Cellular Automata (CA) for Simulations of Complex Geothermal Reservoirs

The exploitation of geothermal power is a renewable energy source with great potential in future. But the exploration and development of deep geothermal energy is connected with high cost and risk. These require a reliable functionality of geological heat exchanger. However the geothermal reservoirs are really complicated as phenomena and concrete downhole data are not completely discovered at present. In order to simulate them, complex modelings combined with different time scale are necessary. Recently, the cellular automata (CA) method is being developed and widely used for solving many complex problems in different fields. Here we introduced CA method combined with Navier-Stoke equation and heat transfer; the domains of reservoirs are initially discretized into many lattice cells. The different cell type and their physical properties (e.g. water cell, porous cell, etc.) are introduced. Thermodynamically correct computation and computing fluid flow in different formations are performed. The paper will give some computational results, showing the efficiency and accuracy of this method, in order to complete the phenomena of complex geothermal problem. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of uncompressible fluid flow through a porous media

Recently, a great interest has been focused for investigations about transport phenomena in disordered systems. One of the most treated topics is fluid flow through anisotropic materials due to the importance in many industrial processes like fluid flow in filters, membranes, walls, oil reservoirs, etc. In this work is described the formulation of a 2D mathematical model to simulate the fluid flow behavior through a porous media (PM) based on the solution of the continuity equation as a function of the Darcy’s law for a percolation system; which was reproduced using computational techniques reproduced using a random distribution of the porous media properties (porosity, permeability and saturation). The model displays the filling of a partially saturated porous media with a new injected fluid showing the non-defined advance front and dispersion of fluids phenomena.     

Joint Distributions for Interacting Fluid Queues

Motivated by recent traffic control models in ATM systems, we analyse three closely related systems of fluid queues, each consisting of two consecutive reservoirs, in which the first reservoir is fed by a two-state (on and off) Markov source. The first system is an ordinary two-node fluid tandem queue. Hence the output of the first reservoir forms the input to the second one. The second system is dual to the first one, in the sense that the second reservoir accumulates fluid when the first reservoir is empty, and releases fluid otherwise. In these models both reservoirs have infinite capacities. The third model is similar to the second one, however the second reservoir is now finite. Furthermore, a feedback mechanism is active, such that the rates at which the first reservoir fills or depletes depend on the state (empty or nonempty) of the second reservoir.The models are analysed by means of Markov processes and regenerative processes in combination with truncation, level crossing and other techniques. The extensive calculations were facilitated by the use of computer algebra. This approach leads to closed-form solutions to the steady-state joint distribution of the content of the two reservoirs in each of the models.     

放/吸热流体在两个充满多孔材料的竖直平行板之间作不稳定的自然对流

在一个由两块无限竖直平行板组成的管道中,充满着多孔的介质材料,使用Darcy模型(Brinkman模型的推广)的动量方程,连同能量方程,计算不可压缩、粘性、放/吸热流体在该管道中的不稳定自然对流,即Couette流动.流动是由于边界平板有不对称的加热,以及作加速运动所引起.选用合理的无量纲参数,对控制方程进行简化,通过Laplace变换进行解析求解,得到闭式的速度和温度分布曲线解,随后导出表面摩擦力和传热率.发现在竖直管道中的不同剖面,流体的流动及温度分布曲线随着时间而增加,且在运动平板附近更高.特别是,流体的速度和温度随着平板间距的增加而增加,但是,表面摩擦力和热传导率随着平板间距的增加而减小.     

Effects of thermal radiation on the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface

In the present investigation we have analyzed the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface. The effects of thermal radiation are carried out for two cases of heat transfer analysis known as (1) Prescribed exponential order surface temperature (PEST) and (2) Prescribed exponential order heat flux (PEHF). The highly nonlinear coupled partial differential equations of Jeffrey fluid flow along with the energy equation are simplified by using similarity transformation techniques based on boundary layer assumptions. The reduced similarity equations are then solved analytically by the homotopy analysis method (HAM). The convergence of the HAM series solution is obtained by plotting (^h/_2p)\hbar-curves for velocity and temperature. The effects of physical parameters on the velocity and temperature profiles are examined by plotting graphs.     

Finite difference simulation of elastic wave propagation through 3D heterogeneous multiscale media based on locally refined grids

To simulate the interaction of seismic waves with microheterogeneities (like cavernous/fractured reservoirs), a finite difference technique based on grids locally refined in time and space is used. These grids are used because the scales of heterogeneities in the reference medium and in the reservoir are different. Parallel computations based on domain decomposition of the target area into elementary subdomains in both the reference medium (a coarse grid) and the reservoir (a fine grid) are performed. Each subdomain is assigned to a specific processor unit, which forms two groups: one for the reference medium, and the other for the reservoir. The data exchange between the groups within a processor unit is performed by non-blocking iSend/iReceive MPI commands. The data exchange between the two groups is performed simultaneously with coupling the coarse and a fine grids, and is controlled by a specially chosen processor unit. The results of a numerical simulation for a realistic model of fracture corridors are presented and discussed.     

