Poroelastic Effects of Chemical Loading in Consolidation of Residual Soils

Poroelastic Effects of Chemical Loading in Consolidation of Residual Soils | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Poroelastic Effects of Chemical Loading in Consolidation of Residual Soils Dada Irheren, Michael Ebie Onyia, Fidelis Onyebuchi Okafor This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3992157/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract An analytical chemo-hydromechanical model of residual soils was presented in this study. The modeling procedure utilized a unique phenomenological approach, which highlighted some of the macro-mechanical influences responsible for the behavior of residual soils. The macromechanical analysis yielded an extended poroelastic theory. The developed model was applied in solving a typical civil engineering problem of soil consolidation. For a chemical loading treatment, the problem was using requisite boundary conditions. Thereafter, the mathematical software “Mathematica” was used for solving the ensuing diffusion equations using physico-chemical properties typical of a residual soil sample. Even though the chemical loading was instantaneous, it took some time before its effect could be felt everywhere according to the poroelastic theory. From obtained results, the evolution of the generated pore pressure showed increasing segments of the soil returning to their initial state after the passage of the pore pressure front. The vertical displacement showed a slight increase or elevation of the soil surface. The flux of water flow in the soil was initially positive before turning negative, which can be explained by initial successful penetration of the infiltrating liquid until it met increasingly tortuous paths as result of lower permeability. The water volume (per unit area) leaving the soil after chemical loading demonstrated that a portion of the water flux associated with the infiltrating liquid ends up leaving the soil through the sides. poroelasticity chemical loading soil consolidation residual soils Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Introduction The properties of residual soils can vary greatly from those of transported soils, as they often do not conform to the traditional definition of soil grains or particle size. Instead, they are typically composed of aggregates or weathered minerals that can break down and become more compact when manipulated or compacted. This means that what may initially appear to be coarse sandy gravel in its natural state can actually break down into fine sandy silt when excavated, mixed, and compacted (Rahman et al., 2018). The relationship between the permeability and granulometry of a transported soil, as seen in the well-known Hazen formula, is not always applicable to residual soils. In these cases, the permeability is largely determined by the soil's micro- and macro-fabric, jointing, and any remaining or added features, such as slickensides, termite channels, or other biological channels. This same principle can also be observed in properties like particle unit weight or specific gravity Gs, where the value of Gs can vary depending on the level of comminution of the sample. Generally, smaller particles have a higher Gs value due to their increased solidity. Another difference between residual and transported soils is the classification scheme for classifying soils. Residual soils possess distinct traits that are not fully accounted for in traditional soil classification methods intended for transported soils like the Unified Soil Classification System (USCS). The unique clay mineralogy found in certain residual soils presents features that do not align with the standard characteristics of the soil group it belongs to, as recognized in current systems such as the USCS (Wibawa et al., 2018 ). Existing soil classification systems, such as the USCS, are not sufficient to describe the range of materials present in a soil mass due to weathering. Depending on the level of weathering, the in situ materials can vary from soil to soft rock. These systems are designed to classify transported soils and focus on properties in a remoulded state, which is not representative of residual soils. The properties of these soils are heavily influenced by the fabric and structural characteristics inherited from the original rock mass, or formed as a result of weathering. The impact of oven-drying and air-drying on soil properties is widely recognized, although it is typically minimal for soils that have been transported. The primary consequence of drying is a decrease in the reported proportion of clay particles (finer than 2 µm). Residual soils, which are formed through slow decomposition in an anaerobic setting, are especially susceptible to changes in properties when exposed to drying and air. This process can lead to dehydration of clay minerals and irreversible alterations in their properties. Even with prolonged maturation after re-wetting, air-drying at normal temperatures can cause irreversible changes in residual soils. In addition to the well-documented effects on index properties, drying also impacts the composition, compressibility, and shear strength characteristics of residual soils (Blight & Leong, 2012 ). The presence of individual particle assemblages (IPAs) in residual soils makes dispersion an essential requirement for particle size analysis. This was demonstrated by Rodriguez ( 2005 ) through an extreme instance of lateritic soil particle size analysis. Without dispersion, the particle size distribution resembles that of silty sand with no clay content. However, with dispersion, it transforms into sandy clay with a significantly high clay content of 56%. This serves as a clear indication of the significance of dispersion when dealing with residual soils, as opposed to transported soils (Duarte & Rodrigues, 2018). The structure and mineralogy of residual soils can be highly diverse. These variations can have a significant impact on their engineering properties, especially when specific microstructural or mineralogical traits are identified. It is crucial to keep in mind that residual soils may exhibit much better engineering behavior than what is indicated by index tests on remoulded soil. This is particularly true when considering correlations between index tests and the engineering behavior of transported soils, as noted by Zhang et al. ( 2007 ). Residual soil research has been undertaken by a number of researchers in the past, including Townsend ( 1985 ), Duarte ( 2002 ), Weslye (2010), Blight & Leong ( 2012 ), Wibawa et al. ( 2018 ), Duarte & Rodrigues (2018) etc. However, these studies have largely been far in between and comparatively small when compared to the more ubiquitous transported soils (Blight & Leong, 2012 ). The few available research have mostly studied only the geological (Vaughan, 1988 ; Duarte, 2002 ), engineering (Brand & Phillipson, 1985 ) and geotechnical (Townsend, 1985 ; Duarte, 2002 ) characteristics of residual soils in general. There appears a dearth of research specifically targeted towards modeling of residual soil. Therefore, in this study, the poroelastic effects of chemical loading in consolidation of residual soils will be investigated. Since chemical loading with an aqueous liquid can be useful in understanding the effect of dehydration on residual soils. This represents a break from the usual modeling studies that basically consider only hydraulic and mechanical loading conditions. Model Formulation Macromechanical Stress-Strain Relations The porochemoelastic expression for a chemohydromechanical treatment can be given as, $$\frac{{\sigma }_{kk}}{3}=Ke-\alpha p-{\alpha }_{c}{{\beta }_{h}^{s}C}_{r}$$ 1 Introducing the deviatoric stress component to account for both volumetric and shear responses, $$\frac{{\sigma }_{kk}}{3}+\left({\sigma }_{ij}-\frac{1}{3}{\delta }_{ij}{\sigma }_{kk}\right)=Ke-\alpha p-{\alpha }_{c}{\beta }_{h}^{s}{C}_{r}+2G\left({e}_{ij}-\frac{1}{3}{\delta }_{ij}e\right)$$ 2 Simplifying, $${\sigma }_{ij}=Ke-\alpha p-{\alpha }_{c}{\beta }_{h}^{s}{C}_{r}+2G\left({e}_{ij}-\frac{1}{3}{\delta }_{ij}e\right)$$ 3 $${\sigma }_{ij}=\left(K-\frac{2G}{3}\right){\delta }_{ij}e+2G{e}_{ij}-\alpha p-{\alpha }_{c}{{\beta }_{h}^{s}C}_{r}$$ 4 In addition, the equation for pore pressure can be expressed as Cheng ( 2016 ): $$p=M\left(\zeta +\alpha e+{\beta }_{c}{C}_{r}\right)$$ 5 Where \(\zeta\) is the variation in fluid content \(\alpha\) is Biot’s effective stress parameter \({\beta }_{c}\) is a new parameter quantifying the contribution of chemical strain to the total pore pressure Balance Laws The balance laws account for the conservation of mass, force and energy in consonance with classical thermodynamic laws. Force Equilibrium Following Mehrabian et al. ( 2020 ), the compressive geologic stresses and the related physico-chemical interactions far outweigh the effects of body forces like gravity, so that the following equilibrium equation applies. $${\sigma }_{ij,j}=0$$ 6 Which reduces to: $$\frac{\partial {\sigma }_{zz}}{\partial z}=0$$ 7 Or for a one-dimensional problem, $$\frac{d{\sigma }_{zz}}{dz}=0$$ 8 Fluid Mass Balance The fluid mass balance ensures the conservation of the total fluid moving into and out of the fluid-filled porous medium at all times. Mathematically, this can be expressed as (Kanfar et al. ,2017): \(\) $$\frac{\partial \zeta }{\partial t}+\nabla .\overrightarrow{q}=0$$ 9 Solute Mass Balance While the solute mass balance accounts for the movement of solute species, which is primarily responsible for chemical interactions. According to Cheng ( 2016 ), the expression for solute mass balance can be written as: $$\varphi \frac{\partial {C}_{k}}{\partial t}+\nabla .{\overrightarrow{J}}_{s}=0$$ 10 Where \({C}_{d}\) is the infiltrating liquid solute (salt) molar concentration \({J}_{s}\) is the diffusive mass flux (mass crossing per unit area of the porous medium per unit time) of chemical species. Dividing both sides by the soil solute concentration, we have: $$\frac{\varphi }{{C}_{s}}\frac{\partial {C}_{d}}{\partial t}+\frac{\nabla .{\overrightarrow{J}}_{s}}{{C}_{s}}=0$$ 11 Or $$\varphi \frac{\partial {C}_{r}}{\partial t}+\frac{1}{{C}_{s}}\nabla .{\overrightarrow{J}}_{s}=0$$ 12 Where \({C}_{r}=\frac{{C}_{d}}{{C}_{s}}\) Transport Laws The transport laws represent the mathematical expressions relating the possible types of fluxes in the physico-chemical system under study to their respective driving forces. The occurrence of multiple irreversible transport processes, such as heat conduction, electrical conduction, and mass diffusion, in a thermodynamic system can lead to interference and impact the system's overall behavior (Gao et al., 2021 ). Following Onsager’s non-equilibrium thermodynamics, the different fluxes can be expressed as follows: $$\overrightarrow{q}=-\kappa \nabla p+{C}_{s}{k}_{pc}\nabla {C}_{r}$$ 13 $${\overrightarrow{J}}_{s}={k}_{cp}\nabla p-{C}_{s}{\left(1-\mathfrak{R}\right)D}_{c}\nabla {C}_{r}$$ 14 The symmetry of the coefficients in the above equation is particularly noticeable. κ, the permeability coefficient (mobility) of Darcy’s law, and R, the reflection coefficient representing the strength of the counter flow induced by the osmotic pressure gradient, are both observed. The reflection coefficient is a dimensionless parameter, with a range of 0 to 1 (0 < \(\mathfrak{R}\) < 1,), where 0 represents a chemically inert rock with no selective filtration of solute and solvent, and 1 represents the ideal ion exclusion membrane where the concentration gradient drives a full osmotic pressure. The second term on the right hand side of the equation does not induce a solvent flow in the lower bound case, while the upper bound case exerts a full osmotic pressure due to the concentration gradient. Fluid Diffusion Equation Substituting the fluid flux equation into the fluid balance equation we have, $$\frac{\partial \zeta }{\partial t}+\nabla .(-\kappa p+{C}_{s}{k}_{pc}\nabla {C}_{r})=0$$ 15 Where \({k}_{pc}=\mathfrak{R}\kappa R{T}_{o}\) $$\frac{\partial \zeta }{\partial t}-\kappa {\nabla }^{2}p+{C}_{s}{k}_{pc}{\nabla }^{2}{C}_{r}=0$$ 16 But, recall that, $$p=M\left(\zeta +\alpha e+{\beta }_{c}{C}_{r}\right)$$ And making \(\zeta\) the subject, $$\zeta =\frac{p}{M}-\left(\alpha e+{\beta }_{c}{C}_{r}\right)$$ 17 Substituting the above equation into the fluid diffusion equation, $$\frac{1}{M}\frac{\partial p}{\partial t}-\alpha \frac{\partial e}{\partial t}-{\beta }_{c}\frac{\partial {C}_{r}}{\partial t}-\kappa {\nabla }^{2}p+{C}_{s}{k}_{pc}{\nabla }^{2}{C}_{r}=0$$ 18 Multiplying both sides by \(M\) and rearranging, $$\frac{\partial p}{\partial t}-\kappa {M\nabla }^{2}p+{C}_{s}{k}_{pc}M{\nabla }^{2}{C}_{r}=\alpha M\frac{\partial e}{\partial t}+{\beta }_{c}M\frac{\partial {C}_{r}}{\partial t}$$ 19 The above equation is quite symbolic, showing the fact that the fluxes serve as the driving force, while the terms on the RHS depict their respective source terms. So, similar to the aforementioned transport laws, the above equation also highlights the effects of chemical interactions on the total pore pressure (hydraulics) of the system, involving osmotic effects (Gao et al., 2021 ). Solute Diffusion Equation Substituting the solute flux equation into the solute balance equation we have, $$\varphi \frac{\partial {C}_{r}}{\partial t}+\frac{1}{{C}_{s}}\nabla .\left\{{k}_{cp}\nabla p-{C}_{s}{\left(1-\mathfrak{R}\right)D}_{c}\nabla {C}_{r}\right\}=0$$ 20 $$\varphi \frac{\partial {C}_{r}}{\partial t}+\frac{1}{{C}_{s}}\left\{{k}_{cp}{\nabla }^{2}p-{C}_{s}{\left(1-\mathfrak{R}\right)D}_{c}{\nabla }^{2}{C}_{r}\right\}=0$$ 21 $$\varphi \frac{\partial {C}_{r}}{\partial t}+\frac{{k}_{cp}}{{C}_{s}}{\nabla }^{2}p-{\left(1-\mathfrak{R}\right)D}_{c}{\nabla }^{2}{C}_{r}=0$$ 22 Where \({k}_{cp}=\left(1-\mathfrak{R}\right){C}_{d}\kappa\) And \(\mathfrak{R}\) is the osmotic reflection coefficient., \({C}_{d}\) is the infilterating liquid solute concentration and k is the soil permeability Similarly, the above equation highlights the effects of hydraulic interactions on the chemical stress as shown by the fluxes terms on the LHS (Cheng, 2016 ). The above equations in the two unknowns, \(p\left(z,t\right) and {C}_{r}\left(z,t\right)\) suffice for a well-posed mathematical problem, since the stresses and strains are equally well defined in terms of the variables. In this study, Wolfram’s Mathematica software will be utilized in solving the obtained partial differential equations above, given the appropriate constants. The list of parameters and their values which is gotten from the literature is given in Table 3.1 below. In this case study, a residual soil sample is assumed with the relevant properties. Table 3.1 Physico-chemical constants for a typical residual soil used for numerical analysis Parameter Symbol Value Source Shear modulus (Pa) G 1853.21 Liu & Abousleiman ( 2018 ) Biot’s effective stress parameter (-) \(\alpha\) 0.812939 Calculated Biot effective chemical stress coefficient (-) \({\alpha }_{c}\) 0.2673 Calculated Soil solute concentration (mol/m 3 ) \({C}_{s}\) 2.5 \(\times\) 10 3 Cheng ( 2016 ) Infiltrating fluid solute concentration (mol/m 3 ) \({C}_{d}\) 5 \(\times\) 10 3 Cheng ( 2016 ) Matrix bulk modulus (Pa) K 5.34995 \(\times\) 10 8 Calculated Effective solid bulk modulus (Pa) \({K}_{\alpha }\) 2.86 \(\times\) 10 9 Cheng ( 2016 ) Bulk modulus of porosity (Pa) \({K}_{\varphi }\) 1.49 \(\times\) 10 9 Cheng ( 2016 ) Bulk modulus of microinhomogeneity (Pa) \({K}_{\psi }\) 0 Cheng ( 2016 ) Biot’s modulus (Pa) \(M\) 6 \(\times\) 10 9 Cheng ( 2016 ) Permeability coefficient (m 4 /N.s) \(\kappa\) 3 \(\times\) 10 −2 Cheng ( 2016 ) Chemo-osmotic coefficient \({k}_{pc}\) 6.0 \(\times\) 10 −11 Calculated Chemical strain parameter \({\beta }_{c}\) 0.95 Calculated Chemo-hydraulic parameter (N/m 2 ) \({\beta }_{h}^{s}\) 1.0 \(\times\) 10 −4 Calculated Matrix porosity \(\varphi\) 0.3 Ezendiokwere et al. ( 2021 ) Reflection coefficient \(\mathfrak{R}\) 0.3 Cheng ( 2016 ) Apparent mass diffusion coefficient (m 2 /s) \({D}_{c}\) 6.0 \(\times\) 10 −9 Cheng ( 2016 ) Application in Soil Consolidation Governing Equations To solve mathematical problems, we can remove variables from the constitutive relations, equilibrium and continuity equations, and flux laws to create a set of partial differential equations that can be used to solve initial and boundary value problems. In our research, we presume that the hyperfiltration effect is minimal, indicating that the movement of solute concentration is influenced solely by the concentration gradient and not the pressure gradient. This is the same as assuming that k cp = 0. This particular model is widely used for practical purposes, and the majority of available analytical solutions are based on this model. Navier Equation The Navier-type equation for porochemoelasticity is attained by substituting the constitutive Eq. (3.64) into the static equilibrium Eq. (3.66), which produces: $$G{\nabla }^{2}\overrightarrow{u}+\left(K+\frac{G}{3}\right)\nabla (\nabla .