Page 262 - Fundamentals of Enhanced Oil and Gas Recovery

250 Alireza Keshavarz et al.

Eq. (8.15) reveals that rather than pressure gradient, diffusion is dependent on con-
centration gradient. Fick’s second law could be solved in terms of fractional adsorp-
tion/desorption:

N !
M t 6 X 1 2 2 Dt
5 1 2 exp 2n π (8.16)
M N π 2 n 2 r 2
n51 p
where M t is the amount of adsorbed/desorbed gas at time t, M N is the total amount
of adsorbed/desorbed gas at equilibrium condition, D is the diffusion coefficient, r p is
the mean radius of coal particle radius.
In order to estimate diffusion coefficient measure, D, at a given time, the experi-
mental data is inserted into Eq. (8.16), thereby D is calculated. Since in this equation
micropores are assumed to be monosized, this model is referred to as “unipore diffu-
sion model.” Although unipore models have been applied to coals, they were proved
to fit data only over restricted time intervals [44,50,51]. Thus, predicting gas flow in
heterogeneous pore structures requires the consideration of different pore sizes in dif-
fusion models.

8.4.2.2 Bidisperse Model
Considering the fact that the coal structure is highly heterogeneous, the unipore model
usually does not predict the diffusion coefficient precisely [52 54]. Ruckenstein et al.
proposed a bidisperse diffusion model to more realistically describe the pore size distri-
bution and consequently more accurately predict the diffusion behavior [55]. Bidisperse
model limits pore size distribution to two sizes: macropore and micropore. In this
model, the adsorbent contains microporous spherical particles separated by inter-particle
macropores. The bidisperse model was applied to coal for the first time by Smith and
Williams [56,57]. In this approach, coal matrix is assumed as a double porosity medium
with two distinct pore sizes, macropores indicating fast diffusion [Eq. (8.17)]and micro-
pores characterized by slow diffusion, as observed in Eq. (8.18) [58,59].

N !
M at 6 X 1 2 2 D a t
5 1 2 exp 2n π (8.17)
M aN π 2 n 2 r 2
n51 pa
N !
M it 6 X 1 2 2 D i t
5 1 2 exp 2n π (8.18)
M iN π 2 n 2 r 2
n51 pi
where M at and M it are the gas adsorption/desorption amount from macropores and
micropores at time t, respectively; M aN and M iN are the total amount of adsorbed/
desorbed gas in macropores and micropores at equilibrium condition, respectively; D a
and D i are the macropores’ and micropores’ diffusion coefficients, respectively.