# Semianalytical Solution for Thermal Consolidation of Viscoelastic Marine Clay with the Fractional Order Derivative

Authors: Minjie Wen; Lichen Li; Xinchen Qiu; Yi Tian; Kuihua Wang; Kaifu Liu; Wenbing Wu
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

The deformation property of marine clay under a heat source has received considerable attention in the geotechnical literature. In this paper, a three-parameter fractional order derivative model is introduced into the thermo-hydro-mechanical coupling governing equations with thermal filtration and thermo-osmosis to simulate viscoelastic characteristics of marine clay. The excess pore pressure, temperature increment, and displacement of marine clay are derived by using the Laplace transform method, and the semianalytical solution for the one-dimensional thermal consolidation in the time domain is derived by using a numerical inversion of the inverse Laplace transform. The influence of the order of the fractional derivative, material parameters, and phenomenological coefficient on thermal consolidation is investigated based on the present solutions. It is shown that the influence of the fractional derivative parameter on the excess pore pressure and displacement of marine clay depends on the properties of soil mass, and the temperature increment has an obvious effect on the thermal filtration and thermo-osmosis process.


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## Body

## 1. Introduction

Over the past decades, more and more engineering activities have caused the change in the temperature field of the surrounding soil layers, which inevitably affects the physical and mechanical properties of the strata. These engineering activities involve a broad range of civil engineering topics such as deep geological disposal of radioactive waste [1], deep drilling and excavation [2, 3], extraction of geothermal energy [4–6], energy piles [7, 8], ground improvement using prefabricated vertical thermal drain [9–11], oil and gas pipelines [12], and frictional heating-induced large-scale landslides [13]. This huge engineering demand has stimulated scholars to pay their attention on the thermo-hydro-mechanical coupling theory of porous media, especially the deformation properties of marine clay under a heat source [14].Biot originally proposed the constitutive equations considering the thermal effect and established the theoretical framework for the thermodynamics of saturated porous media [15]. Following Biot’s research, a number of scholars have investigated the thermal consolidation of elastic saturated porous media by considering the influence of the thermal effect. Booker and Savvidou studied the thermal consolidation of saturated soil under the action of the deep buried spherical and point heat source [16, 17]. Bai [18] derived an analytical solution for the wave response of porous materials under cyclic thermal load [18]. Smith and Booker [19] established the linear thermoelastic theory of homogeneous isotropic materials by using the Green function method [19]. Lu et al. [20] investigated the thermal-mechanical coupling response of saturated porous media under simple harmonic heat and load based on the generalized thermoelastic theory [20]. Ai and Wang [21] studied the axisymmetric thermal consolidation of layered elastic saturated porous media under the action of the heat source [21]. Furthermore, Ai et al. [22] studied the axisymmetric thermal consolidation of layered transversely isotropic porous media [22]. Wen et al. [23] studied the thermal dynamic response of a lined circular tunnel in saturated elastic porous media with the help of a fractional order derivative [23].Although the above-mentioned scholars have greatly promoted the development of the thermo-hydro-mechanical coupling theory of porous media, they neglect thermal filtration (the influence of the excess pore pressure gradient on heat flux) and thermo-osmosis (the influence of the temperature gradient on water flux). In reality, by conducting an experiment on thermo-osmosis of water through kaolinite, Srivastava and Avasthi [24] found that the water flux associated with thermo-osmosis in compacted kaolinite can reach 10−8 m/s under a temperature gradient of 20°C/m [24]. Considering the influence of thermal filtration and thermo-osmosis, Zhou et al. [25] presented a fully coupled thermo-hydro-mechanical model [25]. Considering the influence of seepage velocity on thermal diffusion and the temperature gradient on seepage velocity, Liu et al. [26] established a fully dynamic coupled thermo-hydro-mechanical model and investigated the influence of the thermal and water permeability coefficient of saturated porous media around cylindrical holes and spherical cavities on their thermodynamic response, respectively [26, 27]. Yang et al. [25] proposed a refined mathematical model to study the coupled effect of thermo-osmosis in saturated porous media [28]. All these references utilized the elastic theory to simulate stress-strain relationships in saturated porous media, which may not be applicable to the marine clay covering two-thirds of the Earth.With the continuous development of marine development all over the world, more and more attention has been paid to the deformation characteristics of marine clay. Liu et al. [29] studied the one-dimensional consolidation of viscoelastic marine soft soil under load varying with depth and time [29]. Liu et al. [14] investigated the effect of viscosity on the one-dimensional thermal consolidation of marine soft soil [14]. Wang and Wang [30] studied the rheology and thermal consolidation of layered saturated soft soil under force and thermal load [30]. Although these works have promoted the understanding of the deformation characteristics of marine clay, the existing research is still far from being applied to the engineering applications in marine clay due to its complex rheological properties. In order to reasonably consider the rheological characteristics of soil, many scholars introduced the fractional constitutive model to study the consolidation characteristics of soil [31–35]. However, to the authors’ knowledge, there is no report on the thermal consolidation of viscoelastic marine soil with a fractional order derivative.In light of the above, the objective of this paper is to investigate the one-dimensional thermal consolidation of viscoelastic marine clay with the fractional order derivative. By introducing a three-parameter fractional order derivative model to consider thermal filtration and thermo-osmosis, the solutions of excess pore pressure, temperature increment, and displacement of viscoelastic marine clay are obtained by using the Laplace transform method and its numerical inverse transform. Based on the present solutions, the influence of the order of the fractional derivative, material parameters, and phenomenological coefficient on the thermal consolidation of viscoelastic marine clay is investigated.