\overrightarrow{u})-\alpha \nabla p-{\alpha }_{c}{\nabla C}_{r}=0$$ 23 Dffusion Equations Assuming the chemo-hydraulic coupling is weak, then \({k}_{pc}\) and \({k}_{cp}\) become negligible such that the above pressure-based diffusion equations can be further reduced to the following for a one-dimensional problem like soil consolidation: $$\frac{\partial p}{\partial t}-\kappa M\frac{{\partial }^{2}p}{\partial {z}^{2}}=\alpha M\frac{\partial {e}_{zz}}{\partial t}+{\beta }_{c}M\frac{\partial {C}_{r}}{\partial t}$$ 24 However, the constitutive stress equation can equally be expressed as (Cheng, 2016 ): $${\sigma }_{zz}=\left(K+\frac{4G}{3}\right){e}_{zz}-\alpha p-{\alpha }_{c}{\beta }_{h}^{s}{C}_{r}$$ 25 Making \({e}_{zz}\) the subject, we have: $${\sigma }_{zz}+\alpha p+{\alpha }_{c}{\beta }_{h}^{s}{C}_{r}=\left(\frac{3K+4G}{3}\right){e}_{zz}$$ 26 $${e}_{zz}=\frac{3\left({\sigma }_{zz}+\alpha p+{\alpha }_{c}{{\beta }_{h}^{s}C}_{r}\right)}{3K+4G}$$ 27 Substituting the above expression for strain in the diffusion equation, we have: $$\frac{\partial p}{\partial t}-\kappa M\frac{{\partial }^{2}p}{\partial {z}^{2}}=\frac{3\alpha M}{3K+4G}\left(\frac{\partial {\sigma }_{zz}}{\partial t}+\alpha \frac{\partial p}{\partial t}+{\alpha }_{c}{\beta }_{h}^{s}\frac{\partial {C}_{r}}{\partial t}\right)+{\beta }_{c}M\frac{\partial {C}_{r}}{\partial t}$$ 28 Collecting like terms, $$\frac{\partial p}{\partial t}-\frac{3{\alpha }^{2}M}{3K+4G}\frac{\partial p}{\partial t}-\kappa M\frac{{\partial }^{2}p}{\partial {z}^{2}}=\frac{3\alpha M}{3K+4G}\frac{\partial {\sigma }_{zz}}{\partial t}+\frac{3\alpha M{\alpha }_{c}{\beta }_{h}^{s}}{3K+4G}\frac{\partial {C}_{r}}{\partial t}+{\beta }_{c}M\frac{\partial {C}_{r}}{\partial t}$$ 29 \(\left(1-\frac{3{\alpha }^{2}M}{3K+4G}\right)\frac{\partial p}{\partial t}-\kappa M\frac{{\partial }^{2}p}{\partial {z}^{2}}=\frac{3\alpha M}{3K+4G}\frac{\partial {\sigma }_{zz}}{\partial t}+\left(\frac{3\alpha M{\alpha }_{c}{\beta }_{h}^{s}}{3K+4G}+{\beta }_{c}M\right)\frac{\partial {C}_{r}}{\partial t}\) ` (30) Similarly, the diffusion equation for solute transport can be reduced to: $$\varphi \frac{\partial {C}_{r}}{\partial t}-{\left(1-\mathfrak{R}\right)D}_{c}\frac{{\partial }^{2}{C}_{r}}{\partial {z}^{2}}=0$$ 31 Initial and Boundary Conditions \(p\left(z,t\right),{C}_{r}(z,t)=0\) for t = 0 (32) \(p\left(z,t\right),{C}_{r}(z,t)=0\) for z = 0 (33) \(\frac{\partial p\left(z,t\right)}{\partial z},\frac{\partial {C}_{r}\left(z,t\right)}{\partial z}=0\) for z=h (34) The initial condition is that there is an equilibrium between the pore pressure and solute concentration. The second condition states that the water pressure is zero when a load is applied due to the high permeability of the slab compared to the soil. The third condition specifies that there is no water escaping through the bottom. These conditions can be classified as the initial and boundary conditions, with the initial condition indicating that there is no change in water content when a load is applied, as the water can only escape at a finite rate. Analytical Solution Soil consolidation is a uniaxial stress problem associated with the following relations, assuming the axis to be z: \({u}_{z}={u}_{z}(z,t)\) ; \(p=p(z,t)\) ; \({C}_{r}={C}_{r}(z,t)\) ; \({u}_{x}={u}_{y}=0\) (35) Which also leads to: \({e}_{zz}={e}_{zz}(z,t)\) ; \(\zeta =\zeta (z,t)\) ; \({e}_{xx}={e}_{yy}={e}_{xy}={e}_{xz}={e}_{yz}=0\) (36) As a result of the equations above, the Navier equations can be simplified to: $$\left(K+\frac{4G}{3}\right)\frac{{\partial }^{2}{u}_{z}}{\partial {z}^{2}}-\alpha \frac{\partial p}{\partial z}-{\alpha }_{c}{\beta }_{h}^{s}\frac{\partial {C}_{r}}{\partial z}=0$$ 37 The constitutive equations also become: $${\sigma }_{zz}=\left(K+\frac{4G}{3}\right){e}_{zz}-\alpha p-{\alpha }_{c}{\beta }_{h}^{s}{C}_{r}$$ 38 $${\sigma }_{xx}={\sigma }_{yy}=\left(K+\frac{4G}{3}\right){e}_{zz}-\alpha p-{\alpha }_{c}{{\beta }_{h}^{s}C}_{r}$$ 39 $${\sigma }_{xy}={\sigma }_{xz}={\sigma }_{yz}=0$$ 40 $$p=M\left(\zeta +\alpha e+{\beta }_{c}{C}_{r}\right)$$ 41 Using the above relations, the equilibrium equation further reduces to: $$\frac{\partial {\sigma }_{zz}}{\partial z}=0$$ 42 Thereby, revealing the following functional dependencies: \({\sigma }_{zz}={\sigma }_{zz}\left(t\right)\) ; \({\sigma }_{xx}={\sigma }_{xx}(z,t)\) ; \({\sigma }_{yy}={\sigma }_{yy}(z,t)\) (43) And the diffusion equations become: $$\frac{\partial p}{\partial t}-\frac{\kappa M(3K+4G)}{\left(3K+4G\right)+3{\alpha }^{2}M}\frac{{\partial }^{2}p}{\partial {z}^{2}}=\frac{3\alpha M}{\left(3K+4G\right)+3{\alpha }^{2}M}\frac{\partial {\sigma }_{zz}}{\partial t}+\left(\frac{3\alpha M{\alpha }_{c}{\beta }_{h}^{s}}{3K+4G}+{\beta }_{c}M\right).\left[\frac{(3K+4G)}{\left(3K+4G\right)+3{\alpha }^{2}M}\right]\frac{\partial {C}_{r}}{\partial t}$$ 44 $$\frac{\partial {C}_{r}}{\partial t}-\frac{{\left(1-\mathfrak{R}\right)D}_{c}}{\varphi }\frac{{\partial }^{2}{C}_{r}}{\partial {z}^{2}}=0$$ 45 Chemical Loading This describes the problem of applying the load through the introduction of a chemical potential possibly by an infiltrating fluid, which creates changes in solute (salt) concentration in the soil, while maintaining all other variables at a constant, the above scenario results in the subsequent set of boundary conditions \(t={0}^{+}\) : \({\sigma }_{zz}=0\) ; \(p=0\) ; \({C}_{r}=R\) ; at \(z=0\) (46) Due to the above boundary condition, the diffusion equations reduces to: $$\frac{\partial p}{\partial t}-\frac{\kappa M(3K+4G)}{\left(3K+4G\right)+3{\alpha }^{2}M}\frac{{\partial }^{2}p}{\partial {z}^{2}}=\left(\frac{3\alpha M{\alpha }_{c}{\beta }_{h}^{s}}{3K+4G}+{\beta }_{c}M\right).\left[\frac{(3K+4G)}{\left(3K+4G\right)+3{\alpha }^{2}M}\right]\frac{\partial {C}_{r}}{\partial t}$$ 47 $$\frac{\partial {C}_{r}}{\partial t}-\frac{{\left(1-\mathfrak{R}\right)D}_{c}}{\varphi }\frac{{\partial }^{2}{C}_{r}}{\partial {z}^{2}}=0$$ 48 The equations above are decoupled from the Navier equation, and can be resolved independent of it. To transform from partial differential equations to ordinary differential equations, we utilize Laplace transform on the diffusion equations. $$s\stackrel{\sim}{p}-c\frac{{d}^{2}\stackrel{\sim}{p}}{d{z}^{2}}=sd{\stackrel{\sim}{C}}_{r}$$ 49 $$s{\stackrel{\sim}{C}}_{r}-a\frac{{d}^{2}{\stackrel{\sim}{C}}_{r}}{d{z}^{2}}=0$$ 50 Where $$c=\frac{\kappa M(3K+4G)}{\left(3K+4G\right)+3{\alpha }^{2}M}$$ $$d=\left(\frac{3\alpha M{\alpha }_{c}{\beta }_{h}^{s}}{3K+4G}+{\beta }_{c}M\right).\left[\frac{(3K+4G)}{\left(3K+4G\right)+3{\alpha }^{2}M}\right]$$ $$a=\frac{{\left(1-\mathfrak{R}\right)D}_{c}}{\varphi }$$ The above equations can be rearranged like this: $$c\frac{{d}^{2}\stackrel{\sim}{p}}{d{z}^{2}}+sd{\stackrel{\sim}{C}}_{r}-s\stackrel{\sim}{p}=0$$ 51 $$a\frac{{d}^{2}{\stackrel{\sim}{C}}_{r}}{d{z}^{2}}-s{\stackrel{\sim}{C}}_{r}=0$$ 52 $$\frac{{d}^{2}\stackrel{\sim}{p}}{d{z}^{2}}-\frac{s}{c}\stackrel{\sim}{p}+s\frac{d}{c}{\stackrel{\sim}{C}}_{r}=0$$ 53 $$\frac{{d}^{2}{\stackrel{\sim}{C}}_{r}}{d{z}^{2}}-\frac{s}{a}{\stackrel{\sim}{C}}_{r}=0$$ 54 Furthermore, the solute diffusion equation can be independently solved due to the absence of a coupling term between both diffusion equations. From Mathematica, the solution to the above simultaneous equations in Laplace form becomes: $$\stackrel{\sim}{p}=-\frac{ad{e}^{-\frac{\sqrt{s}z}{\sqrt{a}}-\frac{\sqrt{s}z}{\sqrt{c}}}\left({e}^{\frac{\sqrt{s}z}{\sqrt{a}}}-{e}^{\frac{\sqrt{s}z}{\sqrt{c}}}-{e}^{\frac{2\sqrt{s}z}{\sqrt{a}}+\frac{\sqrt{s}z}{\sqrt{c}}}+{e}^{\frac{\sqrt{s}z}{\sqrt{a}}+\frac{2\sqrt{s}z}{\sqrt{c}}}\right)R}{2(a-c)}$$ 55 $${\stackrel{\sim}{C}}_{r}=\frac{1}{2}{e}^{-\frac{\sqrt{s}z}{\sqrt{a}}}(1+{e}^{\frac{2\sqrt{s}z}{\sqrt{a}}})R$$ 56 On expanding and simplifying the above equations respectively, we have: $$\stackrel{\sim}{p}=\frac{Rad{e}^{-\frac{\sqrt{s}z}{\sqrt{a}}}}{2(a-c)}+\frac{Rad{e}^{\frac{\sqrt{s}z}{\sqrt{a}}}}{2(a-c)}-\frac{Rad{e}^{-\frac{\sqrt{s}z}{\sqrt{c}}}}{2(a-c)}-\frac{Rad{e}^{\frac{\sqrt{s}z}{\sqrt{c}}}}{2(a-c)}$$ 57 $${\stackrel{\sim}{C}}_{r}=\frac{R}{2}\left({e}^{\frac{\sqrt{s}z}{\sqrt{a}}}+{e}^{-\frac{\sqrt{s}z}{\sqrt{a}}}\right)$$ 58 The pressure solution above can be transformed to: $$\stackrel{\sim}{p}={\stackrel{\sim}{p}}_{1}+{\stackrel{\sim}{p}}_{2}+{\stackrel{\sim}{p}}_{3}+{\stackrel{\sim}{p}}_{4}$$ 59 Where $${\stackrel{\sim}{p}}_{1}=\frac{Rad{e}^{-\frac{\sqrt{s}z}{\sqrt{a}}}}{2(a-c)}$$ $${\stackrel{\sim}{p}}_{2}=\frac{Rad{e}^{\frac{\sqrt{s}z}{\sqrt{a}}}}{2(a-c)}$$ $${\stackrel{\sim}{p}}_{3}=-\frac{Rad{e}^{-\frac{\sqrt{s}z}{\sqrt{c}}}}{2(a-c)}$$ $${\stackrel{\sim}{p}}_{4}=-\frac{Rad{e}^{\frac{\sqrt{s}z}{\sqrt{c}}}}{2(a-c)}$$ Such that $${\mathcal{L}}^{-1}\left(\stackrel{\sim}{p}\right)={\mathcal{L}}^{-1}\left({\stackrel{\sim}{p}}_{1}+{\stackrel{\sim}{p}}_{2}+{\stackrel{\sim}{p}}_{3}+{\stackrel{\sim}{p}}_{4}\right)$$ 60 $${\mathcal{L}}^{-1}\left({\stackrel{\sim}{p}}_{1}\right)={\mathcal{L}}^{-1}\left(\frac{Rad{e}^{-\frac{\sqrt{s}z}{\sqrt{a}}}}{2(a-c)}\right)$$ 61 $$=\frac{Rdz\sqrt{a}{e}^{-\frac{{z}^{2}}{4at}}}{4(a-c)\sqrt{\pi }{t}^{3/2}} \text{i}\text{f} \frac{z}{\sqrt{a}}>0$$ 62 $${\mathcal{L}}^{-1}\left({\stackrel{\sim}{p}}_{2}\right)={\mathcal{L}}^{-1}\left(\frac{Rad{e}^{\frac{\sqrt{s}z}{\sqrt{a}}}}{2(a-c)}\right)$$ 63 $$=-\frac{Rzd\sqrt{a}{e}^{-\frac{{z}^{2}}{4at}}}{4(a-c)\sqrt{\pi }{t}^{3/2}} \text{i}\text{f} \frac{z}{\sqrt{a}}<0$$ 64 $${\mathcal{L}}^{-1}\left({\stackrel{\sim}{p}}_{3}\right)={\mathcal{L}}^{-1}\left(-\frac{Rad{e}^{-\frac{\sqrt{s}z}{\sqrt{c}}}}{2(a-c)}\right)$$ 65 $$=-\frac{ad{e}^{-\frac{{z}^{2}}{4ct}}Rz}{4\left(a-c\right)\sqrt{c}\sqrt{\pi }{t}^{3/2}} \text{i}\text{f} \frac{z}{\sqrt{c}}>0$$ 66 $${\mathcal{L}}^{-1}\left({\stackrel{\sim}{p}}_{4}\right)={\mathcal{L}}^{-1}\left(-\frac{Rad{e}^{\frac{\sqrt{s}z}{\sqrt{c}}}}{2(a-c)}\right)$$ 67 $$=\frac{ad{e}^{-\frac{{z}^{2}}{4ct}}Rz}{4(a-c)\sqrt{c}\sqrt{\pi }{t}^{3/2}} \text{i}\text{f} \frac{z}{\sqrt{c}}<0$$ 68 Therefore, $$p={p}_{1}+{p}_{2}+{p}_{3}+{p}_{4}$$ 69 Where $$p={\mathcal{L}}^{-1}\left(\stackrel{\sim}{p}\right)$$ $${p}_{1}={\mathcal{L}}^{-1}\left({\stackrel{\sim}{p}}_{1}\right)$$ $${p}_{2}={\mathcal{L}}^{-1}\left({\stackrel{\sim}{p}}_{2}\right)$$ $${p}_{3}={\mathcal{L}}^{-1}\left({\stackrel{\sim}{p}}_{3}\right)$$ $${p}_{4}={\mathcal{L}}^{-1}\left({\stackrel{\sim}{p}}_{4}\right)$$ Similarly, the solution for the concentration ratio can be presented as: $${\stackrel{\sim}{C}}_{r}=\frac{R}{2}\left({\stackrel{\sim}{C}}_{r1}+{\stackrel{\sim}{C}}_{r2}\right)$$ 70 Where $${\stackrel{\sim}{C}}_{r1}={e}^{\frac{\sqrt{s}z}{\sqrt{a}}}$$ $${\stackrel{\sim}{C}}_{r2}={e}^{-\frac{\sqrt{s}z}{\sqrt{a}}}$$ Hence, $${\mathcal{L}}^{-1}\left({\stackrel{\sim}{C}}_{r}\right)=\frac{R}{2}\left[{\mathcal{L}}^{-1}\left({\stackrel{\sim}{C}}_{r1}\right)+{\mathcal{L}}^{-1}\left({\stackrel{\sim}{C}}_{r2}\right)\right]$$ 71 $${\mathcal{L}}^{-1}\left({\stackrel{\sim}{C}}_{r1}\right)={\mathcal{L}}^{-1}\left({e}^{\frac{\sqrt{s}z}{\sqrt{a}}}\right)$$ 72 $$=-\frac{z{e}^{-\frac{{z}^{2}}{4at}}}{2\sqrt{a}\sqrt{\pi }{t}^{3/2}} \text{i}\text{f} \frac{z}{\sqrt{a}}<0$$ 73 $${\mathcal{L}}^{-1}\left({\stackrel{\sim}{C}}_{r2}\right)={\mathcal{L}}^{-1}\left({e}^{-\frac{\sqrt{s}z}{\sqrt{a}}}\right)$$ 74 \(=-\frac{z{e}^{-\frac{{z}^{2}}{at}}}{\sqrt{a}\sqrt{\pi }{t}^{3/2}}\) (if \(\frac{z}{\sqrt{a}}0\) ) (75) Therefore, $${C}_{r}=\frac{R}{2}\left({C}_{r1}+{C}_{r2}\right)$$ 76 Where $${C}_{r}={\mathcal{L}}^{-1}\left({\stackrel{\sim}{C}}_{r}\right)$$ $${C}_{r1}={\mathcal{L}}^{-1}\left({\stackrel{\sim}{C}}_{r1}\right)$$ $${C}_{r2}={\mathcal{L}}^{-1}\left({\stackrel{\sim}{C}}_{r2}\right)$$ Considering the nature of the above solutions, they are all contingent on the values of either \(\frac{z}{\sqrt{a}}\) or \(\frac{z}{\sqrt{c}}\) being positive or negative. The expressions can only be negative only when z is negative, since the square root of a real number cannot be negative. Given the geometry of the problem at hand, in which the positive direction points into the ground, negative z (corresponding to the opposite direction pointing away from the soil and into the sky) is not admissible. Hence, the final solutions for both the pressure and the concentration ratio will only contain solutions dependent on positive values for both \(\frac{z}{\sqrt{a}}\) and \(\frac{z}{\sqrt{c}}\) . Consequently, $$p={p}_{1}+{p}_{3}$$ 77 $$p=\frac{Rdz\sqrt{a}{e}^{-\frac{{z}^{2}}{4at}}}{4(a-c)\sqrt{\pi }{t}^{3/2}}-\frac{ad{e}^{-\frac{{z}^{2}}{4ct}}Rz}{4\left(a-c\right)\sqrt{c}\sqrt{\pi }{t}^{3/2}}$$ 78 And $${C}_{r}={\frac{R}{2}C}_{r2}$$ 79 $${C}_{r}=\frac{R{e}^{-\frac{{z}^{2}}{4at}}z}{4\sqrt{a}\sqrt{\pi }{t}^{3/2}}$$ 80 Recall that, $${\sigma }_{zz}=\left(K+\frac{4G}{3}\right){e}_{zz}-\alpha p-{\alpha }_{c}{{\beta }_{h}^{s}C}_{r}$$ Such that with \({\sigma }_{zz}\) =0, we have $${e}_{zz}=\left(\frac{3}{3K+4G}\right)\left(\alpha p+{\alpha }_{c}{{\beta }_{h}^{s}C}_{r}\right)$$ 81 By integrating the equation above with respect to z, we can find the displacement u z , which is given by: $${u}_{z}\left(z,t\right)=\left(\frac{3}{3K+4G}\right)\left\{\frac{Rd\alpha \sqrt{a}(-2a{e}^{-\frac{{z}^{2}}{4at}}t+2\sqrt{a}\sqrt{c}{e}^{-\frac{{z}^{2}}{4ct}}t)}{4(a-c)\sqrt{\pi }{t}^{3/2}}-\frac{{\alpha }_{c}R\sqrt{a}{e}^{-\frac{{z}^{2}}{4at}}}{2\sqrt{\pi }\sqrt{t}}\right\}$$ 82 Another intriguing aspect is the chemical-induced water flow. Based on $$\overrightarrow{q}=-\kappa \nabla p+{C}_{s}{k}_{pc}\nabla {C}_{r}$$ 83 We have for the z direction, $${q}_{z}(z,t)=-\frac{\kappa Rd{e}^{-\frac{\left(a+c\right){z}^{2}}{4act}}\left(2a{c}^{3/2}{e}^{\frac{{z}^{2}}{4ct}}t-{c}^{3/2}{e}^{\frac{{z}^{2}}{4ct}}{z}^{2}+{a}^{3/2}{e}^{\frac{{z}^{2}}{4at}}\left(-2ct+{z}^{2}\right)\right)}{8\sqrt{a}\left(a-c\right){c}^{3/2}\sqrt{\pi }{t}^{5/2}}+\frac{{RC}_{s}{k}_{pc}}{2}\left(\frac{{e}^{-\frac{{z}^{2}}{4at}}}{2\sqrt{a}\sqrt{\pi }{t}^{3/2}}-\frac{{e}^{-\frac{{z}^{2}}{4at}}{z}^{2}}{4{a}^{3/2}\sqrt{\pi }{t}^{5/2}}\right)$$ 84 The water volume (per unit area) leaving the soil is $$V\left(t\right)={\int }_{0}^{t}{q}_{z}\left(0,t\right)dt$$ 85 $$V\left(t\right)={\int }_{0}^{t}\left[-\frac{\kappa Rd\left(2a{c}^{3/2}t+{a}^{3/2}\left(-2ct\right)\right)}{8\sqrt{a}\left(a-c\right){c}^{3/2}\sqrt{\pi }{t}^{5/2}}+\frac{{RC}_{s}{k}_{pc}}{2}\left(\frac{1}{2\sqrt{a}\sqrt{\pi }{t}^{3/2}}\right)\right]dt$$ 86 $$={\int }_{0}^{t}\left[-\frac{\kappa Rd\left(2a{c}^{3/2}t+{a}^{3/2}\left(-2ct\right)\right)}{8\sqrt{a}\left(a-c\right){c}^{3/2}\sqrt{\pi }{t}^{5/2}}\right]dt+{\int }_{0}^{t}\left[+\frac{{RC}_{s}{k}_{pc}}{2}\left(\frac{1}{2\sqrt{a}\sqrt{\pi }{t}^{3/2}}\right)\right]dt$$ 87 $$V\left(t\right)=-\frac{\sqrt{a}dR\kappa }{2\left(\sqrt{a}+\sqrt{c}\right)\sqrt{c}\sqrt{\pi }\sqrt{t}}-\frac{R{C}_{s}{k}_{pc}}{2\sqrt{a}\sqrt{\pi }\sqrt{t}}$$ 88 Meanwhile, matrix bulk modulus, Biot effective stress parameter and Biot effective chemical stress parameter were earlier derived from the effective solid bulk modulus, bulk modulus of porosity and bulk modulus of microinhomogeneity. They are expressed as: $$K=\frac{-{\left(1-\varphi \right)}^{2}{K}_{\varphi }\left\{\left(1-\varphi \right){K}_{\alpha }{K}_{\varphi } -{{K}_{\psi }}^{2}\right\}}{2{K}_{\psi }{K}_{\varphi }\left(1-\varphi \right){{-K}_{\varphi }}^{2}{\left(1-\varphi \right)}^{2}{{-K}_{\psi }}^{2}-\left(1-\varphi \right){K}_{\alpha }{K}_{\varphi }}$$ 89 $$\alpha =\frac{-\left\{{{K}_{\psi }}^{2}-{K}_{\psi }{K}_{\varphi }\left(1-\varphi \right)+\left(1-\varphi \right){K}_{\alpha }{K}_{\varphi }+{{\varphi K}_{\varphi }}^{2}{\left(1-\varphi \right)}^{2}-{\varphi K}_{\psi }{K}_{\varphi }\left(1-\varphi \right)\right\}}{2{K}_{\psi }{K}_{\varphi }\left(1-\varphi \right){{-K}_{\varphi }}^{2}{\left(1-\varphi \right)}^{2}{{-K}_{\psi }}^{2}-\left(1-\varphi \right){K}_{\alpha }{K}_{\varphi }}$$ 90 $${\alpha }_{c}=\frac{\{{K}_{\psi }{K}_{\varphi }\left(1-\varphi \right)-{{K}_{\varphi }}^{2}{\left(1-\varphi \right)}^{2}\}}{2{K}_{\psi }{K}_{\varphi }\left(1-\varphi \right){{-K}_{\varphi }}^{2}{\left(1-\varphi \right)}^{2}{{-K}_{\psi }}^{2}-\left(1-\varphi \right){K}_{\alpha }{K}_{\varphi }}$$ 91 Results and Discussion Figure 2 shows the evolution of generated pore pressure within the residual soil sample with passage of time for chemical loading. At t = 0s, the pore pressure can be noticed be initially zero, since prior to this point, no chemical load in the form of solute concentration differences has been introduced possibly by an infiltrating liquid. But at t = 0.5s, a negative pore pressure was can be observed, before a sharp increase in pore pressure as the effect of the chemical loading is felt. The generated pore pressure