Explicit Rate–Time Solutions for Modeling Matrix–Fracture Flow of Single Phase Gas in Dual-Porosity Media

One of the widely used methods for modeling matrix–fracture fluid exchange in naturally fractured reservoirs is dual porosity approach. In this type of modeling, matrix blocks are regarded as sources/sinks in the fracture network medium. The rate of fluid transfer from matrix blocks into fracture medium may be modeled using shape factor concept (Warren and Root, SPEJ 3:245–255, 1963); or the rate–time solution is directly derived for the specific matrix geometry (de Swaan, SPEJ 16:117–122, 1976). Numerous works have been conducted to study matrix–fracture fluid exchange for slightly compressible fluids (e.g. oil). However, little attention has been taken to systems containing gas (compressible fluid). The objective of this work is to develop explicit rate–time solutions for matrix–fracture fluid transfer in systems containing single phase gas. For this purpose, the governing equation describing flow of gas from matrix block into fracture system is linearized using pseudopressure and pseudotime functions. Then, the governing equation is solved under specific boundary conditions to obtain an implicit relation between rate and time. Since rate calculations using such an implicit relation need iterations, which may be computationally inconvenient, an explicit rate–time relation is developed with the aid of material balance equation and several specific assumptions. Also, expressions are derived for average pseudopressure in matrix block. Furthermore, simplified solutions (originated from the complex general solutions) are introduced applicable in infinite and finite acting flow periods in matrix. Based on the derived solutions, expressions are developed for shape factor. An important observation is that the shape factor for gas systems is the same as that of oil bearing matrix blocks. Subsequently, a multiplier is introduced which relates rate to matrix pressure instead of matrix pseudopressure. Finally, the introduced equations are verified using a numerical simulator.     

Analysis of One-Dimensional Advection–Diffusion Model with Variable Coefficients Describing Solute Transport in a Porous medium

This work presents the analytical solution and temporal moments of one-dimensional advection–diffusion model with variable coefficients. Two case studies along with the two different sets of boundary conditions are considered at the inlet and outlet of the domain. In the first case, a time-dependent solute dispersion in the homogeneous domain along uniform flow is taken into account, whereas in the second case, due to inhomogeneity of domain, velocity is taken spatially dependent and the dispersion is assumed proportional to the square of the velocity. The Laplace transform is used to obtain the analytical solutions. The analytical temporal moments are derived from the Laplace domain solutions. To verify the correctness of the analytical solutions, a high-resolution second-order finite volume scheme is applied. Different case studies are considered and discussed. Both analytical and numerical results are in good agreement with each other.     

Incompressible 3D Navier–Stokes Equations as a Limit of a Nonlinear Parabolic System

In this paper, we consider the Cauchy problem for a nonlinear parabolic system ${u^\epsilon_t - \Delta u^\epsilon + u^\epsilon \cdot \nabla u^\epsilon + \frac{1}{2}u^\epsilon\, {\rm div}\, u^\epsilon - \frac{1}{\epsilon}\nabla\, {\rm div}\, u^\epsilon = 0}$ in ${\mathbb {R}^3 \times (0,\infty)}$ with initial data in Lebesgue spaces ${L^2(\mathbb {R}^3)}$ or ${L^3(\mathbb {R}^3)}$ . We analyze the convergence of its solutions to a solution of the incompressible Navier?CStokes system as ${\epsilon \to 0}$ .     

