Page 279 - Fundamentals of Gas Shale Reservoirs

PULSE‐DECAY PERMEABILITY MEASUREMENT TEST 259

kf A 1(/ V 1/ V ) (Coppens, 1999). The main parameter of the permeability
s 1 u d , (11.34) model, the intrinsic permeability, and the remaining unknown
1
( Lc )
g parameters may be estimated using the pulse‐decay experi-
mental data. Most accurate estimates are achieved by
2
f 1 1 , (11.35) performing several pulse‐decay experiments at different
( ab) mean pressures. To obtain reliable results, the lower and
upper bounds of physical parameters should be strictly
where θ is the first solution to
1 defined as well. With these considerations, the following
optimization problem is formulated and solved to estimate
((ab ))
tan . (11.36) the permeability model parameters,
( 2 ab )
J
The plot of ∆m on a log scale versus time yields a straight Minimize fX() ( m m ) 2
D
line at late‐transient times. Equation 11.34 implies that the j 1 D j , Ddata j , (11.38)
slope of the line s is related to permeability,
1 st .. X X X ,
l u
sLc
k 1 g . (11.37) where ∆m is the pseudo‐pressure decline estimate, ∆m
Ddata,j
D,j
fA 1 (/ V u 1/ V ) is the pseudo‐pressure decline experimental data, X is the
1
d
vector of unknown permeability parameters, and J is the
Equation 11.37 calculates the permeability at average number of data points. Subscripts i and u denote lower and
core sample pressure at late‐transient time. Figure 11.14b upper parameter bounds, respectively. The dimensionless
compares the modified late‐transient analytical and pseudo‐pressure difference in the objective function may be
numerical solutions assuming Darcy permeability (i.e., substituted from either numerical or analytical solutions.
k = k ). Figure 11.14c compares the same solutions assuming Equation 11.38 is classified as constrained and nonlinear
D
the apparent permeability function (Darabi et al., 2012). optimization problem (Borwein and Lewis, 2000) and can be
Both figures show that the late‐transient analytical solution solved with any appropriate optimization algorithm (e.g.,
is in good agreement with the numerical solution. Therefore, Mehrotra, 1992). One of the following two methods may be
under typical pulse‐decay conditions, that is, high pressure used to estimate the permeability parameters:
and small pressure difference across core sample, the analyt-
ical solution may be used to estimate apparent permeability 1. With the analytical pseudo‐pressure solution: the
for shale samples. However, if the pressure difference across objective function in Equation 11.38 is formulated
the core is large or the initial core pressure is low, the with the analytical solution (Equation 11.32). The opti-
analytical solution may result in a significant error in the mization problem is then solved to find the best match
permeability estimate. for the late‐transient pressure‐decay behavior and
subsequently estimate the permeability parameters.
2. With an iterative numerical solution‐optimization
11.8.2 Estimation of Permeability Parameters with the
Pulse‐Decay Experiment method: for each optimization step, the numerical
solution is first obtained; then the optimization is
Any pressure‐dependent permeability model has two or performed and a new set of fitting parameters is gener-
more physical or fitting parameters to be estimated. Using ated, which is used by the numerical simulator in the
independent experimental methods often help to better next step. This procedure is repeated until the gradient
estimate the physical parameters that are involved in the per- of the objective function with respect to all fitting
meability model. For example, if we select the apparent per- parameters is less than a specified tolerance.
meability function (APF) as the permeability model, then
appropriate estimations for tortuosity, tangential momentum The choice of estimation method is essentially independent
accommodation coefficient (TMAC), and the fractal from the permeability model; however, the advantage of the
dimension of the pore surface are required before the numerical method is that it uses all the recorded pressure‐
intrinsic permeability can be determined. Tortuosity may be decay data to obtain the best parameter estimates, whereas
estimated using SEM and AFM imaging. TMAC can be the analytical method only uses the late‐transient pressure
determined from the oil drop experiments of Millikan, the data. Therefore, the numerical method tends to yield more
rotating cylinder method, the spinning rotor gauge method, reliable estimates, especially if the number of available data
molecular beam techniques, and flow through microchan- points is limited. Figure 11.15 presents an example of perme-
nels (Agrawal and Prabhu, 2008). Fractal dimensions of ability parameters estimation using the APF based on the
surfaces can be determined from small‐angle X‐ray scattering analytical and numerical procedures.