A hierarchy of models for simulating experimental results from a 3D heterogeneous porous medium

In this work we examine the dispersion of conservative tracers (bromide and fluorescein) in an experimentally-constructed three-dimensional dual-porosity porous medium. The medium is highly heterogeneous ($\sigma_Y^2=5.7$), and consists of spherical, low-hydraulic-conductivity inclusions embedded in a high-hydraulic-conductivity matrix. The bi-modal medium was saturated with tracers, and then flushed with tracer-free fluid while the effluent breakthrough curves were measured. The focus for this work is to examine a hierarchy of four models (in the absence of adjustable parameters) with decreasing complexity to assess their ability to accurately represent the measured breakthrough curves. The most information-rich model was (1) a direct numerical simulation of the system in which the geometry, boundary and initial conditions, and medium properties were fully independently characterized experimentally with high fidelity. The reduced models included; (2) a simplified numerical model identical to the fully-resolved direct numerical simulation (DNS) model, but using a domain that was one-tenth the size; (3) an upscaled mobile-immobile model that allowed for a time-dependent mass-transfer coefficient; and, (4) an upscaled mobile-immobile model that assumed a space-time constant mass-transfer coefficient. The results illustrated that all four models provided accurate representations of the experimental breakthrough curves as measured by global RMS error. The primary component of error induced in the upscaled models appeared to arise from the neglect of convection within the inclusions. Interestingly, these results suggested that the conventional convection-dispersion equation, when applied in a way that resolves the heterogeneities, yields models with high fidelity without requiring the imposition of a more complex non-Fickian model.Comment: 27 pages, 9 Figure

The role of advection and dispersion in the rock matrix on the transport of leaking CO2-saturated brine along a fractured zone

CO2 that is injected into a geological storage reservoir can leak in dissolved form because of brine displacement from the reservoir, which is caused by large-scale groundwater motion. Simulations of the reactive transport of leaking CO2aq along a conducting fracture in a clay-rich caprock are conducted to analyze the effect of various physical and geochemical processes. Whilst several modeling transport studies along rock fractures have considered diffusion as the only transport process in the surrounding rock matrix (diffusive transport), this study analyzes the combined role of advection and dispersion in the rock matrix in addition to diffusion (advection-dominated transport) on the migration of CO2aq along a leakage pathway and its conversion in geochemical reactions. A sensitivity analysis is performed to quantify the effect of fluid velocity and dispersivity. Variations in the porosity and permeability of the medium are found in response to calcite dissolution and precipitation along the leakage pathway. We observe that advection and dispersion in the rock matrix play a significant role in the overall transport process. For the parameters that were used in this study, advection-dominated transport increased the leakage of CO2aq from the reservoir by nearly 305%, caused faster transport and increased the mass conversion of CO2aq in geochemical reactions along the transport pathway by approximately 12.20% compared to diffusive transport.Peer ReviewedPostprint (author's final draft

Natural convection in a porous trapezoidal enclosure with magneto-hydrodynamic effect

The effects of magnetic force, acting vertically downward on natural convection within a porous trapezoidal enclosure saturated with an electrically conducting fluid have been investigated numerically. The bottom wall of the enclosure is subjected to a constant hot temperature and the top wall experiences a constant cold temperature whereas the remaining sidewalls are kept adiabatic. The physical problems are represented mathematically by different sets of governing equations along with the corresponding boundary conditions. By using Galerkin weighted residual method of finite element formulation, the non-dimensional governing equations are discritized. For natural convection in a porous medium the influential parameters are the modified Rayleigh number Ram, the fluid Rayleigh number Raf , the inclination angle of the sidewalls of the cavity γ, the rotational angle of the enclosure Φ and the Hartmann number Ha, through which different thermo-fluid characteristics inside the enclosure are obtained. In the present study, the obtained results are presented in terms of streamlines, isotherms and average Nusselt number along the hot wall. The result shows that with increasing Ha, the diffusive heat transfer become prominent even though the modified Rayleigh number increases. Optimum heat transfer rate is obtained at higher values of Ram in the absence of magnetic force

Weakly Nonlinear Stability Analysis of a Nanofluid in a Horizontal Porous Layer Using a Multidomain Spectral Collocation Method

In this chapter, we present a weakly nonlinear stability analysis of the flow of a nanofluid in a porous medium with stress-free boundary conditions. Some previous studies have investigated cross-diffusion in a nanofluid layer although in most cases these studies mostly deal with linear stability analysis. It is important to study the nonlinear stability in flows subject to cross-diffusion due to the wide range of applications where such flows arise such as in hydrothermal growth, compact heat exchanges, the solidification of binary mixtures, geophysical systems, solar pond, etc. Here we consider flow between parallel plates with an applied magnetic field and zero nanoparticle flux at the boundaries. A truncated Fourier series is introduced reducing the flow equations to a Lorenz-type system of nonlinear evolution equations. The multidomain spectral method is used to solve the equations that describe the growth of the convection amplitudes. The solutions are obtained as sets of trajectories in the phase space. Some interesting spiral trajectories and their sensitivity to the Rayleigh number are given

The kinetics of ice-lens growth in porous media

We analyse the growth rate of segregated ice (ice lenses) in freezing porous media. For typical colloidal materials such as soils we show that the commonly-employed Clapeyron equation is not valid macroscopically at the interface between the ice lens and the surrounding porous medium owing to the viscous dynamics of flow in premelted films. This gives rise to an ‘interfacial resistance’ to flow towards the growing ice which causes a significant drop in predicted ice-growth (heave) rates and explains why many previous models predict ice-growth rates that are much larger than those seen in experiments. We derive an explicit formula for the ice-growth rate in a given porous medium, and show that this only depends on temperature and on the external pressures imposed on the freezing system. This growth-rate formula contains a material-specific function which can be calculated (with a knowledge of the of the geometry and material of the porous medium), but which is also readily experimentally-measurable. We apply the formula to plate-like particles, and obtain good agreement with previous experimental data. Finally we show how the interfacial resistance explains the observation that the maximum heave rate in soils occurs in medium-grained particles such as silts, while heave rates are smaller for fine- and coarse- grained particles

On thermal convective instability in rotating fluids.

