Electromagnetic Waves’ Impact on Hydraulic Conductivity of Granular Soils

Abstract

Electromagnetic (EM) waves, traditionally used for purposes such as geophysical characterization, impact properties to be measured. This paper describes the effects of radio frequency (RF) waves on the hydraulic conductivity of glass beads and natural sand. A series of tests was conducted using a customized, rigid-wall, cylindrical permeameter inside a resonant cavity made of Plexiglas covered with electrically conductive transparent films. Constant-head ASTM-D2434 tests were performed to measure the samples’ hydraulic conductivity. RF stimulation was performed using a magnetically coupled loop antenna at various frequencies and input RF-power levels. The hydraulic conductivity of both natural sand and glass-bead samples increased with RF stimulation. Furthermore, the measurement of the electric field component of RF waves was also performed to illustrate the pattern of the electric field, as well as evaluate RF’s impact on the hydraulic conductivity tests. The electric field was numerically simulated and validated against experimentally measured electric fields. A finite-difference numerical model was developed in MATLAB to analyze the seepage flow, which was then validated against the experimental results. An optimization scheme was then used to develop a governing equation for RF’s impact on hydraulic conductivity.

1. Introduction

Seismic-induced liquefaction is a result of a sudden increase in the excess pore-water pressure (EPWP) in loose, water-saturated non-cohesive soils. This phenomenon has been observed in many different earthquakes, such as the earthquakes in 1964, Niigata, Japan [1]; 1999, Adapazari, Turkey [2]; and 2011, Christchurch, New Zealand [3]. Liquefaction leads to a reduction in effective stress and a decrease in contact force between the soil grains, often resulting in the loss of shear strength, and thus, lateral spreading, slope instability, and foundation and building damage.

Research on soils susceptible to liquefaction, such as sand and silty sand, has demonstrated that liquefaction depends upon parameters such as peak acceleration, groundwater level, and the soil’s grain size, relative density, cyclic shear strength, plasticity, and degree of water saturation [1]. All these parameters, however, are linked to the water dissipation rate. The inability of fluids to fully dissipate in the short period of an earthquake leads to a rapid buildup of excess pore-water pressure (EPWP), resulting in liquefaction. Nonetheless, if all soil properties remain constant, liquefaction depends upon the fluid properties. Using a viscous fluid (k = 0.0021 cm/s) in a centrifuge test, Sharp et al. [4] successfully demonstrated a decrease in the thickness of the liquified layer and settlement with an increase in permeability. Ganainy et al. [5] obtained a lower EPWP ratio ($r_u$), which is the ratio between the excess pore-water pressure and initial effective stress, and a smaller thickness of liquefied soil and shear strain when the tests were conducted using water than when they were conducted using more viscous fluids (the hydraulic conductivity was 25 times lower than in water).

Electromagnetic, acoustic, and seismic waves have traditionally been used for geophysical detection, characterization, and even remediation [6,7,8]. However, because electrostatic sources can align individual water molecules and magnetic fields can float droplets of water [9], water dipoles oscillate in alternating electromagnetic (EM) fields. The vibration of water molecules enhances different transport mechanisms inside the soil, therefore altering hydraulic conductivity. In another work, Azad et al. [10] conducted a study using radio frequency (RF) waves to alter hydraulic conductivity of sandy soils. The hydraulic conductivity increased as much as 14% when an RF stimulation test was conducted on natural sand at a frequency of 153 MHz with an RF-power input of 20 Watts. The increase in hydraulic conductivity was justified by the RF energy absorbed by the water molecules, which decreased the viscosity and led to an increase in hydraulic conductivity. Azad et al. [10] also tested the effects on clay and noticed a decrease in the hydraulic conductivity of clay [10]. This is important because using electromagnetic waves can reduce soil’s susceptibility to liquefaction by reducing the hydraulic conductivity related to liquefaction. A reduction of even the slightest amount of generated EPWP can mitigate liquefaction. Therefore, the objectives of this study were to (1) investigate the effect of RF waves on the hydraulic conductivity of sandy soils and (2) determine whether this could be sufficiently large and fast enough to reduce excessive pore-water pressure (EPWP) during earthquakes to mitigate liquefaction. During earthquakes, EPWP is generated within fine sands or silty sands due to a low dissipation rate. To achieve these stated goals, the effect of RF waves at varying powering levels and frequencies on the hydraulic conductivity of non-cohesive soils was studied, and we measured the increase in the EPWP within non-cohesive soils during unstimulated and RF-stimulated tests at different RF wave frequencies and power levels.

2. Theoretical Background

2.1. Hydraulic Conductivity

Water seeps through the void spaces between soil grains that form interconnected paths. Seepage flow and its velocity are measured based on and depend on hydraulic conductivity. Water flows through the soil from one point to another when there is a difference in the total head (also known as the hydraulic head). In this paper, Darcy’s Law (Equation (1)) was considered the governing equation of seepage flow.

$$Q=vA=(-k\frac{dh}{dL})A$$

(1)

where Q = discharge (m³/s), h = total head (m), dL = flow-path length (m), A = cross-sectional area of the soil specimen (m²), k = hydraulic conductivity of the soil (m/s), v = discharge or Darcy’s velocity (m/s), and $\frac{dh}{dL}$ = hydraulic gradient.

