Poroelastic Inclusions Research Articles

Electromagnetic field generated by acoustic wave scattering at a poroelastic inclusion located in a fluid

The study of elastic wave propagation in poroelastic media is of considerable scientific interest because of some effects that do not appear in monophasic elastic or viscoelastic media. In the case of interconnected pores containing a moving fluid in a poroelastic medium, there exist two types of longitudinal waves. In heterogeneous porous media, the energy of elastic oscillations is redistributed due to mutual transformation of elastic waves of different types at the inclusion boundaries. The filtration flow that appears there results in the case of polar fluids in a displacement of electric double layers at the pore boundaries, and, as a consequence, in generation of an electromagnetic field.We consider the problem of the scattering of an elastic wave by a spherical poroelastic inclusion in a fluid. The solution was obtained for sufficiently low frequencies so that the dispersion of the elastic wave on individual grains of the porous media may be neglected and the averaged equations of the mechanics of continuous media (at the so-named mesoscale) may be used. In order to describe the elastic wave propagation, we have used the Pride system of electrokinetic equations. For the solution of our problem, we have applied the decomposition of the potentials of acoustic and electromagnetic waves into spherical harmonics. We have obtained the dependencies of the generated electromagnetic field on the inclusion porosity and the fluid properties in pores. The results were obtained for fully open pore interface at the inclusion boundary.

Recent Volumetric Deformations of Fault Zones

—Volumetric (non-shear) deformations of fault zones measured from leveling data are analyzed. It is shown that volumetric deformations within fault zones lead to symmetric local surface subsidence. Examples of anomalous subsidence identified in the Ashgabat fault zone (North Kopet Dag) and in the fault zones of the underground gas storage facility are presented. A mechanism describing the formation of these anomalous deformations based on the models of poroelastic inclusion and poroelastic inhomogeneity is proposed. The problem about surface displacements of elastic halfspace with a poroelastic region in the form of vertical extended rectangular prism simulating active fragment of a fault zone is solved. Analytical formulas for surface-displacement estimation are derived for two variants of poroelastic models: a model of inclusion and a model of heterogeneity. The comparison of the two models based on the case study of the Ashgabat fault has shown that the mechanism of anomalous deformations with poroelastic heterogeneity model much better describes the observed displacements than poroelastic inclusion model. The analysis of anomalous surface displacements at an underground gas storage facility indicates that local subsidence occurs during both gas withdrawal and injection. It is established that the formation of anomalous subsidence in the first case follows the poroelastic inclusion model, and in the second case it corresponds to the poroelastic heterogeneity model. This means that local subsidence in a fault zone during gas withdrawal is linearly related to the change in reservoir pressure. Anomalous subsidence during gas injection is nonlinearly linked to the change in reservoir pressure. This is a striking example of induced deformations in the fault zones when temporal variations of poroelastic parameters within a fault zone in the setting of external quasistatic loading induce local perturbation in the stress-strain state of rocks in the vicinity of fault.

The method of fundamental solutions for the elastic wave scattering in a double-porosity dual-permeability medium

For materials that have a matrix porosity at the microscopic scale, and an interconnecting fracture system at the mesoscopic scale, the double-porosity dual-permeability model provides a better prediction of seismic wave attenuation than the single porosity model, due to the flow exchange mechanism between these different size pores. The current work offers for the first time a numerical solution tool based on the method of fundamental solutions for problems of 3-D wave scattering and dynamic stress concentration around scatterers, such as cavities and embedded inhomogeneous inclusions, in an infinite double-porosity dual-permeability medium. The method of fundamental solutions is based on the spherical wave potentials representing three compressional waves and one shear wave. The accuracy and stability of the method are tested against other available results, and are demonstrated to be excellent. Wave scattering and dynamic stress concentration problems due to spherical cavity and poroelastic inclusion are then investigated for incident plane P1 and SV waves in the frequency domain. The research shows that the seismic responses are sensitive to the incident frequency, the boundary drainage condition, and material properties such as porosity.

