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In this paper, Frenkel introduced the theory of wave propagation in saturated porous media. A second-type compressional wave, also known as the slow wave, was predicted for the first time, based on theoretical analysis. The seismoelectric phenomena observed in the field were explained as the result of second-type wave propagation. This paper was originally published in 1944 in the Journal of Physics, Vol. 111, No. 5, pp. 230-241. It is republished with permission by the Schmidt Institute of Earth Physics, Russian Academy of Sciences, courtesy of Professor Alexander O. Gliko, Director of the Institute. The paper was prepared with typographical corrections by Alexander Cheng.

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... The purpose of this scientific research is to advance the quantitative and qualitative description of coupled physical processes in porous rocks. The contributions presented here are distributed across four different disciplines: micromechanics (Eshelby, 1957;Nemat-Nasser and Hori, 2013;Kachanov and Sevostianov, 2018), geophysics (Mavko and Nur, 1975;O'Connell and Budiansky, 1978), computational mechanics and computational poroelasticity (Charney et al., 1950;Harlow and Welch, 1965;Virieux, 1986;Frenkel, 1944;Biot, 1962a) and the theory of non-linear instabilities (e.g., bifurcation theory) which are useful to describe spontaneous earthquake nucleation (Poincaré, 1885;Rudnicki and Rice, 1975;Poliakov et al., 1994;Sulem and Vardoulakis, 1995;Holzapfel, 2000;Yabuno, 2021). I used different analytical and numerical methods in my research. ...

... The two-phase medium is represented by an elastic solid matrix (skeleton) saturated with a compressible viscous fluid. The dynamic response of such an isotropic two phase medium results in two longitudinal waves (Figure 1.4) and one shear wave, as predicted by Frenkel (Frenkel, 1944). The wave of the first kind is a true longitudinal P wave (primary wave or fast P-wave) where the solid matrix motion and the fluid particle velocity are in-phase. ...

... The two phase medium is represented by an elastic solid matrix (skeleton) saturated with a compressible viscous fluid. The dynamic response of such an isotropic two phase medium results in two longitudinal waves and one shear wave, as predicted by Frenkel (Frenkel, 1944) (see also Pride and Garambois (2005)). The wave of the first kind (fast wave) is a true longitudinal wave where the solid matrix motion and the fluid particle velocity are in-phase. ...

... The interaction of seismoacoustic and electromagnetic fields at the subsurface structural interfaces or in ion-conducting media was described back in the 1940s. The first to pay attention to these phenomena in rocks was the Soviet scientist Ivanov A. G. Later, the theoretical model was developed by Soviet and American researchers in 1944, Frenkel J. I. [1] and Biot M. A. in 1956 [2], respectively, who described the effect of the electromagnetic (EM) field generation when the crystalline rock was exposed to elastic mechanical vibrations. ...

... Problem (4) is a well-known Frenkel equation governing the propagation of the longitudinal pore pressure waves in fluid-saturated poroelastic rock [1]. It is formulated with respect to the perturbations of the pore pressure p = p(x, z, t), arising in response to mechanical deformation, described here in terms of volume strain ϑ(x, z, t). ...

... In [1,4,5,8,13,14], an attempt was made to theoretically and experimentally substantiate the SE effect, described by Frenkel, on the basis of the accumulated modern knowledge in the field of the electrokinetic theory of electrolytes, linking the rheological, elastic parameters of the medium (including the porosity and fluid saturation of rocks) with the magnitude of the induced electric field. As an experimental check of the SE effect, a physical model based on the measurement of the SE effect in the ultrasonic range is proposed. ...

The article is devoted to the study of the potential possibilities of using molecular-electronic sensors of seismic waves for field work using the seismoelectric method to explore the hydrocarbon deposits. The introduction provides an analytical review of the current state of research based on data from science magazines and patents. It is shown that at present, seismoelectric effects are at the stage of experimental implementation into the practice of field work for oil and gas geophysical prospecting. Further in the article, theoretical estimates and results of mathematical modeling of the manifestation of seismoelectric (SE) phenomena in the regions of hydrocarbon anomalies are presented, numerical estimates of the values of the seismic and secondary electromagnetic fields are given. The analysis of the results (on a tank and real gas condensate field) showed that the use of molecular-electronic geophones, which have a higher sensitivity and operate in a wider frequency range (up to 0.1 Hz), allows one to obtain higher signal-to-noise ratio. Thus, it has been experimentally established that the use of molecular sensors for recording seismic electric effects when searching for deposits is more preferable when carrying out field work.

... These signals are commonly interpreted in terms of two main underlying mechanisms: (a) electrokinetic effect (Fitterman 1978;Frenkel 1944;Martner and Sparks 1959;Pride 1994;Pride and Garambois 2005;Gershenzon et al. 2014;Surkov and Pilipenko 2015), or (b) geomagnetic inductive effect (Knopoff 1955;Kaliski 1960;Guglielmi 1986;Gorbachev and Surkov 1987;Guglielmi and Ruban 1990). The co-seismic phenomena should be distinguished from the long-lasting abnormal offset caused by piezomagnetic (or tectonomagnetic) effects which can occur several weeks around the main shock (Johnston 1978). ...

... Electrokinetic (EK) effect is due to the charge separation in a wet crust. Walls of pores and cracks absorb negative ions from crust fluid, while positive ions remain in the fluid (Frenkel 1944;Bockris and Reddy 1970;Sparnaay 1972). The crust deformations caused by seismic waves induce the crust fluid movement, whereas the movement of fluid ions produces the EK current. ...

... For low-frequency disturbances, a local relationship between the pore fluid pressure P f and the volume deformation of a medium u V is established (Frenkel 1944): ...

We analyze theoretically two possible sources of co-seismic electromagnetic response to the propagation of various types of seismic waves caused either by the elec-trokinetic phenomena or geomagnetic inductive effect. The differences between these two generation mechanisms have been examined for different types of seismic waves (P, S, and Rayleigh-Love). Theoretical relationships describing the dependence of the co-seismic signal amplitude, polarization and apparent impedance on the earthquake seismic moment and magnitude have been derived as a function of distance. We indicate an observational possibility to discriminate seismo-electrokinetic and seismo-magnetic effects and to estimate their contribution into a recorded co-seismic electromagnetic signal. Magnitudes and polarization of these signals are shown to depend strongly on the type of seismic wave and local crust parameters, such as streaming potential coupling coefficient, conductivity, inhomogeneity, etc. Co-seismic electromagnetic signals, though not directly applied for earthquake prediction, contain a useful information on local crustal phenomena, and can be used to identify ''sensitive'' zones perspective for the monitoring of precursory electromagnetic disturbances.

... There are two basic physical mechanisms of these coseismic phenomena: the seismoelectric effect in porous water-saturated media and the perturbation of the Earth's magnetic field due to the motion of the conductive ground [Surkov et al, 2018]. The seismoelectric effect builds up as a result of electric charge separation caused by the electro-chemical processes in the electrolyte solution, which is contained in the underground fluid [Frenkel, 1944]. The surfaces of cracks and pores can adsorb ions of certain polarity (more frequently the negative-charged ions) from the electrolyte solution whereas the underground fluid accumulates the excess ions of opposite/positive sign. ...

