Journal of Engineering Mechanics

Frenkel was a brilliant Russian physicist who made fundamental contributions to the theory of liquid, the theory of metal, the drop theory, defects in crystals, and excitons, among many other subjects in physics, astronomy, and the material sciences. In geophysics, he was the first to predict the existence of the “second-type wave” in liquid saturated porous media. This article is a brief biography and reviews his scientific accomplishments.

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. III, 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.

The Biot–Frenkel model of fluid-infiltrated material dynamics in its original linear form is adequate to solve many problems for soils and rocks mechanics saturated with fluids. The main interest to it was connected with applications in oil/gas production and the exploration industry and also in civil engineering and ecology. The model was developed and generalized successfully for the first half century. The input of the Russian writers was essential. A corresponding review of their published results is given in this paper.

Frenkel’s 1944 theory of porous media acoustics is carefully scrutinized. After some manipulation, Frenkel’s equations are seen to have nearly identical form to Biot’s 1962 equations. The only difference is that Frenkel includes an extraneous fluid-pressure gradient in his bulk force balance. Frenkel also makes a slight error in the development of his effective poroelastic moduli that prevents him from being the first to obtain the so-called “fluid-substitution” relations. Outside of these two small problems, Frenkel’s analysis can be considered correct. He limits his treatment of electroseismic phenomena to explain the electric field that accompanies a compressional seismic wave in a homogeneous material. He predicts that the electric field in a compressional wave is directly proportional to the particle acceleration and this has been verified by the recent field measurements of Garambois and Dietrich. However, such electric fields are only a small part of the total electroseismic response of the earth. Accordingly, some of the additional phenomena not discussed by Frenkel are also presented and discussed.

One of the most conspicuous aspects of the Biot–Frenkel theory is the existence of the slow compressional wave. An overview is given of the various experimental techniques that were/are used to study its appearance and properties.

Understanding of ultrasonic wave propagation in bones is essential for further development of related techniques in clinical practice. As any other saturated porous medium, bone is characterized by different forms of longitudinal wave propagation, either undrained waves or fast and (Frenkel–Biot) slow compressional waves. We here study the wave propagation in the framework of poromicromechanics. A continuum micromechanics model allows for the prediction of the anisotropic poroelastic properties, Biot’s coefficients, and moduli, from tissue-specific composition data, on the basis of tissue-independent (“universal”) elastic properties of the elementary components of all bones. These poroelastic properties enter the governing equations for wave propagation in anisotropic porous media. They allow for the prediction of undrained, fast and slow waves, as is verified by comparison of model results with experimental findings.

Four methods of upscaling coupled equations at the microscale to equations valid at the mesoscale and/or macroscale for fluid-saturated and partially saturated porous media will be discussed, compared, and contrasted. The four methods are: (1) effective medium theory, (2) mixture theory, (3) two-scale and multiscale homogenization, and (4) volume averaging. All these methods have advantages for some applications and disadvantages for others. For example, effective medium theory, mixture theory, and homogenization methods can all give formulas for coefficients in the up-scaled equations, whereas volume averaging methods give the form of the up-scaled equations but generally must be supplemented with physical arguments and/or data in order to determine the coefficients. Homogenization theory requires a great deal of mathematical insight from the user in order to choose appropriate scalings for use in the resulting power-law expansions, while volume averaging requires more physical insight to motivate the steps needed to find coefficients. Homogenization often is performed on periodic models, while volume averaging does not require any assumption of periodicity and can therefore be related very directly to laboratory and/or field measurements. Validity of the homogenization process is often limited to specific ranges of frequency—in order to justify the scaling hypotheses that must be made—and therefore cannot be used easily over wide ranges of frequency. However, volume averaging methods can quite easily be used for wide band data analysis. So, we learn from these comparisons that a researcher in the theory of poroelasticity and its generalizations needs to be conversant with two or more of these methods to solve problems generally.

A series of triaxial compression experiments have been conducted to investigate the effects of induced stress on the anisotropy developed in dynamic elastic and poroelastic parameters in rocks. The measurements were accomplished by utilizing an array of piezoelectric compressional and shear wave sensors mounted around a cylindrical sample of porous Berea sandstone. Three different types of applied states of stress were investigated using hydrostatic, triaxial, and uniaxial strain experiments. During the hydrostatic experiment, where an isotropic state of stress was applied to an isotropic porous rock, the vertical and horizontal acoustic velocities and dynamic elastic moduli increased as pressure was applied and no evidence of stress induced anisotropy was visible. The poroelastic moduli (Biot’s effective stress parameter, $\alpha$ ) decreased during the test but also with no evidence of anisotropy. The triaxial compression test involved an axisymmetric application of stress with an axial stress greater than the two constant equal lateral stresses. During this test a marked anisotropy developed in the acoustic velocities, and in the dynamic elastic and poroelastic moduli. As axial stress increased the magnitude of the anisotropy increased as well. The uniaxial strain test involved axisymmetric application of stresses with increasing axial and lateral stresses but while maintaining a zero lateral strain condition. The uniaxial strain test exhibited a quite different behavior from either the triaxial or hydrostatic tests. As both the axial and lateral stresses were increased, an anisotropy developed early in the loading phase but then was effectively “locked in” with little or no change in the magnitude of the values of the acoustic velocities, or the dynamic elastic and poroelastic parameters as stresses were increased. These experimental results show that the application of triaxial states of stress induced significant anisotropy in the elastic and poroelastic parameters in porous rock, while under the uniaxial strain condition the poromechanics, Biot’s effective stress parameter, exhibited the largest variation among the three test conditions.

