Acoustic waves in unsaturated soils

Authors

  • Wei-Cheng Lo,

    Corresponding author
    1. Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan, Taiwan
    • Correspondence author: W.-G. Lo, Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan, Taiwan. (lowc@mail.ncku.edu.tw)

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  • Garrison Sposito

    1. Department of Civil and Environmental Engineering, University of California, Berkeley, California, USA
    2. Earth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California, USA
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Abstract

[1] Seminal papers by Brutsaert (1964) and Brutsaert and Luthin (1964) provided the first rigorous theoretical framework for examining the poroelastic behavior of unsaturated soils, including an important application linking acoustic wave propagation to soil hydraulic properties. Theoretical developments during the 50 years that followed have led Lo et al., (2005) to a comprehensive model of these phenomena, but the relationship of its elasticity parameters to standard poroelasticity parameters measured in hydrogeology has not been established. In the present study, we develop this relationship for three key parameters, the Gassman modulus, Skempton coefficient, and Biot-Willis coefficient by generalizing them to an unsaturated porous medium. We demonstrate the remarkable result that well-known and widely applied relationships among these parameters for a porous medium saturated by a single fluid are also valid under very general conditions for unsaturated soils. We show further that measurement of the Biot-Willis coefficient along with three of the six elasticity coefficients in the model of Lo et al. (2005) is sufficient to characterize poroelastic behavior. The elasticity coefficients in the model of Lo et al. (2005) are sensitive to the dependence of capillary pressure on water saturation and its viscous-drag coefficients are functions of relative permeability, implying that hysteresis in the water retention curve and hydraulic conductivity function should affect acoustic wave behavior in unsaturated soils. To quantify these as-yet unknown effects, we performed numerical simulations for Dune sand at two representative wave excitation frequencies. Our results show that the acoustic wave investigated by Brutsaert and Luthin (1964) propagates at essentially the same speed during imbibition and drainage, but is attenuated more during drainage than imbibition. Overall, effects on acoustic wave behavior caused by hysteresis become more significant as the excitation frequency increases.

1. Introduction

1.1. Acoustic Waves and Poroelasticity

[2] The theory of poroelasticity, a term coined by Geertsma [1966], aims to provide a fundamental mathematical framework to describe the time-dependent coupling between applied stress and pore fluid pressure that occurs when acoustic waves pass through a porous medium. It is based on foundations laid down more than 70 years ago in a famous series of papers by Biot [1941, 1956, 1962], who was motivated initially by the classic geomechanical problem of soil consolidation. Biot [1956, 1962] presented a theory of dynamic poroelasticity that provides a general continuum-scale description of wave propagation through an elastic porous medium containing a single fluid, with wave attenuation arising from fluid viscosity. In addition to the dilatational and shear waves like those observed in wave propagation through an isotropic elastic solid, Biot [1956, 1962] correctly predicted the existence of a second dilatational wave (“Biot slow wave”) caused by the coupled out-of-phase motions of the elastic porous framework and the viscous pore fluid.

[3] Extending the approach of Biot [1956], Brutsaert [1964] developed the first rigorous theoretical basis for modeling the effects of partial water saturation, such as occurs routinely in soils, on elastic wave behavior in unconsolidated porous media. As was the case in the pioneering study of Brutsaert [1964], most attempts to model acoustic wave propagation in porous media containing two immiscible fluids [Garg and Nayfeh, 1986; Tuncay and Corapcioglu, 1997; Wei and Muraleetharan, 2002] have neglected the possibility of coupling among the fluid phases and solid framework caused by differences between their accelerations (“inertial drag”), as opposed to coupling caused by differences between their velocities (“viscous drag”). Inertial drag was taken into account in the Biot-inspired model of elastic wave behavior in unsaturated porous media developed by Berryman et al. [1988], but their model was simplified significantly by imposing the condition that the capillary pressure between the two immiscible pore fluids remains constant during acoustic wave excitation.

[4] Santos et al. [1990] appear to have been the first to construct a model of wave propagation through elastic porous media containing two immiscible fluids, one wetting and one nonwetting (e.g., water and air), that accounts for both capillary pressure fluctuations and inertial drag. Since fluctuating capillary pressure allows the pore fluids to experience distinct time-dependent pressure changes, a dilatational mode in addition to the two predicted by the Biot [1962] model is expected. On the basis of numerical simulations, Santos et al. [2004] proposed that this third mode involves the nonwetting fluid moving out-of-phase with the wetting fluid while it moves in-phase with the solid framework. This proposal was elucidated by Lo et al. [2010], who demonstrated numerically that excitation of the third dilatational mode actually does not affect the solid framework significantly, but instead causes the two pore fluids to move out-of-phase, which is consistent with this mode arising from capillary pressure fluctuations.

