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A central issue in the theoretical treatment of a multiphase system is the proper mathematical description of momentum transfer across fluid-solid and fluid-fluid interfaces. Although recent studies have advanced our knowledge on modeling the coupling behavior between a porous framework and the fluids permeating it, the effect of viscous resistance caused by two-fluid flow on elastic wave behavior in unsaturated porous media still remains elusive. In the present study, the theoretical model developed for describing immiscible two-phase fluid flows in a deformable porous medium related to harmonic wave perturbation is generalized to account for viscous cross coupling due to relative velocity between two adjacent fluids. The corresponding dispersion relations whose coefficients feature all elasticity, inertial-drag, and viscous-drag parameters are then precisely formulated, in a physical context characterizing three compressional waves and one shear wave. To evaluate quantitatively this as-yet unknown effect, numerical calculations are conducted to solve the dispersion relations for Columbia fine sandy loam bearing an oil–water mixture as a function of water saturation and excitation frequency. Our results show that the phase speed and attenuation coefficient of the P3 wave which has the smallest speed is strongly sensitive to the presence of viscous cross coupling, as expected since this wave is attributed primarily to the out-of-phase motion of the two pore fluids. Viscous cross coupling also exerts an impact on the attenuation coefficient of the shear wave and the P1 wave whose speed is greatest, which exhibits two opposite trends at different ranges of low and high water contents. Relative differences in these wave attributes are principally independent of excitation frequency. A sensitivity analysis is carried out to assess how changes in viscous cross coupling affect these differences, revealing that some of them become more significant as viscous cross coupling is stronger. [ABSTRACT FROM AUTHOR]

Soil consolidation, involving time-dependent coupling between deformation of a porous medium and interstitial fluid flows within it, is of relevance to many subsurface engineering problems. A comprehensive model of poroelasticity for consolidation in unsaturated soils has been recently developed by Lo et al. (2014), but it still remains elusive how variations in soil texture and water content affect consolidation behavior, and the underlying parameters deriving this behavior. In the current study, a boundary-value problem is first setup corresponding to two symmetric semi-permeable drainage conditions, and then solved analytically for describing the excess pore air and water pressures along with the total settlement in response to time-invariant external loading using the Laplace transform. These solutions are numerically calculated for unsaturated soils with eleven texture classes as a function of three initial water saturations as representative examples. Our results reveal that the excess pore water pressure and time-dependent total settlement are indeed significantly sensitive to both soil texture and initial water saturation. We demonstrate that the coefficient of consolidation for water and its loading efficiency are two important physical parameters controlling consolidation behavior. With respect to the same soil texture, the coefficient of consolidation for water increases with an increase in initial water saturation, taking a value approximately four to five orders of magnitude greater in saturated soils than that in unsaturated ones. For a given initial water saturation, the rate of dissipation of excess pore water pressure is smallest in clay, followed by silty clay, silty clay loam, sandy clay, clay loam, silt loam, loam, sandy clay loam, sandy loam, loamy sand, and sand. A comparative study shows that in the early stage of consolidation, unsaturated soils bear smaller excess pore water pressure, but its dissipation is completed faster in saturated soils. The loading efficiency for water exhibits a concave upward relationship with initial water saturation in silty clay and clay, whereas a positively-correlated relationship between the efficiency and initial water saturation is observed in other soil textures. Unlike saturated soils, a considerable amount of total settlement is shown to occur immediately after application of external loading. [ABSTRACT FROM AUTHOR]

Abstract: Seismic stimulation, the application of low-frequency stress-pulsing to the boundary of a porous medium containing water and a non-aqueous fluid to enhance the removal of the latter, shows great promise for both contaminated groundwater remediation and enhanced oil recovery, but theory to elucidate the underlying mechanisms lag significantly behind the progress achieved in experimental research. We address this conceptual lacuna by formulating a boundary-value problem to describe pore-pressure pulsing at seismic frequencies that is based on the continuum theory of poroelasticity for an elastic porous medium permeated by two immiscible fluids. An exact analytical solution is presented that is applied numerically using elasticity parameters and hydraulic data relevant to recent proof-of-principle laboratory experiments investigating the stimulation-induced mobilization of trichloroethene (TCE) in water flowing through a compressed sand core. The numerical results indicated that significant stimulation-induced increases of the TCE concentration in effluent can be expected from pore-pressure pulsing in the frequency range of 25–100Hz, which is in good agreement with what was observed in the laboratory experiments. Sensitivity analysis of our numerical results revealed that the TCE concentration in the effluent increases with the porous medium framework compressibility and the pulsing pressure. Increasing compressibility also leads to an optimal stimulation response at lower frequencies, whereas changing the pulsing pressure does not affect the optimal stimulation frequency. Within the context of our model, the dominant physical cause for enhancement of non-aqueous fluid mobility by seismic stimulation is the dilatory motion of the porous medium in which the solid and fluid phases undergo opposite displacements, resulting in stress-induced changes of the pore volume. [Copyright &y& Elsevier]

