Dry Rock Moduli Research Articles

Integrated Petrophysical Evaluation and Rock Physics Modeling of Broom Creek Deep Saline Aquifer for Geological CO2 Storage

Fossil fuels, such as coal and hydrocarbons, are major drivers of global warming and are primarily responsible for worldwide greenhouse gas emissions, including carbon dioxide CO2. The storage of CO2 in deep saline reservoirs is acknowledged as one of the top practical and promising methods to reduce CO2 emissions and meet climate goals. The North Dakota Industrial Commission (NDIC) recently approved the fourth Class VI permit for a carbon capture and storage project in the Williston basin of North Dakota for the geological CO2 storage in the Broom Creek formation. The current research aimed to conduct a comprehensive petrophysical characterization and rock physics modeling of the Broom Creek deep saline reservoir to unravel the mineralogical distribution and to understand the variations in petrophysical and elastic properties across the formation. This study utilized geophysical well logs, routine core analysis, and advanced core analysis to evaluate the Broom Creek formation. Multimineral petrophysical analysis calibrated with X-ray diffraction results reveals that this formation primarily comprises highly porous clean sandstone intervals with low-porosity interspersed with dolomite, anhydrite, and silt/clay layers. The formation exhibits varying porosities up to 0.3 and Klinkenberg air permeabilities up to ∼2600 mD. The formation water resistivity using Archie’s equation is approximately 0.055 ohm-m at 150 °F, corresponding to around 63,000 ppm NaCl salinity, which is consistent with prior data. The pore throat distribution in the samples from clean sandstone intervals is primarily situated in the macro-mega scales. However, the presence of anhydrite and dolomite impedes both porosity and pore throat sizes. The accurate prediction of effective elastic properties was achieved by developing a rock physics template. Dry rock moduli were modeled using Hill’s average, while Berryman’s self-consistent scheme was employed for modeling saturated moduli.

Numerical validation of Gassmann’s equations

Derived several decades ago, Gassmann’s equations continue to be widely used in applied geophysics. Gassmann’s equations allow us to calculate the elastic moduli of a fully saturated rock from dry rock moduli knowing the porosity, fluid bulk modulus, and bulk modulus of the solid grains. These equations are treated as exact in the scientific community, but there is a lack of comprehensive numerical validation. Furthermore, recently several publications have appeared in the literature postulating a logical error in the derivation of Gassmann’s equations. Therefore, I develop a numerical validation of Gassmann’s equations. For that, I use a 3D finite-element approach to resolve the conservation of linear momentum that is coupled with the stress-strain relations for the solid phase and the quasistatic linearized compressible Navier-Stokes momentum equation for the fluid phase. Finally, a convergence study validating the correctness of Gassmann’s equations for a particular yet arbitrarily chosen “generic” pore geometry is presented. The arbitrary model geometry is simple as compared with real rocks; however, it is sufficiently complex with elements resembling wider pore bodies and narrower pore throats to, in general, validate Gassmann’s equations. MATLAB routines to reproduce the presented results are provided.

Direct inversion for the equivalent pore aspect ratio based on the theory of ellipsoid modelling

AbstractPore aspect ratio, together with porosity, is a structural parameter that represents the geometric property of rock reservoirs. We have adopted the theory of ellipsoid modelling in material mechanics to derive the dry rock modulus. Based on this derivation, the Gassmann equation, which is a constitutional equation for a fully saturated rock model, can be linearized in terms of the pore structure parameters and the elastic parameters. We have established a relationship between the seismic reflection coefficient and the pore structure parameters, where the equivalent pore aspect ratio and porosity are two key parameters. Based on this relationship, the equivalent pore aspect ratio can be inverted directly from seismic reflection data instead of being converted from the intermediate parameters of conventional seismic inversion. Therefore, seismic inversion is a simultaneous inversion in which seven parameters are inverted: the elastic moduli (matrix bulk modulus and shear modulus, fluid bulk modulus), the densities (matrix and fluid densities) and the pore structure parameters (the equivalent pore aspect ratio and porosity). We applied this direct inversion scheme to a carbonate reservoir to predict the fracture development zone with low porosity and low aspect ratio.