thereafter reached a peak value of more than 0.03MPa. The time lag before the chemical effect was felt can be attributed to poroelastic effects occasioned by the tortuosity of the porous soil material. Such that a finite amount of time is needed for the migration of the pore pressure field. In addition, the initially negative pore pressure can be explained by the fact that a soil segment at a depth of 5m would normally be expected to lie above the water table. Normally, soil found above the water table is expected to have a pore pressure below the atmospheric pressure (negative pore pressure), while soil below the water table is expected to have pore pressures above the atmospheric pressure (positive pore pressure). This explains the initially negative generated pore pressures in the consolidating soil before the effects of chemical load were felt. Beyond 0.5s, what can be noticed is a simultaneous reduction in the peak pore pressure value and increment in the portion of the consolidating soil under negative pore pressures. This shows increasing segments of the soil returning to their initial state after the passage of the pore pressure front. This can be traced to the downward movement of the chemical effects under osmotic pressure and its accompanying increase in pore pressure. The net effect is further penetration of the generated pore pressure field in deeper segments of the soil. This scenario continued to played out even after 5s, 20s. But after 60s, most of the soil segment has returned to negative pore pressure regime after the passage of the pore pressure front. Particularly, after 100s, all the soil segments have returned to negative pore pressures after the pore pressure front has successfully passed. Figure 3 shows the evolution of solute concentration ratio with passage of time for chemical loading. The concentration ratio as presented is the ratio of the solute concentration of the infiltrating liquid to the solute concentration of the soil. Such that solute concentration ratios greater than 1 represent situations where the infiltrating liquid has solute concentration higher than the solute concentration of the soil, which gives rise to positive osmotic pressures. While solute concentration ratios less than one represent situations where the solute concentration of the soil is higher than the solute concentration of the infiltrating liquid, which gives rise to negative osmotic pressures. The criteria excludes negative solute concentration ratios, since the ratio of two positive real numbers cannot be negative. From the figure, it can be seen that at t = 0.5s, the solute concentration ratio increased from zero, peaked at a depth of about 1.5m, before gradually reducing to zero again as the depth increased. Like earlier described, the increasing solute concentration ratio is synonymous with increasing quantities of the infiltrating liquid over the corresponding quantity of the in situ pore fluid at each depth. This description directly follows from the definition of the solute concentration described above. This observation shows that even though the chemical loading was instantaneous, it took some time before its effect could be felt everywhere according to the poroelastic theory, gauging by the concentration front depicted in the figure at t = 0.5s. The same pattern of curve can equally be noticed for t = 1s and t = 5s, but with an increasingly wider base as time progresses. Such that at t = 5s, the right leg of the inverted v-shape of the curve was no longer visible. This clearly shows a net chemical concentration front advancing under osmotic pressure. However, the strength of the chemical difference seemed to be waning as time passaged. This position is supported by the decreasing peak of the inverted v-curve, which decreased from 0.025 to 0.0025 after just five seconds. After 20s, only the left leg of the inverted v-curve can be seen in the figure, with further decrease in the peak value. This shows that the chemical difference front has clearly advanced beyond the 5m depth currently considered for analysis. This scenario continued that for t = 60s and t = 100s, only the straight line segments of the inverted v-curve could be observed, which also depicts an active and advancing front. Figure 4 shows the evolution of vertical displacement with passage of time for chemical loading. From the figure, a quick response can be observed as soon as the infiltration started, which can be seen as negative displacement recorded at t = 0.5s. Based on the geometry of the current application, positive displacements are displacements that occur in the direction of increasing depth, while negative displacements occur away from the soil. Consequently, the initial negative displacement recorded in the figure was a slight increase or elevation of the soil surface experienced as the infiltrating liquid that gave rise to the chemical loading penetrated the pore spaces of the soil in addition to the in situ pore fluid. Since the load is purely chemical, without mechanical stress as depicted in the boundary conditions, the soil is free to briefly expand to accommodate the extra pore fluid. But this increase in height was mainly felt to a depth of about 3m at t = 0.5s, with the depth of largest displacement occurring less than a meter (1m) deep. But as time progressed, the displacement affected deeper sections of the soil, while the depth of largest displacement also moved deeper. Such that after 5s, the negative displacement has reached a depth of about 5m and the depth of largest displacement at about 1.5m deep. This observation clearly shows the case of an advancing concentration difference field moving to deeper sections of the soil. The above description is in agreement with the scenario already described for both the generated pore pressure and solute concentration ratio. After 20s, a clear inflexion can be seen to be developing as areas previously under negative displacement are relaxed after the chemical effect front passed. This position is supported by the v-shaped curve appearing at t = 20s, in which the depth of largest displacement had reached 3m. As time progressed further, the depth of largest displacement went deeper and areas behind the moving chemical loading front were furthered relaxed. Although relaxation took place afterwards, there was a residual displacement, possibly accounting for the remnant portions of the infiltrating left behind under capillary pressure. Figure 5 shows the evolution of the water flux induced by the chemical effect with passage of time for chemical loading. In transport phenomena, flux is frequently defined as the rate of flow of a property per unit area. Its dimension is usually [quantity].[time] −1 .[area] −1 , which effectively translates to [m/s] as presented in the figure. The scenario depicted in the figure can be described as oppositely related to the vertical displacement as it offers the explanation for the evolution of the vertical displacement with time. From the figure, it can be deduced that the flux of water flow in the soil was initially positive before turning negative. This can be explained by initial successful penetration of the infiltrating liquid until it meets increasingly tortuous paths as result of lower permeability. The initial reaction is that of rejection as more fluid could not successfully penetrate the pores, which is seen as a sharp increase in the flux noticed first at t = 0.5s. This possibly explains the initial negative displacement as a result of successful penetration. The pattern described above can be noticed for the subsequent timeframes reported. Consequently, it can be seen that as time progressed, the volume of positive flux gradually reduces and its pressure head is instead utilized in forcefully penetrating the porous structure of the soil. This is now seen in the gradual increase in flux after the initial decrease in flux. The gradual increase in flux represents increasingly successful penetration of the deeper sections, while the flux (flow) remained positive for shallower depths already penetrated. In addition, the depth of most negative flux equally moved with further penetration of the soil by the infiltrating liquid. This observation can be noticed after 60s, 100s and even 150s when it became further developed. The relatively longer duration can be traced to the tortuosity of the residual soil under study. Figure 6 shows the water volume (per unit area) leaving the soil after chemical loading. The figure shows that a portion of the water flux associated with the infiltrating liquid ends up leaving the soil through the sides. It can be deduced that the largest volume of water leaves from the shallower depths of the soil compared to the deeper sections. This is because since the infiltrating liquid penetrates from the top, it would take some time before it gets to the deeper sections of the soil due to porosity and permeability differences as depth increases. Hence, a decreasingly lower volume of water would be lost as the infiltrating liquid travels to the deeper sections of the soil. The loss of water experienced is actually expected as the application discussed is soil consolidation. Conclusion The evolution of the generated pore pressure showed increasing segments of the soil returning to their initial state after the passage of the pore pressure front. This can be traced to the downward movement of the chemical effects under osmotic pressure and its accompanying increase in pore pressure. The net effect is further penetration of the generated pore pressure field in deeper segments of the soil. Even though the chemical loading was instantaneous, it took some time before its effect could be felt everywhere according to the poroelastic theory. This clearly shows a net chemical concentration front advancing under osmotic pressure. The vertical displacement showed a slight increase or elevation of the soil surface experienced as the infiltrating liquid that gave rise to the chemical loading penetrated the pore spaces of the soil in addition to the in situ pore fluid. The flux of water flow in the soil was initially positive before turning negative, which can be explained by initial successful penetration of the infiltrating liquid until it met increasingly tortuous paths as result of lower permeability. Consequently, it can be seen that as time progressed, the volume of positive flux gradually reduces and the available pressure head is instead utilized in forcefully penetrating the porous structure of the soil. The volume (per unit area) of water leaving the soil after chemical loading demonstrated that a portion of the water flux associated with the infiltrating liquid ends up leaving the soil through the sides. This loss of water is actually expected as the application discussed is soil consolidation. Declarations Competing interests: We hereby declare that we do not have any conflicts of interest. Funding: None of the authors received financial support for this research. Author Contribution The study's conception and design were the result of collaboration among all authors. I.D. conducted data collection, simulation, and analysis, with supervision provided by both O.M.E. and O.F.D. The final manuscript was read and approved by all authors. 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Wesley LD (2010) Geotechnical engineering in residual soils. Wiley, Hoboken, USA. Wibawa YS, Sugiarti K, Soebowo E (2018) Characteristics and engineering properties of residual soil of volcanic deposits. Paper presented at the IOP Conference Series: Earth and Environmental Sciences Zhang G, Whittle AJ, Nikolinakou AM, Germaine JT (2007) Characterization and engineering properties of the old alluvium in Puerto Rico. Paper presented at the 2nd Int. Workshop on Characterization & Engineering Properties of Natural Soils, Singapore. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3992157","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":275125444,"identity":"4475f9a2-e372-4374-9477-f6b608331739","order_by":0,"name":"Dada Irheren","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5UlEQVRIiWNgGAWjYBACNgYeMM3YwJDA+ADI4OEjRQuzAUgLG2F7EFrYJCCGEAB80mcPPq6ouCPb3578rPJrjp0MGwPzw0c38DmMLy/Z8MyZZ8Yzzjwzuy27LRnoMDZj4xx8Wnh4zCQb2w4nNtxIMLstuY0ZqIWHTZqAFvOfjf8OJ86/kf6tWHJbPVFazBgbGw4nbriRY8b4cdthorQYSzYcO2y88cybYmnGbcd52JgJ+EW+h8fwY0PNYdl5x9M3fvy5rdqen7354WN8WlAAMziOmIlVDgKMP0hRPQpGwSgYBSMGAACOF0Zl+q1DIQAAAABJRU5ErkJggg==","orcid":"","institution":"University of Nigeria, Nsukka","correspondingAuthor":true,"prefix":"","firstName":"Dada","middleName":"","lastName":"Irheren","suffix":""},{"id":275125445,"identity":"764c6bea-b6dc-4506-b35f-46ed22cf80c4","order_by":1,"name":"Michael Ebie Onyia","email":"","orcid":"","institution":"University of Nigeria, Nsukka","correspondingAuthor":false,"prefix":"","firstName":"Michael","middleName":"Ebie","lastName":"Onyia","suffix":""},{"id":275125446,"identity":"2ae29f64-422f-4f51-859f-c44de20852a8","order_by":2,"name":"Fidelis Onyebuchi Okafor","email":"","orcid":"","institution":"University of Nigeria, Nsukka","correspondingAuthor":false,"prefix":"","firstName":"Fidelis","middleName":"Onyebuchi","lastName":"Okafor","suffix":""}],"badges":[],"createdAt":"2024-02-26 23:14:43","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3992157/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3992157/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":51791380,"identity":"8c846bf1-7e00-446a-a517-b256b621f1cc","added_by":"auto","created_at":"2024-02-29 05:46:19","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":102530,"visible":true,"origin":"","legend":"\u003cp\u003eComposite level of fabric organization in residual soil (Blight \u0026amp; Leong, 2012)\u003c/p\u003e","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-3992157/v1/d226e3b068967dd9efa9ff47.png"},{"id":51791763,"identity":"f7b450be-5fc4-42c3-91cb-6099cab1857a","added_by":"auto","created_at":"2024-02-29 05:54:19","extension":"jpeg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":411716,"visible":true,"origin":"","legend":"\u003cp\u003eEvolution of generated pore pressure with passage of time for chemical loading after (a) 0.5s (b) 1s (c) 5s (d) 20s (e) 60s (f) 100s\u003c/p\u003e","description":"","filename":"floatimage2.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3992157/v1/865b545da813b8ce49334536.jpeg"},{"id":51791383,"identity":"e310c199-7171-4675-b248-174dfb5328bc","added_by":"auto","created_at":"2024-02-29 05:46:19","extension":"jpeg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":397474,"visible":true,"origin":"","legend":"\u003cp\u003eEvolution of solute concentration ratio with passage of time for chemical loading after (a) 0.5s (b) 1s (c) 5s (d) 20s (e) 60s (f) 100s\u003c/p\u003e","description":"","filename":"floatimage3.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3992157/v1/d628634c339e7596b7ecac99.jpeg"},{"id":51791382,"identity":"c66a92e4-009b-498f-9a30-ecbf5044aee3","added_by":"auto","created_at":"2024-02-29 05:46:19","extension":"jpeg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":405367,"visible":true,"origin":"","legend":"\u003cp\u003eEvolution of vertical displacement with passage of time for chemical loading after (a) 0.5s (b) 1s (c) 5s (d) 20s (e) 60s (f) 100s\u003c/p\u003e","description":"","filename":"floatimage4.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3992157/v1/24c0bab2cd342a81a4848b21.jpeg"},{"id":51791385,"identity":"cd8dfd80-6d3e-4f53-950e-539deec80612","added_by":"auto","created_at":"2024-02-29 05:46:19","extension":"jpeg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":350715,"visible":true,"origin":"","legend":"\u003cp\u003eEvolution of the water flux induced by the chemical effect with passage of time after chemical loading after (a) 0.5s (b) 5s (c) 20s (d) 60s (e) 100s (f) 150s\u003c/p\u003e","description":"","filename":"floatimage5.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3992157/v1/083da372e92cdba627f497a2.jpeg"},{"id":51791381,"identity":"a1e66bea-66f0-4630-bfbf-f7e6829122b0","added_by":"auto","created_at":"2024-02-29 05:46:19","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":9531,"visible":true,"origin":"","legend":"\u003cp\u003eWater volume (per unit area) leaving the formation after chemical loading\u003c/p\u003e","description":"","filename":"floatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-3992157/v1/aaee744ef2f810bde6bd4c21.png"},{"id":51846168,"identity":"449a8256-9d63-4ed4-979d-7a4a308b3aba","added_by":"auto","created_at":"2024-03-01 07:11:18","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":717450,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3992157/v1/f71ecdcb-aa42-4fd9-9d7a-63e2f1303e09.