arXiv subjects

Alex Hansen

Publications and source records attributed to Alex Hansen.

At least 37 records · Page 2Linked to original sources

arXiv2024

Influence of the imposed flow rate boundary condition on the flow of Bingham fluid in porous media

The flow of yield stress fluids in porous media presents interesting complexity due to the interplay between the non-linear rheology and the heterogeneity of the medium. A remarkable consequence is that the number of flow paths increases with the applied pressure difference and is responsible for a non-linear Darcy law. Previous studies have focused on the protocol where the pressure difference is imposed. Here we consider instead the case of imposed flow rate, $Q$. In contrast to Newtonian fluids, the two types of boundary conditions have an important influence on the flow field. Using a two-dimensional pore network model we observe a boundary layer of merging flow paths of size $\ell(Q) \sim Q^{-μ/δ}$ where $μ= 0.42 \pm 0.02$ and $δ\simeq 0.63 \pm 0.05$. Beyond this layer the density of the flow paths is homogeneous and grows as $Q^μ$. Using a mapping to the directed polymer model we identify $δ$ with the roughness exponent of the polymer. We also characterize the statistics of non-flowing surfaces in terms of avalanches pulled at one end.

physics.flu-dyn↗

arXiv2023

Disorder-induced non-linear growth of viscously-unstable immiscible two-phase flow fingers in porous media

The immiscible displacement of a fluid by another one inside a porous medium produces different types of patterns depending on the capillary number Ca and viscosity ratio M. At high Ca, viscous fingers resulting from the viscous instability between fluid-fluid interfaces are believed to exhibit the same Laplacian growth behavior as viscously-unstable fingers observed in Hele-Shaw cells by Saffman and Taylor [1], or as diffusion limited aggregates (DLA) [2]. I.e., the interface velocity depends linearly on the local gradient of the physical field that drives the growth process (for two-phase flow, the pressure field). However, steady-state two-phase flow in porous media is known to exhibit a regime for which the flow rate depends as a non-linear power law on the global pressure drop, due to the disorder in the capillary barriers at pore throats. A similar nonlinear growth regime was also evidenced experimentally for viscously-unstable drainage in two-dimensional porous media 20 years ago [3]. Here we revisit this flow regime using dynamic pore-network modeling, and explore the non-linearity in the growth properties. We characterize the previously-unstudied dependencies of the statistical finger width and nonlinear growth law's exponent on Ca, and discuss quantitatively, based on theoretical arguments, how disorder in the capillary barriers controls the growth process' non-linearity, and why the flow regime crosses over to Laplacian growth at sufficiently high Ca. In addition, the statistical properties of the fingering patterns are compared to those of Saffman-Taylor fingers, DLA growth patterns, and the results from the aforementioned previous experimental study.

physics.flu-dyn↗

arXiv2023

Immiscible two-phase flow in porous media: Effective rheology in the continuum limit

It is becoming increasingly clear that there is a regime in immiscible two-phase flow in porous media where the flow rate depends of the pressure drop as a power law with exponent different than one. This occurs when the capillary forces and viscous forces both influence the flow. At higher flow rates, where the viscous forces dominate, the flow rate depends linearly on the pressure drop. The question we pose here is what happens to the linear regime when the system size is increased. Based on analytical calculations using the capillary fiber bundle model and on numerical simulations using a dynamical network model, we find that the non-linear regime moves towards smaller and smaller pressure gradients as the system size grows.

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↗

arXiv2023

Transition from viscous fingers to foam during drainage in heterogeneous porous media

We investigate the behavior of drainage displacements in heterogeneous porous media finding a transition from viscous fingering to foam-like region. A pore network model incorporating the formation of blobs is adopted to study this phenomenon. By imposing a pressure difference between the inlet and outlet, we observe that the displacement pattern undergoes a significant transition from a continuous front of growing viscous fingers to the emergence of foam, which develops and propagates until breakthrough. This transition occurs at a specific distance from the inlet, which we measure and analyze as a function of the viscosity ratio and the capillary number, demonstrating that it follows a non-trivial power-law decay with both the parameters. Moreover, we discuss the relationship between the evolution of the total flow rate and the local pressure drop, showing that the foam developed reduces global mobility. We observe that foam is formed from the fragmentation of viscous fingers beneath the front, and this instability mechanism is connected with fluctuations of the local flow rate, which we analyze both in the viscous fingering region and in the foam region.

cond-mat.soft↗

arXiv2023

Effective rheology of immiscible two-phase flow in porous media consisting of random mixtures of grains having two types of wetting properties