The hydraulic conductivity of soils is a function (Equation (2)) of the properties of both the porous medium (i.e., intrinsic permeability) and the fluid (unit weight and viscosity) [11].

$$k=\frac{K\gamma }{\mu }=\frac{Kg}{v}$$

(2)

where K = intrinsic permeability of the soil medium (m²), which depends on the shape of the openings and the mean pore diameter [12] of the soil; g = gravitational acceleration (m/s²); $\gamma$ = unit weight of water (N/m³); $\mu$ = absolute or dynamic viscosity of water (Pa·s); $v=\frac{\mu }{g}$ = kinematic viscosity (m²/s); and ρ = density of water (kg/m³). The typical values of the intrinsic permeability and hydraulic conductivity of silty sands range from 10⁻³ to 10⁰ cm/s and from 10⁻⁹ to 10⁻⁶ cm/s, respectively.

This research work hypothesizes that by impacting the permeating fluid, in this case water, the hydraulic conductivity can be altered without the need to change the intrinsic permeability via costly and permanent methods of changing the soil environment, e.g., compaction.

2.2. Liquefaction

Liquefaction occurs when the shear strength of saturated loose soils is lost due to a sudden increase in the pore-water pressure (PWP, u) beyond the hydrostatic PWP—referred to as excess pore-water pressure (EPWP, u_e)—due to the rapid dynamic load of an earthquake. During rapid loading, there is a sudden increase in EPWP, thus decreasing the amount of time taken for PWP to dissipate. As a result, the effective vertical stress of the soil decreases to zero, leading to liquefaction. The vanishing effective vertical stress leads to a loss of friction, the sole source of shear strength in non-cohesive soils, such as sand.

Liquefaction susceptibility depends upon a number of factors, including geologic and compositional properties, as well as the state of the soil, since volume-change behavior influences the rise in EPWP. Liquefaction is dominant in soils that have shallow groundwater depths. Moreover, reclaimed lands with loose soil are prone to liquefaction [1]. Liquefaction is not common in all types of soil. Fine-grained soils such as clay are cohesive, and very coarse-grained soils, though frictional, are highly permeable; hence, both are less susceptible to liquefaction. However, liquefaction of non-plastic silts has been observed [1]. Non-plastic and cohesionless silts with dimensions that are equal in all directions (bulk shape) are prone to liquefaction [1]. In the case of the Adalpazari earthquake, the layer of sand containing 30% non-plastic fine sand and classified as silty sand (SM) was considered liquefiable under moderate levels of ground shaking [2] and liquefied in areas where soil treatment was not completed. Liquefaction of soil also depends upon the gradation of soil. Well-graded soils are less prone to any increase in EPWP than poorly graded or uniformly graded soils. An increase in the pore-water pressure during earthquakes also depends upon the density of the soil and its initial stress condition [1].

An EPWP ratio parameter, $r_u=\frac{∆u}{\sigma \prime }$, has been defined to illustrate the path toward liquefaction, where $\sigma \prime$ = initial effective stress, and $∆u=u_e$ = EPWP increase during an earthquake. When r_u = 1, liquefaction occurs, and when r_u = 0.25 to 0.70, partial liquefaction occurs [5].

EPWP generation has a significant effect on the shear strength, stability, and settlement characteristics of soil deposits, even if the soil does not completely liquefy [13]. Therefore, even a small decrease in EPWP can reduce the potentially hazardous effects of liquefaction.

2.3. Electromagnetic Waves

Alternating electric fields generate magnetic fields, and vice versa, alternating magnetic fields generate electric fields. Electromagnetic (EM) waves are formed when an electric and magnetic field alternate perpendicular to each other and to the direction of wave propagation. These orthogonal oscillations are governed by Maxwell’s equations. Maxwell’s equations (Equations (3)–(6)) are a set of four equations, written in either integral or differential form, stating the relationships between the fundamental electromagnetic quantities. The fundamental quantities are the electric flux density, $\stackrel{⃑}{D}$ (C/m²), the magnetic flux density, $\stackrel{⃑}{B}$ (Wb/m²), the electric field intensity, $\stackrel{⃑}{E}$ (V/m), the magnetic field intensity, $\stackrel{⃑}{H}$ (A/m), the current density, $\stackrel{⃑}{J}$ (A/m²), and the electric charge density, $\rho$ (C/m³) [14]. For general time-varying fields, Maxwell’s equation can be written as follows.

$$\stackrel{⃑}{\nabla }·\stackrel{⃑}{D}=\rho$$

(3)

$$\stackrel{⃑}{\nabla }·\stackrel{⃑}{B}=0$$

(4)

$$\stackrel{⃑}{\nabla }\times \stackrel{⃑}{E}=-\frac{\partial \stackrel{⃑}{B}}{\partial t}$$

(5)

$$\stackrel{⃑}{\nabla }\times \stackrel{⃑}{H}=\stackrel{⃑}{J}+\frac{\partial \stackrel{⃑}{D}}{\partial t}$$

(6)

