# Transient Two-Phase Flow in Porous Media: A Literature Review and Engineering Application in Geotechnics

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## Abstract

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## 1. Introduction

## 2. Soil Suction and Soil Water Retention Behaviour

#### 2.1. Multiphase Physics and Definition of Soil Suction

_{t}= total soil suction (kPa), R = universal constant [8.31432 J/(mol K)], T = absolute temperature (K), v

_{w}= partial molar volume of liquid water (1.8 cm

^{3}/mol at 20 °C), RH = relative humidity [RH = u

_{v}/u

_{v}

_{0}, u

_{v}= partial pressure of vapour (kPa), u

_{v}

_{0}= pressure of water vapour over a flat surface of pure water at the same temperature (kPa)], ψ

_{m}= matric suction (kPa), ψ

_{o}= osmotic suction (kPa), u

_{a}= air pressure (kPa), u

_{w}= water pressure (kPa), σ

_{s}= surface tension of soil water (N/m), θ = contact angle of water-air interface, r = pore radius (m), C = molar concentration of solute in the pore solution (mol/m

^{3}), and B

_{i}= viral coefficients [68].

#### 2.2. Soil Water Retention Curve

#### 2.3. Soil Water Retention Function

_{min}= the smallest pore radius; ψ = the soil suction (kPa); ψ

_{max}= the maximum soil suction (kPa); σ

_{s}= surface tension of soil water (N/m); θ = contact angle of water-air interface; δ = dummy variable of soil matric suction [15]. This derivation opened a gate for studying SWRC of deformable soil, where the variation of PSD can be measured using Mercury Intrusive Porosimetry (MIP) [65].

^{6}kPa.

#### 2.4. Problems in Soil Water Retention Behaviour

- The SWRC variation of deformable soil because of hydro-mechanical loading;
- Hysteresis of SWRC additionally involving stiffness variation of deformable soil caused by hysteresis and densification of collapsing soil induced by hysteresis;
- The time-dependent change of SWRC because of wetting phase reconfiguration at transient state.

## 3. Steady-State and Transient Two-Phase Flow Seepage Theories

#### 3.1. Richards Model

_{c}= the soil suction head or negative water pressure head (m); z = the gravitational potential in elevation (m); ψ

_{m}= matric suction (kPa); ρ

_{w}= soil water density (kg/m

^{3}); g = gravitational accelerator (m/s

^{2}); u

_{a}and u

_{w}= soil gas and water pressure (kPa); and, here, u

_{a}is assumed to be zero (u

_{a}= 1 atm).

_{s}= the hydraulic conductivity of saturated soil REV (m/s); k

_{r}= k

_{unsat}/k

_{s}the relative hydraulic conductivity; k

_{unsat}= the hydraulic conductivity of unsaturated soil REV; h = the total water head (m).

_{w}is neglected because of incompressible fluid assumption. This Equation is the entire formularization of the groundwater flow equation for saturated (k

_{r}= 1 and S = 1) and unsaturated soil (0 ≤ k

_{r}≤ 1 and 0 ≤ S ≤ 1). By ignoring different terms in Equation (9), different types of groundwater PDE can be derived. The details could be sourced from Bear [7]. For steady-state and unsteady-state multiphase flow in rigid porous media, Equation (9) can be written in the following forms:

_{m}(h

_{c}) is the specific capillary moisture capacity (m

^{−1}), other notations are the same as above. Equation (10) is for steady-state flow conditions in the unsaturated soil, while Equation (11) is for the transient flow conditions.

#### 3.2. Hydraulic Conductivity Function

_{m}(h

_{c}). Until recent years, those two demands promoted the development of SWRF and Hydraulic Conductivity Function (HCF). However, since estimating HCF from SWRF, those two demands could be merged into SWRF only. Hence, a summary of HCF frequently applied in numerical simulation is listed in Table 3.

_{e}-based HCF is often considered for numerical solving Richard’s Equation. Leij, et al. [98] investigated the performance of a large number of HCFs against 346 S

_{e}(h

_{c})-K(h

_{c}) and 557 S

_{e}(h

_{c})-K(S

_{e}) datasets and recommended using Equation (19) for HCF and Equation (5) for SWRF. However, the hysteresis of S

_{e}-based HCF was questioned by Childs [99] and Lu and Likos [1]: different hydraulic loading paths might not guarantee the same hydraulic conduits in one saturation. Moreover, those HCFs seem only satisfied with sandy soil with negligible soil deformation during the drying-wetting process. As Equation (19) is determined from PSD [17], soil deformation induced by purely hydraulic loading might result in the non-uniqueness of HCF. Thus, HCF encounters the same problem with SWRF for deformable soil. Moreover, Li, Luo, Li, Liu, Tan, Chen and Cai [50] pointed out that the time-dependence of SWRC and HCF coexist. In summary, the three issues previously mentioned for SWRC, including deformation, hysteresis, and time-dependence, could also be encountered for HCF.

#### 3.3. Green-Ampt Model

_{e}). An initial water content (θ

_{i}) is assigned to the dry zone. The wetting front infiltration rate, therefore, can be derived using Darcy law as

_{0}= water head of ponding water above top soil surface (m); h

_{s}= capillary suction head at wetting front (m); and z = depth of wetting front (m). Solving this ordinary differential equation gives cumulative infiltration displacement as

_{0}), effective hydraulic conductivity (k), and initial water content of dry zone (θ

_{i}) are known.

#### 3.4. Buckley and Leverett Model

_{c}(S

_{w}) and relative permeability-saturation K

_{r,i}(S

_{i}) constitutive relationships in petroleum engineering because of the nonwetting phase fluid considered.

## 4. Review of Conventional Experiments in Continuum Scale

#### 4.1. The Conventional Experiments of Soil Water Retention Curve

^{6}kPa. However, this method is quite time-consuming because it usually takes ~7–10 days to reach the following equilibrium condition after the current one [1].

^{®}Umweltanalytische Mess-Systeme GmbH) and corresponding diagram are separately shown in Figure 5a,b. After inserting a tensiometer into a soil specimen, the water in the glass shaft will be absorbed into the soil by soil matric suction. As a result, this soil matric suction can be transferred to a negative pressure vacuuming the water in the glass shaft. This vacuum pressure, finally, is measured by the pressure sensor at the other end. This method only measures soil matric suction because the HAE ceramic tip is not solute impermeable. The accuracy of tensiometer measurement depends on the apparatus itself and users’ prudent installation. The key to measuring suction precisely is to set good contact between unsaturated soil and the HAE ceramic tip [60]. The measuring range of a conventional tensiometer is usually 0 to 100 kPa. Recently, the high-suction tensiometer measuring range has been increased up to 1500 kPa. Toll, et al. [115] reviewed the characteristics of newly developed high suction tensiometers, finding that the pressure transducer range (highest is 15 MPa) is much higher than the HAE ceramic tip (highest is 1.5 MPa). The suction range is constrained by the highest Air Entry Value (AEV) of this tip. Compared with all other methods, tensiometers have the following advantages:

- Fast and ease of installation (ensure tip contact condition)
- Applicability in both laboratory and field (in situ) condition
- Short responding time (less than 1 min for low suction capacity micro-tensiometer)
- Long-term measurement (ensure no drain out of water in the shaft)

#### 4.2. The Conventional Experiments of Hydraulic Conductivity Function

_{unsat}) is the hydraulic conductivity of unsaturated soil varying with suction or water content. It can be calculated by saturated hydraulic conductivity (k

_{s}) and timed by relative hydraulic conductivity (k

_{r}). In Section 3, “Steady-state and transient two-phase seepage theory”, the prediction of k

_{r}from Soil Water Retention Function (SWRF) was already introduced. However, this approach is just an approximation of k

_{unsat}to reduce experimental complexity and time consumption. Therefore, the most reliable result is still based on direct experimental measurement. Here, the experiments for measuring k

_{unsat}will be summarised and discussed.

_{unsat}using ATT is relatively straightforward. Through applying suction by varying air chamber pressure and measuring outflow from the pressure chamber, the k

_{unsat}can be calculated by the Darcy equation and then plotted against the corresponding average suction of soil specimen. ASTM D7664-10 [108] offers two subcategories of ATT: rigid wall (odometer) type ATT in Figure 6b and flexible wall (triaxial cell) type ATT in Figure 6a. The former is more suitable for coarse-grain soil, and the latter is more applicable for deformable soil containing fines. However, as the discussion of ATT shortages in the previous content in terms of air diffusion, air compressing specimen, and impedance of HAE ceramic disk with bubble expanding after pressure drop, ATT can only be used for an approximation of the Hydraulic Conductivity Function (HCF) of unsaturated soil under steady-state flow conditions [108]. According to ASTM D7664-10 [108], the air-water seepage is controlled by applying air pressure on the specimen top and back pressure (external water pressure) at the bottom underneath the HAE ceramic disk. It is recommended to insert tensiometers at different specimen depths, but soil moisture sensors with previously tested SWRC could replace them. The volumetric water content variation in a specific time interval can be integrated to calculate the water flux in this period. The hydraulic gradient can be provided between each pair of depths in the soil specimen where soil suction and water content are measured. With water flux and the chamber’s cross-section area and hydraulic gradient, the k

_{unsat}can be calculated by the Darcy equation. More details can be sourced from ASTM D7664-10 [108].

_{unsat}is calculated by the Darcy equation under various suctions cautiously controlled by ATT. Then, the water content can be measured by the destructive method (the oven drying method) or the nondestructive method (e.g., outflow, soil moisture sensors). For the destructive method, more identical specimens have to be prefabricated. In this way, the HCF can be finally plotted with either a suction or water content basis. The other is the constant flow rate method developed by Olsen, et al. [118]. The core of the experimental setup is similar to the constant head method (suction controlled by ATT). Still, a highly complex steady-state flow control system is required to generate a constant flux in a triaxial cell [118]. After measuring suction, water content, and controlled flux, the k

_{unsat}is also calculated by the Darcy equation.

