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Revisiting the Drainage Relative Permeability Measurement by Centrifuge Method Using a Forward–backward Modeling Scheme

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Abstract

Measurement of drainage relative permeability by the centrifuge method was first introduced by Hagoort (SPE J. 29(3):139–150, 1980). It has been shown that capillary end effects can cause error in the measurement of relative permeability if a minimum rotational speed is not honoured. To determine the minimum rotational speed that makes the capillary end effect negligible, ω min, we propose that the value of capillary-gravity number, N cg, should be of the order of 10−2 or smaller. This conclusion is based on the use a Forward–backward scheme consisting of a forward numerical simulator developed for centrifuge experiments and applying Hagoort’s method as a backward model. The article presents the use of this Forward–backward scheme as a powerful tool for error analysis such as determining the impact of capillary end effects. By using this loop, we first determine ω min for specific core and fluid properties. Later, we generalize the ω min calculations by using the definition of N cg as a “rule of thumb” for designing relative permeability experiments by centrifuge method. We also demonstrate another use of this loop for controlling the quality of the experimental data.

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Abbreviations

g c :

Average centrifugal acceleration [LT−2]

k rw :

Relative permeability of the wetting phase, fraction

\({k^\prime_{\rm rw}}\) :

Derivative of the wetting phase relative permeability, fraction

k rwD :

Relative permeability of the wetting phase, normalized

\({k_{\rm rw}^o }\) :

End point value of the wetting phase relative permeability, fraction

L :

Length of the core sample [L]

n :

Corey exponent, dimensionless

n :

Number of data points in error calculations

N B :

Bond number, dimensionless

N cg :

Capillary-gravity number, dimensionless

N p :

Pore volume cumulative production, dimensionless

P c :

Capillary pressure [ML−1T−2]

P cth :

Threshold capillary pressure [ML−1T−2]

P cD :

Capillary pressure, dimensionless

r m :

Distance of center of the rotation to middle of the core [L]

r o :

Distance of center of the rotation to the core inlet [L]

S wD :

Wetting phase saturation, normalized

S wDe :

Wetting phase saturation at core exit, normalized

t :

Time [T]

t D :

Time, dimensionless

x :

Length in x-direction [L]

x D :

Length in x-direction, dimensionless

β(t):

Time function of the rotational speed, dimensionless

\({\phi}\) :

Porosity, fraction

Δρ :

Density difference [ML−3]

ω :

Rotational speed (rpm)

ω 1 :

Initial rotational speed (rpm)

ω min :

Required rotational speed in relative permeability experiments (rpm)

μ w :

Wetting phase viscosity [ML−1T−1]

σ :

Surface tension [MT−2]

N B :

 = kΔρω 2 r m/σ

N cg :

\({=\dfrac{P_{\rm cth}}{\Delta \rho gL},N_{\rm cg} =\dfrac{P_{\rm cth}}{\Delta \rho \omega^{2}r_{\rm m} L}}\)

S wD :

\({=\dfrac{S_{\rm w} -S_{\rm wir}}{1-S_{\rm wir} -S_{\rm gc}}}\)

t D :

\({=\left( {\dfrac{kk_{\rm rw}^o}{{\rm \mu}_w}\Delta \rho \omega_{1}^{2} r_o}\right)\dfrac{t}{L\phi \left( {1-S_{\rm wir} -S_{\rm gc}}\right)}}\)

β (t ):

\({=\,\left\{ \begin{array}{l@{\quad}l} 0 & {\rm t} < 0 \\ 1 & 0\leq {\rm t} \leq {\rm t}_1 \\ \left(\frac{\omega_j}{\omega_1} \right)^{2} & {\rm t}_{j-1} \leq {\rm t} \leq {\rm t}_{\rm j} \end{array} \right.}\)

P cD :

\({=\dfrac{P_c}{P_{\rm cth}}}\)

k rwD :

\({=\dfrac{k_{\rm rw}}{k_{\rm rw}^o}}\)

x D :

\({=\dfrac{x}{L}}\)

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  1. University of Calgary, Calgary, Canada

    Majid Saeedi & Mehran Pooladi-Darvish

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  1. Majid Saeedi
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  2. Mehran Pooladi-Darvish
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Correspondence to Mehran Pooladi-Darvish.

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Saeedi, M., Pooladi-Darvish, M. Revisiting the Drainage Relative Permeability Measurement by Centrifuge Method Using a Forward–backward Modeling Scheme. Transp Porous Med 86, 49–71 (2011). https://doi.org/10.1007/s11242-010-9605-9

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