A squirt-flow theory to model wave anelasticity in rocks containing compliant microfractures

Wave propagation in rocks induces fluid pressure gradients and flow between stiff intergranular pores and compliant microfractures. This, in turn, leads to wave anelasticity, i.e., attenuation and velocity dispersion. Three known squirt-flow models attempt to explain this phenomenon. Based on these theories, we reformulate a new one, where we derive the dry-rock moduli on the assumption that the boundary fluid pressure is constant. The new model improves the predictions of the other models, e.g., a better description at high frequencies and the inclusion of permeability effects. The numerical example shows the response due to the characteristic squirt-flow length and fluid viscosity on wave propagation. The reformulated model is applied to experiments made on water-saturated tight sandstones, where the squirt-flow length is determined, showing that it increases with increasing permeability and porosity at full saturation.

An Accurate Jacobian Matrix with Exact Zoeppritz for Elastic Moduli of Dry Rock

Seismic velocities are related to the solid matrices and the pore fluids. The bulk and shear moduli of dry rock are the primary parameters to characterize solid matrices. Amplitude variation with offset (AVO) or amplitude variation with incidence angle (AVA) is the most used inversion method to discriminate lithology in hydrocarbon reservoirs. The bulk and shear moduli of dry rock, however, cannot be inverted directly using seismic data and the conventional AVO/AVA inversions. The most important step to accurately invert these dry rock parameters is to derive the Jacobian matrix. The combination of exact Zoeppritz and Biot–Gassmann equations makes it possible to directly calculate the partial derivatives of seismic reflectivities (PP-and PS-waves) with respect to dry rock moduli. During this research, we successfully derive the accurate partial derivatives of the exact Zoeppritz equations with respect to bulk and shear moduli of dry rock. The characteristics of these partial derivatives are investigated in the numerical examples. Additionally, we compare the partial derivatives using this proposed algorithm with the classical Shuey and Aki–Richards approximations. The results show that this derived Jacobian matrix is more accurate and versatile. It can be used further in the conventional AVO/AVA inversions to invert bulk and shear moduli of dry rock directly.

Rock physics modeling and 4D seismic feasibility study in one of the Iranian carbonate reservoirs

Rock Physics is one of the vital steps of any Reservoir Geophysics studies to estimate the reservoir parameters. To do this, it is necessary to quantify the effect of reservoir parameters on the velocity and density of the saturated rocks. Carbonate reservoirs contain different heterogeneities such as the presence of pore types and irregular distribution, making it difficult to estimate the reservoir properties in these reservoirs. Various rock physics models have been provided to estimate the P and S-wave velocities. However, each of these models shows some degree of limitations, especially in carbonate hydrocarbon reservoirs with different pore types. In this paper, an integrated algorithm was introduced to estimate the dry rock modulus that include the pore type, as well as the pressure effect. The effect of porosity and pore type was considered by employing Xu and White's (1995) equation. The pressure effect was added using MacBeth's (2004) methodology. Finally, the fluid effect was included utilizing Gassmann's (1951) equation. During the time-lapse seismic feasibility study for our case study, it is observed that the impact of pore type is as important as the quantity of the porosity, and even more. The variation of pore type from the fracture to the stiff pores (increasing the pore aspect ratio), can double the p-wave velocity. In addition, it is concluded that increasing the Aspect Ratio (the ratio between the minor and major axes of an ellipsoidal pore) of the pores from 0.0.07 to 0.8, could significantly decrease the pressure and saturation sensitivity of the elastic properties from 17% to 2%. Without proper treating of pore type and pressure impact, as an example, a feasible four dimensional (4D) seismic project can be concluded non-feasible and vice versa. In this paper, utilizing the introduced methodology, one dimensional (1D) feasibility study was carried out for one of Iranian Carbonate reservoirs. We could replicate the velocity and density logs with acceptable deviation from the observed logs that are better than the original Gassmann's method. Subsequently, the injection and production scenarios were generated to estimate the impedance variation in this reservoir. We observed that the reservoir pressure increase of 1500 psi and water saturation increase of 60% could change the impedance by −2% and +7%, respectively. However, the reservoir pressure drop has negligible impact on the impedance in this field. In overall, it makes this reservoir feasible for a time-lapse seismic project to monitor production and injection.

