## Introduction

Flow of immiscible fluids in porous media is of special importance in soil science as well as in chemical, environmental, construction, and petroleum industries. In particular, the immiscible flow of nonaqueous phase liquids (NAPLs) commonly occurs in various subsurface conditions, groundwater remediation operations, and in oil recovery processes. In such systems, capillary pressure plays a crucial role in determining the motion of fluids within the porous media.1–8 In addition, temperature variation may affect such flows where the hydraulic properties of the porous media such as hydraulic conductivity and water retention are temperature dependent.9–12 Conventional approaches for modeling of immiscible two-phase flow in porous media involve the use of an extended version of Darcy's law3, 13, 14 for multiphase flows in conjunction with the use of constitutive relationships between capillary pressure, saturation, and relative permeability (*P*^{c}-*S*-*K*_{r}) based on quasi-static conditions.15, 16 These *P*^{c}-*S*-*K*_{r} relationships are highly nonlinear in nature and depend on flow hydrodynamics (dynamic/static) conditions, capillary and viscous forces, contact angles, grain size distribution, surface tension, boundary conditions, fluid properties, and length scales of observation. The conventional relations of the capillary pressure are functions of wetting-phase saturation (*P*^{c}-*S* relationships) and differences between the average pressures for oil/nonwetting and the water/wetting,15, 16 which can be mathematically expressed as

where *P*_{nw} is the average pressure for oil/nonwetting phase, *P*_{w} is the average pressure for water or wetting phase, and *S* refers to the water or wetting-phase saturation. Steady-state *P*^{c}-*S*-*K*_{r} relationships have been previously studied by a large number of authors, some of whom, including17–22 argue that they are insufficient to characterize the behavior of two-phase flows in porous media fully, as they fail to capture the dynamic flow. Hassanizadeh and Gray17, 18 proposed a generalized *P*^{c}-*S* relationship with the inclusion of dynamic capillary pressure as shown below

where τ is a dynamic coefficient, ∂*S*/∂*t* is the time derivative of saturation, *P*^{c,dyn} is the dynamic capillary pressure, *P*^{c,equ} is the steady-state capillary pressure, and the measurements for all variables is taken at the same saturation value. The dynamic coefficient can be determined from the slope of Eq. 2, which exhibits a linear relationship in the form of a straight line. As reported in the literature, Eq. 2 has been subsequently used by many workers. Hassanizadeh et al.,22 Das et al.,23–25 and Mirzaei and Das26 have reviewed these works in significant detail. Das et al.23 determined the effects of fluid properties on the dynamic coefficient without considering the effects of temperature. This work extends the work by Das et al.23 as discussed below.

In the context of investigating multiphase flows within porous media and the quantification of dynamic coefficient and its dependence on various physical parameters, a thorough review of literature reveals that a majority of the studies conducted in this area are applicable for two-dimensional domain, whereas studies in three-dimensional cases being reasonably limited. In addition, none of the studies reported above has been attempted to quantify the effects of temperature on the dynamic coefficients directly. However, some authors report the effects of temperatures on the two-phase porous flow behavior.5, 27–40

Sinnokrot et al.27 determined the effects of temperature on capillary pressure curves through measurement of drainage and imbibition capillary pressures for three consolidated sandstones and one limestone core. Lo and Mungan41 proposed that oil recovery operations are more efficient at higher temperature. They show that the oil relative permeability increases as the residual oil saturation is decreased. They carried out experiments at room temperature and 300°F for Berea sandstone cores, concluding that the change in relative permeability curves is caused by viscosity reductions. Nutt28 showed that the displacement of oil by other fluids is affected by capillary bundle size (pore size volume), pore size distribution, interfacial force, interfacial tension, and viscosity ratio, besides showing that at higher temperature faster oil recovery rates are observed. Davis29 experimentally investigated the effects of temperatures on the *P*^{c}-*S* relationships for water–oil and water–air modes by using a mixture of hydrocarbon oil in two silica sands with different grain sizes, concluding that the temperature changes have a major effect on the residual wetting and nonwetting phase saturation while having a negligible effect on capillary pressure. Davis29 argued that in water–oil systems, the residual wetting saturation increases as the temperature increases, whereas the residual nonwetting saturation decreases as the temperature increases. Davis29 also reported that an increase in temperature also increases the nonwetting to wetting relative permeability ratio, thereby enhancing the fluids movement by reducing the displacement pressures, which is a major advantage in oil recovery and water remediation techniques.