_{unsat}is eventually calculated by the Darcy equation. Nevertheless, due to the high cost of the geotechnical centrifuge, it is not often available in most geotechnical laboratories. Masrouri, Bicalho and Kawai [117] recommended that the centrifuge method is only suitable for nondeformable soil specimens with a pore structure insensitive to the state of stress because of high net stress applied by a centrifugal operation.

_{c}-based diffusion form. Finally, with previously determined specific capillary moisture capacity (C

_{m}), the k

_{unsat}can be calculated by the notation in Equation (13). However, this experiment also has the same limitations for SWRC and steady-state HCF using ATT. Lu and Likos [1] summarised the six assumptions of the multistep outflow methods:

- Each suction increase interval must be small enough, so k
_{unsat}can be assumed as a constant in this interval (which requires meticulous suction control); - The relation between soil suction and water content is linear (but, in fact, it is not only nonlinear but also hysteresis);
- HAE ceramic disk does not cause any hydraulic resistivity (but it is a significant impedance, especially for high permeable sandy soil);
- Flow is just one dimension;
- The gravity effect can be ignored;
- The testing specimen is homogeneous and nondeformable (which is only available for sandy soil).

_{unsat}. Masrouri, Bicalho and Kawai [117] commented on this method because it owns simplicity and is good at mass control, but there are few reliable results compared with other methods. ASTM D7664-10 [108] also records this method as one transient flow method in the standard but mentions its limitations.

_{unsat}by measuring the unsaturated soil diffusivity D(θ). The soil water content along advancing profile, distance, and the corresponding time interval can be recorded during soil water invading through the unsaturated soil. The Boltzmann variable (λ = xt

^{1/2}), which is a function of both invading displacement (x) and duration (t), can be plotted against corresponding water content as a soil water content function of Boltzmann variable. The diffusivity D(θ) can be calculated from the Boltzmann transformation of the 1D Richards model (θ based diffusion equation). It needs integration of D(θ) function against the Boltzmann variable. With the addition of the previous measured SWRC, the k

_{unsat}can be finally calculated by the notation in Equation (13).

_{unsat}. Moreover, it is a good setup for investigating the dynamic effects in SWRC. However, Masrouri, Bicalho and Kawai [117] pointed out that this method lacks stress state and volumetric measurements. Therefore, sandy soil is often selected to conduct soil column tests of IPM to avoid deformation-induced stress variations.

## 5. Limitations of Conventional Theories and Experimental Methods

#### 5.1. The Experimental Exploration of Dynamic Nonequilibrium Effects

- The dynamic effects are not significant in fine-textured soil;
- The higher rate of water content variation, the more significant the dynamic effects;
- The dynamic effects are more significant in coarse-textured sand;
- The dynamic effects in primary drainage curves are more significant than the dynamic effects in main drainage curves.

#### 5.2. The Theoretical Paradoxes of Transient Two-Phase Flow Seepage

#### 5.3. The Physical Causes for Dynamic Nonequilibrium Effects

_{s}= surface tension (N/m); r = capillary tube radius (m); θ = contact angle; μ = dynamic viscosity (kg/ms or Pa·s); D = diffusivity (m

^{2}/s); t = advancing period (s). Equation (27) is the famous Lucas–Washburn (LW) equation for the mathematical formulation of dynamic capillary flow at the beginning of the twentieth century. There is an obvious overestimation of physics in this equation, which is the content angle is assumed to be constant during a moisture diffusion process. The Young–Laplace equation was assumed to be held under transient capillary flow but is not actually. Hoffman [136] stated that four forces control the capillary flow: viscous, inertial, liquid-gas interfacial, and liquid-gas-solid juncture. This set of combining forces dominates capillary flow depending on the testing system and flow rate. Thus, an additional term was added upon static capillary pressure to form the total capillary pressure [146]. This term was experimentally determined as K(Ca)

^{x}in the function of capillary number Ca, and K and x are empirical coefficients. In order to continually use Hagen–Poiseuille’s law as a basis for two-phase flow in a capillary tube, some studies focus on verification of dynamic capillary pressure or dynamic capillary contact angles against Ca, so that the physically validated dynamic term can be added into Hagen–Poiseuille’s law [136,139,140,142,147]. However, the coefficients of such an empirical relationship are more determined by experimental results for the single capillary tube, which varies from study to study. It somehow constrains upscaling to a continuum-scale seepage equation for engineering application in natural porous media.

^{1/2}) and instead gives an x~t

^{2}relation; after viscous drag balancing capillary effect, two-phase flow reach to quasi-steady state obeying the LW equation; finally, flow is eased by gravity. Kim and Kim [137] reviewed recent physic studies on the dynamic capillary rise to find that the power number of advancing time gives different values for different packing beads. They suspected that this is because pores are not fully filled with wetting phase fluids. Moreover, not only for a short instant time-scale, but the LW equation fails to match experimental results for a considerable long enough time [137].

## 6. Advanced Theories of Transient Two-Phase Flow Seepage

#### 6.1. The Theory of Dynamic Fluids Redistribution

_{B}). Based on the experimental phenomenon and this saturation relaxation assumption, the first dynamic theory is proposed as the difference between the equilibrium saturation (S

_{equ}) and dynamic nonequilibrium saturation (S

_{dyn}) is equal to a saturation relaxation term as

_{B}is a fluid redistribution time (s) and t is time (s). So, the capillary pressure-saturation relationship (P

_{c}-S) is a capillary pressure function of S

_{dyn}as

_{c}is dynamic capillary pressure (kPa), other notations are the same as Table 4. Since relative permeability K

_{r}(S) is a function of saturation, therefore, the dynamic relative permeability can be given as

_{r,i}is the relative permeability of fluid phase i = w, n (wetting and nonwetting phase). By adding Equation (28) into the mass balances and two-phase Darcy equations in Table 4, the theory of nonequilibrium two-phase flow seepage was constructed in petroleum engineering.

_{equ}= equilibrium soil water content and τ

_{R}= an equilibrium time constant. The differences from previous petroleum engineering theory include neglecting the nonwetting fluid phase-soil gas and treating soil water redistribution as an independent process from equilibrium transient unsaturated soil water flow described by the Richards model. As Equation (31) does not need to be in Soil Water Retention Function (SWRF) and Hydraulic Conductivity Function (HCF), this theory is much simplified, and a semi-analytical approximation can, therefore, be derived as an asymptotic solution:

_{dyn}= dynamic nonequilibrium soil water content. In this model, Equation (32) approximates θ

_{dyn}, and the Richards model can solve θ

_{equ}. However, they did not develop a systematic method with experiments to split the porosity or total soil water content.

#### 6.2. The Theory of Dual-Fraction with Dynamic Fluids Redistribution

_{s}= hydraulic conductivity of saturated soil, k

_{r}= relative hydraulic conductivity, h = total water head, θ

_{equ}= equilibrium soil water content, θ

_{dyn}= dynamic nonequilibrium soil water content, f

_{equ}= θ

_{equ}/(θ

_{equ}+ θ

_{dyn}) fraction of equilibrium soil water content, f

_{dyn}= θ

_{dyn}/(θ

_{equ}+ θ

_{dyn}) fraction of dynamic nonequilibrium soil water content, and τ

_{D}= soil water equilibration time.

_{D}and f

_{dyn}using an inversion analysis of the dual fraction model against experimental data.

#### 6.3. The Theory of Dual-Porosity and Dual-Permeability

_{m}= hydraulic conductivity for inter-aggregate pore matrix, k

_{im}= hydraulic conductivity for intra-aggregate pore matrix, h

_{m}= water head in inter-aggregate pore matrix, h

_{im}= water head in intra-aggregate pore matrix, θ

_{m}= soil water content in inter-aggregate pore matrix, θ

_{im}= soil water content in intra-aggregate pore matrix, Γ

_{w}= soil water exchange between two domains, w

_{m}= θ

_{m}/(θ

_{m}+ θ

_{im}) fraction of soil water in inter-aggregate pore matrix, and w

_{im}= 1 − w

_{m}= θ

_{im}/(θ

_{m}+ θ

_{im}) fraction of soil water in intra-aggregate pore matrix. In later work, Gerke and van Genuchten [155] also evaluated the first-order water transfer term between two domains as Γ

_{w}= α

_{w}(h

_{m}− h

_{im}), α

_{w}= a first-order mass transfer coefficient. Šimçnek, et al. [159] successfully applied this theory to simulate an upward imbibition experiment. The newly proposed parameters were also determined using inverse modelling. A comprehensive review of the two-domain model simulating the nonequilibrium unsaturated soil water flow can be sourced from Šimůnek, Jarvis, Genuchten and Gärdenäs [149].

#### 6.4. The Theory of Dynamic Nonequilibrium Capillary Pressure

_{c}

^{dyn}= dynamic capillary pressure (Pa), P

_{c}

^{stat}= static capillary pressure (Pa), τ = dynamic coefficient (Pa·s) as the same as the unit of viscosity, S

_{w}= wetting phase saturation, and t is the advancing duration (s). Those two works have commons for deriving the macroscale dynamic capillary pressure. It was achieved by introducing the Gibbs energy for each fluid phase and later concluded that the Helmholtz free energy could not be neglected under the nonequilibrium capillary flow condition [148]. The difference between these two theoretical works is their upscaling methods. Hassanizadeh and Gray [148] used the capillary tube as a unit microscale system and upscale the units by volume averaging method, which is based on the concept of REV [160]. Instead, Kalaydjian [44] used the weight function method. The key findings from those theoretical works were that the specific interfacial area (total interfacial by the volume of porous media REV) and saturation variation play critical roles in Helmholtz energy, which causes the difference between equilibrium and nonequilibrium capillary pressures. Instead of merely considering interface dynamics in single capillary conduit in microscale, the saturation as a variable on REV-scale should also be involved in the REV-scale Helmholtz free energy terms because of local heterogeneity issue in single REV (e.g., ink-bottle effect and Haines jumps [109]).