The Burgers/squirt-flow seismic model of the crust and mantle

Part of the crust shows generally brittle behaviour while areas of high temperature and/or high pore pressure, including the mantle, may present ductile behaviour. For instance, the potential heat source of geothermal fields, overpressured formations and molten rocks. Seismic waves can be used to detect these conditions on the basis of reflection and transmission events. Basically, from the elastic-plastic point of view the seismic properties (seismic velocity, quality factor and density) depend on effective pressure and temperature. Confining and pore pressures have opposite effects on these properties, and high temperatures may induce a similar behaviour by partial melting. In order to model these effects, we consider a poro-viscoelastic model based on the Burgers mechanical element and the squirt-flow model to represent the properties of the rock frame to describe ductility in which deformation takes place by shear plastic flow, and to model local and global fluid flow effects.The Burgers element allows us to model the effects of the steady-state creep flow on the dry-rock frame. The stiffness components of the brittle and ductile media depend on stress and temperature through the shear viscosity, which is obtained by the Arrhenius equation and the octahedral stress criterion. Effective pressure effects are taken into account in the dry-rock moduli by using exponential functions whose parameters are obtained by fitting experimental data as a function of confining pressure. Since fluid effects are important, the density and bulk modulus of the saturating fluids (water at sub- and supercritical conditions) are modeled by using the equations provided by the NIST website. The squirt-flow model has a single free parameter represented by the aspect ratio of the grain contacts. The theory generalizes a preceding theory based on Gassmann (low-frequency) moduli to the more general case of the presence of local (squirt) flow and global (Biot) flow, which contribute with additional attenuation mechanisms to the wave propagation.

The Gassmann–Burgers Model to Simulate Seismic Waves at the Earth Crust And Mantle

The upper part of the crust shows generally brittle behaviour while deeper zones, including the mantle, may present ductile behaviour, depending on the pressure–temperature conditions; moreover, some parts are melted. Seismic waves can be used to detect these conditions on the basis of reflection and transmission events. Basically, from the elastic–plastic point of view the seismic properties (seismic velocity and density) depend on effective pressure and temperature. Confining and pore pressures have opposite effects on these properties, such that very small effective pressures (the presence of overpressured fluids) may substantially decrease the P- and S-wave velocities, mainly the latter, by opening of cracks and weakening of grain contacts. Similarly, high temperatures induce the same effect by partial melting. To model these effects, we consider a poro-viscoelastic model based on Gassmann equations and Burgers mechanical model to represent the properties of the rock frame and describe ductility in which deformation takes place by shear plastic flow. The Burgers elements allow us to model the effects of seismic attenuation, velocity dispersion and steady-state creep flow, respectively. The stiffness components of the brittle and ductile media depend on stress and temperature through the shear viscosity, which is obtained by the Arrhenius equation and the octahedral stress criterion. Effective pressure effects are taken into account in the dry-rock moduli using exponential functions whose parameters are obtained by fitting experimental data as a function of confining pressure. Since fluid effects are important, the density and bulk modulus of the saturating fluids (water and steam) are modeled using the equations provided by the NIST website, including supercritical behaviour. The theory allows us to obtain the phase velocity and quality factor as a function of depth and geological pressure and temperature as well as time frequency. We then obtain the PS and SH equations of motion recast in the velocity–stress formulation, including memory variables to avoid the computation of time convolutions. The equations correspond to isotropic anelastic and inhomogeneous media and are solved by a direct grid method based on the Runge–Kutta time stepping technique and the Fourier pseudospectral method. The algorithm is tested with success against known analytical solutions for different shear viscosities. An example shows how anomalous conditions of pressure and temperature can in principle be detected with seismic waves.

Logging assessment of tight clastic rock reservoir fractures via the extraction of effective pore aspect ratios: A case study of lower Permian strata in the southern Qinshui Basin of eastern China

Research on tight gas enrichment in coal measure strata is contentious worldwide. Gas logging anomalies are very common in the upper Paleozoic coal-bearing strata in the Southern Qinshui Basin of eastern China. The distribution characteristics of these free gases are closely related to the degree of fracture development. The rock pore aspect ratio (α) can reflect the morphology of rock pores and fractures. In this paper, we use full wave logging data to propose a new logging assessment method for tight clastic rock reservoir fractures via the extraction of effective pore aspect ratios. The basic concept is as follows: (1) we extract the rock matrix mineral modulus and dry rock modulus of the rocks using an optimized algorithm, and (2) we invert the effective pore aspect ratio of coal measure strata using the theoretical model of a differential equivalent medium. The research results show that the studied clastic rocks in coal measure strata are very tight, with porosity levels that are generally less than 5%, and sections in which fractures have formed often correspond to relatively low α values of less than 0.1. The microfracture-bearing sections identified based on α inversions correspond well to the sections with fractures at the core scale. Thus, this method can be used to identify fractured sections at the core scale. For sandy mudstone, microfractures mainly exist in sandy intervals, and α is negatively related to rock porosity and positively related to shale content. Regardless of lithology, i.e., sandstone, mudstone or coal, the communication and transport capacity of fracture systems are greater than those of the pore system, and coal-derived gas reservoirs form when there are good gas sources, transportation systems, fracture systems and preservation conditions. Similar to the formation mechanism of fractures in rocks, α is also derived from mechanical mechanisms, and this study shows that the inversion process is effective in characterizing the degree of fracture development in tight clastic rocks in coal measure strata based on elastic rock properties.

