Transient Response of an Unsaturated Single-Layer Seabed Subjected to a Vertical Compressional Wave

Ma, Rong; Shan, Zhendong; Jing, Liping

doi:10.1061/IJGNAI.GMENG-8185

Open access

Technical Papers

May 7, 2024

Transient Response of an Unsaturated Single-Layer Seabed Subjected to a Vertical Compressional Wave

Authors: Rong Ma [email protected], Zhendong Shan https://orcid.org/0000-0001-6622-3627 [email protected], and Liping Jing [email protected]Author Affiliations

Publication: International Journal of Geomechanics

Volume 24, Issue 7

https://doi.org/10.1061/IJGNAI.GMENG-8185

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Abstract

The seismic response of seabed is fundamental for researching the seismic response of marine structures. The one-dimensional transient response of an unsaturated seabed was studied in this paper. An analytical solution for a special case with infinite permeability coefficients and a semianalytical solution for a general case with arbitrary permeability coefficients were developed. The advantage of this semianalytical solution is that it can compute the behavior of unsaturated soils with very low permeability coefficients. During the derivation, the boundary conditions were transformed into homogeneous boundary conditions first, and then the eigenvalues and eigenfunctions were obtained. To obtain solutions in the time domain, the eigenfunction expansion method and the precise time-step integration method were employed. Through numerical examples, the reflection and transmission characteristics of three types of compression waves at the fluid–solid interface were elucidated. The impact of permeability coefficients and water saturation on the seismic response of the seabed was investigated through parametric analysis. When the soil is nearing saturation, even a slight change in saturation can lead to a drastic alteration in the seismic response of the nearly saturated seabed.

Introduction

The dynamic response of the seabed forms the research basis for the earthquake stability of the seabed and the safety of marine structures. In the natural marine environment, based on field and laboratory test results, it is observed that some marine sediments are not completely saturated by water (Gao and Cai 2021; Wang et al. 2018). Typically, the theory of porous media is employed to research the dynamic response of the seabed (Ulker and Rahman 2009; Ulker 2014).

There is an increasing number of articles employing analytical solutions to study the steady-state and transient response of the seabed. Chen et al. (2020a), Wang et al. (2013), Lyu et al. (2014), and Wang et al. (2020) primarily investigated the steady-state response of a saturated porous seabed. Chen et al. (2020b) developed analytical solutions for the steady-state response of a nearly saturated single-layer porous seabed and compared the differences between fully dynamic and semidynamic formulas. The steady-state response can be used to analyze the spectral characteristics of the seabed, but it cannot distinguish different types of compressional waves in the seabed. The transient response not only enables simulation of the actual seismic response but also explicitly distinguishes different types of compressional waves.

For the transient response of porous media, Garg et al. (1974), Simon et al. (1984), de Boer et al. (1993), Gajo and Mongiovi (1995), Schanz and Cheng (2000), Shan et al. (2011), and Zhao et al. (2022) researched the transient response of saturated porous media. Li and Schanz (2011) provided a semianalytical solution for the transient response of one-dimensional unsaturated single-layer porous media. However, none of these studies take into account the influence of the seawater layer.

For the transient response of seabed, Chen et al. (2018) conducted an analytical investigation into the dynamic response and transient liquefaction of a multilayered porous seabed under vertical seismic loading. Shan et al. (2019) developed a one-dimensional transient response solution for a saturated single-layer seabed. Li et al. (2023) analyzed the one-dimensional transient and steady-state responses of a saturated seabed under different fluid–solid interface conditions.

To the best of the authors' knowledge, no analytical solution for the transient response of a one-dimensional unsaturated seabed has been provided. This paper employed the eigenfunction expansion method and the precise time-step integration method to provide an analytical solution for a specific case and a semianalytical solution for a general case. These solutions can clearly distinguish three types of compressional waves and analyze their reflection and transmission properties at the fluid–solid interface. Through numerical examples, the influences of the seawater layer, dynamic permeability coefficients, and saturation on the transient response of the seabed were analyzed.

Mathematic Model

The propagation of transient waves in an unsaturated seabed subjected to a vertical-incidence compressional wave is researched in this paper (Fig. 1).

Fig. 1. Sketch of an unsaturated seabed.

Basic Equations

The seawater is assumed to be an ideal fluid, and the displacement u_F and seawater pressure σ_F have the following forms:

u_{F, z z} = u_{F, t t} / c_{F}^{2}, - σ_{F} = λ_{F} u_{F, z}, c_{F} = \sqrt{λ_{F} / ρ_{F}}

(1)

where c_F = wave velocity; λ_F = bulk modulus of seawater; and ρ_F = seawater density.