_{s}= hydraulic conductivity of saturated soil, k

_{r}= relative hydraulic conductivity, θ = soil water volumetric content, h

_{dyn}= total dynamic nonequilibrium water head, h

_{c,dyn}= total dynamic nonequilibrium soil suction head (Gibbes energy in head), z = elevation potential head, h

_{c,equ}= equilibrium soil suction head (static capillary pressure in head), h

_{equ}= total equilibrium water head (total hydrostatic water head), τ/nρ

_{w}g∙∂θ/∂t = dynamic nonequilibrium overpressure head (Helmholtz free energy in head), τ = dynamic coefficient, n = porosity, ρ

_{w}= soil water density, g = gravitational accelerator, and t = time. In this theory, the material coefficients for Helmholtz free energy were reduced to the dynamic coefficient only. All other previously defined SWRF and HCF can still be applied to the simplified theory. It also alleviates the experimental effort so that the dynamic coefficient can be directly determined using a soil column test or pressure cell apparatus without a ceramic disk underneath.

## 7. Novel Experimental and Numerical Contributions in Multiscale

- Macroscale 1D soil column experiments supported by pressure and moisture sensors with inversion analysis of the Richards and modified models;
- Microscale physical pore network model supported by imaging technique;
- Microscale numerical experiments using Pore Network Model (PNM, solving Poiseuille form capillary flow equation in artificial pore network), Direct Numerical Simulation (DNS, solving Navier–Stoke equation in artificial beads package) and Lattice Boltzmann Method (LBM, solving discretised Boltzmann equation in virtual particles package).

#### 7.1. The State-of-Art of the Continuum-Scale Experiments

#### 7.2. The Influential Factors in Dynamic Nonequilibrium Effect

_{stauffer}= the constant proposed by Stauffer [37] with a value of 0.1 for the air-water system, n = the porosity, K = the intrinsic permeability for the saturated soil, µ = the dynamic viscosity of soil water, ρ = the density of soil water, g = gravitational accelerator, and α

_{BC}and n

_{BC}= the fitting parameters of the SWRF in Equation (3). By far, the influential factors to dynamic effects have still been continuously studied by Das and Mirzaei [166] on saturation dependency, Abidoye and Das [34] on scale effect, Hanspal and Das [41] on temperature effect, Goel and O’Carroll [39] on fluid viscous effects, Mirzaei and Das [35] on local heterogeneity influence, Manthey, Hassanizadeh and Helmig [40] on permeability dependency, O’Carroll, Mumford, Abriola and Gerhard [122] on wettability dependency, and Mirzaei and Das [167] on hysteretic dynamic effects, etc.

**The saturation dependency:**The dynamic coefficient was originally defined as a constant independent of saturation. However, the linear relationship between dynamic coefficient and saturation was firstly found by O’Carroll, Phelan and Abriola [28] using multistep outflow experiments. Later, Sakaki, O’Carroll and Illangasekare [30] also explored the saturation dependency in dynamic effects on primary drainage and main imbibition. This work found that the dynamic coefficient increases with decreasing wetting phase saturation. Moreover, the linear relationship between them was updated to become a log-linear relationship. This log-linear relationship was later reconfirmed by another recent experimental contribution on hysteretic dynamic effects from Zhuang, Hassanizadeh, Qin and de Waal [47]. Meanwhile, Das and Mirzaei [166] used a 1D soil column set up to study the dynamic coefficient and found that dynamic coefficient is not a linear function of saturation but a nonlinear function in which dynamic coefficient can only be treated as a constant within high saturation 70–100%. When saturation is lower than 60%, the dynamic coefficient increases nonlinearly with saturation [166]. Conflictingly, the most recent soil column test results from Luo, Kong, Ji, Shen, Lu, Xin, Zhao, Li and Barry [161] showed the uniqueness of dynamic coefficient-saturation relationships on both drainage and imbibition for specific sand under a given period of water table fluctuation.

**The scale effect:**Abidoye and Das [34] applied dimensional analysis to nine parameters (gravity g, isotropic intrinsic permeability K, bubbling pressure ψ

_{AEV}, the domain volume representing domain scale V, fluid density ρ, fluid viscosity μ, saturation S, porosity n, pore size distribution index of SWRF in Equation (3) n

_{BC}), which are reported as essential variables in the determination of dynamic coefficient, to derive a nonlinear relationship between two dimensionless groups as

_{1}= the first dimensionless groups, ∏

_{3}= the third dimensionless groups, and a and b are fitting coefficients. As for the experimental results of Das and Mirzaei [166], a = 9 × 10

^{−14}and b = 1.31, other notations are given in the previous content. Prediction from Equation (43) shows good agreement with the experimental results not only from Das and Mirzaei [166] but also Bottero [168]. Hence, this might be the first mathematical form to quantify the dynamic coefficient impacted by nine other essential variables, including the domain scale effect. Bottero, Hassanizadeh, Kleingeld and Heimovaara [33] and Abidoye and Das [34] investigated nonequilibrium capillary effects at various scales, thereby concluding that the dynamic coefficient increases with the observation scale. The same conclusion was again validated by Abidoye and Das [169] using an artificial neural network modelling approach. Later, Goel, et al. [170] further studied the scale dependency of dynamic relative permeability. This work found no observing scale dependency of dynamic wetting phase relative permeability, but the dynamic nonwetting phase relative permeability slightly increases with domain size decrease. In addition, the location dependency was also unveiled in this work. According to the data from Goel, Abidoye, Chahar and Das [170], with the measuring zone moving from top to bottom, the dynamic wetting relative permeability decrease but the dynamic nonwetting phase relative permeability increase. As a result, Goel, Abidoye, Chahar and Das [170] concluded that the location dependency varies according to the location of the fluid injection point.

**The temperature effect:**Hanspal and Das [41] carried out a numerical simulation of unsteady capillary flow in porous media between 20 °C and 80 °C. Their results showed that dynamic coefficients were nonlinear functions of temperature and saturation, and the dynamic coefficient increases with a temperature increase [41]. Meanwhile, Civan [171] also investigated the temperature effect and reconfirmed the conclusion from Hanspal and Das [41]. However, in spite of this numerical exploration of the temperature effect, any experimental investigations on the temperature dependency were quite rare and less found in the literature by far.

**The fluid viscosity dependency:**Goel and O’Carroll [39] experimentally studied the variation of dynamic coefficient impacted by the viscosity of non-wetting phase fluids using a 1D sand column drainage test. In their study, three important points are mentioned: (1) there is a delay response of the tensiometer due to permeability of ceramic cup, which implies using a high conductivity tensiometer to reduce response postponement during the dynamic experiment; (2) dynamic coefficient decreases for non-wetting phase fluids having smaller viscosity; (3) their work is the first experimental study used to validate previous numerical experiments for viscosity effect and provided accurate data against some contradictory conclusions from numerical studies. The primary purpose of their work is to validate the conclusion that the dynamic coefficient can be enlarged with increasing the effective viscosity (µ

_{eff}= µ

_{n}S

_{n}+ µ

_{w}S

_{w}) proposed by Barenblatt, et al. [172]. Li, et al. [173] also drew a similar conclusion for the fractured tight reservoir that the dynamic coefficient is proportional to the effective relative viscosity (µ

_{ew}= µ

_{eff}/µ

_{w}) defined by them. Other studies did not consider those newly defined terms. Instead, the viscosity ratio (µ

_{n}/µ

_{w}) and wetting phase viscosity were often applied. Joekar Niasar, Hassanizadeh and Dahle [38] concluded the stronger dynamic capillary effects with a larger viscosity ratio. Later, Abbasi, et al. [174] numerically explored the viscosity dependency of dynamic capillary effect and found the dynamic coefficient increasing with increasing wetting phase viscosity, which is consistent with Equation (42) experimentally determined by Stauffer [37]. Moreover, Goel, Abidoye, Chahar and Das [170] investigated the viscous effects on dynamic relative permeability, in which they found the dynamic coefficient increase with mobility ratio (M = K

_{rw}µ

_{n}/K

_{rn}µ

_{w}) decrease.

**The local heterogeneity influence:**Mirzaei and Das [35] conducted a numerical study on micro-scale heterogeneities influencing the dynamic multiphase flow in porous media. In this study, different distributions and intensities of micro-scale heterogeneities were generated in the solving domain to study dynamic coefficients changed by those two influential factors. This study demonstrates that the dynamic coefficient is dependent on flow conditions and domain geometry. To be concise, the dynamic coefficient also increases with the higher intensity of heterogeneity. Other numerical studies from Helmig, et al. [175] and Abidoye and Das [169] also drew similar conclusions.

**The permeability dependency:**Manthey, Hassanizadeh and Helmig [40] and Mirzaei and Das [35] numerically and experimentally studied the transient two-phase flow in homogeneous and heterogeneous porous media at the continuum scale. A common research finding revealed amongst those works was that the dynamic coefficient is reversely proportional to the intrinsic permeability of porous media. This conclusion highly agrees with Stauffer [37] in Equation (42). Later, the same conclusion was confirmed again by Li, et al. [176] in dealing with rocks of the deep reservoir with different permeabilities. They also recommended the non-negligibility of dynamic effects for low permeable rock (K < 9.87 × 10

^{−14}m

^{2}). Furthermore, the fractures as local heterogeneity in rocks also affect the dynamic coefficient. Salimi and Bruining [177] developed an upscaling model to study the dynamic effects in fractured porous media. It revealed that the dynamic effect enlarged with a higher seepage speed of fracture flow. There was low efficiency of oil recovery by water flooding for this scenario. Later, Tang, Lu, Zhan, Wenqjie and Ma [56] revisited the same research objective using numerical simulation, concluding the dynamic coefficient of fractured porous media is higher than that of unfractured porous media. Similar findings were repeatedly confirmed in later studies for fractured tight reservoirs [50,163,173].

