This paper describes a new technique to decrease the computational times of thermal simulations. Effectively, thermal processes are based on the displacement of a thermal front (combustion front, steam chamber interface), around which most fluid flows will take place. Thus, we propose a dynamic gridding approach, to keep a fine scale representation around the thermal front, and a coarser grid away from the front, thus leading to cheaper computations. We will first describe the principles of this dynamic gridding. Simulations will start with an original fine grid, but will reamalgamate its cells, while keeping some regions (for example around wells) always finely gridded. The gridding will then identify the moving front through large gradients of specific properties (temperatures, fluid saturations and compositions). In the front vicinity, it will de-amalgamate the originally amalgamated cells, and later on re-amalgamate them once the front has passed. Amalgamated cells are assigned up-scaled properties, this upscaling being based upon classical averaging techniques. We will illustrate this dynamic gridding technique with simulation examples, as it has been successfully implemented in a thermal simulator, STARS, a product of Computer Modelling Group Ltd (CMG). Using examples on combustion and SAGD simulations, we will show that it can divide the CPU time of thermal simulations by a factor of 2 to 3, without loss of accuracy. Introduction Reservoir flow must be represented accurately when modelling processes such as combustion and SAGD (steam-assisted gravity drainage) in a reservoir simulator. These thermal processes involve convective, diffusive and dispersive flows of fluids and energy, which lead to the formation of fluid banks and fronts moving in the reservoir. Some of these fronts represent interfaces between mobilized oil, which is hot and has had its viscosity reduced, and the more viscous oils which are as yet untouched by heat. Other fronts occur between phases, such as where a leading edge of hot combustion gas moves into an uncontacted oil. These interfaces are thin when compared to the typical cell sizes used to model EOR processes in a simulator, so there will always be problems in properly representing important fluid physics near interfaces. For instance, the choice made for upscaling could depend on the fronts being generated by a process and where they are positioned in the upscaled reservoir cell, while the use of fine scale computational cells throughout the reservoir would be prohibitively expensive. A technique has been presented(1) to address these problems. It suggests using dynamic grid refinement and amalgamation to choose an appropriate cell size near important regions, while using larger cells elsewhere. The ongoing simulation is reviewed periodically and the cells are re-sized depending on the current fluid distribution. The technique is applicable to simulators using sparse matrix solvers and only involves regenerating pointers and properties at selected times during the simulation. Dynamic grid refinement and amalgamation can result in obtaining a several fold decrease in run time while leaving the results unchanged. User-specified thresholds are used to control when to do grid amalgamation or de-amalgamation. The methods described in this paper will be based on differences in property values between amalgamated cells and their neighbours, or differences among values in a finely divided region. The properties chosen will be designed to find fronts, and include saturations and various compositions. Temperature related thresholds will also be used to give an "early warning" for the leading edge of a front. Pressure related differences are not considered, as different pressure levels do not cause difficulties unless they result in front movement, which would be trapped by the thresholds just described The thresholds should be relatively small so that a buffer region of finer cells is maintained around regions of high activity. This choice sacrifices some speed, but maintains accuracy. Note that the simulator requires some kind of efficient adaptive (or fully) implicit formulation to make good use of these techniques, as smaller cells could have high throughputs that require implicitness without resort to small time steps.