EM Waves Impact Soil Media

Soils have three phases: (i) solid grains with voids filled with (ii) air and/or (iii) fluids that have different physical and dielectric properties. A dielectric is a material that can be polarized by an EM wave. Water has high dielectric permittivity (≈80). Oscillations of individual water molecules can induce a net change in the movement and flow of water through a porous medium without altering the properties of the medium itself [10]. When under the influence of an electric field, water molecules can start to reorient parallel to the direction of the electrostatic field [15]. As reorientation occurs, the hydrogen bond starts weakening, thus decreasing the viscosity of the molecule. A decrease in viscosity could result in an increase in hydraulic conductivity. As electromagnetic waves travel through soil, the soil absorbs their energy. Hence, the skin depth (a measure of the penetration depth of EM waves before they lose half their energy) varies based on the waves’ frequency and how lossy the soil is, i.e., lossy soils are ones with a higher dielectric constant (e.g., water-saturated soils) or higher electric conductivity (e.g., clay) that absorb EM energy and have a shorter skin depth.

3. Materials and Methods

3.1. Experimental Setup and Testing Procedures

The following are the details of a series of tests that evaluated the impact of RF stimulation on the hydraulic conductivity of granular soils, which include measurement details of seepage due to RF wave generation and of hydraulic conductivity.

3.1.1. Seepage Measurements

The flow was simulated experimentally using a rigid-wall constant-head test, according to ASTM D2434 [16], where the flow rate of water was high, and a constant head could be maintained through a continuous supply of water. For a homogeneous soil sample, if there were no EM waves, then hydraulic conductivity could be assumed to be spatially constant and found as follows (Equation (7)).

$$k=-\frac{V}{t}\frac{L}{A∆h}$$

(7)

where $∆$h = hL = hydraulic head loss across the soil medium (m), A = cross-sectional area of soil specimen (m²), L = length of soil specimen (m), t = time of water collection (s), and V = volume of water collected (m³).

3.1.2. RF Wave Setup

To supply the power and RF electric field intensity for these tests, a magnetically coupled loop antenna was inserted into the cavity parallel to the magnetic lines. The loop antenna was made of an RG-8 coaxial cable. The RF signal was generated using an Agilent Model #E4400B signal generator. An amplifier was used to amplify the generated signal. To maximize power output and reduce harmful reflections back into the amplifier, the impedance of the load (setup) should match that of the source (50 Ω). The impedance was measured using a vector network analyzer (VNA) manufactured by Agilent Technologies Inc. (Model # Agilent-N9320A). The impedance was then matched to the 50 Ω impedance of the RF source using a matching network made of a series of variable capacitors. Two dual-directional couplers were also used in the network to monitor the forward power into the device under testing (DUT) and the reflected power back into the amplifier. Figure 1 shows a schematic of the setup used for launching RF into the DUT.

All these tests were performed on a resonant cavity filled with two types of media, one filled with glass beads and the other filled with a natural sand sample. The saturated glass-bead specimen was prepared using the wet-pluviation method [17]. The glass beads used in this study were Class-A Ballotini impact beads with a specific gravity of 2.46 g/cm³.

Table 1. Properties of glass beads (recreated from [7,18]).

Potters Designation	U.S. Sieve Number	Maximum Size (in.)	Minimum Size (in.)	Maximum Size (µm)	Minimum Size (µm)	Minimum % of Round Beads
Class A	20–30	0.0331	0.0234	850	600	65

displays the properties of the beads.

A customized experimental setup to measure RF-stimulated hydraulic conductivity was developed here. Glass beads and natural sand were used as the coarse-grained samples in this test. The natural sand was classified as SW, according to the USCS classification system.

To perform the hydraulic conductivity tests, a customized, rigid-wall, cylindrical permeameter was constructed using acrylic material (see [19] for more details). However, prior to performing the test using the customized permeameter, the customized device needed to be calibrated. Hence, the unstimulated tests were performed using both a standard permeameter (6.35 cm (2.5 inch) diameter and 30.48 cm (12 inch) height), as suggested by D2434 [16], and a customized (152 mm diameter and 140 mm height) permeameter to calibrate the customized permeameter. Figure 2a displays a schematic of the customized, rigid-wall, cylindrical permeameter setup. The glass beads used in this test were prepared via dry-pluviation. The total density of the glass-bead sample was measured to be 14.72 kN/m³. In the case of natural sand, natural sand was poured into the permeameter in three layers. Each layer was compacted using 50 blows with a standard compaction hammer. The density of the natural sand was measured to be 18.25 kN/m³. The depth of the soil specimen in the customized permeameter was 110 mm (see [19] for more details). Figure 2a shows a schematic of the setup and its dimensions, and Figure 2b shows the setup for the RF-stimulated hydraulic conductivity test. The customized permeameter was placed inside a $490 \mathrm{m}\mathrm{m}\times 390 \mathrm{m}\mathrm{m}\times 390 \mathrm{m}\mathrm{m}$ resonant cavity. RF stimulation was performed on both the glass-bead and natural sand samples at frequencies of 498, 632, and 710 MHz at a single power level of 10 Watts, and for the 726 MHz case, at power levels of 10, 25, and 40 Watts (see [19] for more details). The frequency range and power levels were selected based on the instrument and equipment availability, and the frequencies were selected such that oscillations induced by RF waves onto free water molecules are not relaxed.

