Compliant Pores Research Articles

Carbonate rock physics model using different approaches to estimate rock frame stiffness

Deepwater carbonate reservoirs in the Brazilian pre-salt have emerged as significant hydrocarbon plays in the region and globally. Seismic monitoring of these reservoirs is crucial to provide information for well placement assessments, to reduce uncertainty in reservoir models, and to enhance model-based decision making. To integrate seismic data quantitatively, a petroelastic model (PEM) is required to estimate elastic properties. The modeling of the rock frame stiffness is an essential part in the study of PEM. However, this becomes more complex for carbonate rocks given their diverse pore-types. One way for estimating stiffness is to model each distinct pore-type and include them into the overall rock frame (different inclusion models). In this research, an alternative approach is explored to estimate the rock frame stiffness. A proxy model is employed, and its coefficients are subsequently calibrated with well-log information. Two PEMs were developed using different approaches for rock frame stiffness modeling. The first was inspired by inclusion models and incorporated two pore types (compliant and stiff pores) for the modeling process. The second considered a mathematical regression model as a proxy to relate reservoir porosity to the rock frame stiffness. These two PEMs were tested on the Brazilian pre-salt carbonate rocks of the Barra Velha (BVE) formation using well-log data from three wells and core sample measurements for one well. The accuracy of the PEMs’ performances was assessed by measures (such as, mean absolute error and root mean square error) and visual comparisons of their predictions. The results showed that both PEMs predicted velocities (compressional and shear) within an acceptable match with the actual well-log data. Similarly, when tested on the core samples, the PEMs produced comparable results, which indicates the validity of the proxy, not only at the well-log scale but at the core scale. The visual comparison and the accuracy analyses of the results for all three wells confirmed that the two PEMs had comparable predictions. This is significant given that the proxy simplifies the modeling process and facilitates the representation of various pore-types in the prediction of elastic properties in carbonate rocks. Overall, the proxy for the rock frame stiffness modeling is a computationally less expensive model compared to the inclusion models which involve modeling of various pore-types. It also offers a straightforward model which enables the quantitative integration of seismic data in a multidisciplinary team of geosciences and petroleum engineers (for instance, early engagement of petroleum engineers in 3D/4D seismic history matching).

Experimental investigation of pore-filling substitution effect on frequency-dependent elastic moduli of Berea sandstone

SUMMARY Based on both forced oscillation and ultrasonic pulse transmission methods, we investigated solid pore infill influences on rock elastic moduli in a broad frequency range $[ {1 - 3000,\,\,{{10}^6}} ]$ Hz for different differential pressures. For a Berea sandstone sample, filled sequentially by solid (${22\,\,^{\rm{o}}}{\rm{C}}$), quasi-solid (${26\,\,^{\rm{o}}}{\rm{C}}$) and liquid (${34\,\,^{\rm{o}}}{\rm{C}}$) octadecane, a frequency-dependence was found for the Poisson's ratio, Young's modulus and bulk modulus, nevertheless, these elastic parameters were strongly suppressed by increasing pressures. Experimental measurements showed that shear wave velocity and modulus of solid-octadecane-filled samples are significantly larger than those of the dry and liquid-octadecane-filled ones, implying the potential stiffening effects related to solid infill in compliant pores. A three porosity structure model, which describes the solid stiffening effects related to equant, compliant and the intermediate pores with aspect ratios larger than those of compliant pores but much less than those of stiff pores, was used to compare against the experimentally measured elastic properties for octadecane pore infill, together with several other fluid/solid substitution theories. The agreement between experimental measurements and theoretical predictions is reasonably good for the sandstone tested, providing that the three porosity model can be applied for pressure- and frequency-dependent elastic moduli estimations for a viscoelastic pore-infill-saturated sandstone. Evaluating the combined squirt flow mechanism responsible for the observed moduli dispersion and attenuation is of great importance to reduce potential errors in seismic AVO inversion and 4-D seismic monitoring of gas-hydrate or bitumen-saturated reservoir, especially for reservoir rocks with complex microstructures and heterogeneous pore types.

