Attenuation mechanisms in fractured fluid-saturated porous rocks: a numerical modelling study

Seismic attenuation mechanisms receive increasing attention for the characterization of fractured formations because of their inherent sensitivity to the hydraulic and elastic properties of the probed media. Attenuation has been successfully inferred from seismic data in the past, but linking these estimates to intrinsic rock physical properties remains challenging. A reason for these difficulties in fluid-saturated fractured porous media is that several mechanisms can cause attenuation and may interfere with each other. These mechanisms notably comprise pressure diffusion phenomena and dynamic effects, such as scattering, as well as Biot's so-called intrinsic attenuation mechanism. Understanding the interplay between these mechanisms is therefore an essential step for estimating fracture properties from seismic measurements. In order to do this, we perform a comparative study involving wave propagation modelling in a transmission set-up based on Biot's low-frequency dynamic equations and numerical upscaling based on Biot's consolidation equations. The former captures all aforementioned attenuation mechanisms and their interference, whereas the latter only accounts for pressure diffusion phenomena. A comparison of the results from both methods therefore allows to distinguish between dynamic and pressure diffusion phenomena and to shed light on their interference. To this end, we consider a range of canonical models with randomly distributed vertical and/or horizontal fractures. We observe that scattering attenuation strongly interferes with pressure diffusion phenomena, since the latter affect the elastic contrasts between fractures and their embedding background. Our results also demonstrate that it is essential to account for amplitude reductions due to transmission losses to allow for an adequate estimation of the intrinsic attenuation of fractured media. The effects of Biot's intrinsic mechanism are rather small for the models considered in this study.     

Measurements of the elastic and anelastic properties of sandstone flooded with supercritical CO2

The aim of this study was to investigate the effects of supercritical CO₂ (scCO₂) injection on the elastic and anelastic properties of sandstone at seismic and ultrasonic frequencies. We present the results of the low‐frequency and ultrasonic experiments conducted on water‐saturated sandstone (Donnybrook, Western Australia) flooded with scCO₂. The sandstone was cut in the direction perpendicular to a formation bedding plane and tested in a Hoek triaxial pressure cell. During the experiments with scCO_2, the low‐frequency and ultrasonic systems and the pump dispensing scCO₂ were held at a temperature of 42°C. The elastic parameters obtained for the sandstone with scCO₂ at seismic (0.1 Hz–100 Hz) and ultrasonic (～0.5 MHz) frequencies are very close to those for the dry rock. The extensional attenuation was also measured at seismic frequencies for the dry, water‐saturated, and scCO₂‐injected sandstones. The applicability of Gassmann's fluid substitution theory to obtained results was also tested during the experiments.     

Laboratory measurements of seismic attenuation and Young's modulus dispersion in a partially and fully water‐saturated porous sample made of sintered borosilicate glass

We measured the extensional‐mode attenuation and Young's modulus in a porous sample made of sintered borosilicate glass at microseismic to seismic frequencies (0.05–50 Hz) using the forced oscillation method. Partial saturation was achieved by water imbibition, varying the water saturation from an initial dry state up to ～99%, and by gas exsolution from an initially fully water‐saturated state down to ～99%. During forced oscillations of the sample effective stresses up to 10 MPa were applied. We observe frequency‐dependent attenuation, with a peak at 1–5 Hz, for ～99% water saturation achieved both by imbibition and by gas exsolution. The magnitude of this attenuation peak is consistently reduced with increasing fluid pressure and is largely insensitive to changes in effective stress. Similar observations have recently been attributed to wave‐induced gas exsolution–dissolution. At full water saturation, the left‐hand side of an attenuation curve, with a peak beyond the highest measured frequency, is observed at 3 MPa effective stress, while at 10 MPa effective stress the measured attenuation is negligible. This observation is consistent with wave‐induced fluid flow associated with mesoscopic compressibility contrasts in the sample's frame. These variations in compressibility could be due to fractures and/or compaction bands that formed between separate sets of forced‐oscillation experiments in response to the applied stresses. The agreement of the measured frequency‐dependent attenuation and Young's modulus with the Kramers–Kronig relations and additional data analyses indicate the good quality of the measurements. Our observations point to the complex interplay between structural and fluid heterogeneities on the measured seismic attenuation and they illustrate how these heterogeneities can facilitate the dominance of one attenuation mechanism over another.     