**The wettability dependency:**O’Carroll, Mumford, Abriola and Gerhard [122] implemented two-phase multistep outflow experiments with inverse modelling to study the wettability dependency of dynamic effects. In addition to experimental exploration, O’Carroll, Mumford, Abriola and Gerhard [122] derived a microscale capillary advancing equation (Equation (44)). It is in the same form of macroscale dynamic capillary pressure equation from Hassanizadeh, Celia and Dahle [22], using Washburn [145] equation coupled with the current development of dynamic interfacial physics:

_{s}= surface tension, and θ = static contact angle. By comparing Equation (44) with Equation (36), it is obvious to see that when the macroscale dynamic capillary equation is applied to a single capillary tube, the dynamic coefficient can be seen as

**The hysteretic dynamic effects:**Mirzaei and Das [167] experimentally study the dynamic coefficient variation for primary drainage and main imbibition using 1D soil column experiments. Their result confirmed that the hysteresis nature of SWRC also happens to dynamic SWRC, and the dynamic coefficient is different for various hydraulic loading paths. However, only primary drainage and main imbibition data are available in this study. Thus, it provides the first vision to look into the hysteresis nature of dynamic SWRC. A similar experiment was also carried out by Sakaki, O’Carroll and Illangasekare [30] for one drainage-imbibition cycle. In this work, the theories of dynamic capillary pressure and fluid phase redistribution were both studied from their experimental data. They found the differences in dynamic coefficient (τ) and redistribution time (τ

_{B}) between the drainage and imbibition process. Beforehand, Chen [179] already partially investigated hysteresis in dynamic effects for primary drainage, secondary drainage, and the main imbibition using a small flow cell setup, concluding the differences in a dynamic coefficient. Moreover, they found higher dynamic coefficients for the main imbibition than primary and secondary imbibition. The same trends also occurred for dynamic effects in relative permeability. Zhuang, Hassanizadeh, Qin and de Waal [47] experimentally revisited the hysteretic dynamic capillary effects. They further investigated scanning drainage curves beyond prior studies merely on primary drainage and main imbibition paths. Specifically, Zhuang, Hassanizadeh, Qin and de Waal [47] reconfirmed the log-linear relationship between dynamic coefficient and saturation and further found the nonuniqueness of dynamic coefficient for various hydraulic loading paths. Recently, Luo, Kong, Ji, Shen, Lu, Xin, Zhao, Li and Barry [161] revisited the hysteretic dynamic effects using a soil column test under fluctuated water table dynamics. This work experimentally determined the flow rate-dependent hysteretic dynamic coefficients, decreasing with an increasing fluctuation rate [161]. In summary, the cycling wetting and drying paths affecting dynamic coefficient is worth to be deeply studied in the future. There is also a need of identifying if the concept of the dynamic coefficient is a practical and straightforward approach for quantifying the discrepancy between static and dynamic SWRC with hysteresis.

**The pressure boundary conditions effects:**The pressure boundary conditions usually include continuously fluctuating hydraulic head, one-step or multistep in/outflow. Using an instrumented soil column test and full-scale embankment model, Scheuermann, Montenegro, et al. [180] investigated the so defined “hydraulic ratcheting” effect in soil water retention behaviour. Using the Spatial Time Domain Technique (Spatial TDR) and tensiometers, Scheuermann, et al. [180] found the matric suctions lost at minimal water contents for transient infiltration. Moreover, cumulative water content storage was first explored after several cyclic imbibition and drainage processes. This “ratcheting” effect was later verified by numerical studies using multiphase Lattice Boltzmann simulation [181,182]. However, other similar soil column setups from Cartwright, et al. [183] and Cartwright [184] presented no “ratcheting” effect but only hysteresis in static scanning curves. Those works also found no dynamic effects for both drainage and imbibition. The most recent experimental work from Luo, Kong, et al. [161] verified the dynamic effects for soil column tests under the fluctuated water table but had no confirmation of the “ratcheting” effect.

**Acoustic excitation effects:**Regardless of the mainstream reviewed above, some special boundary conditions raised research interests and are worth to be studied. One contribution from Lo, et al. [186] navigated the investigations in the dynamic response of soil water retention behaviour to a novel orientation. Instead of exclusively diving into studying hydraulic boundary conditions, Lo, Yang, Hsu, Chen, Yeh and Hilpert [186] applied acoustic excitations to transient drainage tests on Ottawa sand. Based on their experimental findings, Lo, Yang, Hsu, Chen, Yeh and Hilpert [186] concluded that the acoustic excitations insignificantly influence the static soil water retention curves but have more apparent effects on dynamic ones with the draining flow rate increases. Additionally, they experimentally determined that the dynamic coefficient decreases with frequency increases [186]. The hypothesised mechanism proposed by Lo, Yang, Hsu, Chen, Yeh and Hilpert [186] refers to the dynamic contact angle of capillarity varied with excitation frequency. Nonetheless, this work is merely an outstanding commencement of studying transient two-phase seepage under complex environmental conditions and still needs theoretical and experimental development in multiscale.

**Summary:**Although early studies and current experimental works offer many insights into dynamic SWRC on the macroscale, each study merely focuses on a few influential factors affecting the dynamic coefficient. Moreover, there are still many conflicts of research findings amongst those studies. Therefore, it is worthful to reinvestigate those conflicting conclusions in the future. On the other hand, Sakaki, O’Carroll and Illangasekare [30] suggested that future investigation should be focused on identifying extreme conditions on which the negligibility of the dynamic term can be determined. Moreover, the hysteretic behaviour of the dynamic coefficient (τ) or redistribution time factor (τ

_{B}) needs further investigation. Finally, the influential factors for dynamic capillary effects still demand more experimental and numerical efforts to enhance the clarity and reinforce the principles concluded in prior works.

#### 7.3. The Validations of Advanced Theories against Experiments

**The validation of the Richards model:**To the authors’ best knowledge, validating the Richards model could be sourced since the earlier 1960s. The invalidity of this theory was found in terms of dynamic nonequilibrium effects. Since Gardner [119] developed the method to calculate the diffusivity and hydraulic conductivity inversely using pressure plate outflow data, Rawlins and Gardner [123] later found the velocity-dependent diffusivity. In another straightforward interpretation, if one unsaturated soil hydraulic diffusivity in the function of moisture or soil suction is adopted, this theory will fail to simulate unsaturated soil water seepage for a different seepage velocity. Liakopoulos [24] also validated this model against a series of transient seepage tests using soil column setup and eventually concluded the failure as well. Nevertheless, instead of merely ending on the seepage speed dependence, Liakopoulos [24] further deducted the cause of failure was due to the inability to model soil water inertial using the two-phase Darcy seepage equation. Due to lacking a physical basis for modelling nonequilibrium scenario, the inversion of the Richards model somehow turned into a fitting process to determine the hydraulic properties. Therefore, new theories of fluid redistribution [151] and dynamic soil suction [37] were experimentally explored and mathematically developed for modern theoretical validations.

**The validation of fluid redistribution theory:**The aforementioned fluid redistribution theory in Section 6 was initially developed in the 1970s by Barenblatt [151] in petroleum engineering. Due to the complexity of integrating wetting phase fluid redistribution term into both capillary pressure or dimensionless one as Leverett J function-saturation and relative permeability-saturation, very few works on theoretical verification could be found in the literature, except the theoretical developer. Barenblatt and Vinnichenko [45] and Barenblatt, Garcia Azorero, De Pablo and Vazquez [172] continued to correct and modify this theory for modelling oil-water displacement in porous media. Finally, they successfully verified their model against the counter current seepage tests implemented by Zhou, et al. [187]. Later, Schembre and Kovscek [29] also applied this theory to simulate spontaneous countercurrent imbibition. The experimental results of spontaneous imbibition in diatomite (high porosity and ultra-low permeability) from Le Guen and Kovscek [188] and Zhou, et al. [189] were selected to compare with the simulation outcomes. Due to adding the fluid redistribution time, Schembre and Kovscek [29] could successfully simulate unsteady-state imbibition. Moreover, the shifting between nonequilibrium and static constitutive relationships could be eliminated by involving this term. However, this theory has been less validated in the latest decade because of overlooking dynamic capillary pressure from the physical perspective. The dynamic capillarity effects were more selected for modelling two-phase flow seepage through rock specimens even in petroleum engineering works. In fact, Sakaki, O’Carroll and Illangasekare [30] already proved the transferability between the fluid redistribution time and dynamic capillary coefficient.

**The validation of dual-porosity and dual-permeability theory:**Other theories such as dual-porosity and dual-permeability indeed can be applied to simulate the nonequilibrium effects in unsaturated soil water flow. The validations of those theories can be sourced from many. For instance, one outstanding contribution is from Šimůnek, et al. [159]. However, the physical nature of those theories is to resolve the transient seepage in macroscale heterogeneous porous media (structured porous media), which is beyond the scope of this review. For interests in transient seepage in structured porous media (e.g., soil with aggregate and clay content, rock with fissures and fractures), comprehensive reviews from Šimůnek, Jarvis, Genuchten and Gärdenäs [149] and Jarvis [190] may help to advance the understanding.

**The validation of the thermodynamic-based theory:**By far, the mainstream of modelling transient two-phase flow in porous media is based on the comprehensive theory developed by Hassanizadeh and Gray [43] and the simplified version given by Hassanizadeh, Celia and Dahle [22]. This thermodynamic-based theory applies to both soil hydrology and petroleum engineering.

