Summary
We present formulation and numerical solution of two-phase multicomponent diffusion and natural convection in porous media. Thermal diffusion, pressure diffusion, and molecular diffusion are included in the diffusion expression from thermodynamics of irreversible processes.
The formulation and the numerical solution are used to perform initialization in a 2D cross section. We use both homogeneous and layered media without and with anisotropy in our calculations. Numerical examples for a binary mixture of C1/C3 and a multicomponent reservoir fluid are presented. Results show a strong effect of natural convection in species distribution. Results also show that there are at least two main rotating cells at steady state: one in the gas cap, and one in the oil column.
Introduction
Proper initialization is an important aspect of reliable reservoir simulations. The use of the Gibbs segregation condition generally cannot provide reliable initialization in hydrocarbon reservoirs. This is caused, in part, by the effect of thermal diffusion (caused by the geothermal temperature gradient), which cannot be neglected in some cases; thermal diffusion might be the main phenomenon affecting compositional variation in hydrocarbon reservoirs, especially for near-critical gas/condensate reservoirs (Ghorayeb et al. 2003).
Generally, temperature increases with increasing burial depth because heat flows from the Earth's interior toward the surface. The temperature profile, or geothermal gradient, is related to the thermal conductivity of a body of rock and the heat flux.
Thermal conductivity is not necessarily uniform because it depends on the mineralogical composition of the rock, the porosity, and the presence of water or gas. Therefore, differences in thermal conductivity between adjacent lithologies can result in a horizontal temperature gradient. Horizontal temperature gradients in some offshore fields can be observed because of a constant water temperature (approximately 4°C) in different depths in the seabed floor.
The horizontal temperature gradient causes natural convection that might have a significant effect on species distribution (Firoozabadi 1999). The combined effects of diffusion (pressure, thermal, and molecular) and natural convection on compositional variation in multicomponent mixtures in porous media have been investigated for single-phase systems (Riley and Firoozabadi 1998; Ghorayeb and Firoozabadi 2000a).The results from these references show the importance of natural convection, which, in some cases, overrides diffusion and results in a uniform composition. Natural convection also can result in increased horizontal compositional variation, an effect similar to that in a thermogravitational column (Ghorayeb and Firoozabadi 2001; Nasrabadi et al. 2006).
The combined effect of convection and diffusion on species separation has been the subject of many experimental studies. Separation in a thermogravitational column with both effects has been measured widely (Schott 1973; Costeseque 1982; El Mataaoui 1986). The thermogravitational column consists of two isothermal vertical plates with different temperatures separated by a narrow space. The space can be either without a porous medium or filled with a porous medium. The thermal diffusion, in a binary mixture, causes one component to segregate to the hot plate and the other to the cold plate. Because of the density gradient caused by temperature and concentration gradients, convection flow occurs and creates a concentration difference between the top and bottom of the column. Analytical and numerical models have been presented to analyze the experimental results (Lorenz and Emery 1959; Jamet et al. 1992; Nasrabadi et al. 2006). The experimental and theoretical studies show that the composition difference between the top and bottom of the column increases with permeability until an optimum permeability is reached. Then, the composition difference declines as permeability increases. The process in a thermogravitational column shows the significance of the convection from a horizontal temperature gradient.
Direct Collocation with Reproducing Kernel Approximation for Two-Phase Coupling System in a Porous Enclosure