# What is the Correct Definition of Average Pressure?

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s11242-009-9490-2

- Cite this article as:
- Korteland, S., Bottero, S., Hassanizadeh, S.M. et al. Transp Porous Med (2010) 84: 153. doi:10.1007/s11242-009-9490-2

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## 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.

### Keywords

Porous mediaVolume averagingMacroscale pressureAverage pressureCapillary pressure### List of Symbols

*Latin Symbols*

- \({\hat{\boldsymbol{e}}_{f}}\)
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^{−1}T^{−2}]*P*_{c}Capillary pressure [ML

^{−1}T^{−2}]*P*_{d}Entry pressure [ML

^{−1}T^{−2}]- Δ
*P* nonwetting phase bottom boundary overpressure [ML

^{−1}T^{−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 [L

^{3}]*V*_{α}Volume of integration of phase

*α*[L^{3}]*z*Position vector [L]

- \({\langle z \rangle}\)
Centroid of averaging volume [L]

*z*_{f}Position of the infiltration front in the domain [–]

*Greek Symbols*

*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^{−1}T^{−2}]

*Special Notations*

- \({\langle \rangle}\)
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 [ ]

*Subscripts*

*α*Phase, either wetting (w) or nonwetting (n)

- n
nonwetting phase

- w
Wetting phase