Analysis of diffusion process in fractured reservoirs using fractional derivative approach

The fractal geometry is used to model of a naturally fractured reservoir and the concept of fractional derivative is applied to the diffusion equation to incorporate the history of fluid flow in naturally fractured reservoirs. The resulting fractally fractional diffusion (FFD) equation is solved analytically in the Laplace space for three outer boundary conditions. The analytical solutions are used to analyze the response of a naturally fractured reservoir considering the anomalous behavior of oil production. Several synthetic examples are provided to illustrate the methodology proposed in this work and to explain the diffusion process in fractally fractured systems.     

Analytical solutions for movement of cold water thermal front in a heterogeneous geothermal reservoir

In the present study an analytical model has been presented to describe the transient temperature distribution and advancement of the thermal front generated due to the reinjection of heat depleted water in a heterogeneous geothermal reservoir. One dimensional heat transport equation in porous media with advection and longitudinal heat conduction has been solved analytically using Laplace transform technique in a semi infinite medium. The heterogeneity of the porous medium is expressed by the spatial variation of the flow velocity and the longitudinal effective thermal conductivity of the medium. A simpler solution is also derived afterwards neglecting the longitudinal conduction depending on the situation where the contribution to the transient heat transport phenomenon in the porous media is negligible. Solution for a homogeneous aquifer with constant values of the rock and fluid parameters is also derived with an aim to compare the results with that of the heterogeneous one. The effect of some of the parameters involved, on the transient heat transport phenomenon is assessed by observing the variation of the results with different magnitudes of those parameters. Results prove the heterogeneity of the medium, the flow velocity and the longitudinal conductivity to have great influence and porosity to have negligible effect on the transient temperature distribution.     

Complex analytical solutions for flow in hydraulically fractured hydrocarbon reservoirs with and without natural fractures

Reservoir drainage towards producer wells in a hydraulically and naturally fractured reservoir is visualized by using an analytical streamline simulator that plots streamlines, time-of-flight contours and drainage contours based on complex potentials. A new analytical expression is derived to model the flow through natural fractures with enhanced hydraulic conductivity. Synthetic examples show that in an otherwise homogeneous reservoir even a small number of natural fractures may severely affect streamline patterns and distort the drainage contours. Multiple parallel natural fractures result in a drainage region that is narrower in the direction normal to the natural fractures while the drainage reach is larger in the natural fracture direction. Reservoirs with numerous natural fractures are shown to be characterized by more tortuous drainage patterns than reservoirs without natural fractures. Finally, the analytical flow model for naturally fractured reservoirs is applied to a natural analog of flow into hydraulic fractures. The tendency of the injected fluid to stay confined to the fracture network as opposed to matrix flow is entirely controlled by the hydraulic conductivity contrast between the fracture network and the matrix.     

A comparative study of the steady state and the transient liquid crystal technique for determining the heat transfer coefficients in a bend

For the investigation of internal and external heat transfer thermochromic liquid crystals (TLC's) have become a very commonly used tool. Today, mainly two different experimental methods with TLC's are in use: The steady state TLC method measuring the wall temperature according to a constant heat flux produced by a heater foil and the transient TLC method measuring the time response of the wall temperature due to a sudden change of the fluid temperature. In the present paper the two different measurement techniques are compared for a very complex 3D flow inside a 180°‐turn of a two‐pass cooling system. From theoretical considerations, the general difference in the heat transfer because of the different wall boundary conditions is shown.     

Spectral Lanczos decomposition method for solving single-phase fluid flow porous media

A three-dimensional well model (r ? θ ? z) for the simulation of single-phase fluid flow in porous media is developed. Rather than directly solving the 3-D parabolic PDE (partial differential equation) for fluid flow, the PDE is transformed to a linear operator problem that is defined as u = f( A ) σ , where A is a real symmetric square matrix and σ is a vector. The linear operator problem is solved by using the spectral Lanczos decomposition method. This formulation gives continuous solutions in time. A 7-point finite difference scheme is used for the spatial discretization. The model is useful for well testing problems as well as for the simulation of the wireline formation tester tool behavior in heterogeneous reservoirs. The linear operator formulation also permits us to obtain solutions in the Laplace domain, where the wellbore storage and skin can be incorporated analytically. The infinite-conductivity (uniform pressure) wellbore condition is preserved when mixed boundary conditions, such as partial penetration, occur. The numerical solutions are compared with the analytical solutions for fully and partially penetrated wells in a homogeneous reservoir. © 1994 John Wiley & Sons, Inc.     