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Poroelastic Effects of Chemical Loading in Consolidation of Residual Soils","fulltext":[{"header":"Introduction","content":"\u003cp\u003eThe properties of residual soils can vary greatly from those of transported soils, as they often do not conform to the traditional definition of soil grains or particle size. Instead, they are typically composed of aggregates or weathered minerals that can break down and become more compact when manipulated or compacted. This means that what may initially appear to be coarse sandy gravel in its natural state can actually break down into fine sandy silt when excavated, mixed, and compacted (Rahman et al., 2018).\u003c/p\u003e \u003cp\u003eThe relationship between the permeability and granulometry of a transported soil, as seen in the well-known Hazen formula, is not always applicable to residual soils. In these cases, the permeability is largely determined by the soil's micro- and macro-fabric, jointing, and any remaining or added features, such as slickensides, termite channels, or other biological channels. This same principle can also be observed in properties like particle unit weight or specific gravity Gs, where the value of Gs can vary depending on the level of comminution of the sample. Generally, smaller particles have a higher Gs value due to their increased solidity.\u003c/p\u003e \u003cp\u003eAnother difference between residual and transported soils is the classification scheme for classifying soils. Residual soils possess distinct traits that are not fully accounted for in traditional soil classification methods intended for transported soils like the Unified Soil Classification System (USCS). The unique clay mineralogy found in certain residual soils presents features that do not align with the standard characteristics of the soil group it belongs to, as recognized in current systems such as the USCS (Wibawa et al., \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). Existing soil classification systems, such as the USCS, are not sufficient to describe the range of materials present in a soil mass due to weathering. Depending on the level of weathering, the in situ materials can vary from soil to soft rock. These systems are designed to classify transported soils and focus on properties in a remoulded state, which is not representative of residual soils. The properties of these soils are heavily influenced by the fabric and structural characteristics inherited from the original rock mass, or formed as a result of weathering.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe impact of oven-drying and air-drying on soil properties is widely recognized, although it is typically minimal for soils that have been transported. The primary consequence of drying is a decrease in the reported proportion of clay particles (finer than 2 \u0026micro;m). Residual soils, which are formed through slow decomposition in an anaerobic setting, are especially susceptible to changes in properties when exposed to drying and air. This process can lead to dehydration of clay minerals and irreversible alterations in their properties. Even with prolonged maturation after re-wetting, air-drying at normal temperatures can cause irreversible changes in residual soils. In addition to the well-documented effects on index properties, drying also impacts the composition, compressibility, and shear strength characteristics of residual soils (Blight \u0026amp; Leong, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2012\u003c/span\u003e). The presence of individual particle assemblages (IPAs) in residual soils makes dispersion an essential requirement for particle size analysis. This was demonstrated by Rodriguez (\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2005\u003c/span\u003e) through an extreme instance of lateritic soil particle size analysis. Without dispersion, the particle size distribution resembles that of silty sand with no clay content. However, with dispersion, it transforms into sandy clay with a significantly high clay content of 56%. This serves as a clear indication of the significance of dispersion when dealing with residual soils, as opposed to transported soils (Duarte \u0026amp; Rodrigues, 2018).\u003c/p\u003e \u003cp\u003eThe structure and mineralogy of residual soils can be highly diverse. These variations can have a significant impact on their engineering properties, especially when specific microstructural or mineralogical traits are identified. It is crucial to keep in mind that residual soils may exhibit much better engineering behavior than what is indicated by index tests on remoulded soil. This is particularly true when considering correlations between index tests and the engineering behavior of transported soils, as noted by Zhang et al. (\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2007\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eResidual soil research has been undertaken by a number of researchers in the past, including Townsend (\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e1985\u003c/span\u003e), Duarte (\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2002\u003c/span\u003e), Weslye (2010), Blight \u0026amp; Leong (\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2012\u003c/span\u003e), Wibawa et al. (\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2018\u003c/span\u003e), Duarte \u0026amp; Rodrigues (2018) etc. However, these studies have largely been far in between and comparatively small when compared to the more ubiquitous transported soils (Blight \u0026amp; Leong, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2012\u003c/span\u003e). The few available research have mostly studied only the geological (Vaughan, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e1988\u003c/span\u003e; Duarte, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2002\u003c/span\u003e), engineering (Brand \u0026amp; Phillipson, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e1985\u003c/span\u003e) and geotechnical (Townsend, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e1985\u003c/span\u003e; Duarte, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2002\u003c/span\u003e) characteristics of residual soils in general. There appears a dearth of research specifically targeted towards modeling of residual soil.\u003c/p\u003e \u003cp\u003eTherefore, in this study, the poroelastic effects of chemical loading in consolidation of residual soils will be investigated. Since chemical loading with an aqueous liquid can be useful in understanding the effect of dehydration on residual soils. This represents a break from the usual modeling studies that basically consider only hydraulic and mechanical loading conditions.\u003c/p\u003e"},{"header":"Model Formulation","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eMacromechanical Stress-Strain Relations\u003c/h2\u003e \u003cp\u003eThe porochemoelastic expression for a chemohydromechanical treatment can be given as,\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\frac{{\\sigma }_{kk}}{3}=Ke-\\alpha p-{\\alpha }_{c}{{\\beta }_{h}^{s}C}_{r}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIntroducing the deviatoric stress component to account for both volumetric and shear responses,\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\frac{{\\sigma }_{kk}}{3}+\\left({\\sigma }_{ij}-\\frac{1}{3}{\\delta }_{ij}{\\sigma }_{kk}\\right)=Ke-\\alpha p-{\\alpha }_{c}{\\beta }_{h}^{s}{C}_{r}+2G\\left({e}_{ij}-\\frac{1}{3}{\\delta }_{ij}e\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eSimplifying,\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$${\\sigma }_{ij}=Ke-\\alpha p-{\\alpha }_{c}{\\beta }_{h}^{s}{C}_{r}+2G\\left({e}_{ij}-\\frac{1}{3}{\\delta }_{ij}e\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$${\\sigma }_{ij}=\\left(K-\\frac{2G}{3}\\right){\\delta }_{ij}e+2G{e}_{ij}-\\alpha p-{\\alpha }_{c}{{\\beta }_{h}^{s}C}_{r}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn addition, the equation for pore pressure can be expressed as Cheng (\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2016\u003c/span\u003e):\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$p=M\\left(\\zeta +\\alpha e+{\\beta }_{c}{C}_{r}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\zeta\\)\u003c/span\u003e \u003c/span\u003e is the variation in fluid content\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\alpha\\)\u003c/span\u003e \u003c/span\u003e is Biot\u0026rsquo;s effective stress parameter\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\beta }_{c}\\)\u003c/span\u003e \u003c/span\u003e is a new parameter quantifying the contribution of chemical strain to the total pore pressure\u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eBalance Laws\u003c/h3\u003e\n\u003cp\u003eThe balance laws account for the conservation of mass, force and energy in consonance with classical thermodynamic laws.\u003c/p\u003e\n\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\n\u003ch2\u003eForce Equilibrium\u003c/h2\u003e\n\u003cp\u003eFollowing Mehrabian et al. (\u003cspan class=\"CitationRef\"\u003e2020\u003c/span\u003e), the compressive geologic stresses and the related physico-chemical interactions far outweigh the effects of body forces like gravity, so that the following equilibrium equation applies.\u003c/p\u003e\n\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ6\" class=\"mathdisplay\"\u003e$${\\sigma }_{ij,j}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWhich reduces to:\u003c/p\u003e\n\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ7\" class=\"mathdisplay\"\u003e$$\\frac{\\partial {\\sigma }_{zz}}{\\partial z}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eOr for a one-dimensional problem,\u003c/p\u003e\n\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ8\" class=\"mathdisplay\"\u003e$$\\frac{d{\\sigma }_{zz}}{dz}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec6\" class=\"Section3\"\u003e\n\u003ch2\u003eFluid Mass Balance\u003c/h2\u003e\n\u003cp\u003eThe fluid mass balance ensures the conservation of the total fluid moving into and out of the fluid-filled porous medium at all times. Mathematically, this can be expressed as (Kanfar \u003cem\u003eet al.\u003c/em\u003e,2017):\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ9\" class=\"mathdisplay\"\u003e$$\\frac{\\partial \\zeta }{\\partial t}+\\nabla .\\overrightarrow{q}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\n\u003ch2\u003eSolute Mass Balance\u003c/h2\u003e\n\u003cp\u003eWhile the solute mass balance accounts for the movement of solute species, which is primarily responsible for chemical interactions. According to Cheng (\u003cspan class=\"CitationRef\"\u003e2016\u003c/span\u003e), the expression for solute mass balance can be written as:\u003c/p\u003e\n\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ10\" class=\"mathdisplay\"\u003e$$\\varphi \\frac{\\partial {C}_{k}}{\\partial t}+\\nabla .{\\overrightarrow{J}}_{s}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWhere\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({C}_{d}\\)\u003c/span\u003e \u003c/span\u003e is the infiltrating liquid solute (salt) molar concentration\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({J}_{s}\\)\u003c/span\u003e \u003c/span\u003e is the diffusive mass flux (mass crossing per unit area of the porous medium per unit time) of chemical species.\u003c/p\u003e\n\u003cp\u003eDividing both sides by the soil solute concentration, we have:\u003c/p\u003e\n\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ11\" class=\"mathdisplay\"\u003e$$\\frac{\\varphi }{{C}_{s}}\\frac{\\partial {C}_{d}}{\\partial t}+\\frac{\\nabla .{\\overrightarrow{J}}_{s}}{{C}_{s}}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eOr\u003c/p\u003e\n\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ12\" class=\"mathdisplay\"\u003e$$\\varphi \\frac{\\partial {C}_{r}}{\\partial t}+\\frac{1}{{C}_{s}}\\nabla .{\\overrightarrow{J}}_{s}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({C}_{r}=\\frac{{C}_{d}}{{C}_{s}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\n\u003ch2\u003eTransport Laws\u003c/h2\u003e\n\u003cp\u003eThe transport laws represent the mathematical expressions relating the possible types of fluxes in the physico-chemical system under study to their respective driving forces. The occurrence of multiple irreversible transport processes, such as heat conduction, electrical conduction, and mass diffusion, in a thermodynamic system can lead to interference and impact the system's overall behavior (Gao et al., \u003cspan class=\"CitationRef\"\u003e2021\u003c/span\u003e). Following Onsager\u0026rsquo;s non-equilibrium thermodynamics, the different fluxes can be expressed as follows:\u003c/p\u003e\n\u003cdiv id=\"Equ13\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ13\" class=\"mathdisplay\"\u003e$$\\overrightarrow{q}=-\\kappa \\nabla p+{C}_{s}{k}_{pc}\\nabla {C}_{r}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e13\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ14\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ14\" class=\"mathdisplay\"\u003e$${\\overrightarrow{J}}_{s}={k}_{cp}\\nabla p-{C}_{s}{\\left(1-\\mathfrak{R}\\right)D}_{c}\\nabla {C}_{r}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e14\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe symmetry of the coefficients in the above equation is particularly noticeable. \u0026kappa;, the permeability coefficient (mobility) of Darcy\u0026rsquo;s law, and R, the reflection coefficient representing the strength of the counter flow induced by the osmotic pressure gradient, are both observed. The reflection coefficient is a dimensionless parameter, with a range of 0 to 1 (0 \u0026lt; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\mathfrak{R}\\)\u003c/span\u003e\u003c/span\u003e \u0026lt; 1,), where 0 represents a chemically inert rock with no selective filtration of solute and solvent, and 1 represents the ideal ion exclusion membrane where the concentration gradient drives a full osmotic pressure. The second term on the right hand side of the equation does not induce a solvent flow in the lower bound case, while the upper bound case exerts a full osmotic pressure due to the concentration gradient.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e\n\u003ch2\u003eFluid Diffusion Equation\u003c/h2\u003e\n\u003cp\u003eSubstituting the fluid flux equation into the fluid balance equation we have,\u003c/p\u003e\n\u003cdiv id=\"Equ15\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ15\" class=\"mathdisplay\"\u003e$$\\frac{\\partial \\zeta }{\\partial t}+\\nabla .(-\\kappa p+{C}_{s}{k}_{pc}\\nabla {C}_{r})=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e15\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{pc}=\\mathfrak{R}\\kappa R{T}_{o}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cdiv id=\"Equ16\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ16\" class=\"mathdisplay\"\u003e$$\\frac{\\partial \\zeta }{\\partial t}-\\kappa {\\nabla }^{2}p+{C}_{s}{k}_{pc}{\\nabla }^{2}{C}_{r}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e16\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eBut, recall that,\u003c/p\u003e\n\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equa\" class=\"mathdisplay\"\u003e$$p=M\\left(\\zeta +\\alpha e+{\\beta }_{c}{C}_{r}\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eAnd making \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\zeta\\)\u003c/span\u003e\u003c/span\u003e the subject,\u003c/p\u003e\n\u003cdiv id=\"Equ17\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ17\" class=\"mathdisplay\"\u003e$$\\zeta =\\frac{p}{M}-\\left(\\alpha e+{\\beta }_{c}{C}_{r}\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e17\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eSubstituting the above equation into the fluid diffusion equation,\u003c/p\u003e\n\u003cdiv id=\"Equ18\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ18\" class=\"mathdisplay\"\u003e$$\\frac{1}{M}\\frac{\\partial p}{\\partial t}-\\alpha \\frac{\\partial e}{\\partial t}-{\\beta }_{c}\\frac{\\partial {C}_{r}}{\\partial t}-\\kappa {\\nabla }^{2}p+{C}_{s}{k}_{pc}{\\nabla }^{2}{C}_{r}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e18\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eMultiplying both sides by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(M\\)\u003c/span\u003e\u003c/span\u003e and rearranging,\u003c/p\u003e\n\u003cdiv id=\"Equ19\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ19\" class=\"mathdisplay\"\u003e$$\\frac{\\partial p}{\\partial t}-\\kappa {M\\nabla }^{2}p+{C}_{s}{k}_{pc}M{\\nabla }^{2}{C}_{r}=\\alpha M\\frac{\\partial e}{\\partial t}+{\\beta }_{c}M\\frac{\\partial {C}_{r}}{\\partial t}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e19\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe above equation is quite symbolic, showing the fact that the fluxes serve as the driving force, while the terms on the RHS depict their respective source terms. So, similar to the aforementioned transport laws, the above equation also highlights the effects of chemical interactions on the total pore pressure (hydraulics) of the system, involving osmotic effects (Gao et al., \u003cspan class=\"CitationRef\"\u003e2021\u003c/span\u003e).