[5] Lo et al. [2005] developed a comprehensive Eulerian model of acoustic wave behavior in an elastic porous medium permeated by two immiscible fluids as an exact counterpart of the Lagrangian approach taken by Santos et al. [1990]. The Lo et al. [2005] model includes all previous poroelasticity approaches as special cases. Invoking plane harmonic wave excitation and homogeneous boundary conditions, these authors also demonstrated the existence of three dilatational waves, that with the highest phase speed (P1) corresponding to the compressional wave that propagates through any elastic medium (“Biot fast wave”), followed by one of lower phase speed (P2) that corresponds to the Biot slow wave, and finally a slowest third wave (P3) associated with capillary pressure fluctuations and having a phase speed dependent on the slope of the relationship between capillary pressure and wetting fluid saturation. At seismic frequencies, the P2 and P3 waves are diffusive, with low phase speed and high attenuation; however, they are propagating waves when the excitation frequency lies in the ultrasound régime [Santos et al., 1990].

1.2. Connecting Theory to Experiment

[6] The poroelasticity model of Lo et al. [2005], although comprehensive, contains six linear elasticity coefficients that remain somewhat abstract, with their relation to standard poroelasticity parameters [Wang, 2000] not yet established. Although Lo et al. [2005] developed expressions for their elasticity coefficients in terms of poroelasticity and hydraulic parameters for the individual phases in a porous medium containing two fluids, these expressions were designed to provide mathematical generality more than physical insight. In the present paper, we remedy this situation by formulating the stress-strain relationships in the model of Lo et al. [2005] in terms of two directly measurable pressure variables, the confining pressure and the average fluid pore pressure. This reformulation permits three standard poroelasticity parameters [Wang, 2000], Gassman's modulus (“undrained bulk modulus”), Skempton's coefficient (“undrained pore pressure buildup coefficient”), and the Biot-Willis coefficient (“effective stress coefficient”), to be generalized to apply to a porous medium saturated by two fluids, with their connection to the elasticity coefficients in the linear stress-strain relations of Lo et al. [2005] specified precisely.

[7] Building on the pioneering efforts of Brutsaert and Luthin [1964] to relate the speed of the P1 wave to water content, a number of experimental studies have been undertaken to establish the relationship between acoustic wave behavior and soil hydraulic parameters [e.g., Flammer et al., 2001; Lu and Sabatier, 2009]. Lo et al. [2007b] examined the effect of soil texture on the phase speed and attenuation of the P1 wave under varying water content using numerical simulations based on the model of Lo et al. [2005] and experimental data on the water retention curve and hydraulic conductivity function reported by Rawls et al. [1992]. The phase speed exhibited a maximum at low-water content that grew more pronounced as the texture became more clayey, while the attenuation was relatively greater as the texture became increasingly sandy. The water content dependence of the phase speed was shown to be very sensitive to changes in the capillary pressure with water content, implying the importance of the matric potential, a point also noted nearly 50 years ago by Brutsaert and Luthin [1964].

[8] But it is well known that matric potential and water content have a path-dependent relationship which differs during imbibition (wetting) from that during drainage (drying) [Warrick, 2003; Jury et al., 2004]. This hysteresis effect has not received attention as to its impact on the behavior of acoustic waves in an unsaturated porous medium. Since the elasticity coefficients for a porous medium filled by two fluids are associated with changes in capillary pressure with respect to water saturation and the viscous-drag coefficients are functions of relative permeability [Santos et al., 1990; Tuncay and Corapcioglu, 1996; Lo et al., 2005], imbibition and drainage processes should influence the speed and attenuation of acoustic waves differently. To illustrate the effect of hysteresis in the water retention curve on these properties of the three dilatational waves, we performed numerical simulations at two representative seismic frequencies, 10 and 200 Hz, for unsaturated Dune sand using the boundary-curve approach of Kool and Parker [1987].

