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Algorithm and code are presented which solve the dispersion equation for cylindrical poroelastic structures. The algorithm is based on the spectral method, which discretizes the underlying wave equations with the help of spectral differentiation matrices and solves the corresponding equations as a generalized eigenvalue problem. The results are illustrated for the case of a fluid-saturated free cylinder with open- and closed-pore boundary conditions on its surface. The computed dispersion curves are in good agreement with analytical results, which confirms the accuracy of the method.

A detailed analysis of the relationship between elastic waves in inhomogeneous, porous media and the effect of wave-induced fluid flow is presented. Based on the results of the poroelastic first-order statistical smoothing approximation applied to Biot's equations of poroelasticity, a model for elastic wave attenuation and dispersion due to wave-induced fluid flow in 3-D randomly inhomogeneous poroelastic media is developed. Attenuation and dispersion depend on linear combinations of the spatial correlations of the fluctuating poroelastic parameters. The observed frequency dependence is typical for a relaxation phenomenon. Further, the analytic properties of attenuation and dispersion are analyzed. It is shown that the low-frequency asymptote of the attenuation coefficient of a plane compressional wave is proportional to the square of frequency. At high frequencies the attenuation coefficient becomes proportional to the square root of frequency. A comparison with the 1-D theory shows that attenuation is of the same order but slightly larger in 3-D random media. Several modeling choices of the approach including the effect of cross correlations between fluid and solid phase properties are demonstrated. The potential application of the results to real porous materials is discussed.

Numerical computation from high‐resolution 3‐D microtomographic images of rocks (known as digital rock physics) has the potential to predict elastic properties more accurately. However, successful examples are limited to samples with simple structure and mineralogy. The physical size of sample is often too small to present heterogeneities at a larger scale and the image resolution is insufficient to characterize the details of rocks. Also, the grayscale values of different minerals in microtomographic images are often similar, and previous attempts to segment them as separate phases are not very successful. Here, we propose a practical digital rock physics workflow for somewhat more complex and ubiquitous rocks, namely, sandstones that contain mostly quartz and a small fraction of dispersed clay (known as arenites). Based on a set of images, we obtain a suite of postcomputation corrections to compensate for the effects of sample size and resolution of the microtomographic images. Furthermore, we build a segmentation workflow that effectively detects feldspar and clay minerals, despite their grayscale similarity to quartz. A moduli‐porosity trend is derived from the subsamples of the original digital images. Bulk moduli agree well with the ultrasonic measurements on the dry samples at 40 MPa. Shear moduli remain overestimated, which is likely caused by poor knowledge of the mineral stiffness. We compensate for this effect using a heuristic correction to the matrix moduli. The final version of the workflow provides accurate elastic moduli trends with porosity and clay content based on only two samples of Bentheimer sandstone. Key Points: Effects of microtomographic image scanning parameters on computed effective moduli are investigated and correctedDispersive clay and feldspar are segmented as separate phasesA rock physics template is built with images from only two Bentheimer sandstone miniplugs [ABSTRACT FROM AUTHOR]

