Effect of various physical properties on the reflection coefficients of inhomogeneous waves at the stress-free surface of partially saturated soils induced by obliquely incident fast P-wave

M.Kumri ,M.S.Brk ,A.Singh ,M.Kumr

a Department of Mathematics,M P College for Women,Mandi Dabwali,125104,Sirsa,Haryana,India

b Department of Mathematics,Indira Gandhi University Meerpur,Rewari,122503,Haryana,India

c Department of Mathematics,Dr.B R Ambedkar Govt.College,Dabwali,125104,Sirsa,Haryana,India

Keywords:Plane harmonic wave Inhomogeneous Partially saturated soils Reflection coefficients Energy shares

ABSTRACT Ghasemzadeh and Abounouri [1]developed a mathematical model of partially saturated soils that is solved using the potential method,which decomposes elastodynamics equations into two standard wave equations,a scalar wave equation for scalar potential and a vector wave equation for vector potential.In such a medium,four waves exist three longitudinal and one shear.Each fluid phase tortuous path is taken into account in this model.The inertial coupling between solid and fluid particles is considered.Furthermore,both open-pore and sealed-pore boundaries are explored to investigate the reflection phenomenon at the surface of partially saturated soils.For both boundaries,the reflection coefficients of inhomogeneous waves at a partially saturated soil surface are found as a non-singular set of linear equations.All waves (both reflected and incident) in partially saturated soils are pronounced as inhomogeneous due to viscosity in pore fluids (i.e.,distinct directions of attenuation and propagation).The energy shares of reflected waves are determined using an energy matrix.A numerical example is used to determine the reflection coefficients and the distribution of incident energy among the various reflected waves.The effect of different physical features on reflection coefficients and incident energy partitioning is illustrated graphically.The conservation of incident energy at the surface of partially saturated soils is mathematically confirmed at all angles of incidence.

1.Introduction

Seismic wave characteristics are affected by various parameters,including Poisson’s ratio,pore characteristics,matric suction,water saturation,and pore structure porosity.The study of seismic waves is essential in various scientific engineering challenges,such as detecting hydrocarbons via acoustic sensing in geophysics,probing the ocean floor in hydroacoustics,and pressure variations at the ocean bottom.

A significant rise in experimental and theoretical investigations to grasp the behaviour of unsaturated soils has been noted in recent years.Soils that are partially saturated have three phases say solid,liquid,and gas.Fredlund and his colleagues originated the critical theories for partially saturated soils in the 1970s.The constitutive relations of unsaturated soils for volume change are initiated by Fredlund and Morgenstern [2].Fredlund and Hasan[3]and Lloret and Alonso [4]proposed a generic framework for one-dimensional consolidation in which the water and air phases are treated as continuous.This formulation is based on the simultaneous solution of two continuous equations,one for the air phase and the other for the water phase,to obtain air and water pressures at any moment.Fredlund and Morgenstern [2]constitutive relations were implemented into Fredlund and Hasan model.Dakshanamurthy and Fredlund [5]used an uncoupled strategy to investigate the three-dimensional consolidation problem.Conte[6]used the approach presented by Fredlund and his coworkers to generate the differential equations characterizing unsaturated soils for uncoupled and coupled consolidation and then solved them.Conte et al.[7]used unsaturated soil’s physical and constitutive properties to construct analytical formulas for longitudinal and shear wave velocities (VP and VS).These wave features have been explored theoretically.Ghasemzadeh and Abounouri [1]examined the wave propagation parameters (i.e.,velocity/ inherent attenuation) in unsaturated soils using the potential approach.They only looked at how water saturation and frequency affected the velocity and intrinsic attenuation of two fast waves (P1 andSV).However,they did not consider the effect of the inhomogeneity parameter on these wave propagation characteristics.To derive the propagation velocities of seismic waves,most researchers use the Helmholtz decomposition theorem to solve porous media’s governing equations (i.e.,differential equations).However,Arqub and Shawagfeh [8]suggested and investigated an effective replicating kernel algorithm for numerical solutions of time-fractional partial differential equations in porous media by employing Dirichlet boundary conditions.Recently,Barak et al.[9]solved the governing equations proposed by Ghasemzadeh and Abounouri [1]in terms of Christoffel equations.They used the finite and non-dimensional inhomogeneity parameter to investigate the propagation of four inhomogeneous waves in unsaturated soils.They have also assessed how the inhomogeneity parameter,water saturation,porosity,permeability viscosity,and frequency affected the attenuation and velocity of four waves.