Grant and Salehzadeh31 carried out calculations of several models that incorporate the temperature effects on wetting coefficients and capillary pressure functions. They also considered temperature effects on liquid–gas phase interfacial tension and liquid–solid interfacial tension (surface tension). They found that linear relations of temperature and liquid–gas interfacial tension fit well with the reference interfacial tension data from Haar et al.42 They noted that the capillary pressure function sensitivity on temperature is mostly due to capillarity. Gatmiri and Delage32 later numerically studied coupled thermodynamic flow behavior of saturated porous media using the finite element method. A mathematical model was developed to deal with thermal variations in saturated porous media. A new concept called thermal void ratio state surface was introduced to include the thermal effects and the stress state level influence on volume changes. Variation of water permeability, water and solid unit weight due to thermal effects, and pore pressure changes were included. They concluded that the variation of permeability tensor in their equation incorporates the effects of temperature changes on dynamic viscosity, thus affecting the capillary pressure–saturation relationships. She and Sleep33 studied the temperature dependence of capillary pressure–saturation relationships for water–perchloroethylene (PCE) and water–air systems in silica sand with temperatures ranging from 20 to 80°C. They reported that for water–PCE system the irreducible water saturation increases and the residual nonwetting saturation decreases as the temperature increases. This result was quite similar to the results of Sinnokrot et al.27 and Davis29 even though they used different fluid pair (water-CHEVRON 15 white oil) and different sand domain (consolidated natural sandstone). In terms of the capillary pressure, She and Sleep33 concluded that it decreases when the temperature increases. However, the authors showed that there are other effects besides interfacial tension and contact angles that can have a significant role in the temperature dependence of capillary pressure–saturation relationships. Grant and Salehzadeh31 modified Laplace's equation to determine the capillary pressure functions. Then, by relating Parker and Lenhard43 hysteretic *P*^{c}-*S*_{w} relationships with equations from Grant and Salehzadeh,31 temperature dependence of the *P*^{c}-*S*_{w} relationships was established.

Narasimhan and Lage34 investigated temperature dependency on viscosity and hence on the global pressure drop of the flowing fluid in porous media. They showed that with an increase in temperature the viscosity and the global pressure drop across the domain decrease. Muralidhar and Sheorey35 investigated the displacement of oil and the saturation pattern, which lead to viscous fingering under isothermal and nonisothermal water flow conditions. Investigations were carried out for both homogenous and heterogeneous domains with high- and low-permeability values with water injection temperatures of 50 and 100°C. They conclude that there are three forms that can occur in the presence of both viscous and capillary forces; stable displacement, viscous fingering, and capillary fingering and the fact that higher temperature affects the water saturation in the reservoir and reduces the capillary pressure. They show that the oil–water flow has a strong dependency on the permeability distribution.

Grant36 proposed a modified mathematical model that takes into account the effects of temperature on capillary pressure–saturation relationships in homogeneous and heterogeneous porous media by using one-parameter and two-parameter models. Grant's31, 35, 36 models are modified forms of the Van Genuchten44 equation and one-parameter equation model presented by Grant and Salehzadeh.31 The model takes into account the interfacial tension as a function of temperature. It was determined that the two-parameter model is better than the one-parameter model for a system with a larger temperature range. Grant verified his extended model with temperatures ranging from 273.16 to 448 K. Hanyga and Jianfei37 investigated the dependency of thermal effects on immiscible two-phase flow and implicitly stated that the capillary pressure–saturation relationships with thermodynamic consideration in nonisothermal systems are significantly different with the same relationships in isothermal systems. Schembre et al.40 carried out experiments that predicted the relative permeability of heavy-oil *in situ* aqueous phase saturation phase profiles during high-temperature imbibition. In their study, they compared experimental data with the simulated results and used two different nonwetting phases. They carried out experiments at 120 and 180°C. Their results showed that the oil saturation and the water relative permeability decrease when the temperatures increase, resulting in a decrease in capillary pressure.

As it can be inferred from the studies reported above, although there are a number of studies that suggest the importance of temperature effects on *P*^{c}-*S* relationships, very little information concerning the temperature dependency of the dynamic effects on capillary pressure relationships can be found. The major concern of this work is therefore to address this issue by carrying out numerical simulations of a two-phase flow system (PCE–water) in homogeneous porous domain. The study is important as we need to understand how quickly or slowly a two-phase flow system reaches equilibrium, to determine the range of validity of dynamic capillary pressure theory17, 18 and to obtain values of the dynamic coefficient (τ) at different temperatures. These effects are determined in our simulations through the use of constitutive relations for fluid and material properties (interfacial tension, residual saturation, viscosity, density, and relative permeability) as a function of temperature. A large number of simulations have been conducted to demonstrate how the combined effects of dynamic flow (dependent on different pressure boundary conditions) and temperature variations can affect the capillary pressure relationships. Results presented in this article correspond to temperatures ranging between 20 and 80°C for two different materials namely coarse and fine sands.