... Acoustic and electrokinetic phenomena in porous water-saturated media are related through a set of coupled equations for acoustic and electrodynamic parameters [Frenkel, 1944;Pride, 1994;Pride and Garambois, 2005;Hu and Gao, 2011]. When considering the seismic problem, the electroosmotic phenomenon can be neglected. ...

... The equations related the volumetric strain of porous medium and the variation of pore fluid overpressure (with respect to hydrostatic pressure) have been derived by [Frenkel, 1944]. These equations include the second order time-derivatives which can be neglected in the low frequency region. ...

Seismoelectric effect is studied in the framework of classical Lamb's problem with impulse or time-variable mechanical action on an elastic porous half-space. Radiation of elastic waves gives rise to pressure variation of groundwater fluid contained in pores and cracks. This causes the generation of telluric electric fields and currents due to the seismoelectric effect. A diffusion type equation is applied to describe the variations of the pore pressure and telluric electric field. Particular emphasis has been placed on the properties of seismoelectric signals caused by Rayleigh wave propagation since this wave has maximal amplitude at a considerable distance from the seismic source. For practical purposes and geophysical application, the co-seismic phenomena related to seismoelectric effect are examined in more detail.

... The research on the seismoelectric effect of water-wetted porous media is quite mature. In 1944, Frenkel proposed a coupling equation for the seismoelectric effect of porous media [4]. However, this theory has some drawbacks, because it ignores the effect of the solid phase and suggests that the streaming potential is only generated by the longitudinal wave of the acoustic field. ...

... The existing studies on the seismoelectric effect of porous media mainly focus on water-wetted porous media [4][5][6][7][8][9][10][11][12][13][14], while there are few studies on the seismoelectric effect of oil-wetted porous media containing an oil-water two-phase flow. Based on the above, we used the capillary model [24] and adopted the multiphase flow model [10,11], in which oil and water are assumed to be immiscible, continuous and distributed in parallel, which makes it a more suitable option than that used by Jackson. ...

In oil reservoirs, if oil mainly has wettability in the solid phase, such as in carbonate reservoirs, the medium is oil-wetted. For oil-wetted porous media containing an oil and water two-phase flow, there are electric double layers at both the oil–solid interface and the oil–water interface, which can stimulate the seismoelectric effect. To date, most of the studies on the seismoelectric effects of porous media have mainly focused on water-wetted porous media, however, there are few reported studies on cases of oil-wetted porous media, especially on oil-wetted porous media containing an oil–water two-phase flow. In this paper, we adopted the oil-wetted pore model, in which oil and water are assumed to be immiscible, and each phase is continuous and distributed in parallel. We also considered the influence of the electric double layer at both the oil–solid interface and the oil–water interface on the seismoelectric effect. It was concluded that the seismoelectric effect of oil-wetted porous media containing a two-phase flow is mainly caused by the electric double layer at the oil–water interface, while the effect of the electric double layer at the oil–solid interface can be ignored. We regarded the two-phase flow as an equivalent fluid, and then we derived a governing equation of the seismoelectric effect and proposed the flux-averaging method to derive the electrokinetic coupling coefficients under the excitation of a steady acoustic field and a time-harmonic acoustic field. We also investigated the effects of formation parameters, namely, water saturation, pore size, water viscosity and porosity, on the seismoelectric effect, which can provide a theoretical reference for the study of seismoelectric logging in oil-wetted porous formations containing a two-phase flow.

... It was found that Seismic waves produce changes in the medium that can change electric resistivity and/or produce an EM field (Blau & Statham, 1936). Several physical mechanisms have been suggested for electrical currents excitation by seismic waves in a porous medium (Ivanov, 1939) (Frenkel, 1944) (Neishtadt, Eppelbaum, & Levitski, 2006) : ...

... The SE effect of the second kind was first suggested by Ivanov (1939) and explained by Frenkel (1944). They suggested that seismic waves in porous media can cause the fluid in the pores to flow. ...

The Seismic-Electric (SE) effect couples the seismic and electromagnetic (EM) fields, and
responds to attributes such as porosity, water saturation and dissolved minerals in the fluid, that are not resolved by other geophysical methods. A SE prospecting has the potential to detect and locate shallow sub-surface targets such as ground water, archaeological items, forensic evidence, geological faults, voids and sinkhole. This work addresses the need to improve the capability of the SE method to detect interfaces with low water content, in arid areas in Israel. There are several physical processes that convert seismic waves into electrical current which were calculated to be very small. These currents were estimated to produce electric field of E~10 nV/m. Measuring these signal using ground electrodes was made possible using a recording system that was specially designed, built, tested and implemented. Two full scale experiments were carried-out, one in the Judea desert and the other on the dunes of Haifa-bay, having underground targets and incorporating both geophones and electrodes for detecting the signals. A seismic wave was excited using a hammer impact on a metal plate lying on the ground. The seismic waves were detected by the geophones and theSeismic-Electric signal by the electrodes. In both experiments a SE response from the underground targets were detected. Since the ground electrodes respond to the seismic waves passing through them ("co-seismic response"), there are times when the SE response cannot be resolved from the co-seismic one. A SE time window was identified in which the SE response is detectable free of any other response. It is clear that even in arid places or with low water content in the ground, there is a measurable SE response that can detect near surface underground targets. It is suggested to use multi-sensors surveys in order to increase signal to noise ratio and get location of the targets.

... The Frenkel-Biot [23,24] and Gassman [25] theories for seismological low-frequency range were used to substantiate the applicability of the derived equations of motion for the case of porous saturated soil. The procedure for passing from the solution for SH-waves to the solution for P-waves is based on solving the wave equations using the P-wave velocity instead of S-wave velocity and degradation curves of constrained moduli for saturated soils instead of shear moduli according to [12]. ...

... However, there is a need to make sure that this approach is correct. To describe the joint relative motion of a solid soil matrix and pore fluid, the Frenkel-Biot theory [23,24] is overwhelmingly used. The theory of poroelastic wave propagation is valid under the assumptions about a bound and isotropic porous medium, the pore scale is much smaller in comparison with the wavelength and small deformations that ensure linear elastic material behavior. ...

The paper is devoted to the problem of numerical modeling of earthquake response of porous saturated soil deposits to seismic waves propagation. Site-specific earthquake response analysis is a necessary and important component of seismic hazard assessment. Accounting for the complex structure of porous saturated soils, i.e., the content in them, in addition to the solid matrix, pore water, gas mixture and ice, is especially important for the water areas in the zones of continuous or sparse permafrost, as well as the massive release of bubble gas from bottom sediments. The purpose of this study is to introduce an algorithm and its Matlab implementation for numerical modeling of the nonlinear response of porous saturated soil deposits to vertical P- and SH-waves propagation. The presented MatNERApor package consists of a set of Matlab scripts and functions. The package was tested and verified using the records of vertical seismic arrays of the Kik-net network. In addition, the records of local earthquakes obtained by ocean bottom seismographs in the Laptev Sea in 2019–2020 were used to demonstrate the effect of the water layer above the seabed sites on the reduction of vertical motions spectra. The results of the calculations showed good agreement with the data obtained from real seismic records, which justifies the correctness of the theoretical basis of the presented algorithm and its software implementation.