The model of unsaturated porous media given by Zienkiewicz et al. (1999) and Li et al. (1990) is employed to simulate the unsaturated sediment. The two types of fluids in unsaturated soil are called wetting fluid and nonwetting fluid. The basic equations are listed as follows:

σ_{z} = M u_{t t} + C u_{t}, σ = K u_{z}

(2)

K u_{z z} - M u_{t t} - C u_{t} = 0

(3)

where

\begin{aligned} σ & = {\begin{matrix} σ \\ - p_{w} \\ - p_{n} \end{matrix}}, u = {\begin{matrix} u \\ w \\ v \end{matrix}}, M = [\begin{matrix} ρ & ρ_{w} & ρ_{n} \\ ρ_{w} & ρ_{w} / (n s_{w}) & 0 \\ ρ_{n} & 0 & ρ_{n} / (n s_{n}) \end{matrix}] \\ C & = [\begin{matrix} 0 & 0 & 0 \\ 0 & 1 / k_{w} & 0 \\ 0 & 0 & 1 / k_{n} \end{matrix}], \\ K & = [\begin{matrix} λ + 2 μ + α^{2} (Q_{w} s_{w} + Q_{n} s_{n}) & α Q_{w} & α Q_{n} \\ α Q_{w} & Q_{w} / s_{w} & 0 \\ α Q_{n} & 0 & Q_{n} / s_{n} \end{matrix}] \end{aligned}

\begin{aligned} α & = 1 - K_{b} / K_{s}, 1 / Q_{w} = (α - n) / K_{s} + n / K_{w}, \\ 1 / Q_{n} = (α - n) / K_{s} + n / K_{n} \\ ρ & = n s_{w} ρ_{w} + n s_{n} ρ_{n} + (1 - n) ρ_{s}, s_{w} + s_{n} = 1, K_{b} = λ + 2 μ / 3 \\ k_{w} & = k_{w f} / ρ_{w} g, k_{n} = k_{n f} / ρ_{n} g \end{aligned}

(4)

where subscripts w and n = wetting and nonwetting fluids, respectively; σ = total stress; p_w and p_n = fluid pressures; u = solid displacement; w and v = relative fluid displacements of the wetting and nonwetting fluids, respectively; n = porosity; s_w and s_n = saturations of sediment; K_b, K_s, K_w, and K_n = bulk moduli of the solid skeleton, the solid particles, the wetting fluid, and the nonwetting fluid, respectively; λ and μ = Lamé parameters of the solid skeleton; ρ, ρ_s, ρ_w, and ρ_n = densities of the unsaturated seabed, the solid grains, the wetting fluid, and the nonwetting fluid, respectively; k_w and k_n = dynamic permeability coefficients; and k_wf and k_nf = Darcy permeability coefficients.

Boundary, Interface, and Initial Conditions

As shown in Fig. 1, the boundary conditions can be listed as follows:

σ_{F} (- h) = f_{1} (t), u (H) = f_{2} (t), w (H) = 0, v (H) = 0

(5)

The interface continuity conditions (Deresiewicz and Skalak 1963; Shan et al. 2019) between the seawater and the unsaturated seabed are considered as follows:

\begin{aligned} σ (0, t) = - σ_{F} (0, t), p_{w} (0, t) = σ_{F} (0, t), p_{n} (0, t) = σ_{F} (0, t) \\ u_{F} (0, t) = u (0, t) + w (0, t) + v (0, t) \end{aligned}

(6)

In this research, the initial conditions are assumed to be zero, i.e.,

u_{F, t}^{} = 0

,

u_{F}^{} = 0

,

u_{, t}^{} = 0

, and

u = 0

.

Analytical/Semianalytical Solutions

The eigenfunction expansion method and the precise time step integration method are employed to give the solutions for the one-dimensional transient response.

Eigenvalues and Eigenfunctions

Employing the method in Shan et al. (2011), the displacements of the model are rewritten into two parts:

\begin{aligned} u_{F} (z) & = u_{F}^{d} (z) + u_{F}^{s} (z) \\ u (z) & = u^{d} + u^{s} = {\begin{matrix} u^{d}, w^{d}, v^{d} \end{matrix}}^{T} + {\begin{matrix} u^{s}, w^{s}, v^{s} \end{matrix}}^{T} \end{aligned}

(7)

where

u_{F}^{d} (z)

and

u^{d} (z)

= dynamic solutions; and

u_{F}^{s} (z)

and

u^{s} (z)

= static solutions. The static solutions are chosen to satisfy the boundary conditions and transform the inhomogeneous boundary conditions into homogeneous:

\begin{aligned} u_{F}^{s} (z) & = {- λ_{F}^{- 1} z + \frac{{(α - 1)}^{2} H}{(λ + 2 μ)} + \frac{s_{w} H}{Q_{w}} + \frac{s_{n} H}{Q_{n}}} f_{1} (t) + f_{2} (t), \\ u^{s} (z) & = \frac{α - 1}{(λ + 2 μ)} (z - H) f_{1} (t) + f_{2} (t), \\ w^{s} (z) & = - {\frac{α^{2} - α}{(λ + 2 μ)} + \frac{1}{Q_{w}}} s_{w} (z - H) f_{1} (t), \\ v^{s} (z) & = - {\frac{α^{2} - α}{(λ + 2 μ)} + \frac{1}{Q_{n}}} s_{n} (z - H) f_{1} (t) \end{aligned}