Investigating the influence of mineralogy and pore shape on the velocity of carbonate rocks: Insights from extant global data sets

Using a variety of recent public-domain data sets comprising porosity, velocity (P- and S-waves), and, in most cases, mineralogy and petrographic data, I created an extensive global data set and evaluated the importance of mineralogy and pore type on the elastic properties behavior of carbonate core plugs. Results from this investigation clearly illuminated the potential for overinterpreting elastic properties behavior as a function of pore type(s) when mineralogy was not explicitly included in the analysis. Rock-physics analysis using a combination of heuristic and theoretical models illustrated that mineralogy exerted a significant additional variation on velocity at a given porosity. Failure to account for mineralogy exacerbated inferences about the effect of pore type(s) made using a comparison of P-wave velocity to an inappropriate empirical model (Wyllie) that did not account for pore shape(s). In this analysis, extreme variability in carbonate velocity was observed in only portions of two data sets, when mineralogy was explicitly considered and robust models that accounted for inclusion (pore) shape were used. Results from this analysis resulted in a recommended workflow, including a rock-physics template and dry-rock modulus diagnostics, for the evaluation of lab-based carbonate rock-physics data. The workflow was amenable to further integration with well-based data and other core-based petrophysical measurements (e.g., electrical properties).

Elastic moduli of dry rocks containing spheroidal pores based on differential effective medium theory

Abstract Differential effective medium (DEM) theory is applied to determine the elastic properties of dry rock with spheroidal pores. These pores are assumed to be randomly oriented. The ordinary differential equations for bulk and shear moduli are coupled and it is more difficult to obtain accurate analytical formulae about the moduli of dry porous rock. In this paper, we derive analytical solutions of the bulk and shear moduli for dry rock from the differential equations by applying an analytical approximation for dry-rock modulus ratio, in order to decouple these equations. Then, the validity of this analytical approximation is tested by integrating the full DEM equation numerically. The analytical formulae give good estimates of the numerical results over the whole porosity range. These analytical formulae can be further simplified under the assumption of small porosity. The simplified formulae for spherical pores (i.e., the pore aspect ratio is equal to 1) are the same as Mackenzie's equations. The analytical formulae are relatively easy to analyze the relationship between the elastic moduli and porosity or pore shapes, and can be used to invert some rock parameters such as porosity or pore aspect ratio. The predictions of the analytical formula for the sandstone experimental data show that the analytical formulae can accurately predict the variations of elastic moduli with porosity for dry sandstones. The effective elastic moduli of these sandstones can be reasonably well characterized by spheroidal pores, whose pore aspect ratio was determined by minimizing the error between theoretical predictions and experimental measurements. For sandstones the pore aspect ratio increases as porosity increases if the porosity is less than 0.15, whereas the pore aspect ratio remains relatively stable (about 0.14) if the porosity is more than 0.15.

Digital rock physics: Effect of fluid viscosity on effective elastic properties

Abstract This paper is concerned with the effect of pore fluid viscosity on effective elastic properties using digitized rocks. We determine a significant velocity dispersion in wave propagation simulations by the variation of the pore fluid viscosity. Several attenuation regimes are considered which may contribute to this observation. Starting point is a virtual rock physics approach. Numerical simulations of effective transport and effective mechanical properties are applied to statistically representative rock samples. The rock microstructure is imaged by 3D X-ray tomography. Permeability values were estimated through Lattice-Boltzmann flow simulations. The dry rock moduli and the tortuosity are derived by dynamic wave propagation simulations. We apply a displacement-stress rotated staggered finite-difference grid technique to solve the elastodynamic wave equation. An accurate approximation of a Newtonian fluid is implemented in this technique by using a generalized Maxwell body. We give a practical description of how to use this approach and discuss the application limits. Additionally, we show the simulated signature of a theoretically predicted slow S-wave.