3.1.3. Electric Field Mapping

The electric field inside the cavity was mapped using a vertical monopole probe. Using an RG-58, 18 GHz precision-test cable, the monopole probe was connected to a spectrum analyzer to measure the electric field at the corresponding location of the probe inside the cavity. The probe was moved to various locations throughout the specimen via a number of holes drilled into the top plate of the resonant cavity. These holes were drilled on a $2 \mathrm{c}\mathrm{m}\times 2 \mathrm{c}\mathrm{m}$ grid in the X–Y plane. A glass tube was also used to case the monopole antenna to be inserted vertically into the cavity [20]. Respective X- and Y-coordinates for the probe were decided based on the location of the hole on the top plate, and the depth was decided based on the Z-coordinate (see [19] for more details).

Electric field measurements for the RF-stimulated hydraulic conductivity tests were performed at frequencies of 710 MHz and 726 MHz. The location of the measurement probe along the X-direction was fixed at distances of 2 cm and 4 cm from the centrally placed source (monopole antenna). A total of seven measurement points along the Y-axis were selected, which were separated at 2 cm intervals. Figure 3 presents a schematic top view of the customized permeameter with the location of the depth slices. Measurements were recorded in dBm using the spectrum analyzer at a frequency generated by the signal generator. The electric field was also numerically simulated using the RF module of COMSOL Multiphysics and validated against the experimentally measured electric field (see [19] for more details).

3.2. Three-Dimensional Numerical Forward Model of Seepage Simulated by RF Waves

To study the impact of EM waves on hydraulic conductivity, the spatial and temporal relationships between seepage and electric fields need to be studied. Experimental measurements do not provide a complete three-dimensional image of the two. To acquire a complete picture, experimentally validated numerical models are needed. The following is an explanation of how the numerical simulations were generated and experimentally validated.

3.2.1. Unstimulated Seepage

The 3D numerical model of seepage flow developed in this research simulates a saturated medium where flow is governed by the conservation of mass and—if water density is constant—volume (Equation (8)).

$$\frac{\partial \rho }{\partial t}+\stackrel{⃑}{\nabla }·\left(\rho \stackrel{⃑}{v}\right)=0$$

(8)

where ρ = fluid density (m³/s), t = time (s), and $\stackrel{⃑}{V}$ = seepage flow velocity vector (m/s). In the case of incompressible fluids, the density of the fluid is constant. Therefore, the conservation of mass equation (Equation (8)) can be simplified to the conservation of volume equation (Equation (9)).

$$\stackrel{⃑}{\nabla }·\stackrel{⃑}{v}=0$$

(9)

Applying Darcy’s law, $\stackrel{⃑}{v}=-k\stackrel{⃑}{i}$, where in 3D, $\stackrel{⃑}{i}=\stackrel{⃑}{\nabla }h$ $=\frac{\partial h}{\partial x}\stackrel{⃑}{i}+\frac{\partial h}{\partial y}\stackrel{⃑}{j}+\frac{\partial h}{\partial z}\stackrel{⃑}{k}$. Equation (9) can now be rewritten in the matrix form, as shown in Equation (10).

$$\stackrel{⃑}{\nabla }·\stackrel{⃑}{v}=-\left\{\begin{array}{ccc}\frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\end{array}\right\}\left[\begin{array}{ccc}{k}_{xx}& k_{xy}& k_{xz}\\ k_{yx}& k_{yy}& k_{yz}\\ k_{zx}& k_{zy}& k_{zz}\end{array}\right]\left\{\begin{array}{c}\frac{\partial }{\partial x}\\ \frac{\partial }{\partial y}\\ \frac{\partial }{\partial z}\end{array}\right\}h=0$$

(10)

where $k=\left[\begin{array}{ccc}{k}_{xx}& k_{xy}& k_{xz}\\ k_{yx}& k_{yy}& k_{yz}\\ k_{zx}& k_{zy}& k_{zz}\end{array}\right]$ is the hydraulic conductivity tensor.

A 3D numerical model of seepage flow was then developed in the MATLAB interface using the finite-difference method. The water-saturated soil specimens were 15 cm in diameter and 11 cm in height. The entire grid used to model the flow was a rectangular-cubic specimen with dimensions of $15 \mathrm{c}\mathrm{m}\times 15 \mathrm{c}\mathrm{m}\times 11 \mathrm{c}\mathrm{m}$. This rectangular-cubic domain was discretized into 15 nodes along both the X and Y axes. The top view of the discretized domain has a circular cross-section and rectangular mesh, which is illustrated in Figure 4. The top and bottom surfaces of the cylindrical specimen were modeled as Dirichlet boundary conditions with known hydraulic heads. However, the circumference of the permeameter was an impermeable boundary and was modeled using the Neuman boundary conditions (see [19] for more details).

