Abstract
Relative permeability (k_{r}) data are the key factors for describing the behaviour of the multiphase flow in porous media. During the k_{r} measurements of lowpermeability rocks, high capillary pressure can cause a significant liquid holdup at the core outlet. This liquid holdup, which is known as capillary end effect (CEE), is the main difficulty for laboratory measurements of relative permeability (k_{r}) for tight and shale rocks. In this paper, a novel method is proposed to correct the CEE during the steadystate relative permeability (SSk_{r}) measurements. The integrity of the proposed method is evaluated by a set of artificially generated data and the experimental SSk_{r} data of an Eagle Ford shale sample. It is shown that accurate k_{r} data can be obtained using the proposed technique. This technique can be used to estimate reliable k_{r} data without any saturation profile measurement equipment, such as CT scan or MRI.
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Abbreviations
 A :

Area
 K :

Absolute permeability
 q :

Flow rate
 S :

Saturation
 F :

Liquid/gas flow rate ratio
 P :

Pressure
 L :

Length
 x :

Distance
 F :

Liquid/gas flow rate ratio
 \( S_{\text{o}}^{*} \) :

Wetting phase (oil) saturation
 \( \overline{S}_{\text{o}} \) :

Average wetting phase (oil) saturation
 IFT:

Interfacial tension
 μ :

Viscosity
 g:

Gas
 o:

Oil
 c:

Capillary pressure
 or:

Residual oil
 gr:

Residual gas
 ro:

Oil relative permeability
 rg:

Gas relative permeability
 out:

Outlet
 Exp:

Experimental
 CEE:

Capillary end effect
 unaf:

Unaffected
 t:

Total
 r:

Relative
 CEE:

Capillary end effect
 SS:

Steady state
 LGR:

Liquid/gas flow rate ratio
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Acknowledgements
This study was conducted as a part of the Unconventional Gas and Gascondensate Recovery Project at HeriotWatt University. This research project is sponsored by Daikin, Dong Energy, Ecopetrol/Equion, ExxonMobil, GDF, INPEX, JXNippon, Petrobras, RWE, SaudiAramco and TOTAL, whose contribution is gratefully acknowledged.
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Appendix 1
Appendix 1
Gupta and Maloney (2016) proposed the intercept technique to correct CEE during relative permeability measurements. The intercept method was proposed to correct CEE errors from both pressure and saturation measurements for each LGR. In this technique, several measurements of rate versus pressure drop are required at the same LGR. The obtained trends in pressure drop versus rate and saturation versus rate will be used to correct the data for each single LGR. To correct the pressure data, one can start with Darcy’s equation as follows
where \( \Delta P_{\text{theoritical without CEE}} \) is the pressure drop across the core without any capillary contribution to the pressure drop. \( \Delta P_{\text{theoritical without CEE}} \) can be expressed as the difference between the experimental pressure drop across the core \( \Delta P_{\text{Exp}} \) and the pressure drop resulting from the \( \Delta P_{\text{CEE}} \). Therefore,
Rearranging Eq. A2 gives
Using the above concept, Gupta and Maloney proposed to obtain \( \Delta P_{\text{CEE}} \) from the intercept of the plot of laboratorymeasured pressure drop across the core (\( \Delta P_{\text{Exp}} \)) and the injected total flow rate (\( q_{t } ). \)
To correct the saturation data, they used the following overall saturation balance equation for a given fractional flow condition:
where \( x_{i} \) is the length of CEE region, \( S_{\text{w, avg}} \) is average water saturation, \( S_{\text{w, CEE}} \) is average saturation of CEE region, and \( S_{\text{w, true}} \) is actual water saturation as shown in Fig. 10.
They defined CEE length factor as \( \beta = \frac{{x_{i} }}{L} \) and proposed the following equation as a reliable concept to correct the saturations.
Rearranging Equation A4 and using Equation A5 gives the expression
Based on Equation A6, CEEcorrected saturation (i.e. \( S_{\text{w, true}} \)) is the intercept of the plot of (\( \frac{1}{{\left( {1  \beta } \right)}} S_{\text{w, avg}} ) \) and (\( \frac{\beta }{{\left( {1  \beta } \right)}} \)).
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Nazari Moghaddam, R., Jamiolahmady, M. SteadyState Relative Permeability Measurements of Tight and Shale Rocks Considering Capillary End Effect. Transp Porous Med 128, 75–96 (2019). https://doi.org/10.1007/s11242019012368
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Keywords
 Relative permeability
 Shale rock
 Capillary end effect
 Unconventional reservoirs
 Steady state