Abstract
Exponential four- and five-parameter equations are proposed for gas/oil drainage and water/oil imbibition proposed for gas/oil drainage and water/oil imbibition relative permeability curves. These equations match the experimentally determined curves, in particular at and near their initial points and endpoints, better than standard Corey et al. and polynomial approximations. Some of these parameters have a physical meaning; the others can be determined by nonlinear regression on the experimental data points, and can be adjusted to represent pseudorelative permeability curves. The proposed equations are particularly suitable to describe gas percolation in numerical model simulation of percolation in numerical model simulation of dissolved- gas-drive reservoirs.
Introduction
In computations concerning the behavior of two-phase flow in porous media, the results may depend strongly on the shape of the relative permeability curves used. Algebraic equations are usually employed to reproduce experimentally determined relative permeability curves, or to approximate them when there are no experimental data.
Relations proposed by Corey et al., which are based on bundle-of-capillaries model, are usually employed for gas/oil drainage relative permeability curves. The Wyllie and Gardner model, consisting of a bundle of capillaries cut and rejoined along their axis with related entrapment of the wetting phase, was used by Land to obtain the relations usually employed for water/oil imbibition relative permeability curves.
As demonstrated elsewhere. these "classical" relations fail to match the actual behavior of the experimentally determined relative permeability curves, in particular at their initial points and endpoints. particular at their initial points and endpoints. Proposed Relations Proposed Relations Gas/Oil Drainage. The following equations have been found to reproduce very accurately the experimentally determined gas/oil drainage relative permeability curves, including their behavior at the initial points and endpoints.
kro = exp (-ARLg),.......(1a)
krg = exp (BRg-M),.......(1b)
where A, B, L, and M are positive numbers, and
Sg - SgcRg = 1 - Siw - Sg (2a)
with the constraint
Sg - Sgc = 0 for S g is less thanSgc..(2b)
Eqs. 1a and 1b are four-parameter equations, the parameters being A, L, Sgc, and Siw,. for kro(So), and B, parameters being A, L, Sgc, and Siw,. for kro(So), and B, M, Sgc, and Siw, for krg(Sg). Only Sgc and Siw have a physical meaning; for statistically homogeneous physical meaning; for statistically homogeneous reservoir zones, the average values of Sgc and Siw, can be evaluated by a normalization technique described elsewhere.
The values of the empirical coefficients A, L and B, M are determined by nonlinear regression on the sets of experimental data points. If a regression process is applied to Eqs. 1a and 1b, the minimization of the variance may cover up large relative errors in kr, calc/kr, act in the neighborhood of kr = 0.
To avoid this, the logarithmic form of Eqs. 1a and 1b,
-1n Kro + ARLg...........(3a)
and
-1n Krg = BRg-M,........ (3b)
is used to evaluate the coefficients A, L and B, M. In this case, the variance of the error of the estimate is
(4)
which ensures a good match with the relative permeability curves also for k, values near to zero. permeability curves also for k, values near to zero. An example of the matching obtained by this procedure is shown in Fig. 1. procedure is shown in Fig. 1. Water/Oil Imbibition. The following equations have been found to reproduce very accurately the experimentally determined relative permeability curves, including their behavior at the initial points and endpoints.
K*ro = exp (ARL2),.......(5a)
K*rw = exp (-BRw-M)......(5b)
where A, B, L, and M are positive numbers, and
Sw - SiwRw = 1-Sor-Sw (6)
KroK*ro = Kro(Siw).........(7a)
andKrwK*rw = .....(7b)Krw(Sor)
SPEJ
P. 275
Gas-Oil Relative Permeability and Residual Oil Saturation as Related to Displacement Instability and Dimensionless Numbers