Generalized analytical model of Transient linear flow in heterogeneous fractured liquid-rich tight reservoirs with non-static properties

The industry is increasingly reliant on rate-transient analysis (RTA) to extract valuable information about the reservoir and hydraulic fractures. However, the application of current, commercially-available RTA models can lead to incorrect estimates of reservoir/fracture properties, potentially causing costly mistakes to be made in capital planning and reserve estimation. The root cause of these errors is that currently-available analytical solutions used in RTA models largely ignore reservoir heterogeneities, and assume static reservoir properties.In this work, a new transient linear flow is rigorously modeled in unconventional reservoirs with (1) pressure-dependent rock and fluid properties and (2) both continuous and discontinuous (heterogeneous) porosity and permeability. To achieve this, new transformations of pseudo-pressure, pseudo-time and pseudo-distance are first introduced to reduce the temporal and spatial non-linear diffusivity equation to that with approximately constant coefficients. Both a Laplace-domain solution and approximate analytical solution to the diffusivity equation are verified against a series of fine-grid numerical simulations for the assumption of fractal-based reservoir heterogeneity (over a wide range of stress-dependent rock and fluid properties). The results indicate that reservoir heterogeneity can result in nonlinear square-root-of-time plots. Further, rock and fluid pressure-dependencies act to decrease the slope of the square-root-of-time plot and affect reservoir/fracture property evaluations.Three liquid-rich shale (LRS) field examples in North America are analyzed to demonstrate the practical applicability of the new RTA models. Additional value of new RTA models over the sophisticated numerical simulation is to provide us an improved backforward-analysis workflow that can be used to quantify both effective fracture half-length and non-uniform permeability distribution around the fractures.The major contribution of this work is the introduction of a new analytical model for evaluating the transient linear flow period for the cases of arbitrary reservoir heterogeneity and non-static reservoir properties. This new approach is particularly useful for evaluating the effectiveness of hydraulic fracturing operations by extracting the spatial variability of reservoir quality within the stimulated reservoir volume (SRV).     

Modelado numérico del proceso de soldadura FSW incorporando una técnica de estimación de parámetros

Numerical models of heat transfer and fluid flow used in the simulation of the friction-stir welding (FSW) process have contributed to the understanding of the process. However, there are some input model parameters that cannot be easily determined from fundamental principles or the welding conditions. As a result, the model predictions are not always in agreement with experimental results. In this work, the Levenberg-Marquardt (LM) method is used in order to perform a non-linear estimation of the unknown parameters present in the heat transfer and fluid flow models, by adjusting the temperatures results obtained with the models to temperature experimental measurements. These models are implemented in a general-purpose software that uses a numerical formulation developed from the finite element method (FEM). The unknown parameters are: the friction coefficient and the amount of adhesion of material to the surface of the tool, the heat transfer coefficient on the bottom surface and the amount of viscous dissipation converted into heat. The obtained results show an improvement in the numerical model predictions from the incorporation of parameter estimation techniques.     

Hitting probabilities and hitting times for stochastic fluid flows

Recently there has been considerable interest in Markovian stochastic fluid flow models. A number of authors have used different methods to calculate quantities of interest. In this paper, we consider a fluid flow model, formulated so that time is preserved, and derive expressions for return probabilities to the initial level, the Laplace–Stieltjes transforms (for arguments with nonnegative real part only) and moments of the time taken to return to the initial level, excursion probabilities to high/low levels, and the Laplace–Stieltjes transforms of sojourn times in specified sets. An important feature of our results is their physical interpretation within the stochastic fluid flow environment, which is given. This allows for further implementation of our expressions in the calculation of other quantities of interest.Novel aspects of our treatment include the calculation of probability densities with respect to level and an argument under which we condition on the infimum of the levels at which a “down–up period” occurs.Significantly, these results are achieved with techniques applied directly within the fluid flow model, without the need for discretization or transformation to other equivalent models.     

Generalized model and optimum performance of an irreversible thermal Brownian microscopic heat pump

A generalized model of an irreversible thermal Brownian microscopic heat pump is established in this paper. It is composed of Brownian particles which are moving in a periodic sawtooth potential with external forces and contacting with alternating hot and cold reservoirs along the space coordinate. The generalized irreversible Brownian heat pump model incorporates heat flows driven by both the potential and kinetic energies of the particles as well as the heat leakage between the hot and cold reservoirs. This paper derives the expressions for heating load, power input and coefficient of performance (COP) of the Brownian heat pump. The optimum performance of the generalized heat pump model is analyzed by using the theory of finite time thermodynamics (FTT). Effects of the design parameters, i.e., the external force, the heat leakage coefficient, barrier height of the potential, asymmetry of the sawtooth potential and heat reservoir temperature ratio on the performance of the Brownian heat pump are discussed in detail. The performance of the Brownian heat pump depends strictly on the design parameters. Through the proper choice of these parameters, the Brownian heat pump can operate in the optimal regimes. The optimum COP performance and the fundamental optimal relations between COP and heating load are studied by detailed numerical examples. It is shown that due to the heat leakage between the heat reservoirs and heat flow via the change of kinetic energy of the particles, both the heating load and COP performances of the Brownian heat pump will decrease. The effective ranges of the external force and barrier height of the potential in which the Brownian motor system can operate as a heat pump are further determined.     

