Abstract
Use of a correct definition of average pressure is important in numerical modeling of oil reservoirs and aquifers, where the simulated domain can be very large. Also, the average pressure needs to be defined in the application of pore-network modeling of (two-phase) flow in porous media, as well as in the (theoretical) upscaling of flow equations. Almost always the so-called intrinsic phase-volume average operator, which weighs point pressure values with point saturation values, is employed. Here, we introduce and investigate four other potentially plausible averaging operators. Among them is the centroid-corrected phase-average pressure, which corrects the intrinsic phase-volume average pressure for the distance between the centroid of the averaging volume and the phase. We consider static equilibrium of two immiscible fluids in a homogeneous, one-dimensional, vertical porous medium domain under a series of (static) drainage conditions. An important feature of static equilibrium is that the total potential (i.e., the sum of pressure and gravity potentials) is constant for each phase over the whole domain. Therefore, its average will be equal to the same constant. It is argued that the correct average pressure must preserve the fact that fluid potentials are constant. We have found that the intrinsic phase-volume average pressure results in a gradient in the total phase potential, i.e., the above criterion is violated. In fact, only the centroid-corrected operator satisfies this criterion. However, at high saturations, use of the centroid-corrected average can give rise to negative values of the difference between the average nonwetting and wetting phase pressures. For main drainage, differences among various averaging operators are significantly less because both phases are present initially, such that the difference between the centroids of phases, and the middle of the domain are relatively small.
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Abbreviations
- \({\hat{\boldsymbol{e}}_{f}}\) :
-
Unit vector in the direction of flow [–]
- g :
-
Gravitational acceleration [LT−2]
- H :
-
Height of the domain [L]
- P α :
-
Pressure in phase α[ML−1T−2]
- P c :
-
Capillary pressure [ML−1T−2]
- P d :
-
Entry pressure [ML−1T−2]
- ΔP :
-
nonwetting phase bottom boundary overpressure [ML−1T−2]
- S α :
-
Saturation for phase α [–]
- S e :
-
Effective saturation [–]
- S wr :
-
Residual wetting phase saturation [–]
- S nr :
-
Residual nonwetting phase saturation [–]
- V :
-
Volume of integration [L3]
- V α :
-
Volume of integration of phase α[L3]
- z :
-
Position vector [L]
- \({\langle z \rangle}\) :
-
Centroid of averaging volume [L]
- z f :
-
Position of the infiltration front in the domain [–]
- η α :
-
Indicator function for phase α [–]
- λ:
-
Brooks–Corey pore-size distribution parameter [–]
- ρ α :
-
Density of phase α[ML−3]
- ρ :
-
Density ratio [–]
- \({\Phi_{\alpha}}\) :
-
Potential of phase α[ML−1T−2]
- \({\langle \rangle}\) :
-
Potential-based average operator
- \({\langle \rangle^{i}}\) :
-
Intrinsic phase-volume average operator
- \({\langle \rangle^{sp}}\) :
-
Simple phase-average operator
- [ ]1 :
-
Centroid-corrected phase-average operator
- \({\langle \rangle^{s}}\) :
-
Simple average operator
- . . .′:
-
(prime sign) Dimensionless form of variable [ ]
- α :
-
Phase, either wetting (w) or nonwetting (n)
- n:
-
nonwetting phase
- w:
-
Wetting phase
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Acknowledgments
The authors would like to thank Jan Martin Nordbotten for his advice regarding the centroid-corrected phase-average pressure. Rainer Helmig, Michael Celia, and Helge Dahle are gratefully acknowledged for many fruitful discussions. The first author would like to thank the Utrecht University master program System Earth Modeling for providing a scholarship to visit Stuttgart University. Authors are members of the International Research Training Group NUPUS, financed by the German Research Foundation (DFG) and the Netherlands Organisation for Scientific Research (NWO), and thank DFG (GRK 1398) and NWO (DN 81-754) for their support.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Korteland, S., Bottero, S., Hassanizadeh, S.M. et al. What is the Correct Definition of Average Pressure?. Transp Porous Med 84, 153–175 (2010). https://doi.org/10.1007/s11242-009-9490-2
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DOI: https://doi.org/10.1007/s11242-009-9490-2