**Summary:**Many studies have validated all newly proposed theories against experimental outcomes. Due to more complicated constitutive relationships by adding additional dynamic terms, the numerical solution seems the only method to solve the theory mathematically. Most studies on validating each theory concluded that the newly proposed theory outperformed and outcompeted the conventional theory of two-phase flow in porous media and the Richards model. However, some numerical solutions did not match the experimental observations very well. Furthermore, the theory has been expanded to simulate versatility (e.g., various types of drainage, imbibition, hysteresis, etc.), whereas the fingering flow replication might be still debatable, referring to numerical stability. As a result, the numerical solutions of advanced theories are still worth to be applied to the model validations with laboratory experiments and specific geophysical scenarios with thermal, chemical, and mechanical coupling.

#### 7.4. The State-of-Art of the Micromodels

**Earlier studies of transient two-phase seepage using micromodel:**Most earlier works on micromodel focused on steady-state flow, which is not the objective of this review. To the best of the authors’ knowledge, Tsakiroglou, Theodoropoulou, et al. [124] presented the earliest study on nonequilibrium effects using micromodel, followed by another continuous work on transient and steady-state flow [204]. Tsakiroglou, Theodoropoulou, et al. [124] carried out drainage experiments by percolating paraffin oil-water flow through a hydrophilic planar glass-etched pore network in their earlier work. They also conducted numerical simulations solving thermodynamic-based two-phase flow theory. The parameters in the continuum-scale model were determined by inversion analysis with a Bayesian estimator. In addition to the continuum-scale study, they also explored the microscopic percolation theory on these experiments. As a result, they determined the relation between macroscopic and microscopic parameters in percolation theory: dynamic coefficient to the capillary number and other scaling coefficients, etc. In this study, Tsakiroglou, Theodoropoulou and Karoutsos [124] concluded on the dynamic effects in capillarity and relative permeability depending on the capillary number (10

^{−5}–10

^{−8}), flow regime (capillary or viscous dominating transient flow), flow pattern (capillary or viscous finger), pore network size, etc. In addition, this flow pattern altered from more clustered patterns to shaper driving front with the capillary number increasing. Moreover, Tsakiroglou, Theodoropoulou and Karoutsos [124] determined the decreasing relation between dynamic coefficient and capillary number and finally emphasised on applying flow rate-dependent constitutive relationships for simulating transient two-phase seepage flow dominated by viscous effects.

**Recent studies of transient two-phase seepage using micromodel:**One recently representative work on multiphase flow in a transparent micromodel was given by Karadimitriou, Hassanizadeh, Joekar-Niasar and Kleingeld [51]. Their work originally started from Karadimitriou and Hassanizadeh [210], in which a review of micromodels for two-phase flow studies is given. In this work, fabrication methods, including the Hele-Shaw cell, Lithography, and Wet-etching technique, as well as visualizing processes, such as the camera, camera coupling with a microscope, and laser-induced fluorescence, were reviewed to show their advantages and drawbacks. Karadimitriou, Joekar-Niasar, et al. [211] started with a glass-etched micromodel for a two-phase flow experiment based on this experimental method review. Their experimental result agreed with numerical pore network simulation developed by Joekar-Niasar and Hassanizadeh [150] for a single SWRC.

_{c}-S

_{w}-A

_{nw}) at the static condition.

#### 7.5. The State-of-Art of the Pore-Scale Simulations

**The Pore Network Model (PNM) application:**The representative work on PNM of two-phase flow and validating thermodynamic basis theory using PNM were conducted by Joekar-Niasar and Hassanizadeh [150]. The PNM has two types: PNM for static conditions and the Dynamic Pore Network Model. The pore matrix can be artificially generated into different forms, such as a pure pore throat network or tubes coupled with large pores to study non-wetting phase trapping mechanisms. In an early study of PNM from Joekar-Niasar, et al. [216], the PNM was generated using the sphere-tube network, and the Poiseuille Law calculated the flow rate of each pore throat. After upscaling by averaging variables on a tube scale, they found the uniqueness of P

_{c}-S

_{w}-A

_{nw}surface and k

_{rw}-S

_{w}-A

_{nw}surface under static conditions. So this work confirmed that coupling specific interfacial area with the original constitutive curve to form a 3D parabolic surface can reduce the hysteresis between boundary curves (primary drying and wetting) [216].

_{c}-S

_{w}-A

_{nw}surface can only be generated from PNM when there is no tapping in PNM [217].

_{c}-S

_{w}-A

_{nw}) exists for non-equilibrium drainage and imbibition but the difference between dynamic surface and static surface increases with both viscous forces and capillary number increase. Later, Sweijen, et al. [219] upgraded the angular DPNM to the tetrahedra DPNM using the Discrete Element Method (DEM). Then, they carried out the DPNM simulation through the genuine tetrahedra pore unit formed by an assembly of granular particles. The same as Joekar-Niasar and Hassanizadeh [150], Sweijen, Hassanizadeh, Chareyre and Zhuang [219] also applied the static capillarity into pore-scale modelling and more focused on generating macroscale dynamic effects by regional heterogeneity. Finally, they concluded that regional heterogeneity-induced fingering flow formed dynamic capillarity effects in macroscale.

**The Lattice Boltzmann Method (LBM) application:**Pore-scale simulation is a method in which two-phase fluids flow are simulated by solving either the Boltzmann equation (LBM) or the Navier–Stokes equation (DNS) in a domain with artificially packed beads. Compared to DNPM, both LBM and DNS are computationally expensive and applied to a small domain. The Boltzmann equation is a statistical equation used to describe the streaming and collision of fluid molecules in space. The Boltzmann equation is more fundamental and focuses on particle statistics than the Navier–Stokes equation describing momentum conservation with external force action for a single Control Volume (CV). The Boltzmann equation is

^{1}(

**x**,

**p**,t) is the probability of finding one molecule with given position

**x**and momentum

**p**in time t, Γ

^{+}is molecules invading into aiming portion of molecules, and Γ

^{−}are molecules escaping out of a targeting region of molecules [221]. The last term of the right-hand side describes the particle collision, and the other two terms form particle streaming. This nonlinear partial differential equation has no analytical solution, so the continuum equation is discretised into lattices. Thus, this discretised Boltzmann equation can be solved numerically, thereby defined as the Lattice Boltzmann Method (LBM). Sukop [222] interpreted the Boltzmann equation and solutions for single and two-phase flow in detail. Initially, solving the discretised Boltzmann equation could only simulate the single-phase fluid in a domain. With the addition of three forces: an adhesive force between wetting phase particles and solid surface, a repulsive force between non-wetting phase particles and solid surface, and another repulsive force between two immiscible fluid phases’ particles, the interface can be generated between two fluids in LBM. Shan and Chen [223] invented this method of two-phase generation, so it is also named Shan–Chen Lattice Boltzmann Method (SCLBM).

_{c}-S

_{w}-A

_{nw}constitutive relationship proposed by Hassanizadeh and Gray [148], Porter, et al. [227] applied SCLBM in a 2D domain and validated SCLBM against experimental results from a computed microtomography (CMT). Their LBM results did not only show good agreement with experiments, but they also confirmed the non-hysteretic constitutive surface. However, the uniqueness of the P

_{c}-S

_{w}-A

_{nw}surface was later challenged by Galindo-Torres, et al. [228]. They demonstrated the non-uniqueness of the parabolic constitutive surface using SCLBM by applying two-phase displacement in different principal directions. Most of the recent SCLBM studies focus on experimental validating SCLBM and studying impact factors (pore geometry, viscosity of fluid, wettability, fracture features, etc.) influencing SWRC and HCF under steady-state flow conditions [229,230,231,232,233]. Only very few looked into the transient two-phase flow in porous media.

_{2}) sequestration in porous media using colour-fluid LBM and quantified the transient effect by Capillary number (Ca). This Ca-dependent behaviour agreed with the micromodel experimental result on constitutive surface varying with Ca from Karadimitriou, Hassanizadeh, Joekar-Niasar and Kleingeld [51]. However, they also did not provide any information on SWRC, HCF, and P

_{c}-S

_{w}-A

_{nw}surface. Instead, they determine a unique relationship between non-wetting phase saturation and specific interfacial length (specific interfacial area in the 2D domain) under different Ca. So far, this new proposed relationship has not been validated by any experiments yet but a relationship given by colour-fluid LBM. Other dynamic multiphase flow studies using SCLBM only investigated capillary tubes rather than packing beads [236,237]. As the object was only a capillary tube, they only studied the dynamic contact angle varying with Ca. Therefore, there are few LBM studies on the transient effect of SWRC and HCF, whereas LBM can reproduce the dynamic capillary pressure using Shan and Chen [223] inter-particle potential method.

**The Direct Numerical Simulation (DNS) application:**The newest method for pore-scaled two-phase simulation is achieved by numerically solving the Navier–Stokes (N-S) equation with interfacial tracking by Volume of Fluid (VOF). This method is also defined as DNS. One representative work was given by Ferrari and Lunati [238]. The significance of this study is that a force resulting from surface tension is accounted into a body force portion of the N-S equation. In addition, this surface tension force is not determined by the static Young–Laplace equation but calculated from a dynamic capillary equation having Helmholtz free energy, which transforms back to the Young–Laplace equation when the infinitesimal increment of Helmholtz free energy is minimised to zero (following equilibrium approached). This utilization agrees with a thermodynamic basis theory developed by Hassanizadeh and Gray [148]. Compared to LBM using a virtual lattice unit and needing case-specific calibration, DNS coupling with fluid function accounting Helmholtz free energy can directly use physical parameters to simulate dynamic capillary pressure and inertia effect [238]. This requires less effort for model parameters calibration. Using this DNS model, Ferrari and Lunati [238] demonstrated the invalidity of Darcy law due to neglecting viscous effects and trapping.

- if the characteristic time of the applied boundary flow is larger than redistribution time, Darcy’s law is still applicable;
- otherwise, dynamic should be considered.

