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Register a new userA time-space Hausdorff fractal model for non-Fickian transport in porous media
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This paper presents a time-space Hausdorff derivative model for depicting solute transport in aquifers or water flow in heterogeneous porous media. In this model, the time and space Hausdorff derivatives are defined on non-Euclidean fractal metrics with power law scaling transform which, respectively, connect the temporal and spatial complexity during transport. The Hausdorff derivative model can be transformed to an advection-dispersion equation with time- and space-dependent dispersion and convection coefficients. This model is a fractal partial differential equation (PDE) defined on a fractal space and differs from the fractional PDE which is derived for non-local transport of particles on a non-fractal Euclidean space. As an example of applications of this model, an explicit solution with a constant diffusion coefficient and flow velocity subject to an instantaneous source is derived and fitted to the breakthrough curves of tritium as a tracer in porous media. These results are compared with those of a scale-dependent dispersion model and a time-scale dependent dispersion model. Overall, it is found that the fractal PDE based on the Hausdorff derivatives better captures the early arrival and heavy tail in the scaled breakthrough curves for variable transport distances. The estimated parameters in the fractal Hausrdorff model represent clear physical mechanisms such as linear relationships between the orders of Hausdorff derivatives and the transport distance. The mathematical formulation is applicable to both solute transport and water flow in porous media.
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Many theoretical and experimental results show that solute transport in heterogeneous porous media exhibits multi-scaling behaviors. To describe such non-Fickian diffusions, this work provides a distributed order Hausdorff diffusion model to describe the tracer transport in porous media. This model is proved to be equivalent with the diffusion equation model with a nonlinear time dependent diffusion coefficient. In conjunction with the structural derivative, its mean squared displacement (MSD) of the tracer particles is explicitly derived as a dilogarithm function when the weight function of the order distribution is a linear function of the time derivative order. This model can capture both accelerating and decelerating anomalous and ultraslow diffusions by varying the weight parameter c. In this study, the tracer transport in water-filled pore spaces of two-dimensional Euclidean is demonstrated as a decelerating sub-diffusion, and can well be described by the distributed order Hausdorff diffusion model with c = 1.73. While the Hausdorff diffusion model can accurately fit the sub-diffusion experimental data of the tracer transport in the pore-solid prefractal porous media.
Transport of viscous fluid through porous media is a direct consequence of the pore structure. Here we investigate transport through a specific class of two-dimensional porous geometries, namely those formed by fluid-mechanical erosion. We investigate the tortuosity and dispersion by analyzing the first two statistical moments of tracer trajectories. For most initial configurations, tortuosity decreases in time as a result of erosion increasing the porosity. However, we find that tortuosity can also increase transiently in certain cases. The porosity-tortuosity relationships that result from our simulations are compared with models available in the literature. Asymptotic dispersion rates are also strongly affected by the erosion process, as well as by the number and distribution of the eroding bodies. Finally, we analyze the pore size distribution of an eroding geometry. The simulations are performed by combining a high-fidelity boundary integral equation solver for the fluid equations, a second-order stable time stepping method to simulate erosion, and new numerical methods to stably and accurately resolve nearly-touching eroded bodies and particle trajectories near the eroding bodies.
We present and derive a novel double-continuum transport model based on pore-scale characteristics. Our approach relies on building a simplified unit cell made up of immobile and mobile continua. We employ a numerically resolved pore-scale velocity distribution to characterize the volume of each continuum and to define the velocity profile in the mobile continuum. Using the simplified unit cell, we derive a closed form model, which includes two effective parameters that need to be estimated: a characteristic length scale and a ratio of waiting times RD that lumps the effect of stagnant regions and escape process. To calibrate and validate our model, we rely on a set of pore-scale numerical simulation performed on a 2D disordered segregated periodic porous medium considering different initial solute distributions. Using a Global Sensitivity Analysis, we explore the impact of the two effective parameters on solute concentration profiles and thereby define a sensitivity analysis driven criterion for model calibration. The latter is compared to a classical calibration approach. Our results show that, depending on the initial condition, the mass exchange process between mobile and immobile continua impact on solute profile shape significantly. By introducing parameter RD we obtain a flexible transport model capable of interpreting both symmetric and highly skewed solute concentration profiles. We show that the effectiveness of the calibration of the two parameters closely depends on the content of information of calibration dataset and the selected objective function whose definition can be supported by of the implementation of model sensitivity analysis. By relying on a sensitivity analysis driven calibration, we are able to provide a good interpretation of the concentration profile evolution independent of the given initial condition relying on a unique set of effective parameter values.
We study the transport of inertial particles in water flow in porous media. Our interest lies in understanding the accumulation of particles including the possibility of clogging. We propose that accumulation can be a result of hydrodynamic effects: the tortuous paths of the porous medium generate regions of dominating strain/vorticity, which favour the accumulation/dispersion of the inertial particles. Numerical simulations show that essentially two accumulation regimes are identified: for low and for high flow velocities. When particles accumulate in high-velocity regions, at the entrance of a pore throat, a clog is formed. The formation of a clog significantly modifies the flow, as the partial blockage of the pore causes a local redistribution of pressure. This redistribution can divert the upstream water flow into neighbouring pores. Moreover, we show that accumulation in high velocity regions occurs in heterogeneous media, but not in homogeneous media, where we refer to homogeneity with respect to the distribution of the pore throat diameters.
We investigate the elastoviscoplastic flow through porous media by numerical simulations. We solve the Navier-Stokes equations combined with the elastoviscoplastic model proposed by Saramito for the stress tensor evolution. In this model, the material behaves as a viscoelastic solid when unyielded, and as a viscoelastic Oldroyd-B fluid for stresses higher than the yield stress. The porous media is made of a symmetric array of cylinders, and we solve the flow in one periodic cell. We find that the solution is time-dependent even at low Reynolds numbers as we observe oscillations in time of the unyielded region especially at high Bingham numbers. The volume of the unyielded region slightly decreases with the Reynolds number and strongly increases with the Bingham number; up to 70% of the total volume is unyielded for the highest Bingham numbers considered here. The flow is mainly shear dominated in the yielded region, while shear and elongational flow are equally distributed in the unyielded region. We compute the relation between the pressure drop and the flow rate in the porous medium and present an empirical closure as function of the Bingham and Reynolds numbers. The apparent permeability, normalized with the case of Newtonian fluids, is shown to be greater than 1 at low Bingham numbers, corresponding to lower pressure drops due to the flow elasticity, and smaller than 1 for high Bingham numbers, indicating larger dissipation in the flow owing to the presence of the yielded regions. Finally we investigate the effect of the Weissenberg number on the distribution of the unyielded regions and on the pressure gradient.
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