One-Dimensional Matrix-Fracture Transfer in Dual Porosity Systems with Variable Block Size Distribution

Most of the developed models for fractured reservoirs assume ideal matrix block size distribution. This assumption may not be valid in reality for naturally fractured reservoirs and possibly lead to errors in prediction of production from the naturally fractured reservoirs especially during a transient period or early time production from the matrix blocks. In this study, we investigate the effect of variable block size distribution on one- dimensional flow of compressible fluids in fractured reservoirs. The effect of different matrix block size distributions on the single phase matrix-fracture transfer is studied using a recently developed semi-analytical approach. The proposed model is able to simulate fluid exchange between matrix and fracture for continuous or discrete block size distributions using probability density functions or structural information of a fractured formation. The presented semi-analytical model demonstrates a good accuracy compared to the numerical results. There have been recent attempts to consider the effect of variable block size distribution in naturally fractured reservoir modeling for slightly compressible fluids with a constant viscosity and compressibility. The main objective of this study is to consider the effect of variable block size distribution on a one-dimensional matrix-fracture transfer function for single-phase flow of a compressible fluid in fractured porous media. In the proposed semi-analytical model, the pressure variability of viscosity and isothermal compressibility is considered by solving the nonlinear partial differential equation of compressible fluid flow in the fractured media. The closed form solution provided can be applied to flow of compressible fluids with variable matrix block size distribution in naturally fractured gas reservoirs.     

Identification of Fluid Dynamics in Forced Gravity Drainage Using Dimensionless Groups

A number of forced gravity drainage experiments have been conducted using a wide range of the physical and operational parameters, wherein the type, length, and permeability of the porous medium as well as oil viscosity and injection rate were varied. Results indicate that an increase in the Bond number has a positive effect on oil recovery whereas the capillary number has an opposite effect. These trends were observed over a two-order of magnitude change in the value of the dimensionless groups. Furthermore, it was found that use of each number alone is insufficient to obtain a satisfactory correlation with recovery. A combined dimensionless group is proposed, which combines the effect of all the three forces. Recoveries from all the experiments conducted in this study show a very good correlation with the proposed group. The exponent of the Bond number in the proposed group is larger than the capillary number suggesting a larger importance for the former. We then show that the same group provides a good correlation for recovery from additional experiments conducted in this study (in the presence of connate water) with that of another set of experiments in the literature.     

Dual Porosity and Dual Permeability Modeling of Horizontal Well in Naturally Fractured Reservoir

This article is the first investigation on the dual permeability flow issue for horizontal well-production in a naturally fractured dual-porosity reservoir. Based on the inter-porosity flow from matrix system to fracture system and treating the media directly connected with horizontal wellbore as matrix and fracture systems, we established a model of horizontal well-production and then solved the model using some modern mathematical methods, such as Laplace integral transformation, separation of variables, eigenvalue, and eigenfunction. Later in the article, we obtained the standard log–log type curves using numerical simulation and analyzed the transient flow behavior thoroughly, which showed it is dual porosity and dual permeability flow behavior. The numerical simulation results showed that there are obvious differences between dual permeability and single permeability models. The dual permeability flow behavior accelerates energy supplement during production and reduces the classical matrix-fracture (V-shaped) response. We also showed that type curves characteristics are affected by external boundary conditions, the parameter κ, ω _f and λ _mf, etc. The research results show that our model would be a good semi-analytical model supplied to users. Because the single permeability modeling ignores the direct fluid supply from matrix to wellbore, we recommend using the dual permeability modeling to make well testing and rate decline interpretation in real case studies.     

基于传递函数法的兰姆波解析模拟

论文研究了兰姆波传播的解析模型.薄压电传感器(PZT)被用于兰姆波的激发.为模拟PZT的激励,采用PZT边缘的分布式点源等效PZT的激励,建立了解析模型的边界条件.结合三维傅里叶变换,在频率波数域求解了三维波动方程,得到了在单位周期激励下兰姆波的频域响应.结合传递函数法,可以得到任意激励下兰姆波的频域响应.利用傅里叶逆变换,进一步求解了兰姆波在时间-空间域的全局波场.为了验证解析模型,将解析模拟与实验进行了比对.试验测量值与解析模拟结果相吻合.     

