Radim Ciz

Generalization of Gassmann equations for porous media saturated with a solid material

Gassmann equations predict effective elastic properties of an isotropic homogeneous bulk rock frame filled with a fluid. This theory has been generalized for an anisotropic porous frame by Brown and Korringa’s equations. Here, we develop a new model for effective elastic properties of porous rocks — a generalization of Brown and Korringa’s and Gassmann equations for a solid infill of the pore space. We derive the elastic tensor of a solid-saturated porous rock considering small deformations of the rock skeleton and the pore infill material upon loading them with the confining and pore-space stresses. In the case of isotropic material, the solution reduces to two generalized Gassmann equations for the bulk and shear moduli. The applicability of the new model is tested by independent numerical simulations performed on the microscale by finite-difference and finite-element methods. The results show very good agreement between the new theory and the numerical simulations. The generalized Gass-mann model intro...

Finite-difference modeling of wave propagation on microscale: A snapshot of the work in progress

Digital rock methodology combines modern microscopic imagingwithadvancednumericalsimulationsofthephysicalproperties of rocks. Modeling of elastic-wave propagation directly from rock microstructure is integral to this technology. We survey recent development of the rotated staggered grid RSG finite-difference FD method for pore-scale simulation of elastic wavepropagationindigitalrocksamples,includingthedynamic elastic properties of rocks saturated with a viscous fluid. Examination of the accuracy of this algorithm on models with known analytical solutions provide an additional accuracy condition for numerical modeling on the microscale. We use both the elastic and viscoelastic versions of the RSG algorithm to study gas hydratesandtosimulatepropagationofBiot’sslowwave.Weapply RSG method ology to examine the effect of gas hydrate distributions in the pore space of a rock. We compare resulting P-wave velocities with experimentally measured data, as a basis for buildinganeffective-mediummodelforrockscontaininggashydrates. We then perform numerical simulations of Biot’s slow wave in a realistic 3D digital rock model, fully saturated with a nonviscous fluid corresponding to the high-frequency limit of poroelasticity, and placed inside a bulk fluid.The model clearly demonstrates Biot’s slow curve when the interface is open between the slab and bulk fluid.We demonstrate slow wave propagation in a porous medium saturated with a viscous fluid by analyzing an idealized 2D porous medium represented alternating solid and viscous fluid layers. Comparison of simulation results withtheexactsolutionforthislayeredsystemshowsgoodagreementoverabroadfrequencyrange.

Modeling elastic wave velocities and attenuation in rocks saturated with heavy oil

Although properties of bulk heavy oil can be approximated by an appropriate viscoelastic model, only a few attempts to model properties of rocks saturated with heavy oil have been reported. Rock-physics models used for rocks saturated with conventional fluids are inapplicable to those saturated with heavy oil because its viscoelastic rheology invalidates the main assumptions of the Gassmann and Biot theories. We estimate viscoelastic properties of mixtures of rock and heavy oil by considering (1) a system of layers of a solid and a viscoelastic medium and (2) by computing Hashin-Shtrikman (HS) bounds for this system. These two methods give approximate bounds for the frequency- and temperature-dependent velocities and attenuation coefficients in rocks saturated with heavy oil. We also propose a method to compute a realistic estimate of these properties that lie between those bounds. This estimate is based on a self-consistent equivalent-medium approach known as coherent-potential approximation. In a more general form, this approximation can be used for approximate fluid substitution for heavy oil. This approach gives frequency-dependent velocities and attenuation values that are qualitatively consistent with experimental observations.

Stress-dependent anisotropy in transversely isotropic rocks: Comparison between theory and laboratory experiment on shale

Understanding the effect of stress and pore pressure on seismic velocities is important for overpressure prediction and for 4D reflection seismic interpretation. A porosity-deformation approach (originally called the piezosensitivity theory) and its anisotropic extension describe elastic moduli of rocks as nonlinear functions of the effective stress. This theory assumes a presence of stiff and compliant parts of the pore space. The stress-dependent geometry of the compliant pore space predominantly controls stress-induced changes in elastic moduli. We show how to apply this theory to a shale that is transversely isotropic (TI) under unloaded conditions. The porosity-deformation approach shows that components of the compliance tensor depend on exponential functions of the principal components of the effective stress tensor. In the case of a hydrostatic loading of a TI rock, only the diagonal elements of this tensor, expressed in contracted notation, are significantly stress dependent. Two equal shear compo...