\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e\n\u003ch2\u003eSolute Diffusion Equation\u003c/h2\u003e\n\u003cp\u003eSubstituting the solute flux equation into the solute balance equation we have,\u003c/p\u003e\n\u003cdiv id=\"Equ20\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ20\" class=\"mathdisplay\"\u003e$$\\varphi \\frac{\\partial {C}_{r}}{\\partial t}+\\frac{1}{{C}_{s}}\\nabla .\\left\\{{k}_{cp}\\nabla p-{C}_{s}{\\left(1-\\mathfrak{R}\\right)D}_{c}\\nabla {C}_{r}\\right\\}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e20\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ21\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ21\" class=\"mathdisplay\"\u003e$$\\varphi \\frac{\\partial {C}_{r}}{\\partial t}+\\frac{1}{{C}_{s}}\\left\\{{k}_{cp}{\\nabla }^{2}p-{C}_{s}{\\left(1-\\mathfrak{R}\\right)D}_{c}{\\nabla }^{2}{C}_{r}\\right\\}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e21\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ22\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ22\" class=\"mathdisplay\"\u003e$$\\varphi \\frac{\\partial {C}_{r}}{\\partial t}+\\frac{{k}_{cp}}{{C}_{s}}{\\nabla }^{2}p-{\\left(1-\\mathfrak{R}\\right)D}_{c}{\\nabla }^{2}{C}_{r}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e22\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{cp}=\\left(1-\\mathfrak{R}\\right){C}_{d}\\kappa\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003eAnd \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\mathfrak{R}\\)\u003c/span\u003e\u003c/span\u003e is the osmotic reflection coefficient., \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({C}_{d}\\)\u003c/span\u003e\u003c/span\u003e is the infilterating liquid solute concentration and k is the soil permeability\u003c/p\u003e\n\u003cp\u003eSimilarly, the above equation highlights the effects of hydraulic interactions on the chemical stress as shown by the fluxes terms on the LHS (Cheng, \u003cspan class=\"CitationRef\"\u003e2016\u003c/span\u003e).\u003c/p\u003e\n\u003cp\u003eThe above equations in the two unknowns, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(p\\left(z,t\\right) and {C}_{r}\\left(z,t\\right)\\)\u003c/span\u003e\u003c/span\u003e suffice for a well-posed mathematical problem, since the stresses and strains are equally well defined in terms of the variables. In this study, Wolfram\u0026rsquo;s Mathematica software will be utilized in solving the obtained partial differential equations above, given the appropriate constants. The list of parameters and their values which is gotten from the literature is given in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3.1\u003c/span\u003e below. In this case study, a residual soil sample is assumed with the relevant properties.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab1\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 3.1\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003ePhysico-chemical constants for a typical residual soil used for numerical analysis\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eParameter\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eSymbol\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eValue\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eSource\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eShear modulus (Pa)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eG\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1853.21\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eLiu \u0026amp; Abousleiman (\u003cspan class=\"CitationRef\"\u003e2018\u003c/span\u003e)\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eBiot\u0026rsquo;s effective stress parameter (-)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\alpha\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.812939\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eCalculated\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eBiot effective chemical stress coefficient (-)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\alpha }_{c}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.2673\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eCalculated\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eSoil solute concentration (mol/m\u003csup\u003e3\u003c/sup\u003e)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({C}_{s}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e2.5\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\times\\)\u003c/span\u003e\u003c/span\u003e10\u003csup\u003e3\u003c/sup\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eCheng (\u003cspan class=\"CitationRef\"\u003e2016\u003c/span\u003e)\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eInfiltrating fluid solute concentration (mol/m\u003csup\u003e3\u003c/sup\u003e)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({C}_{d}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e5\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\times\\)\u003c/span\u003e\u003c/span\u003e10\u003csup\u003e3\u003c/sup\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eCheng (\u003cspan class=\"CitationRef\"\u003e2016\u003c/span\u003e)\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eMatrix bulk modulus (Pa)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eK\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e5.34995\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\times\\)\u003c/span\u003e\u003c/span\u003e10\u003csup\u003e8\u003c/sup\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eCalculated\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eEffective solid bulk modulus (Pa)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K}_{\\alpha }\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e2.86\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\times\\)\u003c/span\u003e\u003c/span\u003e10\u003csup\u003e9\u003c/sup\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eCheng (\u003cspan class=\"CitationRef\"\u003e2016\u003c/span\u003e)\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eBulk modulus of porosity (Pa)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K}_{\\varphi }\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.49\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\times\\)\u003c/span\u003e\u003c/span\u003e10\u003csup\u003e9\u003c/sup\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eCheng (\u003cspan class=\"CitationRef\"\u003e2016\u003c/span\u003e)\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eBulk modulus of microinhomogeneity (Pa)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K}_{\\psi }\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eCheng (\u003cspan class=\"CitationRef\"\u003e2016\u003c/span\u003e)\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eBiot\u0026rsquo;s modulus (Pa)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(M\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e6\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\times\\)\u003c/span\u003e\u003c/span\u003e10\u003csup\u003e9\u003c/sup\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eCheng (\u003cspan class=\"CitationRef\"\u003e2016\u003c/span\u003e)\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003ePermeability coefficient (m\u003csup\u003e4\u003c/sup\u003e/N.s)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\kappa\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\times\\)\u003c/span\u003e\u003c/span\u003e10\u003csup\u003e\u0026minus;2\u003c/sup\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eCheng (\u003cspan class=\"CitationRef\"\u003e2016\u003c/span\u003e)\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eChemo-osmotic coefficient\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{pc}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e6.0\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\times\\)\u003c/span\u003e\u003c/span\u003e10\u003csup\u003e\u0026minus;11\u003c/sup\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eCalculated\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eChemical strain parameter\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{c}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.95\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eCalculated\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eChemo-hydraulic parameter (N/m\u003csup\u003e2\u003c/sup\u003e)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{h}^{s}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.0\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\times\\)\u003c/span\u003e\u003c/span\u003e10\u003csup\u003e\u0026minus;4\u003c/sup\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eCalculated\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eMatrix porosity\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\varphi\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.3\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eEzendiokwere et al. (\u003cspan class=\"CitationRef\"\u003e2021\u003c/span\u003e)\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eReflection coefficient\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\mathfrak{R}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.3\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eCheng (\u003cspan class=\"CitationRef\"\u003e2016\u003c/span\u003e)\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eApparent mass diffusion coefficient (m\u003csup\u003e2\u003c/sup\u003e/s)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({D}_{c}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e6.0\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\times\\)\u003c/span\u003e\u003c/span\u003e10\u003csup\u003e\u0026minus;9\u003c/sup\u003e\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eCheng (\u003cspan class=\"CitationRef\"\u003e2016\u003c/span\u003e)\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e\n\u003ch2\u003eApplication in Soil Consolidation\u003c/h2\u003e\n\u003cdiv id=\"Sec12\" class=\"Section3\"\u003e\n\u003ch2\u003eGoverning Equations\u003c/h2\u003e\n\u003cp\u003eTo solve mathematical problems, we can remove variables from the constitutive relations, equilibrium and continuity equations, and flux laws to create a set of partial differential equations that can be used to solve initial and boundary value problems. In our research, we presume that the hyperfiltration effect is minimal, indicating that the movement of solute concentration is influenced solely by the concentration gradient and not the pressure gradient. This is the same as assuming that k\u003csub\u003ecp\u003c/sub\u003e= 0. This particular model is widely used for practical purposes, and the majority of available analytical solutions are based on this model.\u003c/p\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec13\" class=\"Section2\"\u003e\n\u003ch2\u003eNavier Equation\u003c/h2\u003e\n\u003cp\u003eThe Navier-type equation for porochemoelasticity is attained by substituting the constitutive Eq.\u0026nbsp;(3.64) into the static equilibrium Eq.\u0026nbsp;(3.66), which produces:\u003c/p\u003e\n\u003cdiv id=\"Equ23\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ23\" class=\"mathdisplay\"\u003e$$G{\\nabla }^{2}\\overrightarrow{u}+\\left(K+\\frac{G}{3}\\right)\\nabla (\\nabla .\\overrightarrow{u})-\\alpha \\nabla p-{\\alpha }_{c}{\\nabla C}_{r}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e23\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec14\" class=\"Section3\"\u003e\n\u003ch2\u003eDffusion Equations\u003c/h2\u003e\n\u003cp\u003eAssuming the chemo-hydraulic coupling is weak, then \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{pc}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{cp}\\)\u003c/span\u003e\u003c/span\u003e become negligible such that the above pressure-based diffusion equations can be further reduced to the following for a one-dimensional problem like soil consolidation:\u003c/p\u003e\n\u003cdiv id=\"Equ24\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ24\" class=\"mathdisplay\"\u003e$$\\frac{\\partial p}{\\partial t}-\\kappa M\\frac{{\\partial }^{2}p}{\\partial {z}^{2}}=\\alpha M\\frac{\\partial {e}_{zz}}{\\partial t}+{\\beta }_{c}M\\frac{\\partial {C}_{r}}{\\partial t}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e24\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eHowever, the constitutive stress equation can equally be expressed as (Cheng, \u003cspan class=\"CitationRef\"\u003e2016\u003c/span\u003e):\u003c/p\u003e\n\u003cdiv id=\"Equ25\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ25\" class=\"mathdisplay\"\u003e$${\\sigma }_{zz}=\\left(K+\\frac{4G}{3}\\right){e}_{zz}-\\alpha p-{\\alpha }_{c}{\\beta }_{h}^{s}{C}_{r}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e25\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eMaking \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({e}_{zz}\\)\u003c/span\u003e\u003c/span\u003e the subject, we have:\u003c/p\u003e\n\u003cdiv id=\"Equ26\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ26\" class=\"mathdisplay\"\u003e$${\\sigma }_{zz}+\\alpha p+{\\alpha }_{c}{\\beta }_{h}^{s}{C}_{r}=\\left(\\frac{3K+4G}{3}\\right){e}_{zz}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e26\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ27\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ27\" class=\"mathdisplay\"\u003e$${e}_{zz}=\\frac{3\\left({\\sigma }_{zz}+\\alpha p+{\\alpha }_{c}{{\\beta }_{h}^{s}C}_{r}\\right)}{3K+4G}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e27\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eSubstituting the above expression for strain in the diffusion equation, we have:\u003c/p\u003e\n\u003cdiv id=\"Equ28\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ28\" class=\"mathdisplay\"\u003e$$\\frac{\\partial p}{\\partial t}-\\kappa M\\frac{{\\partial }^{2}p}{\\partial {z}^{2}}=\\frac{3\\alpha M}{3K+4G}\\left(\\frac{\\partial {\\sigma }_{zz}}{\\partial t}+\\alpha \\frac{\\partial p}{\\partial t}+{\\alpha }_{c}{\\beta }_{h}^{s}\\frac{\\partial {C}_{r}}{\\partial t}\\right)+{\\beta }_{c}M\\frac{\\partial {C}_{r}}{\\partial t}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e28\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eCollecting like terms,\u003c/p\u003e\n\u003cdiv id=\"Equ29\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ29\" class=\"mathdisplay\"\u003e$$\\frac{\\partial p}{\\partial t}-\\frac{3{\\alpha }^{2}M}{3K+4G}\\frac{\\partial p}{\\partial t}-\\kappa M\\frac{{\\partial }^{2}p}{\\partial {z}^{2}}=\\frac{3\\alpha M}{3K+4G}\\frac{\\partial {\\sigma }_{zz}}{\\partial t}+\\frac{3\\alpha M{\\alpha }_{c}{\\beta }_{h}^{s}}{3K+4G}\\frac{\\partial {C}_{r}}{\\partial t}+{\\beta }_{c}M\\frac{\\partial {C}_{r}}{\\partial t}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e29\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv class=\"BlockQuote\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\left(1-\\frac{3{\\alpha }^{2}M}{3K+4G}\\right)\\frac{\\partial p}{\\partial t}-\\kappa M\\frac{{\\partial }^{2}p}{\\partial {z}^{2}}=\\frac{3\\alpha M}{3K+4G}\\frac{\\partial {\\sigma }_{zz}}{\\partial t}+\\left(\\frac{3\\alpha M{\\alpha }_{c}{\\beta }_{h}^{s}}{3K+4G}+{\\beta }_{c}M\\right)\\frac{\\partial {C}_{r}}{\\partial t}\\)\u003c/span\u003e \u003c/span\u003e` (30)\u003c/p\u003e\n\u003c/div\u003e\n\u003cp\u003eSimilarly, the diffusion equation for solute transport can be reduced to:\u003c/p\u003e\n\u003cdiv id=\"Equ30\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ30\" class=\"mathdisplay\"\u003e$$\\varphi \\frac{\\partial {C}_{r}}{\\partial t}-{\\left(1-\\mathfrak{R}\\right)D}_{c}\\frac{{\\partial }^{2}{C}_{r}}{\\partial {z}^{2}}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e31\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec15\" class=\"Section2\"\u003e\n\u003ch2\u003eInitial and Boundary Conditions\u003c/h2\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(p\\left(z,t\\right),{C}_{r}(z,t)=0\\)\u003c/span\u003e \u003c/span\u003e for t\u0026thinsp;=\u0026thinsp;0 (32)\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(p\\left(z,t\\right),{C}_{r}(z,t)=0\\)\u003c/span\u003e \u003c/span\u003e for z\u0026thinsp;=\u0026thinsp;0 (33)\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\frac{\\partial p\\left(z,t\\right)}{\\partial z},\\frac{\\partial {C}_{r}\\left(z,t\\right)}{\\partial z}=0\\)\u003c/span\u003e \u003c/span\u003e for z=h (34)\u003c/p\u003e\n\u003cp\u003eThe initial condition is that there is an equilibrium between the pore pressure and solute concentration. The second condition states that the water pressure is zero when a load is applied due to the high permeability of the slab compared to the soil. The third condition specifies that there is no water escaping through the bottom. These conditions can be classified as the initial and boundary conditions, with the initial condition indicating that there is no change in water content when a load is applied, as the water can only escape at a finite rate.