Vladimir Chugunov

Particle-size segregation and diffusive remixing in shallow granular avalanches

Segregation and mixing of dissimilar grains is a problem in many industrial and pharmaceutical processes, as well as in hazardous geophysical flows, where the size-distribution can have a major impact on the local rheology and the overall run-out. In this paper, a simple binary mixture theory is used to formulate a model for particle-size segregation and diffusive remixing of large and small particles in shallow gravity-driven free-surface flows. This builds on a recent theory for the process of kinetic sieving, which is the dominant mechanism for segregation in granular avalanches provided the density-ratio and the size-ratio of the particles are not too large. The resulting nonlinear parabolic segregation–remixing equation reduces to a quasi-linear hyperbolic equation in the no-remixing limit. It assumes that the bulk velocity is incompressible and that the bulk pressure is lithostatic, making it compatible with most theories used to compute the motion of shallow granular free-surface flows. In steady-state, the segregation–remixing equation reduces to a logistic type equation and the ‘S’-shaped solutions are in very good agreement with existing particle dynamics simulations for both size and density segregation. Laterally uniform time-dependent solutions are constructed by mapping the segregation–remixing equation to Burgers equation and using the Cole–Hopf transformation to linearize the problem. It is then shown how solutions for arbitrary initial conditions can be constructed using standard methods. Three examples are investigated in which the initial concentration is (i) homogeneous, (ii) reverse graded with the coarse grains above the fines, and, (iii) normally graded with the fines above the coarse grains. Time-dependent two-dimensional solutions are also constructed for plug-flow in a semi-infinite chute.

Analysis of Water Injection in Fractured Reservoirs Using a Fractional-Derivative-Based Mass and Heat Transfer Model

This research proposes a numerical scheme for evaluating the effect of cold-water injection into a geothermal reservoir. A fractional heat transfer equation (fHTE) is derived based on the fractional advection–dispersion equation (fADE) that describes non-Fickian dispersion in a fractured reservoir. Numerical simulations are conducted to examine the applicability of the fADE and the fHTE in interpreting tracer and thermal responses in a fault-related subsidiary structure associated with fractal geometry. A double-peak is exhibited when the surrounding rocks have a constant permeability. On the other hand, the peak in the tracer response gradually decreases when the permeability varies with distance from the fault zone according to a power law, which can be described by the fADE. The temperature decline is more gradual when the permeability of surrounding rocks varies spatially than when they have a constant permeability. The fHTE demonstrates good agreement with the temperature profiles for the different permeabilities of surrounding rocks. The retardation parameters in the fADE and the fHTE increase with the permeability of the surrounding rocks. The orders of the temporal fractional derivatives in the fADE and the fHTE vary with the permeability patterns.

The effect of non-Fickian diffusion into surrounding rocks on contaminant transport in a fractured porous aquifer

Solute transport in a fractured porous confined aquifer is modelled by using an equation with a fractional-in-time derivative of order γ, which may vary from 0 to 1. Accounting for non-Fickian diffusion into the surrounding rock mass, which is modelled by a fractional spatial derivative of order α, leads to the introduction of an additional fractional-in-time derivative of order α/(1+α) in the equation for solute transport. Closed-form solutions for solute concentrations in the aquifer and surrounding rocks are obtained for an arbitrary time-dependent source of contamination located at the inlet of the aquifer. Based on these solutions, different regimes of contaminant transport in aquifers with various physical properties are modelled and analysed.

Heat flow rate at a bore-face and temperature in the multi-layer media surrounding a borehole

Abstract Assessment of the heat either delivered from high temperature rocks to the borehole or transmitted to the rock formation from circulating fluid is of crucial importance for a number of technological processes related to borehole drilling and exploitation. Normally the temperature fields in the well and surrounding rocks are calculated numerically by finite difference method or analytically by applying the Laplace-transform method. The former approach requires tedious and, in certain cases, non-trivial numerical computations. The latter method leads to rather bulky formulae that are inconvenient for further numerical evaluation. Moreover, in previous studies where the solution is obtained analytically, the heat interaction of the circulating fluid with the formation was treated on the condition of constant bore-face temperature. In the present study the temperature field in the rock formation disturbed by the heat flow from the borehole is modeled by a heat conduction equation, assuming the Newton model for the convective heat transfer on the bore-face, with boundary conditions that account for the thermal history of the borehole exploitation. The problem is solved analytically by the generalized heat balance integral method. Within this method the approximate solution of the heat conduction problem is sought in the form of a finite sum of functions that belong to a complete set of linearly independent functions defined at the finite interval bounded by the radius of thermal influence and that satisfy the homogeneous boundary conditions on the bore-face. In the present study first and second order approximations are obtained for the composite multi-layer domain. The numerical results illustrate that the second approximation is in a good agreement with the exact solution. The only disadvantage of this solution is that it depends on the radius of thermal influence, which is an implicit function of time and can only be found numerically by iterative algorithms. In order to eliminate this complication, in this study an approximate explicit formula for the radius of thermal influence and new close-form approximate solution are proposed on the basis of the approximate solution obtained by the integral-balance method. Employing the non-liner regression method the coefficients for this simplified solution are obtained. The accuracy of the approximate solution is validated by comparison with the exact analytical solution found by Carslaw and Jaeger for the homogeneous domain.

