Relative Permeability Correlation for Mixed-Wet Reservoirs
- A. Kjosavik (Statoil) | J.K. Ringen (Statoil) | S.M. Skjaeveland (Stavanger U. College)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 2002
- Document Type
- Journal Paper
- 49 - 58
- 2002. Society of Petroleum Engineers
- 5.1 Reservoir Characterisation, 4.3.4 Scale, 5.3.2 Multiphase Flow, 5.5 Reservoir Simulation, 5.3.1 Flow in Porous Media, 5.2.1 Phase Behavior and PVT Measurements, 5.4.7 Chemical Flooding Methods (e.g., Polymer, Solvent, Nitrogen, Immiscible CO2, Surfactant, Vapex), 1.2.3 Rock properties, 5.3.4 Reduction of Residual Oil Saturation, 5.5.2 Core Analysis
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A two-phase relative permeability correlation for mixed-wet rock is presented and validated. It includes provisions for bounding drainage and imbibition processes and scanning hysteresis loops, and is inferred from a capillary pressure correlation.
The well-known Corey-Burdine relative permeabilities were developed for water-wet rock from a Brooks-Corey power-law capillary pressure correlation and a bundle-of-tubes network model. We have extended this correlation to mixed-wet rock and now propose the ensuing relative permeability correlation for mixed-wet reservoirs. The functional form is symmetric with respect to fluid-dependent properties, because neither fluid has precedence in a mixed-wet environment. It reverts to the standard Corey-Burdine correlation for the completely water- or oil-wet cases, and exhibits the following characteristics in agreement with reported experiments: first, water-wet behavior at low water saturations and oil-wet behavior at low oil saturations; second, an inverted S-shape oil relative permeability curve with an inflection point; and, third, closed hysteresis scanning loops.
Earlier,1 we presented a capillary pressure correlation for mixed-wet reservoirs and suggested an extension of the Corey-Burdine2,3 relative permeability relationship from water-wet to mixed-wet conditions. We now develop this idea further and include hysteresis logic. The main design constraints are:
The functional form is symmetric with respect to oil and water. That is, the functional form is invariant to interchange of subscript o with subscript w.
The hysteresis loops are closed.4
The hysteresis loops of the capillary pressure and the relative permeabilities form a consistent set.5,6
Imbibition oil relative permeability curves may exhibit a characteristic inverted "S" shape.7-13
The validity of the relative permeability correlation and the hysteresis scheme is tested on published relative permeability measurements4 and simultaneously measured hysteretic relative permeability and capillary pressure curves.5 The hysteresis scheme is easy to program and could replace the Killough14 scheme in numerical reservoir simulators.
There is wide acceptance of the view that most reservoirs are mixed-wet, and network models15 allow for this fact. However, to incorporate mixed-wet rock properties into a numerical reservoir flow simulator, validated correlations are required.16-18
Review of Capillary Pressure Correlation
The relative permeability correlation is inferred from the capillary pressure correlation,1 and a review is given here. A sketch of the capillary pressure curve correlation for mixed-wet rock is shown in Fig. 1. It is an extension of the Brooks and Corey19,20 correlation for primary drainage of a completely water-wet reservoir, which may be written as
where cwd = the entry pressure, 1/awd = the pore size distribution index,2 and SwR = residual (or irreducible) water saturation.
For primary imbibition of a completely oil-wet rock (i.e., the process of reducing the oil saturation from So = 1), the capillary pressure may also be represented by Eq. 1, with subscript w replaced by o. For the intermediate cases, the capillary pressure correlation is the sum of the two extremes
where aw, ao, and cw = positive constants, while co = a negative constant. There is one set of constants for imbibition and another for drainage. We use the term "drainage" if Sw is decreasing, and "imbibition" if Sw is increasing, irrespective of wettability preference.
Hysteresis Loop Logic.
The design constraints follow from experimental evidence21-26:
A saturation reversal on the primary drainage curve, before reaching the residual water saturation SwR (Fig. 2), spawns an imbibition scanning curve aiming at a residual oil saturation determined by Land's trapping relation.
Reversal from the primary drainage curve at SwR starts an imbibition scanning curve down to SoR. This curve is labeled (b) in Fig. 1 and is the bounding imbibition curve.
The secondary drainage curve, labeled (c) in Fig. 1, is defined by a reversal from the bounding imbibition curve at SoR. Together, the bounding imbibition and the secondary (bounding) drainage curves constitute the closed bounding hysteresis loop.
All drainage scanning curves that emerge from the bounding imbibition curve scan back to SwR (Fig. 3), and all reversals from the bounding drainage curve scan to SoR (Fig. 4).
A scanning curve originating from Sw[j], the j'th reversal saturation, will trace back to Sw[j-1] and form a closed scanning loop, unless a new reversal occurs.
If a scanning curve tracing back from Sw[j] reaches Sw[j-1] before any new reversal (i.e., forms a closed scanning loop), the process shunts to the path of the [j-2] reversal as if the [j-1] reversal had not occurred, Fig. 5.
The shape of a scanning loop is similar to the bounding loop because the a and c parameters are constants for a given rock-fluid system.
All properties of the j'th scanning curve are labeled by [j]. The capillary pressure is denoted by pca[j], where a denotes the process type and is either i for imbibition or d for drainage. By convention, j is an odd number for imbibition and even number for drainage, 0 denoting the primary drainage process. The asymptotes of the scanning curves are denoted by SwR[j] and SoR[j], and the water reversal saturation is denoted by Sw[j]. The smallest, "global" residual saturations of the largest bounding hysteresis loop are denoted by SwR and SoR.
All scanning curves are modeled by the same constants a and c as the bounding curves. As an example of the notation, the primary drainage capillary pressure is denoted by pcd, and its value at the first reversal, Sw, is given by pcd (Sw).
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