The ocean floor is usually covered with marine silt in natural ocean habitats.Natural marine sediment comprises a matrix(or skeleton) of solid particles whose pores are filled with one or more fluids.A porous solid with a solid frame and pore fluid is described as such a medium.The majority of natural sedimentary materials,on the other hand,have been widely reported to be partially saturated.An oil reservoir,for example,contains natural gas (inviscid),oil (viscous),and solid ingredients.Mineral and fossil fuel exploration and extraction,modern warfare,the development of marine civil engineering proposals,and other scientific inquiries necessitate understanding the geology beneath the ocean’s surface.It is well known that different fluids (liquid and gas) in the rocks impact the attribute of wave propagation and all related events.The reflection coefficient is a valuable measure for quantitatively describing the geoacoustic features of seabed sediment and its sub-bottom structure.The process of seismic reflection is the most effective way of determining the layout of the sedimentary layers underneath and the ocean floor.The oil and gas industries use the seismic reflection technique in the detection/recognition of gas and oil reservoirs placed beneath the earth’s surface.Generally,a seismic wave is sent deep into the ground in the seismic reflection technique,bouncing back.The recorded signals of seismic waves on the earth’s surface (in seismic exploration) depend on the propagation attributes (velocity/attenuation) of elastic waves.However,these propagation characteristics rely on the numerous physical critics of constituents present inside the rocks and rely on the rock parameters.Pugin et al.[10]investigated the hydrogeological parameters of aquifers in glacial sediments at two different sites in Canada using seismic reflection ofSandPwaves.In recent years,many researchers have looked at the reflection of inhomogeneous waves at the surface of highly complex materials,including a three-phase porous solid [11],swelling poroelastic solid[12],double-porosity solid [13],two-phase dissipative poroelastic solid [14],and double porosity and dual-permeability (DPDP) materials [15].Singh et al.[16]investigated the relationships between reflection coefficients and energy ratio expressions for different reflected waves.They visually examined the effect of impedance characteristics on all reflected waves at various angles of incidence.The reflection and transmission of plane waves at an interface between two separate transversely isotropic micropolar piezoelectric half-spaces are examined by Singh et al.[16,17].They discovered relationships between amplitude ratios of reflected and transmitted waves,influenced by material properties and incidence angle.They depict the micropolar piezoelectric effects on amplitude ratios and the square root of energy ratios as a function of angle of incidence graphically.Zhou et al.[18]have determined the accurate equation of plane-wave reflection/transmission coefficients relying on the interconnections between displacements,seismic wave functions,and stress in porous media for the incidence of fast P-waves.In the framework of a three-phase-lag model,Sheokand et al.[19]explored the plane wave propagation in a homogeneous,fiber-reinforced orthotropic thermoelastic rotating half-space.They used MATLAB programming to calculate the reflection coefficients,which are then graphed to highlight the impacts of rotation,fiber reinforcement,and phase lag parameters.However,there is no study in the relevant literature on the reflection of waves (inhomogeneous) that considers the tortuosity of fluid phases as a function of matric suction.

The governing equations introduced by Ghasemzadeh and Abounouri [1]are solved in the current problem using the potential method.In such a medium,there are four waves (three longitudinal and one shear).The incidence of a fastP1wave is taken into account.In the medium,four reflected waves (one shear and three longitudinal) are produced due to the incident wave.For both boundaries (i.e.,open-pore and sealed-pore),the reflection coeffi-cients of inhomogeneous waves at the surface of a partially saturated soil are found as a non-singular set of linear equations.All waves (both reflected and incident) in partially saturated soils are pronounced as inhomogeneous due to viscosity in pore fluids (i.e.,distinct directions of attenuation and propagation).The energy shares of reflected waves are determined using an energy matrix.The reflection coefficients and partition of incident entropy are computed using a numerical example.The effect of incidence direction,pore features,matric suction,water saturation,Poisson’s ratio,and inhomogeneity angle is graphically evaluated on the partition of incident energy and reflection coefficients.The conservation of incident energy at the surface of partially saturated soils is mathematically verified for each angle of incidence.