## 8. Engineering Applications in Unsaturated Soil Mechanics

#### 8.1. Observation of Transient Effects in Natural Slopes

#### 8.2. Transient Effects Coupled in Unsaturated Soil Effective Stress

_{a}= pore air pressure, u

_{w}= pore water pressure, and χ = Bishop factor depends on effective soil saturation (S

_{e}). In most cases, pore air pressure is assumed at atmospheric pressure as a zero reference pressure so that the pore water pressure can be defined as the soil suction when it turns to be negative pore water pressure. Equation (47) has been experimentally validated by Fredlund and Morgenstern [59] and also derived in the textbook of Lu and Likos [1] with spherical packing assumption. Although there are some other forms of unsaturated soil effective stress considering solute suction (osmotic suction) [249,250], the Bishop effective stress form is still the most generic equation used in Geotechnics.

_{s}= suction stress proposed by Lu and Likos [60], ψ

_{m}= static soil matric suction (equilibrium soil suction), k

_{nw}= a material constant, a = specific immiscible fluids interfacial area, A

^{nw}= Helmholtz free energy of interfaces, n = porosity, Γ

^{nw}= interfacial mass density and others as same as notations in Equation (47).

_{s}) can be determined, subsequently calculating the effective stress (σ′) as well. In this way, the conventional hysteretic Soil Water Retention Functions (SWRF) will not be required to give the Bishop χ factor. Nikooee, Habibagahi, Hassanizadeh and Ghahramani [251] successfully validated this derivation against the Soil Water Retention Curves (SWRC) of kaolin, silt, and mixtures of sand and clay samples. For performance in terms of suction stress-soil suction-saturation constitutive surfaces, the work from Nikooee, Habibagahi, Hassanizadeh and Ghahramani [251] deserves to be followed up on.

_{e}= (S − S

_{r})/(1 − S

_{r}), was accepted for sandy soil in other hydro-mechanical coupling studies [253,254].

#### 8.3. Transient Effects Coupled in Unsaturated Soil Shear Strength

#### 8.4. Transient Effects Coupled in Unsaturated Soil Deformation

_{0}) conditions. As a result, Zou, Saad and Grondin [257] finally concluded that more significant settlements (shrinkage) could be generated by transient effects in comparison to calculating deformation without such effects.

#### 8.5. Discussion on Practical Engineering Application

## 9. Conclusions

- (1)
- The conventional theory of transient two-phase flow in porous media is still valid for transient flow conditions only if the instantaneous equilibrium condition can be achieved by carefully controlling the boundary conditions of geotechnical tests.
- (2)
- There are still many research questions left for instantaneous equilibrium two-phase flow in porous media in terms of deformation coupled with hysteresis, the high suction range for soft soil, hysteretic hydraulic properties for problematic soil, etc.
- (3)
- The conventional experiments determine the soil water retention and hydraulic properties under the assumption of instantaneous equilibrium. However, when such an assumption is violated, the inversion analysis can still produce velocity-dependent hydraulic diffusivity. Lacking a physical basis, this scenario becomes a fitting process rather than a physical and experimental characterization.
- (4)
- Earlier experimental observations of dynamic nonequilibrium effects under transient two-phase flow conditions fostered the theoretical development for modelling such effects. Thus, it is expected to advance theories with more experimental findings in multiscale.
- (5)
- The four advanced theories have their advantages and disadvantages. The soil water redistribution model, dual-fraction model with soil water redistribution, and dual-porosity and dual-permeability model can simulate transient effects in multiscale. However, according to this listing sequence of models, the theoretical complexity increases with more parameters in the governing equations. Except that the soil water equilibration time can be straightforwardly determined using instrumented soil column test, other parameters can only be given by inverse modelling.
- (6)
- Compared to the previous three theories in soil hydrology, the thermodynamic-based theory can be applied to simulate both soil moisture dynamics and nonequilibrium soil suction under transient two-phase flow conditions. Therefore, it has a unique advantage for estimating the mechanical properties of unsaturated soil, whereas others neglect this critical application.
- (7)
- Modern continuum-scale experimental methods often incorporate soil suction and moisture sensors into a soil column with various hydraulic boundary conditions. In petroleum engineering, core flooding tests can be implemented with non-destructed measuring methods for fluid saturation (e.g., CT scan, Gamma-ray, outflow, etc.). Nevertheless, pore pressure transducers are irreplaceable for determining dynamic capillary pressure.
- (8)
- Recent continuum-scale experimental and numerical studies focus on investigating influential factors of dynamic nonequilibrium effects in terms of dynamic capillary coefficient. The influential factors mainly include properties of porous media, fluid properties, multiphase physical properties, etc. However, many conflicting conclusions were drawn amongst those studies for each influential factor, therefore, needing more experimental revisits. Furthermore, the hysteretic and extreme conditions for dynamic effects should be continuously investigated using experimental and numerical methods in the future.
- (9)
- All advanced theories have been validated successfully by many experimental studies on specific hydraulic loading paths. However, there are failure cases, which might be due to numerical solution, parameter selection, etc. So, it is still worth casting the numerical tools constantly in order to improve the accuracy of numerical solutions. Moreover, the fingering flow simulated by the thermodynamic-based theory considering hysteresis could be due to conditionally numerical instability, which is quite debatable for mathematicians and numerical modellers.
- (10)
- The pore-scale modelling transient two-phase flow displacement can be achieved by the physical micromodel, numerical Pore-Network Model (PNM), and Computational Fluid Dynamic (CFD) method. Each is a powerful tool to investigate transient two-phase flow from pore-scale up to Representative Elementary Volume (REV) scale or even several REVs. Still, the physical micromodel is not often available in every geotechnical laboratory and is not generic. On the other hand, the Dynamic PNM (DPNM) can simulate the dynamic effects with lower computational expenses than CFD. However, the interfacial dynamics was not counted at pore-scale. Therefore, only local heterogeneities induced dynamic/transient effects can be reproduced by DPNM.
- (11)
- CFD methods include multiphase Lattice Boltzmann Method (LBM), Direct Numerical Simulation (DNS) coupled with Volume of Fluid (VoF) or Level-Set (LS), and Smoothed Particle Hydrodynamics (SPH). Only a few numerical experiments using those methods already replicated dynamic effects in dynamic capillarity, capillary-viscous and inertial dominating flow conditions. Although those numerical methods have been implemented to study steady-state flow conditions in flow patterns and regimes, the contribution to dynamic effects under transient flow conditions are still rare and, therefore, strongly urge pursuit in the future.
- (12)
- The observation of dynamic nonequilibrium effects in natural slopes was firstly reported in the recent literature. It partially supports the application of transient effects for engineering practices. However, the in-situ data are still insufficient to prove the importance of considering dynamic effects in modelling transient two-phase flow seepage. It is expected to receive more contributions from the field or large-scale observations than smaller-scale findings using the pressure or flow cell tests.
- (13)
- The dynamic/transient effects have been coupled into unsaturated soil effective stress, soil suction characteristic stress, subsequently in shear strength, and pore-elastic consolidation. Those studies proposed the theoretical frameworks for hydro-mechanical coupling with transient effects and succeeded in simulating nonequilibrium transient water flow in unsaturated soil. However, the validation can only be given to the transient seepage, while the mechanical prediction cannot be verified by any experimental results yet.
- (14)
- There are many challenges in setting up an experimental apparatus to test the mechanical properties of unsaturated soil under transient flow conditions. An inevitable drawback is the irreplaceability of tensiometers in detecting nonequilibrium soil suction. The insertions of those sensors will cause mechanical properties variation in terms of deformation. Moreover, deformation-induced nonuniqueness of soil water retention curve will dramatically increase complexity for additional coupling with transient effects. Therefore, those reasons constrain hydro-mechanical investigation when transient effects are considered. However, it seems still applicable for shear strength estimation and extreme equilibrium analysis when the soil pore matrix varies insignificantly.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The illustration of soil matric suction: (

**a**) capillary pressure and Young–Laplace equation; (

**b**) ink-bottle effect leading to imbibition and hysteresis in scanning loops, reprinted from Ref. [69].

**Figure 2.**An instance of Soil Water Retention Curve (SWRC), reprinted from Ref. [70].

**Figure 3.**The illustration of 1D vertical Green-Ampt infiltration model, reprinted from Ref. [101].

**Figure 4.**Standard Axis Translation Techniques: (

**a**) a sketch of the hanging column method; (

**b**) a sketch of the pressure chamber method, reprinted from Ref. [110].

**Figure 5.**An example of a tensiometer: (

**a**) UMS T5 Tensiometer; (

**b**) a diagram of T5, reprinted from Ref. [116].

**Figure 6.**k

_{unsat}measurement: (

**a**) unsaturated soil triaxial; (

**b**) constant head method, reprinted from Ref. [6].

**Figure 7.**k

_{unsat}measurement using transient flow experiments: (

**a**) the soil column tests setup; (

**b**) an illustration of soil column tests, reprinted from Ref. [121].

**Figure 8.**The manifestations of nonequilibrium transient two-phase flow: (

**a**) the dynamic effects in drainage SWRC, reprinted from Ref. [21]; (

**b**) the dynamic effects in imbibition SWRC, reprinted from Ref. [29]; (

**c**) the speed-dependent diffusivity against water content, reprinted from Ref. [123]; (

**d**) the dynamic effects in relative permeability, reprinted from Ref. [124].

**Figure 9.**The instrumented soil column and core flooding tests for investigating dynamic nonequilibrium effects for transient two-phase flow in porous media: (

**a**) short soil column, reprinted from Ref. [30]; (

**b**) large soil column, reprinted from Ref. [161]; (

**c**) core flooding test illustration, reprinted from Ref. [162]; (

**d**) core flooding test picture, reprinted from Ref. [163].

**Figure 10.**Two instances of micromodels applied to study dynamic nonequilibrium effects under transient two-phase displacement process using visualization techniques (dark and light in the flow conduits indicating two immiscible fluids): (