Group theoretic methods and similarity solutions of the Savage-Hutter equations.

We consider the spatially one-dimensional time dependent system of equations, obtained by Savage and Hutter, which describes the gravity-driven free surface flow of granular avalanches. All similarity solutions of this system are found by means of group analysis. The family of solutions which are invariant to stretching transformations is investigated in greater detail. Explicit solutions are constructed in three cases and their physical interpretation is given.

Reactive rimming flow of non-Newtonian fluids

The steady and non-steady flows of a liquid polymer treated as a non-Newtonian fluid on the inner surface of a horizontal rotating cylinder are investigated. Since the Reynolds number is small and the liquid film is thin, a simple lubrication approximation is applied. Governing equations for non-steady Power-Law and Ellis fluids are solved numerically and the time of transition from non-steady to steady-state mode for various model parameters and flow conditions are defined. The stabilization effect of a chemical reaction within the polymeric fluid (reactive flow) is examined.

Derivation of Fractional Differential Equations for Modeling Diffusion in Porous Media of Fractal Geometry

Utilizing the double-porosity approach it is assumed that porous medium is constituted by two groups of pores in such way that major mass transport takes place mostly along the network of larger pores (group 1) and these larger (stem) pores are surrounded by the medium formed by the dead-end porous of fractal geometry (group 2). Solving analytically equation for the stem pore from the group 1 and accounting for the mass exchange with pores from the group 2, it is proved that diffusion in a stem pore should be described by the fractional differential equation. Based on this result, equation of mass flux that models non-Fickian diffusion in complex fractal medium is proposed. Applying this generalized form of mass transfer equation for modeling the contaminant transport in fractured porous aquifer leads to a fractional order differential equation, where mass exchange between blocks and fractures is modeled by the temporal fractional derivatives. This equation is solved analytically.

Numerical Simulation of Non‐Fickian Diffusion and Advection in a Fractured Porous Aquifer

A computer program, which enables us the calculation of the non‐Fickian diffusion in a fractured porous media, has been developed. The conventional mathematical model of solute transport in a rock is based on the Fick’s law. In general, rock masses contain a number of preexisting fractures. In the fractured porous media, the conventional model tends to predict smaller solute travel distance than that in the actual transport process. In contrast, the non‐Fickian diffusion model, which is described as a fractional advection‐dispersion equation, can provide realistic representation of actual fluid flow in the heterogeneous media. We provide a numerical solution of the fractional advection‐dispersion equation by using implicit‐finite difference method. The numerical results obtained for one dimensional fractional advection‐dispersion equation using the computer program was shown to be in a good agreement with the analytical solution.

Linear long wave propagation over discontinuous submerged shallow water topography

The dynamics of an isolated long wave passing over underwater obstacles are discussed in this paper within the framework of linear shallow water theory. Areas of practical application include coastal defense against tsunami inundation, harbor protection and erosion prevention with submerged breakwaters, and the construction and design of artificial reefs to use for recreational surfing. Three sea-floor configurations are considered: an underwater shelf, a flat sea-floor with a single obstacle, and a series of obstacles. A piecewise continuous coefficient is used to model the various sea-floor topographies. A simple and easily implementable numerical scheme using explicit finite difference methods is developed to solve the discontinuous partial differential equations. The numerical solutions are verified with the exact analytical solutions of linear wave propagation over an underwater shelf. The scope of this simplified approach is determined by comparison of its results to those of another numerical solution and wave transmission and reflection coefficients from experimental data available in the literature. The efficacy of approximating more complicated continuous underwater topographies by piecewise constant distributions is determined. As an application, a series of underwater obstacles is implemented.

Influence of underwater barriers on the distribution of tsunami waves

Solitary wave propagation over underwater shelves and bumps is examined using straightforward analytical methods. Explicit solutions for wave propagation are obtained. Using the nonlinear shallow-water equations, it was found that propagation of small amplitude long waves can be well described by a linear approximation. The effects of topographical variety and proportion of underwater barriers (steps, bumps, multiple bumps) on the incident wave are demonstrated using linear wave theory. At a step, the incident wave is shown to be more strongly reflected for increased barrier size. The incident wave also transmits an amplified wave with smaller wavelength onto the obstacle. After propagating off of a bump, the wave experiences an amplitude decay. The decay rate is shown to be exponential with a variable number of bumps. Accounting for the presence of the small parameter, which represents the wave amplitude/water depth ratio, the nonlinear shallow-water equations were solved by the method of asymptotic expansions. Using the method of renormalization, a uniformly valid solution was obtained accounting for nonlinear effects in the vicinity of the sharp depth change. Far-field comparisons of the constructed solutions with the associated Riemann waves show good accuracy of the obtained solutions. Over an infinitely long shelf, the amplified transmitted wave breaks.