Comput Geosci (2009) 13:227–234 DOI 10.1007/s10596-008-9109-7 ORIGINAL PAPER Towards a thermodynamics of immiscible two-phase steady-state flow in porous media Alex Hansen · Thomas Ramstad Received: 5 May 2008 / Accepted: 4 September 2008 / Published online: 1 October 2008 © Springer Science + Business Media B.V. 2008 Abstract We propose that steady-state two-phase flow in porous media may be described through a formalism closely resembling equilibrium thermodynamics. This leads to a Monte Carlo method that will be highly efficient in studying two-phase flow under steady-state conditions numerically. Keywords Porous media · Two-phase flow · Steady-state flow PACS 47.20.Ft · 47.56.+r · 47.54.-r · 89.75.Fb 1 Introduction Immiscible two-phase flow in porous media has fas- cinated physicists for decades. In the 1980s, beautiful photographs of displacement patterns appeared and, simultaneously, the theoretical tools for quantitatively analyzing them based on fractal analysis were devel- oped [1]. Common for all these patterns was that they represented snapshots of a system undergoing change. This work was partially supported by the Norwegian Research Council through grants nos. 154535/432 and 180296/S30. A. Hansen (B ) Department of Physics, NTNU, 7491 Trondheim, Norway e-mail: Alex.Hansen@ntnu.no T. Ramstad Numerical Rocks AS, Stiklestadveien 1, 7041 Trondheim, Norway e-mail: thomas@numericalrocks.com That is, the porous medium typically would be prepared containing only one of the fluids. The second fluid, immiscible with the first, would then be pumped into the medium and the invasion patterns recorded. Hence, the focus was on transients. Two-phase flow in porous media is also at the core of a vast range of engineering applications ranging from petroleum technology to ground water management [2, 3]. In these engineering applications, an important aspect would be to determine the macroscopic flow parameters that would be used as input in, e.g., reser- voir simulators [4]. A basic assumption behind this de- termination is the assumption that parameters change slowly. On the microscopic scale, this assumption essen- tially translates into steady-state conditions. A concrete example of this is the use of relative permeability to describe the transport properties of two-fluid systems. The very concept of relative permeability measured against saturation of a given representative elementary volume (REV) only makes sense if the fractional flow entering the REV equals the fractional flow leaving it. In the laboratory, the standard way of measuring relative permeability is to steadily flood the core until pressures no longer change. This does not ensure a homogenous distribution of fluids inside the core but it is steady state. Hence, we are in a situation where most basic re- search, done, e.g., within the physics community, has been focused on transient phenomena, such as pattern formation during flooding. Practical applications would dictate an emphasis on the steady state, i.e., when the system has stopped changing on the macroscopic scale. There has, however, been some work investigating steady-state flow at the microscopic level, especially by the Payatakes group [5–9].