## 2. Mathematical Modeling

Considering the effects of thermal filtration and thermo-osmosis, a fully coupled thermo-hydro-mechanical model was proposed by Zhou et al. [25]. On this basis, a three-parameter fractional order derivative model, consisting of two springs and one dashpot in parallel, is introduced into the thermo-hydro-mechanical coupling governing equations. The constitutive relationship of the fractional order derivative model is expressed as [36](1)1+τεαDασ′z,t=1+τσαDαλ+2G∂uz,t∂z.Here, σ′z,t denotes the vertical effective stress; λ and G are Lame constants; z represents the depth below the ground surface; t is time; τε, and τσ are material parameters; and α is the fractional order parameter which can be obtained by the triaxial test and parameter inversion method, and 0<α<1; uz,t is the displacement in the z direction. Dα=dα/dtα denotes α order Riemann–Liouville fractional derivative, and it is defined as [36](2)Dαxt=1Γ1−αddt∫0txτt−ταdτ.Here, Γφ=∫0∞tφ−1e−φdt is the Gamma function.As shown in Figure1, both the top and bottom surfaces are set as permeable. Instantaneous compressive stress q0 and temperature T1 are applied at the top surface, and a constant temperature T0 is maintained at the bottom surface.Figure 1
Mathematical model.The viscoelastic characteristics of marine clay are simulated by the three-parameter fractional order derivative model, the equilibrium equation is given by [25](3)ξp+K′βT−σ′=q0.Here, ξ=1−K′/Ks depends on the compressibility of the soil grains; K′ represents the drained bulk modulus of the soil medium; Ks denotes the bulk modulus of the soil grains; β is the coefficient of volumetric expansion of the soil medium and T is the temperature increment; and p is excess pore pressure.Substituting equation (1) into equation (3) yields(4)ξp+K′βT−1+τσαDα1+τεαDαλ+2G∂u∂z=q0.The fluid mass balance is governed by(5)k∂2p∂z2+Sw∂2T∂z2=c1∂2u∂z∂t−c2∂T∂t+c3∂p∂t.Here, k denotes the coefficient of permeability; Sw is a phenomenological coefficient associated with the influence of the thermal gradient on the water flux (thermo-osmosis); the parameters c1, c2, and c3 are expressed as c1=1−K′/Ks, c2=naw+1−nas−βK′/Ks, and c3=n/βw−1−n/Ks, respectively; n is the porosity; aw and as are volumetric thermal expansion coefficients of the pore water and soil grains, respectively; and βw is the bulk modulus of pore water.The thermal energy balance equation is given as(6)Cv∂T∂t−T0K′β∂2u∂z∂t=K−T0awβwSw∂2T∂z2+T0Sw−awβwk∂2p∂z2,where Cv is the volumetric specific heat of the soil medium; Cv=1−nρsCs+nρwCw; ρs is the density of soil grains; ρw is the density of water; Cs and Cw denote the gravimetric specific heats of soil grains and the pore water, respectively; and K=1−nλs+nλw is the thermal conductivity of the soil medium, in which λs and λw are the thermal conductivities of the soil grain and the pore water.The initial conditions of the problem are as follows:(7)pz,0=q0,uz,0=0,Tz,0=0.The boundary conditions of the problem are as follows:(8)p0,t=0.(9)pH,t=0.(10)uH,t=0.(11)T0,t=T1.(12)TH,t=0.


