Poroelastic effects on seismic monitoring in partially saturated loosely deposited sands

Introduction: A better understanding of the relationship between water saturation and seismic characteristics of loosely deposited sand is crucial for assessing the liquefaction potential of open-pit mine dumps. The classic elastic models, based on Gassmann’s theory, are commonly used to constrain water saturation from seismic data. However, while effective at low frequencies, these models fail to capture macroscopic wave-induced fluid flow and poroelastic effects at higher frequencies, which may hinder the application of laboratory results to field conditions. In contrast, Biot’s poroelastic theory enables seismic wave propagation modeling across a broad frequency range.

Methods: To evaluate poroelastic effects in seismic monitoring, we compare elastic and poroelastic models in terms of wave velocities, Poisson’s ratio, P-wave travel time, and surface wave dispersion across different frequencies. Our study incorporates numerical simulations and experimental data to assess the extent of discrepancies between these models.

Results: Our findings show that the poroelastic model predicts seismic wave velocities with up to a 6% difference compared to the elastic model. While minimal differences are observed in field-scale surveys, the discrepancies become more significant in pilot plant experiments and ultrasonic measurements as frequency increases. These results highlight the influence of poroelastic effects, which are not captured adequately by elastic models at higher frequencies.

Discussion: The observed frequency-dependent discrepancies suggest that elastic models may be insufficient for high-frequency seismic applications. This underscores the need for improved methodologies that integrate poroelastic effects to enhance the scalability of laboratory findings to field conditions.

1 Introduction

Open-pit lignite mining operations often create mine dumps composed of loosely deposited sands with high porosity and permeability. The risk of liquefaction in these dumps increases when water levels rise due to rainfall or other factors, posing significant environmental and geotechnical challenges. The degree of saturation in these sands is crucial in assessing the liquefaction potential (Molina-Gómez et al., 2023). Therefore, accurate estimation of the spatial distribution of saturation in loosely deposited sands is essential to map the risk of liquefaction.

Seismic methods provide a non-invasive technique that can cover large areas quickly and efficiently to explore the saturation distribution in loosely deposited sediments. These methods can complement other techniques such as time-domain reflectometry (Fenta and Szanyi, 2021), remote sensing (Mohanty, 2013; Plati et al., 2020), and ground-penetrating radar (Moghadas et al., 2010) to obtain a comprehensive understanding of subsurface conditions (Milani et al., 2015; Solazzi et al., 2021). Successful monitoring of saturation distribution requires proper modeling of the physical properties of porous media, which provides a quantitative link between the physical properties of the media and the seismic response. Loosely deposited sand, being a highly porous medium, necessitates that the physical model accounts for poroelastic effects due to the interaction between soil particles and pore fluid, which impacts wave propagation. A common approach is to use the Gassmann theory (Gassmann, 1951; Wood, 1955) in the elastic model to emphasize fluid effects (Morency et al., 2011), which, based on the low-frequency assumption, is widely used in practical applications such as shallow sediment exploration (Bachrach and Nur, 1998; Bachrach et al., 1998; 2000; Accaino et al., 2023), reservoir engineering (Chen and Zhang, 2017; Wang et al., 2022), and geofluid discrimination (Zong et al., 2015), as it simplifies the modeling of seismic wave propagation in porous media.

However, the elastic model is invalid in the high-frequency range as it cannot model the macroscopic wave-induced fluid flow, i.e., poroelastic effects in the high-frequency range. Meanwhile, seismic methods often rely on empirical correlations established in the laboratory for seismic inversion at the field scale. Correlating lab seismic data with field data provides a necessary database for constructing seismic models of specific geological areas. The frequency used in the lab is usually in the ultrasonic range (high-frequency range), where the mode of poroelastic effects differs from the seismic frequency range (low-frequency range). In this context, the poroelastic model (Biot, 1956a; Biot, 1956b) may be more appropriate for predicting the behavior of porous media across a wide frequency range, which also can incorporate more detailed information during parameterization and inversion compared to the elastic model. This has led to its use in various applications, including seismic exploration (Morency et al., 2011; Anthony and Vedanti, 2020; Alajmi et al., 2023), earthquake engineering (Meng et al., 2022), and soil dynamics (Chen et al., 2020). Furthermore, the poroelastic model has been extended and modified to account for complex phenomena such as anisotropy (Huang et al., 2022; Guo et al., 2022), heterogeneity (Wenzlau, 2009), and nonlinear behavior (Yang et al., 2021). Despite its advantages, the application of the poroelastic model in monitoring shallow sedimentary layers remains limited as the complexity of the environmental conditions, e.g., stiffness and saturation strongly depend on the depth in typical soils (Crane, 2013).