(8)

Substituting Eqs. (7) and (8) into Eqs. (1), (3), (5) and (6) and the initial conditions, we have

\begin{aligned} u_{F, z z}^{d} & = c_{F}^{- 2} u_{F, t t}^{d} + c_{F}^{- 2} u_{F, t t}^{s} \\ {K u}_{, z z}^{d} - {M u}_{, t t}^{d} - {C u}_{, t}^{d} & = S (t), S (t) = {M u}_{, t t}^{s} + {C u}_{, t}^{s} \end{aligned}

(9)

Boundary conditions:

σ_{F}^{d} (- h) = 0, u^{d} (H) = 0, w^{d} (H) = 0, v^{d} (H) = 0

(10)

Interface conditions:

\begin{aligned} σ^{d} (0, t) & = - σ_{F}^{d} (0, t), p_{w}^{d} (0, t) = σ_{F}^{d} (0, t), p_{n}^{d} (0, t) = σ_{F}^{d} (0, t) \\ u_{F}^{d} (0, t) & = u^{d} (0, t) + w^{d} (0, t) + v^{d} (0, t) \end{aligned}

(11)

Initial conditions:

\begin{aligned} u_{F, t}^{d} (z, 0) & = - u_{F, t}^{s} (z, 0), u_{F}^{d} (z, 0) = - u_{F}^{s} (z, 0) \\ u_{, t}^{d} (z, 0) & = - u_{, t}^{s} (z, 0), u^{d} (z, 0) = - u^{s} (z, 0) \end{aligned}

(12)

The characteristic equations of Eq. (9) have the following form:

u_{F, z z}^{d} - c_{F}^{- 2} u_{F, t t}^{d} = 0, {K u}_{, z z}^{d} - {M u}_{, t t}^{d} = 0

(13)

through which the eigenvalues and eigenfunctions can be obtained. Assume the solutions of Eq. (13) have the following forms:

u_{F}^{d} = U_{F}^{d} \exp (- i ω t), u_{}^{d} = U_{}^{d} \exp (- i ω t), w^{d} = W^{d} \exp (- i ω t), v^{d} = V^{d} \exp (- i ω t)

(14)

where ω > 0 = eigenvalue. Substituting Eq. (14) into Eq. (13), Eq. (13) can be transformed into ordinary differential equations. Using the state-space method, the solutions of Eq. (13) can be obtained as follows:

X_{F} (z) = Z_{F} (z) X_{F} (- h), X (z) = Z (z) X (0)

(15)

where

\begin{aligned} X_{F} (z) & = {- \sum_{F}^{d}, U_{F}^{d}}^{T}, Z_{F} (z) = \exp [N_{F} (z + h)], N_{F} = [\begin{matrix} 0 & - λ_{F} ω^{2} c_{F}^{- 2} \\ λ_{F}^{- 1} & 0 \end{matrix}] \\ X (z) & = {\sum^{d}, - P_{w}^{d}, - P_{n}^{d}, U_{}^{d}, W^{d}, V^{d}}^{T}, Z (z) = \exp [N z], N = [\begin{matrix} 0 & - ω^{2} M \\ N_{21} & 0 \end{matrix}] \end{aligned}

N_{21} = \frac{1}{λ + 2 μ} [\begin{matrix} 1 & - α s_{w} & - α s_{n} \\ - α s_{w} & (λ + 2 μ + s_{w} α^{2} Q_{w}) s_{w} / Q_{w} & a^{2} s_{w} s_{n} \\ - α s_{n} & α^{2} s_{n} s_{w} & (λ + 2 μ + s_{n} α^{2} Q_{n}) s_{n} / Q_{n} \end{matrix}]

(16)

where

\sum_{F}^{d}

,

\sum^{d}

,

P_{w}^{d}

, and

P_{n}^{d}

= eigenfunctions of

σ_{F}^{d}

, σ^d,

p_{w}^{d}

, and

p_{n}^{d}

, respectively.

By the transfer matrix method and the interface conditions [Eq. (11)], we obtain the following relationship:

\begin{aligned} X (z) & = V (z) {\begin{matrix} \begin{matrix} X_{F} (- h) \\ W^{d} (0) \end{matrix} \\ V^{d} (0) \end{matrix}}, \\ V (z) & = Z (z) Y {\begin{matrix} Z_{F} (0) & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}}, Y = [\begin{matrix} 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & - 1 & - 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \end{aligned}

(17)

Substituting boundary conditions [Eq. (10)] into the first equation of Eq. (17), we obtain the following equation:

[\begin{matrix} v_{42} (H) & v_{43} (H) & v_{44} (H) \\ v_{52} (H) & v_{53} (H) & v_{54} (H) \\ v_{62} (H) & v_{63} (H) & v_{64} (H) \end{matrix}] {\begin{matrix} U_{F}^{d} (- h) \\ W^{d} (0) \\ V^{d} (0) \end{matrix}} = 0