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec16\" class=\"Section2\"\u003e\n\u003ch2\u003eAnalytical Solution\u003c/h2\u003e\n\u003cp\u003eSoil consolidation is a uniaxial stress problem associated with the following relations, assuming the axis to be z:\u003c/p\u003e\n\u003cdiv id=\"Sec17\" class=\"Section3\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({u}_{z}={u}_{z}(z,t)\\)\u003c/span\u003e\u003c/span\u003e; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(p=p(z,t)\\)\u003c/span\u003e\u003c/span\u003e; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({C}_{r}={C}_{r}(z,t)\\)\u003c/span\u003e\u003c/span\u003e; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({u}_{x}={u}_{y}=0\\)\u003c/span\u003e\u003c/span\u003e (35)\u003c/p\u003e\n\u003cp\u003eWhich also leads to:\u003c/p\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec18\" class=\"Section2\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({e}_{zz}={e}_{zz}(z,t)\\)\u003c/span\u003e\u003c/span\u003e; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\zeta =\\zeta (z,t)\\)\u003c/span\u003e\u003c/span\u003e; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({e}_{xx}={e}_{yy}={e}_{xy}={e}_{xz}={e}_{yz}=0\\)\u003c/span\u003e\u003c/span\u003e (36)\u003c/p\u003e\n\u003cp\u003eAs a result of the equations above, the Navier equations can be simplified to:\u003c/p\u003e\n\u003cdiv id=\"Equ31\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ31\" class=\"mathdisplay\"\u003e$$\\left(K+\\frac{4G}{3}\\right)\\frac{{\\partial }^{2}{u}_{z}}{\\partial {z}^{2}}-\\alpha \\frac{\\partial p}{\\partial z}-{\\alpha }_{c}{\\beta }_{h}^{s}\\frac{\\partial {C}_{r}}{\\partial z}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e37\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe constitutive equations also become:\u003c/p\u003e\n\u003cdiv id=\"Equ32\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ32\" class=\"mathdisplay\"\u003e$${\\sigma }_{zz}=\\left(K+\\frac{4G}{3}\\right){e}_{zz}-\\alpha p-{\\alpha }_{c}{\\beta }_{h}^{s}{C}_{r}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e38\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ33\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ33\" class=\"mathdisplay\"\u003e$${\\sigma }_{xx}={\\sigma }_{yy}=\\left(K+\\frac{4G}{3}\\right){e}_{zz}-\\alpha p-{\\alpha }_{c}{{\\beta }_{h}^{s}C}_{r}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e39\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ34\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ34\" class=\"mathdisplay\"\u003e$${\\sigma }_{xy}={\\sigma }_{xz}={\\sigma }_{yz}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e40\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ35\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ35\" class=\"mathdisplay\"\u003e$$p=M\\left(\\zeta +\\alpha e+{\\beta }_{c}{C}_{r}\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e41\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eUsing the above relations, the equilibrium equation further reduces to:\u003c/p\u003e\n\u003cdiv id=\"Equ36\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ36\" class=\"mathdisplay\"\u003e$$\\frac{\\partial {\\sigma }_{zz}}{\\partial z}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e42\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThereby, revealing the following functional dependencies:\u003c/p\u003e\n\u003cdiv id=\"Sec19\" class=\"Section3\"\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{zz}={\\sigma }_{zz}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{xx}={\\sigma }_{xx}(z,t)\\)\u003c/span\u003e\u003c/span\u003e; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{yy}={\\sigma }_{yy}(z,t)\\)\u003c/span\u003e\u003c/span\u003e (43)\u003c/p\u003e\n\u003cp\u003eAnd the diffusion equations become:\u003c/p\u003e\n\u003cdiv id=\"Equ37\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ37\" class=\"mathdisplay\"\u003e$$\\frac{\\partial p}{\\partial t}-\\frac{\\kappa M(3K+4G)}{\\left(3K+4G\\right)+3{\\alpha }^{2}M}\\frac{{\\partial }^{2}p}{\\partial {z}^{2}}=\\frac{3\\alpha M}{\\left(3K+4G\\right)+3{\\alpha }^{2}M}\\frac{\\partial {\\sigma }_{zz}}{\\partial t}+\\left(\\frac{3\\alpha M{\\alpha }_{c}{\\beta }_{h}^{s}}{3K+4G}+{\\beta }_{c}M\\right).\\left[\\frac{(3K+4G)}{\\left(3K+4G\\right)+3{\\alpha }^{2}M}\\right]\\frac{\\partial {C}_{r}}{\\partial t}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e44\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ38\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ38\" class=\"mathdisplay\"\u003e$$\\frac{\\partial {C}_{r}}{\\partial t}-\\frac{{\\left(1-\\mathfrak{R}\\right)D}_{c}}{\\varphi }\\frac{{\\partial }^{2}{C}_{r}}{\\partial {z}^{2}}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e45\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec20\" class=\"Section2\"\u003e\n\u003ch2\u003eChemical Loading\u003c/h2\u003e\n\u003cp\u003eThis describes the problem of applying the load through the introduction of a chemical potential possibly by an infiltrating fluid, which creates changes in solute (salt) concentration in the soil, while maintaining all other variables at a constant, the above scenario results in the subsequent set of boundary conditions \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(t={0}^{+}\\)\u003c/span\u003e\u003c/span\u003e:\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{zz}=0\\)\u003c/span\u003e \u003c/span\u003e; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(p=0\\)\u003c/span\u003e\u003c/span\u003e; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({C}_{r}=R\\)\u003c/span\u003e\u003c/span\u003e; at \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(z=0\\)\u003c/span\u003e\u003c/span\u003e (46)\u003c/p\u003e\n\u003cp\u003eDue to the above boundary condition, the diffusion equations reduces to:\u003c/p\u003e\n\u003cdiv id=\"Equ39\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ39\" class=\"mathdisplay\"\u003e$$\\frac{\\partial p}{\\partial t}-\\frac{\\kappa M(3K+4G)}{\\left(3K+4G\\right)+3{\\alpha }^{2}M}\\frac{{\\partial }^{2}p}{\\partial {z}^{2}}=\\left(\\frac{3\\alpha M{\\alpha }_{c}{\\beta }_{h}^{s}}{3K+4G}+{\\beta }_{c}M\\right).\\left[\\frac{(3K+4G)}{\\left(3K+4G\\right)+3{\\alpha }^{2}M}\\right]\\frac{\\partial {C}_{r}}{\\partial t}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e47\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ40\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ40\" class=\"mathdisplay\"\u003e$$\\frac{\\partial {C}_{r}}{\\partial t}-\\frac{{\\left(1-\\mathfrak{R}\\right)D}_{c}}{\\varphi }\\frac{{\\partial }^{2}{C}_{r}}{\\partial {z}^{2}}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e48\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe equations above are decoupled from the Navier equation, and can be resolved independent of it. To transform from partial differential equations to ordinary differential equations, we utilize Laplace transform on the diffusion equations.\u003c/p\u003e\n\u003cdiv id=\"Equ41\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ41\" class=\"mathdisplay\"\u003e$$s\\stackrel{\\sim}{p}-c\\frac{{d}^{2}\\stackrel{\\sim}{p}}{d{z}^{2}}=sd{\\stackrel{\\sim}{C}}_{r}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e49\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ42\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ42\" class=\"mathdisplay\"\u003e$$s{\\stackrel{\\sim}{C}}_{r}-a\\frac{{d}^{2}{\\stackrel{\\sim}{C}}_{r}}{d{z}^{2}}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e50\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWhere\u003c/p\u003e\n\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equb\" class=\"mathdisplay\"\u003e$$c=\\frac{\\kappa M(3K+4G)}{\\left(3K+4G\\right)+3{\\alpha }^{2}M}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equc\" class=\"mathdisplay\"\u003e$$d=\\left(\\frac{3\\alpha M{\\alpha }_{c}{\\beta }_{h}^{s}}{3K+4G}+{\\beta }_{c}M\\right).\\left[\\frac{(3K+4G)}{\\left(3K+4G\\right)+3{\\alpha }^{2}M}\\right]$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equd\" class=\"mathdisplay\"\u003e$$a=\\frac{{\\left(1-\\mathfrak{R}\\right)D}_{c}}{\\varphi }$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe above equations can be rearranged like this:\u003c/p\u003e\n\u003cdiv id=\"Equ43\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ43\" class=\"mathdisplay\"\u003e$$c\\frac{{d}^{2}\\stackrel{\\sim}{p}}{d{z}^{2}}+sd{\\stackrel{\\sim}{C}}_{r}-s\\stackrel{\\sim}{p}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e51\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ44\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ44\" class=\"mathdisplay\"\u003e$$a\\frac{{d}^{2}{\\stackrel{\\sim}{C}}_{r}}{d{z}^{2}}-s{\\stackrel{\\sim}{C}}_{r}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e52\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ45\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ45\" class=\"mathdisplay\"\u003e$$\\frac{{d}^{2}\\stackrel{\\sim}{p}}{d{z}^{2}}-\\frac{s}{c}\\stackrel{\\sim}{p}+s\\frac{d}{c}{\\stackrel{\\sim}{C}}_{r}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e53\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ46\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ46\" class=\"mathdisplay\"\u003e$$\\frac{{d}^{2}{\\stackrel{\\sim}{C}}_{r}}{d{z}^{2}}-\\frac{s}{a}{\\stackrel{\\sim}{C}}_{r}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e54\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eFurthermore, the solute diffusion equation can be independently solved due to the absence of a coupling term between both diffusion equations.\u003c/p\u003e\n\u003cp\u003eFrom Mathematica, the solution to the above simultaneous equations in Laplace form becomes:\u003c/p\u003e\n\u003cdiv id=\"Equ47\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ47\" class=\"mathdisplay\"\u003e$$\\stackrel{\\sim}{p}=-\\frac{ad{e}^{-\\frac{\\sqrt{s}z}{\\sqrt{a}}-\\frac{\\sqrt{s}z}{\\sqrt{c}}}\\left({e}^{\\frac{\\sqrt{s}z}{\\sqrt{a}}}-{e}^{\\frac{\\sqrt{s}z}{\\sqrt{c}}}-{e}^{\\frac{2\\sqrt{s}z}{\\sqrt{a}}+\\frac{\\sqrt{s}z}{\\sqrt{c}}}+{e}^{\\frac{\\sqrt{s}z}{\\sqrt{a}}+\\frac{2\\sqrt{s}z}{\\sqrt{c}}}\\right)R}{2(a-c)}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e55\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ48\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ48\" class=\"mathdisplay\"\u003e$${\\stackrel{\\sim}{C}}_{r}=\\frac{1}{2}{e}^{-\\frac{\\sqrt{s}z}{\\sqrt{a}}}(1+{e}^{\\frac{2\\sqrt{s}z}{\\sqrt{a}}})R$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e56\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eOn expanding and simplifying the above equations respectively, we have:\u003c/p\u003e\n\u003cdiv id=\"Equ49\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ49\" class=\"mathdisplay\"\u003e$$\\stackrel{\\sim}{p}=\\frac{Rad{e}^{-\\frac{\\sqrt{s}z}{\\sqrt{a}}}}{2(a-c)}+\\frac{Rad{e}^{\\frac{\\sqrt{s}z}{\\sqrt{a}}}}{2(a-c)}-\\frac{Rad{e}^{-\\frac{\\sqrt{s}z}{\\sqrt{c}}}}{2(a-c)}-\\frac{Rad{e}^{\\frac{\\sqrt{s}z}{\\sqrt{c}}}}{2(a-c)}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e57\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ50\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ50\" class=\"mathdisplay\"\u003e$${\\stackrel{\\sim}{C}}_{r}=\\frac{R}{2}\\left({e}^{\\frac{\\sqrt{s}z}{\\sqrt{a}}}+{e}^{-\\frac{\\sqrt{s}z}{\\sqrt{a}}}\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e58\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe pressure solution above can be transformed to:\u003c/p\u003e\n\u003cdiv id=\"Equ51\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ51\" class=\"mathdisplay\"\u003e$$\\stackrel{\\sim}{p}={\\stackrel{\\sim}{p}}_{1}+{\\stackrel{\\sim}{p}}_{2}+{\\stackrel{\\sim}{p}}_{3}+{\\stackrel{\\sim}{p}}_{4}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e59\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWhere\u003c/p\u003e\n\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Eque\" class=\"mathdisplay\"\u003e$${\\stackrel{\\sim}{p}}_{1}=\\frac{Rad{e}^{-\\frac{\\sqrt{s}z}{\\sqrt{a}}}}{2(a-c)}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equf\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equf\" class=\"mathdisplay\"\u003e$${\\stackrel{\\sim}{p}}_{2}=\\frac{Rad{e}^{\\frac{\\sqrt{s}z}{\\sqrt{a}}}}{2(a-c)}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equg\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equg\" class=\"mathdisplay\"\u003e$${\\stackrel{\\sim}{p}}_{3}=-\\frac{Rad{e}^{-\\frac{\\sqrt{s}z}{\\sqrt{c}}}}{2(a-c)}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equh\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equh\" class=\"mathdisplay\"\u003e$${\\stackrel{\\sim}{p}}_{4}=-\\frac{Rad{e}^{\\frac{\\sqrt{s}z}{\\sqrt{c}}}}{2(a-c)}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eSuch that\u003c/p\u003e\n\u003cdiv id=\"Equ52\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ52\" class=\"mathdisplay\"\u003e$${\\mathcal{L}}^{-1}\\left(\\stackrel{\\sim}{p}\\right)={\\mathcal{L}}^{-1}\\left({\\stackrel{\\sim}{p}}_{1}+{\\stackrel{\\sim}{p}}_{2}+{\\stackrel{\\sim}{p}}_{3}+{\\stackrel{\\sim}{p}}_{4}\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e60\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ53\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ53\" class=\"mathdisplay\"\u003e$${\\mathcal{L}}^{-1}\\left({\\stackrel{\\sim}{p}}_{1}\\right)={\\mathcal{L}}^{-1}\\left(\\frac{Rad{e}^{-\\frac{\\sqrt{s}z}{\\sqrt{a}}}}{2(a-c)}\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e61\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ54\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ54\" class=\"mathdisplay\"\u003e$$=\\frac{Rdz\\sqrt{a}{e}^{-\\frac{{z}^{2}}{4at}}}{4(a-c)\\sqrt{\\pi }{t}^{3/2}} \\text{i}\\text{f} \\frac{z}{\\sqrt{a}}\u0026gt;0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e62\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ55\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ55\" class=\"mathdisplay\"\u003e$${\\mathcal{L}}^{-1}\\left({\\stackrel{\\sim}{p}}_{2}\\right)={\\mathcal{L}}^{-1}\\left(\\frac{Rad{e}^{\\frac{\\sqrt{s}z}{\\sqrt{a}}}}{2(a-c)}\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e63\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ56\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ56\" class=\"mathdisplay\"\u003e$$=-\\frac{Rzd\\sqrt{a}{e}^{-\\frac{{z}^{2}}{4at}}}{4(a-c)\\sqrt{\\pi }{t}^{3/2}} \\text{i}\\text{f} \\frac{z}{\\sqrt{a}}\u0026lt;0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e64\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ57\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ57\" class=\"mathdisplay\"\u003e$${\\mathcal{L}}^{-1}\\left({\\stackrel{\\sim}{p}}_{3}\\right)={\\mathcal{L}}^{-1}\\left(-\\frac{Rad{e}^{-\\frac{\\sqrt{s}z}{\\sqrt{c}}}}{2(a-c)}\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e65\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ58\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ58\" class=\"mathdisplay\"\u003e$$=-\\frac{ad{e}^{-\\frac{{z}^{2}}{4ct}}Rz}{4\\left(a-c\\right)\\sqrt{c}\\sqrt{\\pi }{t}^{3/2}} \\text{i}\\text{f} \\frac{z}{\\sqrt{c}}\u0026gt;0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e66\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ59\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ59\" class=\"mathdisplay\"\u003e$${\\mathcal{L}}^{-1}\\left({\\stackrel{\\sim}{p}}_{4}\\right)={\\mathcal{L}}^{-1}\\left(-\\frac{Rad{e}^{\\frac{\\sqrt{s}z}{\\sqrt{c}}}}{2(a-c)}\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e67\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ60\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ60\" class=\"mathdisplay\"\u003e$$=\\frac{ad{e}^{-\\frac{{z}^{2}}{4ct}}Rz}{4(a-c)\\sqrt{c}\\sqrt{\\pi }{t}^{3/2}} \\text{i}\\text{f} \\frac{z}{\\sqrt{c}}\u0026lt;0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e68\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eTherefore,\u003c/p\u003e\n\u003cdiv id=\"Equ61\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ61\" class=\"mathdisplay\"\u003e$$p={p}_{1}+{p}_{2}+{p}_{3}+{p}_{4}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e69\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWhere\u003c/p\u003e\n\u003cdiv