Comparison of elastic properties of tectonic coals along the face and butt cleat directions

AbstractCoalbed methane is a hot spot for gas exploration at present, and fractures are widely developed in the coals. However, despite being essential for a number of geophysical applications such as reservoir prediction and hydraulic fracturing, the influence of fractures on the elastic properties and anisotropy of coals is still poorly understood. Therefore, three groups of cylindrical coals were drilled along the face and butt cleat directions to study the effects of pressure and fracture on the elastic properties and anisotropy of tectonic coals. The velocity along the face cleat direction is less sensitive to the pressure than that along the butt cleat direction due to the difference in compliant pore and fracture content. Coals display a strong directional dependence in petrophysical and elastic properties. The permeabilities and velocities of samples along the butt cleat direction are much lower than those along the face cleat direction. The fracture densities are quantitatively characterized by the theoretical model. The values along the face cleat direction are much smaller than those along the butt cleat direction. The influence of fractures on the elastic properties and anisotropy is comprehensively studied by theoretical and laboratory methods. The mineralogy and fracture effects are important factors to the elastic anisotropy of coals, the latter can be a dominant factor to the coal elastic anisotropy. The results can contribute to the understanding of the cleat system of coalbed methane reservoir and can provide a critical rock physics basis for micro‐seismic fracture monitoring and reservoir prediction.

Laboratory experiments and theoretical study of pressure and fluid influences on acoustic response in tight rocks with pore microstructure

AbstractWave‐induced fluid flow is considered to be a major source of seismic attenuation and dispersion in porous rocks. From the physical description of partially saturated reservoirs, numerous analytical solutions based on upscaling homogenization theories have been employed to calculate equivalent frequency‐dependent poroelastic media. Nevertheless, dispersion and attenuation predictions are often not reasonably consistent with laboratory and field measurements in a broad frequency range, particularly due to influences of biphasic fluids and their distribution, presence of heterogeneities on various length scales, and pore microstructure. We investigate the role of pore microstructure on pressure and fluid saturation dependence of elastic velocities in tight sandstones. Previous work points out that differentiating the impacts of heterogeneities at various scales on dispersion within seismic exploration and sonic frequencies can be very difficult. In practice, this is because fluid‐related dispersion mechanisms are impossible to be independent. Thus, it is important for a theoretical and more quantitative analysis of the relative contribution of interrelated energy dissipation processes through a better understanding of combined influences due to the presence of microscopic and mesoscopic heterogeneities. Based on microscopic squirt flow and mesoscopic flow in a partially saturated medium, we develop a poroelastic model that allows evaluating the overall frequency‐dependent dispersion via considering a random distribution of fluid heterogeneities as well as the broadly distributed aspect ratio of compliant pores. Experimental validation of the model is accomplished via a comprehensive comparison of predictions with measurements of partially saturated velocities versus pressure and fluid for sandstones with specific pore microstructures.

A seismic-reflection-based approach for determining the hydraulic permeability of rocks in a subsurface region

Abstract. The hydraulic permeability of rocks characterizes fluid mobility in the pores of underground reservoirs. Accurate information on this quantity is important for various geotechnology applications with urgent social, environmental and political significance (e.g., the exploration, optimization and monitoring of underground disposals of gases, fluids and radioactive wastes; safe underground engineering operations; and the exploration and development of geothermal and hydrocarbon energy). Because of the high resolution and large penetration range of seismic waves as well as the ability of seismic exploration to operate from the Earth's surface, a seismic-reflection-based method for characterizing underground permeability is very important. The elastic properties of rocks of a given lithology are strongly influenced by the compliant pore space. However, the hydraulic permeability of such rocks can be controlled by compliant pores, stiff pores or both types of pores. Using stress-dependent rock-physics experiments, we derive models of elastic properties (e.g., seismic velocities) as functions of compliant and stiff pore-space geometries. Simultaneously analyzing the stress-dependent behavior of the porosity and of the permeability, we identify which part of the pore space controls the permeability – compliant and/or stiff pore space – and derive a corresponding model. These models are used to directly characterize stiff and compliant pore spaces from seismic velocities and to calculate the underground permeability.