... Recently, many researchers have shown the remarkable effect of porosity parameters on the phase and group velocity of surface waves. However, the foundation theory of elastic surface wave in fluid-saturated porous media was introduced by Frenkel [13] and Biot [2] in the middle of the 1900s. Later, highly appreciated literature has been published on the theory and applications of elastic surface wave in fluid-saturated porous media by Edelman [12], Ke et al. [20], Zhang et al. [41], Kundu et al. [22], Manna et al. [28] and many more. ...

... Using the stress-displacement relation (8) in Eq. (13), the equation of motion can be written as ...

This paper investigates the dynamic behavior of material strength due to the traction and propagation of torsional surface waves. Dynamic behavior of material strength depends on non-homogeneity, aeolotropy, irregularity, porosity, and internal prestress in an elastic body. The geometry of the model is constructed by a local-elastic, highly non-homogeneous finite thickness layer over a trigonometric variation of non-homogeneous aeolotropic porous half-space. The surface boundaries of both the free surface and interface are considered corrugated boundaries. The separation variable technique is used to solve the equation of motions (hyperbolic type PDEs), and the Bessel function is adopted in the displacement components of torsional waves for both the media. Because of highly non-homogeneity in the upper layer, the equation of motion comprises a special form of differential equation called Whittaker differential equation, and the solution (as displacement) of the equation is obtained in the form of Whittaker functions. The quantitative prediction of a differential equation is one of the significant tasks in the study of wave propagation, thus, the dispersion equation of the torsional surface waves is introduced analytically with the help of traction-free and continuity conditions. The validity of the final dispersion equation is described using several particular cases. To study the intensity of the torsional surface waves in the model, traction components have been computed along the coordinate axes, which helps to detect the material fracture in the medium. MATLAB software is used to study the dynamic behavior of the phase velocity, under the influence of prestressing irregular boundary surface, non-homogeneity, and porosity parameters in the media. By the numerical simulations, some marvelous changes in the propagation of the torsional surface waves due to the irregularity at the interface and free surface are obtained. The work may be useful to study the behavior of torsional surface waves in the micro-structure material and smart material composite structures used in the aerospace industry and seismology.

... At present, the theory and application of seismoelectric logging is an important research and development subject. Frenkel carried out a theoretical study on the seismoelectric effect and proposed the wave transmission equations for porous media [2]. Based on Biot's acoustic theory of porous media, the Pride theory of the seismoelectric coupling of porous media was obtained [3,4]. ...

... At the same time, the gain setting of the chip is simple and a single resistor can be used to control the gain. The corresponding gain formula is given by Equation (2). According to this equation, the gain reaches the value of 11 when R G is set to 4.99 kΩ. ...

To increase the accuracy of reservoir evaluation, a new type of seismoelectric logging instrument was designed. The designed tool comprises the invented sonde-structured array complex. The tool includes several modules, including a signal excitation module, data acquisition module, phased array transmitting module, impedance matching module and a main system control circuit, which are interconnected through high-speed tool bus to form a distributed architecture. UC/OS-II was used for the real-time system control. After constructing the experimental measurement system prototype of the seismoelectric logging detector, its performance was verified in the laboratory. The obtained results showed that the consistency between the multi-channel received waveform amplitude and benchmark spectrum was more than 97%. The binary phased linear array transmitting module of the instrument can realize 0° to 20° deflection and directional radiation. In the end, a field test was conducted to verify the tool’s performance in downhole conditions. The results of this test proved the effectiveness of the developed seismoelectric logging tool.

... There were many studies on the seismoelectric effect in porous media. In 1944, Frenkel first proposed the coupling equations of the seismoelectric effect in porous media [1], but this theory has some limitations: it ignores the influence of solid skeleton acceleration on the streaming current and suggests that only longitudinal waves can generate a streaming potential. In 1953, Packard proposed the capillary pore model and studied the microscopic mechanism of the seismoelectric effect [2]. ...

The seismoelectric effect is the fundamental basis for seismoelectric logging. Most of the existing theories for the seismoelectric effect are based on the Pride theory, which adopts the assumption of a thin electric double layer and uses the volume-averaging method to derive the seismoelectric coupling equations; hence, the obtained electrokinetic coupling coefficient is not applicable to large-Debye-length cases. In addition, the Pride theory neglects the change in seepage velocity with the radial position of the pore when calculating the streaming current, which leads to an inaccurate reflection of the influence of pore size on the electrokinetic coupling coefficient. In this study, we proposed a flux-averaging method to solve the effective net residual charge density of porous media and further derived the electrokinetic coupling coefficient expressed by the effective net residual charge density. We also investigated the effect of formation parameters and compared the results with those calculated using the Pride theory. Since the proposed method is not limited by the thin electric double layer assumption, it is suitable for both small- and large-Debye-length cases. Moreover, we also carried out flume experiments to investigate the influence of salinity, where both thin and thick electric double layer cases were studied. The comparison between the results of the experiment and simulation verified the correctness of the proposed method. Furthermore, the proposed method took into account the variation in seepage velocity with pore location when solving for the streaming current; therefore, the influence of the pore size on the electrokinetic coefficient can be described more accurately.

... The phenomenon of seismic-to-electromagnetic energy conversion along with its counterpart have been known for a long time by the geophysical community, as well as the existence of several theoretical models and experiments aiming to describe it, i.e., (Frenkel, 1944;Neev and Yeatts, 1989;Thompson and Gist, 1993). The work of Pride (1994), presenting a closed set of equations modeling the propagation of coupled mechanical and electromagnetic perturbations in an isotropic porous medium saturated with an electrolyte, created a long-standing wave of interest of many research groups around the world. ...

... The phenomenon of seismic-to-electromagnetic energy conversion along with its counterpart have been known for a long time by the geophysical community, as well as the existence of several theoretical models and experiments aiming to describe it, i.e., (Frenkel, 1944;Neev and Yeatts, 1989;Thompson and Gist, 1993). The work of Pride (1994), presenting a closed set of equations modeling the propagation of coupled mechanical and electromagnetic perturbations in an isotropic porous medium saturated with an electrolyte, created a long-standing wave of interest of many research groups around the world. ...

In this paper a set of equations governing the electromagnetic/acoustic coupling in partially-saturated porous rocks in the low-frequency regime is derived. The equations are obtained by volume averaging of fundamental electromagnetic and mechanical equations valid at the pore-scale, following the same procedure as the one developed in the seminal paper of S. Pride for porous media where the fluid electrolyte fully saturates the pore space. In the present approach it is assumed that the porous rock is partially saturated with a wetting-fluid electrolyte (water) and a non-wetting fluid (air). We also assume that an electromagnetic/mechanical coupling exists at the water-solid and water-air contact surfaces through adsorbed excess charges balanced by mobile ions in the water. The capillary pressure perturbations are assumed to be negligible. The governing equations thus derived are similar to the ones obtained by Pride with the main difference that the various coefficients, including the electrokinetic coupling coefficient and electric conductivity appearing in the transport equations are functions of the water saturation and depend on electrical and topological properties of both electric double layers. Onsager reciprocity is also demonstrated.