## 3. Solutions to Governing Equations

Laplace transform is introduced to solve governing equations (4)–(6), and the following Laplace transform equation is defined:(13)sγf¯s=∫0∞∂γft∂tγe−stdt.Applying Laplace transforms to equations (4)–(6), one can obtain transformed governing equations as(14)ξp¯+K′βT¯−1+τσαsα1+τεαsαλ+2G∂u¯∂z=q0s.(15)k∂2p¯∂z2+Sw∂2T¯∂z2=sc1∂u¯∂z−sc2T¯+c3sp¯−q0.(16)CvsT¯−T0K′βs∂u¯∂z=K−T0awβwSw∂2T¯∂z2+T0Sw−awβwk∂2p¯∂z2.Here, s is the Laplace transform parameter; p¯=∫0∞e−stpdt, T¯=∫0∞e−stTdt, and u¯=∫0∞e−studt.The Laplace transform of equations (8)–(12) with respect to the time variable t is derived as(17)p¯z=0=0.(18)p¯z=H=0.(19)u¯z=H=0.(20)T¯z=0=T1s.(21)T¯z=H=0.Substituting equation (15) into equation (16) yields(22)a1∂2T¯∂z2+a2∂2p¯∂z2+a3sT¯+a4sp¯−a4q0=0.Here, a1=−T0K′βSw/c1−K+T0awβwSw; a2=−T0K′βk/c1−T0Sw−awβwk; a3=Cv−T0K′βc2/c1; and a4=T0K′βc3/c1.Substituting equation (15) into equation (14) gives(23)b1∂2T¯∂z2+b2∂2p¯∂z2+b3sT¯+b4sp¯+b5q0=0,where b1=1+τσαsα/1+τεαsαλ+2GSw/c1; b2=1+τσαsα/1+τεαsαλ+2Gk/c1; b3=1+τσαsα/1+τεαsαλ+2Gc2/c1−K′β; b4=−ξ−1+τσαsα/1+τεαsαλ+2Gc3/c1; and b5=1+τσαsα/1+τεαsαλ+2Gc3/c1+1.Further transformation of equations (22) and (23) leads to(24)T¯=h1s∂2p¯∂z2+h2p¯+h3q0s,where h1=a2b1−a1b2/a1b3−a3b1; h2=a4b1−a1b4/a1b3−a3b1; and h3=−a4b1+a1b5/a1b3−a3b1.Substituting equation (24) into equation (22) yields(25)g1∂4p¯∂z4+g2s∂2p¯∂z2+g3s2p¯+sa1a3h3−a4q0=0,where g1=h1; g2=h2+a2/a1+a3h1/a1; and g3=a3h2+a4/a1.The general solution of equation (25) is derived as(26)p¯=A1eγ1z+A2e−γ1z+B1eγ2z+B2e−γ2z−a3h3−a4a1g3sq0,where A1, A2, B1, B2 are undetermined coefficients.γ12=−g2−g22−4g1g3/2g1s;γ22=−g2+g22−4g1g3/2g1sSubstituting equation (26) into equation (24) gives(27)T¯=h1γ12s+h2A1eγ1z+A2e−γ1z+h1γ22s+h2B1eγ2z+B2e−γ2z−h2a3h3−a4a1g3sq0+h3q0s.Substituting equation (26) into equations (17), (18), (20), and (21), the undermined coefficients can be obtained