Origins of pressure dependent permeability in unconventional hydrocarbon reservoirs

Unconventional hydrocarbon assets represent a rapidly expanding proportion of North American oil and gas production. Similar to the incipient phase of conventional oil production at the turn of the twentieth century, there are ample opportunities to improve production efficiency. In this work we demonstrate that pressure dependent permeability degradation exhibited by unconventional reservoir materials is due to the mechanical response of a few commonly encountered microstructural constituents. In particular, the mechanical response of unconventional reservoir materials may be conceptualized as the superposed deformation of matrix (or ~ cylindrical/spherical), and compliant (or slit) pores. The former are representative of pores in a granular medium or a cemented sandstone, while the latter represent pores in an aligned clay compact or a microcrack. As a result of this simplicity, we demonstrate that permeability degradation is accounted for through a weighted superposition of conventional permeability models for these pore architectures. This approach permits us to conclude that the most severe pressure dependence is due to imperceptible bedding parallel delamination cracks in the oil bearing argillaceous (clay-rich) mudstones. Finally, we demonstrate that these delaminations tend to populate layers that are enriched with organic carbon. These findings are a basis for improving recovery factors through the development of new completion techniques to exploit, then mitigate pressure dependent permeability in practice.

Depth-variant pore type modeling in a pre-salt carbonate field offshore Brazil

Brazilian pre-salt carbonates encompass more than 70% of the total oil and gas produced in the country nowadays and yet, present several challenges such as heterogeneous composition in mineralogy with the presence of Mg-clays, a complex pore network and diagenetic processes, e.g., dolomitization, silicification and cementation. Rock physics provides a powerful route to understand the elastic behavior of rocks by connecting geology and geophysics. It is known that pore type determination is crucial to understand this behavior. In this paper, we propose a workflow that integrates several methods to obtain depth-variant distribution of pore types and their respective volumes for seven wells in the carbonate-bearing Barra Velha Formation interval. We compared the modeling results with thin sections, performed sensitivity analysis with several parameters (e.g., mineral content, saturation, different aspect ratios) to verify the impact of each one and, analyzed results with hydraulic flow units to search for favorable porosity-permeability scenarios and their relationship with the different pore types. Results suggest that the key parameters impacting the elastic behavior are mineralogy and pore types. Also, compliant pore type may act as connectors between pores with bigger storage capacity, such as reference and stiff pore types. The proposed workflow contributes to understand these complex carbonates, by providing a feasible path to obtain three pore type distributions for each depth point. Calibration with fluid data and especially mineralogy, is essential for the predictions to be as trustworthy as possible and should be applied for each well with available information.

Effects of different pore shapes on the pressure and frequency dependence of velocities of oil-saturated tight sandstone

The pressure dependence of elastic parameters of rocks is mainly controlled by the geometry of the pore space. In general, the compliant-stiff pore structure model can be used to reasonably describe this pressure dependence. However, our experiment measurements revealed that for tight sandstone rock with complex pore structures, the contribution of the compressibility of the stiff pores to the elastic modulus is significant. The dual porosity is not sufficient to explain the variation of ultrasonic velocity with pressure. For this reason, we adopted a triple pore structure to divide the rock pore space into equant pores, intermediate pores and compliant pores. Our laboratory measurement and model results show that this pore space division can better describe the pressure dependence of the elastic moduli of rocks. The low-frequency stress–strain measurements show that the fluid-saturated tight sandstone has obvious dispersion in the seismic frequency band, which is primarily attributed to the squirt flow effect. In order to study the pressure and frequency dependence of the elastic moduli of tight sandstone, we retrieved the geometric parameters of the pore structure from the pressure variation of the ultrasonic velocities under dry conditions. Based on this complex pore structure and the extension of the squirt flow model, we constructed an elaborate rock physics model to explain the pressure and frequency dependence of velocity. The model does not require adjustable parameters, and all parameters are measured and calculated by the laboratory, which improves the accuracy of theoretical modelling. The modified squirt flow model can be used to describe dispersion and attenuation in a wide frequency band, and fit well with the velocity measurements in both the low-frequency range and the ultrasonic frequency range under different pressures. Therefore, this rock physics model could be applied in the extraction of pore microstructure and fluid properties provided elastic moduli or velocities can be estimated accurately.