Forced oscillation measurements of seismic wave attenuation and stiffness moduli dispersion in glycerine-saturated Berea sandstone

Fluid pressure diffusion occurring on the microscopic scale is believed to be a significant source of intrinsic attenuation of mechanical waves propagating through fully saturated porous rocks. The so-called squirt flow arises from compressibility heterogeneities in the microstructure of the rocks. To study squirt flow experimentally at seismic frequencies the forced oscillation method is the most adequate, but such studies are still scarce. Here we present the results of forced hydrostatic and axial oscillation experiments on dry and glycerine-saturated Berea sandstone, from which we determine the dynamic stiffness moduli and attenuation at micro-seismic and seismic frequencies (0.004–30 Hz). We observe frequency-dependent attenuation and the associated moduli dispersion in response to the drained–undrained transition (∼0.1 Hz) and squirt flow (>3 Hz), which are in fairly good agreement with the results of the corresponding analytical solutions. The comparison with very similar experiments performed also on Berea sandstone in addition shows that squirt flow can potentially be a source of wave attenuation across a wide range of frequencies because of its sensitivity to small variations in the rock microstructure, especially in the aspect ratio of micro-cracks or grain contacts.     

Research note: Seismic attenuation due to wave‐induced fluid flow at microscopic and mesoscopic scales

Wave‐induced fluid flow at microscopic and mesoscopic scales arguably constitutes the major cause of intrinsic seismic attenuation throughout the exploration seismic and sonic frequency ranges. The quantitative analysis of these phenomena is, however, complicated by the fact that the governing physical processes may be dependent. The reason for this is that the presence of microscopic heterogeneities, such as micro‐cracks or broken grain contacts, causes the stiffness of the so‐called modified dry frame to be complex‐valued and frequency‐dependent, which in turn may affect the viscoelastic behaviour in response to fluid flow at mesoscopic scales. In this work, we propose a simple but effective procedure to estimate the seismic attenuation and velocity dispersion behaviour associated with wave‐induced fluid flow due to both microscopic and mesoscopic heterogeneities and discuss the results obtained for a range of pertinent scenarios.     

Effect of fracture fill on frequency‐dependent anisotropy of fractured porous rocks

In fractured reservoirs, seismic wave velocity and amplitude depend on frequency and incidence angle. Frequency dependence is believed to be principally caused by the wave‐induced flow of pore fluid at the mesoscopic scale. In recent years, two particular phenomena, i.e., patchy saturation and flow between fractures and pores, have been identified as significant mechanisms of wave‐induced flow. However, these two phenomena are studied separately. Recently, a unified model has been proposed for a porous rock with a set of aligned fractures, with pores and fractures filled with two different fluids. Existing models treat waves propagating perpendicular to the fractures. In this paper, we extend the model to all propagation angles by assuming that the flow direction is perpendicular to the layering plane and is independent of the loading direction. We first consider the limiting cases through poroelastic Backus averaging, and then we obtain the five complex and frequency‐dependent stiffness values of the equivalent transversely isotropic medium as a function of the frequency. The numerical results show that, when the bulk modulus of the fracture‐filling fluid is relatively large, the dispersion and attenuation of P‐waves are mainly caused by fractures, and the values decrease as angles increase, almost vanishing when the incidence angle is 90° (propagation parallel to the fracture plane). While the bulk modulus of fluid in fractures is much smaller than that of matrix pores, the attenuation due to the “partial saturation” mechanism makes the fluid flow from pores into fractures, which is almost independent of the incidence angle.     

Seismic time–frequency analysis as a robust method to estimate the fluid saturation: A case study of carbonates reservoir,Iran

The propagation of seismic waves through a saturated reservoir compresses the fluid in the pore spaces. During this transition, parts of seismic energy would be attenuated because of intrinsic absorption. Rock physics models make the bridge between the seismic properties and petrophysical reality in the earth. Attenuation is one of the significant seismic attributes used to describe the fluid behaviour in the reservoirs. We examined the core samples using ultrasonic experiments at the reservoir conditions. Given the rock properties of the carbonate reservoir and experiment results, the patchy saturation mechanism was solved for substituted fluid using the theory of modulus frequency. The extracted relationship between the seismic attenuation and water saturation was used in time–frequency analysis. We performed the peak frequency method to estimate the Q factor in the Gabor domain and determined the water saturation based on the computed rock physics model. The results showed how the probable fault in the reservoir has stopped the fluid movement in the reservoir and caused touching the water‐bearing zone through drilling.     