Quantifying the effects of pore‐filling materials on elastic properties of porous rocks is of considerable interest in geophysical practice. For rocks saturated with fluids, the Gassmann equation is proved effective in estimating the exact change in seismic velocity or rock moduli upon the changes in properties of pore infill. For solid substance or viscoelastic materials, however, the Gassmann theory is not applicable as the rigidity of the pore fill (either elastic or viscoelastic) prevents pressure communication in the pore space, which is a key assumption of the Gassmann equation. In this paper, we explored the elastic properties of a sandstone sample saturated with fluid and solid substance under different confining pressures. This sandstone sample is saturated with octadecane, which is a hydrocarbon with a melting point of 28°C, making it convenient to use in the lab in both solid and fluid forms. Ultrasonically measured velocities of the dry rock exhibit strong pressure dependency, which is largely reduced for the filling of solid octadecane. Predictions by the Gassmann theory for the elastic moduli of the sandstone saturated with liquid octadecane are consistent with ultrasonic measurements, but underestimate the elastic moduli of the sandstone saturated with solid octadecane. Our analysis shows that the difference between the elastic moduli of the dry and solid‐octadecane‐saturated sandstone is controlled by the squirt flow between stiff, compliant, and the so‐called intermediate pores (with an aspect ratio larger than that of compliant pore but much less than that of stiff pores). Therefore, we developed a triple porosity model to quantify the combined squirt flow effects of compliant and intermediate pores saturated with solid or viscoelastic infill. Full saturation of remaining stiff pores with solid or viscoelastic materials is then considered by the lower embedded bound theory. The proposed model gave a reasonable fit to the ultrasonic measurements of the elastic moduli of the sandstone saturated with liquid or solid octadecane. Comparison of the predictions by the new model to other solid substitution schemes implied that accounting for the combined effects of compliant and intermediate pores is necessary to explain the solid squirt effects. [ABSTRACT FROM AUTHOR]

Time-lapse ultrasonic measurements constitute a tool to establish and calibrate rock physics models for surface seismic monitoring of partially saturated rocks. This workflow requires one to take into account seismic dispersion caused by frequency-dependent wave-induced fluid flow. We develop a theory of squirt flow in rocks saturated with a viscoelastic material containing isolated gas patches between compliant intergranular contacts. This model is valid for the entire frequency range, from seismic to ultrasonic. In the limit of full saturation the derived equations reduce to the Gassmann equations in the low-frequency regime and traditional squirt theory in the high-frequency regime. The model prediction of ultrasonic velocities versus saturation matches with experimental observations. [ABSTRACT FROM AUTHOR]

ABSTRACT A major cause of attenuation in fluid-saturated media is the local fluid flow (or squirt flow) induced by a passing wave between pores of different shapes and sizes. Several squirt flow models have been derived for isotropic media. For anisotropic media however, most of the existing squirt flow models only provide the low- and high-frequency limits of the saturated elastic properties. We develop a new squirt flow model to account for the frequency dependence of elastic properties and thus gain some insight into velocity dispersion and attenuation in anisotropic media. In this paper, we focus on media containing aligned compliant pores embedded in an isotropic background matrix. The low- and high-frequency limits of the predicted fluid-saturated elastic properties are respectively consistent with Gassmann theory and Mukerji-Mavko squirt flow model. Results are also expressed in terms of Thomsen anisotropy parameters. It turns out that the P-wave anisotropy parameter ε tends to zero in the high-frequency limit, whereas the δ parameter remains the only indicator of P-S⊥ anisotropy. The S-wave anisotropy parameter γ is not affected by the presence of fluid and remains the same for all frequency ranges. A new definition for attenuation anisotropy parameters is also proposed to quantify the attenuation anisotropy. In the most important case of liquid saturation, analytical expressions are derived for elastic properties, velocity anisotropy parameters, quality factors, and attenuation anisotropy parameters. A companion paper considers the case of cracks with an ellipsoidal distribution of orientations resulting from the application of anisotropic stress. [ABSTRACT FROM AUTHOR]

ABSTRACT Estimating the impact of solid pore fill on effective elastic properties of rocks is important for a number of applications such as seismic monitoring of production of heavy oil or gas hydrates. We develop a simple model relating effective seismic properties of a rock saturated with a liquid, solid, or viscoelastic pore fill, which is assumed to be much softer than the constituent minerals. A key feature of the model is division of porosity into stiff matrix pores and compliant crack-like pores because the presence of a solid material in thin voids stiffens the rock to a much greater extent than its presence in stiff pores. We approximate a typical compliant pore as a plane circular interlayer surrounded by empty pores. The effect of saturation of the stiff pores is then taken into account using generalized Gassmann's equations. The proposed model provides a good fit to measurements of the shear stiffness and loss factor of the Uvalde heavy-oil rock at different temperatures and frequencies. When the pore fill is solid, the predictions of the scheme are close to the predictions of the solid squirt model recently proposed by Saxena and Mavko. At the same time, the present scheme also gives a continuous transition to the classic Gassmann's equations for a liquid pore fill at low frequencies and the squirt theory at high frequencies. [ABSTRACT FROM AUTHOR]