id=\"Equi\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equi\" class=\"mathdisplay\"\u003e$$p={\\mathcal{L}}^{-1}\\left(\\stackrel{\\sim}{p}\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equj\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equj\" class=\"mathdisplay\"\u003e$${p}_{1}={\\mathcal{L}}^{-1}\\left({\\stackrel{\\sim}{p}}_{1}\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equk\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equk\" class=\"mathdisplay\"\u003e$${p}_{2}={\\mathcal{L}}^{-1}\\left({\\stackrel{\\sim}{p}}_{2}\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equl\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equl\" class=\"mathdisplay\"\u003e$${p}_{3}={\\mathcal{L}}^{-1}\\left({\\stackrel{\\sim}{p}}_{3}\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equm\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equm\" class=\"mathdisplay\"\u003e$${p}_{4}={\\mathcal{L}}^{-1}\\left({\\stackrel{\\sim}{p}}_{4}\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eSimilarly, the solution for the concentration ratio can be presented as:\u003c/p\u003e\n\u003cdiv id=\"Equ62\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ62\" class=\"mathdisplay\"\u003e$${\\stackrel{\\sim}{C}}_{r}=\\frac{R}{2}\\left({\\stackrel{\\sim}{C}}_{r1}+{\\stackrel{\\sim}{C}}_{r2}\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e70\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWhere\u003c/p\u003e\n\u003cdiv id=\"Equn\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equn\" class=\"mathdisplay\"\u003e$${\\stackrel{\\sim}{C}}_{r1}={e}^{\\frac{\\sqrt{s}z}{\\sqrt{a}}}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equo\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equo\" class=\"mathdisplay\"\u003e$${\\stackrel{\\sim}{C}}_{r2}={e}^{-\\frac{\\sqrt{s}z}{\\sqrt{a}}}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eHence,\u003c/p\u003e\n\u003cdiv id=\"Equ63\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ63\" class=\"mathdisplay\"\u003e$${\\mathcal{L}}^{-1}\\left({\\stackrel{\\sim}{C}}_{r}\\right)=\\frac{R}{2}\\left[{\\mathcal{L}}^{-1}\\left({\\stackrel{\\sim}{C}}_{r1}\\right)+{\\mathcal{L}}^{-1}\\left({\\stackrel{\\sim}{C}}_{r2}\\right)\\right]$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e71\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ64\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ64\" class=\"mathdisplay\"\u003e$${\\mathcal{L}}^{-1}\\left({\\stackrel{\\sim}{C}}_{r1}\\right)={\\mathcal{L}}^{-1}\\left({e}^{\\frac{\\sqrt{s}z}{\\sqrt{a}}}\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e72\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ65\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ65\" class=\"mathdisplay\"\u003e$$=-\\frac{z{e}^{-\\frac{{z}^{2}}{4at}}}{2\\sqrt{a}\\sqrt{\\pi }{t}^{3/2}} \\text{i}\\text{f} \\frac{z}{\\sqrt{a}}\u0026lt;0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e73\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ66\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ66\" class=\"mathdisplay\"\u003e$${\\mathcal{L}}^{-1}\\left({\\stackrel{\\sim}{C}}_{r2}\\right)={\\mathcal{L}}^{-1}\\left({e}^{-\\frac{\\sqrt{s}z}{\\sqrt{a}}}\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e74\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(=-\\frac{z{e}^{-\\frac{{z}^{2}}{at}}}{\\sqrt{a}\\sqrt{\\pi }{t}^{3/2}}\\)\u003c/span\u003e \u003c/span\u003e (if \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{z}{\\sqrt{a}}\u0026lt;0\\)\u003c/span\u003e\u003c/span\u003e) or \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{e}^{-\\frac{{z}^{2}}{4at}}z}{2\\sqrt{a}\\sqrt{\\pi }{t}^{3/2}}\\)\u003c/span\u003e\u003c/span\u003e (if \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{z}{\\sqrt{a}}\u0026gt;0\\)\u003c/span\u003e\u003c/span\u003e) (75)\u003c/p\u003e\n\u003cp\u003eTherefore,\u003c/p\u003e\n\u003cdiv id=\"Equ67\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ67\" class=\"mathdisplay\"\u003e$${C}_{r}=\\frac{R}{2}\\left({C}_{r1}+{C}_{r2}\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e76\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWhere\u003c/p\u003e\n\u003cdiv id=\"Equp\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equp\" class=\"mathdisplay\"\u003e$${C}_{r}={\\mathcal{L}}^{-1}\\left({\\stackrel{\\sim}{C}}_{r}\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equq\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equq\" class=\"mathdisplay\"\u003e$${C}_{r1}={\\mathcal{L}}^{-1}\\left({\\stackrel{\\sim}{C}}_{r1}\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equr\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equr\" class=\"mathdisplay\"\u003e$${C}_{r2}={\\mathcal{L}}^{-1}\\left({\\stackrel{\\sim}{C}}_{r2}\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eConsidering the nature of the above solutions, they are all contingent on the values of either \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{z}{\\sqrt{a}}\\)\u003c/span\u003e\u003c/span\u003e or \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{z}{\\sqrt{c}}\\)\u003c/span\u003e\u003c/span\u003e being positive or negative. The expressions can only be negative only when z is negative, since the square root of a real number cannot be negative. Given the geometry of the problem at hand, in which the positive direction points into the ground, negative z (corresponding to the opposite direction pointing away from the soil and into the sky) is not admissible. Hence, the final solutions for both the pressure and the concentration ratio will only contain solutions dependent on positive values for both \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{z}{\\sqrt{a}}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{z}{\\sqrt{c}}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eConsequently,\u003c/p\u003e\n\u003cdiv id=\"Equ68\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ68\" class=\"mathdisplay\"\u003e$$p={p}_{1}+{p}_{3}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e77\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ69\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ69\" class=\"mathdisplay\"\u003e$$p=\\frac{Rdz\\sqrt{a}{e}^{-\\frac{{z}^{2}}{4at}}}{4(a-c)\\sqrt{\\pi }{t}^{3/2}}-\\frac{ad{e}^{-\\frac{{z}^{2}}{4ct}}Rz}{4\\left(a-c\\right)\\sqrt{c}\\sqrt{\\pi }{t}^{3/2}}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e78\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eAnd\u003c/p\u003e\n\u003cdiv id=\"Equ70\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ70\" class=\"mathdisplay\"\u003e$${C}_{r}={\\frac{R}{2}C}_{r2}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e79\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ71\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ71\" class=\"mathdisplay\"\u003e$${C}_{r}=\\frac{R{e}^{-\\frac{{z}^{2}}{4at}}z}{4\\sqrt{a}\\sqrt{\\pi }{t}^{3/2}}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e80\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eRecall that,\u003c/p\u003e\n\u003cdiv id=\"Equs\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equs\" class=\"mathdisplay\"\u003e$${\\sigma }_{zz}=\\left(K+\\frac{4G}{3}\\right){e}_{zz}-\\alpha p-{\\alpha }_{c}{{\\beta }_{h}^{s}C}_{r}$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eSuch that with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{zz}\\)\u003c/span\u003e\u003c/span\u003e=0, we have\u003c/p\u003e\n\u003cdiv id=\"Equ72\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ72\" class=\"mathdisplay\"\u003e$${e}_{zz}=\\left(\\frac{3}{3K+4G}\\right)\\left(\\alpha p+{\\alpha }_{c}{{\\beta }_{h}^{s}C}_{r}\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e81\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eBy integrating the equation above with respect to z, we can find the displacement u\u003csub\u003ez\u003c/sub\u003e, which is given by:\u003c/p\u003e\n\u003cdiv id=\"Equ73\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ73\" class=\"mathdisplay\"\u003e$${u}_{z}\\left(z,t\\right)=\\left(\\frac{3}{3K+4G}\\right)\\left\\{\\frac{Rd\\alpha \\sqrt{a}(-2a{e}^{-\\frac{{z}^{2}}{4at}}t+2\\sqrt{a}\\sqrt{c}{e}^{-\\frac{{z}^{2}}{4ct}}t)}{4(a-c)\\sqrt{\\pi }{t}^{3/2}}-\\frac{{\\alpha }_{c}R\\sqrt{a}{e}^{-\\frac{{z}^{2}}{4at}}}{2\\sqrt{\\pi }\\sqrt{t}}\\right\\}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e82\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eAnother intriguing aspect is the chemical-induced water flow. Based on\u003c/p\u003e\n\u003cdiv id=\"Equ74\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ74\" class=\"mathdisplay\"\u003e$$\\overrightarrow{q}=-\\kappa \\nabla p+{C}_{s}{k}_{pc}\\nabla {C}_{r}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e83\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWe have for the z direction,\u003c/p\u003e\n\u003cdiv id=\"Equ75\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ75\" class=\"mathdisplay\"\u003e$${q}_{z}(z,t)=-\\frac{\\kappa Rd{e}^{-\\frac{\\left(a+c\\right){z}^{2}}{4act}}\\left(2a{c}^{3/2}{e}^{\\frac{{z}^{2}}{4ct}}t-{c}^{3/2}{e}^{\\frac{{z}^{2}}{4ct}}{z}^{2}+{a}^{3/2}{e}^{\\frac{{z}^{2}}{4at}}\\left(-2ct+{z}^{2}\\right)\\right)}{8\\sqrt{a}\\left(a-c\\right){c}^{3/2}\\sqrt{\\pi }{t}^{5/2}}+\\frac{{RC}_{s}{k}_{pc}}{2}\\left(\\frac{{e}^{-\\frac{{z}^{2}}{4at}}}{2\\sqrt{a}\\sqrt{\\pi }{t}^{3/2}}-\\frac{{e}^{-\\frac{{z}^{2}}{4at}}{z}^{2}}{4{a}^{3/2}\\sqrt{\\pi }{t}^{5/2}}\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e84\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe water volume (per unit area) leaving the soil is\u003c/p\u003e\n\u003cdiv id=\"Equ76\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ76\" class=\"mathdisplay\"\u003e$$V\\left(t\\right)={\\int }_{0}^{t}{q}_{z}\\left(0,t\\right)dt$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e85\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ77\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ77\" class=\"mathdisplay\"\u003e$$V\\left(t\\right)={\\int }_{0}^{t}\\left[-\\frac{\\kappa Rd\\left(2a{c}^{3/2}t+{a}^{3/2}\\left(-2ct\\right)\\right)}{8\\sqrt{a}\\left(a-c\\right){c}^{3/2}\\sqrt{\\pi }{t}^{5/2}}+\\frac{{RC}_{s}{k}_{pc}}{2}\\left(\\frac{1}{2\\sqrt{a}\\sqrt{\\pi }{t}^{3/2}}\\right)\\right]dt$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e86\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ78\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ78\" class=\"mathdisplay\"\u003e$$={\\int }_{0}^{t}\\left[-\\frac{\\kappa Rd\\left(2a{c}^{3/2}t+{a}^{3/2}\\left(-2ct\\right)\\right)}{8\\sqrt{a}\\left(a-c\\right){c}^{3/2}\\sqrt{\\pi }{t}^{5/2}}\\right]dt+{\\int }_{0}^{t}\\left[+\\frac{{RC}_{s}{k}_{pc}}{2}\\left(\\frac{1}{2\\sqrt{a}\\sqrt{\\pi }{t}^{3/2}}\\right)\\right]dt$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e87\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ79\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ79\" class=\"mathdisplay\"\u003e$$V\\left(t\\right)=-\\frac{\\sqrt{a}dR\\kappa }{2\\left(\\sqrt{a}+\\sqrt{c}\\right)\\sqrt{c}\\sqrt{\\pi }\\sqrt{t}}-\\frac{R{C}_{s}{k}_{pc}}{2\\sqrt{a}\\sqrt{\\pi }\\sqrt{t}}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e88\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eMeanwhile, matrix bulk modulus, Biot effective stress parameter and Biot effective chemical stress parameter were earlier derived from the effective solid bulk modulus, bulk modulus of porosity and bulk modulus of microinhomogeneity. They are expressed as:\u003c/p\u003e\n\u003cdiv id=\"Equ80\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ80\" class=\"mathdisplay\"\u003e$$K=\\frac{-{\\left(1-\\varphi \\right)}^{2}{K}_{\\varphi }\\left\\{\\left(1-\\varphi \\right){K}_{\\alpha }{K}_{\\varphi } -{{K}_{\\psi }}^{2}\\right\\}}{2{K}_{\\psi }{K}_{\\varphi }\\left(1-\\varphi \\right){{-K}_{\\varphi }}^{2}{\\left(1-\\varphi \\right)}^{2}{{-K}_{\\psi }}^{2}-\\left(1-\\varphi \\right){K}_{\\alpha }{K}_{\\varphi }}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e89\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ81\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ81\" class=\"mathdisplay\"\u003e$$\\alpha =\\frac{-\\left\\{{{K}_{\\psi }}^{2}-{K}_{\\psi }{K}_{\\varphi }\\left(1-\\varphi \\right)+\\left(1-\\varphi \\right){K}_{\\alpha }{K}_{\\varphi }+{{\\varphi K}_{\\varphi }}^{2}{\\left(1-\\varphi \\right)}^{2}-{\\varphi K}_{\\psi }{K}_{\\varphi }\\left(1-\\varphi \\right)\\right\\}}{2{K}_{\\psi }{K}_{\\varphi }\\left(1-\\varphi \\right){{-K}_{\\varphi }}^{2}{\\left(1-\\varphi \\right)}^{2}{{-K}_{\\psi }}^{2}-\\left(1-\\varphi \\right){K}_{\\alpha }{K}_{\\varphi }}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e90\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ82\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ82\" class=\"mathdisplay\"\u003e$${\\alpha }_{c}=\\frac{\\{{K}_{\\psi }{K}_{\\varphi }\\left(1-\\varphi \\right)-{{K}_{\\varphi }}^{2}{\\left(1-\\varphi \\right)}^{2}\\}}{2{K}_{\\psi }{K}_{\\varphi }\\left(1-\\varphi \\right){{-K}_{\\varphi }}^{2}{\\left(1-\\varphi \\right)}^{2}{{-K}_{\\psi }}^{2}-\\left(1-\\varphi \\right){K}_{\\alpha }{K}_{\\varphi }}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e91\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e"},{"header":"Results and Discussion","content":"\u003cp\u003eFigure \u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e shows the evolution of generated pore pressure within the residual soil sample with passage of time for chemical loading. At t\u0026thinsp;=\u0026thinsp;0s, the pore pressure can be noticed be initially zero, since prior to this point, no chemical load in the form of solute concentration differences has been introduced possibly by an infiltrating liquid. But at t\u0026thinsp;=\u0026thinsp;0.5s, a negative pore pressure was can be observed, before a sharp increase in pore pressure as the effect of the chemical loading is felt. The generated pore pressure thereafter reached a peak value of more than 0.03MPa. The\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003etime lag before the chemical effect was felt can be attributed to poroelastic effects occasioned by the tortuosity of the porous soil material. Such that a finite amount of time is needed for the migration of the pore pressure field.\u003c/p\u003e \u003cp\u003eIn addition, the initially negative pore pressure can be explained by the fact that a soil segment at a depth of 5m would normally be expected to lie above the water table. Normally, soil found above the water table is expected to have a pore pressure below the atmospheric pressure (negative pore pressure), while soil below the water table is expected to have pore pressures above the atmospheric pressure (positive pore pressure). This explains the initially negative generated pore pressures in the consolidating soil before the effects of chemical load were felt. Beyond 0.5s, what can be noticed is a simultaneous reduction in the peak pore pressure value and increment in the portion of the consolidating soil under negative pore pressures.\u003c/p\u003e \u003cp\u003eThis shows increasing segments of the soil returning to their initial state after the passage of the pore pressure front. This can be traced to the downward movement of the chemical effects under osmotic pressure and its accompanying increase in pore pressure. The net effect is further penetration of the generated pore pressure field in deeper segments of the soil. This scenario continued to played out even after 5s, 20s. But after 60s, most of the soil segment has returned to negative pore pressure regime after the passage of the pore pressure front. Particularly, after 100s, all the soil segments have returned to negative pore pressures after the pore pressure front has successfully passed.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e shows the evolution of solute concentration ratio with passage of time for chemical loading. The concentration ratio as presented is the ratio of the solute concentration of the infiltrating liquid to the solute concentration of the soil. Such that solute concentration ratios greater than 1 represent situations where the infiltrating liquid has solute concentration higher than the solute concentration of the soil, which gives rise to positive osmotic pressures. While solute concentration ratios less than one represent situations where the solute concentration of the soil is higher than the solute concentration of the infiltrating liquid, which gives rise to negative osmotic pressures. The criteria excludes negative solute concentration ratios, since the ratio of two positive real numbers cannot be negative.