(18)

where v_ij(z) = element of matrix

V (z)

. If Eq. (18) has a nonzero solution, the determinant of the coefficient matrix should be equal to zero, i.e.,

\begin{aligned} v_{42} (H) v_{53} (H) v_{64} (H) + v_{43} (H) v_{54} (H) v_{62} (H) \\ + v_{44} (H) v_{52} (H) v_{63} (H) - v_{44} (H) v_{53} (H) v_{62} (H) \\ - v_{43} (H) v_{52} (H) v_{64} (H) - v_{42} (H) v_{54} (H) v_{63} (H) = 0 \end{aligned}

(19)

This is a characteristic equation of ω and a transcendental equation with infinite positive roots labeled as ω_k (k = 1, 2, 3, …) from the smallest to the largest. A nonzero solution to Eq. (18) has the following form:

\begin{aligned} U_{F}^{d} (- h) & = \frac{v_{44 k} (H) v_{53 k} (H) - v_{43 k} (H) v_{54 k} (H)}{v_{43 k} (H) v_{52 k} (H) - v_{42 k} (H) v_{53 k} (H)} \\ W^{d} (0) & = \frac{v_{42 k} (H) v_{54 k} (H) - v_{44 k} (H) v_{52 k} (H)}{v_{43 k} (H) v_{52 k} (H) - v_{42 k} (H) v_{53 k} (H)}, V^{d} (0) = 1 \end{aligned}

(20)

where k corresponds to eigenvalue ω_k. Therefore, the eigenfunctions can be solved as follows:

\begin{aligned} U_{F k}^{d} & = z_{F 22 k} (z) U_{F}^{d} (- h), U_{k}^{d} = {\begin{matrix} U_{k}^{d} (z) \\ W_{k}^{d} (z) \\ V_{k}^{d} (z) \end{matrix}} \\ = [\begin{matrix} v_{42 k} (z) & v_{43 k} (z) & v_{44 k} (z) \\ v_{52 k} (z) & v_{53 k} (z) & v_{54 k} (z) \\ v_{62 k} (z) & v_{63 k} (z) & v_{64 k} (z) \end{matrix}] {\begin{matrix} U_{F}^{d} (- h) \\ W^{d} (0) \\ 1 \end{matrix}} \end{aligned}

(21)

Solutions in the Time Domain

By using the method of undetermined coefficients, the solution to Eq. (9) can be assumed to have the following form:

u_{F}^{d} (z, t) = \sum_{k = 1}^{\infty} U_{F k}^{d} (z) Ω_{k} (t), u_{}^{d} (z, t) = \sum_{k = 1}^{\infty} U_{k}^{d} (z) Ω_{k} (t)

(22)

where the eigenfunctions

U_{F k}^{d} (z)

and

U_{k}^{d} (z)

are shown in Eq. (21), and the unknown function Ω_k(t) can be obtained by the initial condition [Eq. (12)]. Substituting Eq. (22) into Eq. (9) and using the orthogonality relationship of eigenfunctions (Shan et al. 2019), we obtain an ordinary differential equation in terms of Ω_k(t):

\begin{aligned} Ω_{p, t t} (t) + \sum_{q = 1}^{\infty} c_{p q} Ω_{p, t} (t) + ω^{2} Ω_{p} (t) = \bar{S} (t) \\ c_{p q} = G_{p}^{- 1} \int_{0}^{H} W_{p}^{d} W_{q}^{d} k_{w}^{- 1} + V_{p}^{d} V_{q}^{d} k_{n}^{- 1} d z \\ \bar{S} (t) = - G_{p}^{- 1} \int_{0}^{H} {[U_{p}^{d}]}^{T} S (t) d z - G_{p}^{- 1} \int_{- h}^{0} U_{F p}^{d} λ_{F} c_{F}^{- 2} u_{F, t t}^{s} (t) d z \end{aligned}

(23)

Similarly, the initial conditions [Eq. (12)] can be rewritten as follows:

\begin{aligned} Ω_{p, t} (0) & = - \frac{1}{G_{p}} \int_{- h}^{0} U_{F p}^{d} (z) λ_{F} c_{F}^{- 2} u_{F, t}^{s} (z, 0) d z \\ - \frac{1}{G_{p}} \int_{0}^{H} {[U_{p}^{d} (z)]}^{T} {M u}_{, t}^{s} (z, 0) d z \\ Ω_{p} (0) & = - \frac{1}{G_{p}} \int_{- h}^{0} U_{F p}^{d} (z) λ_{F} c_{F}^{- 2} u_{F}^{s} (z, 0) d z \\ - \frac{1}{G_{p}} \int_{0}^{H} {[U_{p}^{d} (z)]}^{T} M u^{s} (z, 0) d z \end{aligned}

(24)

1.