In this regard, some scholars have established the relationship between the seismic characteristics and the relevant properties of shallow loose media in the lab based on the poroelastic model, e.g., Emerson and Foray (2006); Whalley et al. (2012); Linneman et al. (2021). However, the measurement of seismic wave characteristics in lab sample tests provides only a single data point, which cannot be directly compared to field survey results, e.g., P-wave travel time curvestravel-time curves, and consider both the pressure dependence of the initial elastic modulus and the saturation distribution under practical conditions. Lab small-scale physical models using non-contact ultrasound techniques offer new perspectives for monitoring the saturation distribution in the shallow loose media (Bodet et al., 2010; 2014; Pasquet et al., 2016). Although some small-scale physical experiments have demonstrated the applicability of the poroelastic model in the ultrasonic frequency range (e.g., Barrière et al., 2012), there is still lacking However, a comprehensive theoretical model for partially saturated seismic wave propagation in partially saturated, loosely deposited sands to establish a connection between is still lacking, hindering the extrapolation of small-scale experimental results to field-scale observations. This limitation arises due to the frequency-dependent nature of poroelastic effects, as different frequency ranges are employed in small-scale physical experiments and field-scale observations. Therefore, it is necessary to compare the results between the elastic model and applications. To address this, we used the elastic model as a baseline and compared its predictions with those of the poroelastic model to investigate highlight the impact of poroelastic effects on seismic monitoring data in across different frequency ranges. Additionally, the theoretical model must account for the depth dependence of both the initial elastic modulus and the saturation distribution.

This paper aims to numerically explore poroelastic effects on seismic monitoring data in loosely deposited sands. To this end, we combine the poroelastic model saturated with two immiscible fluids and the Hertz-Mindin model, considering the saturation-depth profile, to get profiles of $V_{\text{p}}$ and $V_{\text{s}}$ which allow for the simulation of seismic monitoring datasets with different groundwater level, i.e., Poisson’s ratio, P-wave travel time and surface-wave dispersion. To evaluate the impact of poroelastic effects on seismic monitoring, we conducted three different scales of monitoring at varying water levels: field survey (seismic frequency), pilot plant survey (sonic frequency), and ultrasonic measurements (ultrasonic frequency). The elastic model serves as the baseline for comparison with the poroelastic model, highlighting poroelastic signatures across different frequency ranges.

2 Methodology

2.1 Background

In a typical open-pit mine dump scenario consisting of loosely deposited sands, as Figure 1a, the vadose zone exhibits varying degrees of saturation ranging from dry to partially saturated, while below the groundwater level, the sediment is fully saturated (Bachrach and Mukerji, 2012). Moreover, due to the complex seismic response of partially saturated loosely deposited sands, a small-scale physical model can be employed to validate relevant theories and media models. For instance, Pasquet et al. (2016) used a lab-scale controlled physical model (PM) consisting of glass beads (GBs), employing the ultrasonic technology to test the sensitivity of seismic monitoring results to different water table levels (Figures 1b, c).

Figure 1

Figure 1. (a) A typical loosely deposited sand scenario and the saturation-depth profile. Small-scale physical model with different water level$(z_{\text{wt}})$ and capillary fringe$(z_{\text{cap}})$(Pasquet et al., 2016): (b) PM-W1 with $z_{\text{wat}}=145\text{\hspace{0.17em}mm}$ and $z_{\text{wat}}=100\text{\hspace{0.17em}mm}$; (c) PM-W2 with $z_{\text{wat}}=85\text{\hspace{0.17em}mm}$ and $z_{\text{wat}}=50\text{\hspace{0.17em}mm}$.

As the model exhibits anisotropy only in the vertical direction, we consider a one-dimensional column of loosely deposited sand with properties dependent on the depth $z$, such as saturation, density, stiffness, and permeability. This model can be discretized into a stack of $n$ homogeneous and isotropic layers with thickness $h_j$, where $j=1,\dots ,n$. The $V_{\text{p},j}$ and $V_{\text{s},j}$ values for each layer can be obtained through plane wave solution (Morency and Tromp, 2008) of elastic and poroelastic models.

2.2 Calculation of $V_{\mathrm{p}}$ and $V_{\mathrm{s}}$

In this section, we present the poroelastic model considering the depth dependence of stiffness and saturation. For completeness and reference, a detailed summary of the derivation process of plane wave solution is presented in supplemental materials.

2.2.1 Governing equations of poroelastic model

Wave propagation in porous media (Biot, 1956a; Biot, 1956b) is formulated in terms of the primary variables solid skeleton displacement $\mathbf{u}$, relative fluid displacement $\mathbf{w}$ and pore pressure $p$. The equilibrium equation of the porous media is formulated, as presented in Equations 1–4.

