Understanding Oil and Gas Flow Mechanisms in Shale Reservoirs Using SLD–PR Transport Model

More and more attention has been paid to the oil and gas flow mechanisms in shale reservoirs. The solid–fluid interaction becomes significant when the pores are in the nanoscale. The interaction changes the fluid’s physical properties and leads to different flow mechanisms in shale reservoirs from those in conventional reservoirs. By using a Simplified Local Density–Peng Robinson transport model, we consider the density and viscosity profiles, which result from solid–fluid interaction. Gas rarefaction effect is negligible at high pressure, so we assume it is viscous flow. Considering the density- and viscosity-changing effects, we proposed a slit permeability model. The velocity profiles are obtained by this newly established model. This proposed model is validated by matching the density profile and velocity profile from molecular dynamic simulation. Then, the effects of pressure and pore size on gas and oil flow mechanisms are also studied in this work. The results show that both gas and oil exhibit enhanced flow rates in nanopores. Gas-phase flow in nanopores is dominated by the density-changing effect (adsorption), while the oil-phase flow is mainly controlled by the viscosity-changing effect. Both gas and oil permeability quickly decrease to the Darcy permeability when the slit aperture becomes large. The results reported in this work are representative and should significantly help us understand the mechanisms of oil and gas flow in shale reservoirs.     

Transport of Polymer Particles in Oil–Water Flow in Porous Media: Enhancing Oil Recovery

Transport in Porous Media - 3D printing with powders offers the most analogous method to the natural way in which clastic reservoir rocks are formed, resulting in pore network textures and...     

On the Existence and Stability of Time Periodic Solution to the Compressible Navier–Stokes Equation on the Whole Space

The existence of a time periodic solution of the compressible Navier–Stokes equation on the whole space is proved for a sufficiently small time periodic external force when the space dimension is greater than or equal to 3. The proof is based on the spectral properties of the time-T-map associated with the linearized problem around the motionless state with constant density in some weighted L ^∞ and Sobolev spaces. The time periodic solution is shown to be asymptotically stable under sufficiently small initial perturbations and the L ^∞ norm of the perturbation decays as time goes to infinity.     

Multi-scale Fractured Coal Gas–Solid Coupling Model and Its Applications in Engineering Projects

With coal mining entering the geological environment of “high stress, rich gas, strong adsorption and low permeability,” the difficulty of joint coal and gas extraction clearly augments, the risk of solid–gas coupling dynamic disasters greatly increases, and the underlying mechanisms become more complex. In this paper, based on the characteristics of coal’s multi-scale structure and spatiotemporal variation, the multi-scale fractured coal gas–solid coupling model (MSFM) was built. In this model, the interaction between coal matrix and its fractures and the mechanical characteristics of gas-bearing coal were considered, as well as their coupling relationship. By MATLAB software, the stress–damage–seepage numerical computation programs were developed, which were applied into Comsol Multiphysics to simulate gas flow caused by coal mining. The simulation results showed the spatial variability of coal elastic modulus and cross-flow behaviors of coal seam gas, which were superior to the results of traditional gas–solid coupling model. And the numerical results obtained from MSFM were closer to the measured results in field, while the computation results of traditional model were slightly higher than the measured results. Furthermore, the MSFM in a large scale was verified by field engineering project.     

Complex Ginzburg–Landau Equation with Absorption: Existence,Uniqueness and Localization Properties

In this paper we study the time-dependent complex Ginzburg–Landau equation with a nonlinear absorbing term in ${\Omega \times(0,T),\, \Omega }$ open bounded set in ${\mathbb{R}^{n}}$ . We prove global existence and uniqueness of solutions for the initial and boundary-value problem and study the properties of localization and extinction of solutions in some special cases.     

Near-Critical Gas–Liquid Flow in Porous Media: Monovariant Model, Analytical Solutions and Convective Mass Exchange Effects

We examine a class of hydrocarbon reservoirs whose thermodynamic state remains close to the critical point during the all period of reservoir exploitation. Such a situation is typical for the so-called gas–condensate systems, in which the liquid phase is formed from gas when pressure decreases. Due to proximity to critical point, the mixture contains many components which are neutral with respect to the phase state. This determines a low thermodynamic degree of freedom of the system. As the results, the mathematical flow model allows a significant reduction in the number of conservation equations, whatever the number of chemical components. In the vicinity of a well, the system may be reduced to one transport equation for saturation. This nonlinear model yields exact analytical solutions when the flow is self-similar. In more general case of flow, we develop partially linearized solutions which are shown to be sufficiently exact. The spectrum of examined cases covers the flow in a medium with a sharp heterogeneity and a sharp variation in the flow rate. A significant relative gas flow past liquid gives rise to a convective mass exchange phenomenon which appears highly different from that observed in static. In the case of a medium discontinuity, the convective mass exchange gives rise to a phenomenon of condensate saturation billow formation. A sharp variation in the flow rate leads to a hysteretic behavior of the saturation field.     