The computational code was developed based on the central finite-difference method and then used to calculate the spatial variations in the hydraulic head within the soil using the hydraulic conductivity values obtained for the unstimulated tests. Equation (10) was used to compute the flow rate and discharge through the soils for the steady state unstimulated tests. However, this only solved the hydraulic head inside the specimen, while the test was performed in the unstimulated condition.

3.2.2. RF-Stimulated Seepage

The hydraulic head under RF stimulation was assumed to be k’, and $\stackrel{⃑}{V}=-k^{\prime }\stackrel{⃑}{\nabla }h$ for flow was rewritten as Equation (11) [19].

$$\stackrel{⃑}{\nabla }·\stackrel{⃑}{V}=-\left\{\begin{array}{ccc}\frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\end{array}\right\}\left[\begin{array}{ccc}{k}_{xx}^{\prime }& k_{xy}^{\prime }& k_{xz}^{\prime }\\ k_{yx}^{\prime }& k_{yy}^{\prime }& k_{yz}^{\prime }\\ k_{zx}^{\prime }& k_{zy}^{\prime }& k_{zz}^{\prime }\end{array}\right]\left\{\begin{array}{c}\frac{\partial }{\partial x}\\ \frac{\partial }{\partial y}\\ \frac{\partial }{\partial z}\end{array}\right\}h=0$$

(11)

where $k^{\prime }=\left[\begin{array}{ccc}{k}_{xx}^{\prime }& k_{xy}^{\prime }& k_{xz}^{\prime }\\ k_{yx}^{\prime }& k_{yy}^{\prime }& k_{yz}^{\prime }\\ k_{zx}^{\prime }& k_{zy}^{\prime }& k_{zz}^{\prime }\end{array}\right]$ is the RF-stimulated hydraulic conductivity tensor.

Since the application of RF waves demonstrated altering hydraulic conductivity, it was proposed that RF-stimulated hydraulic conductivity (k’) and the flow discharge (Q_st) in the RF-stimulated tests are functions of the electric field, E. Hence,

$$k^{\prime }=f\left(E\right) \& Q_{st}=f\left(E\right)$$

(12)

Additionally,

$$s=\frac{E^2}{Z_o} \mathrm{or} E=\sqrt{s{Z}_o}$$

(13)

$$P=s\times a$$

(14)

where s = RF-power density (W/m²), Z_o = characteristic impedance of the free space (Ω), P = RF-power (Watts), and a = area enclosing each node (m²).

From Equations (12)–(14), it was proposed that the hydraulic conductivity and the flow discharge in the RF-stimulated tests are functions of the power density, and hence, power. The results showed an increase in hydraulic conductivity, with the increase in RF-power level in coarse-grained media. A nonlinear, but proportional, relation was assumed between the RF-stimulated hydraulic conductivity, k’, and RF-power, P. Based on Equations (12)–(14), the RF-stimulated hydraulic conductivity and RF-power can be correlated using Equation (15) [19].

$$k^{\prime }=k+\beta \sqrt{P}$$

(15)

where β is a constant value, and k is the unstimulated hydraulic conductivity (cm/s). Therefore, the increase in k’ at each discretized node in the soil specimen domain could be correlated to the increase in the RF-power at that node, while calculating the hydraulic head at the specific node. Equation (15) can be redefined as Equation (16) [19].

$$k^{\prime }=k+\beta \sqrt{s\times a}$$

(16)

A similar computational code in the MATLAB interface was developed to solve Equations (11) and (16). The RF-power density at each node was obtained from the RF forward model generated using COMSOL Multiphysics. The RF-power densities in the form of a 3D matrix were exported from the COMSOL model into the MATLAB interface. Afterward, an optimization scheme was implemented to find $\beta$ by minimizing a cost function equal to the difference between the numerically simulated and experimentally measured values of the RF-stimulated discharge, i.e., Q_sim,st and Q_exp,st, respectively. In other words, the only parameter that was varied to minimize the cost function was $\beta$. No constraint was implemented in the optimization as the cost function was minimized without any constraints on $\beta$. The COMSOL code was used to export the RF-power densities at a frequency of 726 MHz at power levels of 10, 25, and 40 Watts to be imported into the optimization scheme (see [19] for more details).

4. Results

This section shows and discusses the results of the experimental and numerical simulations of the electric field and seepage, as well as the optimization used to find the governing equation of the impact of RF waves on hydraulic conductivity.

4.1. Electric Field Measurement and Validation of Numerical Simulation

The electric field pattern within the cavity numerically simulated using COMSOL was validated against experimentally measured values of the Z component of the electric field. This was because the vertically located monopole probe was vertically polarized, i.e., its measurement was dominated by the Z component of the electric field. The experimental measurements were performed at two frequencies, 498 MHz and 632 MHz. As seen in Figure 5, the experimental measurements were performed near where the pore-pressure transducer was installed within the sample. The location of Slice 1 was +4 cm away from the transducer along the X-axis, and Slice 2 was −4 cm along the X-axis. The electric field measurements were performed using a glass-cased monopole probe inserted into the cavity through the top plate down to the desired depth. Figure 6 shows maps of the numerically simulated (left) experimentally measured (right) electric field, E_z, on Slices 1 and 2, respectively. The electric field measurements were normalized to the maximum electric field measured on the corresponding slice (see [19] for more details).

