimating Elastic Parameters of Framework and Pore Fluid from CLYJ Saturated Limestone Samples

AbstractIn this paper, numerical method which is dominated by inversion is used to analyze CLYJ saturated limestone samples. Elastic parameters of rock framework and pore fluid are estimated from the overall measured density and velocity data in different porosities based on the effective inversion method. A critical porosity model is used, and elastic parameters are calculated after grouping. The estimating process includes grouping, calculating, counting and evaluating. Firstly, three samples are chosen as a group and there are N different ways to group. The samples are grouped in both integral way and individual way. Secondly, for each group the component elastic parameters are calculated based on the critical porosity model and N groups of results are obtained. Third, the results are analyzed to estimate the elastic parameters of samples. At last, the results are evaluated through comparisons of statistic parameters between the estimated values and measured ones, such as maximum, minimum, mean, and standard deviation. The results indicate that the method to estimate the elastic parameters of rock framework and pore fluid is effective and reliable. The component elastic parameters reflect the elastic properties of CLYJ limestone samples, and enrich the contents of these samples.

Modulus ratio of dry rock based on differential effective-medium theory

There are two points of view with regard to the relationship between the dry-framework modulus ratio of bulk to shear modulus and porosity. One view, assumed by most empirical models such as the critical porosity model, is that the dry-framework modulus ratio is independent of porosity. However, this assumption is sometimes in disagreement with experiments. Another view supported by the differential effective-medium (DEM) theory is that the dry-framework modulus ratio depends on porosity. Ordinary differential equations (ODE) for bulk and shear moduli are coupled, and there is no analytical formula for the modulus ratio of dry porous rock. By assuming that the difference between the polarization factors of dry inclusions for bulk and shear modulus within the ODE has an almost linear relationship with the effective modulus ratio, an analytical solution for the dry-rock modulus ratio is derived from the ODE. The above linear relationship characterized by intercept [Formula: see text] and gradient [Formula: see text] is confirmed by the numerical results calculated from the analytical formulas for the three specific pore shapes: spherical pores, needle-shaped pores, and penny-shaped cracks. Then the validity of this assumption is tested by integrating the full DEM equation numerically. The analytical approximation gives a good estimate of the numerical results for the cases of the three given pore shapes. The dry-frame modulus ratio is monotonic with respect to porosity, and its monotonicity depends on the mineral-grain modulus ratio and the pore shape. The analytical formula can be simplified further to some popular models such as the critical porosity model by setting constants [Formula: see text] and [Formula: see text] to specific values. Predictions of the analytical formula compare favorably with real and synthetic sandstone data and granite data measured in the laboratory.

The role of rock physics for the Enfield 4D seismic monitoring project

The Enfield rock physics model was constructed to enable 4D feasibility studies and interpretation of the 2007 Enfield 4D seismic monitor survey. The rock physics model links reservoir static and dynamic parameters to impedances, using log data from five wells in the field, laboratory core measurements taken from cores on Enfield and neighbouring fields, and theoretical rock models from the literature. The reservoir is modelled by a sand-shale mix: sand properties are described using a modified critical porosity model whereas shale properties are generated from log data averaging. The dynamic properties in the model include saturation and pressure. Saturation is modelled using Gassmann’s formula assuming homogeneous mixing. The reservoir sand velocity-pressure relationship is described by an empirical model fitted to dry core plug measurements. An assessment of the effect of uncertainty is included for both the saturation and pressure elements of the model. The resultant rock physics model was used before the acquisition of the seismic monitor survey to assess the likelihood of detecting a 4D seismic signal only 7 months after production start-up. Our modelling results indicate that the strong pressure build-up around the water injectors would result in a detectable 4D seismic signal and this prediction is confirmed by the successful 4D seismic monitor data acquired in 2007. The rock physics model has been validated against the 4D monitor data and is being used to quantify the 4D interpretation, linking the observed 4D response back to predicted pressure and saturations changes in the field.