**a**) the glass-etched micromodel, reprinted from Ref. [204]; (

**b**) the polydimethylsiloxane (PDMS) micromodel, reprinted from Ref. [205].

**Figure 11.**The hydro-mechanical coupling process considers dynamic effects, reprinted from Ref. [257].

SWRF Authors | Fitting Functions | |
---|---|---|

Gardner [71] | ${S}_{e}=\frac{1}{1+{\alpha}_{G}{\psi}^{{n}_{G}}}$ | (2) |

Brooks [13] | $\begin{array}{l}{S}_{e}=\{\begin{array}{c}\begin{array}{cccc}1& & & \psi <{\psi}_{AEV}\end{array}\\ \begin{array}{cccc}{\left(\frac{{\alpha}_{BC}}{\psi}\right)}^{{n}_{BC}}& & \psi >{\psi}_{AEV}& \end{array}\end{array}\\ {\alpha}_{BC}={\psi}_{AEV}\end{array}$ | (3) |

Van Genuchten [14] | ${S}_{e}={\left[\frac{1}{1+{\alpha}_{VG}{\psi}^{{n}_{VG}}}\right]}^{{m}_{VG}}$ | (4) |

Fredlund and Xing [15] | ${S}_{e}=\left[1-\frac{\mathrm{ln}(1+\frac{\psi}{{\psi}_{r}})}{\mathrm{ln}(1+\frac{{10}^{6}}{{\psi}_{r}})}\right]\frac{1}{{\left\{\mathrm{ln}\left[e+{(\frac{\psi}{{\alpha}_{FX}})}^{{n}_{FX}}\right]\right\}}^{{m}_{FX}}}$ | (5) |

**Table 2.**Richards model forms on different state variable basis [12].

Forms | PDE | |
---|---|---|

h_{c} base | $\frac{\partial}{\partial x}\left[{k}_{s}{k}_{r}({h}_{c})\frac{\partial {h}_{c}}{\partial x}\right]+\frac{\partial}{\partial y}\left[{k}_{s}{k}_{r}({h}_{c})\frac{\partial {h}_{c}}{\partial y}\right]+\frac{\partial}{\partial z}\left[{k}_{s}{k}_{r}({h}_{c})\frac{\partial ({h}_{c}+z)}{\partial z}\right]=C({h}_{c})\frac{\partial {h}_{c}}{\partial t}$ | (12) |

θ base | $\frac{\partial}{\partial x}\left[D(\theta )\frac{\partial \theta}{\partial x}\right]+\frac{\partial}{\partial y}\left[D(\theta )\frac{\partial \theta}{\partial y}\right]+\frac{\partial}{\partial z}\left[D(\theta )\frac{\partial \theta}{\partial z}\right]+\frac{\partial k(\theta )}{\partial z}=\frac{\partial \theta}{\partial t}$ where diffusivity $D(\theta )=\frac{k(\theta )}{{C}_{m}(\theta )};k(\theta )={k}_{s}{k}_{r}(\theta );{C}_{m}(\theta )=\frac{\partial \theta}{\partial {h}_{c}}$ | (13) |

Mixing | $\frac{\partial}{\partial x}\left[{k}_{s}{k}_{r}({h}_{c})\frac{\partial {h}_{c}}{\partial x}\right]+\frac{\partial}{\partial y}\left[{k}_{s}{k}_{r}({h}_{c})\frac{\partial {h}_{c}}{\partial y}\right]+\frac{\partial}{\partial z}\left[{k}_{s}{k}_{r}({h}_{c})\frac{\partial ({h}_{c}+z)}{\partial z}\right]=\frac{\partial \theta}{\partial t}$ | (14) |

Model Authors | Model Equations | Notations | |
---|---|---|---|

Childs and Collis-George [95] | ${k}_{r}=\frac{{\displaystyle {\int}_{{\theta}_{r}}^{\theta}\frac{\theta -x}{\psi {(x)}^{2}}dx}}{{\displaystyle {\int}_{{\theta}_{r}}^{{\theta}_{s}}\frac{{\theta}_{s}-x}{\psi {(x)}^{2}}dx}}$ | (15) | S_{e} based statistical model; Fredlund, Xing and Huang [16] rewrote it into continuum form. |

Gardner [96] | ${k}_{r}=\mathrm{exp}({\alpha}_{G}{h}_{c})$ | (16) | Simplifying analytical solution derivation but having poor-fitting performance; h_{c} based empirical model. |

Brooks [13] | ${k}_{r}=\{\begin{array}{c}\begin{array}{cccc}1& & & {h}_{c}<{h}_{c}{}_{AEV}\end{array}\\ \begin{array}{cccc}{\left(\frac{{\alpha}_{BC}}{{h}_{c}}\right)}^{2+3{n}_{BC}}& & {h}_{c}>{h}_{c}{}_{AEV}& \end{array}\end{array}$ | (17) | h_{c} based empirical model; h_{cAEV} = ψ_{AEV}/ρ_{w}g = air entry value in water head. |

Brooks [13] | ${k}_{r}={S}_{e}{}^{3+2/{n}_{bc}}$ | (18) | S_{e} based empirical model; BC SWRF inserted into Equation (18). |

Mualem [17] | ${k}_{r}={S}_{e}{}^{0.5}{\left[\frac{\left({\displaystyle {\int}_{0}^{{\theta}_{w}}\frac{d{\theta}_{w}}{{h}_{c}}}\right)}{\left({\displaystyle {\int}_{0}^{{\theta}_{s}}\frac{d{\theta}_{w}}{{h}_{c}}}\right)}\right]}^{2}$ | (19) | Statistical model; Requiring well-developed SWRF inserted into Equation (19). |

Van Genuchten [14] | ${k}_{r}={S}_{e}{}^{0.5}[1-{(1-{S}_{e}{}^{1/{m}_{VG}})}^{{m}_{VG}}]$ | (20) | m_{VG} = 1 − 1/n_{VG}; S_{e} based model; VG SWRF inserted into Eqaution (20). |

Fredlund, Xing and Huang [16] | ${k}_{r}=\frac{{\displaystyle {\int}_{\psi}^{{\psi}_{r}}\frac{\theta (y)-\theta (\psi )}{{y}^{2}}{\theta}^{\prime}(y)dy}}{{\displaystyle {\int}_{{\psi}_{aev}}^{{\psi}_{r}}\frac{\theta (y)-{\theta}_{s}}{{y}^{2}}{\theta}^{\prime}(y)dy}}$ | (21) | Insert Fredlund and Xing [15] SWRF into Equation (21); Suction (ψ) based. |

**Table 4.**The full governing equations for transient two-phase flow in porous media under equilibrium.

Equations | Forms | |
---|---|---|

Mass balance of phase i | $\frac{\partial {\rho}_{i}n{S}_{i}}{\partial t}+\nabla ({\rho}_{i}{q}_{i})=0$ | (24) |

Momentum balance of phase i | ${q}_{i}=-\frac{{K}_{r,i}({S}_{i})K}{{\mu}_{i}}(\nabla {P}_{i}-{\rho}_{i}g)$ | (25) |

Capillary pressure and Leverett J function | ${P}_{c}({S}_{w})=({P}_{n}-{P}_{w})={\sigma}_{s}\sqrt{\frac{n}{K}}J({S}_{w})$ | (26) |

_{i}= density of fluid phase i; n = porosity of a designated REV of porous media; S

_{i}= saturation of fluid phase i; t = time; q

_{i}= volumetric flux for fluid phase i in unit area (Darcy flux); K

_{r,i}(S

_{i}) = relative permeability for fluid phase i; K = intrinsic permeability of a designated REV of porous media; µ

_{i}= dynamic viscosity of fluid phase i; g = gravational accelerator; P

_{i}= pressure of fluid phase i; P

_{c}(S

_{w}) = capillary pressure in the function of saturation S

_{w}; σ

_{s}= surface tension at two-phase interface; J(S

_{w}) = dimensionless P

_{c}, also defined as the Leverett J function by Buckley and Leverett [8].

**Table 5.**A simplified system of advanced theory for dynamic two-phase flow in porous media [43].

Equations | Forms | |
---|---|---|

Mass balance phase | $\frac{\partial {\rho}_{i}n{S}_{i}}{\partial t}+\nabla ({\rho}_{i}{q}_{i})=0$ | (37) |

Momentum balance | ${q}_{i}=-\frac{{S}_{i}{}^{2}K}{{\mu}_{i}}\left[(\nabla {P}_{i}-{\rho}_{i}g)+\frac{{\lambda}_{ii}}{{a}_{wn}}\nabla {a}_{wn}+\frac{{\Omega}_{i}}{{S}_{i}}\nabla {S}_{i}\right]$ | (38) |

Dynamic capillary pressure | ${P}_{c}{}^{dyn}-{P}_{c}{}^{stat}=({P}_{n}-{P}_{w})-{P}_{c}{}^{stat}=-\tau \frac{\partial {S}_{w}}{\partial t}$ | (39) |

Equation of state | ${P}_{c}{}^{stat}={P}_{c}{}^{stat}({S}_{w},{a}_{wn},T)$ | (40) |

_{i}= the density of fluid phase i, n = the porosity, t = time, S

_{i}= the saturation of fluid phase i, q

_{i}= the volumetric flux in unit area (discharge velocity), K

_{i}= the intrinsic permeability of porous media, µ

_{i}= dynamic viscosity of fluid phase i, g = gravitational accelerator, P

_{i}= the pressure of fluid phase i, a

_{wn}= the specific interfacial area (interfacial area per REV), λ

_{ii}, Ω

_{i}= the material coefficients (S

_{w}·λ

_{wn}=S

_{n}·λ

_{nw}), P

_{c}

^{dyn}= dynamic capillary pressure, P

_{c}

^{stat}= static capillary pressure, τ = dynamic coefficient, T = the absolute temperature.

**Table 6.**The entire theoretical framework for modelling unsaturated soil elastic deformation in coupled with the thermodynamic-based theory of transient soil gas-water flow in unsaturated soil.