Effects of petrophysical parameters on attenuation and dispersion of seismic waves in the simplified poroelastic theory: Comparison between the Biot's theory and viscosity‐extended theory

The simplified macro‐equations of porous elastic media are presented based on Hickey's theory upon ignoring effects of thermomechanical coupling and fluctuations of porosity and density induced by passing waves. The macro‐equations with definite physical parameters predict two types of compressional waves (P wave) and two types of shear waves (S wave). The first types of P and S waves, similar to the fast P wave and S wave in Biot's theory, propagate with fast velocity and have relatively weak dispersion and attenuation, while the second types of waves behave as diffusive modes due to their distinct dispersion and strong attenuation. The second S wave resulting from the bulk and shear viscous loss within pore fluid is slower than the second P wave but with strong attenuation at lower frequencies. Based on the simplified porous elastic equations, the effects of petrophysical parameters (permeability, porosity, coupling density and fluid viscosity) on the velocity dispersion and attenuation of P and S waves are studied in brine‐saturated sandstone compared with the results of Biot's theory. The results show that the dispersion and attenuation of P waves in simplified theory are stronger than those of Biot's theory and appear at slightly lower frequencies because of the existence of bulk and shear viscous loss within pore fluid. The properties of the first S wave are almost consistent with the S wave in Biot's theory, while the second S wave not included in Biot's theory even dies off around its source due to its extremely strong attenuation. The permeability and porosity have an obvious impact on the velocity dispersion and attenuation of both P and S waves. Higher permeabilities make the peaks of attenuation shift towards lower frequencies. Higher porosities correspond to higher dispersion and attenuation. Moreover, the inertial coupling between fluid and solid induces weak velocity dispersion and attenuation of both P and S waves at higher frequencies, whereas the fluid viscosity dominates the dispersion and attenuation in a macroscopic porous medium. Besides, the heavy oil sand is used to investigate the influence of high viscous fluid on the dispersion and attenuation of both P and S waves. The dispersion and attenuation in heavy oil sand are stronger than those in brine‐saturated sandstone due to the considerable shear viscosity of heavy oil. Seismic properties are strongly influenced by the fluid viscosity; thus, viscosity should be included in fluid properties to explain solid–fluid combination behaviour properly.     

Micro X‐Ray computed tomography imaging and ultrasonic velocity measurements in tetrahydrofuran‐hydrate‐bearing sediments

Naturally occurring gas hydrates contain significant amounts of natural gas that might be produced as an energy resource in the foreseeable future. Thus, it is necessary to understand the pore‐space characteristics of hydrate reservoirs, particularly the pore‐scale distribution of the hydrate and its interaction with the sediment. Four end‐member models for hydrate distribution in the pore space are pore filling, sediment‐frame component, envelope cementing, and contact cementing. The goal of this study is to compare the models with pore‐scale hydrate distributions obtained in laboratory‐formed hydrates. Our results verify hydrate pore‐scale distributions by direct, visual observations that were previously implied by indirect, elastic property measurements. Laboratory measurements were conducted using tetrahydrofuran as a guest molecule since tetrahydrofuran hydrate can be used as a proxy for naturally occurring hydrates. We performed micro X‐ray computed tomography to obtain information about the distribution of hydrate in the pore space of synthetic sediment (glass beads). We also made ultrasonic velocity measurements on the same samples. Micro X‐ray computed tomography images and ultrasonic velocity measurements both indicate that the tetrahydrofuran hydrate forms in the pore space with a part of the hydrate bridging the grains without touching the grain surfaces. These hydrate‐bearing sediments appear to follow a pore‐filling model with a portion of the hydrate becoming a load‐bearing part of the sediment frame.     

Porosity measurement of heavy oil sands

Porosity measurement is an important step for petrophysical characterization of reservoir rocks, as porosity is essential for analysing elastic properties and estimating the capacity of rocks to hold hydrocarbon resources. Commonly used porosity measurement methods fail to work on heavy oil sands, because of the loose fragile sand frame and solid-like irreducible heavy oil. We develop three methods to measure the porosity of heavy oil sands, which are feasible and affordable in most laboratories. Method 1 involves calculating the bulk volume of the sample by measuring its physical dimensions directly. Method 2 uses calibrated sample weight and volume (after removing the protecting foil cover and metal clips). Method 3 first applies pressurizing on samples and then measures the weight and volume of the bare samples. The measurement results of the three methods are then compared and method 3 is determined as the most reliable, which is also verified by the porosity log. Finally, an analysis of the potential sources of errors associated with method 3 is performed.     