Summary: A theoretical model for the analysis of the reflection and refraction of obliquely incident elastic waves upon the interface between two semi-infinite porous elastic half-spaces saturated by different fluid mixtures is developed in the present study based on the poroelasticity theory of and the normal coordinates derived by for describing the modes of dilatory motion. The amplitude and energy ratios of the reflected and refracted waves generated from either an incident P1 wave (the first dilatational wave) or an incident SV wave (the shear wave polarized in the vertical plane) are in turn theoretically determined for the first time with respect to the angle of incidence. As a representative example, a numerical simulation is conducted for Lincoln sand permeated by an air–water mixture in the lower half-space and Columbia fine sandy loam permeated by an air–water mixture in the upper half-space. Our numerical results indicate that regardless of the type of pore fluid mixtures and porous media, the sum of the energy ratio of the reflected and refracted waves is always equal to unity, a result that indeed can not be achieved if the normal coordinates for dilatory motional modes is not taken into account as to represent the Helmholtz potential of the reflected and refracted waves. In addition, their amplitude and energy ratios are shown to be significantly affected by the angle of incidence. It is also revealed that as a SV wave is incident upon the interface, a critical angle of 31° and 33° can be found for the reflected and refracted P1 waves respectively, while the occurrence of the critical angle is not observed for the case of an incident P1 wave. [ABSTRACT FROM AUTHOR]

Summary: A theoretical analysis for the dynamic response of a semi-infinite fluid-bearing porous medium to external harmonic loading is presented in this study based on the decoupled poroelasticity equations of . A corresponding initial and boundary value problem is formulated and the analytical solution for the induced pore pressure and total dilatational stress is determined using the technique of Laplace transforms. To investigate the quantitative impact of inertial effect on the poroelastic response, the problem is also solved analytically in the diffusive model (i.e. inertial terms are ignored). Comparison of the analytical solutions obtained from two different models shows that as inertial effect is accounted for, the response undergoes in a dynamic manner but lags behind the loading by a physical factor equal to the travel time necessary for pressure wave to reach the location that is prescribed. A numerical study is then conducted for water-containing Columbia fine sandy loam at lower excitation frequencies as a representative example. Our numerical results reveal that the dynamic model yields a cyclic response for the induced pore pressure and total dilatational stress with respect to depth, but the diffusive model fails to predict this attribute. Lastly, we find in the dynamic model that effective stress may take on a positive value at some depths due to the existence of time lag in the response of pore fluid to external loading so that the solid skeleton needs to sustain excess fluid pressure. This positive value is crucial for the phenomenon of liquefaction if the loading is substantial enough. [Copyright &y& Elsevier]

Abstract: Numerical simulations of dilatational waves in an elastic porous medium containing two immiscible viscous compressible fluids indicate that three types of wave occur, but the modes of dilatory motion corresponding to the three waves remain uncharacterized as functions of relative saturation. In the present paper, we address this problem by deriving normal coordinates for the three dilatational waves based on the general poroelasticity equations of Lo et al. 2005 . The normal coordinates provide a theoretical foundation with which to characterize the motional modes in terms of six connecting coefficients that depend in a well defined way on inertial drag, viscous drag, and elasticity properties. Using numerical calculations of the connecting coefficients in the seismic frequency range for an unconsolidated sand containing water and air as a representative example relevant to hydrologic applications, we confirm that the dilatational wave whose speed is greatest corresponds to the motional mode in which the solid framework and the two pore fluids always move in phase, regardless of water saturation, in agreement with the classic Biot theory of the fast compressional wave in a water-saturated porous medium. For the wave which propagates second fastest, we show, apparently for the first time, that the solid framework moves in phase with water, but out of phase with air [Mode (III)], if the water saturation is below about 0.8, whereas the solid framework moves out of phase with both pore fluids [Mode (IV)] above this water saturation. The transition from Mode (III) to Mode (IV) corresponds to that between the capillarity-dominated region of the water retention curve and the region reflecting air-entry conditions near full water saturation. The second of the two modes corresponds exactly to the slow compressional wave in classic Biot theory, whereas the first mode is possible only in a two-fluid system undergoing capillary pressure fluctuations. For the wave which has the smallest speed, the dilatational mode is dominated by the motions of the two pore fluids, which are always out of phase, a result that is consistent with the proposition that this wave is caused by capillary pressure fluctuations. [Copyright &y& Elsevier]