2.Basic equations

The equations of motion [1]in the absence of body force are as follows.

wherewi,vi,anduiclassify the displacement components of water,gas,and solid particles,respectively.The three phases of a composite medium,air,water,and solid,are classified by the indicesa,w,ands,respectively.A dot over the variable illustrates the partial time derivative.Theτ’s describe effective tortuosity,while theρ’s specify the material densities.φa(=φ(1-Sw))andφw(=φSw)denote the volume fractions of air and water phases,respectively.Swidentifies the water saturation,andφrepresents the porosity of the soil.The inertial coupling of two fluid phases with solid is represented by the parametersbwandba,which are given by

The equations of fluid momentum [20]are as follows.

whereτa(τw) describes the effective tortuosity of the air (water)phase.

The composite medium stress-strain relations are given by

whereδijdenotes a Kronecker symbol,Grepresents the shear modulus,andχdenotes the effective stress variable.Appendix A contains the other elastic coefficients used in the preceding equations.By substituting (4)-(8) into equations (1)-(3),the equations of motion of unsaturated soils can be expressed in vectorial form as

wherea11=H+G+φwW+φaM,a12=φw[χL+(1-χ)C]-φ2wL-φaφwC,a13=φa[χC+(1-χ)N]-φ2aN-φaφwC,a21=φwW,a22=-φ2wL,a23=-φwφaC,a31=φaM,a32=-φwφaC,a33=-φ2aN.

3.Plane harmonic wave

A plane wave has a constant disturbance across all points of a plane drawn perpendicular to the propagation direction.A plane harmonic wave is a wave whose wavefront (profile) is either a sine or a cosine curve.

The three displacement vectors are expressed as using the standard Helmholtz resolution of a vector.

The equations (9)-(11) are stated in terms of displacement potentials (φ0,φ1,φ2;ψ0,ψ1,ψ2) as

wherea*11=a11+G.

For time harmonic vibrations (～e-ιωt) potentials(φ0,φ1,φ2),the relations (15)-(17) transform to

whereΛs=ρs(1-φ)+(τw-1)φwρw+(τa-1)φaρa+(bw+ba),Λw=τwφwρw+

The equations (16) and (17) are evaluated to yield the following two relations.

whereA1=a22a33-a23a32,B1=-a22Λa-a33Λw,C1=ΛaΛw,

A2=a31a23-a21a33,B2=a23ra+a21Λa-a33rw,C2=Λarw,

A3=a21a32-a31a22,B3=a32rw+a31Λw-a22ra,C3=Λwra.

On using the above relations into equation (21),we get

where

Decompose the differential Eq.(26) into three Helmholtz equations,which are as follows.

It means that three longitudinal waves with phase velocitiesϑk,(k=1,2,3)propagate in unsaturated soils.These velocities are computed using the equation,which is provided by

The velocitiesϑ1,ϑ2,ϑ3pertain to the longitudinal waves classified asP1,P2,P3waves in descending order of real parts.The potential function for aggregate dilatation in the given medium is as follows.

When we employed the above equation in relations (24)-(25),it produces

where

The equations (18)-(20) for time-harmonic vector potentials are solved similarly to scalar potentials.

We get a Helmholtz equation by solving the preceding relations,which is given by

It proves the presence of a shear wave propagating atϑ4 velocity.

4.Displacements

Solid and fluid phase displacement components in thexzplane are

5.Formulation of the problem

Partially saturated soils take up the regionz＞0,defined by the horizontal planez=0,in a rectangular Cartesian coordinate system (x,y,z).In the present problem,we have considered the wave motion in thexz-plane.As a result,all quantities are unaffected by they-coordinate.Withθ0(angle of incidence),a plane harmonic inhomogeneous fastP1wave moving withϑ0(velocity of incident wave) andω(frequency) is made incident atz=0.As a result,four inhomogeneous waves (reflected) are generated.

6.Boundary conditions

In the discipline of wave propagation theory,determining the suitable boundary conditions to solve a specific problem is critical.We have looked at two different scenarios in this problem:open surface pores and perfectly sealed surface pores.Discharge of fluid is allowed out of aggregate in the case of fully opened pores boundary surfaces,but it is not allowed out of a total in totally sealed pores boundary surfaces.As a result,all components of the stress of each constituent of partially saturated soils should vanish for fully open surface pores,and the displacement components of fluid particles should be unrestricted.While aggregate normal and tangential strains should disappear separately and constraints on the displacement components of fluid particles are implemented for fully sealed surface pores.Finally,proper boundary conditions are established.