Geomechanical Characterization of the Wufeng and Longmaxi Shales: Mechanical and Velocity Anisotropy, Static and Dynamic Moduli

Summary The Wufeng and Longmaxi shales of Sichuan Basin, Southwest China have been the primary targets for shale gas development. Because hydraulic fracturing and seismic interpretation require detailed characterization of formation mechanical properties, a sufficient understanding of anisotropy and elastic behavior in Wufeng and Longmaxi shales is necessary. In this study, we conducted Brazilian and triaxial tests and ultrasonic velocity measurements to characterize tensile and compressive strengths and P- and S-wave velocities, respectively. Shale samples were cored at a range of orientations relative to bedding and tested at multiple confining pressures, which allowed a detailed study of mechanical and velocity anisotropy, static and dynamic moduli. Our experimental work shows that Wufeng and Longmaxi shales possess similar compressive strength and associated anisotropy with other shale formations but apparently weaker tensile strength anisotropy and velocity anisotropy. These two shales also exhibit much lower static moduli than dynamic values, which are interpreted to be caused by compliant pores such as microcracks and fractures. Comparison between Wufeng and Longmaxi shales reveals distinct levels of heterogeneity. Wufeng shale shows more pronounced heterogeneity regarding measured tensile and compressive strengths as well as elastic moduli. These general characteristics of Wufeng and Longmaxi shales provide valuable first-order understanding regarding anisotropy, heterogeneity, and elastic behavior. Utilizing this understanding could help improve hydraulic fracture design and seismic data interpretation.

Understanding carbonates from Saudi Arabia: Laboratory measurements of elastic properties. Part I

Laboratory measurements of elastic properties are presented for carbonate samples from the Saudi Arabian Arab formation. Compressional and shear wave velocities (at 1 MHz) and the quasi-static strains of 37 carbonate rock samples were measured as functions of the saturating fluid and the confining pressure. Furthermore, P- and S-wave velocities of the saturated samples were measured at constant differential pressure of 15 MPa. The lithology, mineralogy, porosity, pore type and pore size distribution of each sample were obtained using a combination of thin section and scanning electron microscopy, helium porosimetry and mercury intrusion porosimetry. The samples showed a high sensitivity to the applied differential pressure especially for the high porosity samples. Due to the lack of closing microcracks and compliant pores in low porosity samples, the travel times show slight changes with the confining pressure whereas the high porosity samples showed a remarkable reduction of the travel time with an increase in confining pressure for both P- and S-waves. We found that the sample physically deformed at pressure above 25 MPa. Evidence of inelastic deformation were observed in a few samples even at a differential pressure of 25 MPa. Both P- and S-wave velocities show a visible deviation with respect to the conventional velocity-porosity inverse relationships, with large variabilities in the velocities at a given porosity. The tight samples only measured a 3% average increase in velocity over the pressure range, whereas the porous samples measured a 20% average increase in velocity over the same pressure range. Despite the nature of porosity, the measured velocities showed a general increase with pressure. Also, all measured samples showed an increase in the bulk modulus of the rock after water saturation with 60% and less than 1% increases. There was no trivial correlation between porosity and the magnitude of the bulk modulus increase. Indeed, the shear modulus of the rock should be insensitive to any changes in water saturation, however, we observed some changes. The presence of anhydrite reduced both P- and S-wave velocities and had a significant influence on the velocities of carbonate rocks at any given porosity. A second paper will build on the results and will consist of predicting and validating suitable rock physics models using the measured carbonate data. The data that support findings of this study are openly available in ERA: Education and Research Archive University of Alberta at https://doi.org/10.7939/R3091B

Role of pressure and pore microstructure on seismic attenuation and dispersion of fluid-saturated rocks: laboratory experiments and theoretical modelling

SUMMARY Understanding the effects of pressure and rock microstructure on seismic elastic properties of fully saturated rocks is of considerable importance in a range of geophysical applications, especially at seismic frequency range. A recently proposed theoretical model of squirt attenuation and dispersion can be used to interpret the stress and frequency dependence of elastic properties on the basis of a triple porosity structure. The poroelastic model requires the knowledge of a variety of pore microstructure parameters, in particular, the compliant pores with a discrete distribution of aspect ratio. We performed laboratory measurements of (compressional and shear wave) velocity dispersion and attenuation, associated with the pressure-related closure of compliant pores on three dry and wet sandstones, to verify the effects of squirt flow arising from compressibility heterogeneities in the rock microstructure on the pressure dependence of dynamic elastic moduli and attenuation. Ultrasonic velocities experimentally measured on dry rocks were applied to extract pressure-dependent pore aspect distribution of compliant pores and the effective porosity of three types of pores with distinct aspect ratios, via fitting of the poroelastic model to the pressure dependence of elastic compressibilities. Under the assumption of frequency-independent dry elastic properties, inferred velocities and the associated attenuation of the saturated rocks from the forced oscillation experiments, which are still scarcely investigated, are in fairly good agreement with the predictions of the squirt model of three porosity types at seismic frequencies. The Gassmann's relation was found, nevertheless, underpredicts the ultrasonic saturated velocity measurements. The results validate applicability of the recently developed squirt model to account for dispersion and attenuation of phase velocities at varying effective pressures.

