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This section discusses the impact of vertical variations in permeability and the effect of gravity on simple 2D reservoir situations in which the areal effects are ignored. Gravity effects always are present because for any potential waterflood project, oil always is less dense than water, even more so after the gas is included that is dissolved in the oil at reservoir conditions. The discussion below does not include the Pc effects on vertical saturation distributions. Through countercurrent imbibition, Pc effects help to counteract nonequilibrium water/oil saturation distributions. The mathematics of including Pc effects makes the problems too complicated for inclusion here.
The motivation for high-performance computing in reservoir simulation has always existed. From the earliest simulation models, computing resources have been severely taxed simply because the level of complexity desired by the engineer almost always exceeded the speed and memory of the hardware. The high-speed vector processors such as the Cray of the late 1970s and early 1980s led to orders-of-magnitude improvement in speed of computation and led to production models of several hundred thousand cells. The relief brought by these models, unfortunately, was short-lived. The desire for increased physics of compositional modeling and the introduction of geostatistically/structurally based geological models led to increases in computational complexity even beyond the large-scale models of the vector processors. Tens of millions of cells with complete reservoir parameters now became available for use by the engineer.
A Monte Carlo model is, in principle, just a worksheet in which some cells contain probability distributions rather than values. Thus, one can build a Monte Carlo model by converting a deterministic worksheet with the help of commercial add-in software. Practitioners, however, soon find that some of their deterministic models were constructed in a way that makes this transition difficult. Redundancy, hidden formulas, and contorted logic are common features of deterministic models that encumber the resulting Monte Carlo model. Likewise, presentation of results from probabilistic analysis might seem no different from any other engineering presentation (problem statement, summary and conclusions, key results, method, and details).
Just as there are shortcomings of deterministic models that can be avoided with probabilistic models, the latter have their associated pitfalls as well. Adding uncertainty, by replacing single estimate inputs with probability distributions, requires the user to exercise caution on several fronts. Without going into exhaustive detail we offer a couple of illustrations. First, the probabilistic model is more complicated. It demands more documentation and more attention to logical structure.
For the effects of gravity segregation to be important, however, the well spacing may need to be prohibitively large or the producing rate may need to be prohibitively low. In such reservoirs, the vertical permeability is not high enough to permit much gravity segregation. The likely role of gravity segregation can be measured in terms of a gravity number, Ng. Ng is defined as the ratio of the time it takes a fluid to move from the drainage radius to the wellbore to the time it takes a fluid to move from the bottom of the reservoir to the top.
If least-squares linear regression is used to compute N in Step 5, an equation analogous to Eq. 17 is used (where Eow is substituted for Eowf). This solution method is iterative because the material-balance error must be minimized. This calculation is carried out with a trial-and-error method or a minimization algorithm. Least-squares linear regression and minimization algorithms have become standard features in commercial spreadsheets.
Reservoir simulation is a widely used tool for making decisions on the development of new fields, the location of infill wells, and the implementation of enhanced recovery projects. It is the focal point of an integrated effort of geosciences, petrophysics, reservoir, production and facilities engineering, computer science, and economics. Geoscientists using seismic, well-log, outcrop analog data and mathematical models are able to develop geological models containing millions of cells. These models characterize complex geological features including faults, pinchouts, shales, and channels. Simulation of the reservoir at the fine geologic scale, however, is usually not undertaken except in limited cases.
Streamline simulation provides an alternative to cell-based grid techniques in reservoir simulation. Streamlines represent a snapshot of the instantaneous flow field and thereby produce data such as drainage/irrigation regions associated with producing/injecting wells and flow rate allocation between injector/producer pairs that are not easily determined by other simulation techniques. Streamline-based flow simulation differentiates itself from cell-based simulation techniques such as finite-differences and finite-elements in that phase saturations and components are transported along a flow-based grid defined by streamlines (or streamtubes) rather than moved from cell-to-cell. This difference allows streamlines to be extremely efficient in solving large, heterogeneous models if key assumptions in the formulation are met by the physical system being simulated (see below). Specifically, large relates to the number of active grid cells.
The linear equation solver is an important component in a reservoir simulator. It is used in the Newton step to solve the discretized nonlinear partial differential equations. These equations describe mass balances on the individual components treated in the model. For nonisothermal problems, an energy balance is added to the system. The matrix problem involves solving Ax b, where A is typically a large sparse matrix, b is the right-side vector, and x is the vector of unknowns.
The aim of gridding in reservoir simulation is to turn the geological model of the field into a discrete system on which the fluid flow equations can be solved. The basic structure of an oil reservoir is a set of geological horizons representing bedding planes. The reservoir may contain faults, at which the strata are displaced. It is usually possible to identify many more layers in the geological model than it is practical to include in reservoir flow simulation, so some upscaling of rock properties will normally be carried out. Even after this process, the geology to be represented is rarely homogeneous at the scale of the simulation grid.