... The existence of these waves was proven experimentally in the work (Plona, 1980). In many cases, the propagation of elastic waves in such media is followed by generation of electromagnetic radiation (Frenkel, 1944;Grobbe et al., 2020;Ivanov, 1939;Schakel et al., 2012). Among the reasons of such phenomena are the electrokinetic effects associated with the displacement of the ions in double electric layers present near the boundaries separating two phases of the medium. ...

Elastic wave propagation in porous media containing a movable fluid is frequently accompanied by electromagnetic field generation. Such field can be produced, for instance, by piezoelectric effect or have electrokinetic origin. In the present work, we consider the case of electrolyte-filled pores; thus, the generated electromagnetic field is associated with the electrokinetic effects. In this case, the magnitude of the electric current generated by the elastic wave is proportional to the magnitude of the relative displacement in the fluid and the solid phases. The presence of in-homogeneities in a porous medium can result in appearance of additional fluid flow with respect to the solid phase; thus, an additional electric field is generated. In our work, the magnitude of the electromagnetic field generated during the scattering of an elastic wave on a spherical porous inclusion in a porous matrix is calculated. The inclusion can differ from the matrix by the properties of the fluid in the pores or the properties of the elastic skeleton, such as porosity or permeability. The solution of the dynamic equations of electrokinetics is obtained by using the hypothesis that the inclusion size is much larger than the characteristic size of the pores. We have obtained the dependencies of the amplitude of the generated electromagnetic field on the inclusion parameters and the incident wave frequency. It was shown that the analysis of the additional electromagnetic field generated during the scattering of the elastic wave can provide some useful information about the structure of porous media.

... The poroelasticity theory is widely used in geomechanics, biophysics and other fields of science and technology. The Frenkel-Biot theory is a closed system of second order differential equations with respect to displacement vectors of an elastic porous body and fluid displacements [5,12]. This system describes the propagation of seismic waves in a porous medium and in the isotropic case contains four independent elastic parameters. ...

In a reversible hydrodynamic approximation, a closed system of second order dynamic equation with respect to the displacement vector of an elastic porous body and pore pressure has been obtained. The Cauchy problem for the obtained system of poroelasticity equations in the stationary case is considered. The Carleman formula for the Cauchy problem under consideration has been constructed.

... relatively low fluid salinity ∕ =0) the seismoelectric effect (Parkhomenko and Gaskarov, 1971;Ageeva et al., 1999). This phenomenon generates an electric signal propagating almost at the same seismic velocity (Ivanov, 1939;Frenkel, 1944). The measurements, as shown in the experimental setup presented in Fig. 1, are performed via the use of electrodes. ...

Seismoelectric (SE) measurements present low voltages signals that are difficult to detect and acquire in a quantitative way. One of the main reasons is the lack of information about the impedance seen by the electronic acquisition system while the SE effect is taking place. Knowing its nature (real or complex) and values are required to better design the electronic acquisition system, to show its limitations and also to better estimate the SE voltage in modulus and phase, i.e. being insensitive to the influence of the electronic acquisition system and contact losses linked to electrode coupling within the medium. There are electronic models that represent the electrodes used to measure these signals but no models representing the whole system output impedance during SE experiments. This is crucial because the model values change depending on many factors such as soil composition. Thus, to give an answer to the remaining questions we propose a novel electronic model of SE measurements which components values are obtained while a SE signal is propagating. We present a new theoretical-experimental method to retrieve the impedance of SE signal sources and better SE signal estimation. To validate the model we performed experiments and simulations. The results obtained present a goodness of fit of 90% between the model and the experimental measurements. They also show that the SE measurements impedance is complex and that even 1 m coaxial cable between the electrodes and the acquisition system affects considerably the SE signal. Hence, to avoid these issues we propose the use of active electrodes that include the preamplifier in the electrode itself.

... The main approach to the study of wave processes in liquid-saturated porous media is based on the Frenkel-Biot theory [8,9]. The model, which describes the processes of deformation of an elastic porous medium and the flow of fluid in it, is macroscopic, i.e., assumes that the space is filled with an enclosing poroelastic two-phase medium, and the phases corresponding to the porous solid and the liquid contained in the pores are present at each point in space. ...

This article presents an algorithm for the numerical solution of an initial-boundary value problem for a symmetric t-hyperbolic system of partial differential equations. This problem is based on continual filtration model, which describes the propagation of seismic waves in a poroelastic medium saturated with a fluid characterized by such physical parameters as the propagation velocities of longitudinal P- (fast and slow) and transverse S-waves, the density of the medium materials, and porosity. The system of linearized equations of saturated porous media is formulated in terms of physical variables of the velocity–stress tensor of the porous matrix and the velocity–pressure of the saturating fluid in the absence of energy dissipation. The solution is implemented numerically using an explicit finite difference upwind scheme built on a staggered grid to avoid the appearance of oscillations in the solution functions. The program code implementing parallel computing is developed in the high-performance Julia programming language. The possibility of using the approach is demonstrated by the example of solving the problem of propagation of seismic waves from a source located in the formation. Computational experiments based on real data from oil reservoirs have been implemented, and dynamic visualization of solutions consistent with the first waves arrival times has been obtained.

... To analyze the mechanisms of seismoelectric phenomena and the characteristics of the abovementioned two types of signals, Pride [6] proposed a series of equations to describe the conversion of the seismic and EM waves for a homogeneous poroelastic isotropic medium. Depending on the electrokinetic coupling coefficient, Pride combined Biot's poroelastodynamic equations [10], [11] with Maxwell' equations to account for the propagation of the EM field [6], [12], [13]. Moreover, Revil and Linde [14] derived a model to describe the propagation of seismoelectric waves in the unsaturated porous medium. ...

Considering the viscoelastic anisotropy and electrical depression characteristics of the complex geological media, we introduce the generalized standard linear solid (GSLS) model to describe the relaxation effect of the solid skeleton and the Cole–Cole model to describe the frequency dependence of electric conductivity. The seismoelectric model of the poro-viscoelastic anisotropic medium was constructed, and the corresponding wave and diffusion equations in the time domain were derived. We then analyze the characteristics of seismoelectric wavefields in viscoelastic transversely isotropic (TI) media with a homogenous model, a two-layer model, and a layered model with depression. Results show that the TI anisotropy, viscosity of fluid, and tilt angle all have significant effects on the propagation of seismoelectric waves in the homogenous model. The strong attenuation of seismoelectric waves in the two-layer model shows the validity of the relaxed skeleton and frequency-dependent conductivity used in our approach, which could also effectively capture the reflection and transmission phenomena in the seismic and associated electromagnetic (EM) fields in the layered model with depression.

... As is well known, the first and second kind of compressional waves exist in saturated porous media, which are predicted by the Biot theory [13,14]. The wave mode, dispersion relation [15,16], and transfer function [17] are proposed. ...

Propagation of pore pressure and stress in water-saturated elastic porous media is theoretically investigated when considering the Darcy-Brinkman law. The wave mode, phase velocity, phase lag, damping factor, and characteristic frequency are found from the updated mathematic model. The Brinkman term describes the fluid viscous shear effects and importantly contributes to the dispersion relation and wave damping. The coincidence of the properties of Biot waves of the first and second kinds occurs at a characteristic frequency, which is remarkably influenced by the Brinkman term. A key finding is that, compared to the Darcy-Brinkman law, Darcy’s law overestimates the phase velocity, damping, and phase lag of the first wave, while underestimates the phase velocity, damping, and phase difference of the second wave. The introduction of the Darcy-Brinkman law yields an improved description of the damping of the compressional wave modes in saturated porous media.