as(28)A1=−sϕ1−c22ϕ01−e−γ1H+T1e−γ1Hsc11−c22eγ1H−e−γ1H,A2=sϕ1−c22ϕ01−eγ1H+T1eγ1Hsc11−c22eγ1H−e−γ1H,B1=sϕ1−c11ϕ01−e−γ2H+T1e−γ2Hsc11−c22eγ2H−e−γ2H,B2=−sϕ1−c11ϕ01−eγ2H+T1eγ2Hsc11−c22eγ2H−e−γ2H,where c11=h1γ12/s+h2, c22=h1γ22/s+h2, φ0=−a3h3−a4/a1g3sq0, and φ1=−h2a3h3−a4/a1g3s−h3/sq0.Then, equation (26) can be rewritten as(29)p¯=−θ1sinhγ1z−T1−θ1sinhγ1H−z−T0sinhγ1zh1γ12−γ22sinhγ1H+θ2sinhγ2z+θ2−T1sinhγ2H−z−T0sinhγ2zh1γ12−γ22sinhγ2H+ϕ0.where, θ1=sφ1−φ0c22 and θ2=sφ1−φ0c11.Substituting equations (26) and (27) into equation (14) yields(30)∂u¯∂z=χK′βh1γ12s+h2+ξA1eγ1z+A2e−γ1z+χK′βh1γ22s+h2+ξB1eγ2z+B2e−γ2z+χq0−ξa3h3−a4a1g3s−K′βh2a3h3−a4a1g3s+K′βh3−1s,where χ=1/λ+2G1+τϵαsα/1+τσαsα.Integrating both sides of equation (30) from zero to infinity gives(31)u¯=χγ1K′βh1γ12s+h2+ξA1eγ1z−A2e−γ1z+χγ2K′βh1γ22s+h2+ξB1eγ2z−B2e−γ2z+χq0−ξa3h3−a4a1g3s−K′βh2a3h3−a4a1g3s+K′βh3−1sz+D1.Substituting equation (31) into equation (19), the coefficient D1 can be obtained as(32)D1=−χγ1K′βh1γ12s+h2+ξA1eγ1H−A2e−γ1H−χγ2K′βh1γ22s+h2+ξB1eγ2H−B2e−γ2H−χq0−ξa3h3−a4a1g3s−K′βh2a3h3−a4a1g3s+K′βh3−1sH.Through the above derivation, the analytical expressions of excess pore pressure, temperature increment, and displacement can be obtained accordingly. However, it is difficult to directly obtain the analytical solutions in the transformed space for some formulas are difficult to integrate. Therefore, it is necessary to utilize the numerical method to program and analyze the solution process in this paper. Currently, there exist many Laplace inverse transform methods, among which the Crump’s numerical inversion results of Laplace transform is the most accurate [10].Assuming thatF (s) is the Laplace transform of function F (t), Crump’s inversion algorithm of the Laplace inverse transform can be written as(33)Ft≈eatT∗12Fa+∑m=1500ReFa+mπiT∗cosmπtT∗−ImFa+mπiT∗sinmπtT∗,where T∗>t/2. If Ft<Meαt, the error ς≤Meυe−2T∗a−υ.