Forced imbibition into a limestone: measuring P‐wave velocity and water saturation dependence on injection rate

Quantitative interpretation of time‐lapse seismic data requires knowledge of the relationship between elastic wave velocities and fluid saturation. This relationship is not unique but depends on the spatial distribution of the fluid in the pore‐space of the rock. In turn, the fluid distribution depends on the injection rate. To study this dependency, forced imbibition experiments with variable injection rates have been performed on an air‐dry limestone sample. Water was injected into a cylindrical sample and was monitored by X‐Ray Computed Tomography and ultrasonic time‐of‐flight measurements across the sample. The measurements show that the P‐wave velocity decreases well before the saturation front approaches the ultrasonic raypath. This decrease is followed by an increase as the saturation front crosses the raypath. The observed patterns of the acoustic response and water saturation as functions of the injection rate are consistent with previous observations on sandstone. The results confirm that the injection rate has significant influence on fluid distribution and the corresponding acoustic response. The complexity of the acoustic response —‐ that is not monotonic with changes in saturation, and which at the same saturation varies between hydrostatic conditions and states of dynamic fluid flow – may have implications for the interpretation of time‐lapse seismic responses.     

Generalized anisotropy parameters and approximations of attenuations and velocities in viscoelastic media of arbitrary anisotropy type – theoretical and experimental aspects*

Seismic anisotropy in geological media is now widely accepted. Parametrizations and explicit approximations for the velocities in such media, considered as purely elastic and moderately anisotropic, are now standards and have even been extended to arbitrary types of anisotropy. In the case of attenuating media, some authors have also recently published different parametrizations and velocity and attenuation approximations in viscoelastic anisotropic media of particular symmetry type (e.g., transversely isotropic or orthorhombic). This paper extends such work to media of arbitrary anisotropy type, that is to say to triclinic media. In the case of homogeneous waves and using the so‐called ‘correspondence principle’, it is shown that the viscoelastic equations (for the phase velocities, phase slownesses, moduli, wavenumbers, etc.) are formally identical to the corresponding purely elastic equations available in the literature provided that all the corresponding quantities are complex (except the unit vector in the propagation direction that remains real). In contrast to previous work, the new parametrization uses complex anisotropy parameters and constitutes a simple extension to viscoelastic media of previous work dealing with non‐attenuating elastic media of arbitrary anisotropy type. We make the link between these new complex anisotropy parameters and measurable parameters, as well as with previously published anisotropy parameters, demonstrating the usefulness of the new parametrization. We compute the explicit complete directional dependence of the exact and of the approximate (first and higher‐order perturbation) complex phase velocities of the three body waves (qP, qS1 and qS2). The exact equations are successfully compared with the ultrasonic phase velocities and phase attenuations of the three body waves measured in a strongly attenuating water‐saturated sample of Vosges sandstone exhibiting moderate velocity anisotropy but very strong attenuation anisotropy. The approximate formulas are checked on experimental data. Compared to the exact solutions, the errors observed on the first‐order approximate velocities are small (<1%) for qP‐waves and moderate (<10%) for qS‐waves. The corresponding errors on the quality factor Q are moderate (<6%) for qP‐waves but critically large (up to 160%) for the qS‐waves. The use of higher‐order approximations substantially improves the accuracy, for instance typical maximum relative errors do not exceed 0.06% on all the velocities and 0.6% on all the quality factors Q, for third‐order approximations. All the results obtained on other rock samples confirm the results obtained on this rock. The simplicity of the derivations and the generality of the results are striking and particularly convenient for practical applications.     

Seismic attenuation and velocity dispersion in fractured rocks: The role played by fracture contact areas

The presence of fractures in fluid‐saturated porous rocks is usually associated with strong seismic P‐wave attenuation and velocity dispersion. This energy dissipation can be caused by oscillatory wave‐induced fluid pressure diffusion between the fractures and the host rock, an intrinsic attenuation mechanism generally referred to as wave‐induced fluid flow. Geological observations suggest that fracture surfaces are highly irregular at the millimetre and sub‐millimetre scale, which finds its expression in geometrical and mechanical complexities of the contact area between the fracture faces. It is well known that contact areas strongly affect the overall mechanical fracture properties. However, existing models for seismic attenuation and velocity dispersion in fractured rocks neglect this complexity. In this work, we explore the effects of fracture contact areas on seismic P‐wave attenuation and velocity dispersion using oscillatory relaxation simulations based on quasi‐static poroelastic equations. We verify that the geometrical and mechanical details of fracture contact areas have a strong impact on seismic signatures. In addition, our numerical approach allows us to quantify the vertical solid displacement jump across fractures, the key quantity in the linear slip theory. We find that the displacement jump is strongly affected by the geometrical details of the fracture contact area and, due to the oscillatory fluid pressure diffusion process, is complex‐valued and frequency‐dependent. By using laboratory measurements of stress‐induced changes in the fracture contact area, we relate seismic attenuation and dispersion to the effective stress. The corresponding results do indeed indicate that seismic attenuation and phase velocity may constitute useful attributes to constrain the effective stress. Alternatively, knowledge of the effective stress may help to identify the regions in which wave induced fluid flow is expected to be the dominant attenuation mechanism.     

