A Coupled Model for Three-Phase Capillary Pressure and Relative Permeability
- Odd Steve Hustad (SINTEF Petroleum Research)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 2002
- Document Type
- Journal Paper
- 59 - 69
- 2002. Society of Petroleum Engineers
- 4.1.5 Processing Equipment, 5.4 Enhanced Recovery, 4.1.2 Separation and Treating, 5.4.2 Gas Injection Methods, 5.1.1 Exploration, Development, Structural Geology, 1.6.9 Coring, Fishing, 1.2.3 Rock properties, 4.6 Natural Gas, 4.3.4 Scale, 5.5 Reservoir Simulation, 5.4.1 Waterflooding, 5.2.1 Phase Behavior and PVT Measurements
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A fully coupled formulation for three-phase capillary pressure and relative permeability is presented. The formulation incorporates hysteresis and miscibility on both capillary pressure and relative permeability, simultaneously. Consistency is ensured for all three two-phase boundary conditions through the use of two-phase data and the application of normalized saturations.
Simulation examples of water-alternating-gas (WAG) injection are demonstrated using water-, mixed-, and oil-wet capillary pressure and relative permeability curves, both exhibiting hysteresis. The examples represent typical core-flooding scales in the centimeter to meter range. Discussions and illustrations are given demonstrating the importance of the functional dependency and the extrapolation of two-phase capillary pressure values to three-phase flow properties.
Many papers have been written addressing the problem of three-phase relative permeability and capillary pressure. Literature review of various models is well documented elsewhere. See Refs. 1 through 8, along with their references. Likewise, for documentation of experimental observations, see Refs. 9 through 17.
One three-phase formulation that has been suggested is such that each phase's property is dependent on all three saturations.18 This formulation is extended here to include hysteresis and miscibility, where hysteresis may be applied to any of the two-phase properties.
Models are usually developed to address specific reservoir needs for the modeling of recovery processes, such as gas injection after waterflooding, WAG injection, or pressure blowdown after waterflooding. These models may be generalized to be applicable for other processes and reservoirs. When considering three-phase flow, models usually have been developed for relative permeability to oil, often assumed to be the intermediate wetting phase. These models combine two-phase data in various forms, making them dependent on two saturations, gas and water. Relative permeabilities to gas and water are usually made dependent on each phase's saturation, because they are often considered as the nonwetting and wetting phases, respectively. Gas-oil and oil-water capillary pressures are dependent on gas and water saturations, respectively. It is usually assumed that gas-water capillary pressure is the sum of the other two capillary pressures.19 Hysteresis has also been suggested for both relative permeability and capillary pressure.4,7,8,14,20
The motivation for developing an alternative three-phase formulation for reservoir simulation lies first of all in the difficulty of reproducing coreflooding results with reservoir simulators. Commonly used formulations like those of Stone21 have been successfully applied, although some questions may be raised when applying the same formulation to more than one process having convective flow with and without vaporization.22 Ref. 22 presents two gas displacement experiments in vertical cores after waterflooding and their associated numerical modeling. The modeling of the first experiment, in which equilibrium gas to oil is injected, achieved an acceptable match of the recoveries with minor adjustments to Stone's first model. However, applying the capillary pressure and relative permeability model and data to model the similar second experiment, dry gas injection, did not result in satisfactory recoveries. In fact, the water recovery (wetting phase) was the one most in error.
Simulations of coreflooding experiments have demonstrated that the saturation profiles are strongly dependent on the capillary forces and on the boundary conditions at the inlet and outlet of the core.18,22
Oak et al.17 concluded that relative permeability depends on fluid saturation and on the saturation history. The proposed model should therefore have capillary pressure and relative permeability values that are saturation history-dependent. That is, the properties assigned to the capillary pressure and relative permeability for a particular set of three-phase saturations should not be unique, but dependent on the saturation history leading up to the three-phase saturation condition.
An example in which modeling problems may arise for the state-of-the-art modeling of three phases is when a first-contact miscible injection process causes an oil-water system to become a gas-water system. The problem relates to the labeling of the hydrocarbon phase. Discontinuous relative permeability to hydrocarbon phase may occur when the phase label changes. Consistency may also be violated if incorrect gas-water capillary pressure and endpoint saturation values are applied.
The use of normalized saturation to model scanning two-phase capillary pressure curves have been successfully applied by Kleppe et al.14 Their model is based on the fact that the scanning capillary pressure curves lie between two limiting scanning curves and converge at the same endpoint saturation.16
The proposed model should ensure that capillary pressure and relative permeability are consistent and continuous for flow processes that go from any two-phase state to another two-phase state by way of the three-phase state.
Another motivation for this model is that the user should apply the "true" two-phase data, while the three-phase formulation should incorporate these data in a manner that most correctly estimates the three-phase properties.
It has been observed from centrifuge measurements that different fluid pairs will result in different endpoint saturations.22 A distinction should therefore be made between the terms "residual" and "endpoint" saturations. The endpoint saturation is defined here as the smallest obtainable saturation where the phase is still continuous. The endpoint saturation also represents the saturation where the relative permeability becomes zero. For saturation values at and below the endpoint saturation, the phase is discontinuous.
Residual saturation, on the other hand, represents the saturation where a phase becomes immobile. This condition may arise for different capillary pressure values, depending on the phase's pressure gradient or how high (or low) capillary pressure is achieved by the flow conditions. The relative permeability need not be zero at this condition, but for no flow to occur the phase pressure gradients (phase potentials) must be zero. By these definitions, the residual saturation is therefore greater than or equal to the endpoint saturation.
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