$\rho \ddot{\mathbf{u}}+{\rho }_{\text{f}}\ddot{\mathbf{w}}=\left(H-G+{\alpha }^{2}M\right)\nabla \left(\nabla \cdot \mathbf{u}\right)+G{\nabla }^2\mathbf{u}+\alpha M\nabla \left(\nabla \cdot \mathbf{w}\right)(1)$

where ${\rho }_{\text{f}}$, ${\rho }_{\text{s}}$ and $\rho$ are the densities of fluid, solid skeleton, and porous media, respectively, with $\rho =\varphi {\rho }_{\text{f}}+(1-\varphi ){\rho }_{\text{s}}$; $\varphi$ is the porosity of the porous media; $\ddot{\mathbf{u}}$ and $\ddot{\mathbf{w}}$ are the acceleration of the solid skeleton and fluid (relative to solid), respectively; $H$ and $\mu$ are the P-wave modulus and shear modulus, respectively, with $H=K+3/4G$; $K$ and $G$ are bulk modulus and shear modulus of the porous media, respectively; $\alpha$ is Biot’s coefficient, with $\alpha =1-K/K_{\text{s}}$ (Biot and Willis, 1957; Zienkiewicz, 1982); $K_{\text{s}}$ is the bulk modulus of solid grain; $M$ is the inverse of the storage coefficient, with $M={(\varphi /K_{\text{f}}+(\alpha -\varphi )/K_{\text{s}})}^{-1}$ (Chang and Yoon, 2018).

The equilibrium equation of fluid is written as

${\rho }_{\text{f}}\ddot{\mathbf{u}}+m\ddot{\mathbf{w}}+bF\left(t\right)\ast \dot{\mathbf{w}}=\alpha M\nabla \left(\nabla \cdot \mathbf{u}+M\nabla \left(\nabla \cdot \mathbf{w}\right)\right)(2)$

where $m=\tau {\rho }_{\text{f}}/\varphi$ is the coupling mass and $\tau$ is the tortuosity with $\tau =1+0.5(1/\varphi -1)$; $b={\mu }_{\text{f}}/k_{\text{f}}$ is the viscous damping parameter; ${\mu }_{\text{f}}$ is the viscosity of the fluid and $k_{\text{f}}$ is the permeability of the fluid.

In Biot’s theory, the poroelastic effects is frequency-dependent, which could be characterized by the viscodynamic factor in the frequency domain (Barrière et al., 2012):

$\tilde{F}\left(\omega \right)=\sqrt{1+\frac{2i\omega }{{\omega }_{\text{c}}}}(3)$

where ${\omega }_{\text{c}}$ is the critical frequency in Biot’s theory, given by:

${\omega }_{\text{c}}=\frac{\varphi {\mu }_{\text{f}}}{k_{\text{f}}\tau {\rho }_{\text{f}}}.(4)$

In low-frequency conditions, i.e., $\omega /{\omega }_{\text{c}}\ll 1$, the pressure gradients generated within the fluid are transmitted to the solid via viscous drag, resulting in no relative motion between the fluid and the solid skeleton, called purely elastic wave regime. This behavior is consistent with Gassmann’s theory (Gassmann, 1951), an elastic theory of fluid substitution.

2.2.2 Stiffness modulus of loosely deposited sands

In order to model the wave propagation characteristic in granular material, the Hertz-Mindlin model (Mavko et al., 2009) is employed to assess the stiffness modulus of the soil frame. The bulk and shear modulus are defined in Equations 5, 6, displayed as

$K={\left[\frac{N^2{\left(1-\varphi \right)}^2{G}_{\text{s}}^2}{18{\pi }^2{\left(1-{\nu }_{\text{s}}\right)}^2}{P}_{\text{e}}\right]}^{1/3}(5)$

and

$G=\frac{5-4{\nu }_{\text{s}}}{5\left(2-{\nu }_{\text{s}}\right)}{\left[\frac{3N^2{\left(1-\varphi \right)}^2{G}_{\text{s}}^2}{2{\pi }^2{\left(1-{\nu }_{\text{s}}\right)}^2}{P}_{\text{e}}\right]}^{1/3}(6)$

where the ${\nu }_{\text{s}}$ is the solid grain Poisson’s ratio; the average number of contacts per grain $N=9$; $G_{\text{s}}$ is the shear modulus of the solid grain. The stiffness of loose deposited media is strongly dependent on the effective overburden stress $P_{\text{e}}$, which is governed not only in terms of overburden stress, saturation, and pore pressure but also especially in terms of capillary pressure, which has been observed in many field measurement and laboratory experiments. Thereby, an expression of effective stress considering capillary pressure is proposed in Equation 7 (Solazzi et al., 2021):

$P_{\text{e}}={\sigma }_z-\left(1-S_{\text{ew}}\right)p_{\text{g}}-S_{\text{ew}}\left(p_{\text{g}}-p_{\text{c}}\right)(7)$

where ${\sigma }_z=\rho gz$ is the overburden stress; $p_{\text{g}}={\rho }_{\text{g}}gz$ is the gas pore fluid pressure; $p_{\text{c}}$ is the capillary pressure; The effective saturation of water is defined as $S_{\text{ew}}=(S_{\text{w}}-S_{\text{w,\hspace{0.17em}r}})/(1-S_{\text{w,\hspace{0.17em}r}})$; $S_{\text{w}}$ is the water saturation and $S_{\text{w,r}}$ is the water residual saturation.