Monitoring and volumetric estimation of injected CO2 using 4D seismic,petrophysical data,core measurements and well logging: a case study at Ketzin,Germany

More than 50 000 tons of CO₂ have been injected at Ketzin into the Stuttgart Formation, a saline aquifer, at approximately 620 m depth, as of summer 2011. We present here results from the 1^st repeat 3D seismic survey that was performed at the site in autumn 2009, after about 22 000 tons of CO₂ had been injected. We show here that rather complex time‐lapse signatures of this CO₂ can be clearly observed within a radius of about 300 m from the injection well. The highly irregular amplitude response within this radius is attributed to the heterogeneity of the injection reservoir. Time delays to a reflection below the injection level are also observed. Petrophysical measurements on core samples and geophysical logging of CO₂ saturation levels allow an estimate of the total amount of CO₂ visible in the seismic data to be made. These estimates are somewhat lower than the actual amount of CO₂ injected at the time of the survey and they are dependent upon the choice of a number of parameters. In spite of some uncertainty, the close agreement between the amount injected and the amount observed is encouraging for quantitative monitoring of a CO₂ storage site using seismic methods.     

Seismic dispersion and attenuation in layered porous rocks with fractures of varying orientations

Wave‐induced fluid flow plays an important role in affecting the seismic dispersion and attenuation of fractured porous rocks. While numerous theoretical models have been proposed for the seismic dispersion and attenuation in fractured porous rocks, most of them neglect the wave‐induced fluid flow resulting from the background anisotropy (e.g. the interlayer fluid flow between different layers) that can be normal in real reservoirs. Here, according to the theories of poroelasticity, we present an approach to study the frequency‐dependent seismic properties of more realistic and complicated rocks, i.e. horizontally and periodically layered porous rock with horizontal and randomly orienting fractures, respectively, distributed in one of the two periodical layers. The approach accounts for the dual effects of the wave‐induced fluid flow between the fractures and the background pores and between different layers (the interlayer fluid flow). Because C₃₃ (i.e., the modulus of the normally incident P‐wave) is directly related to the P‐wave velocity widely measured in the seismic exploration, and its comprehensive dispersion and attenuation are found to be most significant, we study mainly the effects of fracture properties and the stiffness contrast between the different layers on the seismic dispersion and attenuation of C₃₃. The results show that the increasing stiffness contrast enhances the interlayer fluid flow of the layered porous rocks with both horizontal and randomly orienting fractures and weakens the wave‐induced fluid flow between the fractures and the background pores, especially for the layered porous rock with horizontal fractures. The modelling results also demonstrate that for the considered rock construction, the increasing fracture density reduces the interlayer fluid flow while improves the dispersion and attenuation in the fracture‐relevant frequency band. Increasing fracture aspect ratio is found to reduce the dispersion and attenuation in the fracture‐relevant frequency band only, especially for the layered porous rock with horizontal fractures.     

Mechanical compaction and ultrasonic velocity of sands with different texture and mineralogical composition