Special case: If the dynamic permeability coefficients k_w and k_n are sufficiently large and the fluids can flow freely within the porous media, we have

C = 0

and c_pq = 0. Therefore, the first equation of Eq. (23) can be rewritten as follows:

Ω_{p, t t} (t) + ω^{2} Ω_{p} (t) = \bar{S} (t)

(25)

The solution of the aforementioned equation has the following form:

Ω_{p} (t) = α_{T 0} (t) Ω_{p} (0) + α_{T 0} (t) \int_{0}^{t} α_{T 1} (- ξ) \bar{S} (ξ) d ξ + α_{T 1} (t) Ω_{p, t} (0) + α_{T 1} (t) \int_{0}^{t} α_{T 0} (- ξ) \bar{S} (ξ) d ξ

(26)

2.

General case: As the dynamic permeability coefficients k_w and k_n are arbitrary, not all c_pq equal zero. Therefore, the first equation of Eq. (23) has infinite equations and should be truncated during numerical calculations. The first N items are chosen to construct the system, and the precise time-integration method is used to solve this type of equation (Zhong and Williams 1994; Lin et al. 1995). The detailed procedure can be found in Shan et al. (2019).

Substituting Ω_k(t) into Eq. (22) and using Eq. (7), the displacements u_F(z, t) in the seawater layer and

u (z, t)

in the unsaturated seabed can be written as follows:

u_{F}^{} (z, t) = \sum_{k = 1}^{\infty} U_{F k}^{d} (z) Ω_{k} (t) + u_{F}^{s}, u (z, t) = \sum_{k = 1}^{\infty} U_{k}^{d} (z) Ω_{k} (t) + u^{s}

(27)

Example Analysis

To evaluate the influence of gas on the transient response of unsaturated seabed, we researched both the soil with water and gas and the soil with water and oil. The material parameters are as follows: h = 10 m, H = 20 m, n = 0.48, ρ_s = 2,700 kg/m³, ρ_w = ρ_F = 1,000 kg/m³, λ_F = 2.25 GPa, λ = 144.7 MPa, μ = 98 MPa, K_s = 11 GPa, K_w = 2.25 GPa, ρ_n = 762 kg/m³ (oil) or 1.1 kg/m³ (gas), K_n = 570 MPa (oil) or 0.145 MPa(gas), and s_w = 0.8. We have verified the analytical/semianalytical solutions by numerical results obtained by the Laplace transform method [Fig. 3(a)]. The Laplace numerical inverse transformation results will diverge if the calculation time is long. Therefore, only the calculation results of the first 0.06 s are given.

Three Types of Compressional Waves

There are three types of compressional waves in unsaturated porous media (Li and Schanz 2011; Albers 2009): P1, P2, and P3, from fast to slow. The amplitude of the compressional waves in a porous medium (where the nonwetting fluid is gas) is too small for analysis. However, the characteristics of compressional waves can be effectively analyzed by using porous media with oil as the nonwetting fluid. The bottom surface is subjected to a single sine load; thus f₁(t) = 0 and f₂(t) = sin(2,000πt) (0 ≤ t ≤ 0.001 s) in Eq. (5).

Fig. 2 shows the wave propagation at different times. At t = 0.005 s, three types of compressional waves are excited from the bottom. At t = 0.015 s, the P1 wave arrives at the fluid–solid interface and generates three reflected compressional waves (P11, P12, and P13) in the unsaturated layer and one transmitted wave (F1) in the seawater layer. At t = 0.02 s, the F1 wave reflects from the free fluid surface and generates one reflected F1 wave. At t = 0.025 s, the P2 wave arrives at the fluid–solid interface and generates three reflected compressional waves (P21, P22, and P23) in the unsaturated seabed and one transmitted wave (F2) that meets the F1 wave. At t = 0.0275 s, the F1 wave in the seawater layer arrives at the fluid–solid interface and generates one reflected wave (F1f wave) and three transmitted compressional waves (F11, F12, and F13). The P11 wave is reflected from the bottom of the unsaturated layer, which generates one reflected wave P11.

Fig. 2. Distribution of displacements in the fluid/unsaturated seabed system at different times: (a) t = 0.005 − 0.0225 s; and (b) t = 0.025 − 0.0375 s.

The wave propagating from the unsaturated layer to the seawater layer generates three reflected compressional waves and one transmitted compressional wave. The wave propagating from the seawater layer to the unsaturated layer will generate three transmitted compressional waves and one reflected compressional wave.

Permeability Coefficients

The effects of the seawater layer and the permeability coefficients on the transient response of porous media are analyzed in this section. The bottom surface is subjected to a step load; thus, f₁(t) = 0 and f₂(t) = 1 in Eq. (5).

