Analytical Investigation of Transient Wave Propagation in One-Dimensional Unsaturated Poroelastic Materials

As a dynamic response, the wave propagation phenomenon usually varies with diﬀerent media. In this study, the dynamic response of unsaturated poroelastic materials to an impulse load has been analytically investigated. The governing equations, in the Laplace domain, of the unsaturated poroelastic soil in terms of the variables u s (solid displacement), p f (pore ﬂuid pressure), and p a (pore air pressure) will be derived. These equations will be simpliﬁed in a one-dimensional form. The solutions that include the dynamic response to the vertical displacement of solid particles and to the variations of ﬂuid and air pressures will be provided which are applicable for an arbitrary loading form. The solutions were validated with the results for saturated soil presented in the literature. The eﬀect of material parameters on dynamic response was analysed through a series of parameter studies. It was found that increasing the porosity or ﬂuid saturation eﬀectively increases the amplitude of ﬂuid and air pressures as well as the wave velocity. Although increasing the ﬂuid saturation leads to solid displacement gradually decreases gradually, it results in increasing the amplitude of ﬂuid and air pressure. The ﬂuid saturation increasing above 0.9 results in wave travels faster signiﬁcantly. The variation of ﬂuid intrinsic permeability has little inﬂuence on the soil dynamic response until it reaches a high level. The ﬁndings of this study can help for better understanding of one-dimensional wave propagation in unsaturated soil.


Introduction
e soil in the civil engineering field is known as a mixture of solid particles, fluids, and gases, and it is commonly treated as an ideal elastic and fluid-filled porous medium, namely, saturated poroelastic material [1,2].
In the past few years, the load-induced saturated soil response has been widely investigated by analytical method. For example, Philippacopoulos [3] calculated the explicit general solutions of displacements and stresses of a homogeneous half-space model in the wavenumber-frequency domain via Helmholtz decompositions and Fourier transforms. Yazdi et al. [4] analysed the dynamic response of a horizontal layer model to a disturbance induced by a strip footing.
ey found an exact solution in the frequency domain and proposed a finite element scheme for complex geometries. Jin and Liu [5] investigated the motion of a three-dimensional poroelastic half-space produced by a horizontal buried loading using integral transform techniques. In addition to the analytical method, numerical methods have also been extensively used in load-induced saturated soil response problems. For instance, Cheng et al. [6] presented a singular integral equation technique, called the boundary element method, for solving dynamic poroelasticity problems. Moreover, other methods can be exemplified by the stiffness matrix method proposed by Mesgouez and Lefeuve-Mesgouez [7] and Wang and Rokhlin [8]; transmission and reflection matrix method by Pei et al. [9], and generalized transfer matrix method by Liu and Zhao [10].
Regarding the governing equations, in practice, the soil is considered as a partially saturated multiphase material consisting of a solid, a pore fluid (water or oil), and the air. e governing equations of a linear elastic porous medium permeated by two immiscible Newtonian fluids were established by Tuncay and Corapcioglu [11]. Based on the unsaturated soil model theories, load-induced dynamic response in unsaturated soil has been also explored. For example, in order to solve concentrated force source problem, the classical Green's functions were derived by Gatmiri and Jabbari [12] in both frequency and time domains using the Kupradze method. Lo [13] described the propagation and attenuation of Rayleigh waves, originated from a loading source, along the free surface of the unsaturated soil layer using the potential functions decomposition. Xu [14] investigated Lamb problem in half-space and obtained the dynamic response of unsaturated soils using integral transform method. Zhang et al. [15], by means of integral transform techniques, proposed Green functions in the transformed domain for an arbitrary internal harmonic loading buried in unsaturated soil.
In general, due to interactions between different media phases, or coupling characteristics of media, the analytical solutions are limited to a few simple cases. Actually, the numerical methods are widely applicable for dynamic response in complicated media. However, during the numerical calculations, it is necessary to evaluate the numerical results, where some accurate analytical solutions are necessary to be sought. By means of Laplace transform, Boer et al. [16] studied the dynamic response of saturated soil column with infinite length and incompressible grains and fluid. Schanz and Cheng [17] presented the dynamic response expression of saturated soil column with infinite length under dynamic load using the integral transformation and convolution quadrature method. ese problems can be described in Figure 1.
eir work had established a benchmark for numerical calculation solving dynamic response, namely, wave propagation, in saturated media. Note that in the previous theoretical investigations, the results cannot be utilized in the one-dimensional unsaturated media. erefore, using analytical methods, the transient dynamic response in a one-dimensional unsaturated soil column under dynamic load excitation is presented in the paper. First, the governing equations based on u s -p f -p a (solid displacement-pore fluid pressure-pore air pressure) form were derived clearly. en, the analytical solution of an impulse loading boundary condition was deduced in the Laplace domain for the case of one-dimensional soil column in order to provide a benchmark for the corresponding numerical solutions. In view of those results, the solutions in the time domain can be extended for arbitrary loading history using a conventional numerical method. Finally, the effects of different parameters on the dynamic response in unsaturated soil columns were discussed rationally.