... The seismoelectrical effect was studied by pioneering authors in the 1930 s (e.g., Thompson 1939;Frenkel 1944) and it remains a highly active research field (e.g., Pride & Garambois 2005;Revil et al. 2015;Jouniaux & Zyserman 2016;Jougnot & Solazzi 2021). To model the seismoelectric effect, one normally uses the electrokinetic coupling coefficient C EK (ω), which is a frequency dependent parameter that relates the measured electrical potential difference with the fluid pressure difference driving the pore fluid flow. ...

Seismoelectric signals are generated by electrokinetic coupling from seismic wave propagation in fluid-filled porous media. This process is directly related to the existence of an electrical double layer at the interface between the pore fluid and minerals composing the pore walls. The seismoelectric method attracts the interest of researchers in different areas, from oil and gas reservoir characterization to hydrogeophysics, due to the sensitivity of the seismoelectric signals to medium and fluid properties. In this work, we propose a physically-based model for the dynamic streaming potential coupling coefficient (SPCC) by conceptualizing a porous medium as a bundle of tortuous capillaries characterized by presenting different pore size distributions (PSD). The results show that the dynamic streaming potential coupling coefficient is a complex function depending on the properties of pore fluid, mineral-pore fluid interfaces, microstructural parameters of porous media and frequency. Parameters influencing the dynamic SPCC are investigated and explained. In particular, we show that the PSD affects the transition frequency as well as the shape of the SPCC response as a function of frequency. The proposed model is then compared with published data and previous models. It is found that the approach using the lognormal distribution is in very good agreement with experimental data as well as with previous models. Conversely, the approach that uses the fractal distribution provides a good match with published data for sandstone samples but not for sand samples. This result implies that the fractal PSD may not be pertinent for the considered sand samples, which exhibit a relatively narrow distribution of pore sizes. Our proposed approach can work for any PSD, for example, including complex ones such as double porosity or inferred from direct measurements. This makes the proposed models more versatile than models available in literature.

... The seismoelectrical effect was studied by pioneering authors in the 1930 s (e.g., Thompson 1939;Frenkel 1944) and it remains a highly active research field (e.g., Pride & Garambois 2005;Revil et al. 2015;Jouniaux & Zyserman 2016;Jougnot & Solazzi 2021). To model the seismoelectric effect, one normally uses the electrokinetic coupling coefficient C EK (ω), which is a frequency dependent parameter that relates the measured electrical potential difference with the fluid pressure difference driving the pore fluid flow. ...

Seismoelectric signals are generated by electrokinetic coupling from seismic wave propagation in fluid-filled porous media. This process is directly related to the existence of an electrical double layer at the interface between the pore fluid and minerals composing the pore walls. The seismoelectric method attracts the interest of researchers in different areas, from oil and gas reservoir characterization to hydrogeophysics, due to the sensitivity of the seismoelectric signals to medium and fluid properties. In this work, we propose a physically-based model for the dynamic streaming potential coupling coefficient (SPCC) by conceptualizing a porous medium as a bundle of tortuous capillaries characterized by presenting different pore size distributions (PSD). The results show that the dynamic streaming potential coupling coefficient is a complex function depending on the properties of pore fluid, mineral-pore fluid interfaces, microstructural parameters of porous media and frequency. Parameters influencing the dynamic SPCC are investigated and explained. In particular, we show that the PSD affects the transition frequency as well as the shape of the SPCC response as a function of frequency. The proposed model is then compared with published data and previous models. It is found that the approach using the lognormal distribution is in very good agreement with experimental data as well as with previous models. Conversely, the approach that uses the fractal distribution provides a good match with published data for sandstone samples but not for sand samples. This result implies that the fractal PSD may not be pertinent for the considered sand samples, which exhibit a relatively narrow distribution of pore sizes. Our proposed approach can work for any PSD, for example, including complex ones such as double porosity or inferred from direct measurements. This makes the proposed models more versatile than models available in literature.

... In this chapter, we present the principles of the electrokinetic, following Pride's theory, [105,110]. The electrokinetic theory was first introduced by Frenkel [109,56], then it has been generalized to Pride's model. The conversions have been observed on the field and in laboratory experiments, [23,117]. ...

We consider the time-harmonic waves propagation in conducting poroelastic media. In poroelastic materials, whichare composed of an elastic solid frame and pores filled with fluid, wave propagation is described by Biot’s model. Ingeophysical media, due to the polarization of the fluid in the pores, one can observe the creation of elecromagneticwaves and even conversions between electromagnetic and seismic fields. They are the electrokinetic effects and aremodeled using Pride’s equations, a coupling between Maxwell’s and Biot’s equations. The electrokinetic couplinghas been observed in natural geophysical media both in laboratory and on the field. The converted waves are veryinteresting because they are heavily sensitive to the medium properties, and the seismoelectric conversions could forexample help to locate interfaces in the material that seismic waves cannot detect. The characterization of poroelasticor conducting poroelastic media is complex and involves many physical parameters, some of which depend non-linearlyon the frequencies. In addition, the seismic and electromagnetic speeds are significantly different, which is complicatedto handle for time domain simulations. Hence, we have chosen to solve the equations in the frequency domain andto use a Fourier transform to generate the seismograms in time domain. The main drawback to this is that we mustinvert one global linear system for each frequency, and this has a large computational cost because of the complexityof the equations and hence the high number of unknowns. In this work, we focus on the development and validation ofa Hybridizable Discontinuous Galerkin method for Pride’s model in the frequency regime. We validate the numericalmethod in two dimensions in circular geometry thanks to analytical solutions that we have constructed. Using theseanalytical solutions, we show in particular that the numerical method has an optimal order of convergence. In addition,to extend the method to infinite domains, we propose new radiation boundary condition for poroelastic equations andelectrokinetic equations. We have also implemented Perfectly Matched Layers and we have compared the performancesof the two methods. Finally, we show that the code we have developed is capable of modeling electrokinetic conversionsin the time domain. It is worth noting that we also provide details on the development of a HDG formulation for theporoelastic equations both in 2D and 3D.

... Biot's theory does not take this local viscous fluid flow into account. Biot's theory attributes the vicious interaction between the solid skeleton and the fluid to be the attenuation mechanism, and it predicts the existence of the slow P-wave, investigated first on seismoelectric waves by Frenkel (1944). Although this viscous interaction has a significant attenuation on the slow P-wave, it has much less effect on the fast P-wave, especially in the seismic band of frequency (Dai et al., 1995;Zhang et al., 2014). ...