2. Generalized Poroelasticity Parameters

2.1. Generalized Biot-Willis Coefficient

[9] Consider an elastic porous medium containing a solid framework and two immiscible fluids in the pore space, one wetting and one nonwetting, and let ps designates the mean principal dilatational (incremental gauge) stress in the solid phase, while pξ designates the (incremental gauge) pressure in a fluid phase ξ, with ξ = 1 referring to the nonwetting fluid, henceforth termed “air,” and ξ = 2 referring to the wetting fluid, henceforth termed “water.” The confining pressure acting on a volume element of this porous medium, pc, and the average fluid pressure in its pores, pf, may be defined, respectively, by the equations [Pride et al., 1992; Wang, 2000; Pride, 2005; Lo et al., 2005]

display math(1)
display math(2)

where ϕ signifies porosity, S1 denotes the relative saturation of air, normalized to the porosity by inline image, and θ1 expresses the volume fraction of air.

[10] Lo et al. [2005] showed that the linear stress-strain relationships for the porous medium can be expressed quite generally as:

display math(3)
display math(4)
display math(5)

where e and εξ indicate the dilatations (volumetric strains) of the solid phase and the fluid phases (ξ= 1, 2), respectively, and aij (i, j = 1, 2, 3) are the elasticity coefficients, with the cross terms being symmetric, i.e., aij =aji. Substitution of equation (1) into equation (2) yields the relationships:

display math(6)
display math(7)

for the confining pressure and average pore fluid pressure, respectively.

[11] Closer examination of equations (29) and (30) in the model of Lo et al. [2005] shows that the coefficients in equation (3) are not entirely independent. For example (for details, see Appendix A),

display math(8)
display math(9)

where inline image is the Biot-Willis coefficient [Biot and Willis, 1957; Wang, 2000], also known as the effective stress coefficient [Kumpel, 1991; Wang, 2000], and Kb and Ks are the bulk moduli of the porous medium framework and solid phase, respectively. Putting equation (4) into equation (3), one may derive simpler expressions for pc and pf:

display math(10)
display math(11)

[12] The form of equation (5) lends itself to using a fluid variable initially introduced by Biot [1962], then generalized by Berryman et al. [1988], the linearized increment of fluid content: inline image. It quantifies the fractional volume of fluid entering (ζ > 0) or leaving (ζ < 0) a volume element attached to the solid framework in response to an applied stress or a change in fluid pressure [Wang, 2000]. With the introduction of the variables ζ1 and ζ2, equation (5) can be rearranged to have the form:

display math(12)
display math(13)

[13] It follows immediately from equation (6) that the generalized Biot-Willis coefficient can be interpreted physically as the change in confining pressure induced by a change in average pore pressure under the constraint of no dilatation of the solid phase:

display math(14)

thus extending the definition of α given by Pride [2005] for a porous medium containing a single fluid to one containing two immiscible fluids.

2.2. Generalized Gassmann Modulus and Skempton Coefficient

[14] Two standard poroelasticity parameters, Gassmann's modulus and Skempton's coefficient, are routinely used to describe the response of a porous medium undergoing undrained deformation wherein the pore fluids are prevented from escaping or entering the deforming medium [Berryman, 1999; Wang, 2000; Pride, 2005]. As generalized to a porous medium permeated by two fluids, this condition stipulates that the increment of fluid content for each pore fluid must be zero, i.e., inline image, or equivalently, inline image. The volumetric strain of the solid e for undrained deformation can be expressed [Wang, 2000]:

display math(15)

where Ku, termed the “undrained bulk modulus,” is also known as Gassmann's modulus [Carcione et al., 2005]. It may be defined for a porous medium bearing two fluids by the equation:

display math(16)

[15] Imposition of the undrained condition on equation (6) simplifies them to:

display math(17)
display math(18)

from which it follows with the help of equations (8.2) and (9.1) that the generalized Ku can be related to the elasticity coefficients aij as:

display math(19)

and, correspondingly, equation (6.1) takes on the more physical form:

display math(20)

[16] Skempton's coefficient B, also known as the “undrained pore pressure buildup coefficient” [Kumpel, 1991; Wang, 2000], describes the response of pore pressure to a change in the confining pressure applied to an undrained porous medium. For a porous medium filled by two fluids, we give this definition as:

display math(21)

[17] The relationship of B to Ku can be obtained by combining equation (9) with equation (21) to yield:

display math(22)

[18] Consequently, the linear stress-strain relationship in equation (6.2) can be recast into the form:

display math(23)

