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Techniques described in this page are classic methods for describing immiscible displacement assuming equilibrium between injected gas and displaced oil phases while accounting for differing physical characteristics of the fluids, the effects of reservoir heterogeneities, and injection/production well configurations. Included are modifications to typical displacement equations, evaluating sweep efficiency, and calculating performance. In simple calculations, the reservoir is treated in terms of average properties for volume of rock, and production performance is described on the basis of an average well. Black-oil-type reservoir simulation models use essentially these same techniques but, by means of 1D, 2D, or 3D cell arrays, account for areal and vertical variations in rock and fluid properties, well-to-well gravity effects, and individual well characteristics. More complex compositional models account for nonequilibrium conditions between injected and displaced fluids and can be used to describe individual well streams in terms of the compositions of the produced fluids.
Geostatistical reservoir-modeling technologies depart from traditional deterministic modeling methods through consideration of spatial statistics and uncertainties. Geostatistical models typically examine closely the numerous solutions that satisfy the constraints imposed by the data. Using these tools, we can assess the uncertainty in the models, the unknown that inevitably results from never having enough data. Reservoir characterization encompasses all techniques and methods that improve our understanding of the geologic, geochemical, and petrophysical controls of fluid flow. It is a continuous process that begins with the field discovery and all the way through to the last phases of production and abandonment.
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.
This article discusses the basic concepts of single-component or constant-composition, single phase fluid flow in homogeneous petroleum reservoirs, which include flow equations for unsteady-state, pseudosteady-state, and steady-state flow of fluids. Various flow geometries are treated, including radial, linear, and spherical flow. Virtually no important applications of fluid flow in permeable media involve single component, single phase 1D, radial or spherical flow in homogeneous systems (multiple phases are almost always involved, which also leads to multidimensional requirements). The applications given in this Chapter are based on a model that includes many simplifying assumptions about the well and reservoir, and are interesting mainly only from a historical perspective See "Reservoir Simulation" for proper treatment of multi-component, multiphase, multidimensional flow in heterogeneous porous media. The simplifying assumptions are introduced here as needed to combine the law of conservation of mass, Darcy's law, and equations of state to obtain closed-form solutions for simple cases. Consider radial flow toward a well in a circular reservoir. Combining the law of conservation of mass and Darcy's law for the isothermal flow of fluids of small and constant compressibility yields the radial diffusivity equation,  In the derivation of this equation, it is assumed that compressibility of the total system, ct, is small and independent of pressure; permeability, k, is constant and isotropic; viscosity, μ, is independent of pressure; porosity, ϕ, is constant; and that certain terms in the basic differential equation (involving pressure gradients squared) are negligible.
The standard fluid flow equations require modification when the reservoir content is gas or multiphase flow is occurring. This article discusses those modifications and the equations to use for gas or multiphase reservoirs. This form of the diffusivity equation is linear, which makes solutions (such as theEi-function solution) much easier to find and which allows us to use superposition in time and space to develop solutions for complex flow geometries and for variable rate histories from simple, single-well solutions. Other forms of the equation for flow of gases must be developed because the equation of state for a slightly compressible liquid will not be applicable. This equation can be partially linearized by introducing the pseudopressure transformation,  Eq. 4 has the same form as the diffusivity equation for slightly compressible liquids, with pressure replaced by pseudopressure, pp. However, this equation is nonlinear because the product μct is a strong function of pressure.
The three primary functions of a drilling fluid depend on the flow of drilling fluids and the pressures associated with that flow. These functions includes: The transport of cuttings out of the wellbore, prevention of fluid influx, and the maintenance of wellbore stability. If the wellbore pressure exceeds the fracture pressure, fluids will be lost to the formation. If the wellbore pressure falls below the pore pressure, fluids will flow into the wellbore, perhaps causing a blowout. It is clear that accurate wellbore pressure prediction is necessary. To properly engineer a drilling fluid system, it is necessary to be able to predict pressures and flows of fluids in the wellbore.
To extract fluid types or saturations from seismic, crosswell, or borehole sonic data, we need a procedure to model fluid effects on rock velocity and density. Numerous techniques have been developed. However, Gassmann's equations are by far the most widely used relations to calculate seismic velocity changes because of different fluid saturations in reservoirs. The importance of this grows as seismic data are increasingly used for reservoir monitoring. Gassmann's formulation is straightforward, and the simple input parameters typically can be directly measured from logs or assumed based on rock type.
Elastic waves are comprised of compressional (or P-waves) and shear (or S-waves). In compressional waves, the particle motion is in the direction of propagation. In shear waves, the particle motion is perpendicular to the direction of propagation. Understanding the velocity of these waves provide valuable information about the rocks and fluids through which they propagate. Stress strain relationships in rocks considered only the static elastic deformation of materials.
Both the computation of classical statistical measures (e.g., mean, mode, median, variance, standard deviation, and skewness), and graphic data representation (e.g., histograms and scatter plots) commonly are used to understand the nature of data sets in a scientific investigation--including a reservoir study. A distinguishing characteristic of earth-science data sets (e.g., for petroleum reservoirs), though, is that they contain spatial information, which classical statistical descriptive methods cannot adequately describe. Spatial aspects of the data sets, such as the degree of continuity--or conversely, heterogeneity--and directionality are very important in developing a reservoir model. Analysis of spatially rich data is within the domain of geostatistics (spatial statistics), but a foundation in classical statistics and probability is prerequisite to understanding geostatistical concepts. Sampling also has proved invaluable in thousands of studies, but it, too, can lead to statistical insufficiencies and biases.
Many important applications of fluid flow in permeable media involve 1D, radial flow. These applications are based on a model that includes many simplifying assumptions about the well and reservoir. These assumptions are introduced as needed to combine the law of conservation of mass, Darcy's law, and equations of state to achieve our objectives. Consider radial flow toward a well in a circular reservoir. Combining the law of conservation of mass and Darcy's law for the isothermal flow of fluids of small and constant compressibility yields the radial diffusivity equation,  ....................(8.1)