Figure 6 shows reasonable agreement between the experimentally measured and numerically simulated electric fields on each slice. Since the monopole probe was not calibrated, the experimentally measured electric field does not represent the actual strength of the electric field within the cavity and just represents the electric field pattern. Both the numerically simulated and experimentally measured electric field patterns have their maximum at similar depths, i.e., the predesigned location of the pore-pressure transducer. As mentioned, the strength of the electric field at any specific location depends on the frequency of the RF waves launched. The effects of the RF waves on physical properties, such as the viscosity of water, may not be linear due to having different field patterns at different frequencies. Hence, a change in any physical property may or may not be linearly correlated with an increase or decrease in frequency (see [19] for more details).

4.2. Hydraulic Conductivity Tests

The RF waves’ effects on hydraulic conductivity at different combinations of powers and frequencies need to be studied. Figure 7 shows the measurement of the unstimulated hydraulic conductivity of the glass-bead sample, performed using a standard permeameter. The test was continued for four hours. The hydraulic conductivity values remained fairly constant during the entire run of the test. The average unstimulated hydraulic conductivity of the glass-bead sample was measured to be $1.391\times {10}^{-2}$ cm/s (see [19] for more details).

A series of RF-stimulated hydraulic conductivity tests were conducted on the glass-bead sample in the customized permeameter. There was a discrepancy of 3.5% between the customized permeameter and the standard permeameter. The discrepancy was calibrated. To calibrate the customized permeameter, the hydraulic conductivity measurements obtained were multiplied with a multiplication factor of 1.035 (see [19] for more details).

The RF-stimulated tests were then conducted at a frequency of 726 MHz and RF-power levels of 10, 25, and 40 Watts. Figure 8 shows the hydraulic conductivity measurements at different RF-power levels. There was no sharp increase in the hydraulic conductivity with the RF stimulation. However, the hydraulic conductivity started to increase gradually with time and eventually attained a peak value. As seen in Figure 8, at an RF-power level of 10 Watts, hydraulic conductivity started to increase from $1.3942\times {10}^{-2}$ cm/s to a peak value of $1.452\times {10}^{-2}$ cm/s after two hours and 30 min and remained constant thereafter. Similar changes were observed at the other RF-power levels of 25 Watts and 40 Watts (Figure 8). At the RF-power level of 25 Watts, the RF stimulation was terminated once the RF-stimulated hydraulic conductivity reached its maximum. After the termination of RF waves, the hydraulic conductivity slightly decreased with time, and then, stabilized at a smaller value, but was still larger than the unstimulated value. Even though the increase in hydraulic conductivity was larger at higher RF-power levels, the slope of the increase was, on average, 0.0225% per hour, and even seemed higher for 10 Watts of power. The average unstimulated hydraulic conductivity before the start of RF stimulation and the peak value of the RF-stimulated hydraulic conductivity are compared in

Table 2. Hydraulic conductivity measurements of the glass-bead sample at a frequency of 726 MHz and various RF-power levels.

Power (Watts)	Average of Unstimulated Hydraulic Conductivity Measured Before RF Stimulation, k (cm/s)	RF-Stimulated Hydraulic Conductivity, Peak Value, k’ (cm/s)	Percentage Change (%)
10	1.3942 × 10−2	1.452 × 10−2	(+) 4.190%
25	1.3911 × 10−2	1.482 × 10−2	(+) 6.864%
40	1.3923 × 10−2	1.514 × 10−2	(+) 8.774%

. As seen in Table 2, the hydraulic conductivity increased with the increase in the RF input power. The increase in the hydraulic conductivity was, however, small compared to the result obtained by [20].

The effect of RF waves on hydraulic conductivity was smaller in glass beads than in natural sand [10]. Hence, a new set of RF-stimulated hydraulic conductivity tests were performed on the natural sand sample. Unstimulated hydraulic conductivity tests were first performed on the natural sand sample in the standard permeameter for calibration purposes [19].

Figure 9 shows the measurement of unstimulated hydraulic conductivity of the natural sand sample performed using the standard permeameter. The average unstimulated hydraulic conductivity of the natural sand sample was measured to be $0.7924\times {10}^{-2}$ cm/s. A series of RF-stimulated hydraulic conductivity tests were then conducted on the natural sand sample in the customized permeameter. There was a discrepancy of 5.1% between the customized and standard permeameters, which was later calibrated. RF-stimulated tests were performed at a frequency of 726 MHz and power levels of 10, 25, and 40 Watts. The conducted tests were similar to the RF-stimulated hydraulic conductivity tests carried out on glass beads. Figure 10 shows the hydraulic conductivity measurements at different RF-power levels. The results show a similar pattern of change in hydraulic conductivity. After RF stimulation started, the hydraulic conductivity of the sand sample started to increase gradually with time and attained a peak value. At an RF-power level of 10 Watts, the hydraulic conductivity started to increase from $0.7933\times {10}^{-2}$ cm/s reached a peak value of $0.881\times {10}^{-2}$ cm/s after four hours and remained constant thereafter. Similar changes were observed at other input powers of 25 and 40 Watts. At 25 Watts, RF stimulation was terminated once the RF-stimulated hydraulic conductivity of sand decreased with time, and then, stabilized at a slightly smaller value, albeit much larger than the original unstimulated one. In the case of natural sand, not only was the increase in hydraulic conductivity larger at higher RF-power levels, but the slope of the increase was also slightly larger for higher RF-power levels, increasing from 0.02% per hour at 10 W to 0.03 and 0.033% per hour at 25 and 40 W (see [19] for more details).