2.2.3 Effective fluid properties

The properties of the fluid mixture with water and gas can be evaluated in Equations 8–13. According to the work of Berryman (Berryman et al., 1988), the effective compressibility of fluid can be represented by the average of water and air compressibility (Wood, 1955):

$\frac{1}{K_{\text{f}}}=\frac{1-S_{\text{w}}}{K_{\text{g}}}+\frac{S_{\text{w}}}{K_{\text{w}}}.(8)$

where $K_{\text{g}}$ is the gas bulk modulus and $K_{\text{w}}$ is the water bulk modulus. Considering the interaction between the pore gas and water, the relative permeability functions depend on saturation (Barrière et al., 2012), specifically

$k_{\text{r\hspace{0.17em}w}}=S_{\text{ew}}^{1/2}{\left[1-{\left(1-{\left(S_{\text{e\hspace{0.17em}w}}\right)}^{1/m_{\text{c}}}\right)}^{m_{\text{c}}}\right]}^2(9)$

and

$k_{\text{r\hspace{0.17em}g}}={\left(1-S_{\text{ew}}\right)}^{1/2}{\left[1-\left(S_{\text{e\hspace{0.17em}w}}^{1/m_{\text{c}}}\right)\right]}^{2m_{\text{c}}}(10)$

were chosen here. $m_{\text{c}}$ is the fitting parameters. Effective permeability is defined as consisting of gas and water relative permeability:

$k_{\text{f}}=k_{\text{f0}}\left(k_{\text{rw}}+k_{\text{rg}}\right)(11)$

where $k_{\text{f0}}$ is the intrinsic permeability. An effective viscosity can be obtained from water and gas viscosity following (Berryman et al., 1988):

${\mu }_{\text{f}}=\frac{{\mu }_{\text{w}}{\mu }_{\text{g}}}{{\mu }_{\text{w}}{k}_{\text{rg}}+{\mu }_{\text{g}}{k}_{\text{rw}}}(12)$

where ${\mu }_{\text{g}}$ is the viscosity of the gas and ${\mu }_{\text{w}}$ is the viscosity of the water. Meanwhile, for the mixture of two-phase fluid, the fluid density is a linear average of each component, given by

${\rho }_{\text{f}}={\rho }_{\text{g}}\left(1-S_{\text{w}}\right)+{\rho }_{\text{w}}{S}_{\text{w}}(13)$

where ${\rho }_{\text{g}}$ and ${\rho }_{\text{w}}$ are densities of gas and water, respectively.

2.2.4 Saturation profile

The saturation profile is employed in this paper to consider the variation of saturation with depth. If the capillary pressure at the groundwater table is set as 0, and the depth at Earth’s surface is set as 0, according to the well-known van Genuchten’s model (Van Genuchten, 1980), the saturation-depth relationship is expressed in Equation 14:

$S_{\text{ew}}=\left\{\begin{array}{cc}{\left\{1+{\left[{\alpha }_{\text{vg}}\left(z_{\text{w}}-z\right)\right]}^{n_{\text{c}}}\right\}}^{-m_{\text{c}}}\hfill & \text{for}z<z_{\text{w}}\hfill \\ 1\hfill & \text{for}z\ge z_{\text{w}}\hfill \end{array}\right.(14)$

where $m_{\text{c}}$ and $n_{\text{c}}$ are the exponent parameters related to the pore size distribution, with $m_{\text{c}}=1-1/n_{\text{c}}$; ${\alpha }_{\text{vg}}$ is the pressure scaling parameter. Afterward, the capillary pressure expression with depth

$p_{\text{c}}={\rho }_{\text{w}}g\left(z_{\text{w}}-z\right)(15)$

could be obtained. From Equation 15, the capillary pressure profile is only related to the groundwater table. Nevertheless, since the contribution of capillary pressure to the effective pressure is also dependent on the saturation profile, this model can still be applied for extensive to more silty and clayey soils.

2.3 Seismic monitoring indicators

There are three indicators employed to assess the poroelastic effects on seismic monitoring in loosely deposited sands, i.e., Poisson’s ratio (Equation 16), P-wave travel time curve (Equations 17–19) and surface wave dispersion curves.

2.3.1 Poisson’s ratio

In the field of near-surface geophysics, borehole probing is a commonly employed technique for detecting groundwater tables. However, its large-scale applicability may be constrained by financial and other factors. Seismic surveys are used to acquire the spatial distribution of $V_p$ and $V_s$ in subsurface structures (Grelle and Guadagno, 2009). The equivalent Poisson’s ratio is then utilized to differentiate between partially saturated and fully saturated soils, aiding in the determination of the groundwater table location (Uyanik, 2011; Pasquet et al., 2015; Flinchum et al., 2020). This method effectively complements borehole measurements. According to the model setup process, the equivalent Poisson’s ratio for each layer can be defined as (Mavko et al., 2009):

$\nu =\frac{V_{p,j}^2-2V_{s,j}^2}{2\left(V_{p,j}^2-V_{s,j}^2\right)}(16)$

where the values of $V_{p,j}$ and $V_{s,j}$ can be calculated through the plane wave analysis presented in Appendix A. In near-surface applications, the P-wave velocity profile can be inverted using the seismic refraction method, while the S-wave velocity profile can be estimated through surface wave dispersion analysis.