Effect of Wetting on Gravity Drainage in Porous Media

A series of benchmark experiments on the effect of the wetting state on the flow properties in porous media were performed, allowing us to relate the wetting properties at the pore scale to the macroscale hydrodynamics. Drainage of n-alkanes (oils) displaced by air in a model porous medium consisting of water-wet sand was studied using gamma-ray densitometry and weight measurements. The enormous advantage of our system is that we know and control the wetting properties perfectly: we can tune the wetting properties by changing the salinity of the water. This allows us to perform porous medium flow experiments for the different wetting states without changing the transport properties (viscosity and density) of the oil. Drainage is found to be more efficient, and consequently oil recovery more important for partial wetting.     

Forced Convection Heat Transfer of Nanofluids in a Porous Channel

This article is concerned with the effects of flow and migration of nanoparticles on heat transfer in a straight channel occupied with a porous medium. Investigation of force convective heat transfer of nanofluids in a porous channel has not been considered completely in the literature and this challenge is generally considered to be an open research topic that may require more study. The fully developed flow and steady Darcy?CBrinkman?CForchheimer equation is employed in porous channel. The thermal equilibrium model is assumed between nanofluid and solid phases. It is assumed that the nanoparticles are distributed non-uniformly inside the channel. As a result the volume fraction distribution equation is also coupled with governing equations. The effects of parameters such as Lewis number, Schmidt number, Brownian diffusion, and thermophoresis on the heat transfer are completely studied. The results show that the local Nusselt number is decreased when the Lewis number is increased. It is observed that as the Schmidt number is increased, the wall temperature gradient is decreased and as a consequence the local Nusselt number is decreased. The effects of Lewis number, Schmidt number, and modified diffusivity ratio on the volume fraction distribution are also studied and discussed.     

Experimental and Modelling Study of Gravity Drainage in a Three-Block System

Transport in Porous Media - Gravity drainage is known as the controlling mechanism of oil recovery in naturally fractured reservoirs. The efficiency of this mechanism is controlled by...     

Probability Density Function Modeling of Multi-Phase Flow in Porous Media with Density-Driven Gravity Currents

A probability density function (PDF) based approach is employed to model multi-phase flow with interfacial mass transfer (dissolution) in porous media. The joint flow statistics is represented by a mass density function (MDF), which is transported in the physical and probability spaces via Fokker?CPlanck equation. This MDF-equation requires Lagrangian evolutions of the random flow variables; these evolutions are stochastic processes honoring the micro-scale flow physics. To demonstrate the concept, we consider an example of immiscible two-phase flow with the non-equilibrium dissolution of single component from one phase into the other-a model for solubility trapping during CO₂ storage in brine aquifer. Since CO₂-rich brine is denser than pure brine, density-driven countercurrent flow is set up in the brine phase. The stochastic models mimicking the physics of countercurrent flow lead to a modeled MDF-equation, which is solved using our recently developed stochastic particle method for multi-phase flow (Tyagi et al. J Comput Phys 227:6696?C6714, 2008). In addition, we derive Eulerian equations for stochastic moments (mean, variance, etc.) and show that unlike the MDF-equation the system of moment equations is not closed. In classical Darcy formulation, for example, the mean concentration equation is closed by neglecting variance. However, with several one- and two-dimensional simulations, it is demonstrated that the PDF and Darcy modeling approaches give significantly different results. While the PDF-approach properly accounts for the long correlation length scales and the concentration variance in density-driven countercurrent flow, the same phenomenon cannot be captured accurately with a standard Darcy model.     

