Pseudofunctions and Extended Dietz Theory for Gravity-Segregated Displacement in Stratified Reservoirs
- Pal Ingsoy (Esso Norge A/S) | Renaud Gauchet (Elf Aquitaine) | Larry W. Lake (U. of Texas)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- February 1994
- Document Type
- Journal Paper
- 67 - 72
- 1994. Society of Petroleum Engineers
- 5.3.1 Flow in Porous Media, 5.5 Reservoir Simulation, 2.4.3 Sand/Solids Control, 4.3.4 Scale, 5.4.1 Waterflooding, 4.1.2 Separation and Treating, 5.1 Reservoir Characterisation, 4.1.5 Processing Equipment, 5.4.7 Chemical Flooding Methods (e.g., Polymer, Solvent, Nitrogen, Immiscible CO2, Surfactant, Vapex), 5.3.2 Multiphase Flow
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This paper develops a new vertical equilibrium (VE) theory that accounts forgravity-segregated displacement in stratified reservoirs and is expressedeither in terms of pseudofunctions or as an extended Dietz model. The theoryincludes reservoir stratification and capillary effects. An example calculationbased on field data compares favorably with the more general numericalsimulation.
Pseudofunctions are used extensively in reservoir simulation to reducecomplexity and decrease computer costs. Desktop recovery calculations also maybe based on the introduction of pseudofunctions. Coats proposed a partiallytheoretically based method that assumes gravity/capillary equilibrium, VE, instratified reservoir segments and introduces pseudoproperties by averaging inthe vertical direction. The method has been verified to some degree forviscously dominated cases (i.e., under conditions of vanishing gravity andcapillary effects). This is somewhat surprising because the theoretical modelis framed on gravity/capillary equilibrium.
In this paper, we focus on displacements with significant gravity effects,with the intention of providing a theoretical basis for applying VEpseudofunctions. Introducing a VE theory that describes a displacement (e.g.,waterflood) as a set of moving isosaturation contours, we derive equations forthe pseudorized problem. The form of these equations allows certain conclusionsabout analytical techniques for estimating recovery to be drawn.
We also show that the pseudorized problem has traveling-wave solutionssimilar to those developed by Dietz and Ekrann. The traveling-wave solutiondescribes a nonchanging front (saturation pattern) propagating through thereservoir, also called a stationary displacement. In contrast to these studies,our analysis of the traveling-wave solution includes the capillary transitionzone.
The theory we present is general enough to account for both gravity under-and override cases and the wetting fluid is not always the denser fluid.Waterflooding a water-wet reservoir, however, serves as our main example, andsome of the results are given for gravity underride only. The extension togravity override is straight forward.
Various methods for assessing waterflood performance have been formulated byassuming a negligibly small capillary transition zone approximated by aninterface that can be described mathematically as a function of space and time.Such methods, called interface models, have a long history in hydrology. Inreservoir engineering, the best-known example is the Dietz theory forhomogeneous 2D reservoirs.
Ekrann extended Dietz's approach to stratified reservoirs, presuming that VEprevailed during the flooding. Ekrann devel oped several interesting results,including an ordinary differential equation describing the shape of thewater/oil interface for the special stationary displacement case. Ekrann alsoderived a critical injection rate for breakdown of the stationary solution.Ingsoy and Skjaeveland modified the theory slightly and presented experimentalresults that seemed to verify major aspects of the inter face model.
The interface and pseudofunction methods may seem to be separate entities.We intend to show that the two approaches can be unified under the VEassumption and that the unified perspective opens new possibilities foraddressing flood performance.
In this section, we outline a VE theory pertaining to the 2D stratifiedreservoir in Figs. 1 and 2. The horizontal upper and lower boundaries areassumed to be impermeable, and injection and production take place throughvertical wells at the left and right edges of the reservoir. The reservoir dipangle may be nonzero. We assume injection takes place at the left side and thatthe production well is located at the right side of Figs. 1 and 2.
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