Scattering of elastic wave from poroelastic inclusions located in a fluid

The problem of scattering of a plane compressional acoustic wave in a fluid from a spherical poroelastic inclusion is considered. The elastic wave propagation in the inclusion is described by the equations of the Biot theory. The wave field in the inclusion consists of fast and slow compressional and shear waves. Outside the inclusion, a scattered spherical compressional wave is formed. The solution for an isolated inclusion is obtained in terms of series of spherical Bessel functions and Legendre polynomials. This solution is used for the calculation of effective wave number of compressional wave propagating in the fluid containing a set of poroelastic inclusions (suspension). For deriving the effective wave number, the theory of multiple scattering is used. It is shown that the effective wave number depends strongly on hydrodynamic permeability of inclusions and fluid properties in the inclusion pore space.

Anisotropic Inhomogeneous Poroelastic Inclusions: With Application to Underground Energy-Related Problems

Understanding the stress change in a reservoir generated by fluid production/injection is important for field development purposes. In this paper, we provide the Eshelby solution for stress and strain distribution inside and outside of an anisotropic poroelastic inhomogeneity due to pore pressure changes inside the inhomogeneity. The term anisotropic inhomogeneity refers to an inhomogeneity with anisotropic poroelastic constants. Some graphical results for strain and stress ratios for different material properties and geometries are presented as well. Anisotropy in elastic properties has been studied extensively in the last century; however, anisotropy in poroelastic properties, despite its potential significant impact in different engineering problems, has not been explored thoroughly. The results show how neglecting the effect of anisotropic poroelastic properties may result in large differences in calculated stresses. Due to the authors' primary interest in geomechanical problems, the discussions and examples are chosen for applications involving fluid withdrawal/injection into hydrocarbon reservoirs.

A framework for assessing the uncertainty in wave energy delivery to targeted subsurface formations

Stress wave stimulation of geological formations has potential applications in petroleum engineering, hydro-geology, and environmental engineering. The stimulation can be applied using wave sources whose spatio-temporal characteristics are designed to focus the emitted wave energy into the target region. Typically, the design process involves numerical simulations of the underlying wave physics, and assumes a perfect knowledge of the material properties and the overall geometry of the geostructure. In practice, however, precise knowledge of the properties of the geological formations is elusive, and quantification of the reliability of a deterministic approach is crucial for evaluating the technical and economical feasibility of the design. In this article, we discuss a methodology that could be used to quantify the uncertainty in the wave energy delivery. We formulate the wave propagation problem for a two-dimensional, layered, isotropic, elastic solid truncated using hybrid perfectly-matched-layers (PMLs), and containing a target elastic or poroelastic inclusion. We define a wave motion metric to quantify the amount of the delivered wave energy. We, then, treat the material properties of the layers as random variables, and perform a first-order uncertainty analysis of the formation to compute the probabilities of failure to achieve threshold values of the motion metric. We illustrate the uncertainty quantification procedure using synthetic data.

Pressurized Poroelastic Inclusions: Short-term and Long-term Asymptotic Solutions

This paper provides semi-analytical, asymptotic short-term and long-term solutions for the volume change and corresponding leak-off volume of a fluid-saturated, three-dimensional poroelastic inclusion, considering fluid exchange with the surrounding poroelastic medium. Considering possibly different material properties and different fluid pressure of hydrocarbon-bearing formations or proppant-filled fractures in comparison to those of the surrounding geological structures, fractures or whole reservoirs can be regarded as inclusions. The problem-solving approach used in our study is inspired by the theory of inclusions and modal decomposition technique previously developed and used to solve several poroelasticity problems. Previous studies on the topic, however, have not incorporated the hydraulic communication between the inclusion and the surrounding medium; therefore, fluid pressure changes in the surrounding rock due to fluid pressure changes in the inclusion were ignored. An example of this problem would be a pressurized stationary fracture, which, depending on pressure, might have fluid exchange with the surroundings. Numerical examples considering inclusions with different aspect ratios and material properties are provided to better describe the significance of fluid exchange.