Fig. 3 illustrates the solid displacements in an unsaturated seabed versus time at z = 2 m (the nonwetting fluid is oil). The seawater layer has an obvious effect on the transient response of the unsaturated seabed. When

k_{w} = k_{n} = 10^{- 5} m^{4} / (Ns)

, the displacement is the smallest one and dissipates fastest. When

k_{w} = k_{n} = 10^{- 5} m^{4} / (Ns)

,

10^{- 6} m^{4} / (Ns)

,

10^{- 7} m^{4} / (Ns)

, and

10^{- 9} m^{4} / (Ns)

, the solid displacement of sediment with a fluid layer is larger than that of sediment without a fluid layer in the initial stage. The dissipation rates of the six cases are different. Fig. 4 shows the solid displacements versus time at z = 2 m (the nonwetting fluid is gas). When

k_{w} = k_{n} = 10^{- 6} m^{4} / (Ns)

,

10^{- 7} m^{4} / (Ns)

, and

10^{- 9} m^{4} / (Ns)

, the solid displacement of sediment with a fluid layer dissipates faster than that of sediment without a fluid layer.

Fig. 3. Solid displacements versus time at point z = 2 m (the nonwetting fluid is oil): (a) k_w → ∞, k_n → ∞; (b) $k_{w} = 10^{- 4} m^{4} / (N s)$ , $k_{n} = 10^{- 4} m^{4} / (N s)$ ; (c) $k_{w} = 10^{- 5} m^{4} / (N s)$ , $k_{n} = 10^{- 5} m^{4} / (N s)$ ; (d) $k_{w} = 10^{- 6} m^{4} / (N s)$ , $k_{n} = 10^{- 6} m^{4} / (N s)$ ; (e) $k_{w} = 10^{- 7} m^{4} / (N s)$ , $k_{n} = 10^{- 7} m^{4} / (N s)$ ; and (f) $k_{w} = 10^{- 9} m^{4} / (N s)$ , $k_{n} = 10^{- 9} m^{4} / (N s)$ .

Fig. 4. Solid displacements versus time at point z = 2 m (the nonwetting fluid is gas): (a) k_w → ∞, k_n → ∞; (b) $k_{w} = 10^{- 4} m^{4} / (N s)$ , $k_{n} = 10^{- 4} m^{4} / (N s)$ ; (c) k_w = 10⁻⁵ m^4/(N s), $k_{n} = 10^{- 5} m^{4} / (N s)$ ; (d) $k_{w} = 10^{- 6} m^{4} / (N s)$ , $k_{n} = 10^{- 6} m^{4} / (N s)$ ; (e) $k_{w} = 10^{- 7} m^{4} / (N s)$ , $k_{n} = 10^{- 7} m^{4} / (N s)$ ; and (f) $k_{w} = 10^{- 9} m^{4} / (N s)$ , $k_{n} = 10^{- 9} m^{4} / (N s)$ .

The dissipation of waves in porous media is primarily governed by the relative fluid displacement and the viscosity between the fluid and solid phases. With a decrease in the dynamic permeability coefficient, the relative fluid displacement experiences a gradual decrease, while simultaneously, the viscosity gradually increases. When k_w and k_n → ∞, no viscosity is present between the fluid and solid phases. As a result, wave dissipation does not occur. Similarly, when the dynamic permeability coefficient is exceedingly small, the relative displacement of the fluid is minimal, resulting in a sluggish dissipation rate of waves. When k_w = k_n = 10⁻⁵ m^4/(Ns) (the nonwetting fluid is oil) and k_w = k_n = 10⁻⁷ m^4/(Ns) (the nonwetting fluid is gas), both the viscosity and the relative fluid displacement are large. Therefore, the wave dissipates very fast.

Saturation

The effect of saturation on wave propagation in seabed saturated with water and gas is studied in this section. Fig. 5(a) shows the displacements versus time at different saturations at point z = 2 m (k_w = k_n = 10⁻⁶ m^4/(Ns)). The amplitude and dissipation of displacement are different for different saturations. Fig. 5(b) shows the variation curve of max displacement with saturation. When the sediment is nearly saturated, a small change in saturation will cause a sharp change in the maximum displacement.

Conclusions

An analytical solution for a special case with infinite dynamic permeability coefficients and a semianalytical solution for a general case with arbitrary dynamic permeability coefficients are developed in this paper. Due to the seawater layer, the wave field of an unsaturated seabed is very complex. When the wave arrives at the seawater-unsaturated seabed interface, three compressional waves in the unsaturated seabed and one compressional wave in the seawater will be generated. The dynamic permeability coefficients and the seawater layer have obvious effects on wave propagation in a seawater/unsaturated seabed system. The effects of saturation and nonwetting fluid types on the transient response of unsaturated seabed are considerable. In particular, when the nonwetting fluid is a gas, even a slight change in saturation can lead to significant variations in the displacement of solids within a nearly saturated seabed.

Data Availability Statement

Some or all data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request, e.g., all the figures in this paper and the code for all simulation models.

Acknowledgments

This work was supported by the National Key R&D Program of China (2021YFC3100700, 2018YFC1504004), the National Natural Science Foundation of China (U2039209, 41874067), and the Heilongjiang Provincial Natural Science Foundation of China (YQ2021D010).