6.1.Open-pore boundary

6.2.Sealed-pore boundary

7.Reflection coefficients

Reflection coefficients are the ratios of reflected wave amplitude to incident wave amplitude.The boundary conditions,which express the continuity of displacements and stresses dependent on the nature of the boundary,are solved to produce reflection coefficients.According to the reflection coefficients principle,the incident wave,reflected waves,and normal to the surface all lie in the same plane.Furthermore,the sine of the angle of reflection is equal to the sine of the angle of incidence.This research aims to investigate the reflection of plane harmonic inhomogeneous waves at the partially saturated soils stress-free surfacez=0.Because pore fluids have viscosity,the examined medium becomes dissipative.Hence,all waves (both incident and reflected) are inhomogeneous (with various propagation and attenuation directions).Both propagation directions and attenuation govern wave propagation in the current medium.As a result,the displacement potentials characterizing reflected wave-particle motions [21]are expressed as

where the amplitudes ofP1,P2,P3,SVwaves (reflected) are specified by arbitrary coefficientsFn,(n=1,2,3,4),respectively.The vectors of attenuationand propagationare expressed as

The propagation of an attenuated wave in a dissipative medium is assumed to be inhomogeneous.It is determined by the propagation vector direction and the attenuation vector deviation from the propagation vector.As a result,an incident wave is characterized byθ0(propagation direction) andγ0(attenuation direction).The angle created by(incident wave propagation vector) with thez-axis in thexz-plane isθ0,and the angle formed bywith(incident wave attenuation vector) isγ0.The complex wavenumberin regards to these angles [22]is represented as

where,for the incident wave of velocityϑ0,we have

Fig.1.Reflection coefficients (|Zj|,j=1 ,2 ,3 ,4) of P1 ,P2 ,P3 ,SV waves respectively; variations with incident direction θ0 and Poisson’s ratio (ν); γ0=45 0 ,φ=0.4 ,ψ=1000 kPa,Sw=0.8 ,ω=2 π×10 kHz.

As a result,the incident wave potential is characterized as follows.

where arbitrary coefficientF0represents the amplitude of incident fastP1wave.

7.1.Amplitudes and phase shifts

The boundary conditions (38)-(39) for displacements and stresses determined from the potentials in (35)-(37) are met by a set of four simultaneous linear non-homogeneous equations,provided by

For complex unknownsZj,the Gauss elimination method is used to solve the Eqs.(46) numerically.The ratios of the amplitudes of commensurate reflected waves to the amplitude of the incident wave are described by the magnitudes of complex unknownsZj,(j=1,2,3,4),and the principle value of arg(Zj) indicates the phase shift of these reflected waves.

Fig.2.Reflection coefficients (|Zj|,j=1 ,2 ,3 ,4) of P1 ,P2 ,P3 ,SV waves respectively; variations with incident direction θ0 and matric suction (ψ); γ0=45 0 ,φ=0.3 ,ν=0.35 ,Sw=0.9 ,ω=2 π×10 kHz.

7.1.1.Foropensurfacepores

where

7.1.2.Forsealedsurfacepores

where

Fig.3.Reflection coefficients (|Zj|,j=1 ,2 ,3 ,4) of P1 ,P2 ,P3 ,SV waves respectively; variations with incident direction θ0 and inhomogeneity angle (γ0); ψ=1000 kPa,φ=0.4 ,ν=0.4 ,Sw=0.9 ,ω=2 π×10 kHz.

7.2.Energy shares

This section examined how distributed the incident wave energy between distinct reflected waves at the stress-free surface of partially saturated soils in two scenarios (i.e.,for fully open surface pores and fully sealed surface pores).According to Achenbach[23],at the surface,z=0 (stress-free) of partially saturated soils,the rate of average energy distributed per unit area is given by

The medium can support the four refracted waves propagation in the current geometry.As a result,the formula for calculating the bulk and interaction energy atz=0 is specified as

In partially saturated soils,the energy matrix (E) specifies the energy shares of all waves as well as interaction energy between two different waves.The diagonal entries of E i.e.,E11,E22,E33,E44specify the energy shares of reflected wavesP1,P2,P3,SV,respectively.The expressiongives the interaction energy generated at the surfacez=0 as a result of interaction between incident and reflected waves.An expressionERR=determines the proportion of interaction energy distributed across all reflected waves.The sum of the energy shares conveyed through diverse reflected waves during reflection process equal to unity at each angle of incidence atz=0 is demonstrated by the relationE11+E22+E33+E44+ERR+EIR=1.The elementsQijin Eq.(48) are obtained after the cancellation of a common factor.