An XFEM-based numerical procedure for the analysis of poroelastic composites with coherent imperfect interface

Composites under consideration consist of a poroelastic matrix in which poroelastic inclusions are embedded via a poroelastic transition layer of small uniform thickness. A prototype of such poroelastic composites is concrete in which aggregates are often bonded to a cement paste through an interfacial transition zone (ITZ). In this work, the poroelastic transition layer is replaced by an interface of zero thickness described by the poroelastic coherent imperfect interface model. According to this model, the displacement and pore pressure are continuous across the interface but the traction and normal fluid flux suffer interfacial jumps. The main objective of this work is to propose an efficient numerical solution to the problem of poroelastic composites with coherent imperfect interface under the regime of undrained saturation. To this end, the weak formulations are first established for the relevant boundary value problems of linear poroelasticity and fluid transport. The extended finite element method (XFEM) and level-set method (LSM) are then used to elaborate a numerical procedure. The accuracy and convergence rate of this procedure are analyzed with the aid of a benchmark problem whose analytical exact solution is derived. The numerical procedure proposed turns out to be efficient in capturing the interfacial discontinuities and computing the elastic and transport fields, and can be applied, for example, to solving the important problem of homogenizing poroelastic composites with coherent imperfect interface.

Interacting double poroelastic inclusions

In this paper, we provide Eshelby solution for applied stress and strain distribution around double inhomogeneous poroelastic inclusions due to pore pressure changes in inclusions. To address the problem, we modified an approximate analytical approach (Moschovidis and Mura, 1975) for poroelastic inclusions. Inhomogeneous Inclusions are finite sub-volumes of a medium, which are made of different materials and may experience different strain status at the same time. This method could have a wide range of applications from rock mechanics problems to tissue mechanics. An application of this model in analyzing earth stress changes around hydrocarbon reservoirs due to fluid withdrawal/injection is discussed at the end of the paper.

Numerical simulation of ambient seismic wavefield modification caused by pore-fluid effects in an oil reservoir

We have modeled numerically the seismic response of a poroelastic inclusion with properties applicable to an oil reservoir that interacts with an ambient wavefield. The model includes wave-induced fluid flow caused by pressure differences between mesoscopic-scale (i.e., in the order of centimeters to meters) heterogeneities. We used a viscoelastic approximation on the macroscopic scale to implement the attenuation and dispersion resulting from this mesoscopic-scale theory in numerical simulations of wave propagation on the kilometer scale. This upscaling method includes finite-element modeling of wave-induced fluid flow to determine effective seismic properties of the poroelastic media, such as attenuation of P- and S-waves. The fitted, equivalent, viscoelastic behavior is implemented in finite-difference wave propagation simulations. With this two-stage process, we model numerically the quasi-poroelastic wave-propagation on the kilometer scale and study the impact of fluid properties and fluid saturation on the modeled seismic amplitudes. In particular, we addressed the question of whether poroelastic effects within an oil reservoir may be a plausible explanation for low-frequency ambient wavefield modifications observed at oil fields in recent years. Our results indicate that ambient wavefield modification is expected to occur for oil reservoirs exhibiting high attenuation. Whether or not such modifications can be detected in surface recordings, however, will depend on acquisition design and noise mitigation processing as well as site-specific conditions, such as the geologic complexity of the subsurface, the nature of the ambient wavefield, and the amount of surface noise.

Imaging with cross-hole seismoelectric tomography

We propose a cross-hole imaging approach based on seismoelectric conversions (SC) associated with the transmission of seismic waves from seismic sources located in a borehole to receivers (electrodes) located in a second borehole. The seismoelectric (seismic-to-electric) problem is solved using Biot theory coupled with a generalized Ohm's law with an electrokinetic streaming current contribution. The components of the displacement of the solid phase, the fluid pressure, and the electrical potential are solved using a finite element approach with Perfect Match Layer (PML) boundary conditions for the seismic waves and boundary conditions mimicking an infinite material for th electrostatic problem. We develop an inversion algorithm using the electrical disturbances recorded in the second borehole to localize the position of the heterogeneities responsible for the SC. Because of the ill-posed nature of the inverse problem (inherent to all potential-field problems), regularization is used to constrain the solution at each time in the SC-time window comprised between the time of the seismic shot and the time of the first arrival of the seismic waves in the second borehole. All the inverted volumetric current source densities are aggregated together to produce an image of the position of the heterogeneities between the two boreholes. Two simple synthetic case studies are presented to test this concept. The first case study corresponds to a vertical discontinuity between two homogeneous sub-domains. The second case study corresponds to a poroelastic inclusion (partially saturated by oil) embedded into an homogenous poroelastic formation. In both cases, the position of the heterogeneity is recovered using only the electrical disturbances associated with the SC. That said, a joint inversion of the seismic and seismoelectric data could improve these results.