References

Albers, B. 2009. “Analysis of the propagation of sound waves in partially saturated soils by means of a macroscopic linear poroelastic model.” Transp. Porous Media 80 (1): 173–192. https://doi.org/10.1007/s11242-009-9360-y.

Google Scholar

Chen, W., G. Chen, D. Jeng, and L. Xu. 2020a. “Ocean bottom hydrodynamic pressure due to vertical seismic motion.” Int. J. Geomech. 20 (9): 06020025. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001802.

Google Scholar

Chen, W., D. Jeng, W. Chen, G. Chen, and H. Zhao. 2020b. “Seismic-induced dynamic responses in a poro-elastic seabed: Solutions of different formulations.” Soil Dyn. Earthquake Eng. 131: 106021. https://doi.org/10.1016/j.soildyn.2019.106021.

Google Scholar

Chen, W.-Y., Z.-H. Wang, G.-X. Chen, D.-S. Jeng, M. Wu, and H.-Y. Zhao. 2018. “Effect of vertical seismic motion on the dynamic response and instantaneous liquefaction in a two-layer porous seabed.” Comput. Geotech. 99: 165–176. https://doi.org/10.1016/j.compgeo.2018.03.005.

Google Scholar

de Boer, R., W. Ehlers, and Z. Liu. 1993. “One-dimensional transient wave propagation in fluid-saturated incompressible porous media.” Arch. Appl. Mech. 63 (1): 59–72. https://doi.org/10.1007/BF00787910.

Google Scholar

Deresiewicz, H., and R. Skalak. 1963. “On uniqueness in dynamic poroelasticity.” Bull. Seismol. Soc. Am. 53 (4): 783–788. https://doi.org/10.1785/BSSA0530040783.

Google Scholar

Gajo, A., and L. Mongiovi. 1995. “An analytical solution for the transient response of saturated linear elastic porous media.” Int. J. Numer. Anal. Methods Geomech. 19 (6): 399–413. https://doi.org/10.1002/nag.1610190603.

Google Scholar

Gao, Z., and H. Cai. 2021. “Effect of total stress path and gas volume change on undrained shear strength of gassy clay.” Int. J. Geomech. 21 (11): 04021218. https://doi.org/10.1061/(ASCE)GM.1943-5622.0002198.

Google Scholar

Garg, S., A. H. Nayfeh, and A. Good. 1974. “Compressional waves in fluid-saturated elastic porous media.” J. Appl. Phys. 45 (5): 1968–1974. https://doi.org/10.1063/1.1663532.

Google Scholar

Li, J., Z. Shan, T. Li, and L. Jing. 2023. “Effects of the fluid–solid interface conditions on the dynamic responses of the saturated seabed during earthquakes.” Soil Dyn. Earthquake Eng. 165: 107728. https://doi.org/10.1016/j.soildyn.2022.107728.

Google Scholar

Li, P., and M. Schanz. 2011. “Wave propagation in a 1-d partially saturated poroelastic column.” Geophys. J. Int. 184 (3): 1341–1353. https://doi.org/10.1111/j.1365-246X.2010.04913.x.

Google Scholar

Li, X., O. Zienkiewicz, and Y. Xie. 1990. “A numerical model for immiscible two-phase fluid flow in a porous medium and its time domain solution.” Int. J. Numer. Methods Eng. 30 (6): 1195–1212. https://doi.org/10.1002/nme.1620300608.

Google Scholar

Lin, J., W. Shen, and F. W. Williams. 1995. “A high precision direct integration scheme for structures subjected to transient dynamic loading.” Comput. Struct. 56 (1): 113–120. https://doi.org/10.1016/0045-7949(94)00537-D.

Google Scholar

Lyu, D.-D., J.-T. Wang, F. Jin, and C.-H. Zhang. 2014. “Reflection and transmission of plane waves at a water–porous sediment interface with a double-porosity substrate.” Transp. Porous Media 103 (1): 25–45. https://doi.org/10.1007/s11242-014-0286-7.

Google Scholar

Schanz, M., and A.-D. Cheng. 2000. “Transient wave propagation in a one-dimensional poroelastic column.” Acta Mech. 145 (1-4): 1–18. https://doi.org/10.1007/BF01453641.

Google Scholar

Shan, Z., D. Ling, and H. Ding. 2011. “Exact solutions for one-dimensional transient response of fluid-saturated porous media.” Int. J. Numer. Anal. Methods Geomech. 35 (4): 461–479. https://doi.org/10.1002/nag.904.

Google Scholar

Shan, Z., D. Ling, Z. Xie, L. Jing, and Y. Li. 2019. “A semianalytical solution for one dimensional transient wave propagation in a saturated single-layer porous medium with a fluid surface layer.” Int. J. Numer. Anal. Methods Geomech. 43 (13): 2184–2199. https://doi.org/10.1002/nag.2978.