ABSTRACT CO2 sequestration projects benefit from quantitative assessment of saturation distribution and plume extent for field development and leakage prevention. In this work, we carry out quantitative analysis of time-lapse seismic by using rock physics and seismic modelling tools. We investigate the suitability of Gassmann's equation for a CO2 sequestration project with 1600 tons of CO2 injected into high-porosity, brine-saturated sandstone. We analyze the observed time delays and amplitude changes in a time-lapse vertical seismic profile dataset. Both reflected and transmitted waves are analyzed qualitatively and quantitatively. To interpret the changes obtained from the vertical seismic profile, we perform a 2.5D elastic, finite-difference modelling study. The results show a P-wave velocity reduction of 750 m/s in the proximity of the injection well evident by the first arrivals (travel-time delays and amplitude change) and reflected wave amplitude changes. These results do not match with our rock physics model using Gassmann's equation predictions even when taking uncertainty in CO2 saturation and grain properties into account. We find that time-lapse vertical seismic profile data integrated with other information (e.g., core and well log) can be used to constrain the velocity-saturation relation and verify the applicability of theoretical models such as Gassmann's equation with considerable certainty. The study shows that possible nonelastic factors are in play after CO2 injection (e.g., CO2-brine-rock interaction and pressure effect) as Gassmann's equation underestimated the velocity reduction in comparison with field data for all three sets of time-lapse vertical seismic profile attributes. Our work shows the importance of data integration to validate the applicability of theoretical models such as Gassmann's equation for quantitative analysis of time-lapse seismic data. [ABSTRACT FROM AUTHOR]

Knowledge about the pressure dependency of elastic and electrical properties is important for a variety of geophysical applications. We present a technique to invert for the stiff and compliant porosity from velocity measurements made as a function of differential pressure on saturated sandstones. A dual porosity concept is used for dry rock compressibility and a squirt model is employed for the pressure and frequency dependent elastic properties of the rocks when saturated. The total porosity obtained from inversion shows satisfactory agreement with experimental results. The electrical cementation factor was determined using the inverted porosity in combination with measured electrical conductivity. It was found that cementation factor increased exponentially with increasing differential pressure during isostatic loading. Elastic compressibility, electrical cementation factor and electrical conductivity of the saturated rocks correlate linearly with compliant porosity, and electrical cementation factor and electrical conductivity exhibit linear correlations with elastic compressibility of the saturated rocks under loading. The results show that the dual porosity concept is sufficient to explain the pressure dependency of elastic, electrical and joint elastic-electrical properties of saturated porous sandstones. [ABSTRACT FROM AUTHOR]

A key objective of stage 2 of the Cooperative Research Centre for Greenhouse Gas Technologies Otway Project is to evaluate the seismic detection limit of greenhouse gas injected into a saline aquifer. For this purpose, injection of a small amount of CO2-rich gas into the Paaratte Formation, a saline aquifer located at a depth of approximately 1.5 km, is planned. Before the injection experiment is undertaken, we assessed the detectability of injected gas with seismic methods in a modeling study. A key objective of this study was to model changes in elastic properties caused by CO2-saturation effects using predictions of reservoir simulations. To this end, we established an elastic property/porosity relation to link the reservoir flow model and the elastic properties of the subsurface. Predicting changes in elastic properties requires suitable velocity-saturation relations. To choose an appropriate velocity-saturation relation, we analyzed the effect of fluid distribution on the time-lapse seismic response by performing 1,5D poroelastic and elastic modeling based on reservoir simulations. The modeling results emphasized the importance of taking the variability of rock properties into account and to carefully estimate dry bulk moduli to adequately represent the sensitivity of rock properties to fluid changes. Furthermore, we determined that the Gassmann-Wood relation was an appropriate velocity-saturation relation at seismic frequencies for the Paaratte Formation. However, changes in acoustic contrasts caused by CO2 saturation between layers below the seismic resolution had to be considered. In this sense, an appropriate velocity-saturation relation also depends on the scale at which we model the seismic response. [ABSTRACT FROM AUTHOR]