Fig.4.Reflection coefficients (|Zj|,j=1 ,2 ,3 ,4) of P1 ,P2 ,P3 ,SV waves respectively; variations with incident direction θ0 and water saturation (Sw); γ0=45 0 ,ψ=10000 kPa,φ=0.3 ,ν=0.4 ,ω=2 π×10 kHz.

8.Numerical example and discussion

8.1.Numerical example

A vast number of parameters are used in the computation of reflection coefficients and energy shares.Then,to investigate the relationship between these seismic wave characteristics and the various qualities.For the computational model of partially saturated soils,chose the sand containing an air-water mixture.The constant elastic values are provided byρs=2650 kgm-3,ρa=1.1 kgm-3,ρw=1000 kgm-3,G=50 MPa,Ka=0.11 MPa,Kw=2.25 GPa,ηa=18×10-6Pa.s,ηw=1×10-3Pa.s,a=10,m=1,n=2,λ=2.2,Sr=0.05,ψr=2000 kPa,andk0=10-11m2.

Fig.5.Reflected waves energy shares and interaction energy as a function of incident direction (θ0) and Poisson’s ratio (ν); γ0=45 0 ,φ=0.4 ,ψ=10000 kPa,Sw=0.9 ,ω=2 π×1000 kHz.

8.2.Numerical discussion

The goal of the above numerical example is to see how different physical attributes (like incident direction,Poisson’s ratio,pore characteristics,water saturation,and matric suction and inhomogeneity angle) affect reflection coefficients and incident energy partitioning between various reflected waves at surfacez=0(stress-free) in partially saturated soils.Figs.1-6 illustrate the reflection coefficients and incident energy distribution at the surfacez=0 with incidence directionθ0∈(0,900).The following is a detailed discussion of figures:

The relationship between absolute values of reflection coeffi-cients and incidence directionθ0for three fixed values of Poisson’s ratio (ν=0.35,0.38,0.4) of soil skeleton is depicted in Fig.1.All the reflection coefficients decrease asνincreases for both boundaries (i.e.,sealed-pore and open-pore) over the whole range ofθ0,except for theP1wave.It means that asνrises,theP1wave can reflect increases at both surfaces (sealed-pore and open-pore).The fluctuations ofνhave a substantial impact on all waves.Furthermore,pore features (i.e.,sealed and open-pore boundaries) have a considerable influence on all waves across the whole range ofθ0except fastSVwave on which it appears after 400.SlowerP2andP3waves have nearly identical variational patterns.The reflection ofP2,P3,andSVwaves at sealed-pore (open-pore) boundary surface is almost non-existent around the grazing (normal) incidence.However,the reflection capability of fasterP1waves is higher at both grazing and normal incidence at both boundaries.

Fig.6.Reflected waves energy shares and interaction energy as a function of incident direction (θ0) and matric suction (ψ); γ0=45 0 ,φ=0.4 ,ν=0.4 ,Sw=0.8 ,ω=2 π×10 0 0 kHz.

For three different values of the matric suction parameter (ψ),Fig.2 depicts the fluctuation of absolute values of distinct reflection coefficients with incidence directionθ0.An increase inψweakens the fasterP1wave for both boundary surfaces,whereas an increase inψstrengthens the slowerSVwave.On all of the waves,there is a strong impact ofψ.For all values ofψ,the variation of theP1wave concerning the incident directionθ0is nearly opposite to theSVwave.Forθ0∈ [00,200),there is no effect ofψon theSVwave.The variation inψat normal incidence (for open-pore boundary) and grazing incidence (for sealed-pore boundary) does not affect the reflection capability of fastP1wave.In contrast,the variation inψat normal (grazing) incidence has no effect on the reflection capability of fastSVwave for both open and sealed-pore(only for a sealed-pore) boundary.

The fluctuation of absolute values of distinct reflection coeffi-cients with incident directionθ0at three fixed values of inhomogeneity direction (γ0=50,450,750) is depicted in Fig.3.Compared to the change inγ0,the variational pattern of fastP1andSVwaves with incident direction is nearly opposite.For the openpore boundary,γ0influences longitudinal wave reflection capabilities afterθ0=600,whereas for the sealed-pore boundary,γ0has a substantial impact on theP1wave in the rangeθ0∈(00,650).