An accurate analytical model for squirt flow in anisotropic porous rocks — Part 1: Classical geometry

Seismic wave propagation in porous rocks that are saturated with a liquid exhibits significant dispersion and attenuation due to fluid flow at the pore scale, so-called squirt flow. This phenomenon takes place in compliant flat pores such as microcracks and grain contacts that are connected to stiffer isometric pores. Accurate quantitative description is crucial for inverting rock and fluid properties from seismic attributes such as attenuation. Up to now, many analytical models for squirt flow were proposed based on simplified geometries of the pore space. These models were either not compared with a numerical solution or showed poor accuracy. We have developed a new analytical model for squirt flow, which is validated against a 3D numerical solution for a simple pore geometry that has been classically used to explain squirt flow; that is why we refer to it as classical geometry. The pore space is represented by a flat cylindrical (penny-shaped) pore whose curved edge is fully connected to a toroidal (stiff) pore. Compared with correct numerical solutions, our analytical model provides very accurate predictions for the attenuation and dispersion across the whole frequency range. This includes correct low- and high-frequency limits of the stiffness modulus, the characteristic frequency, and the shape of the dispersion and attenuation curves. In a companion paper (part 2), we extend our analytical model to more complex pore geometries. We provide as supplemental information MATLAB and symbolic Maple routines to reproduce our main results.

High‐Precision Tracking of Sandstone Deformation From Micro‐CT Images

AbstractElastic properties of sandstones can be highly pressure dependent. However, the pressure‐induced deformation of intact sandstones in X‐ray micro‐Computed Tomography (micro‐CT) images are below the resolving power of conventional image analysis methods. Also, most digital rock physics studies of the elastic properties operate with micro‐CT images acquired at ambient pressure. Here, we perform a comprehensive analysis of an intact sandstone sample with pronounced stress‐sensitivity. A purpose‐built X‐ray‐transparent pressure cell enables scanning micro‐CT images at confining pressures of up to 36 MPa. To detect the small pressure‐induced changes from these images, we design an overall deformation detection workflow with slice‐searching technique. We apply this workflow to a set of micro‐CT images of a 2 × 2 × 2 mm sample scanned at several pressures with a voxel size length of ∼1 μm. The results give the static modulus of 2.78 GPa, which is ∼0.5 times of the dynamic modulus derived from ultrasonic measurements. This ratio is consistent with literature data, showing the accuracy of the deformation detection workflow. Images scanned at different pressures are then segmented into two‐phase labels. The porosity change detected from segmented labels is consistent with the value derived from static modulus using poroelastic theory. The two‐phase segmented images are utilized to simulate the elastic properties with a finite element method. Most of the pressure dependency of elastic moduli shown in ultrasonic measurements is caused by closure of compliant pores at grain contacts, which cannot be resolved by the micro‐CT. The accuracy improvement of the estimated elastic properties from images scanned at higher pressures is negligible.

Seismic Velocity Response to Atmospheric Pressure Using Time-Lapse Passive Seismic Interferometry

ABSTRACT Seismic velocities in the shallow crust down to a few kilometers depth show a remarkable sensitivity to stress perturbations due to the presence of compliant pores, cracks, fractures, and faults. Monitoring temporal changes of seismic velocities can thus provide key insights on dynamic processes affecting the shallow crust such as those related to the atmosphere (rainfall, barometric pressure, and temperature) and those with deeper tectonic and volcanic origins. In this work, we investigate the specific response of the near surface down to 300 m depth to atmospheric pressure variations. We conduct a four month passive seismic monitoring experiment in the desert of Oman using continuous noise recorded at geophones located within five wells. The results show a clear, direct correlation between seismic velocities and barometric pressure variations for monthly transients. At a longer, seasonal temporal scale, seismic velocities are stable, whereas atmospheric pressure shows a clear positive trend. We use the undrained coupled poroelastic theory to model these observations and find that the lack of seasonal velocity changes can be partly explained by the atmospheric pressure that diffuses into the pores with a strong hydraulic diffusivity likely higher than 100 m2/s consistent with the local geology referring to carbonates. Finally, the comparison between the modeled and observed velocity changes leads to estimate a velocity–stress sensitivity on the order of 6.3×10−7 Pa−1 which is consistent with previous studies. Using this result for calibration, we find that a sudden step-change drop of velocity of 0.015% occurring in the beginning of October 2019 and corresponding to a stress perturbation likely larger than 240 Pa affected the entire studied area. This small change could be related to a perturbation at greater depth associated with variations in the production rates within the underlying reservoir.