2.3.2 P-wave travel time curve

In general, the seismic refraction method can be applied to detect the saturation distribution of loosely deposited sands due to the sensitivity of P-wave velocity to saturation (Berryman et al., 2002). In shallow seismic refraction surveys, the P-wave travel time curve includes the direct wave in the near field and the refracted wave in the far field. Consider a layer of thickness $h$ and velocity $V_1$ overlying a uniform halfspace with velocity $V_2$, where $V_1<V_2$. The arrival times at point $x$ for the direct and refracted waves, $t_{\text{dir}}$ and $t_{\text{refr}}$, are given by:

$t_{\text{dir}}=\frac{x}{V_1}(17)$

and

$t_{\text{refr}}=\frac{x}{V_2}+\frac{2h\sqrt{V_2^2-V_1^2}}{V_1{V}_2}.(18)$

In this paper, the 1D model is discretized into a sufficient number of layers with equal thickness $\mathrm{\Delta }h$. For multiple layers, the seismic energy at each interface is refracted according to Snell’s law. The intercept of the refraction travel time is controlled by the root mean square (RMS) velocity of all the layers above the reflector, defined as

$V_{\text{rms}}=\sqrt{\frac{{\sum }_{i=1}^n{V}_i^2\mathrm{\Delta }{t}_i}{{\sum }_{i=1}^n\mathrm{\Delta }{t}_i}}(19)$

where $\mathrm{\Delta }{t}_i$ is the two-layer travel time for seismic energy propagating perpendicular through the $i$-th layer. The $V_1$ in Equation 18 can be replaced by the RMS velocity.

2.3.3 Surface wave dispersion analysis

Surface wave dispersion analysis is a non-invasive geophysical technique used to quantitatively assess frequency-dependent changes in the phase velocities of surface waves (Foti et al., 2011; Mi et al., 2018; Zhang et al., 2020). This method primarily relies on the properties of Rayleigh waves (Elias and Alderton, 2020), which result from the superposition of wave propagation between different subsurface media interfaces. As the wavelength shortens and the depth of penetration increases, wave velocity varies with frequency, exhibiting a dispersive phenomenon. This dispersion is related to the physical properties of the subsurface, such as density, compressibility, and shear resistance. Consequently, analyzing the surface wave dispersion curve provides valuable insights into the characteristics and structure of the subsurface.

Rayleigh waves are primarily driven by S-waves (Flinchum et al., 2020), which typically increase with depth in loosely deposited sands. In surface wave dispersion analysis, the phase velocity at lower frequencies reflects the deeper S-wave velocity structure, while higher frequencies reveal the shallow structure (Flinchum et al., 2020). Consequently, the surface wave dispersion curve can be utilized to invert and obtain the 1D S-wave profile (Pasquet and Bodet, 2017). In this study, we investigate the influence of different models on the surface-wave dispersion curve across various scales. To construct surface-wave dispersion curves, the dispa python library¹ is employed for the forward modeling of surface-wave dispersion.

3 Results and discussion

In this section, we numerically investigate the poroelastic effects on seismic monitoring in loosely deposited sands across various scale applications (corresponding to different frequency ranges). We first study the poroelastic effects on the velocities of P- and S-waves with different saturation degrees in the ultrasonic frequency range, where the macroscopic wave-induced fluid flow happens. Then we explore the poroelastic effects on the Poisson’s ratio, P-wave travel time, and surface wave dispersion with different frequency ranges.

3.1 Poroelastic effects on $V_p$ and $V_s$

In the following, we first validate the model employing two sets of lab sample experimental results and briefly illustrate the influence of poroelastic effects on body wave propagation characteristics with different saturation degrees.

3.1.1 Experiment 1

The first experimental study (Barrière et al., 2012) examines the effects of saturation (ranging from 0.1 to 0.9) on the characteristics of the fast P wave in an unconsolidated porous medium. Wave velocity and attenuation are determined using plane wave analysis, with material parameters listed in Table 1 (Exp.1). Additionally, the energy losses due to friction between sand particles are considered in the energy dissipation analysis. In natural soils and rocks, internal damping, known as hysteretic damping, is independent of frequency. The Constant-Q model is commonly used to characterize this behavior across all frequency ranges (Carcione, 2015). The frequency-dependent stiffness modulus is expressed in Equations 20, 21 (Barrière et al., 2012):

$K^{*}=K\ast {\left(i\frac{\omega }{{\omega }_0}\right)}^{2{\gamma }_K}(20)$

$G^{*}=G\ast {\left(i\frac{\omega }{{\omega }_0}\right)}^{2{\gamma }_G}(21)$

Table 1

Table 1. Physical properties of soils employed in the numerical experiments.

where ${\omega }_0/2\pi =2.0\text{\hspace{0.17em}kHz}$ is the reference frequency of the Constant-Q model. The exponent parameters related to the quality factor for stiffness modulus losses, ${\gamma }_K=0.038$ and ${\gamma }_G=0.062$ in this simulation.