Prediction of sonic velocities in shales from well logs is important for seismic to log ties if the sonic log is absent for a shaly section, for pore pressure anomaly detection, and for data quality control. An anisotropic differential effective medium (DEM) was used to simulate elastic properties of shales from elastic properties and volume fractions of silt and wet clay (a hypothetical composite material that includes all clay minerals and water). Anisotropic elastic coefficients of the wet clay were assumed as a first-order approximation to be linearly dependent on wet clay porosity (WCP). Here, by WCP we mean a ratio of a pore volume occupied by water to a total volume of the wet clay. Effects of silt inclusions on elastic coefficients of shales were taken into account by using the anisotropic differential effective medium model. Silt inclusions were modeled as spherical quartz particles. Simulated elastic coefficients of shales were used to calculate compressional and shear velocities, and these were in a good agreement with the sonic velocities observed on a test data set from an offshore Mid-Norway well penetrating a 500-m vertical section of shale. To further study the elastic properties of wet clays, elastic coefficients calculated from compressional and sonic velocities measured in shales were inverted for vertical profiles of wet clay elastic coefficients. Analysis of these coefficients found that in the well considered, the increase in elastic coefficients of shales was controlled by the increase of silt fraction with depth. Elastic coefficients of wet clay found no increase with depth. The inverted elastic moduli of wet clay found much stronger correlation with WCP than do the moduli of shale. This confirmed the hypothesis that silt fraction is one of the key parameters for the modeling of elastic properties of shale. [ABSTRACT FROM AUTHOR]

Predicting seismic velocities in isotropic fluid-saturated rocks is commonly done using the isotropic Gassmann theory. For anisotropic media, the solution is expressed in terms of stiffness or compliance, which does not provide an intuitive understanding on how the fluid affects wave propagation in anisotropic media. Assuming weak anisotropy, we expressed the anisotropy parameters of transversely isotropic saturated media as a function of the anisotropy parameters in the dry medium, the bulk and shear moduli of the saturated and dry media, the grain and fluid bulk moduli, and the porosity. By deriving an approximation of the anellipticity parameter ŋ, we discovered that if the dry medium was elliptical, the saturated medium was also elliptical but only if the porosity exceeded a certain threshold value. This result can provide a way of differentiating between stress- and fracture-induced anisotropy. [ABSTRACT FROM AUTHOR]

ABSTRACT Up-scaling the elastic properties of digitized rock volumes as obtained from X-ray computer tomography (CT) imaging via computer simulations has the potential to assist and complement laboratory measurements. This computational up-scaling approach remains a challenging task as the overall elastic properties are not only dependent on the elastic properties of individual grains but also on the hardly resolvable pore spaces between adjacent grains such as micro-cracks. We develop a digitized rock image and elastic up-scaling workflow based on general-purpose and widely available software packages. Particular attention is paid to CT image processing including filtering, smoothing and segmentation. A strategy for optimal meshing for subsequent finite-element modelling is also proposed. We apply this workflow to the micro-tomographic image of a well-consolidated, feldspatic sandstone sample and determine the up-scaled bulk and shear moduli. These effective elastic moduli are compared to the moduli inferred from laboratory ultrasound measurements at variable effective stresses (0-70 MPa). We observe that the numerically up-scaled elastic moduli correspond to the moduli at a certain effective stress level (50 MPa), beyond which the effective-stress dependency follows a linear trend. This indicates that the computational up-scaling approach yields moduli as if all compliant (soft) porosity was absent, i.e., microcracks are closed. To confirm this hypothesis, we estimate the amount of soft porosity on the basis of the double-porosity theory (Shapiro, 2003) and find that at 50 MPa the soft porosity is indeed practically zero. We conclude that our computational elastic up-scaling approach yields physically consistent effective moduli even if some geometrical features are below CT resolution. To account for these sub-resolution features either theoretical or additional computational approaches can be used. [ABSTRACT FROM AUTHOR]