Computation of Gas–Dynamic Parameters and Heat Transfer in Supersonic Turbulent Separated Flows near Backward–Facing Steps

Computation results of plane turbulent flows in the vicinity of backward–facing steps with leeward–face angles = 8, 25, and 45° for Mach numbers M_infin = 3 and 4 are presented. The averaged Navier—Stokes equations supplemented by the Wilcox model of turbulence are used as a mathematical model. The boundary–layer equations were also used for the case of an attached flow ( = 8°). The computed and experimental distributions of surface pressure and skin friction, the velocity and pressure fields, and the heat–transfer coefficients are compared.     

Capillarity-Driven Oil Flow in Nanopores: Darcy Scale Analysis of Lucas–Washburn Imbibition Dynamics

Transport in Porous Media - We present gravimetrical and optical imaging experiments on the capillarity-driven imbibition of silicone oils in monolithic silica glasses traversed by 3D networks of...     

Existence and Homogenisation of Travelling Waves Bifurcating from Resonances of Reaction–Diffusion Equations in Periodic Media

The existence of travelling wave type solutions is studied for a scalar reaction diffusion equation in \(\mathbb {R}^2\) with a nonlinearity which depends periodically on the spatial variable. We treat the coefficient of the linear term as a parameter and we formulate the problem as an infinite spatial dynamical system. Using a centre manifold reduction we obtain a finite dimensional dynamical system on the centre manifold with fully degenerate linear part. By phase space analysis and Conley index methods we find conditions on the parameter and nonlinearity for the existence of travelling wave type solutions with particular wave speeds. The analysis provides an approach to the homogenisation problem as the period of the periodic dependence in the nonlinearity tends to zero.     

Oil–water flows through sudden contraction and expansion in a horizontal pipe – Phase distribution and pressure drop

In this paper, change of flow patterns during the simultaneous flow of high viscous oil and water through the sudden contraction and expansion in a horizontal conduit has been studied. It is noted that these sudden changes in cross-section have a significant influence on the downstream phase distribution of lube oil–water flow. The observation suggests a simple technique to establish core flow as well as a way to prevent pipe wall fouling during the transportation of such oil. A number of interesting differences have been noted during low viscous oil–water flow through the same test rigs. While several types of core annular flow are observed for the former case, a wider variety of interfacial distribution characterizes kerosene–water systems. The pressure profiles during the simultaneous flow of lube oil and water through the sudden contraction and expansion are also studied and compared with low viscous oil–water flows. The pressure profiles are found to be independent of liquid viscosity and the loss coefficients are observed to be independent of flow patterns in both the cases.     

Existence and Stability of Time Periodic Solution to the Compressible Navier–Stokes–Korteweg System on $${\mathbb{R}^3}$$

The compressible Navier–Stokes–Korteweg system is considered on \({\mathbb{R}^3}\) when the external force is periodic in the time variable. The existence of a time periodic solution is proved for a sufficiently small external force by using the time-T-map related to the linearized problem around the motionless state with constant density and absolute temperature. The spectral properties of the time-T-map is investigated by a potential theoretic method and an energy method in some weighted spaces. The stability of the time periodic solution is proved for sufficiently small initial perturbations. It is also shown that the \({L^\infty}\) norm of the perturbation decays as time goes to infinity.     

From the Highly Compressible Navier–Stokes Equations to Fast Diffusion and Porous Media Equations,Existence of Global Weak Solution for the Quasi-Solutions

We consider the compressible Navier–Stokes equations for viscous and barotropic fluids with density dependent viscosity. The aim is to investigate mathematical properties of solutions of the Navier–Stokes equations using solutions of the pressureless Navier–Stokes equations, that we call quasi solutions. This regime corresponds to the limit of highly compressible flows. In this paper we are interested in proving the announced result in Haspot (Proceedings of the 14th international conference on hyperbolic problems held in Padova, pp 667–674, 2014) concerning the existence of global weak solution for the quasi-solutions, we also observe that for some choice of initial data (irrotationnal) the quasi solutions verify the porous media, the heat equation or the fast diffusion equations in function of the structure of the viscosity coefficients. In particular it implies that it exists classical quasi-solutions in the sense that they are \({C^{\infty}}\) on \({(0,T)\times \mathbb{R}^{N}}\) for any \({T > 0}\). Finally we show the convergence of the global weak solution of compressible Navier–Stokes equations to the quasi solutions in the case of a vanishing pressure limit process. In particular for highly compressible equations the speed of propagation of the density is quasi finite when the viscosity corresponds to \({\mu(\rho)=\rho^{\alpha}}\) with \({\alpha > 1}\). Furthermore the density is not far from converging asymptotically in time to the Barrenblatt solution of mass the initial density \({\rho_{0}}\).     