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

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

Non-Destructive Techniques to Determine the Effective Stress Coefficient of Sandstone Formations

Abstract The concept of effective stress coefficient, "n," is critical for the study of stress and pressure-dependent behaviour of rocks. This parameter is used to scale the pore fluid pressure for the computation of effective stress and impacts on the accuracy of pore pressure and saturation predictions in reservoirs. However, "n" is relatively unknown and difficult to estimate and, in many cases, is often incorrectly assumed to be equal to one. We present a general review of "n" and give a theoretical proof that "n" is less than one for porous rocks. We also introduce a laboratory methodology to measure the value of "n" employing ultrasonic P-wave velocity (Vp) and attenuation quality factor (Qp). We then confirm the value of "n' experimentally by using this laboratory method to estimate "n" of quartz sandstone. We compare our experimental value of "n" to both theoretical and practical results obtained from other literature that used different estimation methodology. We argue from our experimental results that:"n" is not one but varies from zero to one;there is no single value of "n" for a particular reservoir rock;"n" depends on internal factors (porosity and pore geometry) and external factors (pore pressure and confining pressure) of the rock;velocity derived "n" and quality factor derived "n" are slightly different for the same rock; and,for the quartz sandstone used in this experiment, the velocity derived "n" is smaller and also less sensitive to changes in pressure than quality factor derived "n." Introduction Most of the pioneering work on what is now known as effective stress coefficient was done by Terzaghi(1), who showed that most rocks are porous to some degree due to the pore spaces and micro cracks within them. The effect of these pores on the strength of rocks has widely been documented(1–5). A rock with a few or without pores is considerably stronger than porous ones since pore spaces are unsupported within the rock. With the presence of unsupported spaces, the intact rock between the pores has to take larger loads to sustain an overall applied stress. This important phenomenon has a direct bearing on the effective stress of the rock (Figure 1). Thus, the effective stress (Pe) is not directly equal to the differential pressure (Pd). Many writers(4–12) have highlighted the problem caused by this phenomenon and the resulting uncertainties associated with the estimated pressures and saturation in reservoirs. This uncertainty is increasingly significant for reservoir engineering due to the expanding use of seismic techniques for remote estimation of abnormal pore pressures for drill planning and for monitoring reservoir pore pressure changes during production. For example, Carcione and Helle(13) applied a new technique for estimating reservoir pressures from seismic data to an overpressured gas field in the Norwegian North Sea and concluded that once a reliable velocity field was determined for the reservoir formation, the most important part of the prediction process was the determination of the effective stress coefficients and related dry-rock moduli.

Gas-hydrate concentration estimated from P- and S-wave velocities at the Mallik 2L-38 research well, Mackenzie Delta, Canada

We estimate the concentration of gas hydrate at the Mallik 2L-38 research site using P- and S-wave velocities obtained from well logging and vertical seismic profiles (VSP). The theoretical velocities are obtained from a generalization of Gassmann's modulus to three phases (rock frame, gas hydrate and fluid). The dry-rock moduli are estimated from the log profiles, in sections where the rock is assumed to be fully saturated with water. We obtain hydrate concentrations up to 75%, average values of 37% and 21% from the VSP P- and S-wave velocities, respectively, and 60% and 57% from the sonic-log P- and S-wave velocities, respectively. The above averages are similar to estimations obtained from hydrate dissociation modeling and Archie methods. The estimations based on the P-wave velocities are more reliable than those based on the S-wave velocities.

Acoustic and mechanical response of reservoir rocks under variable saturation and effective pressure.

We investigate the acoustic and mechanical properties of a reservoir sandstone saturated by two immiscible hydrocarbon fluids, under different saturations and pressure conditions. The modeling of static and dynamic deformation processes in porous rocks saturated by immiscible fluids depends on many parameters such as, for instance, porosity, permeability, pore fluid, fluid saturation, fluid pressures, capillary pressure, and effective stress. We use a formulation based on an extension of Biot's theory, which allows us to compute the coefficients of the stress-strain relations and the equations of motion in terms of the properties of the single phases at the in situ conditions. The dry-rock moduli are obtained from laboratory measurements for variable confining pressures. We obtain the bulk compressibilities, the effective pressure, and the ultrasonic phase velocities and quality factors for different saturations and pore-fluid pressures ranging from normal to abnormally high values. The objective is to relate the seismic and ultrasonic velocity and attenuation to the microstructural properties and pressure conditions of the reservoir. The problem has an application in the field of seismic exploration for predicting pore-fluid pressures and saturation regimes.