[19] Finally, by combining equations (20) and (23), one generalizes a well-known relationship [Pride, 2005] among the three variables pc, e, and pf involving the three standard poroelasticity coefficients Ku, α, and B:

display math(24)

which extends this relationship to a porous medium containing two fluids. In the limit of negligible capillary pressure changes or, what is equivalent, low-wave excitation frequency, assumed by Berryman et al. [1988], it is shown in Appendix B that Gassmann's modulus and Skempton's coefficient for a porous medium containing two fluids take on their original forms that apply to a porous medium containing a single fluid. Therefore, neglecting changes in capillary pressure blurs the physical distinction between the two immiscible fluids.

[20] To bring matters full circle, the famous Biot and Willis [1957] “jacketed experiment” [Pride et al., 1992; Wang, 2000; Pride, 2005] may be invoked, in which one supposes that the porous medium is deformed by a change in confining pressure while draining freely. The conditions imposed are:

display math(25)
display math(26)

[21] These conditions, when applied to equation (24), generalize a famous relation among the four parameters Ku, Kb, α, and B [Wang, 2000; Pride, 2005]:

display math(27)

which, as equation (24) shows, is consistent with the basic definition [Pride, 2005]:

display math(28)

now extended to a porous medium containing two fluids. We note that equations (19) and (22) imply that knowledge of the Biot-Willis coefficient, along with the first three of the elasticity coefficients in equation (2), i.e., a11, a12, and a13, is sufficient to characterize poroelastic behavior in an elastic medium saturated by two immiscible fluids.

2.3. Generalized Undrained P-Wave Modulus

[22] Irrespective of whether one or two fluids saturate a porous medium, the phase speed of the P1 wave is equal to the square root of the ratio of H, termed the “undrained P-wave modulus” [Pride, 2005], to the average porous medium density ρ:

display math(29)

where [Lo et al., 2005, 2007a]

display math(30)

[23] G is the shear modulus of the porous medium, and inline image. The second step in equation (30) is obtained after incorporating equation (4). It now follows immediately from equation (19) that equation (30) can be rewritten in the classic Biot model form (see equation (3.14) in Biot [1962]):

display math(31)

which generalizes this well-known expression to a porous medium containing two fluids.

[24] Alternative physically important expressions for H and Ku can be derived by incorporating equations (30d)–(30g) in Lo et al. [2005] into equation (30) to yield:

display math(32)

where N1 and N3 are given by:

display math(33)
display math(34)

with inline image and inline image defining effective air storativity parameters related to changes in the capillary pressure, inline image [Lo et al., 2005]. The parameter M1 is dimensionless and the units of M2 are the same as those of pressure. By virtue of equations (32) and (22), the undrained modulus H can be now expressed:

display math(35)

while equation (31) can be written:

display math(36)

[25] We note in passing that, with equation (35) introduced, the phase speed of the P1 wave defined in equation (29) has the closed-form expression:

display math(37)

which exposes its functional dependence on relative fluid saturation and changes in the capillary pressure. Finally, comparing equation (35) with the classic Biot model expression (see equations (3.6), (3.15), and (3.32) in Biot [1962]), for the modulus H:

display math(38)

reveals that, for a porous medium bearing two fluids, the Biot modulus M [Wang, 2000] is generalized to:

display math(39)

which now incorporates the effects of capillary pressure changes. Thus, the classic Biot model expressions in equations (31) and (38) are generalized to an elastic porous medium containing two immiscible fluids, provided that the elasticity parameters H and M are appropriately generalized as in equations (35) and (39).

3. Effects of Hysteresis

[26] Following a widely endorsed approach [Warrick, 2003], we adopted the methodology of Kool and Parker [1987] using the van Genuchten [1980] model of the water retention curve and the Mualem [1976] model of the hydraulic conductivity function:

display math(40)
display math(41)
display math(42)

where inline image designates the absolute value of the matric potential; inline image expresses the relative permeability of the fluid phase ξ; inline image and inline image denote the saturated (maximum) and residual (minimum) volumetric water contents, respectively; χ is the inverse of the air-entry value; m and n are shape parameters related to soil texture, with inline image; and η represents the pore connectivity parameter, conventionally recommended to take the value of 0.5 for most soils [Mualem, 1976]. Equation (28) constitute the van Genuchten-Mualem (VGM) model [Chen et al., 1999; Warrick, 2003].