Figure 2 illustrates the effect of saturation on the fast P-wave velocity and wave attenuation. From those figures, it can be seen that the poroelastic model agrees better with the experimental data compared to the elastic model, which demonstrates the favorable predictive capability of the poroelastic model for fast P-wave characteristics. This discrepancy is attributed to the inability of Gassmann’s theory to accurately characterize the poroelastic effects at the exciting frequency $(1.8\text{\hspace{0.17em}kHz})$. Besides, the wave velocity exhibits a slight decrease as saturation increases, possibly due to the increased effective density of porous media.

Figure 2

Figure 2. Effect of saturation on fast P wave propagation characteristics (exciting frequency $1.8\text{\hspace{0.17em}kHz}$): velocity of fast P-wave (upper) and inverse quality factor (bottom).

3.1.2 Experiment 2

In the first experiment, the model was studied within a saturation range, from 0.1 to 0.9. However, the properties of fluid undergo significant changes when saturation is close to 1.0 (Whalley et al., 2012). Therefore, it is necessary to investigate the wave propagation characteristics in the near-saturated range (approximately $0.9-1.0$). Gu et al. (2021) conducted a series of laboratory specimen tests using the bender element, where Fujian sand was oven-dried before sample preparation, and all specimens underwent compaction to ensure material homogeneity. Moreover, carbon dioxide flushing ensuring complete displacement of air and back pressure changing were employed to achieve high saturation and vary the saturation levels, respectively. All parameters used in the simulation are listed in Table 1 (Exp.2).

Figure 3 show the effect of saturation on fast P-wave, slow P-wave, and S-wave. The simulation results obtained from the poroelastic model generally align with the measured dataexcept for some discrepanciesattributed to the selected parameters (Gu et al., 2021). , with some discrepancies. These deviations are primarily due to parameter selection. For instance, the shear modulus used in the theoretical prediction was derived from S-wave measurements taken when the soil was nearly saturated (Gu et al., 2021). Additionally, the definition of effective permeability and viscosity influences the critical frequency, contributing to the observed differences. Moreover, the elastic model fails to predict the existence of the slow P-wave as it can not model mascopic wave-induced fluid flow. As saturation is close to 1.0, the velocity of the fast P-wave changes significantly due to sharp variations in the fluid properties. From Figure 3a, it can be observed that the predictive ability of the elastic model is comparable to the poroelastic model with saturation near 99.9%. This may be due to the dominance of changes in fluid properties as opposed to poroelastic effects.

Figure 3

Figure 3. Effect of saturation on body wave velocities (exciting frequency $10\text{\hspace{0.17em}kHz}$): (a) fast P-wave; (b) slow P-wave; (c) S-wave.

For a better understanding of the poroelastic effect on seismic wave velocities in the high-frequency regime where pressure gradients in the fluid only serve to accelerate fluid motion, we compared the model predictions with experimental data in the fully saturated state, as shown in Figure 4. From those figures, it can be found there is a better agreement between the model predictions and the measured data. As shown in Figures 4a, c, there is a significant discrepancy of up to 6% between the elastic model and the poroelastic model, indicating the inability of the elastic model to characterize the poroelastic effect in the high-frequency regime.

Figure 4

Figure 4. Effect of frequency on body wave velocities in full saturation state: (a) fast P-wave; (b) slow P-wave; (c) S-wave.

3.2 Poroelastic effects on seismic monitoring in field surveys

According to Section 2, the dominant frequency of the source, set at $50\text{\hspace{0.17em}Hz}$, establishes a purely elastic wave regime wherein there is no relative motion between the fluid and solid matrix. The static water table depth is assumed as $5\text{\hspace{0.17em}m},10\text{\hspace{0.17em}m},15\text{\hspace{0.17em}m} \mathrm{a}\mathrm{n}\mathrm{d} 20\text{\hspace{0.17em}m}$, yielding four distinct saturation profiles which can be determined by Equation 15 with a specific value ${\alpha }_{\text{vg}}=0.442{\mathrm{m}}^{-1}$, as shown in Figure 5a. The other parameters are listed in Table 2.

Figures 5b–d display the profiles of P-wave, S-wave, and Poisson’s ratio in the low-frequency range, respectively. From those figures, it can be found that there is a general consistency in the predictions of both poroelastic and elastic models, which has been validated through various field measurements, e.g., Bachrach and Nur (1998); Flinchum et al. (2020). It can be noted that both P- and S-wave velocity profiles exhibit a minor inflection point characterized by a localized decrease in velocity near the water table. This phenomenon can be attributed to a sudden increase in the effective density of the medium prior to the full saturation. As shown in Figure 5b, the P-wave velocity undergoes a sudden increase close to the water level. This is due to the abrupt alteration of the effective fluid bulk modulus $K_{\text{f}}$. Besides, Poisson’s ratio also has a sharp increase near the water table, while maintaining relative constancy in both unsaturated and saturated zones. This makes it possible to determine the location of the water table in the shallow subsurface using Poisson’s ratio profile although it falls short in adequately characterizing saturation changes from a continuous water table to the ground surface (Solazzi et al., 2021).