The Hashin-Shtrikman (HS) bounds define the range of bulk and shear moduli of an elastic composite, given the moduli of the constituents and their volume fractions. Recently, the HS bounds have been extended to the quasi-static moduli of composite viscoelastic media. Because viscoelastic moduli are complex, the viscoelastic bounds form a closed curve on the complex plane. We analyze these general viscoelastic bounds for a particular case of a porous solid saturated with a Newtonian fluid. In our analysis, for poroelastic media, the viscoelastic bounds for the bulk modulus are represented by a semicircle and a segment of the real axis, connecting formal HS bounds that are computed for an inviscid fluid. Importantly, viscoelastic bounds for poroelastic media turn out to be independent of frequency. However, because the bounds are quasi-static, the frequency must be much lower than Biot's characteristic frequency. Furthermore, we find that the bounds for the bulk modulus are attainable (realizable). We also find that these viscoelastic bounds account for viscous shear relaxation and squirt-flow dispersion, but do not account for Biot's global flow dispersion, because the latter strongly depends on inertial forces. [ABSTRACT FROM AUTHOR]

Modeling dispersion and attenuation of elastic waves in fluid-saturated rocks due to squirt flow requires the knowledge of a number of geometrical parameters of the pore space, in particular, the characteristic aspect ratio of the pores. These parameters are usually interred by fitting measurements on saturated rocks to model predictions. To eliminate such fitting and thus make the model more predictive, we propose to recover the geometrical parameters of the pore space from the pressure dependency of elastic moduli on dry samples. Our analysis showed that the pressure dependency of elastic properties of rocks (and their deviation from Gassmann's prediction) at ultrasonic frequencies is controlled by the squirt flow between equant, stiff, and so-called intermediate pores (with aspect ratios between 10-3-2 X 10-1) Such intermediate porosity is expected to close at confining pressures of between 200 and 2000 MPa, and thus cannot be directly obtained from ultrasonic experiments performed at pressures below 50 MPa. However, the presence of this intermediate porosity is inferred from the significant linear trend in the pressure dependency of elastic properties of the dry rock and the difference between the bulk modulus of the dry rock computed for spherical pores and the measured modulus at 50 MPa. Moreover, we can infer the magnitude of the intermediate porosity and its characteristic aspect ratio. Substituting these parameters into the squirt model, we have computed elastic moduli and velocities of the water-satnrated rock and compared these predictions against laboratory measmements of these velocities. The agreement is good for a number of clean sandstones, but not unexpectedly worse for a broad range of shaley sandstones. Our predictions showed that dispersion and attenuation caused by the squirt flow between compliant and stiff pores may occur in the seismic frequency band. Confirmation of this prediction requires laboratory measurements of elastic properties at these frequencies. [ABSTRACT FROM AUTHOR]