[27] According to the model of Lo et al. [2005], the elasticity coefficients aij are a function of the matric potential, quantified by equation (28.1), while the viscous coupling coefficients R11 and R22 are dependent on inline image and inline image, quantified by equations (28.2) and (28.3). Thus, there are four adjustable parameters that must be determined from fitting experimental data for each boundary curve, yielding a total of eight fitting parameters ( inline image, inline image, nd, χd) and ( inline image, inline image, nw, χw), where the superscripts d and w denote drainage and imbibition, respectively. Table 1 lists these fitting parameters and the permeability value reported by Kool and Parker [1987] for Dune sand. Material properties of air, water, and sand particles, as well as elasticity parameters for sand, were taken from Lo et al. [2007b] under the assumption that inline image.

Table 1. Fitting Parameters χ, n, inline image, and inline image on the Boundary Curves for Unsaturated Dune Sand [Kool and Parker, 1987]a
Model Assumption inline image inline image inline image inline image inline imagenwnd
  1. a

    Permeability: inline image.

inline image0.3015.473.020.0930.0984.2648.904
inline image0.3015.273.060.1010.1016.7796.779

[28] Figure 1 shows the effect of hysteresis on the phase speed and attenuation coefficient of the P1 wave as a function of water saturation and excitation frequency. It can be seen from Figure 1(a) that the phase speed of the P1 wave is not affected by hysteresis. Figure 1(b) reveals that hysteresis does have an impact on the attenuation coefficient of the P1 wave, however, which is greater during drainage.

Figure 1.

Effect of hysteresis on the P1 wave at excitation frequencies 10 and 200 Hz: (a) phase speed and (b) attenuation coefficient.

[29] Figure 2 shows that the P2 wave has greater speed but lower attenuation during imbibition as compared to drainage. The effective dynamic shear viscosity of the pore fluid, equal to the inverse of the sum of relative mobilities of air and water [Berryman et al., 1988] controls the speed and attenuation of the P2 and P3 waves [Lo et al., 2005]; Figure 2(c) shows its water-content dependence. The phase speed and attenuation coefficient of the P3 wave is described in Figure 3. The graphs of the attenuation coefficient of the P2 wave and the phase speed of the P3 wave do bear a resemblance to that of the effective dynamic shear viscosity. We note also that the differences in speed and attenuation between drainage and imbibition increase with an increase in excitation frequency.

Figure 2.

Effect of hysteresis on the P2 wave at excitation frequencies 10 and 200 Hz: (a) phase speed, (b) attenuation coefficient, and (c) the effective dynamic shear viscosity.

Figure 3.

Effect of hysteresis on the P3 wave at excitation frequencies 10 and 200 Hz: (a) phase speed and (b) attenuation coefficient.

[30] Kool and Parker [1987] suggested that it is possible to reduce the number of required parameters, while still providing a reasonable description of hysteresis in the water retention curve, by setting inline image and inline image. This restriction leaves only the parameter χ to differ on the boundary curves, while the hydraulic conductivity function becomes nonhysteretic. Numerical calculations (data not shown) reveal that these changes in χ alone do not have substantial influence on the phase speed and attenuation coefficient of the P1 and P2 waves, but they do affect those of the P3 wave (Figure 4). It can be seen in Figure 4(a) that, regardless of excitation frequency, the phase speed of the P3 wave for a given water saturation is greater during drainage; however, an opposite trend is observed in Figure 4(b), which shows that during imbibition there is greater attenuation. Since the hydraulic conductivity function is nonhysteretic when n is fixed, the effective dynamic shear viscosity is exactly the same during imbibition and drainage. This in turn suggests that the air-entry value (and, therefore, the matric potential) must be important to the behavior of the P3 wave, which is consistent with its being caused by capillary pressure fluctuations [Santos et al., 1990; Tuncay and Corapcioglu, 1997; Lo et al., 2005].

Figure 4.

Effect of hysteresis on the P3 wave at excitation frequencies 10 and 200 Hz under the assumptions inline image and inline image: (a) phase speed and (b) attenuation coefficient.

