Page 253 - Geology and Geochemistry of Oil and Gas

MODELS OF STATIC GEOLOGIC SYSTEMS 219
If a reservoir is modeled by a bundle of capillary tubes, the rate of flow, q, can be
given by the Hagen–Poiseuille equation
4
q ¼ Npr Dp=8mL c (11.16)
c
3
where q is the volumetric flow rate (cm /sec), N is the number of capillaries, r c is the
2
capillary radius (cm), Dp is the differential pressure across the capillaries (dyn/cm ),
m is the fluid viscosity (P) and L c is the length of capillaries (cm).
The Darcy equation for rate of flow q is
(11.17)
q ¼ kADp=mL c
where q is the volumetric rate of flow, k is the permeability (darcy), A is the total
2
cross-sectional area (cm ), Dp is the differential pressure (atm), m is the fluid viscosity
(cP), and L c is the length of the flow path.
2
If, instead, viscosity is expressed in poises and differential pressure in dyn/cm ,
then
9
q ¼ 9:869 10 kDp=mL c (11.18)
The porosity f of this bundle of capillary tubes may be expressed as the capillary
volume V c per unit of bulk volume, V b :
2
2
f ¼ V c =V b ¼ Npr L c =AL c ¼ Npr =A (11.19)
c c
Thus, the total cross-sectional area A of the bundle of capillary tubes is
2
A ¼ Npr =f (11.20)
c
The average capillary tube radius r c may be found by combining Eqs. 11.16, 11.17,
and 11.20:
r c ¼ 2ð2k=fÞ 1=2 (11.21)
The surface area per unit of pore volume s p is given by:
2 2
s p ¼ N pr c L c =Npr L c ¼ 2=r c (11.22)
c
On substituting the value of capillary tube radius from Eq. 11.21 into Eq. 11.22,
the specific surface area per unit of pore volume, s p , can be expressed as
1=2
s p ¼ ðf=2kÞ (11.23)
For the specific surface area per unit of grain volume of rock consisting of equal-
size spheres, the following formula was proposed (Buryakovsky et al., 2001):
2 2 3
s g ¼ A=V ¼ Npd ð1 fÞ=V sph ¼ 6Npd ð1 fÞ=Nd p ¼ 6ð1 fÞ=d (11.24)
where A is the surface area, V is the bulk volume of rock, V sph is the total volume of
3
spheres (cm ), N is the number of spheres per unit of grain volume, d is the diameter
of sphere (cm), and f is the porosity (fractional).
If spheres (grains) are not equal in size, then the specific surface area per unit of
grain volume of such a rock may be calculated using the following formula:

X
s g ¼ ½6ð1 fÞ=100Š ðC i =d i Þ (11.25)