Structure Theorems for Basic Algebras

A basic finite dimensional algebra over an algebraically closed field $k$ is isomorphic to a quotient of a tensor algebra by an admissible ideal. The category of left modules over the algebra is isomorphic to the category of representations of a finite quiver with relations. In this article we will remove the assumption that $k$ is algebraically closed to look at both perfect and non-perfect fields. We will introduce the notion of species with relations to describe the category of left modules over such algebras. If the field is not perfect, then the algebra is isomorphic to a quotient of a tensor algebra by an ideal that is no longer admissible in general. This gives hereditary algebras isomorphic to a quotient of a tensor algebra by a non-zero ideal. We will show that these non-zero ideals correspond to cyclic subgraphs of the graph associated to the species of the algebra. This will lead to the ideal being zero in the case when the underlying graph of the algebra is a tree

Representations of Thread Quivers

We introduce thread quivers as an (infinite) generalization of quivers, and show that every k-linear (k algebraically closed) hereditary category with Serre duality and enough projectives is equivalent to the category of finitely presented representations of a thread quiver. In this way, we obtain an explicit construction of a new class of hereditary categories with Serre duality.Comment: As accepted by the Proceedings of the London Mathematical Society; some differences in label and page numbering may occur between this version and the published versio

Relative permeability as a stationary process: energy fluctuations in immiscible displacement

Relative permeability is commonly used to model immiscible fluid flow through porous materials. In this work we derive the relative permeability relationship from conservation of energy, assuming that the system to be non-ergodic at large length scales and relying on averaging in both space and time to homogenize the behavior. Explicit criteria are obtained to define stationary conditions: (1) there can be no net change for extensive measures of the system state over the time averaging interval; (2) the net energy inputs into the system are zero, meaning that the net rate of work done on the system must balance with the heat removed; and (3) there is no net work performed due to the contribution of internal energy fluctuations. Results are then evaluated based on direct numerical simulation. Dynamic connectivity is observed during steady-state flow, which is quantitatively assessed based the Euler characteristic. We show that even during steady-state flow at low capillary number ($\mathsf{Ca}\sim1\times10^5$), typical flow processes will explore multiple connectivity states. The residence time for each connectivity state is captured based on the time-and-space average. The distribution for energy fluctuations is shown to be multi-modal and non-Gaussian when terms are considered independently. However, we demonstrate that their sum is zero. Given an appropriate choice of the thermodynamic driving force, we show that the conventional relative permeability relationship is sufficient to model the energy dissipation in systems with complex pore-scale dynamics that routinely alter the structure of fluid connected pathways

Nova genera plantarum...

pt.3, 5 1783-8

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.Comment: 14 pages, 6 figure

Direct Evidence of Salinity Difference Effect on Water Transport in Oil: Pore-Scale Mechanisms

Low salinity water flooding is a common technique for enhancing oil recovery; however, the mechanism behind the low-salinity effect, positive or negative, is still not fully understood. In the proposed mechanisms, osmosis and emulsification are considered as two potential reasons for explaining the oil remobilization, but the specific contributions on the remobilization are not well studied at pore-scale. In this article, we performed a series of microfluidic experiments to investigate the movement of constrained oil between invading low-salinity brine and residual high-salinity brine. We find that various salinity contrasts over oil films cause different water fluxes through the oil and swelling areas of the trapped brine, resulting in the relocation of oil phases within the pore spaces. A higher salinity contrast (1.7-170 g/L salt concentrations) provides a faster water penetration in oil phases. In the presence of an oil-soluble surfactant, spontaneous emulsification occurs at the interface of low-salinity brine/oil, which enhances almost 100 times the water flux in two oil phases (n-heptane and n-dodecane). We directly observe pore-scale spontaneous emulsification at the low-salinity brine/oil interface but not at the high-salinity brine/oil interface. Furthermore, two scenarios for explaining water transport through the oil phase are proposed: water diffusion due to chemical potential gradient and water transport via reverse micelle or microemulsions movement