arXiv subjects

Håkon Pedersen

Publications and source records attributed to Håkon Pedersen.

5 recordsLinked to original sources

arXiv2025

The Co-Moving Velocity and Projective Transformations

In a string of recent papers starting with (Transport in Porous Media, 125, 565 (2018)), a theory of immiscible two-phase flow in porous media based on Euler homogeneity of the total volumetric flow rate has been investigated. The thermodynamic-like theory has an associated statistical mechanics based on a maximum entropy principle. A quantity called the co-moving velocity connects the equations of state of the intensive thermodynamic velocities and the physical seepage velocities of two flowing fluids. The obtained relations have a structure that can be interpreted using affine- and projective geometry. The co-moving velocity can be expressed as a transformation of the saturation using projective duality of points and lines. One obtains an exact constitutive relation depending on a projective invariant, the cross-ratio, which allows the co-moving velocity to be expressed in terms of a simple steady-state advection equation. A kinematic view of the velocity relations is presented, modeled by a well-known non-trivial geometry which turns out to be pseudo-Euclidean. The cross-ratio determines a hyperbolic angle in this space, and can be parametrized in terms of three numbers using a linear fractional transformation. Knowing these parameters, the pore velocity and the derivative of the pore velocity with respect to the saturation, an approximation for the co-moving velocity can be obtained for a range of applied pressures, viscosity ratios and surface tensions. The parametrization is demonstrated using data from a dynamic pore network model and relative permeability data from the literature. This paper only considers the pore areas as extensive variables, however, the geometric principles are general, and the same idea could potentially be used in other systems.

physics.flu-dyn↗

arXiv2025

Geometric structure of parameter space in immiscible two-phase flow in porous media

In a recent paper, a continuum theory of immiscible and incompressible two-phase flow in porous media based on generalized thermodynamic principles was formulated (Transport in Porous Media, 125, 565 (2018)). In this theory, two immiscible and incompressible fluids flowing in a porous medium are treated as an effective fluid, substituting the two interacting subsystems for a single system with an effective viscosity and pressure gradient. In assuming Euler homogeneity of the total volumetric flow rate and comparing the resulting first order partial differential equation to the total volumetric flow rate in the porous medium, one can introduce of a novel velocity that relates the two pairs of velocities. This velocity, the co-moving velocity, describes the mutual co-carrying of fluids due to immiscibility effects and interactions between the fluid clusters and the porous medium itself. The theory is based upon general principles of thermodynamics, and allows for many relations and analogies to draw upon for a two-phase flow systems. The goal of this work is to provide relations between geometric concepts and the variables appearing in the thermodynamics-like theory of two-phase flow. We encounter two interpretations of the velocities of the fluids: as tangent vectors (derivations) acting on functions, or as coordinates on an affine line. The two views are closely related, with the former viewpoint being more useful in relation to the underlying geometrical structure of equilibrium thermodynamics, and the latter being more useful in concrete computations and finding examples of constitutive relations. We apply these straightforward geometric contexts to interpret the relations between the velocities, and from this obtain a general form for the co-moving velocity.

physics.flu-dyn↗

arXiv2023

The impact of wettability on the co-moving velocity of two-fluid flow in porous media

The impact of wettability on the co-moving velocity of two-fluid flow in porous media is analyzed herein. The co-moving velocity, developed by Roy et al. (2022), is a novel representation of the flow behavior of two fluids through porous media. Our study aims to better understand the behavior of the co-moving velocity by analyzing simulation data under various wetting conditions. The simulations were conducted using the Lattice-Boltzmann color-fluid model and evaluated the relative permeability for different wetting conditions on the same rock. The analysis of the simulation data followed the methodology proposed by Roy et al. (2022) to reconstruct a constitutive equation for the co-moving velocity. Surprisingly, it was found that the coefficients of the constitutive equation were nearly the same for all wetting conditions. Based on these results, a simple approach was proposed to reconstruct the oil phase relative permeability using only the co-moving velocity relationship and water phase relative permeability. This proposed method provides new insights into the dependency of relative permeability curves, which has implications for the history matching of production data and solving the associated inverse problem. The research findings contribute to a better understanding of the impact of wettability on fluid flow in porous media and provide a practical approach for estimating relative permeability based on the co-moving velocity relationship, which has never been shown before.

physics.flu-dyn↗

arXiv2022

Parameterizations of Immiscible Two-Phase Flow in Porous Media

A fundamental variable characterizing immiscible two-phase flow in porous media is the wetting saturation, which is the ratio between the pore volume filled with wetting fluid and the total pore volume. More generally, this variable comes from a specific choice of coordinates on some underlying space, the domain of variables that can be used to express the volumetric flow rate. The underlying mathematical structure allows for the introduction of other variables containing the same information, but which are more convenient from a theoretical point of view. We introduce along these lines polar coordinates on this underlying space, where the angle plays a role similar to the wetting saturation. We derive relations between these new variables based on the Euler homogeneity theorem. We formulate these relations in a coordinate-free fashion using differential forms. Lastly, we discuss and interpret the co-moving velocity in terms of this coordinate-free representation.

physics.flu-dyn↗

arXiv2022

The Co-Moving Velocity in Immiscible Two-Phase Flow in Porous Media

We present a continuum (i.e., an effective) description of immiscible two-phase flow in porous media characterized by two fields, the pressure and the saturation. Gradients in these two fields are the driving forces that move the immiscible fluids around. The fluids are characterized by two seepage velocity fields, one for each fluid. Following Hansen et al.\ (Transport in Porous Media, 125, 565 (2018)), we construct a two-way transformation between the velocity couple consisting of the seepage velocity of each fluid, to a velocity couple consisting of the average seepage velocity of both fluids and a new velocity parameter, the co-moving velocity. The co-moving velocity is related but not equal to velocity difference between the two immiscible fluids. The two-way mapping, the mass conservation equation and the constitutive equations for the average seepage velocity and the co-moving velocity form a closed set of equations that determine the flow. There is growing experimental, computational and theoretical evidence that constitutive equation for the average seepage velocity has the form of a power law in the pressure gradient over a wide range of capillary numbers. Through the transformation between the two velocity couples, this constitutive equation may be taken directly into account in the equations describing the flow of each fluid. This is e.g., not possible using relative permeability theory. By reverse engineering relative permeability data from the literature, we construct the constitutive equation for the co-moving velocity. We also calculate the co-moving constitutive equation using a dynamic pore network model over a wide range of parameters, from where the flow is viscosity dominated to where the capillary and viscous forces compete.

physics.flu-dyn↗