Modeling Variable Density Flow and Solute Transport in Porous Medium: 2. Re‐Evaluation of the Salt Dome Flow Problem

Case 5, Level 1 of the international HYDROCOIN groundwater flow modeling project is an example of idealized flow over a salt dome. The groundwater flow is strongly coupled to solute transport since density variations in this example are large (20%).Several independent teams simulated this problem using different models. Results obtained by different codes can be contradictory. We develop a new numerical model based on the mixed hybrid finite elements approximation for flow, which provides a good approximation of the velocity, and the discontinuous finite elements approximation to solve the advection equation, which gives a good approximation of concentration even when the dispersion tensor is very small. We use the new numerical model to simulate the salt dome flow problem.In this paper we study the effect of molecular diffusion and we compare linear and nonlinear dispersion equations. We show the importance of the discretization of the boundary condition on the extent of recirculation and the final salt distribution. We study also the salt dome flow problem with a more realistic dispersion (very small dispersion tensor). Our results are different to prior works with regard to the magnitude of recirculation and the final concentration distribution. In all cases, we obtain recirculation in the lower part of the domain, even for only dispersive fluxes at the boundary. When the dispersion tensor becomes very small, the magnitude of recirculation is small. Swept forward displacement could be reproduced by using finite difference method to compute the dispersive fluxes instead of mixed hybrid finite elements.     

Gas–Dynamic Structures in Supersonic Combustion of Hydrogen in a System of Plane Jets in a Supersonic Flow

Results of numerical and theoretical studies of supersonic diffusive combustion of a system of plane hydrogen jets in a supersonic air flow are described. It is shown that large–scale vortex structures appear in the mixing zone of the system of hydrogen jets and the cocurrent flow. These vortex structures affect the mechanism of turbulent exchange between the fuel and the oxidizer.     

Stress State of a Threaded Joint in a Dental Implant–Bone System*

International Applied Mechanics - The load distribution over the turns of an implant–bone threaded joint is studied. Such a joint is modeled by a rod structure in which the functional axial...     

Stability of the Camassa–Holm Multi-peakons in the Dynamics of a Shallow-Water-Type System

Consideration herein is the stability issue of a variety of superpositions of the Camassa–Holm peakons and antipeakons in the dynamics of the two-component Camassa–Holm system, which is derived in the shallow water theory. These wave configurations accommodate the ordered trains of the Camassa–Holm peakons, the ordered trains of Camassa–Holm antipeakons and peakons as well as the Camassa–Holm multi-peakons. Using the features of conservation laws and the monotonicity properties of the local energy, we prove the orbital stability of these wave profiles in the energy space by the modulation argument.     

Global Solutions for a Coupled Compressible Navier–Stokes/Allen–Cahn System in 1D

In this paper, we investigate a coupled compressible Navier–Stokes/Allen–Cahn system which describes the motion of a mixture of two viscous compressible fluids. We prove the existence and uniqueness of global classical solution, the existence of weak solutions and the existence of unique strong solution of the Navier–Stokes/Allen–Cahn system in 1D for initial data ρ ₀ without vacuum states.     

Advection–Dispersive Mechanisms of Electrolyte Species in Porous Matrix

Advection–dispersive phenomena inside two geometries of porous media, corresponding to a structured and unstructured network of minichannels, are studied from a combination of analytical, numerical, and electrodiffusion techniques. The instantaneous limiting diffusion current, connected to the concentration of electroactive species flowing in the porous matrix, is recorded and measured during flow experimentation. Transport phenomena investigation consists of the step injection of a tracer of higher concentration than the bulk flow in order to characterize the mixing of electrolyte species inside two arrangements of network, so called \(\times \) _network and T_network. The experimental results are supported by 2D-numerical simulations performed in the \(\times \) _network. A pore model is proposed in order to predict the pore velocity, which is used within the resolution of the diffusion–convection balance. The numerical simulations, based on a second-order finite difference scheme, give rise to a good agreement in terms of mixing index and the methodology employed for the numerical injection concentration appears suitable. The numerical experiments are quite-well representative in laminar regime below the critical Reynolds number ( \(Re_{\mathrm{crit}}\approx \,200\) ). Above \(Re_{\mathrm{crit}}\) , the inertial effects are not negligible and the momentum transfer needs to be taken into account. This latter phenomenon is analyzed at the pore-scale in term of local skin friction measured at the channels crossing, then compared and discussed from analytical solutions.     