Abstract: An analytical model for describing the propagation and attenuation of Rayleigh waves along the free surface of an elastic porous medium containing two immiscible, viscous, compressible fluids is developed in the present study based on the poroelastic equations formulated by Lo et al. [Lo WC, Sposito G, Majer E. Wave propagation through elastic porous media containing two immiscible fluids. Water Resour Res 2005;41:W02025]. The dispersion equation obtained is complex-valued due to viscous dissipation resulting from the relative motion of the solid to the pore fluids. As an excitation frequency is stipulated, the dispersion equation that is a cubic polynomial is numerically solved to determine the phase speed and attenuation coefficient of Rayleigh waves in Columbia fine sandy loam permeated by an air–water mixture. Our numerical results show that, corresponding to three dilatational waves, there is also the existence of three different modes of Rayleigh wave in an unsaturated porous medium, which are designated as the R1, R2, and R3 waves in descending order of phase speed, respectively. The phase speed of the R1 wave is non-dispersive (frequency-independent) in the frequency range we examined (10Hz–10kHz) and decreases as water saturation increases, whose magnitude ranges from 20% to 49% of that of the first dilatational wave with respect to water content. However, it is revealed numerically that the R2 and R3 waves are functions of excitation frequency. Given the same water saturation and excitation frequency, the phase speeds of the R2 and R3 waves are found to be approximately 90% of those of the second and third dilatational waves, respectively. The R1 wave has the lowest attenuation coefficient whereas the R3 wave attenuates highest. [Copyright &y& Elsevier]

Summary: The study of the propagation and dissipation of acoustic waves through a fluid-containing porous medium is crucial for the successful application of seismic methods to characterize subsurface hydrological properties. To gain a better understanding of changes in two important acoustic wave characteristics (speed and attenuation) due to the effect of soil texture and excitation frequency, a complex-valued dispersion relation obtained from the Biot theory of poroelasticity was solved numerically for eleven soil texture classes whose pore space is fully saturated by one of two very different fluids, air or water. Two modes of acoustic motion can be demonstrated to exist, known as the Biot fast and slow waves. Five lower excitation frequencies (100–500Hz) were selected for numerical simulation, below which Darcy’s law remains valid for describing porous media flow under wave perturbation. Numerical results show that in the frequency range we examined, the predicted phase speed of the Biot fast wave takes the same value as the Biot reference speed. The variation in speed is not obvious in a water-filled system, but becomes more significant in an air-filled system. When the pore fluid is water, an inverse linear relation exists between the phase speed of the Biot fast wave and porosity. An important physical parameter controlling its attenuation coefficient is intrinsic permeability, which renders a positive impact. A statistical analysis indicates that the attenuation coefficient of the Biot fast wave linearly increases with an increase in intrinsic permeability. In an air-saturated system, the phase speed of the Biot slow wave is found to be quadratically proportional to intrinsic permeability, whereas the attenuation coefficient of the Biot slow wave bears a quadratic relation with the inverse of intrinsic permeability. A study on the influence of pore fluid reveals that the Biot fast wave attenuates more in the water-saturated system than in the air-saturated system; an exception occurs in clay. A reverse trend is observed for the Biot slow wave. As far as the effect of excitation frequency is concerned, the attenuation coefficient of the Biot fast wave is essentially proportional to the square of excitation frequency, whereas the phase speed and attenuation coefficient of the Biot slow wave are both proportional to the square root of excitation frequency. The phase speed of the Biot fast wave is frequency independent. [Copyright &y& Elsevier]

Abstract: The study of the poroelastic behavior of sedimentary materials containing two immiscible fluids in response to either applied stress or pore pressure change in a quasistatic limit, i.e., negligible second time-derivatives, is of great importance to many hydrogelogical problems, e.g., land subsidence caused by withdrawal of subsurface fluids. The poroelasticity models developed for analyzing these problems feature partial differential equations that are coupled in the terms describing viscous damping and strain field. To determine closed-form analytical solutions for induced volumetric strain (dilatation) of the solid framework and its interaction with fluid flows, the choice of normal coordinates whose transformation can be performed to decouple these poroelastic equations is highly desirable. In this paper, we show that normal coordinates for decoupling these equations are real-valued and equal to three different linear combinations of the dilatations of the solid and the fluids (or equivalently, three different linear combinations of two individual fluid pressures and solid dilatation). In contrast to fully saturated porous media, it is found that the viscous damping effect must be represented in normal coordinates in the presence of the second fluid. The resulting decoupled equations representing independent motional modes are a Laplace equation and two diffusion equations, which can be solved analytically under a variety of initial and boundary conditions. Thus, after inverse transformation of normal coordinates is performed, the closed-form analytical solutions for induced solid volumetric strain and excess pore fluid pressures can be obtained simultaneously from our decoupled partial differential equations. [Copyright &y& Elsevier]