The squirt-flow wave attenuation mechanism is implemented in Biot's theory of poroeiasticity in the form of differential equations. All the stiffnesses involved in the stress-strain relation become complex and frequency dependent, which can exactly be expressed in terms of kernels based on the Zener mechanical model. In the time domain, this approach implies time convolutions, which are circumvented by introducing memory variables. The differential equations are consistent with Gassmann's and Mavko-Jizba equations at low and high frequencies, respectively. All the coefficients in the poro-viscoelastic differential equations have a clear physical meaning and can be obtained or estimated from independent measurements. The key additional parameters are the dry-rock bulk modulus at a confining pressure where all the compliant pores are closed, i.e., a hypothetical rock without the soft porosity, the grain-contact aspect ratio and the compliant porosity. We recasted the wave equation in. the particle-velocity/stress formulation and solved it by using a time-splitting technique and the Fourier pseudospectral method to compute the spatial derivatives. The algorithm can be used to obtain synthetic wave fields in inhomogeneous media. [ABSTRACT FROM AUTHOR]

To improve quantitative interpretation of seismic data, we analyze the effect of fluid substitution in a porous and fractured medium on elastic properties and reflection coefficients. This analysis uses closed-form expressions suitable for fluid substitution in transversely isotropic media with a horizontal symmetry axis (HTI). For the HTI medium, the effect of changing porosity and water saturation on (1) P-wave moduli, (2) horizontal and vertical velocities, (3) anisotropic parameters, and (4) reflection coefficients are examined. The effects of fracture density on these four parameters are also studied. For the model used in this study, a 35% increase in porosity lowers the value of P-wave moduli by maximum of 45%. Consistent with the reduction in P-wave moduli, P-wave velocities also decrease by maximum of 17% with a similar increment in porosity. The reduction is always larger for the horizontal P-wave modulus than for the vertical one and is nearly independent of fracture density. The magnitude of the anisotropic parameters of the fractured medium also changes with increased porosity depending on the changes in the value of P-wave moduli. The reflection coefficients at an interface of the fractured medium with an isotropic medium change in accordance with the above observations and lead to an increase in anisotropic amplitude variation with offset (AVO) gradient with porosity. Additionally, we observe a maximum increase in P-wave modulus and velocity by 30% and 8%, respectively, with a 100% increase in water saturation. Water saturation also changes the anisotropic parameters and reflection coefficients. Increase in water saturation considerably increases the magnitude of the anisotropic AVO gradient irrespective of fracture density. From this study, we conclude that porosity and water saturation have a significant impact on the four studied parameters and the impacts are seismically detectable. [ABSTRACT FROM AUTHOR]

A major cause of seismic attenuation in fluid-saturated rocks is the flow of the pore fluid induced by the passing wave. At sonic and ultrasonic frequencies, attenuation appears to be dominated by the local (pore-scale) flow between pores of different shapes and orientations. A simple squirt flow model is developed in which all of the parameters can be independently measured or estimated from measurements. The pore space of the rock is assumed to consist of stiff porosity and compliant (or soft) pores present at grain contacts. The effect of isotropically distributed compliant pores is modeled by considering pressure relaxation in a disk-shaped gap between adjacent grains. This derivation gives the complex and frequency-dependent effective bulk and shear moduli of a rock, in which the compliant pores are liquid saturated and stiff pores are dry. The resulting squirt model is consistent with Gassmann's and Mavko-Jizba equations at low and high frequencies, respectively. The magnitude of attenuation and dispersion given by the model is directly related to the variation of dry bulk modulus with pressure and is relatively independent of fluid properties. [ABSTRACT FROM AUTHOR]

The traditional method of fluid substitution in porous rock requires the total porosity and the elastic modulus of the mineral phase as input and assumes that the fluid reaches instantaneous hydraulic equilibrium throughout the pore space. This assumption may not be appropriate for shaley sediment because of the low permeability of shale and the resulting immobility of the water in it. To address this problem, we propose an alternative method that uses effective porosity instead of total porosity. Effective porosity is lower than total porosity if porous shale is present in the system. A new, composite mineral phase is introduced, which includes the porous water-saturated shale together with the nonporous minerals and whose elastic modulus is an average of those of its components, including the porous shale. This alternative method increases the sensitivity of the elastic properties of sediment-to-pore-fluid changes and therefore may be used as a physics-based theoretical tool to better explain and interpret seismic data during exploration as well as variations in seismic response as hydrocarbon production progresses. [ABSTRACT FROM AUTHOR]