Finite element modelling of the effective elastic properties of partially saturated rocks

Simulation of effective physical properties from microtomographic 3D images of porous structures allows one to relate properties of rocks directly to their microstructure. A static FEM code has been previously used to estimate effective elastic properties of fully saturated monomineralic (quartz) rock under wet and dry conditions. We use the code to calculate elastic properties under partially saturated conditions. The numerical predictions are compared to the Gassmann theory combined with Woods formula (GW) for a mixture of pore fluids, which is exact for a monomineralic macroscopically homogeneous porous medium.Results of the numerical simulations performed for two Boolean sphere pack distributions show significant deviation from the GW limit and depend on the spatial distribution of fluids. This is shown to be a numerical artefact caused by incomplete equilibration of fluid pressure, which is primarily due to insufficient spatial resolution.To investigate the effect of pore-size and pore geometry, we perform FEM simulations for a model with regular pore geometry, where all pore channels have the same size and shape. Accuracy of these simulations increases with the total cross-section area of the channels and the size of individual channels. For the case where the total cross-section of the channels is large enough (on the same order as total porosity), there is a minimum of 4 voxels per channel diameter required for adequate fluid pressure equilibration throughout the pore space. Increasing the spatial resolution of the digital models reduces the discrepancy between the simulations and theory, but unfortunately increases the memory and CPU requirements of the simulations.

Simple expressions for normal incidence reflection coefficients from an interface between fluid-saturated porous materials

Hydrocarbon reservoirs as well as many other sedimentary rocks are fluid-saturated porous materials, whose elastic properties can be described by the theory of poroelasticity (Biot, 1962).

Pore scale numerical modeling of elastic wave dispersion and attenuation in periodic systems of alternating solid and viscous fluid layers

Numerical pore-scale simulation of elastic wave propagation is an emerging tool in the analysis of static and dynamic elastic properties of porous materials. Rotated staggered-grid (RSG) finite difference method has proved to be particularly effective in modeling porous media saturated with ideal fluids. Recently this method has been extended to viscoelastic (Maxwell) media, which allows simulation of wave propagation in porous solids saturated with Newtonian fluids. To evaluate the capability of the viscoelastic RSG algorithm in modeling wave dispersion and attenuation we perform numerical simulations for an idealized porous medium, namely a periodic system of alternating solid and viscous fluid layers. Simulations are performed for a single frequency of 50kHz (for shear waves) and 500kHz (for compressional waves) and a large range of fluid viscosities. The simulation results show excellent agreement with the theoretical predictions. Specifically the simulations agree with the prediction of Biot’s theory...

Influence of microheterogeneity on effective stress law for elastic properties of rocks

Understanding the effective stress coefficient for seismic velocity is important for geophysical applications such as overpressure prediction from seismic data as well as for hydrocarbon production and monitoring using time-lapse seismic measurements. This quantity is still not completely understood. Laboratory measurements show that the seismic velocities as a function of effective stress yield effective stress coefficients less than one and usually vary between 0.5 and 1. At the same time, theoretical analysis shows that for an idealized monomineral rock, the effective stress coefficient for elastic moduli (and therefore also for seismic velocities) will always equal one. We explore whether this deviation of the effective stress coefficient from unity can be caused by the spatial microheterogeneity of the rock. The results show that only a small amount (less than 1%) of a very soft component is sufficient to cause this effect. Such soft material may be present in grain contact areas of many rocks and may explain the variation observed experimentally.

Viscoelastic Modeling of Rocks Saturated With Heavy Oil

While properties of bulk heavy oil can be approximated by an appropriate viscoelastic model, only a few attempts to model properties of rocks saturated with heavy oil have been reported (Eastwood, 1993; Leurer and Dvorkin, 2006). Rock physics for heavy oil is different from rock physics for conventional fluids because its viscoelastic rheology makes Gassmann theory and all its extensions inapplicable in principle. In this paper, we estimate the properties of such rock-heavy oil mixtures by considering (1) a system of layers of solid and a viscoelastic medium and (2) by computing Hashin-Shtrikman bounds for this system. These two methods essentially give approximate bounds for the frequencyand temperature-dependent properties of these rocks. We also propose how to compute a realistic estimate of these properties that would lie between these bounds. This proposed estimate is based on one particular equivalent-medium approach known as coherent potential approximation (CPA) (Berryman, 1980). In a more general form, this approximation can be used for approximate fluid substitution for heavy oil.

Generalization of Gassmann Equations for Porous Rocks Saturated with a Solid Material - Theory and Applications

Gassmann equations predict effective elastic properties of an isotropic homogeneous bulk rock frame filled with a fluid. This theory has been generalized for an anisotropic porous frame by Brown and Korringa equations. Recently, Ciz and Shapiro (2008) developed a generalization of Gassmann equations for porous media saturated with a solid material. In this paper we present an overview of the new theory and compare to numerical simulations of effective elastic properties of porous rocks filled with an additional solid component. Moreover, we model the effective elastic prroperties of clay-sand mixtures and rocks saturated with heavy oils. The latter application we validate by comparing to experimental data. A very good agreement between numerical and experimental data has been obtained. Thus, this new generalized Gassmanns model is directly applicable as useful rock physics model, e.g.: in proper seismic data interpretation.