Biots theory of poroelasticity describes seismic waves propagating through fluid-saturated porous media, so-called two-phase media. The classic Biots theory of poroelasticity considers the wave dissipation mechanism being the friction of relative motion between the fluid in the pores and the solid rock skeleton. However, within the seismic frequency band, the friction has a major influence only on the slow P-wave and has an insignificant influence on the fast P-wave. In order to represent the intrinsic viscoelasticity of the solid skeleton, we incorporate a generalized viscoelastic wave equation into Biots theory for the fluid-saturated porous media. The generalized equation which unifies the pure elastic and viscoelastic cases is constituted by a single viscoelastic parameter, presented as the fractional order of the wavefield derivative in the compact form of the wave equation. The generalized equation that includes the viscoelasticity appropriately describes the dissipation characteristics of the fast P-wave. Plane-wave analysis and numerical solutions of the proposed wave equation reveal that (1) the viscoelasticity in the solid skeleton causes the energy attenuation on the fast P-wave and the slow P-wave at the same order of magnitude, and (2) the generalized viscoelastic wave equation effectively describes the dissipation effect of the waves propagating through the fluid-saturated porous media.

... Seismo electric effects related to electro kinetic potential, dielectric permitivity, pressure gradient, fluid viscosity, and electric conductivty was first reported by Frenkel J [1]. Capillary pressure follows the scaling law at low wetting phase saturation was reported by Li K, et al. [2]. ...

... Although the seismoelectric/electroseismic effect has been known for several tens of years (Ivanov, 1939;Frenkel, 1944), it reached a rapid progress just in the last thirty years after Pride (1994) established a set of governing equations that describe the seismoelectric and electroseismic phenomena by coupling Biot's poroelasticity equations to the full Maxwell electromagnetic (EM) equations. Since then, lots of numerical simulations (e.g., Pride and Haartsen, 1996;Haartsen and Pride, 1997;Garambois and Dietrich, 2002) and field/laboratory experiments (e.g., Butler et al., 1996;Zhu et al., 1999Zhu et al., , 2000Zhu et al., , 2008 have boosted the geophysical research on seismoelectric and electroseismic phenomena. ...

Seismoelectric and electroseismic interface responses resulting from the electrokinetic effect are useful to study the properties of subsurface medium. In this article, we investigate the characteristics of these interfacial signals generated at irregular subsurface interfaces considering hydrocarbon exploration scenarios. Adopting several typical models, the electroseismic and seismoelectric wavefields are calculated using the finite-difference frequency domain method. Besides the well-known electroseismic and seismoelectric signals created at a flat interface, the scattered seismic wave and scattered electromagnetic (EM) wave can also be generated by EM and seismic sources at subsurface scattering points, respectively. When an electric source is applied for excitation, the waveforms recorded by horizontally- or vertically-aligned receiver array indicate that electroseismic interface responses nearly do not change with the source locations and can directly delineate the shapes and morphologies of the corresponding interfaces. Simulations of seismoelectric wavefield show that both the interface seismoelectric responses and scattered EM waves display as flat events in the electric record. It is not easy to distinguish them if we do not know the realistic underground structure. Based on simulations, the electroseismic interface responses seem more promising than the seismoelectric interface responses for imaging the subsurface interfaces.

... The study of physical processes underlying the seismoelectrical signal generation can be tracked back to the late 1930's (e.g., Thomson, 1939;Frenkel, 1944) but it remains an active 3 research subject (e.g., Pride and Garambois, 2005;Revil et al., 2015;Jouniaux and Zyserman, 2016). The most traditional approach to model the seismoelectric conversion is the use of the electrokinetic coupling coefficient, that is, a frequency-dependent parameter that relates a difference in fluid pressure to a difference in electrical potential. ...

The seismoelectric method is based on the capacity of seismic waves to generate measurable modifications of the electrical field in porous media. Even though it combines the advantage of both seismic and geoelectrical methods, it remains largely under-used in hydrogeophysics. Its signal results from an electrokinetic coupling that can be modeled using either the coupling coefficient or the effective excess charge density. The traditional approach is based on the frequency dependent coupling coefficient, which relates the pressure drop with the change in the electrical potential. A more recent approach consists of describing the excess charge that is effectively dragged by water flowing within the pores. In this work, we present a new model for the frequency dependent effective excess charge density. For this, we make use of a mechanistic up-scaling of the electrokinetic coupling in a capillary. This novel flux-averaging approach takes into account inertial effects arising within the pore space to explain the frequency dependence of the effective excess charge density. The presented model is successfully compared to previous models and published data. This new upscaling approach has the potential of fundamentally improving our current understanding of the seismoelectrical signal in more complex environments, such as partially saturated and fractured media.

... The pros of this method is a low computational complexity. Modern computing systems allow scientists to deal with the Biot model [10,11], which is more detailed. Its internal structure produces three seismic waves: two longitudinal waves and one shear wave. ...

The discovery of oil deposits with complex internal structures requires an improvement of available seismic survey methods. The development of modern high-performance computing systems provides an opportunity to use more sophisticated mathematical models. This paper aims to investigate seismic wave propagation in porous fluid-saturated media. The Dorovsky three velocity model was formulated in the two-dimensional case and its numerical solution was obtained with the grid-characteristic method. The computational domain consisted of three layers with different rheology: a water layer, a porous fluid-saturated layer, and an elastic layer. Explicit contact conditions were derived between them and successfully applied with the help of Riemann invariants. The curvature of geological layers was taken into consideration by means of structured hexahedral grids. The time evolution of the spatial distribution of stress tensors and material velocity vectors were calculated and analyzed. These signals contain a mixture of volume, surface, transmitted and reflected waves.

... The pioneering studies of Reuss (1809), Wiedemann (1852), von Helmholtz (1879), Smoluchowski (1903), and Frenkel (1944) were completed in more recent times with the development of the corresponding theory. Thompson and Gist (1993) and Pride (1994) explained the electrokinetic coupling mechanism within the double electrical layer at the solid-fluid interface. ...

... relatively low fluid salinity ∕ =0) the seismoelectric effect (Parkhomenko and Gaskarov, 1971;Ageeva et al., 1999). This phenomenon generates an electric signal propagating almost at the same seismic velocity (Ivanov, 1939;Frenkel, 1944). The measurements, as shown in the experimental setup presented in Fig. 1, are performed via the use of electrodes. ...

We present a general approach to seismoelectric (SE) field modeling based on coupled multiphysics formulation, including the equation of motion of an isotropic elastic medium, Frenkel's equations for the Biot poroelastic model, and Maxwell's equations for the electromagnetic (EM) field assuming diffusion approximation. Based on this approach, we have simulated an extensive dataset, including dynamic patterns of SE-signal components for a set of models differing in a number of geomechanical, pore fluid flow and electrical conductivity properties. Simulated SE-field patterns reveal their sensitivity to structure permeability and pore fluid properties.

Seismo-electric phenomena gained more attention from geophysicists over the last decade.The development of theoretical background and the success of laboratory experimentsas well as limited field applications give a lot of opportunities and hope as a meansfor providing exploration and production data. Seismo-electric effect accommodates anyphenomena which links seismic and electrical energy including seismic to electric conversionas well as electro kinetic in origin.Experiment has been conducted to prove that free ions can be considered to accumulateinside a granite crack containing crude oil which in turn can generate stream oscillatoryelectric current when a seismic wave hit the fracture. As a result, electric potential canbe detected at the mouth of the fracture which intersects the borehole. The environmentalnature of the mouth which is full of fluid facilitates the detection of high resolution seismoelectricsignal by simple electrodes which is made of metal.

