Extension of Stone's Method 1 and Conditions for Real Characteristics in Three-Phase Flow
- F.J. Fayers (BP Research Centre)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- November 1989
- Document Type
- Journal Paper
- 437 - 445
- 1989. Society of Petroleum Engineers
- 5.5 Reservoir Simulation, 5.3.1 Flow in Porous Media, 5.2 Reservoir Fluid Dynamics, 2.5.2 Fracturing Materials (Fluids, Proppant), 5.2.1 Phase Behavior and PVT Measurements, 5.3.4 Reduction of Residual Oil Saturation, 1.10.1 Drill string components and drilling tools (tubulars, jars, subs, stabilisers, reamers, etc)
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This paper presents extensions to Stone's Method I in the definition of the residual oil saturation (ROS) parameter, Sorm, in which linear, quadratic, and cubic forms are compared with measurements. The methods are also examined in terms of the system providing hyperbolic characteristics, and it is shown that small elliptic regions will usually occur in the saturation space. Some of the factors influencing the elliptic regions are analyzed and their significance is discussed.
Three-phase relative permeabilities play a central role in most major reservoir simulators. including both black-oil and EOR applications. Thus, oil reservoirs with a free gas phase under production from a waterdrive, condensate fields with liquid dropout under aquifer invasion. such tertiary processes as multicontact-miscible displacement. development of a middle-phase microemulsion in a surfactant system, and steam/hot-water displacement of heavy oil all entail reliance on empiricisms describing three-phase flow. Compared with the extensive research undertaken on the Buckley-Leverett theory and determination of two-phase relative permeabilities, theoretical and experimental foundations for three-phase flow remain remarkably limited. This paper addresses some of the issues involved.
The most commonly used expressions for three-phase relative permeabilities are the normalized forms of Stone's Methods 1 and 2 given by Aziz and Settari. Method 2 is a principal option in many commercial simulators because it does not entail specification of an ROS parameter, Sorm, which is required in Method 1. Fayers and Matthews reviewed the normalized forms of Methods 1 and 2 and suggested a simple linear interpolation for Sorm, removing the problem of an arbitrary specification. Ref. 2 demonstrates that use of this extension of the normalized form of Stone's Method 1 results in better agreement with the available experimental results than use of Method 2. The results show that the behaviors of both methods are similar for oil relative permeability at larger oil saturations or in the middle of the mobile-oil-saturation range, but more noticeable differences occur as the oil relative permeabilities decrease. It is at these lower oil-saturation values, where the low fractional oil flow rates determine the reservoir's ultimate oil recovery, that the differences in the methods become important. Unfortunately, very few (if any) reliable measurements in this low saturation range exist, nor are the systematics of variation of Sorm, with changes in gas or water saturation amenable to any simple physical arguments. However, the extension of Stone's Method 1 to embrace the simple linear form for Sorm can be taken further to consider quadratic or even cubic forms. We examine some aspects of these extensions in this paper.
In two-phase flow, the general form of two-phase relative permeabilities ensures that the water fractional-flow function will have a monotonically increasing form with a point of inflection. This ensures that, mathematically, the noncapillary displacement equation will have real positive characteristics and gives rise to conditions for formation of the Buckley-Leverett shock front. This result is the unique physical solution to the capillary displacement problem when the capillary pressure tends to zero. In three-phase flow, we do not have established procedures for investigating "method-of-characteristics" solutions to the flow equations, nor do we have any clearly formulated rules for determining the formation and strengths of the shock fronts that may form. The first step is to investigate whether the use of the Stone's relative permeability expressions will necessarily give a hyperbolic system of equations. Arguments on why this mathematical property may be important and tests of its fulfillment are made against a number of experimental data sets in this paper. We find that the three-phase flow problem is not necessarily formally hyperbolic over the complete range of mobile gas and water saturations. The problem becomes nonhyperbolic (i.e., elliptic) in one or more small saturation zones, and the magnitude of the imaginary component of the characteristics is usually quite small in this zone (referred to as an "elliptic region"). Plausible physical arguments are made on why these nonhyperbolic zones may, not be significant from the standpoint of unstable solutions. The existence of nonhyperbolic regions. however, presents some mathematical difficulties in obtaining formal solutions to the noncapillary three-phase displacement problem.
This work was undertaken in parallel to more fundamental mathematical studies elsewhere on the characteristics of the three-phase flow problem. We concentrate on practical expressions of two-phase relative permeabilities and their use in various forms of Stone's Method 1. In particular, we numerically examine whether one of the alternatives for Sorm is preferable, either from its agreement with measurements or from the standpoint of the extent of nonhyperbolic regions. Other forms of correlations for three-phase relative permeabilities have been evaluated in Ref. 4, in which a simple saturation weighted-interpolation appears to give encouraging comparisons with measurements. This method, however, does not have the flexibility of Method 1 in ensuring that the correct behavior for k, is obtained in the important region near Sorm.
Summary of Previous Position on Evaluation of Stone's Methods
The original statement of Stone,s Method 1 was unsatisfactory in some cases because it gave k, values greater than unity; thus, Aziz and Settari's normalized form, expressed in the following equation, is generally preferred.
Swc=connate water saturation, krocw=two-phase oil permeability at So=1-Swc, and krow, krog=two-phase oil permeabilities for the oil/water and oil/gas systems, respectively. Thus, krow=krow(Sw) and krog=krog(Sg).
It is usually assumed that the krg and k,, curves are measured in the presence of interstitial water. Sorm is the ROS to simultaneous displacement by gas and water and has to be selected in a manner determined by the user. An example would be to choose Sorm = 1/2(Sorw+Sorg), where Sorw and Sorg are the ROS values associated with the two-phase curves defined above.
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