4. Conclusions

[31] Three standard poroelasticity parameters, Gassman's modulus, Skempton's coefficient, and the Biot-Willis coefficient, developed for porous media saturated by a single fluid [Wang, 2000], can be generalized directly to unsaturated soils (equations (8.2), (21), and (14)). Moreover, well-known relationships among these parameters also can be generalized to apply unchanged in form to unsaturated soils (equations (17.1) and (31)). The relationship of these poroelasticity parameters to the six Lo et al. [2005] model elasticity coefficients (equations (19) and (22)) shows further that the Biot-Willis coefficient plus just three of the model elasticity coefficients are sufficient to characterize poroelastic behavior. Last, a closed-form analytical expression relating the phase speed of the P1 wave in an unsaturated soil to water content and changes in capillary pressure (equation (37)) allows the generalization of two more well-known relationships among standard elastic moduli to unsaturated soils (equations (31) and (38)). Thus, the model of Lo et al. [2005] preserves for unsaturated soils all of the major relationships among poroelasticity parameters that have been developed and applied for many years to porous media containing a single fluid.

[32] Hysteresis in the water retention curve and hydraulic conductivity function does not impact the speed of the P1 wave, but it is more attenuated during drainage. On the other hand, hysteresis has a major effect on the P2 wave but only minor effects on the P3 wave, although the latter is sensitive to the air-entry pressure. Overall, the effect of hysteresis becomes more apparent as the wave excitation frequency increases.

Appendix A: Derivation of Equation (4)

[33] Combining equations (30h) and (30i) in Lo et al. [2005], one gets:

display math(43)

where K1 and K2 refer to the bulk moduli of the nonwetting fluid (air) and the wetting fluid (water), respectively. In view of equations (30f) and (29a) in Lo et al. [2005], equation (43) can be recast as:

display math(44)

[34] Similarly, using equations (30i) and (30j) in Lo et al. [2005], we may write:

display math(45)

[35] It follows from equations (30g) and (29a) in Lo et al. [2005] that equation (45) can be expressed:

display math(46)

Appendix B: Negligible Capillary Pressure Changes

[36] Berryman et al. [1988] extended the classic Biot [1962] theory of poroelastic behavior to a two-fluid system based on the premise that the wave excitation is of sufficiently low frequency to justify neglect of the effects of capillary pressure changes. Their generalized Biot relationship among the average pore pressure, increment of fluid content, and solid dilatation can be recovered from our equations (20) and (23) as a special case.

[37] With the help of equations (30f) and (30g) in Lo et al. [2005], the coefficients of ζ1 and ζ2 in equation (23) can be put into the form:

display math(47)
display math(48)

[38] Neglect of capillary pressure changes implies inline image and, therefore, inline image since [Lo et al., 2005]:

display math(49)

where inline image designates the capillary pressure. Equation (B1) then become identical and simplify considerably:

display math(50)

where C and M are elasticity coefficients originally defined by Biot [1962] for a porous medium containing a single fluid and extended by Berryman et al. [1988] to one containing two fluids. In deriving equation (B3), we follow Berryman et al. [1988] in defining an average fluid bulk modulus Kf as the harmonic mean of the bulk moduli of fluid phases 1 and 2: inline image. Equation (22) then reduces to:

display math(51)

and equations (20) and (23) become:

display math(52)
display math(53)

[39] These equations are identical to the corresponding Biot [1962] stress-strain relationships for a porous medium containing a single fluid, with equation (B5.2) being the relationship derived by Berryman et al. [1988] for a porous medium containing two fluids under the assumption of negligible capillary pressure changes.

[40] Finally, the poroelasticity parameters Ku and B derived for a porous medium containing two fluids simplify to model expressions derived previously by Johnson [2001] and Berryman [2002] for a porous medium containing a single fluid. Comparing equation (17.1) with equation (B4), we generate the result:

display math(54)

which reproduces equation (14) in Johnson [2001]. Combining equations (22) and (17.1), we find:

display math(55)

[41] Incorporation of equation (B3) into equation (B7) leads to the expression:

display math(56)

which is equivalent to equation (20) in Berryman [2002]. These results further illustrate the blurring of the physical identity of the two fluids when capillary pressure changes become negligible.

Acknowledgments

[42] Gratitude is expressed for financial support to the National Science Council, Taiwan, under contract NSC100–2628-E-006–033. Thanks also to Ernest Majer, Lawrence Berkeley National Laboratory, for introducing the authors to the intriguing problem of modeling poroelastic behavior. Finally, many thanks to Wilfried Brutsaert for years of friendship and an approach to hydrologic modeling that is both beautiful and true.

Ancillary

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