This chapter explores the analogy between the fields of acoustics and electromagnetism. The two-dimensional Maxwell equations are equivalent to the SH-wave equation based on the Maxwell mechanical model. It is shown that Fresnel formulae can be obtained from the reflection and transmission coefficients of shear waves. The layer problem illustrates the analogy with the quantum tunneling and magnetotelluric fields as examples. Using the Debye-Zener analogy, the analogy is extended to three dimensions, where the acoustic and electromagnetic material properties are equivalent. Moreover, an electromagnetic energy-balance equation is obtained from viscoelasticity. Other analogies involve, for instance, Fresnel wave surface, Backus averaging, the Kramers-Kronig dispersion relations, and the reciprocity principle. Useful cross-property relations between elastic-wave velocity and electrical conductivity are discussed. Finally, the wave equation can also describe the behavior of the recently discovered gravitational waves.

We analyze theoretically ultra-low frequency electromagnetic noise caused by deformations of seabed and porous coastal rocks subjected to incident long oceanic waves. A variable pressure on the seabed due to propagation of long gravity waves (LGWs) gives rise to variations in pore pressure gradient followed by groundwater filtration in pores and channels of porous rocks. These processes result in the generation of telluric electric currents in water-saturated porous rock of the seashore due to electrokinetic effect. In the model a displacement of the sea surface in LGWs is described in the "shallow water" approximation. A set of basic equations describing rock strain and electrokinetic effect is solved in quasi-static approximation. The telluric electric field in the porous rocks of coastal zone are found as a function of depth and distance to the coastline at different frequencies of LGWs. The theoretical analysis has shown that telluric electric noise produced by the LGW can exceed the level of natural electric noise during geomagnetically quiet period in a coastal strip about several tens of meters.

Absolute and effective permeability are two very important petrophysical parameters that govern the production of gas from hydrate-bearing sediments. In the present study, an attempt is made to estimate the permeability from well log data using a theoretical approach, which is validated by comparing the obtained results with the core-derived values. The log data of the well NGHP-02-16B in Krishna–Godavari basin is used for the purpose of computing the permeability, and the core data from the same site are used for validation. The absolute permeability in the reservoir estimated using the Timur method ranges from 0.1 to 100 mD, and matches well with the core sample permeability. It is also demonstrated that the hydrate saturation and the existing hydrate morphology in pore spaces of the sediments play a significant role in the computation of effective permeability. The computed P-wave velocities reveal that the hydrates occur within the pore spaces of the sediments with hydrate saturation of 44–90%. The effective permeability of the hydrate-bearing sediments obtained by the Masuda model with a permeability reduction exponent (N = 2.5) agrees well with the core-derived permeability. The coating of the grain surfaces by the interspace hydrate within the pore is confirmed by comparison and normalization of effective permeability obtained from the Masuda model. The present study infers that the Masuda model is the most accurate and can be reliably used in the absence of core data for the computation of permeability of hydrate-bearing sediments in the vicinity of the study area.

Wave spreading in the media with fading is best studied for the conditions of low amplitude flat waves within a boundless and uniform porous medium saturated with a viscous fluid. The factor affecting the wave spreading is various boundaries and rock stratification in the productive reservoirs. Experimental observations showed that in a porous medium saturated with oil, gas, or water, reflection coefficients from various fluids’ contact surfaces are comparable with reflection coefficients by geological boundaries separating different lithologies. As an indicator of efficiency at a certain frequency may serve encompassment radius within which are maintained certain interrelations between the threshold values of vibration parameters—vibratory displacements and vibratory accelerations. Currently, the problem of wave propagation from the vibrating surface of the reservoir matrix into the various media has not yet been satisfactorily solved. Vibration excitation in oil reservoirs is best studied in cases of power load application directly to the land surface.

Thermodynamically, a near‐well wall reservoir rock zone resides in a substantially unbalanced state, in the phase of active energy and mass exchange with the well and reservoir. Among favorable methods of defeating formation damage, it may turn out to be the vibration. Low‐frequency vibrations accelerate by two or three orders of magnitude the mechanical tension relaxation processes. The vibration effect on decolmatation processes caused by contamination of the near‐bottomhole zone by various materials has been studied on physical models. The material of a cement sheath is “adjusting” to the vibration load and no noticeable fatigue damage occurs within any arbitrary long time. This chapter aims to identify major factors determining vibro‐endurance of hardened cement in well conditions. As for the effect of loading frequency, it is most dangerous in cases of vibro‐fatigue as well as vibro‐creep have low frequencies.

The first articulation of the second type of dilatational wave propagating through fluid-saturated geomaterials has been traced to Heinrich’s formulation built on Fillunger’s framework of the mixture theory and Terzaghi’s principle of effective stress. Although this Fillunger–Heinrich theory (FHT) precedes the celebrated Biot’s wave theory and Frenkel’s theory, research has yet to systematically investigate the FHT’s predictive ability. To value the scientific heritage, the original formulation of FHT was first revisited with minor corrections and then reformulated in a dimensionless form. Using the method of separation of variables, an analytical solution was developed for the dimensionless FHT in the context of consolidation under instantaneously applied surcharge and with one-way drainage at the top boundary of the consolidating stratum. The predictive power of FHT was validated against available wave measurements; the proposed solution was verified against the finite-difference method with nonclassical Newmark’s integration schemes. The parametric analysis conducted herein further suggests that FHT can qualitatively interpret observed complex phenomena, including the consolidation delay effect, the top-down progressive pattern, and the initial settlement overestimation. FHT significantly fills the knowledge gap between Terzaghi’s classic theory and Biot’s theory, thus enabling engineers to analyze the one-dimensional dynamic behavior of saturated geo-/poro-materials with incompressible constituents.

The interior of the Earth is quite complex due to the actual geometrical structure and the presence of complex rheological materials, including viscoelastic rocks, porous sediments and the presence of anisotropy. Seismic wavefield forward modeling in such media forms the basis of most wave equation‐based methods for investigating the structure of the Earth and processing and imaging of seismic data, e.g., seismic full waveform inversion. Numerical modeling using Biot's equations that describe the physics of poroelasticity provides a useful framework to investigate wave attenuation and dispersion. Poroelasticity describes the interaction between the deformation of an elastic porous solid and the fluid flow in the porous structure. We present a time‐domain finite‐difference method for seismic wavefield modeling, taking into account both attenuation and anisotropy of poroviscoelastic Earth structure. The mathematical formulations and modeling methods are based on (a) the linear Biot's poroelastic theory and double porosity models in anisotropic media and (b) a relaxation function that uses the generalized standard linear solid model with anisotropic τ parameter for magnitude of attenuation and nearly constant Q model. Using a modified relaxation function that is suggested to describe the anisotropy attenuation, we develop a generalized anisotropic Biot model in the anisotropic, viscoelastic media. In addition to the anisotropic poroelasticity based on Biot theory, we implement the generalized anisotropic Biot model with complex moduli for an effective Biot's model in which the attenuative anisotropic viscoelastic model with the generalized standard linear solid model is used to approximate the attenuation factor function. The method generalizes the linear poroviscoelastic model based on effective Biot theory for seismic wave modeling to the attenuative, anisotropic case. In the anisotropic, poroviscoelastic model, we represent the bulk modulus and shear modulus of the solid frame by the modified relaxation function. We present time‐domain finite‐difference modeling for seismic wavefields in anisotropic, viscoelastic porous media including transversely isotropic media with a vertical, tilted or horizontal symmetry axis (VTI, TTI, and HTI). We also consider the extension of the two new models to nearly constant Q viscoelastic anisotropy.

