Page 294 - Fundamentals of Gas Shale Reservoirs

274 A REVIEW OF THE CRITICAL ISSUES SURROUNDING

found in standard references. Also Equation 12.28, since it Where the relaxation time τ is given by
is a regression, is controlled by clays that hold large amounts
of clay‐bound water (Ws) such as montmorillonite. So the o a , 0 (12.31b)
quantitative significance of this mechanism is difficult to S w S wi a
establish at this time.
The wettability of the organic pores varies as a function of Hence, the relaxation time τ varies with the wetting fluid
maturity. Even if dominantly gas wet, as one expects for mature (water) saturation S , where the parameters (S , τ , and a)
wi
o
w
organic pores, oxygen sites at the pore walls can hold and trap depend on temperature.
water (Hu et al., 2013a, b). For less mature organic material, In Barenblatt et al. (2003) where an effective saturation is
the pore wall can go from mixed wettability to water wet. introduced, it is argued that there exists an effective satura­
Other possible trapping mechanisms are even more tion S such that the relative permeability curves and the
eff
poorly understood, so at this time it is impossible to make a capillary pressure curve evaluated under equilibrium condi­
good estimate on how much frac water will go into the tions at this effective saturation are equal to the relative per­
formation and how much will be mobile. It is, however, clear meability values and capillary pressure value at the actual
that current simulators do not properly handle mobile water nonequilibrium saturation S. They recognize that the
in the matrix, when there are both low permeability and high assumption that the same S works for all the functions may
eff
capillary pressures. not always be true, but assume it for simplicity. In their paper
The assumption that the gas and water are in capillary (Barenblatt et al., 2003), they argue that τ for most of its
equilibrium can lead to unrealistic results. Qin (2007) found range is a constant. To implement this approach in a simu­
for gas in water wet pores that under high draw‐down pres­ lator requires a better understanding of τ.
sures and high gas flow rates, this assumption led to negative For better understanding of the function of τ, Michel
water pressures. et al. (2012, 2013) have explored this for a bundle of capil­
To overcome the classical assumptions of equilibrium, lary tubes where for incompressible fluids of the same vis­
the variations in phase saturations need to be augmented cosity and the transition from one equilibrium state to
with a different formulation. One approach to this would be another can be solved analytically. For this case, the assump­
to introduce the capillary force directly into the flow tion that τ is a constant over most of its range is not satisfied.
equations. This approach has been applied to model the rise Though Michel et al. (2012, 2013) were able to find expres­
of water in capillary tubes (Hamraoui and Nylander, 2002; sions for τ that may be generalizable to the gas/water case,
Schoelkopf et al., 2000; Zhmud et al., 2000). A second further exploratory work is necessary in order to determine
approach is to introduce a relaxation time function that if the capillary nonequilibrium formulation for modeling
relates the nonequilibrium saturation state to the equilibrium multiphase gas‐water flows in very low permeability sys­
state (Andrade et al., 2010, 2011; Barenblatt et al., 2003; tems where the time to reach equilibrium may be signifi­
Hanspal and Das, 2011). This is done by relating the effec­ cantly longer than the simulation time step is feasible. If not,
tive equilibrium fluid saturation S to the instantaneous adding the capillary force directly into the flow equations
e
nonequilibrium fluid saturation S according to the follow­ may be required.
ing relationship (Barenblatt et al., 2003; Hassanizadeh and
Gray, 1993):
12.6 CHARACTERIZATION OF FLUID BEHAVIOR
S S S
S e S XS,, S, gS AND EQUATIONS OF STATE VALID FOR
t t t NANOPOROUS MEDIA
(12.29)
Numerous studies have demonstrated that the properties and
which is usually approximated simply as behavior of fluid systems in nanoporous media deviate from
S those observed in bulk fluids. This has two primary sources.
S S (12.30) The first is in nanoporous media where the interaction of the
e t
fluid molecules with the pore wall cannot be ignored relative
where the relaxation time is denoted by τ and simulation time to intermolecular interactions, only the latter of which is
by t, S is the effective fluid saturation, and S is the instanta­ important in bulk fluids. The second is that the small number
e
neous fluid saturation under nonequilibrium conditions. Civan of molecules in the pore may suppress development of mul­
(2012) proposed the following formulation for correlation of tiple phases or the existence of a true liquid phase. In some
the relaxation time: cases, in hydrocarbon‐bearing shale reservoirs, the modifi­
cation of the fluid properties and behavior may create effects
dS that enhance transport capabilities of the pore fluids. Because
p dyn p e w (12.31a)
c c dt the extent of such modifications depends on pore size in