Abstract: The strong coupling of applied stress and pore fluid pressure, known as poroelasticity, is relevant to a number of applied problems arising in hydrogeology and reservoir engineering. The standard theory of poroelastic behavior in a homogeneous, isotropic, elastic porous medium saturated by a viscous, compressible fluid is due to Biot, who derived a pair of coupled partial differential equations that accurately predict the existence of two independent dilatational (compressional) wave motions, corresponding to in-phase and out-of-phase displacements of the solid and fluid phases, respectively. The Biot equations can be decoupled exactly after Fourier transformation to the frequency domain, but the resulting pair of Helmholtz equations cannot be converted to partial differential equations in the time domain and, therefore, closed-form analytical solutions of these equations in space and time variables cannot be obtained. In this paper we show that the decoupled Helmholtz equations can in fact be transformed to two independent partial differential equations in the time domain if the wave excitation frequency is very small as compared to a critical frequency equal to the kinematic viscosity of the pore fluid divided by the permeability of the porous medium. The partial differential equations found are a propagating wave equation and a dissipative wave equation, for which closed-form solutions are known under a variety of initial and boundary conditions. Numerical calculations indicate that the magnitude of the critical frequency for representative sedimentary materials containing either water or a nonaqueous phase liquid is in the kHz–MHz range, which is generally above the seismic band of frequencies. Therefore, the two partial differential equations obtained should be accurate for modeling elastic wave phenomena in fluid-saturated porous media under typical low-frequency conditions applicable to hydrogeological problems. [Copyright &y& Elsevier]

Macroscopic differential equations of mass and momentum balance for two immiscible fluids in a deformable porous medium are derived in an Eulerian framework using the continuum theory of mixtures. After inclusion of constitutive relationships, the resulting momentum balance equations feature terms characterizing the coupling among the fluid phases and the solid matrix caused by their relative accelerations. These terms, which imply a number of interesting phenomena, do not appear in current hydrologic models of subsurface multiphase flow. Our equations of momentum balance are shown to reduce to the Berryman–Thigpen–Chen model of bulk elastic wave propagation through unsaturated porous media after simplification (e.g., isothermal conditions, neglect of gravity, etc.) and under the assumption of constant volume fractions and material densities. When specialized to the case of a porous medium containing a single fluid and an elastic solid, our momentum balance equations reduce to the well-known Biot model of poroelasticity. We also show that mass balance alone is sufficient to derive the Biot model stress–strain relations, provided that a closure condition for porosity change suggested by de la Cruz and Spanos is invoked. Finally, a relation between elastic parameters and inertial coupling coefficients is derived that permits the partial differential equations of the Biot model to be decoupled into a telegraph equation and a wave equation whose respective dependent variables are two different linear combinations of the dilatations of the solid and the fluid. [Copyright &y& Elsevier]

• The delayed tensiometer response affects the dynamic WRC measurement during drainage. • The response deviated from the ideal exponential decay under unsaturated conditions. • The proposed models described the non-ideal curve and corrected the measured data. In recent years, more studies have been focusing on the transient behaviors of the movement of soil water and changes in matric potential, which require the measurement of the rapid changes in matric potential or capillary pressure. Because of the delayed response, the value of the matric potential measured using a tensiometer may not precisely reflect the actual matric potential during a transient condition. We performed a series of experiments to analyze the effect of grain size, effective saturation, and the hydraulic conductivities of ceramic cup and porous media on the response time of a tensiometer. The outcomes suggested that a decrease of hydraulic conductivity of the ceramic cup significantly increases the response time. Moreover, the response time measured under unsaturated conditions did not follow an ideal exponential decay. This finding was consistent with the existing theories, based on which, we proposed three models to describe the time-varied response function. These models were applied to correct the measured matric potential for dynamic water retention curve (WRC), and the findings were compared with those of other existing methods used for measuring the matric potential. We found that the incorporation of the theory of response time significantly reduced the discrepancy between the matric potential measured using two ceramic cups with different response times during dynamic drainage. We concluded that the delayed response of a tensiometer can have a significant effect on the measurement of the transient matric potential or capillary pressure and can be corrected using response time equations. [ABSTRACT FROM AUTHOR]