Optimum work in real systems with a class of finite thermal capacity reservoirs

The optimum work in real systems with two finite thermal capacity reservoirs is determined. It is obtained by using optimal control theory. It is shown that the temperature of external working fluid changes exponentially with respect to flow velocity and process duration. The analysis proves that the optimum work is different for heat engine mode and heat pump mode. The recently obtained results are compared with those obtained previously. The models and results in this paper provide an approach to improve calculations of energy limits in real systems.     

Application of Smoothed Particle Hydrodynamics for modelling gated spillway flows

Computational models of spillways are important for evaluating and improving dam safety, optimising spillway design and updating operating conditions. Traditionally, scaled down physical models have been used for validation and to collect hydraulic data. Computational fluid dynamics (CFD) models however provide advantages in time, cost and resource reduction. CFD models also provide greater efficiency when evaluating a range of spillway designs or operating conditions. Within the present literature, most studies of computational spillway models utilise a mesh-based method. In this work we use the particle based method of Smoothed Particle Hydrodynamics (SPH) to model weir flow through a four bay, gated, spillway system. Advantages of SPH for such modelling include automatic representation of the free surface flow behaviour due to the Lagrangian nature of the method, and the ability to incorporate complex and dynamic boundary objects such as gate structures or debris. To validate the SPH model, the reservoir water depth simulated is compared with a related physical study. The effect of SPH resolution on the predicted water depth is evaluated. The change in reservoir water level with discharge rates for weir flow conditions is also investigated, with the difference in simulated and experimental water depths found to range from 0.16% to 11.48%. These results are the first quantitative validation of the SPH method to capture spillway flow in three dimensions. The agreement achieved demonstrates the capability of the SPH method for modelling spillway flows.     

Effects of variable viscosity on non-Darcy MHD free convection along a non-isothermal vertical surface in a thermally stratified porous medium

An analysis is performed for non-Darcy free convection flow of an electrically conducting fluid over an impermeable vertical plate embedded in a thermally stratified, fluid saturated porous medium for the case of power-law surface temperature. The present work examines the effects of non-Darcian flow phenomena, variable viscosity, Hartmann–Darcy number and thermal stratification on free convective transport and demonstrates the variation in heat transfer prediction based on three different flow models. The wall effect on porosity variation is approximated by an exponential function. The effects of thermal dispersion and variable stagnant thermal conductivity are taken into consideration in the energy equation. The resulting non-similar system of equations is solved using a finite difference method. Results are presented for velocity, temperature profiles and local Nusselt number for representative values of different controlling parameters.     

Multidomain mixed variational analysis of transport flow through elastoviscoplastic porous media

Multidomain mixed nonlinear transport and flow phenomena through elastoviscoplastic porous media is variationally analyzed. Mixed variational formulations of the poro-mechanical system are established via composition duality methods, determining solvability results on the basis of duality principles. The conformation of the coupled physical system corresponds to constrained transport processes driven by a compressible Darcian flow, in a quasistatic elastoviscoplastic deformable subsurface porous media, modeled variationally by primal evolution mixed transport and consolidation, and dual evolution mixed flow and quasistatic deformation. For parallel computing, non-overlapping multidomain decomposition methods based on variational macro-hybridization, are presented and discussed, providing a natural multi-physics approach for the coupled transport flow and deformation system. For computational realizations, internal variational macro-hybrid mixed semi-discrete approximations are given, as well as primal and dual fully discrete semi-implicit time marching schemes. Furthermore, the corresponding coupled transport-flow-deformation system is concluded and analyzed, proposing natural resolution coupling techniques.     

Mathematical simulation of oil reservoir properties

The study and computational representation of porous media properties are very important for many industries where problems of fluid flow, percolation phenomena and liquid movement and stagnation are involved, for example, in building constructions, ore processing, chemical industries, mining, corrosion sciences, etc. Nevertheless, these kinds of processes present a noneasy behavior to be predicted and mathematical models must include statistical analysis, fractal and/or stochastic procedures to do it. This work shows the characterization of sandstone berea core samples which can be found as a porous media (PM) in natural oil reservoirs, rock formations, etc. and the development of a mathematical algorithm for simulating the anisotropic characteristics of a PM based on a stochastic distribution of some of their most important properties like porosity, permeability, pressure and saturation. Finally a stochastic process is used again to simulated the topography of an oil reservoir.