Wei Cheng Lo

Wave propagation through elastic porous media containing two immiscible fluids

[1] Acoustic wave phenomena in porous media containing multiphase fluids have received considerable attention in recent years because of an increasing scientific awareness of poroelastic behavior in groundwater aquifers. To improve quantitative understanding of these phenomena, a general set of coupled partial differential equations was derived to describe dilatational wave propagation through an elastic porous medium permeated by two immiscible fluids. These equations, from which previous models of dilatational wave propagation can be recovered as special cases, incorporate both inertial coupling and viscous drag in an Eulerian frame of reference. Two important poroelasticity concepts, the linearized increment of fluid content and the closure relation for porosity change, originally defined for an elastic porous medium containing a single fluid, also are generalized for a two-fluid system. To examine the impact of relative fluid saturation and wave excitation frequency (50, 100, 150, and 200 Hz) on free dilatational wave behavior in unconsolidated porous media, numerical simulations of the three possible modes of wave motion were conducted for Columbia fine sandy loam containing either an air-water or oil-water mixture. The results showed that the propagating (P1) mode, which results from in-phase motions of the solid framework and the two pore fluids, moves with a speed equal to the square root of the ratio of an effective bulk modulus to an effective density of the fluid-containing porous medium, regardless of fluid saturation and for both fluid mixtures. The nature of the pore fluids exerts a significant influence on the attenuation of the P1 wave. In the air-water system, attenuation was controlled by material density differences and the relative mobilities of the pore fluids, whereas in the oil-water system an effective kinematic shear viscosity of the pore fluids was the controlling parameter. On the other hand, the speed and attenuation of the two diffusive modes (P2, resulting from out-of-phase motions of the solid framework and the fluids, and P3, the result of capillary pressure fluctuations) were closely associated with an effective dynamic shear viscosity of the pore fluids. The P2 and P3 waves also had the same constant value of the quality factor, and by comparison of our results with previous research on these two dilatational wave modes in sandstones, both were found to be sensitive to the state of consolidation of the porous medium.

Immiscible two-phase fluid flows in deformable porous media

Abstract 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.

The dynamic response of the water retention curve in unsaturated soils during drainage to acoustic excitations

We examined the effects of acoustic excitations on the water retention curve, i.e., the relationship between capillary pressure (PC) and water saturation (SW) in unsaturated porous media, during drainage. The water retention curves were measured under static and dynamic conditions, where water was withdrawn from a sandbox with three different pumping rates, 12.6, 19.7, and 25.2 mL/s. Excitations with frequencies of 75, 100, 125, and 150 Hz were applied. The acoustic excitations had no effect on the static water retention curve but altered the dynamic water retention curve. The acoustic excitations lowered the dynamic PC, especially under the dynamic condition where the pumping rate was 25.2 mL/s and when SW varied between 0.6 and 0.95. The differences between the capillary pressures measured under static and dynamic conditions decreased when acoustic excitations were applied. We link this finding to the change in contact angle induced by the acoustic excitation. The dynamic coefficients, τ, for the dynamic water retention curves that we fitted to the experimental data were smaller with than without acoustic excitations. We attribute the decrease of the dynamic coefficient to the combination of the increase in the permeability and the decline in the air-entry pressure caused by adding acoustic excitations.

Effect of Impermeable Boundaries on the Propagation of Rayleigh Waves in an Unsaturated Poroelastic Half-Space

This study presents an analytical model for describing propagation of Rayleigh waves along the impermeable surface of an unsaturated poroelastic half-space. This model is based on the existence of the three modes of dilatational waves which employ the poroelastic equations developed for a porous medium containing two immiscible viscous compressible fluids (Lo, Sposito and Majer, [13]). In a two-fluid saturated medium, the three Rayleigh waves induced by the three dilatational waves can be expressed as R1, R2, and R3 waves in descending order of phase speed magnitude. As the excitation frequency and water saturation are given, the dispersion equation of a cubic polynomial can be solved numerically to obtain the phase speeds and attenuation coefficients of the R1, R2, and R3 waves. The numerical results show the phase speed of the R1 wave is frequency independent (non-dispersive). Comparatively, the R1 wave speed is 93 ∼ 95% of the shear wave speed, and 28% to 49% of the first dilatational wave speed for selected frequencies between 50Hz & 200Hz and relative water saturation ranging from 0.01 to 0.99. However, the R2 and R3 waves are dispersive at the frequencies examined. The ratios of R2 and R3 wave phase speeds to the second and third dilatational wave speeds fall between 56% and 90%. The R1 wave attenuates the least while the R3 wave has the highest attenuation coefficient. Furthermore, the phase speed of the R1 wave under an impermeable surface approaches 1.01 ∼ 1.37 times of the R1 wave under a permeable boundary. Impermeability has significant influence on the phase speeds and attenuation coefficients of the R1 and R2 waves at high water saturation due to the existence of confined fluids.

Mechanical behavior of the soil-root system

The reinforcement of soil by vegetation depends on the biomechanics, length, and morphology of roots. Often, morphology and length are simplified in root reinforcement models by assuming symmetry above and below the shear plane. For many root systems, however, the presence of root collars and lateral rootlets could be greater beneath the shear plane. This is not taken into account in current models, so the current study derives a set of analytical equations to account for these effects and thereby provide a more realistic model of root reinforcement. To gain a better understanding of the mechanical behavior of the soil-root system, the resulting simulation was also compared with that of the data produced by the existing model in the literature (Waldron and Dakessian, 1981). This comparison shows that as root morphology and length are taken into account, the shear strength increment obtained with the new equations could reach values 1.4 times greater than that of previous models when the shear bond strength is small. In contrast, if the shear bond strength is greater, an increase in the root embedment length only produces a minor effect on the shear strength increment. In addition, we note that if the root system has the same constituents, the influence of roots on the shear strength increment decreases with an increase in the depth of the shear plane. The influence of rooted soils with smaller shear bond strength is found to be more significant than that of rooted soils with greater shear bond strength.

Modeling and Field Results from Seismic Stimulation

Modeling the effect of seismic stimulation employing Maxwell‐Boltzmann theory shows that the important component of stimulation is mechanical rather than fluid pressure effects. Modeling using Biot theory (two phases) shows that the pressure effects diffuse too quickly to be of practical significance. Field data from actual stimulation will be shown to compare to theory.