## 4. Numerical Results and Discussion

Having illustrated and reviewed the obtained theoretical results in the preceding section, we now present some numerical results. Following the works of Liu et al. [14] and Zhou et al. [25], the parameters of marine clay in the calculation can be set as λ=6×106 Pa, G=8×106, n=0.25, Sw=6.0×10−11 m2/s/°C, K′=2.78×106 Pa, k=1.0×10−14 m5/J/s, λw=0.582 J/m/s/°C, λs=3.29 J/m/s/°C, Ks=59×109 Pa, βw=3.3×109 Pa, ρw=1000 kg/m3, ρs=2610 kg/m3, Cw=4186 J/kg/°C, Cs=937 J/kg/°C, as=3.0×10−6°C−1, aw=3.0×10−4°C−1, τε=1×1012Pa⋅s, q0=0 Pa, T1=50°C, and T0=23°C.


### 4.1. Comparative Analysis

The Kelvin viscoelastic model is combined with the thermo-hydro-mechanical coupling governing equations for marine clay, and the influence of viscosity on one-dimensional thermal consolidation was investigated by Liu et al. [14]. In addition, Zhou et al. [25] proposed a fully coupled thermo-hydro-mechanical model with thermal filtration and thermo-osmosis. In this paper, for the purpose of verifying and presenting the differences of the three cases of the fractional derivative viscoelastic model (FDVM), Kelvin viscoelastic model (KVM), and elastic model (EM), the order of the fractional derivative α is assumed to be zero for the comparison of the FDVM with the EM, and the material parameter τεα is assumed to be zero, and α is equal to 1 for the comparison of the FDVM with the KVM. Figures 2–4 illustrate the distribution of excess pore pressure, displacement, and temperature increment against depth among solutions of Zhou et al. [25], Liu et al. [14], and present work. It can be found that the KVM produces the largest excess pore pressure, followed by the results of the FDVM and EM. At depth above 0.6 m, the displacement value produced by the EM is the largest, followed by those of the FDVM and KVM. At larger depth, remarkable increase of the displacement value could be observed for all the three models. For the distribution of temperature, the three models produce similar results.Figure 2
Comparison of excess pore pressure among solutions of Zhou et al. [25], Liu et al. [14], and present work.Figure 3
Comparison of displacement among solutions of Zhou et al. [25], Liu et al. [14], and present work.Figure 4
Comparison of temperature increment among solutions of Zhou et al. [25], Liu et al. [14], and present work.


### 4.2. Influence of the Order of the Fractional Derivativeα

Figures5 and 6 show the influence of the parameter α on the development of excess pore pressure when the values of τσ/τε are 3.0 and 0.5, respectively. When τσ/τε=3.0, the excess pore pressure increases with the increase of α before the peak value. After the peak, the increase of α seems to accelerate the dissipation of the excess pore pressure. When τσ/τε=0.5, an opposite trend could be observed. With increasing α, the peak value of excess pore pressure shows a decreasing trend, and the after-peak dissipation rate is smaller. It can be seen that the influence of the fractional derivative parameter α on the development of excess pore pressure depends on the characteristics of the soil mass.Figure 5
Influence of the order of the fractional derivative on excess pore pressure whenτσ/τε=3.Figure 6
Influence of the order of the fractional derivative on excess pore pressure whenτσ/τε=0.5.Figures7 and 8 show the influence of the parameter α on the displacement development under the same conditions. Similar trends to that of the excess pore pressure can be observed. Another finding is that the influence of α is more obvious at τσ/τε=3.0. It can be seen that the soil property would influence the impact of the fractional derivative parameter on the development of displacement.Figure 7
Influence of the order of the fractional derivative on displacement whenτσ/τε=3.Figure 8
Influence of the order of the fractional derivative on displacement whenτσ/τε=0.5.