Anisotropic dispersion and attenuation due to wave‐induced fluid flow: Quasi‐static finite element modeling in poroelastic solids

Heterogeneous porous media such as hydrocarbon reservoir rocks are effectively described as anisotropic viscoelastic solids. They show characteristic velocity dispersion and attenuation of seismic waves within a broad frequency band, and an explanation for this observation is the mechanism of wave‐induced pore fluid flow. Various theoretical models quantify dispersion and attenuation of normal incident compressional waves in finely layered porous media. Similar models of shear wave attenuation are not known, nor do general theories exist to predict wave‐induced fluid flow effects in media with a more complex distribution of medium heterogeneities. By using finite element simulations of poroelastic relaxation, the total frequency‐dependent complex stiffness tensor can be computed for a porous medium with arbitrary internal heterogeneity. From the stiffness tensor, velocity dispersion and frequency‐dependent attenuation are derived for compressional and shear waves as a function of the angle of incidence. We apply our approach to the case of layered media and to that of an ellipsoidal poroelastic inclusion. In the case of the ellipsoidal inclusion, compressional and shear wave modes show significant attenuation, and the characteristic frequency dependence of the effect is governed by the spatiotemporal scale of the pore fluid pressure relaxation. In our anisotropic examples, the angle dependence of the attenuation is stronger than that of the velocity dispersion. It becomes clear that the spatial attenuation patterns show specific characteristics of wave‐induced fluid flow, implying that anisotropic attenuation measurements may contribute to the inversion of fluid transport properties in heterogeneous porous media.

The dilative intake of poroelastic inclusions an alternative to the Mandel–Cryer effect

This study examines the time-dependent poromechanics behavior of a fluid-saturated spherical inclusion embedded inside a fluid-saturated porous medium with different poroelastic properties. Both media comprise compressible constituents with distinctively defined poroelastic parameters. It is assumed that the inclusion is subjected to a fluid source at the center. The problem is formulated and solved using Biot theory of poromechanics. The contrasts in inclusion and the medium matrix stiffnesses and their respective hydraulic conductivities can be recognized as two competing factors, which affect the inclusion’s rate of expansion during fluid injection. Findings show a certain type of behavior that the inclusion exhibits at the onset of fluid injection when having greater stiffness than the medium matrix, where the inclusion experiences some decrease in the pore pressure. Compared to what announced as the stress redistribution due the Mandel–Cryer effect in earlier researches on dilation of free spheres, this study shows that the associated phenomenon would be likewise attributed to the coupled nature of pressures and deformations in the theory of poroelasticity. However, it is a consequence of the inclusion-matrix stiffness contrast where a dilating free sphere can be regarded as a special case of this new problem. The asymptotic expansions of pressure terms verify the existence of such an effect. The results of this study would put forward very good insight in some engineering applications.

Seismic attenuation due to wave-induced fluid flow in a porous rock with spherical heterogeneities

SUMMARY Most natural porous rocks have heterogeneities at nearly all scales. Heterogeneities of mesoscopic scale—that is, much larger than the pore size but much smaller than wavelength—can cause significant attenuation and dispersion of elastic waves due to wave-induced flow between more compliant and less compliant areas. Analysis of this phenomenon for a saturated porous medium with a small volume concentration of randomly distributed spherical inclusions is performed using Waterman-Truell multiple scattering theorem, which relates attenuation and dispersion to the amplitude of the wavefield scattered by a single inclusion. This scattering amplitude is computed using recently published asymptotic analytical expressions and numerical results for elastic wave scattering by a single mesoscopic poroelastic sphere in a porous medium. This analysis reveals that attenuation and dispersion exhibit a typical relaxation-type behaviour with the maximum attenuation and dispersion corresponding to a frequency where fluid diffusion length (or Biot’s slow wavelength) is of the order of the inclusion diameter. In the limit of low volume concentration of inclusions the effective velocity is asymptotically consistent with the Gassmann theory in the low-frequency limit, and with the solution for an elastic medium with equivalent elastic inclusions (no-flow solution) in the low-frequency limit. Attenuation (expressed through inverse quality factor 1/Q) scales with frequency ω in the lowfrequency limit and with ω −1/2 in the high-frequency limit. These asymptotes are consistent with recent results on attenuation in a medium with a periodic distribution of poroelastic inclusions, and in continuous random porous media.