Displacement of Cr(III)–Partially Hydrolyzed Polyacrylamide Gelling Solution in a Fracture in Porous Media

During waterflooding of a fractured formation, water may channel through the fracture or interconnected network of fractures, leaving a large portion of oil bearing rock unswept. One remedial practice is injection of a gelling solution into the fracture. Such placement of a gelling mixture is associated with leak-off from the fracture face into the adjoining matrix. Design of a gel treatment needs understanding of the flow of gelling mixture in and around the fracture. This flow is addressed here for Cr(III)–partially hydrolyzed polyacrylamide formulation through experiments and conceptual model. A fractured slab was used to develop a lab-model, where the flow along the fracture and simultaneous leak-off into the matrix can be controlled. Also, the fracture and matrix properties had to be evaluated individually for a meaningful analysis of the displacement of gelling solution. During this displacement, the gelling fluid leaked off from the fracture into the matrix as a front, resulting in a decreasing velocity (and pressure gradient) along the fracture. With pressure in the fracture held constant with time, the leak-off rate decreased as the viscous front progressed into the matrix. The drop in leak-off rate was rapid during the initial phase of displacement. A simple model, based on the injection of a viscous solution into the dual continua, could explain the displacement of Cr(III)–polyacrylamide gelling mixture through the fractured slab. This study rules out any major complication from the immature gelling fluid, e.g., build-up of cake layer on the fracture face. The model, due to its simplicity may become useful for quick sizing of gel treatment, and any regression-based evaluation of fluid properties in a fracture for other applications.     

Random Kick-Forced 3D Navier–Stokes Equations in a Thin Domain

We consider the Navier–Stokes equations in the thin 3D domain , where is a two-dimensional torus. The equation is perturbed by a non-degenerate random kick force. We establish that, firstly, when ε ≪ 1, the equation has a unique stationary measure and, secondly, after averaging in the thin direction this measure converges (as ε → 0) to a unique stationary measure for the Navier–Stokes equation on . Thus, the 2D Navier–Stokes equations on surfaces describe asymptotic in time, and limiting in ε, statistical properties of 3D solutions in thin 3D domains.     

EBSD Study of Delamination Fracture in Al–Li Alloy 2090

Aluminum–lithium (Al–Li) alloys offer attractive combinations of high strength and low density for aerospace structural applications. However, a tendency for delamination fracture has limited their use. Identification of the metallurgical mechanisms controlling delamination may suggest processing modifications to minimize the occurrence of this mode of fracture. In the current study of Al–Li alloy 2090 plate, high quality electron backscattered diffraction (EBSD) information has been used to evaluate grain boundary types exhibiting delamination fracture and characterize microtexture variations between surrounding grains. Delamination was frequently observed to occur between variants of the brass texture component, along near-Σ3, incoherent twin boundaries. EBSD analyses indicated a tendency for intense deformation along one side of the fractured boundary. A through-thickness plot of grain-specific Taylor factors showed that delaminations occurred along boundaries with the greatest difference in Taylor factors. Together, these suggest a lack of slip accommodation across the boundary, which promotes significantly higher deformation in one grain, and stress concentrations that result in delamination fracture.     

A New Coal-Permeability Model: Internal Swelling Stress and Fracture–Matrix Interaction

We have developed a new coal-permeability model for uniaxial strain and constant confining-stress conditions. The model is unique in that it explicitly considers fracture–matrix interaction during coal-deformation processes and is based on a newly proposed internal swelling stress concept. This concept is used to account for the impact of matrix swelling (or shrinkage) on fracture-aperture changes resulting from partial separation of matrix blocks by fractures that do not completely cut through the whole matrix. The proposed permeability model is evaluated using data from three Valencia Canyon coalbed wells in the San Juan Basin, where increased permeability has been observed during CH₄ gas production, as well as using published data from laboratory tests. Model results are generally in good agreement with observed permeability changes. The importance of fracture–matrix interaction in determining coal permeability, demonstrated in this study using relatively simple stress conditions, underscores the need for a dual-continuum (fracture and matrix) mechanical approach to rigorously capture coal-deformation processes under complex stress conditions, as well as the coupled flow and transport processes in coal seams.     