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFrom the figure, it can be seen that at t\u0026thinsp;=\u0026thinsp;0.5s, the solute concentration ratio increased from zero, peaked at a depth of about 1.5m, before gradually reducing to zero again as the depth increased. Like earlier described, the increasing solute concentration ratio is synonymous with increasing quantities of the infiltrating liquid over the corresponding quantity of the \u003cem\u003ein situ\u003c/em\u003e pore fluid at each depth. This description directly follows from the definition of the solute concentration described above.\u003c/p\u003e \u003cp\u003eThis observation shows that even though the chemical loading was instantaneous, it took some time before its effect could be felt everywhere according to the poroelastic theory, gauging by the concentration front depicted in the figure at t\u0026thinsp;=\u0026thinsp;0.5s. The same pattern of curve can equally be noticed for t\u0026thinsp;=\u0026thinsp;1s and t\u0026thinsp;=\u0026thinsp;5s, but with an increasingly wider base as time progresses. Such that at t\u0026thinsp;=\u0026thinsp;5s, the right leg of the inverted v-shape of the curve was no longer visible. This clearly shows a net chemical concentration front advancing under osmotic pressure.\u003c/p\u003e \u003cp\u003eHowever, the strength of the chemical difference seemed to be waning as time passaged. This position is supported by the decreasing peak of the inverted v-curve, which decreased from 0.025 to 0.0025 after just five seconds. After 20s, only the left leg of the inverted v-curve can be seen in the figure, with further decrease in the peak value. This shows that the chemical difference front has clearly advanced beyond the 5m depth currently considered for analysis. This scenario continued that for t\u0026thinsp;=\u0026thinsp;60s and t\u0026thinsp;=\u0026thinsp;100s, only the straight line segments of the inverted v-curve could be observed, which also depicts an active and advancing front.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e shows the evolution of vertical displacement with passage of time for chemical loading. From the figure, a quick response can be observed as soon as the infiltration started, which can be seen as negative displacement recorded at t\u0026thinsp;=\u0026thinsp;0.5s. Based on the geometry of the current application, positive displacements are displacements that occur in the direction of increasing depth, while negative displacements occur away from the soil. Consequently, the initial negative displacement recorded in the figure was a slight increase or elevation of the soil surface experienced as the infiltrating liquid that gave rise to the chemical loading penetrated the pore spaces of the soil in addition to the \u003cem\u003ein situ\u003c/em\u003e pore fluid. Since the load is purely chemical, without mechanical stress as depicted in the boundary conditions, the soil is free to briefly expand to accommodate the extra pore fluid.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eBut this increase in height was mainly felt to a depth of about 3m at t\u0026thinsp;=\u0026thinsp;0.5s, with the depth of largest displacement occurring less than a meter (1m) deep. But as time progressed, the displacement affected deeper sections of the soil, while the depth of largest displacement also moved deeper. Such that after 5s, the negative displacement has reached a depth of about 5m and the depth of largest displacement at about 1.5m deep. This observation clearly shows the case of an advancing concentration difference field moving to deeper sections of the soil. The above description is in agreement with the scenario already described for both the generated pore pressure and solute concentration ratio.\u003c/p\u003e \u003cp\u003eAfter 20s, a clear inflexion can be seen to be developing as areas previously under negative displacement are relaxed after the chemical effect front passed. This position is supported by the v-shaped curve appearing at t\u0026thinsp;=\u0026thinsp;20s, in which the depth of largest displacement had reached 3m. As time progressed further, the depth of largest displacement went deeper and areas behind the moving chemical loading front were furthered relaxed. Although relaxation took place afterwards, there was a residual displacement, possibly accounting for the remnant portions of the infiltrating left behind under capillary pressure.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e shows the evolution of the water flux induced by the chemical effect with passage of time for chemical loading. In transport phenomena, flux is frequently defined as the rate of flow of a property per unit area. Its dimension is usually [quantity].[time]\u003csup\u003e\u0026minus;1\u003c/sup\u003e.[area]\u003csup\u003e\u0026minus;1\u003c/sup\u003e, which effectively translates to [m/s] as presented in the figure. The scenario depicted in the figure can be described as oppositely related to the vertical displacement as it offers the explanation for the evolution of the vertical displacement with time. From the figure, it can be deduced that the flux of water flow in the soil was initially positive before turning negative. This can be explained by initial successful penetration of the infiltrating liquid until it meets increasingly tortuous paths as result of lower permeability.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe initial reaction is that of rejection as more fluid could not successfully penetrate the pores, which is seen as a sharp increase in the flux noticed first at t\u0026thinsp;=\u0026thinsp;0.5s. This possibly explains the initial negative displacement as a result of successful penetration. The pattern described above can be noticed for the subsequent timeframes reported. Consequently, it can be seen that as time progressed, the volume of positive flux gradually reduces and its pressure head is instead utilized in forcefully penetrating the porous structure of the soil. This is now seen in the gradual increase in flux after the initial decrease in flux. The gradual increase in flux represents increasingly successful penetration of the deeper sections, while the flux (flow) remained positive for shallower depths already penetrated.\u003c/p\u003e \u003cp\u003eIn addition, the depth of most negative flux equally moved with further penetration of the soil by the infiltrating liquid. This observation can be noticed after 60s, 100s and even 150s when it became further developed. The relatively longer duration can be traced to the tortuosity of the residual soil under study.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e shows the water volume (per unit area) leaving the soil after chemical loading. The figure shows that a portion of the water flux associated with the infiltrating liquid ends up leaving the soil through the sides. It can be deduced that the largest volume of water leaves from the shallower depths of the soil compared to the deeper sections. This is because since the infiltrating liquid penetrates from the top, it would take some time before it gets to the deeper sections of the soil due to porosity and permeability differences as depth increases. Hence, a decreasingly lower volume of water would be lost as the infiltrating liquid travels to the deeper sections of the soil. The loss of water experienced is actually expected as the application discussed is soil consolidation.\u003c/p\u003e"},{"header":"Conclusion","content":"\u003cp\u003eThe evolution of the generated pore pressure showed increasing segments of the soil returning to their initial state after the passage of the pore pressure front. This can be traced to the downward movement of the chemical effects under osmotic pressure and its accompanying increase in pore pressure. The net effect is further penetration of the generated pore pressure field in deeper segments of the soil. Even though the chemical loading was instantaneous, it took some time before its effect could be felt everywhere according to the poroelastic theory. This clearly shows a net chemical concentration front advancing under osmotic pressure. The vertical displacement showed a slight increase or elevation of the soil surface experienced as the infiltrating liquid that gave rise to the chemical loading penetrated the pore spaces of the soil in addition to the \u003cem\u003ein situ\u003c/em\u003e pore fluid.\u003c/p\u003e \u003cp\u003eThe flux of water flow in the soil was initially positive before turning negative, which can be explained by initial successful penetration of the infiltrating liquid until it met increasingly tortuous paths as result of lower permeability. Consequently, it can be seen that as time progressed, the volume of positive flux gradually reduces and the available pressure head is instead utilized in forcefully penetrating the porous structure of the soil. The volume (per unit area) of water leaving the soil after chemical loading demonstrated that a portion of the water flux associated with the infiltrating liquid ends up leaving the soil through the sides. This loss of water is actually expected as the application discussed is soil consolidation.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e \u003ch2\u003eCompeting interests:\u003c/h2\u003e \u003cp\u003eWe hereby declare that we do not have any conflicts of interest.\u003c/p\u003e \u003c/p\u003e\u003ch2\u003eFunding:\u003c/h2\u003e \u003cp\u003eNone of the authors received financial support for this research.\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eThe study's conception and design were the result of collaboration among all authors. I.D. conducted data collection, simulation, and analysis, with supervision provided by both O.M.E. and O.F.D. The final manuscript was read and approved by all authors.\u003c/p\u003e\u003ch2\u003eAcknowledgements:\u003c/h2\u003e \u003cp\u003eThe authors wish to acknowledge the department of civil engineering, University of Nigeria, Nsukka, for providing an enabling environment that made this research possible.\u003c/p\u003e\u003ch2\u003eAvailability of data and materials:\u003c/h2\u003e \u003cp\u003eThe published article contains all data that was produced or examined during the course of this study.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eAkhtar MN (2012) Role of soil mechanics in civil engineering. International Journal of Emerging trends in Engineering and Development, 6: 104\u0026ndash;111\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eBlight GE, Leong EC (2012). Mechanics of residual soils. Balkema, Rotterdam\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eCheng AH-D (2016) Poroelasticity. Springer, Zurich.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eBrand EW,Phillipson HB (1985) In: Technical committee 25 on the properties of Tropical and Residual Soils of the International Society for Soil Mechanics and Foundation Engineering (ed) Sampling and testing of residual soils. A review of international practice. Scorpion Press, Hong Kong\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eDuarte IMR \u0026amp; Rodrgues CMG (2018) Residual soils. In Encyclopedia of engineering geology (pp. 751\u0026ndash;752). Springer,Zurich\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eDuarte IMR (2002) Residual soils of granitoid rocks to south of the Tagus River: Geological and geotechnical characteristics. (Unpublished Doctoral Dissertation, in Portuguese). University of Evora, Portugal\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eEzendiokwere NE, Aimikhe VJ, Dosunmu A, Joel OF (2021) Influence of depth on induced geo-mechanical, chemical, and thermal poromechanical effects. Journal of Petroleum Exploration and Production Technology 11: 2917\u0026ndash;2930\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eGao J, Lin H, Wu B, Deng J, Liu H (2021) Porochemothermoelastic solutions considering fully coupled thermo-hydro-mechanical-chemical processes to analyze the stability of inclined boreholes in chemically active porous media. Computers and Geotechnics, 134: 104019.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKanfar MF, Chen Z, Rahman SS (2017) Analyzing wellbore stability in chemically-active anisotropic formations under thermal, hydraulic, mechanical and chemical loadings. Journal of Natural Gas Science \u0026amp; Engineering, 41: 93\u0026ndash;111\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eLi, WW, Wong KS (2001) Geotechnical properties of old alluvium in Singapore. J. Inst. Engs. 41: 10\u0026ndash;20\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eLiu C, Abousleiman YN (2018) Multiple porosity/multiple permeability inclined wellbore solutions with mudcake effects. SPE Journal, Paper 191136.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMehrabian A, Nguyen VX, Abousleiman YN (2020) Wellbore mechanics and stability in shale, In Dewars T, Heath J, Sanchez M (eds) Shale: Subsurface science and engineering, Geophysical Monograph 245.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eRodriguez TT (2005) Colluvium classification: a geotechnical approach. Dissertation, COPPE/UFRJ, Brazil.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eSchofield AN \u0026amp; Wroth CP (1968) Critical state soil mechanics. McGraw-Hill, London\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eTownsend FC (1985) Geotechnical characteristics of residual soils. J Geotech Eng 111(1): 77\u0026ndash;94\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eVaughan PR (1988) Characterising the mechanical properties of in-situ residual soil. In: Proceedings of the II international conference on geomechanics in tropical soils, Singapore.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eWesley LD (2010) Geotechnical engineering in residual soils. Wiley, Hoboken, USA.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eWibawa YS, Sugiarti K, Soebowo E (2018) Characteristics and engineering properties of residual soil of volcanic deposits. Paper presented at the IOP Conference Series: Earth and Environmental Sciences\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eZhang G, Whittle AJ, Nikolinakou AM, Germaine JT (2007) Characterization and engineering properties of the old alluvium in Puerto Rico. Paper presented at the 2nd Int. Workshop on Characterization \u0026amp; Engineering Properties of Natural Soils, Singapore.\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"poroelasticity, chemical loading, soil consolidation, residual soils","lastPublishedDoi":"10.21203/rs.3.rs-3992157/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3992157/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAn analytical chemo-hydromechanical model of residual soils was presented in this study. The modeling procedure utilized a unique phenomenological approach, which highlighted some of the macro-mechanical influences responsible for the behavior of residual soils. The macromechanical analysis yielded an extended poroelastic theory. The developed model was applied in solving a typical civil engineering problem of soil consolidation. For a chemical loading treatment, the problem was using requisite boundary conditions. Thereafter, the mathematical software \u0026ldquo;Mathematica\u0026rdquo; was used for solving the ensuing diffusion equations using physico-chemical properties typical of a residual soil sample. Even though the chemical loading was instantaneous, it took some time before its effect could be felt everywhere according to the poroelastic theory. From obtained results, the evolution of the generated pore pressure showed increasing segments of the soil returning to their initial state after the passage of the pore pressure front. The vertical displacement showed a slight increase or elevation of the soil surface. The flux of water flow in the soil was initially positive before turning negative, which can be explained by initial successful penetration of the infiltrating liquid until it met increasingly tortuous paths as result of lower permeability. The water volume (per unit area) leaving the soil after chemical loading demonstrated that a portion of the water flux associated with the infiltrating liquid ends up leaving the soil through the sides.\u003c/p\u003e","manuscriptTitle":"Poroelastic Effects of Chemical Loading in Consolidation of Residual Soils","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-02-29 05:46:14","doi":"10.21203/rs.3.rs-3992157/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"3a944dc6-f9f4-49fe-9f34-01f02a095421","owner":[],"postedDate":"February 29th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-03-12T10:07:54+00:00","versionOfRecord":[],"versionCreatedAt":"2024-02-29 05:46:14","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3992157","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3992157","identity":"rs-3992157","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}