Effective medium modeling of pressure effects on the joint elastic and electrical properties of sandstones

Subsurface rocks are subjected to different stress states, and as such, physical properties of rocks including elastic modulus and electrical resistivity are stress-dependent. Joint modeling of the stress-dependence of elastic and electrical properties is important for interpreting coincident elastic and electrical measurements. We present a modeling approach for the pressure-dependence of the joint elastic and electrical properties in brine-saturated sandstones, that links both properties to the same microstructure. This approach combines the differential effective medium (DEM) and the self-consistent approximation (SCA) with the dual-porosity concept to model the pressure-dependence of P-wave velocity and electrical resistivity simultaneously measured in four sandstone samples. There is good agreement between model results and laboratory experimental results for both the elastic and electrical properties. The model parameters optimized to fit the data are: constant stiff and “soft” (or compliant) aspect ratios; a fixed critical porosity; and pressure-dependent stiff and soft porosities. These model-derived pressure-dependent porosities correlate with the observed velocity and resistivity pressure-sensitivity of the rocks, for example, the higher the initial volume fraction of compliant pores, the higher the P-wave velocity pressure sensitivity. The results show that the pressure-dependence of the P-wave velocity and electrical resistivity can be jointly modeled satisfactorily using the dual-porosity concept, where the pore space is simplified to contain a single set of stiff and soft pores.

Poroacoustoelasticity for rocks with a dual-pore structure

Acoustoelasticity describes the interaction of acoustic waves with nonlinear elastic deformations, particularly the change of wave velocity due to initial stresses or strains in a predeformed body. The theory extends the strain energy to cubic terms (third-order elasticity) and allows for finite strains to model deformations at high confining pressures. However, the theory considers equant (stiff) pores but neglects the effects of soft (compliant) pores, such as microfractures, cracks, and grain contacts. Our main contribution is to include these effects. Application of the novel poroacoustoelasticity theory to ultrasonic measurements on carbonate samples at varying confining pressures provides a better fit for the measured data of pressure dependence of wave velocity. We have quantified the contribution of the compliant pores to the nonlinear behavior of the wave velocity and determined the relation between the threshold pressure (beyond which the theories with and without compliant pores yield the same velocity) and porosity and permeability. The extension of poroacoustoelasticity theory by incorporating a dual-pore structure provides better description for stress dependence of wave velocity in fluid-saturated heterogeneous rocks, which can be applicable in further field studies regarding reservoir characterization and in situ stress estimation.

Elastic wave propagation and scattering in prestressed porous rocks

Poro-acoustoelastic theory has made a great progress in both theoretical and experimental aspects, but with no publications on the joint research from theoretical analyses, experimental measurements, and numerical validations. Several key issues challenge the joint research with comparisons of experimental and numerical results, such as digital imaging of heterogeneous poroelastic properties, estimation of acoustoelastic constants, numerical dispersion at high frequencies and strong heterogeneities, elastic nonlinearity due to compliant pores, and contamination by boundary reflections. Conventional poro-acoustoelastic theory, valid for the linear elastic deformation of rock grains and stiff pores, is modified by incorporating a dual-porosity model to account for elastic nonlinearity due to compliant pores subject to high-magnitude loading stresses. A modified finite-element method is employed to simulate the subtle effect of microstructures on wave propagation in prestressed digital cores. We measure the heterogeneity of samples by extracting the autocorrelation length of digital cores for a rough estimation of scattering intensity. We conductexperimental measurements with a fluid-saturated sandstone sample under a constant confining pressure of 65 MPa and increasing pore pressures from 5 to 60 MPa. Numerical simulations for ultrasound propagation in the prestressed fluid-saturated digital core of the sample are followed based on the proposed poro-acoustoelastic model with compliant pores. The results demonstrate a general agreement between experimental and numerical waveforms for different stresses, validating the performance of the presented modeling scheme. The excellent agreement between experimental and numerical coda quality factors demonstrates the applicability for the numerical investigation of the stress-associated scattering attenuation in prestressed porous rocks.