Identification of Molecular Transport Mechanisms in Micro-Porous Hydrotalcite–Silica Membrane

Hydrotalcite (HT) materials have been known to be able to adsorb \(\hbox {CO}_{2}\) even at high temperature. However, HT has not been made into a micro-porous membrane because of its meso-porous nature. In order to form a micro-porous HT membrane, silica was selected as a host matrix due to its ability to retain its micro-porosity. In this paper, a micro-porous hydrotalcite–silica membrane was formed on a meso-porous \(\upgamma \) -alumina layer supported by a macro-porous \(\upalpha \) -alumina substrate. Most of the micro-porosity determined from nitrogen adsorption measurement was found to be either closed or open but not interconnected, whereas most of the meso-porosity (at ca. 11.2 nm) in the micro-porous membrane was open and interconnected, thereby promoting gas flow. Viscous flow mechanism was observed to dominate transport of gases in macro-porous membrane. Knudsen diffusion dominated transport of gases in meso-porous membrane. On the other hand, surface affinity influenced the transport of carbon dioxide through the micro-porous membrane rather significantly. While permeability of pure hydrogen and carbon dioxide were independent of pressure, the permeability of the gases in the binary mixtures decreased with increasing pressure. Both experiment and simulation demonstrated consistent results.     

Measure Valued Solutions of the 2D Keller–Segel System

We develop a theory of global measure-valued solutions for the classical Keller–Segel model. These solutions are obtained considering the limit of solutions of a regularized problem. We also prove that different regularizations yield different limit measures in the case in which classical solutions of the Keller–Segel system are not globally defined in time.     

Transport modeling of sedimenting particles in a turbulent pipe flow using Euler–Lagrange large eddy simulation

A volume-filtered Euler–Lagrange large eddy simulation methodology is used to predict the physics of turbulent liquid–solid slurry flow through a horizontal periodic pipe. A dynamic Smagorinsky model based on Lagrangian averaging is employed to account for the sub-filter scale effects in the liquid phase. A fully conservative immersed boundary method is used to account for the pipe geometry on a uniform cartesian grid. The liquid and solid phases are coupled through volume fraction and momentum exchange terms. Particle–particle and particle–wall collisions are modeled using a soft-sphere approach. Three simulations are performed by varying the superficial liquid velocity to be consistent with the experimental data by Dahl et al. (2003). Depending on the liquid flow rate, a particle bed can form and develop different patterns, which are discussed in light of regime diagrams proposed in the literature. The fluctuation in the height of the liquid-bed interface is characterized to understand the space and time evolution of these patterns. Statistics of engineering interest such as mean velocity, mean concentration, and mean streamwise pressure gradient driving the flow are extracted from the numerical simulations and presented. Sand hold-up calculated from the simulation results suggest that this computational strategy is capable of predicting critical deposition velocity.     

Modeling the mechanics of HMX detonation using a Taylor–Galerkin scheme

Design of energetic materials is an exciting area in mechanics and materials science. Energetic composite materials are used as propellants, explosives, and fuel cell components. Energy release in these materials are accompanied by extreme events: shock waves travel at typical speeds of several thousand meters per second and the peak pressures can reach hundreds of gigapascals. In this paper, we develop a reactive dynamics code for modeling detonation wave features in one such material. The key contribution in this paper is an integrated algorithm to incorporate equations of state, Arrhenius kinetics, and mixing rules for particle detonation in a Taylor–Galerkin finite element simulation. We show that the scheme captures the distinct features of detonation waves, and the detonation velocity compares well with experiments reported in literature.