### 4.3. Influence of Material Parametersτσ/τε

Figures9 and 10 depict the influence of τσ/τε variance on the development of excess pore pressure and displacement with α=0.5. With increasing τσ/τε, both the peak value of the excess pore value and the after-peak dissipation rate are much larger, while the net maximum displacement value decreases. The influence of τσ/τε on the trend of displacement development is less obvious.Figure 9
Influence of the material parameterτσ/τε on excess pore water pressure when α=0.5.Figure 10
Influence of the material parameterτσ/τε on displacement when α=0.5.The influence of embedment depth on the development of excess pore pressure and displacement is presented in Figures11–14. As shown in Figures 11 and 12, the magnitude of excess pore pressure at τσ/τε=3 is much larger than that at τσ/τε=0.5. On the other hand, the development trend of excess pore pressure at different depths seems to be uninfluenced with the variance of τσ/τε, but the peak excess pore pressure decreases with depth. As depicted in Figures 13 and 14, similar trends could be observed for the development of the displacement value. With increasing depth, the displacement value is much smaller.Figure 11
Influence of embedment depth on pore water pressure whenτσ/τε=3.Figure 12
Influence of embedment depth on pore water pressure whenτσ/τε=0.5.Figure 13
Influence of embedment depth on displacement whenτσ/τε=3.Figure 14
Influence of embedment depth on displacement whenτσ/τε=0.5.


### 4.4. Influence of the Phenomenological CoefficientSw

The influence of the phenomenological coefficientSw on the development of excess pore pressure, displacement, and temperature increment is presented in Figures 15–17. It is clear that both the excess pore pressure and displacement increase with increasing Sw. On the other hand, the variance of Sw seems to have no impact on the development of temperature increment. This parametric analysis indicates that the thermal gradient has an obvious effect on the thermal filtration and thermo-osmosis process.Figure 15
Influence of the phenomenological coefficientSw on excess pore pressure.Figure 16
Influence of the phenomenological coefficientSw on displacement.Figure 17
Influence of the phenomenological coefficientSw on temperature increment.


## 4.1. Comparative Analysis

The Kelvin viscoelastic model is combined with the thermo-hydro-mechanical coupling governing equations for marine clay, and the influence of viscosity on one-dimensional thermal consolidation was investigated by Liu et al. [14]. In addition, Zhou et al. [25] proposed a fully coupled thermo-hydro-mechanical model with thermal filtration and thermo-osmosis. In this paper, for the purpose of verifying and presenting the differences of the three cases of the fractional derivative viscoelastic model (FDVM), Kelvin viscoelastic model (KVM), and elastic model (EM), the order of the fractional derivative α is assumed to be zero for the comparison of the FDVM with the EM, and the material parameter τεα is assumed to be zero, and α is equal to 1 for the comparison of the FDVM with the KVM. Figures 2–4 illustrate the distribution of excess pore pressure, displacement, and temperature increment against depth among solutions of Zhou et al. [25], Liu et al. [14], and present work. It can be found that the KVM produces the largest excess pore pressure, followed by the results of the FDVM and EM. At depth above 0.6 m, the displacement value produced by the EM is the largest, followed by those of the FDVM and KVM. At larger depth, remarkable increase of the displacement value could be observed for all the three models. For the distribution of temperature, the three models produce similar results.Figure 2
Comparison of excess pore pressure among solutions of Zhou et al. [25], Liu et al. [14], and present work.Figure 3
Comparison of displacement among solutions of Zhou et al. [25], Liu et al. [14], and present work.Figure 4
Comparison of temperature increment among solutions of Zhou et al. [25], Liu et al. [14], and present work.


