Page 144 - gas transport in porous media

Chapter 8: Gas Injection and Fingering in Porous Media
137
If two miscible fluids, with an initially sharp front separating them, are put in
contact, their subsequent mixing caused by molecular diffusion is described by the
diffusion equation:
∂G i ∂C i
=−D io A (8.2)
∂t ∂x
where G i is the amount (in moles) of fluid i that has diffused across the front at time
t, D io is the effective diffusion coefficient of fluid i in the porous medium, A is the
cross-sectional area for diffusion, and C i is the molar concentration of i at position
x at time t. The diffusion coefficient D io depends, in principle, on the mixture’s
composition, but in reservoir simulations an average diffusion coefficient at 50%
solvent concentration usually yields adequate representation of the diffusive mixing.
Many experimental methods have been developed for measurement of the effective
diffusion coefficient D io involving oil in porous media (see, for example, Reamer and
Sage, 1958; Gavalas et al., 1968; Schmidt et al., 1982; Renner, 1988; Nguyen and
Farouq-Ali, 1995). Experimental data for the effective diffusivity are still necessary
because, despite several decades of research, no accurate theoretical method for
estimating the effective diffusivities of mixtures in porous media is yet available
(Sahimi, 1993a, 1995, 2003). Unfortunately, even the experimental measurements
are generally difficult and very time consuming. Most conventional methods require
composition analysis which is tedious and expensive (Moulu, 1989). Simpler methods
of measuring the effective diffusion coefficients for gas-oil mixtures, which use PVT
cells with no compositional analysis, have also been proposed (Riazi, 1996; Zhang
et al., 2000).
Dispersion is mixing of two miscible fluids flowing in a system, such as a porous
medium. Therefore, unlike diffusion, the flow velocity field plays an important role
in this type of mixing process. Similar to diffusion, mixing by a dispersion process
can decrease the viscosity and density contrasts between the displacing gas and the
displaced fluids, which in most cases is very useful to the displacement process.
Two major mechanisms of dispersive mixing are small- and large-scale variations
of fluid velocities (or, equivalently, the permeabilities), and molecular diffusion,
both of which help mixing of the two miscible fluids. If, for example, a first-contact
miscible solvent is injected into a linear packed-bed column to displace oil from it, the
effluent concentration profile of the solvent will have an S-shape, which is the result
of mixing of the solvent and oil in the packed-bed. Because of this, a transition zone
of solvent/oil mixtures separates a zone of 100% solvent from one which is pure oil.
This mixing, which is in the direction of the macroscopic flow, is called longitudinal
dispersion.
Dispersioncanalsooccurinthedirection(s)perpendiculartothedirectionofmacro-
scopic flow; this is referred to as the transverse dispersion, which occurs when, for
example, a solvent is injected into a stratified porous medium which consists of layers
of different permeabilities parallel to the macroscopic flow. In this case, the solvent