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.Abstract available on the PDF

Linear and Nonlinear Convection in Porous Media between Coaxial Cylinders

In this thesis we develop a mathematical model for describing three-dimensional natural convection in porous media filling a vertical annular cylinder. We apply a linear stability analysis to determine the onset of convection and the preferred convective mode when the annular cylinder is subject to two different types of boundary conditions: heat insulated sidewalls and heat conducting sidewalls. The case of an annular cylinder with insulated sidewalls has been investigated earlier, but our results reveal more details than previously found. We also investigate the case of the radius of the inner cylinder approaching zero and the results are compared with previous work for non-annular cylinders. Using pseudospectral methods we have built a high-order numerical simulator to uncover the nonlinear regime of the convection cells. We study onset and geometry of convection modes, and look at the stability of the modes with respect to different types of perturbations. Also, we examine how variations in the Rayleigh number affects the convection modes and their stability regimes. We uncover an increased complexity regarding which modes that are obtained and we are able to identify stable secondary and mixed modes. We find the different convective modes to have overlapping stability regions depending on the Rayleigh number. The motivation for studying natural convection in porous media is related to geothermal energy extraction and we attempt to determine the effect of convection cells in a geothermal heat reservoir. However, limitations in the simulator do not allow us to make any conclusions on this matter.Master i Anvendt og beregningsorientert matematikkMAMN-MABMAB39

Adomian decomposition solution for propulsion of dissipative magnetic Jeffrey biofluid in a ciliated channel containing a porous medium with forced convection heat transfer

Physiological transport phenomena often feature ciliated internal walls. Heat, momentum and multi-species mass transfer may arise and additionally non-Newtonian biofluid characteristics are common in smaller vessels. Blood (containing hemoglobin) or other physiological fluids containing ionic constituents in the human body respond to magnetic body forces when subjected to external (extra-corporeal) magnetic fields. Inspired by such applications, in the present work we consider the forced convective flow of an electrically-conducting viscoelastic physiological fluid through a ciliated channel under the action of a transverse magnetic field. The flow is generated by a metachronal wave formed by the tips of cilia which move to and fro in a synchronized fashion. The presence of deposits (fats, cholesterol etc) in the channel is mimicked with a Darcy porous medium drag force model. The two-dimensional unsteady momentum equation and energy equation are simplified with a stream function and small Reynolds' number approximation. The effect of energy loss is simulated via the inclusion of viscous dissipation in the energy conservation (heat) equation. The non-dimensional, transformed moving boundary value problem is solved with appropriate wall conditions via the semi-numerical Adomian decomposition method (ADM). The velocity, temperature and pressure distribution are computed in the form of infinite series constructed by ADM and numerically evaluated in a symbolic software (MATHEMATICA). Streamline distributions are also presented. The influence of Hartmann number (magnetic parameter), Jeffrey first and second viscoelastic parameters, permeability parameter (modified Darcy number), and Brinkman number (viscous heating parameter) on velocity, temperature, pressure gradient and bolus dynamics is visualized graphically. The flow is decelerated with increasing with increasing Hartmann number and Jeffery first parameter in the core flow whereas it is accelerated in the vicinity of the walls. Increasing permeability and Jeffery second parameter are observed to accelerate the core flow and decelerate the peripheral flow near the ciliated walls. Increasing Hartmann number elevates pressure gradient whereas it is reduced with permeability parameter. Temperatures are elevated with increasing magnetic parameter, Brinkman number and Jeffery second parameter. Increasing magnetic field is also observed to reduce the quantity of trapped boluses. Increasing permeability parameter suppresses streamline amplitudes. Both the magnitude and quantity of trapped boluses is elevated with an increase in both first and second Jeffery parameters

Characteristics of transient pressure performance of horizontal wells in fractured-vuggy tight fractal reservoirs considering nonlinear seepage

International audienceSince the classical seepage theory has limitations in characterizing the heterogeneity of fractured-vuggy tight reservoirs, well test interpretation results are not consistent with actual production by far. Based on the nonlinear percolation theory, a new nonlinear seepage equation considering the boundary layer and yield stress was derived to describe the seepage characteristics of dense matrix blocks and the stress sensitivity and fractal features of fracture systems were characterized by applying the fractal theory. Thus, the nonlinear model of a horizontal well in a fractured-vuggy tight fractal reservoir was established naturally. Then the finite element method was applied to solve the bottom hole pressure based on the processing of internal boundary conditions. After solving the model, the seepage characteristics of different models were summarized by analyzing the bottom hole pressure dynamic curves and the sensitivity analysis of multiple parameters such the nonlinear parameter and fractal index were conducted. Finally, the practicality of the model was proved through a field application. The results show that the pressure dynamic curves can be divided into nine flow stages and the increase of the nonlinear parameter will cause the intensity of the cross flow from matrix blocks to the fracture system to decrease. The fractal index is irrelevant to the intensity of the cross flow while it decides the upwarping degree of the curve at the middle and late flow stages. On the basis of the results of the field application, it can be concluded that the model fits well with actual production and the application of this model can improve the accuracy of well test interpretation