Håkon Pedersen has submitted the following academic thesis as part of the doctoral work at the Norwegian University of Science and Technology (NTNU):
“Geometry of the Co-Moving Velocity in Immiscible and Incompressible Two-Phase Porous Media Flow”
Assessment Committee
The Faculty of Natural Sciences has appointed the following Assessment Committee to assess the
thesis:
– Professor Bjarne Andresen, Niels Bohr Institute, University of Copenhagen, Denmark
– Dr. Thomas Ramstad, Equinor, Norway
– Dr. Marie-Laure Olivier, Department of Physics, NTNU
Dr. Marie Laure Olivier has been appointed Administrator of the Committee. The Committee recommends that the thesis is worthy of being publicly defended for the PhD degree.
Supervisors:
The doctoral work has been carried out at the Department of Physics, where Professor Alex Hansen has been the candidate’s supervisor. Professor Knut Jørgen Måløy has been the candidate’s co-supervisor.
Public trial lecture:
Time: 25th of April at 10:15
Place: Disputasrommet, Main building, NTNU Gløshaugen
Prescribed subject: “Describe the origin of the thermodynamic length and its applications as well as
its relationship to your own studies”
Public defense of the thesis:
Time: 25th of April at 13:15
Place: Disputasrommet, Main building, NTNU Gløshaugen
Summary of thesis:
This thesis studies a continuum theory of immiscible, incompressible two-phase flow in a homogeneous porous medium. By leveraging degree-1 Euler homogeneity in the total volumetric flow rate through a domain of the porous medium, one can relate the theory to the mathematical framework of thermodynamics. A quantity called the co-moving velocity relates the intensive variables obtained through an Euler relation to the physical seepage velocities of the two flowing fluids. Hence, the comoving velocity relates a set of thermodynamic and continuum equations of state. This quantity, its classification and parametrization is the main topic of this thesis.
Knowledge of the co-moving velocity and the average fluid velocity allows one to compute both physical velocities. For many flow parameters, the co-moving velocity has a relatively simple behaviour compared to the pair of seepage velocities. The quantity has a predictable behaviour in different flow regimes, and is insensitive to the wetting properties of the porous medium.
Seeing as the co-moving velocity relates a set of partial quantities from thermodynamics to measurable velocities, it must depend on the constitutive relations of the two physical fluids. The saturation weighted velocity of the two fluids defines the average velocity as measured in the limit of large length scales. The co-moving velocity then describes an interaction between the two-fluids and between the fluids and the porous medium. This interaction has no associated transport, and is therefore not obtainable from the total volumetric flow rate.
The goal of this thesis is threefold: to present the co-moving velocity through computations from data, provide the first geometric characterizations of this quantity, and obtain a novel parametrization of the co-moving velocity that is applicable to a wide range of flow regimes. The reason for seeking a geometric interpretation is that the theory sits at the intersection of thermodynamics and kinematics, which can be described using the same mathematical framework. This thesis shows how the relations between the velocities can have several geometric interpretations. In terms of the simplest possible description using a single vector space, the relations between the variables take on a simple form, but are difficult to connect to computations. Applying a differential geometric formalism, one can readily classify the quantities of the theory, but in a general manner that presents no clear method of parametrizing the comoving velocity. As it turns out, projective geometry can be used as a framework to obtain such a parametrization, with several advantages compared to other efforts in this regard. This constitutes a concrete application of one of the most well-known types of mathematical spaces to the field of two phase flow in porous media. An attempt has been made to keep the amount of formalism to a minimum, such that the material is presentable to researchers in the fields of porous media and general fluid mechanics.
A central conclusion of the thesis is that the assumption of homogeneity in the total volumetric flow rate in general leads to two related interpretations, each describing different aspects of the same problem. The first interpretation follows the differential formulation often associated to thermodynamics in terms of state functions and differential operators. Homogeneity here distinguishes extensive and intensive quantities. In the end, one obtains a theory with a thermodynamic structure in which the total volumetric flow rate plays the role of a potential associated to some statistical ensemble by extending its role to that of a state function.
The second alternative is to interpret the volumetric flow rate in terms of the quantities that define it, namely the areas and velocities, and apply these to express kinematic relations in terms of motions of mathematical objects like points and lines. These parametrized objects represent the state of the system, and can be viewed as a type of global variant of the differential formulation.
Both interpretations lead to geometry which can look quite different when treated in isolation. The unifying concept is duality between two sets of variables. These two sets describe the same structure, for instance a surface of equilibrium states, by means of primal or dual objects such as vectors and co-vectors, or points and lines. This is the duality described by the areas and pore velocities, independent from the interpretation of the relations as thermodynamic or kinematic.