Moreover, the percentage change in the hydraulic conductivity of natural sand due to RF stimulation was larger than that of the glass beads for all input RF-power levels. The results, however, show a similar pattern of increase and decrease in the hydraulic conductivity of both sand and glass beads during the application and termination of the RF stimulation, respectively (

Table 3. Hydraulic conductivity measurements of the natural sand sample at a frequency of 726 MHz and various RF-power levels.

RF-Power (Watts)	Average of Measured Unstimulated Hydraulic Conductivity Values, k (cm/s)	RF-Stimulated Hydraulic Conductivity, Peak Value, k’ (cm/s)	Percentage Change (%)
10	0.7933 × 10−2	0.881 × 10−2	(+) 11.091%
25	0.7932 × 10−2	0.915 × 10−2	(+) 15.287%
40	0.7928 × 10−2	0.994 × 10−2	(+) 25.386%

). The increase in hydraulic conductivity was also slow and gradual. With a test setup that can only provide up to 40 Watts of RF-power, RF stimulation would not be high enough to change the hydraulic conductivity promptly, even for natural sand [19].

4.3. Electric Field

Measurement and Comparison

The experimental measurements of the electric field pattern for the RF-stimulated tests were performed at two frequencies, 710 MHz and 726 MHz. As seen in Figure 11, the patterns of both the numerically simulated and experimentally measured electric fields matched each other. As mentioned, since the monopole was not calibrated, the experimentally measured electric field measurements do not represent the actual intensity of the electric field within the cavity. The intensity of the electric field at any specific location depends on the frequency of the RF waves launched. The effect of the RF waves on physical properties, such as the viscosity of water, may not be linear due to having different electric field patterns at different frequencies. Hence, the change in any physical property may not be linearly correlated with the change in frequency [19].

4.4. Seepage Flow Numerical Simulation

The unstimulated seepage flow model was developed based on the hydraulic conductivity value of the unstimulated test performed in the lab. The model in MATLAB interface was used to compute the spatial variations in the hydraulic head within the soil. Figure 12 shows contour maps of the hydraulic head on fifteen vertical slices within the sandy sample obtained using the forward model for the unstimulated test. The hydraulic head decreased gradually and uniformly from the bottom (inlet) to the top (outlet), indicating the existence of a uniform gradient along the length of the soil specimen and an upward uniform flow within the homogeneous sand sample.

The experimentally measured flow discharge in the natural sand sample during the unstimulated tests was $9.3\times {10}^{-6} {\mathrm{m}}^3/\mathrm{s}$. Additionally, the finite-difference forward model computed the flow discharge to be $9.23\times {10}^{-6} {\mathrm{m}}^3/\mathrm{s}$. There is a small discrepancy of 1.18% between the experimental and numerical values, and the difference is due to the approximation of the cylindrical walls of the permeameter using a stepwise rectangular cubical wall (see [19] for more details).

To better understand the RF stimulation effect on hydraulic conductivity, the increase in hydraulic conductivity due to RF waves was correlated with the RF-power level. The RF-power density at each node was obtained from the RF forward model generated using COMSOL Multiphysics.

Table 4. Experimentally measured and numerically computed flow discharge in the natural sand sample at various RF-power levels.

RF Input Power (Watts)	${\mathit{Q}}_{\mathit{e}\mathit{x}\mathit{p},\mathit{s}\mathit{t}}({\mathbf{m}}^3/\mathbf{s})$ (Experimental)	$\mathbf{Slope} \left(\mathit{\beta }\right)$ (Numerical Stimulation)	${\mathit{Q}}_{\mathit{s}\mathit{i}\mathit{m},\mathit{s}\mathit{t}}({\mathbf{m}}^3/\mathbf{s})$ (Numerical Value Based on Optimized k′)	$\mathbf{Cos}\mathbf{t} \mathbf{Function} \frac{\left({\mathit{Q}}_{\mathit{e}\mathit{x}\mathit{p},\mathit{s}\mathit{t}}-{\mathit{Q}}_{\mathit{s}\mathit{i}\mathit{m},\mathit{s}\mathit{t}}\right)}{{\mathit{Q}}_{\mathit{e}\mathit{x}\mathit{p},\mathit{s}\mathit{t}}}\times 100\mathbf{\%}$
0	9.34 × 10−6	None	9.23 × 10−6	1.18%
10	10.39 × 10−6	3.65 × 10−8	10.11 × 10−6	2.77%
25	10.78 × 10−6	3.25 × 10−8	10.52 × 10−6	2.47%
40	11.71 × 10−6	3.25 × 10−8	11.42 × 10−6	2.54%

shows the experimental flow discharge, numerically computed flow discharge, and slope ($\beta$) at a frequency of 726 MHz and RF-power levels of 10, 25, and 40 Watts [19].

