Modeling Leaking Gas Plume Migration Page: 3 of 15
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MODELING LEAKING GAS PLUME MIGRATION
two-phase flow states the following relationships between the Darcy velocities of gas and brine and their
respective pressure gradients (Muskat, 1949; Hubbert, 1956):
krg(S)k Opg '1
Ug = g g (1)
- krw(S)k Opw -(2)
uw ~ =Oz g(
Here ug and uw are Darcy velocities, or volumetric fluxes, of the gas and liquid, pg and ,w are the dynamic
viscosities of the fluids, pg and pw, and gg and gw are their pressures and densities, respectively. The brine
volumetric saturation and gravity acceleration are denoted by S and g, and k is the absolute permeability
of the medium. In this derivation, we neglect the compressibility of brine and supercritical gas. Since there
is no sink or source of gas or brine, the flow is countercurrent:
Ug + uw = 0 (3)
To obtain equations in dimensionless form, we introduce dimensionless vertical coordinate ( and time T:
z k=(_ g-g ) t (4)
H p2 H
Here H is the thickness of the reservoir. A dimensionless formulation helps to single out the most important
parameters and to simplify the model by dropping insignificant terms. Silin et al. (2006) have obtained
that the contribution of the capillary pressure in the flow dynamics is negligible if
(s as < 1 (5)
1
where
(w - gg)H k (6)
is an analog of the reciprocal Bond number. In what follows, we assume that condition (5) holds true and
the impact of capillarity can be neglected. Combination of Darcy's law with conservation of mass yields a
hyperbolic equation for brine saturation S:
as f(S) = 0 (7)
where
f(S) = krw(S) (8)
krw(S) pg + 1
krg(S) Pw
is the partial flow function. Equation (7) constitutes a Buckley-Leverett type flow model (Buckley and
Leverett, 1942) in dimensionless form. It is interesting to note that the dimensionless Darcy velocity of
brine
k w
WW = k(gw - gg)9uw ()
equals minus partial flow function:
W = -f (S) (10)
and the gravity acceleration enters the model through the time scale (4) only.3
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Silin, Dmitriy; Patzek, Tad & Benson, Sally M. Modeling Leaking Gas Plume Migration, article, August 20, 2007; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc902218/m1/3/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.