Governing Equations
A partially saturated porous medium is a three-phase material, composing of solid particles, pore fluid, and pore air. Following Zhang et al. [15], by neglecting body forces, the momentum balance equation of unit volume of a threephase porous medium can be expressed as where σ ij is the total stress tensor, ρ s � (1 − n)ρ s , ρ f � nS r ρ f , and ρ a � n(1 − S r )ρ a are the apparent density for solid, fluid, and air phase, respectively, n is porosity, S r is saturation, and ρ m (m � s, f, a) is the absolute mass density of each individual phase. e variables u s i , u f i , and u a i represent the displacement vectors in the solid, fluid, and air phases, respectively. e dot in the superscript represents the differentiation with respect to the time variable (t). Note that repeated indices in the subscript denote the conventional summation. e relative displacement vectors of fluid and air with respect to solid can be defined as And the Laplace transform for function f (t, x, y, z) is given by where s is a complex variable. Substituting equations (2a) and (2b) into equation (1) and performing Laplace transform with the assumption of a stationary initial state, the momentum balance equation in the Laplace domain can be expressed as where the total density ρ of the solid-fluid-air mixture is determined by e elastic solid skeleton is assumed to be isotropic and homogeneous. Hence, the constitutive stress-strain equations for the three-phase porous medium is given as where λ and μ are two Lamb's constants, e � u s j,j is the volumetric strain, and ε ij is the solid strain determined by . e parameter α is the effective stress coefficient given by α denotes the bulk modulus of soil skeleton and K s is the compressibility modulus of the solid particles. e variable p represents the mean pore pressure acting on the solid particles, i.e., p � cp f + (1 − c)p a , c is a parameter related to the effective stress due to matrix suction, p f and p a are the pore fluid and pore gas pressures, respectively, and δ ij is Kronecher's delta.
Taking the derivative of equation (6) and applying Laplace transform lead to According to equations (4) and (7), the following equation can be obtained: For an isotropic medium, the generalized Darcy's law that is usually adopted to describe the motion of fluid and gas air without considering the body forces is where Substituting equations (10a) and (10b) into equations (9a) and (9b) and combining with equations (2a) and (2b) give By applying the Laplace transform defined in equations (3) to (11a) and (11b), the relative displacement vectors of fluid and air with respect to solid can be written as where Substituting equations (12a) and (12b) into equation (8) yields Applying the Laplace transform to the mean pore pressure p and taking the derivative lead to en, substituting equations (14) into (13) results in Neglecting the mass exchange between any two phases, the mass balance of a multiphase medium can be expressed as Based on the soil-water retention curve proposed by Van Genuchten [18], i.e., S e � [1 + (χp c ) d ] − m , the saturation variable (S r ) can be further described as where p c � p a − p f is the matrix suction, S e is the effective water saturation given as S e � S e � (S r − S w0 )/(1 − S w0 ), S w0 is the irreducible saturation, and χ, m, and d are the material parameters of the V-G model [18].
Combining equations (16) and (17), the following simplified equations can be derived: where A 11 ∼A 25 are coefficients which are given in equation (A.1). According to equations (18a) and (18b), p f and p a can be found as z Saturated soil Figure 1: Investigated problem by Boer et al. [16] and Schanz and Cheng [17].
where the coefficients a 11 ∼a 23 are presented in equation Taking the derivative of equations (19a) and (19b) and combining with equations (2a) and (2b) lead to where the coefficients D 1 ∼D 6 are defined in equation (A.3).
Applying the Laplace transform into equations (20a) and (20b) and combining with equations (12a) and (12b) result in erefore, the governing equations of motion for unsaturated soil with u s -p f -p a (solid displacement-pore fluid pressure-pore air pressure) form in the Laplace domain are obtained as equations (15), (21a), and (21b).