Aspects of the cold production process have been studied mechanistically employing a unified framework and coupled geomechanical-fluid flow simulation models. The unified framework is a generalization of that proposed by Papamichos et al1 such that stress concentration causes mechanical weakening, and fluid flow causes erosional mobilization of sand particles. Porosity change is the primary coupling parameter whereby the soil mechanical and strength properties are altered as a continuum damage process, while erosional generation of mobilized sand particles causes fluid porosity increase and oil-sand slurry flow. These ideas are implemented in coupled simulation models using different coupling strategies and different physical assumptions to study various aspects of the cold production process. The first coupled model is a stand-alone finite element model for two-dimensional radial plane strain and single phase fluid flow. Here the geomechanics and fluid flow aspects are fully coupled such that the time dependent (undrained to drained) mechanical stress transfer is strictly preserved. The second coupled model allows a more complete fluid model (including gas ex-solution and water influx), as well as a time dependent sand mobilization mechanism, but at the expense of a weaker (time explicit) geomechanical-fluid flow coupling, through Settari's concept of volume (effective porosity) coupling21. These models are applied to simulations of cold production at the laboratory and field scale. The laboratory scale models are based on the CT experiments of Tremblay et al2, which emphasize the erosional aspects of the cold production process, and its coupling to gas exsolution. The field scale models are based on single well production observations in Alberta bitumen reservoirs, such as Yeung et al44 and include sensitivities to water influx. Additional geomechanical aspects which may be important in deeper reservoirs such as those in Venezuela are also illustrated. Finally implications and recommendations for future simulation modelling at the field scale are provided. Introduction There has been much interest recently in the cold production of heavy oil3,4,5,6,7. This primary depletion process with simultaneous production of sand has been applied with commercial success in the Lloydminster area and elsewhere in Western Canada. Here oil production is critically controlled by the reservoir enhancement due to sand production, through the creation of cavities and wormholes. Extension of this production method to other heavy oil containing areas in the world (e.g. Venezuela) is also contemplated. In other parts of the world, sand production is viewed as a major problem, especially in unconsolidated reservoirs, and often expensive sand control methods have been implemented to limit sand production. A survey in Elf Aquitaine indicates that the issue of sand production has been raised for more than 70% of the active wells in the North Sea. Similar problems have also been reported for producing companies in the Gulf of Mexico, particularly for horizontal wells. In either situation, the challenge is to manage sand production to obtain optimal oil production. Sand production is a direct result of aggressive production, as it is often associated with high drawdown and high production. The understanding of the massive sand production and associated enhanced oil production is important to the field operations. Consequently, a practical tool for predicting the onset of sand production and for evaluating the post-sanding performance is highly desirable. Furthermore, a mechanistic understanding of process parameters can be achieved through the use of tuned mathematical models.
Summary The nature of non-Darcy flow in porous media is analyzed by means of the volumetric averaging approach for a medium modelled by diverging converging capillaries. To clarify the mechanism responsible for the nonlinearity, a physical explanation is deduced for the dispersion term in the averaged momentum equation. With the present periodical model, the numerically obtained microscopic flow fields, in association with the macroscopic coefficients calculated by the average momentum balance, indicated quantitatively that the microscopic inertial phenomena, which lead to distorted pore velocity and pressure fields, is the fundamental reason for the onset of non-Darcy effects as filtration velocity increases. Introduction On the macroscopic scale, fluid flow in a porous medium at low velocity (specific discharge) is generally described by Darcy's law which presents a linear relationship between the driving force, and the filtration velocity, U. However, as the filtration velocity is raised beyond a certain value, numerous experimental observations have confirmed that Darcy's law should be replaced by another time-honored empirical formula, the Forchheimer equation, (1) to account for the nonlinear effects for the medium considered. In this equation, k denotes the Darcy's law permeability, is an experimentally derived parameter called the inertial coefficient, and and represent the density and the dynamic viscosity of the fluid, respectively. Both k and are thought to be material constants in the range of validity of Eq. (1). Proceeded from a consideration that the permeability may alternatively be treated as being velocity dependent, a new form of Forchheimer equation has been proposed: (2) where k = is the velocity dependent permeability and Fo = is called the Forchheimer number which replaces the Reynolds number as a dimensionless criterion to indicate when microscopic effects lead to significant macroscopic inertial effects. It has long been a common interest for many researchers to clarify the physical reason for the onset of the nonlinearity. Early descriptions attributed the nonlinearity to the occurrence of turbulence. However, experiments have indicated that when the macroscopic velocity gradually increases, the nonlinear phenomena appear much before the onset of real turbulence in porous media flow. Thus, it can be concluded firmly that the deviations from Darcy's law are not initiated by changes of flow regimes. A diversity of opinions as to why nonlinearity at high flow rates occurs still exists. In their paper, Hassanizadeh and Gray performed an order of magnitude analysis for the averaged momontum equation and concluded that the microscopic viscous force is the source for the onset of nonlinearity.
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