This study presents the results of experimental compaction while measuring ultrasonic velocities of sands with different grain size, shape, sorting and mineralogy. Uniaxial mechanical compaction tests up to a maximum of 50 MPa effective stress were performed on 29 dry sand aggregates derived from eight different sands to measure the rock properties. A good agreement was found between the Gassmann saturated bulk moduli of dry and brine saturated tests of selected sands. Sand samples with poor sorting showed low initial porosity while sands with high grain angularity had high initial porosity. The sand compaction tests showed that at a given stress well‐sorted, coarse‐grained sands were more compressible and had higher velocities (Vp and Vs) than fine‐grained sands when the mineralogy was similar. This can be attributed to grain crushing, where coarser grains lead to high compressibility and large grain‐to‐grain contact areas result in high velocities. At medium to high stresses the angular coarse to medium grained sands (both sorted sands and un‐sorted whole sands) showed high compaction and velocities (Vp and Vs). The small grain‐to‐grain contact areas promote higher deformation at grain contacts, more crushing and increased porosity loss resulting in high velocities. Compaction and velocities (Vp and Vs) increased with decreasing sorting in sands. However, at the same porosity, the velocities in whole sands were slightly lower than in the well‐sorted sands indicating the presence of loose smaller grains in‐between the framework grains. Quartz‐poor sands (containing less than 55% quartz) showed higher velocities (Vp and Vs) compared to that of quartz‐rich sands. This could be the result of sintering and enlargement of grain contacts of ductile mineral grains in the quartz‐poor sands increasing the effective bulk and shear stiffness. Tests both from wet measurements and Gassmann brine substitution showed a decreasing Vp/Vs ratio with increasing effective stress. The quartz‐rich sands separated out towards the higher side of the Vp/Vs range. The Gassmann brine substituted Vp and Vs plotted against effective stress provide a measure of the expected velocity range to be found in these and similar sands during mechanical compaction. Deviations of actual well log data from experimental data may indicate uplift, the presence of hydrocarbon, overpressure and/or cementation. Data from this study may help to model velocity‐depth trends and to improve the characterization of reservoir sands from well log data in a low temperature (<80–100^o C) zone where compaction of sands is mostly mechanical.     

Research Note: Overpressure prediction from C‐wave seismic data

Recently, mode converted shear waves (C‐waves) have been shown to enable overpressure prediction in media where primary wave acquisition is inhibited by gas and fluid effects – C‐wave moveout is analysed and a long standing relationship between differential stress and primary‐wave (P‐wave) velocity is modified and employed. Though pore‐pressure prediction based on C‐waves is supported by empirical evidence from laboratory and field experiments, a theoretical justification has yet to be developed. In this research note, we provide a supporting algebra for the original relationship between pore pressure and C‐wave velocity.     

Properties of low‐frequency trapped mode in viscous‐fluid waveguides

We derived the velocity and attenuation of a generalized Stoneley wave being a symmetric trapped mode of a layer filled with a Newtonian fluid and embedded into either a poroelastic or a purely elastic rock. The dispersion relation corresponding to a linearized Navier–Stokes equation in a fracture coupling to either Biot or elasticity equations in the rock via proper boundary conditions was rigorously derived. A cubic equation for wavenumber was found that provides a rather precise analytical approximation of the full dispersion relation, in the frequency range of 10^?3 Hz to 10³ Hz and for layer width of less than 10 cm and fluid viscosity below 0.1 Pa· s [100 cP]. We compared our results to earlier results addressing viscous fluid in either porous rocks with a rigid matrix or in a purely elastic rock, and our formulae are found to better match the numerical solution, especially regarding attenuation. The computed attenuation was used to demonstrate detectability of fracture tip reflections at wellbore, for a range of fracture lengths and apertures, pulse frequencies, and fluid viscosity.     

A semi-empirical approach to model pressure dependence of elastic moduli in granular media accounting for variations of coordination-number and Poisson-ratio

The effective medium theory based on the Hertz–Mindlin contact law is the most popular theory to relate dynamic elastic moduli (or elastic velocities) and confining pressure in dry granular media. However, many experimental results proved that the effective medium theory predicts pressure trends lower than experimental ones and over-predicts the shear modulus. To mitigate these mispredictions, several evolutions of the effective medium theory have been presented in the literature. Among these, the model named modified grain contact theory is an empirical approach in which three parametric curves are included in the effective medium theory model. Fitting the parameters of these curves permits to adjust the pressure trends of the Poisson ratio and the bulk modulus. In this paper, we present two variations of the modified grain contact theory model. First, we propose a minor modification in the fitting function for the porosity dependence of the calibration parameters that accounts for non-linearity in the vicinity of the critical porosity. Second, we propose a major modification that reduces the three-step modified grain contact theory model to a two-step model, by skipping the calibration parameter–porosity fit in the model and directly modelling the calibration parameter–pressure relation. In addition to an increased simplicity (the fitting parameters are reduced from 10 to 6), avoiding the porosity fit permits us to apply the model to laboratory data that are not provided with accurate porosity measurements. For this second model, we also estimate the uncertainty of the fitting parameters and the elastic velocities. We tested this model on dry core measurements from literature and we verified that it returns elastic velocity trends as good as the original modified grain contact theory model with a reduced number of fitting parameters. Possible developments of the new model to add predictive power are also discussed.