Analytical Solutions of One-Dimensional
Wave Propagation

General Solutions.
A one-dimensional unsaturated porous soil column under dynamic loading is illustrated in Figure 2, where the length of the column, measured in the vertical (z) direction, is represented by L. e side and the base of the column are assumed to be rigid, frictionless, and impermeable. e vertical normal stress σ z is applied at the top of the column. e vertical displacement (u z ), fluid pressure (p f ), and air pressure (p a ) in the column are still unknown, and their directions at the top boundary are assumed to be normal to the boundary along the vertical (z) direction. e loading function f (t) is an arbitrary temporal function that describes the surface loading acting on the solid skeleton, and P 0 denotes the loading amplitude. For the one-dimensional loading scenario, the governing equations, equations (15), (21a), and (21b), can be expressed as And equations (22a), (22b), and (22c) can be rewritten as where the coefficients E 11 ∼E 33 are defined in equation (A.4).
Note that equations (23a), (23b), and (23c) can be solved as Substituting equation (24) into equations (23a), (23b), and (23c) leads to an eigenvalue problem in terms of ξ as e corresponding characteristic equation is where B 1 ∼B 3 and r can be found in equation (A.5).
Providing that equation (26) yielded three roots, i.e., r i (i � 1 ∼ 3), the eigenvalue problem would have six complex roots ±ξ n (n � 1, 2, 3)(Re(ξ n ) ≥ 0) because of the square root operation. erefore, the complete solution of equations (23a), (23b), and (23c) given by where 6 , C i are a series of constants that need to be determined. e eigenvector relations derived from equation (25) are expressed as

Boundary
Conditions. e solutions to the displacement and pressure in equation (27) involve unknown constants (C i ) which can be determined by the boundary conditions. As aforementioned, the top boundary of the column is permeable for both the fluid and the air.
e impulse function f (t) � δ (t) is adopted herein where δ (t) represents the Dirac distribution. erefore, the boundary conditions are given as Substituting equation (27) into the one-dimensional constitutive relation equation (6) yields Combining equation (27) with equations (12a) and (12b) gives the following relations: Accordingly, a series of linear equations with respect to the unknown constants (C i ) can be derived by substituting equations (30), (31a), and (31b) into equation (29). By substituting the calculated C i into equation (27), the solutions for the vertical displacement of solid and pressures of fluid and air in the column can then be derived in the Laplace domain as where E 0 , δ i , and ψ i (i � 1∼3) are defined in equation (A.6). e response in the time domain can also be calculated using the convolution integral. erefore, the vertical displacement, and pressures of fluid and air can be derived as

Shock and Vibration
where L − 1 is the inverse Laplace transform operator.

Verification of the Solution
e solution calculated for unsaturated soil in the previous section was verified using the degenerated solution compared with the results in one-dimensional saturated soil column presented by Schanz and Cheng [17]. For this purpose, the foregoing coefficients S r , c, κ a , ρ a , and A s are simplified for the saturation state: Also, the coefficients in equations (18a) and (18b) can be rewritten as e coefficients in equations (20a) and (20b) can also be simplified as (36) erefore, the coefficients in equations (23a), (23b), and (23c) can be rewritten as Accordingly, linear equation (25) is reformed as Due to the absence of the air in the saturated soil, the air pressure is equal to zero, i.e., p a � 0, and the coefficients defined in equation (28b) satisfy the relation ζ 1 � ζ 2 � ζ 3 � 0. Finally, the solutions for the vertical displacement of solid and fluid pressure in the saturated scenario can be derived as Figure 3 compares the vertical displacement of solid particles and fluid pressure calculated in this study to the solutions of Schanz and Cheng [17]. e values of the unsaturated soil variables in this paper are listed in Table 1 that are taken from Zhang et al. [15]. As can be seen, the results of this study are remarkably consistent with those of Schanz and Cheng [17].

