Natural Convection and Diffusion in Porous Media

The final topic of this chapter is natural convection and diffusion in porous media with the objective of studying composition variation in hydrocarbon reservoirs. The understanding of irreversible phenomena facilitates such a study the use of the Gibbs sedimentation equation, —Migdz, which has been used by some authors in the literature, is not justified because of entropy production. 

In the formulation of thermal convection in porous media at steady state, it was demonstrated that the horizontal gradient of temperature drives the thermal convection. In fact, the driving force for both thermal convection (that is, the convection due to thermal gradient) and natural convection (that is, the convection due to both thermal gradient and composition gradient) is governed by (dp/dx) at steady state. The expression for (dp/9x) is given by 

When there is only bulk flow and diffusive fluxes are zero, the temperature gradient (dT/dx) is the sole contributor to density gradient (dp/dx). With diffusion, the second term on the right side of Eq. (2.122) becomes effective. The two terms on the right side of Eq. (2.122) may have the same sign, or opposite signs and may have different magnitudes relative to each other. Therefore, convection may enhance composition variation due to the effect of the second term. Such a behavior is not in line with the common belief in the literature that convection always reduces composition variation in hydrocarbon reservoirs. 

We now present thfe equations that describe the combined effect of convection and diffusion in porous media. Let us assume that there are two components in the mixture, and that there is a single phase, either gas or liquid the geometry is a two-dimensional rectangle (that is, X — z). It is also assumed that the temperature field is known. In hydrocarbon reservoirs, temperature data can be measured with modern 

The diffusive mass flux of component 1 is given by the expression 

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