We consider the effective rheology of immiscible two-phase flow in porous media with random mixtures of two types of grains with different wetting properties using a dynamic pore network model under steady-state. Two immiscible fluids A and B flow through the pores between these two types of grains denoted "+" and "-". Fluid A is fully wetting and B is fully non-wetting with respect to "+" grains and opposite with "-" grains. The direction of the capillary forces in the links between two "+" grains is therefore opposite compared to that between two "-" grains, whereas the capillary forces in the links between two opposite types of grains average to zero. For a window of grain occupation probabilities, a percolating regime appears where there is a high probability of having connected paths with zero capillary forces. Due to these paths, no minimum threshold pressure is required to start a flow in this regime. While varying the pressure drop across the porous medium from low to high in this regime, the relation between the volumetric flow rate and the pressure drop goes from being linear to a power law with exponent 2.56 to linear again. Outside the percolation regime, there is a threshold pressure. No linear regime is observed for low pressure drops. When the pressure drop is high enough for there to be flow, we find that the flow rate depends on the excess pressure drop to an exponents around 2.2-2.3. At even higher pressure drops, the relation is linear. We see no change in exponent for the intermediate regime at the percolation critical points where the zero-capillary force paths disappear. We measure the mobility at the percolation threshold at low pressure drops so that the flow rate versus pressure drop is linear. Assuming a power law, the mobility is proportional to the difference between the occupation probability and the critical occupation probability to a power of around 5.7.

physics.flu-dyn↗

arXiv2023

Local Statistics of Immiscible and Incompressible Two-Phase Flow in Porous Media

We consider immiscible and incompressible two-phase flow in porous media under steady-state conditions using a dynamic pore network model. We focus on the fluctuations in a Representative Elementary Area (REA), with the aim to demonstrate that the statistical distributions of the volumetric flow rate and the saturation within the REA become independent of the size of the entire model when the model is large enough. This independence is a necessary condition for developing a local statistical theory for the flow, which in turn opens for the possibility to formulate a description at scales large enough for the typical pore size to be negligible using differential equations.

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

Steady-state two-phase flow of compressible and incompressible fluids in a capillary tube of varying radius

We study immiscible two-phase flow of a compressible and an incompressible fluid inside a capillary tube of varying radius under steady-state conditions. The incompressible fluid is Newtonian and the compressible fluid is an inviscid ideal gas. The surface tension associated with the interfaces between the two fluids introduces capillary forces that vary along the tube due to the variation in the tube radius. The interplay between effects due to the capillary forces and the compressibility results in a set of properties that are different from incompressible two-phase flow. As the fluids move towards the outlet, the bubbles of the compressible fluid grow in volume due to the decrease in pressure. The volumetric growth of the compressible bubbles makes the volumetric flow rate at the outlet higher than at the inlet. The growth is not only a function of the pressure drop across the tube, but also of the ambient pressure. Furthermore, the capillary forces create an effective threshold below which there is no flow. Above the threshold, the system shows a weak non-linearity between the flow rates and the effective pressure drop, where the non-linearity also depends on the absolute pressures across the tube.

physics.flu-dyn↗

arXiv2022

A statistical mechanics for immiscible and incompressible two-phase flow in porous media

We construct a statistical mechanics for immiscible and incompressible two-phase flow in porous media under local steady-state conditions based on the Jaynes maximum entropy principle. A cluster entropy is assigned to our lack of knowledge of, and control over, the fluid and flow configurations in the pore space. As a consequence, two new variables describing the flow emerge: The agiture, that describes the level of agitation of the two fluids, and the flow derivative which is conjugate to the saturation. Agiture and flow derivative are the analogs of temperature and chemical potential in standard (thermal) statistical mechanics. The associated thermodynamics-like formalism reveals a number of hitherto unknown relations between the variables that describe the flow, including fluctuations. The formalism opens for new approaches to characterize porous media with respect to multi-phase flow for practical applications, replacing the simplistic relative permeability theory while still keeping the number of variables tractable.

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↗

arXiv2021

Burst dynamics, up-scaling and dissipation of slow drainage in porous media

We present a theoretical and experimental investigation of drainage in porous media. The study is limited to stabilized fluid fronts at moderate injection rates, but it takes into account capillary, viscous, and gravitational forces. In this theory the work applied on the system, the energy dissipation, the final saturation and the width of the stabilized fluid front can all be calculated if we know the dimensionless fluctuation number, the wetting properties, the surface tension between the fluids, the fractal dimensions of the invasion front and the invading structure, and the exponent describing the divergence of the correlation length in percolation. This theoretical description explains how the Haines jumps' local activity and dissipation relate to dissipation on larger scales.