Seepage Equations | Mathematical Formulations | |
---|---|---|

Governing equation for transient soil water flow in unsaturated soil considering soil skeleton deformation (Zou, Saad and Grondin [257]) | $\begin{array}{l}\frac{\partial {\rho}_{w}{n}_{e}{S}_{w}}{\partial t}+\nabla \left[{\rho}_{w}\left(\frac{{K}_{r}K}{{\mu}_{w}}(\nabla {P}_{c}{}^{dyn}-{\rho}_{w}g)+{n}_{e}{S}_{w}\frac{\partial u}{\partial t}\right)\right]=0\\ \frac{\partial {\rho}_{w}{n}_{e}{S}_{w}}{\partial t}={\rho}_{w}{n}_{e}\frac{\partial {S}_{w}}{\partial {P}_{c}{}^{stat}}\frac{\partial {P}_{c}{}^{stat}}{\partial t}+{\rho}_{w}{S}_{w}\frac{\partial {n}_{e}}{\partial t}\end{array}$ | (52) |

Dynamic capillary pressure (Hassanizadeh and Gray [148]) | ${P}_{c}{}^{dyn}-{P}_{c}{}^{stat}=({P}_{a}-{P}_{w})-{P}_{c}{}^{stat}=-\tau \frac{\partial {S}_{w}}{\partial t}{P}_{a}=0$ | (53) |

Effective porosity-volumetric strain coupling (Zou, Saad and Grondin [257]) | ${n}_{e}=\frac{{n}_{0}+{\epsilon}_{v}}{1+{\epsilon}_{v}}$ | (54) |

Permeability-effective porosity coupling (Modified Kozeny-Carman Wang and Nackenhorst [260]) | $K={K}_{0}\frac{{n}_{e}{}^{3}}{{(1-{n}_{e})}^{2}}\frac{{(1-{n}_{0})}^{2}}{{n}_{0}{}^{3}}$ | (55) |

Dynamic coefficient-porosity and permeability coupling (Stauffer [37]) | $\tau =\frac{{\alpha}_{stauffer}{n}_{e}{\mu}_{w}}{K{n}_{BC}}{\left(\frac{{\alpha}_{BC}}{{\rho}_{w}g}\right)}^{2}$ | (56) |

Soil water retenion fuction(Modified van Genuchten [14]) | ${S}_{e}=\frac{{S}_{w}-{S}_{r}}{1-{S}_{r}}={\left[\frac{1}{1+{\alpha}_{VG}{({P}_{c}{}^{stat})}^{{n}_{VG}}}\right]}^{{m}_{VG}}{m}_{VG}=1-\frac{1}{{n}_{VG}}$ | (57) |

Relative permeability-saturation model (Modified van Genuchten [14]) | ${K}_{r}={S}_{e}{}^{0.5}[1-{(1-{S}_{e}{}^{1/{m}_{VG}})}^{{m}_{VG}}]$ | (58) |

Brooks & Corey to modified van Genuchten fitting parameters transformation (Morel-Seytoux, et al. [261]) | $\begin{array}{l}\frac{{\alpha}_{BC}}{{\rho}_{w}g}=\frac{1}{{\alpha}_{stauffer}}\frac{p+3}{2p(p-1)}\frac{127.8+8.1p+0.0092{p}^{2}}{55.6+7.4p+{p}^{2}}\\ p=3+\frac{2}{{n}_{VG}}\end{array}$ | (59) |

Mechanical equations | Mathematical formulations | |

Unsaturated soil effective stress (Biot and Willis [259]) | $\begin{array}{l}{\sigma}_{ij}\prime ={\sigma}_{ij}-b\left[(1-{S}_{w}){P}_{a}+{S}_{w}{P}_{w}\right]{\delta}_{ij}\\ {P}_{a}=0;b=1-\frac{{\kappa}_{b}}{{\kappa}_{s}}\le 1\end{array}$ | (60) |

The full pore-elastic stress-strain constitutive relationship (Biot [258]) | $\begin{array}{l}{\sigma}_{ij}\prime ={\lambda}_{b}{\epsilon}_{v}{\delta}_{ij}+2{G}_{b}{\epsilon}_{ij}\\ {\epsilon}_{v}={\epsilon}_{xx}+{\epsilon}_{yy}+{\epsilon}_{zz}\\ {\epsilon}_{ij}=\frac{1}{2}\left(\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}}\right)\\ {\lambda}_{b}=\frac{{v}_{b}{E}_{b}}{(1+{v}_{b})(1-2{v}_{b})},{G}_{b}=\frac{{E}_{b}}{2(1+{v}_{b})}\end{array}$ | (61) |

Simplified pore-elastic constitutive relationship under isotropic loading condition (Zou, Saad and Grondin [257]) | $\begin{array}{l}{\epsilon}_{v}=\frac{{\sigma}_{mean}}{{\kappa}_{b}}+{S}_{w}{P}_{c}{}^{dyn}\left(\frac{1}{{\kappa}_{b}}-\frac{1}{{\kappa}_{s}}\right)\\ {\kappa}_{b}=\frac{(3{\lambda}_{b}+2{G}_{b})}{3}\\ {\sigma}_{mean}=\frac{{\sigma}_{xx}+{\sigma}_{yy}+{\sigma}_{zz}}{3}\end{array}$ | (62) |

Nonequilibrium soil suction increase-induced shrinkage by assuming rigid particles(Zou, Saad and Grondin [257]) | $\begin{array}{l}{\epsilon}_{v}={S}_{w}{P}_{c}{}^{dyn}\left(\frac{1}{{\kappa}_{b}}-\frac{1}{{\kappa}_{s}}\right){\kappa}_{s}\to +\infty \\ {\epsilon}_{v}={S}_{w}\left({P}_{c}{}^{stat}-\tau \frac{\partial {S}_{w}}{\partial t}\right)\left(\frac{1}{{\kappa}_{b}}\right)\end{array}$ | (63) |

Mechanical equilibrium equation | ${\sigma}_{ij,j}+{\rho}_{b}b=0$ | (64) |

**Seepage parameters:**ρ

_{w}= the density of soil water, n

_{e}= the effective porosity, S

_{w}= the soil water saturation, t = the time, K = the intrinsic permeability, K

_{r}= the relative permeability, µ

_{w}= the dynamic viscosity of pore water; g = the gravational accelerator; P

_{w}= the pore water pressure, P

_{a}= the pore air pressure assumed zero for unsaturated soil, P

_{c}

^{stat}= the static capillary pressure (equilibrium soil matric suction), P

_{c}

^{dyn}= the dynamic capillary pressure (nonequilibrium soil matric suction), τ = the dynamic coefficient, u = the soil matrix displacement, n

_{0}= the initial effective porosity, ε

_{v}= the volumetric strain, K

_{0}= the initial intrinsic permeability, α

_{stauffer}= the Stauffer coefficient with a value of 0.1 for the air-water system, α

_{BC}and n

_{BC}= the Brooks and Corey fitting parameters in Equation (3) in Table 1, S

_{e}= the effective saturation, S

_{r}= the irreductable saturation, α

_{VG}, n

_{VG}and m

_{VG}= the van Genuchten fitting parameters in Equation (4) in Table 1, and p = a coefficient for parameters transformation between Brooks and Corey and van Genuchten functions;

**M**

**echanical parameters:**σ

_{ij}′ = the effective stress, σ

_{ij}= the total normal stress, b = the Biot’s coefficient, δ

_{ij}= the Kronecker delta (unit diagonal matrix, dimension of matrix depending on 2D or 3D), κ

_{b}= the drained bulk modulus of soil, κ

_{s}= the bulk modulus of solid particles (assumed = +∞ for rigid particles), λ

_{b}= the Lame’s moduli of dry porous media, ε

_{v}= the volumetric strain, ε

_{xx}, ε

_{yy}, and ε

_{zz}= the linear strain on x, y and z directions, G

_{b}= the shear modulus, ε

_{ij}= the strain tensor, u

_{i}or u

_{j}= the displacements on i or j directions (i or j = x, y, z), v

_{b}= the soil Poisson ratio, E

_{b}= Young’s elastic modulus of soil, σ

_{mean}= the mean total normal stress for isotropic loading, σ

_{xx}, σ

_{yy}, and σ

_{zz}= the total normal stress on x, y and z directions, P

_{c}

^{dyn}= the dynamic capillary pressure (nonequilibrium soil matric suction), P

_{c}

^{stat}= the static capillary pressure (equilibrium soil matric suction), τ = the dynamic coefficient, σ

_{ij,j}= the entire stress tensor (dimension of matrix depending on 2D or 3D), ρ

_{b}= the bulk density of soil, and ρ

_{b}b = the total body force including the gravational force (ρ

_{b}g), centrifuge force, Coriolis force for large scale groundwater and geological models, and other field-generated body forces, usually the gravational force exclusively adopted for geotechnical engineering.

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**MDPI and ACS Style**

Yan, G.; Li, Z.; Galindo Torres, S.A.; Scheuermann, A.; Li, L.
Transient Two-Phase Flow in Porous Media: A Literature Review and Engineering Application in Geotechnics. *Geotechnics* **2022**, *2*, 32-90.
https://doi.org/10.3390/geotechnics2010003

**AMA Style**

Yan G, Li Z, Galindo Torres SA, Scheuermann A, Li L.
Transient Two-Phase Flow in Porous Media: A Literature Review and Engineering Application in Geotechnics. *Geotechnics*. 2022; 2(1):32-90.
https://doi.org/10.3390/geotechnics2010003

**Chicago/Turabian Style**

Yan, Guanxi, Zi Li, Sergio Andres Galindo Torres, Alexander Scheuermann, and Ling Li.
2022. "Transient Two-Phase Flow in Porous Media: A Literature Review and Engineering Application in Geotechnics" *Geotechnics* 2, no. 1: 32-90.
https://doi.org/10.3390/geotechnics2010003