The book of abstracts contains the brief description of talks of the participants
of the international conference " Modern problems of applied mathematics
and information technologies al-Khwarizmi 2021". The topics are related to
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education.

The book of abstracts contains the brief description of talks of the participants
of the international conference " Modern problems of applied mathematics
and information technologies al-Khwarizmi 2021". The topics are related to
the scientific heritage of Al-Khwarizmi, theory of algorithms, mathematical modeling
of nonlinear processes, algebra and functional analysis, differential equations and
dynamical systems, ill-posed and inverse problems, mathematical analysis, geometry
and topology, computational mathematics, statistical modeling, artificial intelligence and
digital technology, information security, digital technologies in education, engineering
education.
This collection is intended for specialists in mathematics, applied mathematics and
information technology, university teachers and for PhD, master students.

It is observed that seismic waves generate electromagnetic variations. The excitation mechanisms have been analyzed and formulated by many researchers. Three mechanisms have been mainly considered; stress effects for physical properties of rocks such as the piezoelectricity and the piezomagnetic effects, the electromagnetic induction due to the oscillation in the geomagnetic field, and the electrokinetic phenomena in pore fluids. While the stress effects and the electrokinetic phenomena take place in the crust, the induction process functions in various regions of the earth as the crust, the oceans, and the ionosphere. The characteristics of the electromagnetic signals produced with these mechanisms are also reviewed. It is expected that the electromagnetic signals derived from seismic waves are used to investigate the dynamics and the structures of the earth and planets.

Biot's equations describe the physics of hydromechanically coupled systems establishing the widely recognized theory of poroelasticity. This theory has a broad range of applications in Earth and biological sciences as well as in engineering. The numerical solution of Biot's equations is challenging because wave propagation and fluid pressure diffusion processes occur simultaneously but feature very different characteristic time scales. Analogous to geophysical data acquisition, high resolution and three dimensional numerical experiments lately redefined state of the art. Tackling high spatial and temporal resolution requires a high-performance computing approach. We developed a multi- graphical processing units (GPU) numerical application to resolve the anisotropic elastodynamic Biot's equations that relies on a conservative numerical scheme to simulate, in a few seconds, wave fields for spatial domains involving more than 1.5 billion grid cells. We present a comprehensive dimensional analysis reducing the number of material parameters needed for the numerical experiments from ten to four. Furthermore, the dimensional analysis emphasizes the key material parameters governing the physics of wave propagation in poroelastic media. We perform a dispersion analysis as function of dimensionless parameters leading to simple and transparent dispersion relations. We then benchmark our numerical solution against an analytical plane wave solution. Finally, we present several numerical modeling experiments, including a three-dimensional simulation of fluid injection into a poroelastic medium. We provide the Matlab, symbolic Maple, and GPU CUDA C routines to reproduce the main presented results. The high efficiency of our numerical implementation makes it readily usable to investigate three-dimensional and high-resolution scenarios of practical applications.

The efficient and accurate numerical modeling of Biot's equations of poroelasticity requires the knowledge of the exact stability conditions for a given set of input parameters. Up to now, a numerical stability analysis of the discretized elastodynamic Biot's equations has been performed only for a few numerical schemes. We perform the von Neumann stability analysis of the discretized Biot's equations. We use an explicit scheme for the wave propagation and different implicit and explicit schemes for Darcy's flux. We derive the exact stability conditions for all the considered schemes. The obtained stability conditions for the discretized Biot's equations were verified numerically in one-, two-and three-dimensions. Additionally, we present von Neumann stability analysis of the discretized linear damped wave equation considering different implicit and explicit schemes. We provide both the Matlab and symbolic Maple routines for the full reproducibility of the presented results. The routines can be used to obtain exact stability conditions for any given set of input material and numerical parameters.

The article studies the reflection and refraction of sound waves passing through the “porous medium–gas” interface with oblique incidence from the side of the porous medium. Based on the analysis of the obtained analytical solutions, in the case of an oblique incidence of a “slow” wave at the interface from the side of the porous medium for a certain frequency range at angles of incidence exceeding a certain critical value, the general internal reflection (without reflection) is realized. This is due to the fact that the speed of the “slow” wave for a certain frequency range is less than the speed of sound waves in the gas surrounding the porous medium. In addition, it was found that the porous layer has certain wave properties, possibly slightly weak from the point of view of conventional waveguides.

Numerical modeling of seismic response of soil deposits is usually conducted as part of seismic hazard assessment, preceding facility construction in any tectonically active regions, including offshore sites. A significant feature of subsea soils is their porous and water-saturated structure. Thus, the purpose of the present study is to introduce a procedure for modeling nonlinear behavior of porous, moist soils during SH-wave propagation, to verify it and compare response for synthetic soil profiles with porous medium parameters specific for low moisture onshore and high moisture offshore sites with cohesive and non-cohesive soils. The well-known and approved NERA code was used as a basis and improved to incorporate the Biot and Gassman equations for elastic waves propagation in a fluid-saturated porous solid. The applicability of the presented approach was substantiated for integration into other well-known algorithms. Obtained results showed good agreement between the simulated by different methods and observed spectra. The modeling also showed that the response of cohesive and non-cohesive soils with moisture specific both for onshore and offshore sites is explained by effects of resonances and effect of seismic amplitude saturation, which, in turn, depend on the corresponding value of the layer thickness and S-wave impedance for porous saturated soil layer. The proposed scheme could have significant practical usage for studying the effect of porous medium parameters on the seismic response of the moist soil deposits.

The article considers the observational statistics analysis of seismoelectric effects of the second kind at the Bystryanskoye gas condensate reservoir in Krasnoyarsk Territory. The measurements were carried out by the passive method from 2014 to 2019 on one exploration profile. The authors obtained an averaged profiling curve. Its analysis showed that there exists the repeatability of measurement results, as well as the convergence with the results of mathematical modeling. The collected statistics on the observation points shows the possibility of this method application in carrying out industrial field work in conjunction with the standard seismic exploration.

The efficient and accurate numerical modeling of Biot’s equations of poroelasticity requires the knowledge of the exact stability conditions for a given set of input parameters. Up to now, a numerical stability analysis of the discretized elastodynamic Biot’s equations has been performed only for a few numerical schemes. We perform the von Neumann stability analysis of the discretized Biot’s equations. We use an explicit scheme for the wave propagation and different implicit and explicit schemes for Darcy’s flux. We derive the exact stability conditions for all the considered schemes. The obtained stability conditions for the discretized Biot’s equations were verified numerically in one-, two- and three-dimensions. Additionally, we present von Neumann stability analysis of the discretized linear damped wave equation considering different implicit and explicit schemes. We provide both the Matlab and symbolic Maple routines for the full reproducibility of the presented results. The routines can be used to obtain exact stability conditions for any given set of input material and numerical parameters.

- A G Ivanov

A.G. Ivanov, Bull. Aca. Sci. URSS, série géographique et géophysique, No. 5, 699, 1940.