Solution and asymptotic analysis of a boundary value problem in the spring–mass model of running

We consider the classic spring–mass model of running which is built upon an inverted elastic pendulum. In a natural way, there arises an interesting boundary value problem for the governing system of two nonlinear ordinary differential equations. It requires us to choose the stiffness to ascertain that after a complete step, the spring returns to its equilibrium position. Motivated by numerical calculations and real data, we conduct a rigorous asymptotic analysis in terms of the Poicaré–Lindstedt series. The perturbation expansion is furnished by an interplay of two time scales what has an significant impact on the order of convergence. Further, we use these asymptotic estimates to prove that there exists a unique solution to the aforementioned boundary value problem and provide an approximation to the sought stiffness. Our results rigorously explain several observations made by other researchers concerning the dependence of stiffness on the initial angle of the stride and its velocity. The theory is illustrated with a number of numerical calculations.

     

Existence,Uniqueness and Lipschitz Dependence for Patlak–Keller–Segel and Navier–Stokes in {\mathbb{R}^2} with Measure-Valued Initial Data

We establish a new local well-posedness result in the space of finite Borel measures for mild solutions of the parabolic–elliptic Patlak–Keller–Segel (PKS) model of chemotactic aggregation in two dimensions. Our result only requires that the initial measure satisfy the necessary assumption \({\max_{x \in \mathbb{R}^2} \mu (\{x\}) < 8 \pi}\) . This work improves the small-data results of Biler (Stud Math 114(2):181–192, 1995) and the existence results of Senba and Suzuki (J Funct Anal 191:17–51, 2002). Our work is based on that of Gallagher and Gallay (Math Ann 332:287–327, 2005), who prove the uniqueness and log-Lipschitz continuity of the solution map for the 2D Navier–Stokes equations (NSE) with measure-valued initial vorticity. We refine their techniques and present an alternative version of their proof which yields existence, uniqueness and Lipschitz continuity of the solution maps of both PKS and NSE. Many steps are more difficult for PKS than for NSE, particularly on the level of the linear estimates related to the self-similar spreading solutions.     

A Semi-Analytical Solution for Large-Scale Injection-Induced Pressure Perturbation and Leakage in a Laterally Bounded Aquifer–Aquitard System

A number of (semi-)analytical solutions are available to drawdown analysis and leakage estimation of shallow aquifer–aquitard systems. These solutions assume that the systems are laterally infinite. When a large-scale pumping from (or injection into) an aquifer–aquitard system of lower specific storativity occurs, induced pressure perturbation (or hydraulic head drawdown/rise) may reach the lateral boundary of the aquifer. We developed semi-analytical solutions to address the induced pressure perturbation and vertical leakage in a “laterally bounded” system consisting of an aquifer and an overlying/underlying aquitard. A one-dimensional radial flow equation for the aquifer was coupled with a one-dimensional vertical flow equation for the aquitard, with a no-flow condition imposed on the outer radial boundary. Analytical solutions were obtained for (1) the Laplace-transform hydraulic head drawdown/rise in the aquifer and in the aquitard, (2) the Laplace-transform rate and volume of leakage through the aquifer–aquitard interface integrated up to an arbitrary radial distance, (3) the transformed total leakage rate and volume for the entire interface, and (4) the transformed horizontal flux at any radius. The total leakage rate and volume depend only on the hydrogeologic properties and thicknesses of the aquifer and aquitard, as well as the duration of pumping or injection. It was proven that the total leakage rate and volume are independent of the aquifer’s radial extent and wellbore radius. The derived analytical solutions for bounded systems are the generalized solutions of infinite systems. Laplace-transform solutions were numerically inverted to obtain the hydraulic head drawdown/rise, leakage rate, leakage volume, and horizontal flux for given hydrogeologic and geometric conditions of the aquifer–aquitard system, as well as injection/pumping scenarios. Application to a large-scale injection-and-storage problem in a bounded system was demonstrated.     

Heating and Melting of Asphalt–Paraffin Plugs in Oil‐Well Equipment Using an Electromagnetic Radiation Source Operating in a Periodic Mode

This paper considers the elimination of asphalt–paraffin plugs in wellbore equipment using a highfrequency radiation source which is energized and deenergized periodically. The dynamics of melting of plugs is analyzed numerically. The time of removal of plugs is determined with variation in the offduty ratio of the operating cycle of the highfrequency oscillator and the time of its continuous operation in a cycle.