Fluid Flow in Various Patterns and Implications for EOR Pilot Flooding
- William E. Brigham (Stanford U.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- June 2004
- Document Type
- Journal Paper
- 170 - 174
- 2004. Society of Petroleum Engineers
- 5.1.5 Geologic Modeling, 4.1.5 Processing Equipment, 5.5.8 History Matching, 5.6.5 Tracers, 1.6 Drilling Operations, 4.1.2 Separation and Treating, 4.3.4 Scale, 5.4.1 Waterflooding, 5.1.1 Exploration, Development, Structural Geology, 5.3.2 Multiphase Flow, 5.7.2 Recovery Factors, 5.1 Reservoir Characterisation
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Pilot testing of enhanced oil recovery (EOR) projects in the field is common, for such tests demonstrate the effectiveness of a particular EOR process. Nevertheless, separating the effects on recovery of pattern geometry, pattern size, and reservoir heterogeneities from the EOR process itself is nontrivial. This paper illustrates that geometric and geological factors are evaluated easily at unit mobility ratio. Subsequently, nonunit-mobility-ratio effects can then be predicted. In this fashion, the efficiency of the EOR process is decoupled from geometry and permeability variation.
When an EOR project is considered, questions are asked regarding detailed knowledge of reservoir-scale well-to-well flow patterns. Such questions sometimes arise during waterflood (and are partially answered), but the far greater cost of EOR processes, as compared to waterflooding, puts a greater emphasis on the need for effective definition of the reservoir. The interplay of pattern geometry, heterogeneity, and fluid mobility plays a major role in determining oil recovery during EOR operations. Simulation and history matching are important tools for evaluation of EOR field tests and projects; however, a first step in pilot flooding interpretation is achieved with simple-to-use analytical tools, as detailed in this paper.
Significant details on the analytical approximation of multiphase-flow behavior at the pattern and field scale currently exists in the literature.1-9 For example, Abbaszadeh-Dehghani1 and Abbaszadeh-Dehghani and Brigham2,3 showed that the flow of tracers in various flooding patterns at a mobility ratio, M, of unity generalizes into a series of curves with behavior that depends only on the spacing of the patterns, a , and the hydrodynamic dispersion constant, a. They also showed that the behaviors of these patterns are directly related to each other. Analytical solutions were presented for M=1, and suggestions were made on how to use these ideas at other mobility ratios. The narrative to follow expands these ideas in some detail. It shows how they can be used to improve evaluations of EOR projects to distinguish the geometric and permeability variation effects from the effect of the EOR process itself.
Before proceeding, we make explicit the assumptions underlying the analytical tools discussed. The primary assumption is that the flow-pattern geometry does not change significantly. That is, there are no infill wells drilled, no recompletions, and no conversions of well function. Changing the flow geometry may change any inferences about the heterogeneity of the system.
Secondly, the effect of gravity forces is assumed to remain constant as a function of the injection rate. The techniques presented here will not work well when gravity forces are significant and vary considerably with the injection rate. Additionally, this approach will not work in reservoirs described by dual-porosity-type formulations. In such systems, the recovery history is strongly related to the fracture-matrix transfer function that is a function of time. It will, however, work where fractures are represented explicitly and where there is relatively little capillary imbibition of water.
This article proceeds with a critical review and reinterpretation of areal sweep calculations and correlations for unit and nonunit-mobility-ratio displacements. This sets the stage for a discussion of the interpretation of tracer-breakthrough behavior. The next topic is the relation between heterogeneity and recovery history. With these foundations, a procedure for interpreting EOR pilot tests is given. Conclusions complete the paper.
Results at Unit Mobility Ratio
When the mobility ratio is unity, it is possible to define exact equations for the breakthrough behavior of miscible or immiscible displacements in any balanced homogeneous pattern. It is necessary to do this if one wishes to calculate the produced concentration history of any tracer flowing through a pattern. This concentration history is calculated with a superposition integral, the nature of which varies depending only on the geometry of the pattern. The heart of this calculation depends only on the pattern-breakthrough history, the mixing characteristics, a/a, and any adsorption or reaction of the tracer. The breakthrough fractional-flow, or water-cut, histories are shown in Fig. 13 for unit-mobility-ratio displacements in five-spot, inverted seven-spot, direct line-drive, and staggered line-drive patterns at d/a ratios of 1:1. These pattern geometries are standard and are illustrated elsewhere.4 The line-drive patterns at greater d/a ratios show similar results.1 These breakthrough curves have shapes that resemble each other, with earlier or later breakthrough times indicating the relative efficiency of each pattern. Morgan5 concluded that all water-cut curves collapse into a single curve for a dimensionless pore volume, PVD, defined as
where Vp is the pore volume injected and VpBT is the pore volume injected at breakthrough. The resulting correlated fractional flow or water-cut curve is shown in Fig. 2 for M=1.3
The actual values of the curves of Fig. 1 differ slightly in Fig. 2 depending on the pattern calculated, but these differences are too small to be seen on this graph. A simple empirical equation describes the nondimensionalized curve3
where fD is the fractional flow or water cut of displacing fluid at the producer. Eq. 2 has a maximum error of 2% in fD for all cases in which fD is greater than approximately 0.1. For very early parts of the curve, the correlation is nearly vertical, and the deviation is larger.1
These same general ideas are used to calculate areal sweep efficiencies for these patterns as a function of volume injected. Such a graph is shown in Fig. 3.1 These curves, not surprisingly, show that the staggered line drive is the most efficient and the direct line drive is the least efficient of the patterns shown for a unit-mobility-ratio displacement. In another paper, we illustrate the change in relative pattern performance with mobility ratio.6
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