Parametric Study
In order to investigate the effects of fluid saturation, fluid permeability, and the porosity on the vertical displacement of solid and pressures of fluid and air, a series of parametric studies are conducted. e adopted parameters are given in Table 1, and the boundary conditions are the same as equation (29), and a Heaviside step function with P 0 � 1 is assumed for the time history of loading. Figure 4 shows the distributions of the solid vertical displacement, the fluid and air pressures in a time interval from 0 to 0.8s for cases with fluid saturation value varying from 0.4 to 0.9. As can be seen, when the fluid saturation value is less than 0.9, the vertical solid displacement response for different cases is fairly close to each other whilst when the saturation values reach 0.9, the amplitude of solid displacement tends to reduce. In contrast, the effects of the fluid saturation on the pressures of fluid and air are much more evident. As shown in Figures 3(b) and 3(c), the amplitude of the fluid and air pressure response increases with the increasing fluid saturation value. Figure 5 illustrates the distributions of the solid displacement, and the fluid and air pressures in the time coordinate for cases with fluid saturation value varying from 0.9 to 0.99. Clearly, fluid saturation has an influence on the amplitude of the displacement and pressure responses. As the fluid saturation increases, the amplitude of displacement decreases and pressure responses increase. ese responses have the phenomenon of shortening the cycle of periodic change, that is, the periodic change frequency is increasing.
is indicates that the wave travels faster. Figure 6 shows the distributions of the solid displacement, and the fluid and air pressures with time for cases with fluid intrinsic permeability value varying from 10 −13 to 10 −10 . Clearly, the magnitude of the intrinsic permeability (κ f ) has little impact on the vertical displacement of solid (u z ), the fluid pressure (p f ), and the air pressure (p a ), except for the case with κ f equal to 10 −9 which manifests slight deviation from the other cases when the magnitude of the displacement and pressures approach the maximum and minimum values. Figure 7 shows the distributions of the solid displacement, and the fluid and air pressures with time for cases with porosity value varying from 0.35 to 0.55. It can be seen that the increase in porosity (n) insignificantly affects the amplitude of the solid displacements (u z ) but it will result in a slight increase in the frequency. On the other hand, as the porosity increases, the amplitude of the fluid and air pressures will decrease accordingly while the corresponding frequency will increase.

Summary and Conclusions
is paper studies the dynamic response caused by an impulse load in unsaturated poroelastic materials for the case of onedimensional space using the analytical method. e governing equations in the form of u s -p f -p a (solid displacement-pore fluid pressure-pore air pressure) were derived, and the analytical solutions for a one-dimensional poroelastic column in the Laplace transform domain were calculated, which were transformed to the time domain via the inverse Laplace transform using a numerical method. e solution has a concise closed form. e solutions were verified by comparing to the degeneration results for saturated soil with those reported by Schanz and Cheng [17]; and the effects of three important parameters, i.e., fluid saturation, fluid permeability, and porosity, were investigated.
From the present parametric analysis, some findings of this study may be summarized as follows: (i) e increase in fluid saturation (S r ) larger than 0.9 or porosity (n) has an evident effect on the amplitude of the fluid and air pressures as well as the wave velocity (ii) e amplitude of solid displacement (u z ) gradually decreases when the fluid saturation (S r ) increases. In 8 Shock and Vibration contrast, the amplitude of the fluid and air pressure response increases (iii) e fluid saturation (S r ) increasing above 0.9 results in wave travels faster significantly (iv) e variation of the fluid intrinsic permeability (κ f ) has little influence on the soil dynamic response until it reaches a high order of magnitude Note that this study only considered the permeable boundary conditions for both the fluid and air phases, and further research on different types of boundary conditions seems necessary. e paper only discusses a type of unsaturated soil, without considering other practical case. e conclusions have yet to be thoroughly verified. However, for the unsaturated poroelastic soil, by means of the solutions found in this study, the accuracy of different numerical methods applied to the dynamic problem can be evaluated directly [19,20]. Bulk modulus of soil skeleton K f , K s : Compressibility modulus of the fluid and solid grains, respectively L: Column length M: 1/A 11 P 0 : Loading amplitude S r : Fluid saturation S e : Effective water saturation S w0 : Irreducible saturation U, P f , P a : Amplitude vector of solid displacement, fluid, and air, respectively Lower case a 11 , a 12 , . . ., a 23 : Calculation coefficients e: Volumetric strain of solid grains g: Gravity acceleration k f , k a : Intrinsic permeability of fluid and air, respectively m, d: V-G model parameter n: Porosity p: Mean pore pressure p c : Matrix suction p f , p a : Pore fluid pressure and air pressure, respectively q