Thermodynamically Consistent Limiting Forced Convection Heat Transfer in a Asymmetrically Heated Porous Channel: An Analytical Study

Fluid transport and the associated heat transfer through porous media is of immense importance because of its numerous practical applications. In view of the widespread applications of porous media flow, the present study attempts to investigate the forced convective heat transfer in the limiting condition for the flow through porous channel. There could be many areas, where heat transfer through porous channel attain some limiting conditions, thus, the analysis of limiting convective heat transfer is far reaching. The primary aim of the present study is focused on the limiting forced convection analysis considering the flow of Newtonian fluid between two asymmetrically heated parallel plates filled with saturated porous media. Utilizing a few assumptions, which are usually employed in the literature, an analytical methodology is executed to obtain the closed-form expression of the temperature profile, and in the following the expression of the limiting Nusselt numbers. The parametric variations of the temperature profile and the Nusselt numbers in different cases have been shown highlighting the influential role of different performance indexing parameters, like Darcy number, porosity of the media, and Brinkman number of forced convective heat transfer in porous channel. In doing so, the underlying physics of the transport characteristics of heat has been delineated in a comprehensive way. Moreover, a discussion has been made regarding an important feature like the onset of point of singularity as appeared on the variation of the Nusselt number from the consideration of energy balance in the flow field, and in view of second law of thermodynamics.     

Impact of a Porosity-Dependent Retention Function on Simulations of Porous Flow

Transport in Porous Media - Numerical models of flow in unsaturated porous media employ a range of functions to account for capillary effects. In general, these retention functions are assigned at...     

Modeling of Liquid Fuel Injection, Evaporation and Mixing in a Gas Turbine Burner Using Large Eddy Simulations

The interaction of turbulence, temperature fluctuation, liquid fuel transport, mixing and evaporation is studied by using Large Eddy Simulations (LES). To assess the accuracy of the different components of the methods we consider first isothermal, single phase flow in a straight duct. The results using different numerical methods incorporating dynamic Sub-Grid-Scale (SGS) models are compared with DNS and experimental data. The effects of the interactions among turbulence, temperature fluctuation, spray transport, evaporation and mixing of the gaseous fuel are studied by using different assumptions on the temperature field. It has been found that there are strong non-linear interactions among temperature-fluctuation, evaporation and turbulent mixing which require additional modeling if not full LES is used. The developed models and methods have been applied to a gas turbine burner into which liquid fuel is injected. The dispersion of the droplets in the burner is described. This revised version was published online in July 2006 with corrections to the Cover Date.     

Optimization of Forced Convection Heat Transfer in a Composite Porous Medium Channel

The optimization of heat transfer for forced convection in a composite porous channel was studied. We investigated the question where should one place, in the core or in the sheath, the material with high permeability and high-thermal conductivity and where should one place the material with low permeability and low-thermal conductivity, to maximize heat transfer from the walls. We also investigated the optimal heat transfer situation when one has the freedom to vary the relative volumes of the core and the sheath.     

Assessment of an Evaporation Model in CFD Simulations of a Free Liquid Pool Fire Using the Mass Transfer Number Approach

In the present paper an evaporation model is implemented and assessed in a Computational Fluid Dynamics (CFD) code named ISIS. First, the influence of the cell size and time step on the temperature field is studied via simulations with a prescribed fuel Mass Loss Rate (MLR). Then, the evaporation model is assessed using predictive simulations. The experimental scenario is a 30 cm-diameter heptane pool fire. The average fuel Mass Loss Rate Per Unit Area (MLRPUA) is predicted within 5.5% deviation from the experimental value. In addition, an analysis of the temperature and heat fluxes at the surface of the liquid, the mass transfer coefficient and the temperature inside the liquid is performed.     

Analysis of Laminar Flow and Forced Convection Heat Transfer in a Porous Medium

The flow of an incompressible Newtonian fluid confined in a planar geometry with different wall temperatures filled with a homogenous and isotropic porous medium is analyzed in terms of determining the unsteady state and steady state velocities, the temperature and the entropy generation rate as function of the pressure drop, the Darcy number, and the Brinkman number. The one-dimensional approximate equation in the rectangular Cartesian coordinates governing the flow of a Newtonian fluid through porous medium is derived by accounting for the order of magnitude of terms as well as accompanying approximations to the full-blown three-dimensional equations by using scaling arguments. The one-dimensional approximate energy and the entropy equations with the viscous dissipation consisting of the velocity gradient and the square of velocity are derived by following the same procedure used in the derivation of velocity expressions. The one-dimensional approximate equations for the velocity, the temperature, and the entropy generation rate are analytically solved to determine the velocity, the temperature, and the entropy distributions in the saturated porous medium as functions of the effective process parameters. It is found that the pressure drop, the Darcy number, and the Brinkman number affect the temperature distribution in the similar way, and besides the above parameters, the irreversibility distribution ratio also affects the entropy generation rate in the similar way.     