## 4.2. Influence of the Order of the Fractional Derivativeα

Figures5 and 6 show the influence of the parameter α on the development of excess pore pressure when the values of τσ/τε are 3.0 and 0.5, respectively. When τσ/τε=3.0, the excess pore pressure increases with the increase of α before the peak value. After the peak, the increase of α seems to accelerate the dissipation of the excess pore pressure. When τσ/τε=0.5, an opposite trend could be observed. With increasing α, the peak value of excess pore pressure shows a decreasing trend, and the after-peak dissipation rate is smaller. It can be seen that the influence of the fractional derivative parameter α on the development of excess pore pressure depends on the characteristics of the soil mass.Figure 5
Influence of the order of the fractional derivative on excess pore pressure whenτσ/τε=3.Figure 6
Influence of the order of the fractional derivative on excess pore pressure whenτσ/τε=0.5.Figures7 and 8 show the influence of the parameter α on the displacement development under the same conditions. Similar trends to that of the excess pore pressure can be observed. Another finding is that the influence of α is more obvious at τσ/τε=3.0. It can be seen that the soil property would influence the impact of the fractional derivative parameter on the development of displacement.Figure 7
Influence of the order of the fractional derivative on displacement whenτσ/τε=3.Figure 8
Influence of the order of the fractional derivative on displacement whenτσ/τε=0.5.


## 4.3. Influence of Material Parametersτσ/τε

Figures9 and 10 depict the influence of τσ/τε variance on the development of excess pore pressure and displacement with α=0.5. With increasing τσ/τε, both the peak value of the excess pore value and the after-peak dissipation rate are much larger, while the net maximum displacement value decreases. The influence of τσ/τε on the trend of displacement development is less obvious.Figure 9
Influence of the material parameterτσ/τε on excess pore water pressure when α=0.5.Figure 10
Influence of the material parameterτσ/τε on displacement when α=0.5.The influence of embedment depth on the development of excess pore pressure and displacement is presented in Figures11–14. As shown in Figures 11 and 12, the magnitude of excess pore pressure at τσ/τε=3 is much larger than that at τσ/τε=0.5. On the other hand, the development trend of excess pore pressure at different depths seems to be uninfluenced with the variance of τσ/τε, but the peak excess pore pressure decreases with depth. As depicted in Figures 13 and 14, similar trends could be observed for the development of the displacement value. With increasing depth, the displacement value is much smaller.Figure 11
Influence of embedment depth on pore water pressure whenτσ/τε=3.Figure 12
Influence of embedment depth on pore water pressure whenτσ/τε=0.5.Figure 13
Influence of embedment depth on displacement whenτσ/τε=3.Figure 14
Influence of embedment depth on displacement whenτσ/τε=0.5.


## 4.4. Influence of the Phenomenological CoefficientSw

The influence of the phenomenological coefficientSw on the development of excess pore pressure, displacement, and temperature increment is presented in Figures 15–17. It is clear that both the excess pore pressure and displacement increase with increasing Sw. On the other hand, the variance of Sw seems to have no impact on the development of temperature increment. This parametric analysis indicates that the thermal gradient has an obvious effect on the thermal filtration and thermo-osmosis process.Figure 15
Influence of the phenomenological coefficientSw on excess pore pressure.Figure 16
Influence of the phenomenological coefficientSw on displacement.Figure 17
Influence of the phenomenological coefficientSw on temperature increment.


## 5. Conclusion

In this paper, a three-parameter fractional order derivative model is proposed based on the improved thermo-hydro-mechanical coupling theory by accounting for the viscoelastic behavior of marine clay. One-dimensional thermal consolidation of viscoelastic marine clay is analyzed, considering the thermal filtration and thermo-osmosis process. Solutions of excess pore pressure, temperature increment, and displacement are obtained by using the Laplace transform method and its numerical inverse transform. The influence of the order of the fractional derivative, material parameters, and phenomenological coefficient on the characteristics of thermal consolidation of marine clay is investigated. The performance of the proposed fractional derivative viscoelastic model (FDVM), Kelvin viscoelastic model (KVM), and elastic model (EM) is compared. The calculated excess pore pressure values from the three models are quite different, while the development of temperature increment seems to be uninfluenced by the selected model. For the development of the displacement value, there is an obvious turn of the displacement value for different models at the depth of 0.6 m. Further parametric analysis indicates that the influence of the fractional derivative parameter on the development of excess pore pressure and displacement depends on the properties of the soil mass, and the temperature increment has an obvious effect on the thermal filtration and thermo-osmosis process.



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