The Role of Rock Physics Modelling for the Enfield 4D Seismic

The Enfield rock physics model was constructed to enable interpretation of the 2007 Enfield 4D monitor. The rock physics model links reservoir static and dynamic parameters to impedances. The model was based on a combination of log data from 5 wells in the field, laboratory core measurements taken from core on Enfield and neighbouring fields, and theoretical rock models from the literature. The reservoir is modelled by a sand-shale mix. Sand properties are described using a modified critical porosity model whereas shale properties are generated from log data averaging. The dynamic properties in the model include saturation and pressure. Saturation is modelled using Gassmann?s formula assuming a homogeneous mixing. The reservoir sand velocity-pressure relationship was described by an empirical model fitted to dry core plug measurements. An assessment of the effect of uncertainty was included for both the saturation and pressure elements of the model. The resultant rock physics model was used prior to the acquisition of the monitor to assess the likelihood of measuring a 4D signal only 7 months after production start-up. Our modelling results indicated that the strong pressure build-up around the water injectors would result in a measurable 4D signal and this prediction was fulfilled by the successful 4D survey acquired in 2007. The rock physics model has been validated against the 4D monitor data and is being used to quantify the 4D interpretation, linking the observed 4D response back to predicted pressure and saturations in the field.

An approximation for the Xu‐White velocity model

In 1995, S. Xu and R. E. White described a method for estimating compressional and shear‐wave velocities of shaley sandstones from porosity and shale content. Their model was able to predict the effect of increasing clay content on compressional‐wave velocity observed in laboratory measurements. A key step in the Xu‐White method estimates dry rock bulk and shear moduli for the sand/shale mixture. This step is performed numerically by applying the differential effective medium method to the Kuster‐Toksöz equations for ellipsoidal pores. This step is computationally intensive. Using reasonable assumptions about dry rock elastic properties, this step can be replaced with a set of approximations for dry rock bulk and shear moduli. Numerical experiments show an extremely close match between velocities obtained with these approximations and velocities computed with the differential effective medium method. These approximations simplify the application of the Xu‐White method, and make the method computationally more efficient. They also provide insight into the Xu‐White method. For example, these approximations show how the Xu‐White model is related to the critical porosity model.

Stress sensitivity of sandstones

Our observations made on dry‐sandstone ultrasonic velocity data relate to the variation in velocity (or modulus) with effective stress, and the ability to predict a velocity for a rock under one effective pressure when it is known only under a different effective pressure. We find that the sensitivity of elastic moduli, and velocities, to effective hydrostatic stress increases with decreasing porosity. Specifically, we calculate the difference between an elastic modulus, [Formula: see text], of a sample of porosity ϕ at effective pressure [Formula: see text] and the same modulus, [Formula: see text], at effective pressure [Formula: see text]. If this difference, [Formula: see text], is plotted versus porosity for a suite of samples, then the scatter of ΔM is close to zero as porosity approaches the critical porosity value, and reaches its maximum as porosity approaches zero. The dependence of this scatter on porosity is close to linear. Critical porosity here is the porosity above which rock can exist only as a suspension—between 36% and 40% for sandstones. This stress‐sensitivity pattern of grain‐supported sandstones (clay content below 0.35) practically does not depend on clay content. In practical terms, the uncertainty of determining elastic moduli at a higher effective stress from the measurements at a lower effective stress is small at high porosity and increases with decreasing porosity. We explain this effect by using a combination of two heuristic models—the critical porosity model and the modified solid model. The former is based on the observation that the elastic‐modulus‐versus‐porosity relation can be approximated by a straight line that connects two points in the modulus‐porosity plane: the modulus of the solid phase at zero porosity and zero at critical porosity. The second one reflects the fact that at constant effective stress, low‐porosity sandstones (even with small amounts of clay) exhibit large variability in elastic moduli. We attribute this variability to compliant cracks that hardly affect porosity but strongly affect the stiffness. The above qualitative observation helps to quantitatively constrain P‐ and S‐wave velocities at varying stresses from a single measurement at a fixed stress. We also show that there are distinctive linear relations between Poisson’s ratios (ν) of sandstone measured at two different stresses. For example, in consolidated medium‐porosity sandstones [Formula: see text], where the subscripts indicate hydrostatic stress in MPa. Linear functions can also be used to relate the changes (with hydrostatic stress) in shear moduli to those in compressional moduli. For example, [Formula: see text], where [Formula: see text] is shear modulus and [Formula: see text] is compressional modulus, both in GPa, and the subscripts indicate stress in MPa.