A dynamic elastic model for squirt-flow effect and its application on fluid-viscosity-associated velocity dispersion in reservoir sandstones

Velocity dispersion is a common phenomenon for fluid-charged porous rocks and carries important information on the pore structure and fluid in reservoir rocks. Previous ultrasonic experiments had measured more significant non-Biot velocity dispersion on saturated reservoir sandstones with increasing pore-fluid viscosity. Although wave-induced local squirt-flow effect could in theory cause most of the non-Biot velocity dispersion, its quantitative prediction remains a challenge. Several popular models were tested to predict the measured velocities under undrained conditions, but they either underestimated the squirt-flow effect or failed to simultaneously satisfy P- and S-wave velocity dispersions (especially for higher viscosity fluids). Based on the classic double-porosity theory that pore space is comprised of mainly stiff/Biot’s porosity and minor compliant porosity, an effective “wet frame” was hypothesized to account for the squirt-flow effect, whose compliant pores are filled with a hypothesized fluid with dynamic modulus. A new dynamic elastic model was then introduced by extending Biot theory to include the squirt-flow effect, after replacing the dry-frame bulk/shear moduli with their wet-frame counterparts. In addition to yielding better velocity predictions for P- and S-wave measurements of different fluid viscosities, the new model is also more applicable because its two key tuning parameters (i.e., the effective aspect ratio and porosity of compliant pores) at in situ reservoir pressure could be constrained with laboratory velocity measurements associated with pore-fluid viscosities.

Frequency-dependent attenuation and dispersion caused by squirt flow: Three-dimensional numerical study

Seismic waves may exhibit significant dispersion and attenuation in reservoir rocks due to pore-scale fluid flow. Fluid flow at the microscopic scale is referred to as squirt flow and occurs in very compliant pores, such as grain contacts or microcracks, that are connected to other stiffer pores. We have performed 3D numerical simulations of squirt flow using a finite-element approach. Our 3D numerical models consist of a pore space embedded into a solid grain material. The pore space is represented by a flat cylinder (a compliant crack) whose edge is connected with a torus (a stiff pore). Grains are described as a linear isotropic elastic material, whereas the fluid phase is described by the quasistatic linearized compressible Navier-Stokes momentum equation. We obtain the frequency-dependent effective stiffness of a porous medium and calculate dispersion and attenuation due to fluid flow from a compliant crack to a stiff pore. We compare our numerical results against a published analytical solution for squirt flow and analyze the effects of its assumptions. Previous interpretation of the squirt flow phenomenon based mainly on analytical solutions is verified, and some new physical effects are identified. The numerical and analytical solutions agree only for the simplest model in which the edge of the crack is subjected to zero fluid pressure boundary condition while the stiff pore is absent. For the more realistic model that includes the stiff pore, significant discrepancies are observed. We identify two important aspects that need improvement in the analytical solution: the calculation of the frame stiffness moduli and the frequency dependence of attenuation and dispersion at intermediate frequencies.

Pressure-dependent cross-property relationships between elastic and electrical properties of partially saturated porous sandstones

Successful joint inversion of seismic and electromagnetic survey data to estimate parameters and improve the characterization of the earth’s subsurface relies on the understanding of the cross-property relations between the elastic and electrical properties. Pressure-dependent cross-property relationships of rocks with varying degrees of fluid saturation remain poorly understood even though such conditions are encountered frequently. We investigate this problem by designing and undertaking dedicated laboratory experiments to simultaneously measure the elastic velocity and electrical resistivity in two Berea sandstone samples subject to varying confining pressure and water saturation, using a state-of-the-art joint elastic-electrical measurement system. P- and S-wave velocities are found to increase exponentially with increasing confining pressure at each water saturation, whereas electrical conductivity can either increase or decrease with confining pressure depending on the degree of water saturation. More interestingly, the elastic and electrical properties of the two samples are shown to be systematically correlated as functions of water saturation and the applied confining pressure. The correlations reveal the natural link between the elastic and electrical properties in the two porous sandstones with partial water saturation, and they help to better indicate the fluid flow within compliant pores in addition to the known flow between stiff and compliant pores. The results will form the theoretical basis for the successful joint interpretation of seismic and electromagnetic survey data acquired from partially saturated porous rocks in the subsurface of the earth.