Forced Convection Heat Transfer on a Flat Plate Embedded in Porous Media for Power-Law Fluids

The effect of the presence of an isotropic solid matrix on the forced convection heat transfer rate from a flat plate to power-law non- Newtonian fluid-saturated porous medium, has been investigated. Numerical results are presented for the distribution of velocity and temperature profiles within the boundary layer. The effects of the flow index, first-order and second-order resistance on the velocity, and temperature profiles are discussed. The missing wall values of the velocity and thermal functions are tabulated.     

Effects of Porous Fins on Mixed Convection and Heat Transfer Mechanics in Lid-Driven Cavities: Full Numerical Modeling and Parametric Simulations

Transport in Porous Media - The present study presents a comprehensive analysis of effects of porous fins on mixed convection heat transfer in lid-driven square cavities, where the top lid has the...     

Modeling of Heat and Mass Transfer in a Fractured Porous Medium

While fractured formations are possibly the most important contributors to the production of oil worldwide, modeling fractured formations with rigorous treatments has eluded reservoir engineers in the past. To date, one of the most commonly used fractured reservoir models remains the one that was suggested by Warren and Root nearly four decades ago. In this paper, a new model for fractures embedded in a porous medium is proposed. The model considers the Navier-Stokes equation in the fracture (channel flow) while using the Brinkman equation for the porous medium. Unlike the previous approach, the proposed model does not require the assumption of orthogonality of the fractures (sugar cube assumption) nor does it impose incorrect boundary conditions for the interface between the fracture and the porous medium. Also, the transfer coefficient between the fracture and matrix interface does not need to be specified, unlike the cases for which Darcy's law is used. In order to demonstrate the usefulness of the approach, a two-dimensional model of a fractured formation is developed and numerical simulation runs conducted.

The proposed model is derived through a series of finite element modeling runs for various cases using the Navier-Stokes equation in the channel while maintaining the Brinkman equation in the porous medium. Various cases studied include different fracture orientations, fracture frequencies, and thermal and solutal constraints. The usefulness of the proposed model in modeling complex formations is discussed. Finally, a series of numerical runs also provided validity of the proposed model for the cases in which thermal and solutal effects are important. Such a study of double diffusive phenomena, coupled with forced convection, in the context of fractured formations has not been reported before.     

Effect of an Inserted Porous Layer Located at a Wall of a Parallel Plate Channel on Forced Convection Heat Transfer

A theoretical study is performed on heat and fluid flow in partially porous medium filled parallel plate channel. A uniform symmetrical heat flux is imposed onto the boundaries of the channel partially filled with porous medium. The dimensional forms of the governing equations are solved numerically for different permeability and effective thermal conductivity ratios. Then, the governing equations are made dimensionless and solved analytically. The results of two approaches are compared and an excellent agreement is observed, indicating correctness of the both solutions. An overall Nusselt number is defined based on overall thermal conductivity and difference between the average temperature of walls and mean temperature to compare heat transfer in different channels with different porous layer thickness, Darcy number, and thermal conductivity ratio. Moreover, individual Nusselt numbers for upper and lower walls are also defined and obtained. The obtained results show that the maximum overall Nusselt number is achieved for thermal conductivity ratio of 1. At specific values of Darcy number and thermal conductivity ratio, individual Nusselt numbers approach to infinity since the value of wall temperatures approaches to mean temperature.     

Method of Optimizing Maneuvers of an Orbital Transfer Vehicle in a Strong Central Field of Gravity

International Applied Mechanics - A method of constructing the optimal trajectories of interorbital flights of an orbital transfer vehicle with a low thrust propulsion system is described. Testing...