A New Streamline Method for Evaluating Uncertainty in Small-Scale, Two-Phase Flow Properties
- J.J. Hastings (Imperial College) | A.H. Muggeridge (Imperial College) | M.J. Blunt (Imperial College)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 2003
- Document Type
- Journal Paper
- 32 - 40
- 2003. Society of Petroleum Engineers
- 5.1.5 Geologic Modeling, 1.2.3 Rock properties, 5.6.3 Deterministic Methods, 1.6.9 Coring, Fishing, 4.1.2 Separation and Treating, 5.3.2 Multiphase Flow, 5.5.7 Streamline Simulation, 5.4.1 Waterflooding, 5.5 Reservoir Simulation, 5.5.3 Scaling Methods, 4.3.4 Scale, 5.4.6 Thermal Methods, 4.1.5 Processing Equipment
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This paper presents a new, semianalytical streamline method that allows rapid quantification of the impact of small-scale heterogeneities in permeability and relative permeability on reservoir production and water cut. This is achieved using a new method for upscaling total mobility in 1D, combined with an existing method for fractional flow.1 The properties along each streamline are upscaled analytically to provide effective permeability, porosity, relative permeability, and geometry for that streamline.
The estimation of the oil recovery distribution, resulting from uncertainty about the small-scale heterogeneity, is further sped up by making a separation of scales assumption. The oil recovery distribution can then be calculated from the streamline simulator using only a single pressure solve. A stochastic model of a cross section through a synthetic fluvial reservoir is used to demonstrate the validity of this assumption.
Recent studies2,3 have shown that even small-scale (cm to m) variations in permeability and relative permeability can have a large effect on reservoir performance. Our incomplete knowledge of the exact location and nature of these heterogeneities may represent a significant uncertainty in our predictions of oil recovery.
The usual way of assessing the impact of such uncertainty is to generate multiple equiprobable realizations of the reservoir and then use reservoir simulation to predict the recovery expected from each realization (the Monte Carlo method). Unfortunately, it is not currently possible to represent all length scales of heterogeneities explicitly in conventional field-scale, finite-difference simulations. These are limited typically to grid cells with volumes in the order of 104-105 m3. Streamline simulators have better spatial resolution, but even these are limited to an underlying grid with cells at most ten times smaller. Instead, the average effects of small-scale heterogeneities are usually modeled by using effective permeabilities (e.g., Renard and de Marsily4) and pseudorelative permeabilities (e.g., Barker and Dupouy5) within each simulation gridblock.
However, it is still generally not feasible to evaluate the impact of uncertainty in the small-scale reservoir properties by Monte Carlo simulation. The time taken to generate multiple geological models of the reservoir, upscale the small-scale flow properties for use in a coarse-grid simulation, and then predict production for each realization is usually prohibitive.
This paper presents a novel, semianalytical streamline method that allows the impact of small-scale uncertainties on waterflood performance to be evaluated rapidly, by the Monte Carlo method, using a single realization of the large-scale heterogeneities and multiple realizations of the small-scale heterogeneity. This is achieved by
Making a separation of scales assumption about the large and small scale heterogeneities.
Using a new method for upscaling the average total mobility for flow through a series of different rock-types with different relative permeabilities, combined with an existing method for upscaling the fractional flow.1
The ability to model changes in relative permeability between rock types is vital for capturing the effects of small-scale heterogeneities (e.g., trapped oil in high permeability laminae in crossbedded sandstones).
The method is validated by comparison with conventional streamline simulation techniques using a synthetic, 2D, vertical cross section through a fluvial reservoir.
The method uses a single set of fixed streamlines, with constant, upscaled flow properties, throughout the displacement. The streamline geometries and locations are not updated to account for gravity or changing well conditions, but the injection rate into each is continuously recalculated to model the changing mobility field. This has been shown to be a good approximation in similar streamtube methods.6,7 Fixing the streamlines reduces the computational effort to a single pressure solve to calculate the streamline locations and then a 1D flow simulation along each of the streamlines. Capillary pressure is assumed to be negligible.
The streamline locations, geometries, and properties are calculated as follows:
A geological model is created on a conventional Cartesian grid.
A conventional single-phase pressure solution (in this work we used routines from 3DSL8,9) is used to calculate the total velocity field in the geological model.
The streamlines are determined by tracing velocities through the grid. They are launched from cell faces containing injectors and, at this stage, it is assumed that each streamline carries the same flux (as flow is single-phase). Thus, the single-phase flow rate in each streamline, Q1, is found by dividing the flow rate out of an injection cell face by the number of streamlines leaving that face.
The streamlines are broken into segments in which the reservoir properties (permeability, relative permeability, and porosity) do not change. The segment boundaries are defined by the gridblock boundaries in the geological model.
The cross-sectional area (A) of each segment (i) is calculated from the position and time-of-flight (?t) information output by 3DSL, i.e., Equation 1, where ?Li=the segment length and Q1 =the single-phase flow rate in each streamline.
Finally, each streamline is homogenized using the new 1D upscaling solution (described in detail in the next section) to an equivalent homogeneous streamline with constant porosity, permeability, and relative permeabilities.
The reservoir production and water cut can then be estimated by performing a streamline simulation through the homogenized streamlines. The flow along each streamline as a function of pressure drop is calculated using the upscaled total mobility, while the saturation distribution is calculated using Buckley-Leverett theory.10
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