physics.flu-dyn↗

arXiv2021

Rheology of immiscible two-phase flow in mixed wet porous media: Dynamic pore network model and capillary fiber bundle model results

Immiscible two-phase flow in porous media with mixed wet conditions was examined using a capillary fiber bundle model, which is analytically solvable, and a dynamic pore network model. The mixed wettability was implemented in the models by allowing each tube or link to have a different wetting angle chosen randomly from a given distribution. Both models showed that mixed wettability can have significant influence on the rheology in terms of the dependence of the global volumetric flow rate on the global pressure drop. In the capillary fiber bundle model, for small pressure drops when only a small fraction of the tubes were open, it was found that the volumetric flow rate depended on the excess pressure drop as a power law with an exponent equal to 3/2 or 2 depending on the minimum pressure drop necessary for flow. When all the tubes were open due to a high pressure drop, the volumetric flow rate depended linearly on the pressure drop, independent of the wettability. In the transition region in between where most of the tubes opened, the volumetric flow depended more sensitively on the wetting angle distribution function and was in general not a simple power law. The dynamic pore network model results also showed a linear dependence of the flow rate on the pressure drop when the pressure drop is large. However, out of this limit the dynamic pore network model demonstrated a more complicated behaviour that depended on the mixed wettability condition and the saturation. In particular, the exponent relating volumetric flow rate to the excess pressure drop could take on values anywhere between 1.0 and 1.8. The values of the exponent were highest for saturations approaching 0.5, also, the exponent generally increased when the difference in wettability of the two fluids were larger and when this difference was present for a larger fraction of the porous network.

physics.flu-dyn↗

arXiv2021

Non-Newtonian rheology in a capillary tube with varying radius

The flow through a capillary tube with non-constant radius and where bubbles of yield stress fluid are injected is strongly non-linear. In particular below a finite yield pressure drop, $P_y$, flow is absent, while a singular behaviour is expected above it. In this paper we compute the yield pressure drop statistics and the mean flow rate in two cases: (i) when a single bubble is injected, (ii) when many bubbles are randomly injected in the fluid.

physics.flu-dyn↗

arXiv2021

Role of pore-size distribution on effective rheology of two-phase flow in porous media

The flow of immiscible fluids inside a porous medium shows non-linearity in the form of a power law in the rheological properties of the fluids under steady state flow conditions. However, different experimental and numerical studies have reported different values for the exponent related to this power law. Here we explore how the rheological properties of the two-phase flow in porous media depends on the distribution of the pore sizes and how it affects the power-law exponent. The pore-size distribution controls fluctuation in the pore radii and their density in a porous material. We present two approaches, analytical calculations using a capillary bundle model and numerical simulations using dynamic pore-network modeling. We observe crossover from a non-linear to linear rheology when increasing the flow rate where the non-linear part is highly affected by the pore-size distribution. We have also carried out the study for different saturations of the two fluids.

physics.flu-dyn↗

arXiv2020

Hyperballistic superdiffusion and explosive solutions to the non-linear diffusion equation

By means of a particle model that includes interactions only via the local particle concentration, we show that hyperballistic diffusion may result. This is done by findng the exact solution of the corresponding non-linear diffusion equation, as well as by particle simulations. The connection between these levels of description is provided by the Fokker-Planck equation describing the particle dynamics.

cond-mat.stat-mech↗

arXiv2020

Crack Localization and the Interplay between Stress Enhancement and Thermal Noise

We study the competition between thermal fluctuations and stress enhancement in the failure process of a disordered system by using a local load sharing fiber bundle model. The thermal noise is introduced by defining a failure probability that constitutes the temperature and elastic energy of the fibers. We observe that at a finite temperature and low disorder strength, the failure process, which nucleate in the absence of any thermal fluctuation, becomes spatially uncorrelated when the applied stress is sufficiently low. The dynamics of the model in this limit lies closely to the universality class of ordinary percolation. When applied stress is increased beyond a threshold value, localized fractures appear in the system that grow with time. We identify the boundary between the localized and random failure process in the space of temperature and applied stress, and find that the threshold of stress corresponding to the onset of localized crack growth increases with the increase of temperature.

cond-mat.dis-nn↗

arXiv2020

The Three Extreme Value Distributions: An Introductory Review

The statistical distribution of the largest value drawn from a sample of a given size has only three possible shapes: it is either a Weibull, a Fréchet or a Gumbel extreme value distributions. I describe in this short review how to relate the statistical distribution followed by the numbers in the sample to the associate extreme value distribution followed by the largest value within the sample. Nothing I present here is new. However, from experience, I have found that a simple and compact guide on this matter written for the physics community is missing.

cond-mat.stat-mech↗