## 1. Introduction

[2] When Darcy introduced his well-known equation 150 years ago, it was solely based on one-dimensional flow of water, as a constant density incompressible fluid, in a homogeneous nondeformable porous medium under isothermal conditions. That simple formula has subsequently been “extended” (but actually keeping its original form) to be valid for simultaneous flow of many (possibly compressible) fluids in a heterogeneous anisotropic deformable porous medium under nonisothermal conditions in three dimensions. One major change has been to formulate Darcy's equation in terms of fluid pressure instead of the hydraulic potential. This is particularly relevant for multiphase flow where an auxiliary equation for the difference in pressures of the fluids, often simply referred to as the capillary pressure, is introduced.

[3] The common form of Darcy's law for two fluids reads

Here **q**_{α} denotes the Darcy velocity vector for phase *α*, *k*_{r,α} is relative permeability, *μ*_{α} is viscosity, **K** is the intrinsic permeability tensor, *ρ*_{α} is mass density, *g* the gravitational constant, *z* the vertical coordinate (increasing downward), and [*P*]^{α} is the Darcy-scale fluid pressure. The pressure has been placed in square brackets to emphasize that this is a macroscale quantity. In this multiphase extension, it is not clear what exactly the macroscale pressure is, and in particular how it relates to the pressure at the microscale (pore scale).

[4] Derivations of Darcy's law using volume averaging methods have always involved the assumption that the macroscale pressure is equal to the intrinsic phase average pressure 〈*P*〉^{α} (to be defined later). In all volume-averaging based derivations, many assumptions are made and a number of terms are neglected. In a separate paper [*Nordbotten et al.*, 2007], we have shown that these assumptions can lead to unacceptable restrictions on the range of applicability of Darcy's law. In particular, when there are gradients in fluid distribution and pressure, use of the intrinsic phase average leads to untraditional gravitational forces. This has also been shown in a recent paper by *Gray and Miller* [2004]. In the case of two-phase flow, intrinsic phase average pressure will involve weighting of the microscopic pressure by saturation, as well as porosity. So, even in the case of homogeneous media, because we will in general have gradients in saturation and pressure on all scales, use of the intrinsic phase average pressure will lead to additional terms in Darcy's law. In fact, as we show in section 4.1, use of the intrinsic phase average pressure leads to relative permeability being larger than unity in some flow situations.

[5] These observations lead us to question the definition of macroscale pressure in terms of the microscale pressure. While this question has significant theoretical implications, it is also relevant to many upscaling studies that involve computation of the pressure field at a given scale and the associated inference of parameters at a larger scale (see, e.g., *Barker and Thibeau* [1996] for a review pointing out the ambiguity of pressure definition in upscaling). For example, in dynamic pore-scale network models, the macroscale pressure field for a given fluid is almost always defined as the intrinsic phase pressure (i.e., the average pressure of a fluid in all pore bodies weighted by the pore body volume occupied by that fluid) [see, e.g., *Dahle and Celia*, 1999; *Gielen et al.*, 2004, 2005; *Manthey et al.*, 2005]. This means that results of computations from dynamic pore-scale network models may lead to predictions of unphysical macroscopic behavior. This problem does not appear to a similar extent in static upscaling models, where the pressure in connected phases is essentially constant over the sample (see *Blunt* [2001] for a recent review). We illustrate this in section 4.1, where we study the displacement of a wetting phase by a nonwetting phase in a capillary tube, which is the simplest pore-scale model one can construct. We expand upon the analysis by also analyzing the results obtained using the dynamic pore-scale model of *Nordhaug et al.* [2003]. Our objective in this paper is to develop a definition of macroscopic variables from microscopic variables which gives physically consistent results at the macroscale. The variable we focus on is pressure, but the concepts apply to other upscaled variables as well.