Summary
Many gas-condensate wells show a significant decrease in productivity once the pressure falls below the dew point pressure. A widely accepted cause of this decrease in productivity index is the decrease in the gas relative permeability due to a buildup of condensate in the near wellbore region. Predictions of well inflow performance require accurate models for the gas relative permeability. Since these relative permeabilities depend on fluid composition and pressure as well as on condensate and water saturations, a model is essential for both interpretation of laboratory data and for predictive field simulations as illustrated in this article.
Introduction
Afidick et al.1 and Barnum et al.2 have reported field data which show that under some conditions a significant loss of well productivity can occur in gas wells due to near wellbore condensate accumulation. As pointed out by Boom et al.,3 even for lean fluids with low condensate dropout, high condensate saturations may build up as many pore volumes of gas pass through the near wellbore region. As the condensate saturation increases, the gas relative permeability decreases and thus the productivity of the well decreases. The gas relative permeability is a function of the interfacial tension (IFT) between the gas and condensate among other variables. For this reason, several laboratory studies3–14 have been reported on the measurement of relative permeabilities of gas-condensate fluids as a function of interfacial tension. These studies show a significant increase in the relative permeability of the gas as the interfacial tension between the gas and condensate decreases. The relative permeabilities of the gas and condensate have often been modeled directly as an empirical function of the interfacial tension.15 However, it has been known since at least 194716 that the relative permeabilities in general actually depend on the ratio of forces on the trapped phase, which can be expressed as either a capillary number or Bond number. This has been recognized in recent years to be true for gas-condensate relative permeabilities.8,10 The key to a gas-condensate relative permeability model is the dependence of the critical condensate saturation on the capillary number or its generalization called the trapping number. A simple two-parameter capillary trapping model is presented that shows good agreement with experimental data. This model is a generalization of the approach first presented by Delshad et al.17 We then present a general scheme for computing the gas and condensate relative permeabilities as a function of the trapping number, with only data at low trapping numbers (high IFT) as input, and have found good agreement with the experimental data in the literature. This model, with typical parameters for gas condensates, was used in a compositional simulation study of a single well to better understand the productivity index (PI) behavior of the well and to evaluate the significance of condensate buildup.
Model Description
The fundamental problem with condensate buildup in the reservoir is that capillary forces can retain the condensate in the pores unless the forces displacing the condensate exceed the capillary forces. To the degree that the pressure forces in the displacing gas phase and the buoyancy force on the condensate exceed the capillary force on the condensate, the condensate saturation will be reduced and the gas relative permeability increased. Brownell and Katz16 and others recognized early on that the residual oil saturation should be a function of the ratio of viscous to interfacial forces and defined a capillary number to capture this ratio. Since then many variations of the definition have been published,17–20 with some of the most common ones written in terms of the velocity of the displacing fluid, which can be done by using Darcy's law to replace the pressure gradient with velocity. However, it is the force on the trapped fluid that is most fundamental and so we prefer the following definition:
N c l = | k → → ⋅ ∇ → ϕ l | σ l l ′ , ( 1 )
where definitions and dimensions of each term are provided in the nomenclature. Although the distinction is not usually made, one should designate the displacing phase l ? and the displaced phase l in any such definition. In some cases, buoyancy forces can contribute significantly to the total force on the trapped phase. To quantify this effect, the Bond number was introduced and it also takes different forms in the literature.20 One such definition is as follows:
N B l = k g ( ρ l ′ − ρ l ) σ l l ′ . ( 2 )
For special cases such as vertical flow, the force vectors are collinear and one can just add the scalar values of the viscous and buoyancy forces and correlate the residual oil saturation with this sum, or in some cases one force is negligible compared to the other force and just the capillary number or Bond number can be used by itself. This is the case with most laboratory studies including the recent ones by Boom et al.3,8 and by Henderson et al.10 However, in general the forces on the trapped phase are not collinear in reservoir flow and the vector sum must be used. A generalization of the capillary and Bond numbers was derived by Jin 21 and called the trapping number. The trapping number for phase l displaced by phase l? is defined as follows:
N T l = | k → → ⋅ ( ∇ → ϕ l ′ + g ( ρ l ′ − ρ l ) ∇ → D ) | σ l l ′ . ( 3 )
This definition does not explicitly account for the very important effects of spreading and wetting on the trapping of a residual phase. However, it has been shown to correlate very well with the residual saturations of the nonwetting, wetting, and intermediate-wetting phases in a wide variety of rock types.
The Numerical Simulation Study of the Oil–Water Seepage Behavior Dependent on the Polymer Concentration in Polymer Flooding