For three different amounts of water saturation (Sw=0.01,0.5,0.99),Fig.4 demonstrates how the absolute values of distinct reflection coefficients vary with the incidence directionθ0.Increases inSwdiminish the fasterP1wave for open-pore boundary surfaces,while increases inSwenhance theSVwave for both boundary surfaces.In the full range ofθ0forSw=0.01,pore characteristics (i.e.,sealed-pore vs.open-pore) have essentially little effect on the fastP1andSVwaves.For both boundaries,the slowerP2wave is strengthened atSw=0.99.In the case of all gas or no water,the precise value ofSw=0 is used.However,the exact valueSw=0 caused certain numerical computation singularities in all gas and no water.As a result,Sw=0.01 is used to represent this instance.If the degree of saturation is near one,air appears as discrete bubbles when the pressure in partially saturated soils lowers.Smeulders [20]clearly explains the concept of gas bubbles in liquid-saturated porous media.When the pore space of the medium is shared equally by water and gas,the slowerP3wave is strengthened for both open and sealed-pore boundaries.The variation in water share in the pore space of partially saturated soils has a substantial impact on all waves.

The fluctuation of energy sharing for three distinct values of Poisson’s ratio (ν=0.35,0.38,0.4) of soil skeleton with an incident angle at the stress-free surface of partially saturated soils is shown in Fig.5.At both boundary surfaces,νhas a significant influence on all energy shares.Pore properties have an impact on all energy sharing across the complete range of incident angles.When the value ofνrises,the energy share of the fastP1(SV)wave rises (falls) across the whole incident angle range at both open and sealed-pore boundary surfaces.It suggests that a larger Poisson’s ratio can strengthen (weaken) the fastP1(SV)wave reflection capability for open and sealed-pore boundaries.The ability of slowerP2andP3waves to reflect increases asνrises.

Fig.6 shows the fluctuation in energy sharing with incidence directionθ0for three distinct values of the matric suction parameter (ψ).With an increase inψ,the fasterP1(SV)wave is weakened (strengthened) for both boundary surfaces.On all of the waves,there is a strong impact ofψ.The variation concerning the incident directionθ0ofP1wave is nearly adverse to theSVwave for all values ofψ.The impact ofψis not seen on theSVwave forθ0∈ [00,150).The variation inψat normal incidence (for openpore boundary) and grazing incidence (for sealed-pore boundary)does not affect the reflection capability of fastP1waves.In contrast,the variation inψat normal incidence has no effect on the reflection capability of fastSVwaves for both open and sealed-pore boundaries.

9.Conclusions

The distinct properties of reflection coefficients and energy sharing at both open-pore and sealed-pore boundary surfaces of partially saturated soils with incidence angles are investigated.The effects of Poisson’s ratio,pore features,water saturation,and matric suction are examined on the reflection coefficients and energy shares.Based on a numerical example,the following are the most relevant observations.

1.The increase of matric suction reduces (enlarges) the reflection ability of fastP1(SV)wave for both open and sealed-pore surfaces.

2.The increase of Poisson’s ratio enlarges (reduces) the reflection ability of fastP1(SV)wave for both open and sealed-pore surfaces.

3.A remarkable impact of pore characteristics is seen on the reflection ability of all waves.

4.A remarkable impact of inhomogeneity direction is observed on the fasterP1andSVwaves.

5.The fasterP1wave is weakened with the increase inSwfor open-pore boundary surface,whileSVwave is strengthened with an addition inSwfor both open and sealed-pore boundary surfaces.

6.The conservation of incident energy is observed at each angle of incidence.

Researchers of oil industries and water resources may find the model helpful.This research reveals some details concerning reservoir product quality.The relationship between seismic attributes and rock characteristics in the crust contributes to developing the ideal mathematical model for studying seismic waves/earthquakes.This mathematical model can also evaluate the internal structure of reservoir rocks in the crust and their hydrological features.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A

whereψ(ψr)represents soil matric suction (residual soil suction),edenotes natural log base,θsrepresents the volumetric water content at saturation,aspecifies the air entry value of the soil,mdescribes the residual water content of the soil,andnidentifies the rate of a degree of saturation.

The effective tortuosity of the water(τw)and air(τa)phases and the tortuosity of a porous mediaτare calculated as follows.