Operations including borehole fluid injections are typical for exploration and development of hydrocarbon or geothermic reservoirs. Microseismicity occurring during such operations has a large potential for understanding physics of the seismogenic process as well as in obtaining detailed information about reservoirs at locations as far as several kilometers from boreholes. We propose that the phenomenon of microseismicity triggering by borehole fluid injections is related to the process of the Frenkel–Biot slow wave propagation. In the low-frequency range (hours or days of fluid injection durations) this process reduces to the pore-pressure diffusion. We search for diffusion-related features of induced microseismicity. Two types of such signatures are considered. The first one is related to the geometry of microseismic clouds. Another type of signature is related to the probability of microearthquakes. On this basis we introduce a concept for interpretation of microseismic data which provides a possibility to infer information about hydraulic properties of rocks. Such information can be of significant importance for industrial applications and for understanding physical properties of geological structures.

This study considers the most fundamental problem of multiple scattering in a poroelastic medium. It treats the interaction of a plane compressional elastic wave with a cluster of two of spherical inhomogeneities in a boundless fluid-saturated porous elastic formation. The novel features of Biot classic model for dynamic description of poroelastic material behavior along with the appropriate wave field expansions, the pertinent boundary conditions, and the translational addition theorems for spherical wave functions are employed to develop a closed-form solution in the form of infinite series. The analytical results are illustrated with numerical examples in which a pair of spherical inclusions is insonified by a plane (fast) compressional wave at end-on incidence. The effects of incident wave frequency, proximity of the two inclusions, and inclusion type are examined. Particular attention has been focused on multiple scattering interactions in addition to the slow wave coupling effects which is known to be the primary distinction of the scattering phenomenon in poroelasticity from the classical elastic case. The limiting case involving two elastic spheres submerged in an ideal unbounded fluid medium is considered and fair agreement with a well-known solution is established.

Theoretical analyses on nonlinear pressure waves evolution in porous medium saturated with a liquid containing gas bubbles is carried out. The evolution equations for fast and slow longitudinal modes are derived for slightly nonlinear, disperse, and dissipation processes. The pressure wave distribution in gas bubble liquid-saturated porous media was investigated experimentally. It was revealed that both modes might have oscillating structure induced by bubble oscillation in the wave. It is shown that the wave damping is determined by a combined impact of heat losses due to gas cooling in the bubbles and dissipation due to longitudinal displacement of liquid and porous skeleton, both influenced by the wave. Experimental data on the velocity and structure of fast and slow modes are compared with results of theoretical modeling.

In this paper we present a comparison of the linear wave analysis for four models of poroelastic materials. A nonlinear thermodynamical construction of a two-component model of such materials requires a dependence on the porosity gradient. In the linear version this dependence may or may not be present. Consequently, we may work with the model without a dependence on this gradient which is identical to Biot’s model or we can use the so-called full model. In both cases we can construct simplified models without a coupling between partial stresses introduced by Biot. These simplified models have the advantage that their application to, for instance, surface wave analysis yields much simpler mathematical problems. In the present work we show that such a simplification for granular materials leads to a good qualitative agreement of all four models in ranges of porosity and Poisson’s ratio commonly appearing in geotechnical applications. Quantitative differences depend on the mode of propagation and vary between 10 and 20%. We illustrate the analysis with a numerical example corresponding to data for sands. Simultaneously we demonstrate severe limitations of the applicability of Gassmann relations which yield an instability of models in a wide range of practically important values of parameters.

Fundamental principles of elastic–plastic mechanics of soils and rocks are given on the base of the original publications. The solid friction and dilatancy effects are included in the nonstandard form of nonassociative rule of plastic flow. The resulting hyperbolic system of equations is represented for a plane case. The slip surfaces are assumed to be jump tangential discontinuities of a velocity field. The possibility of limit equilibrium at slip surfaces is accounted for. The attempts to account for grain rotations, permitting study of slip surface structure, are discussed. The Biot–Frenkel model of interpenetrating continua is developed for plastic flow of porous saturated matrix. In this case the solid matrix state is determined by the effective stresses and pore pressure diffusion happens in plastically flowing matrix. To illustrate the theory possibilities, solutions for failure and mass sand flow, driven by the pore pressure gradient, are selected. They are important especially for oil/gas reservoirs with a weak matrix, typical for offshore geology.