Page 43 - gas transport in porous media

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Q, C flow Ho
C = 0
δ (t)
C = C s

Liquid

Figure 3.5. Sketch of one-dimensional evaporation from a liquid trapped in a stagnant region with
external convection in an adjacent region

Another one-dimensional transient diffusion scenario exists if the evaporating liq-
uid is trapped in a stagnant porous region while external convection exists (say, in an
adjacent high-permeability zone). Figure 3.5 illustrates this scenario.
The concentration at the surface of the liquid is a constant, C s , and the concen-
tration at the boundary of the region with advective flow is approximately zero. The
flow rate in the advective region is Q, and the average concentration in the advec-
tive region is denoted as C flow . During the initial periods of this external convective
drying scenario, mobile liquid will be drawn to the interface between the advec-
tive and stagnant regions by capillarity, keeping the evaporating surface stationary
at the interface. When the liquid reaches a residual saturation and becomes immobile,
the evaporating surface will begin to recede into the stagnant region (Ho and Udell,
1992). The distance between the receding evaporation front and the interface between
the advective and stagnant regions is denoted as δ(t).
Assuming that the transport between the receding evaporation front and the advec-
tive zone is governed by diffusion only, and that the recession of the evaporation
front is slow (quasi-steady), the evaporation rate at time, t, can be written as
follows:

dm C s − 0
=−DA =−QC flow (3.24)
dt δ(t)

where A is the cross-sectional area available for diffusion. Equation (3.24) equates
the evaporation rate (i.e., the rate of change in mass of the liquid, m) to the rate of
diffusion through the stagnant region and to the rate of mass advected away. The rate
of change in mass of the liquid can also be written as follows, assuming that the liquid
saturation and porous-media properties are constant:

dm dδ(t)
=−φS l ρ l A (3.25)
dt dt