We analyse compressional wave attenuation in fluid saturated porous material with porous inclusions having different compressibilities and very different spatial scales in comparison with the background. Such a medium exhibits significant attenuation due to wave-induced fluid flow across the interface between inclusion and background. For the representative element containing two layers (one of them representing inclusion), we show that overall wave attenuation is governed by the superposition of two coupled fluid-diffusion processes. Associated with two characteristic spatial scales, we compute two cross-over frequencies that separate three different frequency regimes. At low frequencies inverse quality factor scales with the first power of frequency ω, while at high frequencies the attenuation is proportional to ω−1/2. In the intermediate range of frequencies inverse quality factor scales with ω1/2. These characteristic frequency regimes can be observed in all theoretical models of wave-induced attenuation, but quantitative estimates of their locations have been lacking so far. The potential application of this model is in estimation of the background permeability as well as inclusion scale (thickness) by identifying these frequencies from attenuation measurements. [ABSTRACT FROM AUTHOR]

Fractures in a porous rock can be modelled as very thin and highly porous layers in a porous background. First, a dispersion equation for a P wave propagating in periodically layered poroelastic medium is obtained using propagator matrix approach applied to Biot equations of poroelasticity with periodic coefficients. Then in the limit of low stiffness and thickness this dispersion equation yields an expression for the effective P-wave modulus of the fractured porous material. When both pores and fractures are dry, this material is equivalent to a transversely isotropic elastic porous material with linear–slip interfaces. When saturated with a liquid this material exhibits significant attenuation and velocity dispersion due to wave-induced fluid flow between pores and fractures. In the low-frequency limit the material properties are equal to those obtained by anisotropic Gassmann (or Brown–Korringa) theory applied to a porous material with linear-slip interfaces. At low frequencies inverse quality factor scales with the first power of frequency ω. At high frequencies the effective elastic properties are equal to those for isolated fluid-filled fractures in a solid (non-porous) background, and inverse quality factor scales with . The magnitude of both attenuation and dispersion strongly depends on both the degree of fracturing and background porosity of the medium. The characteristic frequency of the attenuation and dispersion depends on the background permeability, fluid viscosity, as well as fracture density and spacing. [ABSTRACT FROM AUTHOR]

We design a velocity–porosity model for sand-shale environments with the emphasis on its application to petrophysical interpretation of compressional and shear velocities. In order to achieve this objective, we extend the velocity–porosity model proposed by Krief et al., to account for the effect of clay content in sandstones, using the published laboratory experiments on rocks and well log data in a wide range of porosities and clay contents. The model of Krief et al. works well for clean compacted rocks. It assumes that compressional and shear velocities in a porous fluid-saturated rock obey Gassmann formulae with the Biot compliance coefficient. In order to use this model for clay-rich rocks, we assume that the bulk and shear moduli of the grain material, and the dependence of the compliance on porosity, are functions of the clay content. Statistical analysis of published laboratory data shows that the moduli of the matrix grain material are best defined by low Hashin–Shtrikman bounds. The parameters of the model include the bulk and shear moduli of the sand and clay mineral components as well as coefficients which define the dependence of the bulk and shear compliance on porosity and clay content. The constants of the model are determined by a multivariate non-linear regression fit for P- and S-velocities as functions of porosity and clay content using the data acquired in the area of interest. In order to demonstrate the potential application of the proposed model to petrophysical interpretation, we design an inversion procedure, which allows us to estimate porosity, saturation and/or clay content from compressional and shear velocities. Testing of the model on laboratory data and a set of well logs from Carnarvon Basin, Australia, shows good agreement between predictions and measurements. This simple velocity-porosity-clay semi-empirical model could be used for more reliable petrophysical interpretation of compressional and shear... [ABSTRACT FROM AUTHOR]