Google Scholar

Simon, B., O. Zienkiewicz, and D. Paul. 1984. “An analytical solution for the transient response of saturated porous elastic solids.” Int. J. Numer. Anal. Methods Geomech. 8 (4): 381–398. https://doi.org/10.1002/nag.1610080406.

Google Scholar

Ulker, M., and M. Rahman. 2009. “Response of saturated and nearly saturated porous media: Different formulations and their applicability.” Int. J. Numer. Anal. Methods Geomech. 33 (5): 633–664. https://doi.org/10.1002/nag.739.

Google Scholar

Ulker, M. B. C. 2014. “Wave-induced dynamic response of saturated multi-layer porous media: Analytical solutions and validity regions of various formulations in non-dimensional parametric space.” Soil Dyn. Earthquake Eng. 66: 352–367. https://doi.org/10.1016/j.soildyn.2014.08.005.

Google Scholar

Wang, J.-T., F. Jin, and C.-H. Zhang. 2013. “Reflection and transmission of plane waves at an interface of water/porous sediment with underlying solid substrate.” Ocean Eng. 63: 8–16. https://doi.org/10.1016/j.oceaneng.2013.01.028.

Google Scholar

Wang, P., G. Zhang, M. Zhao, and R. Xi. 2020. “Semianalytical solutions for the wave-induced and vertical earthquake-induced responses of a fluid-stratified seabed-bedrock system.” Soil Dyn. Earthquake Eng. 139: 106391. https://doi.org/10.1016/j.soildyn.2020.106391.

Google Scholar

Wang, Y., L. Kong, Y. Wang, M. Wang, and K. Cai. 2018. “Deformation analysis of shallow gas-bearing ground from controlled gas release in Hangzhou bay of China.” Int. J. Geomech. 18 (1): 04017122. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001029.

Google Scholar

Zhao, Y., X. Chen, Z. Shan, D. Ling, and Z. Xiao. 2022. “Semianalytical solution for thetransient response of one-dimensional saturated multilayered soil column.” Int. J. Geomech. 22 (12): 04022232. https://doi.org/10.1061/(ASCE)GM.1943-5622.0002600.

Google Scholar

Zhong, W., and F. Williams. 1994. “A precise time step integration method.” Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci. 208 (6): 427–430. https://doi.org/10.1243/PIME_PROC_1994_208_148_02.

Google Scholar

Zienkiewicz, O. C., A. Chan, M. Pastor, B. Schrefler, and T. Shiomi. 1999. Computational 510 geomechanics. Vol. 613. Chichester, UK: Wiley.

Google Scholar

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International Journal of Geomechanics

Volume 24 • Issue 7 • July 2024

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This work is made available under the terms of the Creative Commons Attribution 4.0 International license, https://creativecommons.org/licenses/by/4.0/.

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Received: Jul 8, 2022

Accepted: Oct 21, 2023

Published online: May 7, 2024

Published in print: Jul 1, 2024

Discussion open until: Oct 7, 2024

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Rong Ma [email protected]

Key Laboratory of Earthquake Engineering and Engineering Vibration, Institute of Engineering Mechanics, China Earthquake Administration, Harbin 150080, PR China; Key Laboratory of Earthquake Disaster Mitigation, Ministry of Emergency Management, Harbin 150080, PR China. Email: [email protected]

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Zhendong Shan https://orcid.org/0000-0001-6622-3627 [email protected]

Key Laboratory of Earthquake Engineering and Engineering Vibration, Institute of Engineering Mechanics, China Earthquake Administration, Harbin 150080, PR China; Key Laboratory of Earthquake Disaster Mitigation, Ministry of Emergency Management, Harbin 150080, PR China (corresponding author). ORCID: https://orcid.org/0000-0001-6622-3627. Email: [email protected]

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Liping Jing [email protected]

Key Laboratory of Earthquake Engineering and Engineering Vibration, Institute of Engineering Mechanics, China Earthquake Administration, Harbin 150080, PR China; Key Laboratory of Earthquake Disaster Mitigation, Ministry of Emergency Management, Harbin 150080, PR China. Email: [email protected]

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Transient Response of an Unsaturated Single-Layer Seabed Subjected to a Vertical Compressional Wave

Abstract

Introduction

Mathematic Model

Basic Equations

Boundary, Interface, and Initial Conditions

Analytical/Semianalytical Solutions

Eigenvalues and Eigenfunctions

Solutions in the Time Domain

Example Analysis

Three Types of Compressional Waves

Permeability Coefficients

Saturation

Conclusions

Data Availability Statement

Acknowledgments

References

Information & Authors

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Authors

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Abstract

Introduction

Mathematic Model

Basic Equations

Boundary, Interface, and Initial Conditions

Analytical/Semianalytical Solutions

Eigenvalues and Eigenfunctions

Solutions in the Time Domain

Example Analysis

Three Types of Compressional Waves

Permeability Coefficients

Saturation

Conclusions

Data Availability Statement

Acknowledgments

References

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