Figure 5

Figure 5. Comparison of results obtained from the elastic and poroelastic models at the field scale ($50\text{\hspace{0.17em}Hz}$ dominant frequency): (a)Saturation profile; (b)P-wave profile; (c)S-wave profile; (d)Poisson’s ratio profile; (e)P-wave travel and(f) surface wave phase velocity as a function of frequency. These results confirm that surface waves are highly sensitive to vadose zone saturation, while P-wave travel times are less affected in field-scale studies. The comparison between the two models supports the use of elastic models for low-frequency field applications.

Table 2

Table 2. Physical properties of soils for application studies.

Figure 5e demonstrate the curves of P-wave travel time with different water tables. It is noticed from this figure the predicted results of the poroelastic model agree with the elastic model. This suggests that both models possess comparable predictive capabilities in the low-frequency range. Comparing the travel time curve among various water tables, it can be seen that there is a similarity in slopes among direct wave travel time curves which suggests that P-wave velocity remains relatively unaffected by saturation levels in the unsaturated region. The difference among the travel-time curves stems from fluctuations in the location of the water table. The slope of refracted wave travel time curves remains consistent, indicating that P-wave velocities near water table depths are basically the same. This is because P-wave velocity is mainly governed by the fluid in shallow loose media consistent with Figure 5c.

Figure 5f shows the surface wave velocity dispersion as a function of frequency simulated by the elastic and poroelastic models. From this figure it is noted that there is a notable agreement concerning surface wave velocity dispersion between two models. It also can be found that surface wave dispersion is significantly influenced by the alterations in the water table. By comparing the outcomes of different water tables, it is found there is notable variability in the low-frequency (high period) range indicating variations in the deep medium structure, e.g., water table location. Furthermore, in the high-frequency range (low period), it is observed discernible differences among the results of different saturation profiles. This suggests surface waves are responsive to changes in saturation of the vadose zone, while this sensitivity is not shown in the P-wave travel time curves.

3.3 Poroelastic effects on seismic monitoring in pilot plant surveys

For intermediate frequencies with regard to the ratio $\omega /{\omega }_{\text{c}}$, as it is the case for a source dominant frequency of $1.0\text{\hspace{0.17em}kHz}$ in the model with the pilot plant surveys, the inertia force and the viscous shearing act simultaneously. The four different saturation profiles are shown in Figure 6a with ${\alpha }_{\text{vg}}=8.84$. The other parameters are listed in Table 2.

Figure 6

Figure 6. Comparison of results obtained from the elastic and poroelastic models at the pilot plant scale ($1\text{\hspace{0.17em}kHz}$ dominant frequency): (a) saturation profile; (b) P-wave profile; (c) S-wave profile; (d) Poisson’s ratio profile; (e) P-wave travel and (f) surface wave phase velocity as a function of frequency. These results indicate that poroelastic effects become more prominent at this scale and emphasize the increasing importance of frequency-dependent poroelastic effects as seismic frequency increases.

Figure 6b, c display the P- and S-wave velocity profiles of elastic and poroelastic models in the intermediate-frequency range. It can be seen from those figures that the poroelastic and elastic modeling results are broadly similar in the low saturation region, while a clear divergence occurs in the high saturation region. This divergence stems from the lower velocity dispersion observed at low saturation and higher at high saturation. Figure 6d shows the Poisson’s ratio profiles with different water table levels. There exists only a minor discrepancy between the outcomes of elastic and poroelastic models. This suggests that the influence of poroelastic effects on Poisson’s ratio profiles remains relatively consistent across low and intermediate-frequency conditions.

Figure 5e describes the P-wave travel time curves for the elastic and poroelastic models with different water table levels. From this figure, it can be seen that the major difference between the two models is in the refraction band, even though it is not significant. An indication of the apparent velocity near the water table is given by the slope of the refraction curve. Therefore, the velocity dispersion in the intermediate-frequency range are not sufficient to cause significant changes in the apparent velocity. Moreover, in the direct band region, both models produce nearly identical predictions due to minimal velocity dispersion in the unsaturated range.

Figure 5f illustrates the surface wave dispersion curves for the elastic and poroelastic models with different water table levels. It can be seen from this figure that there is a significant difference between the predictions of the two models, especially in the low-frequency region (high period). This observation indicates that the elastic model fails to adequately capture the poroelastic effects in the intermediate frequency range. Furthermore, in the high-frequency region (low period), there is also a marked difference between the predictions of the two models. Even though it is not as pronounced as in the low-frequency region, this finding further highlights the limitations of the elastic models.

