Reservoir Simulation (includes associated papers 21606 and 21620 )

Abstract

"... the precision with which variations In reservoir properties can be modeled Is determined by the number of blocks In the model." Introduction In the past 30 years, reservoir simulation has evolved from a research area into one of the most flexible and widely used tools in reservoir engineering. Use of reservoir simulation has grown because of its ability to predict the future performance of oil and gas reservoirs over a wide range of operating conditions. Reservoir simulators use numerical methods and high-speed computers to model multidimensional fluid flow in reservoir rock. Reliable simulators and adequate computing capacity are available to most reservoir engineers, so simulation is usually practical for all reservoir sizes and all types of reservoir performance studies. Although the use of simulation frequently is optional, it may be the only reliable way to predict the performance of a large, complex reservoir, especially if such external considerations as government regulations influence the production schedule. Even for small reservoirs where simple calculations or extrapolations may be adequate, simulation is often faster, cheaper, and more reliable than alternative methods for predicting performance. Modeling Concepts A reservoir simulator models a reservoir as if it were divided into a number of individual blocks (gridblocks). Each block corresponds to a designated location in the reservoir and is assigned properties-porosity, permeability, relative permeability, etc. -believed to be representative of the reservoir at that location. In the simulator, fluids can flow between neighboring blocks at a rate determined by pressure differences between blocks and flow properties assigned to the interfaces between blocks. In essence, the mathematical problem is reduced to a calculation of flow between adjacent blocks. For every block-to-block interface, a set of equations must be solved to calculate the flow of all mobile phases. The equations generally incorporate Darcy's law and the concept of material balance and contain terms describing the permeability "between" blocks, fluid mobilities (relative permeability and viscosity), and rock and fluid compressibilities. Fig. 1 illustrates the most common types of models used in simulation. Models range in complexity from a single block, useful only for classical material-balance calculations, to fully 3D models capable of modeling all major factors that influence reservoir performance. Each gridblock in a model has only one set of properties; there is no variation in any property within a block. For example, phase saturations in a model block will be volumetric averages of the saturations in that part of the reservoir represented by the block. In this respect, a model block can be visualized as a well-stiffed tank (i.e., its contents are homogeneous)connected to adjacent tanks with pipes whose flow capacities are determined by reservoir flow properties. This visualization, although simplistic, demonstrates that the precision with which variations in reservoir properties can be modeled is determined by the number of blocks in the model. A simulator also divides the life of a reservoir into discrete increments. Changes in a reservoir (pressure, saturation, etc.) are computed over each of many time increments, or timesteps. Conditions are defined only at the beginning and end of each timestep; nothing is defined at any intermediate time within a time interval. The accuracy with which reservoir behavior can be calculated generally will be influenced by the length of the timesteps as well as the number of gridblocks. The preceding discussion implies that for any size gridblock and any length of timestep, there will always be abrupt changes in reservoir conditions from one block to the next and from one timestep to the next. Fig. 2 illustrates this point. At the top of Fig. 2 are plan views of a hypothetical two-well reservoir being waterflooded and a four-gridblock simulator model of the reservoir. The two plots show-water-saturation distribution at a given time in both the reservoir and the model. JPT P. 692⁁

Analysis of Some Difference Schemes Used in Numerical Simulation of Production/Injection Wells

“…We rewrite these two schemes from [4], [9], [8] below in a form that is more convenient for analysis. As one of the outcomes of the analysis, we indicate that the approximations proposed in [2]- [10] and included in the contemporary handbooks [11]- [15] have nonremovablle asymptotical error or, in other words, they are not asymptotically exact.…”

“…Similarly, the modified "zero-flux statement" for the h × h squares centered at p F , F = (−1, 1), (−1, −1), (1, −1), become Λ 2 u F = 0. Finally, if for every (i, j) / ∈ L ∪ F, we put Λ 2 u i,j = Λu i,j , then Λ 2 u i,j = 0 for all (i, j), and we arrived at (11).…”

“…The results obtained in [7] are similar to those of [2], except that in [7] no restriction was placed on the position of the well within a grid block. This approach was summarized in [11]- [15] In [4], [8], and [9], the differential problem (1) - (3) is approximated with an 'adjustment' method in two different ways which yield varying accuracies. The coefficients of the difference equations which approximate the differential problem are constructed (adjusted), taking into account the singularity of the solution in the neighborhood of the well.…”

Redes Neurais Aplicadas À Construção De Aproximadores Para Simulação Integrada Entre Reservatório E Sistema De Produção

“…Para se resolver esse conjunto de equações, o simulador utiliza métodos de diferenças finitas, que é um método de resolução de equações diferenciais que se baseia na aproximação de derivadas por diferenças finitas (Mattax, 1990). A modelagem de um sistema de reservatório envolve inúmeras variáveis e inúmeras equações para descrever todo o escoamento do fluido dentro do reservatório.…”

“…Alguns dos modelos de fluxo mais utilizados em simulações de reservatórios são: o modelo black-oil, o modelo composicional, os modelos térmicos e os modelos de fluxo miscível (Mattax et al, 1990).…”