Similarly, the experimentally measured and numerically computed flow discharge with respect to the RF-power levels were plotted in Figure 13. Using the optimization scheme, the cost function was minimized to less than 3%, indicating that the numerically computed values reasonably matched the experimental values. In addition, the β-value at all three power levels was nearly constant.

As the optimization was completed, optimized flow discharge, optimized hydraulic conductivity, and optimized hydraulic heads were obtained. Figure 14 displays contour/color maps of the hydraulic heads obtained from the inverse model at a frequency of 726 MHz and an RF-power level of 25 Watts. Similar to the unstimulated case, the contour/color maps show a uniform decrease in the hydraulic head from the bottom (inlet) to the top (outlet), indicating an upward flow. There was a spatially variable alteration in the hydraulic conductivity in the X, Y, and Z directions, which is a function of the spatially variable electric field power density. This resulted in a change in the hydraulic head within the soil specimen due to the application of RF waves, which is uniform along the X and Y directions and only varies along the Z direction. However, the change is uniform and does not manifest as a distortion in the contour/color maps of the hydraulic head.

As mentioned, the value of hydraulic conductivity and flow discharge changed due to RF stimulation. The model assumed that the scalar value of the RF-power governs the alteration in the hydraulic conductivity. Hence, the variation will be the same for all components of the hydraulic conductivity tensor (i.e., the same slope, $\beta$, in all directions). Figure 15 shows a contour/color map for $k_{zz}^{\prime }$ (Z-component of the RF-stimulated hydraulic conductivity), computed using the inverse model on Slice 7 at a frequency of 726 MHz and RF-power level of 25 Watts. Other components on the k’ tensor can also be plotted. However, the hydraulic conductivity of the soil was assumed to be isotropic, and $\beta$ was assumed to be the same in all directions. Therefore, k’ will be isotropic. Figure 16 shows a contour/color map of the RF-power density at a frequency of 726 MHz and RF-power level of 25 Watts on Slice 7. The RF-power density at each node was obtained from the RF forward model generated using COMSOL Multiphysics (see [19] for more details).

As seen in Figure 15 and Figure 16, the hydraulic conductivity along the Z-direction changed due to the application of RF waves. Additionally, the contour plot of the hydraulic conductivity corresponds to the variation in the power densities due to the linear relationship between the power densities at each node and the RF-stimulated hydraulic conductivity.

The measurement of the temperature of the medium close to the antenna and permeable wall at an RF-power of 30 W showed a minimal increase (<1.6 °C). The temperature variation between the center and boundary of the cylindrical soil sample was even smaller (<0.1 °C). Therefore, even though the temperature change and gradient can cause a minor convective flow or a decrease in the viscosity of water, it is not strong enough to create such a large change in k.

The impact of RF waves on hydraulic conductivity has been studied by a limited number of researchers. The results presented in this paper are consistent with the observations of Azad [20], Acharaya [19], and Azad et al. [8].

5. Conclusions

The work outlined in this paper demonstrates changes in the hydraulic conductivity of glass beads and a natural sand sample due to RF waves of various frequencies.

The electric field inside the cavity was measured at various resonant frequencies and then validated against the numerically simulated electric field using COMSOL Multiphysics. The measured and numerically simulated electric fields reasonably agreed.

At all RF-power levels, for both the glass beads and natural sand, RF waves increased the hydraulic conductivity. The increase in the hydraulic conductivity was gradual with time. Additionally, for both the glass-bead and natural sand samples, the RF-stimulated hydraulic conductivity increased with increasing RF-power levels. However, this percentage increase was observed more in the natural sand sample than in the glass-bead sample. This could be due to the influence of the silt content within the natural sand. When RF stimulation was terminated, the hydraulic conductivity initially stabilized at its peak value. However, as time increased after termination, the hydraulic conductivity slightly decreased and stabilized at a smaller value, albeit larger than the initial unstimulated value. The tests were not continued for a long enough period of time to observe any further decrease.

The measured hydraulic conductivity was then correlated with numerically simulated hydraulic conductivity using the MATLAB interface. The results from the numerical model for the seepage flow showed a uniform decrease in the hydraulic head from the bottom (flow inlet) to the top (flow outlet) due to upward seepage flow through the sample. In the case of RF-stimulated tests, hydraulic heads were computed using an inverse model based on an optimization scheme. After optimization, the optimized RF-stimulated numerically computed hydraulic conductivity showed a similar spatially variable pattern to that of the RF-power density.

This paper describes the impact of RF waves on two specific types of soil at certain frequency ranges and power levels. Hence, to be able to comprehensively study this phenomenon, other soil types and wider ranges of frequency and RF-power level need to be studied in the future. This research also provides a method of changing the soil’s hydraulic conductivity without changing the soil type, packing, and, in turn, intrinsic hydraulic conductivity. Hence, this provides a way of studying the impact of hydraulic conductivity on various events, such as the pore-water pressure increase during liquefaction, without the need to change the soil [20].

6. Patents

An invention disclosure has been submitted and is pending.