3.4 Poroelastic effects on seismic monitoring in ultrasonic measurements

For high frequencies with regard to the ratio $\omega /{\omega }_{\text{c}}$, as is the case for a source dominant frequency of $10\text{\hspace{0.17em}kHz}$ in the model with the ultrasonic measurements, the pressure gradients on the fluid only contribute to accelerating the fluid motion. The four different saturation profiles are shown in Figure 7a with ${\alpha }_{\text{vg}}=88.4$. The other parameters are listed in Table 2.

Figure 7

Figure 7. Comparison of results obtained from the elastic and poroelastic models at the ultrasonic scale ($10\text{\hspace{0.17em}kHz}$ dominant frequency): (a) Saturation profile; (b) P wave profile; (c) S wave profile; (d) Poisson’s ratio profile; (e) P-wave travel and (f) surface wave phase velocity as a function of frequency. These findings reinforce the conclusion that ultrasonic-scale experiments are not directly transferable to field applications and suggest that laboratory-scale measurements may need additional corrections when applied to real scenarios.

Figure 7b, c show the P- and S-wave velocity profiles for the elastic and poroelastic models in the high-frequency range. It is clear from these figures that there is a significant difference between the elastic and poroelastic models in the prediction of body wave velocity profiles. Compared to the intermediate-frequency range, this difference becomes more apparent as the frequency increases. These observations suggest that the poroelastic effects on body waves is frequency-dependent. Figure 7d presents the Poisson’s ratio profiles of the elastic and poroelastic models for different water table conditions. It is worth noting that the results remain consistent regardless of which model or frequency range is chosen, as described in the previous section. A plausible explanation for the wave velocity dispersion is the frequency-dependent nature of the inertial interaction mass (Carcione, 2015). It is apparent that the Poisson’s ratio or $V_{\text{p}}/V_{\text{s}}$ is independent of the mass.

Figure 7e demonstrates the P-wave travel time curves for elastic and poroelastic models with different water table levels. It can be seen there is a significant decrease in the slope of the refractive section. This is due to the larger velocity dispersion in the high-frequency range. Figure 7F shows the surface wave dispersion curves for the elastic and poroelastic models with different water table levels. Compared to the pilot-scale results, the variability in the predictions of both models increases significantly, indicating that the lab results are not directly transferable to field application.

4 Conclusion

In this study, we investigate the impact of poroelastic effects on seismic monitoring results. The poroelastic signature is derived by contrasting elastic and poroelastic models. Based on the obtained results, the following conclusions can be drawn:

(1) A comparison of two sets of laboratory data investigates the poroelastic effects on body wave velocities. Besides, the inability of the elastic model to capture porous elastic effects in the high-frequency range is briefly illustrated. In low saturation conditions, the poroelastic effect exhibits a small frequency dependence, whereas the opposite is true in high saturation conditions, especially in the case of full saturation.

(2) Different frequencies typically correspond to applications at different scales (seismic site surveys, pilot plant surveys, and ultrasonic measurements). The Poisson’s ratio profile is an effective way to detect water table levels at various scales. Besides, the frequency-dependent poroelastic effect mainly affects the refractive wave band and surface wave dispersion curve.

(3) Seismic monitoring results exhibit high sensitivity to changes in water level, suggesting the feasibility of utilizing time-lapse seismic methods to monitor saturation distribution in loosely deposited sands.

These findings highlight the discrepancies in seismic monitoring results across different frequency ranges and can serve as a basis for parameterization across applications at various scales. However, some limitations of this work should be acknowledged. While the poroelastic effects on seismic monitoring results have been confirmed by comparing elastic and poroelastic models, more experimental data with different scales are needed for further validation. Additionally, incorporating the poroelastic effects involve only macroscopic wave-induced fluid flow in this paper. Incorporating the heterogeneous fluid distribution in porous media may provide insight into the poroelastic effects on seismic wave propagationdistribution-induced mesoscopic wave-induced fluid flow in poroelastic effects may provide more realistic predictions, especially for applications in natural sediments and mine tailings. Moreover, future research could explore the use of machine learning techniques to improve poroelastic parameter estimations. Recent studies have demonstrated that deep learning approaches can effectively infer complex subsurface properties from seismic data. Applying such methods to poroelastic modeling could enhance parameter inversion accuracy and reduce computational costs. In addition, the model could be extended to assess geohazards, e.g., liquefaction (Molina-Gómez et al., 2023) and landslides (Li et al., 2024; Fang et al., 2023).

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

TD: Conceptualization, Data curation, Funding acquisition, Investigation, Software, Validation, Visualization, Writing – original draft, Writing – review and editing.

Funding

The author(s) declare that financial support was received for the research and/or publication of this article. The authors thank the support from the China Scholarship Council (CSC) project (No. 201906430070).

Acknowledgments

We thank all reviewers for valuable comments, and the Frontiers Editorial Office for improving the layout of this Research Topic.

Conflict of interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declare that no Generative AI was used in the creation of this manuscript.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/feart.2025.1561168/full#supplementary-material

Footnotes

¹https://github.com/keurfonluu/disba

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