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Reservoir inflow performance

Reservoir inflow performance is the reservoir pressure-rate behavior of an individual well. This article discusses the mathematical estimation of inflow performance.

Mathematical models

Mathematical models describing the flow of fluids through porous and permeable media can be developed by combining physical relationships for the conservation of mass with an equation of motion and an equation of state. This leads to the diffusivity equations, which are used in the petroleum industry to describe the flow of fluids through porous media.

The diffusivity equation can be written for any geometry, but radial flow geometry is the one of most interest to the petroleum engineer dealing with single well issues. The radial diffusivity equation for a slightly compressible liquid with a constant viscosity (an undersaturated oil or water) is ....................(1)

The solution for a real gas is often presented in two forms: traditional pressure-squared form and general pseudopressure form. The pressure-squared form is ....................(2)

and the pseudopressure form is ....................(3)

where the real gas pseudopressure is defined by Al-Hussainy, Ramey, and Crawford as ....................(4)

The pseudopressure relationship is suitable for all pressure ranges, but the pressure-squared relationship has a limited range of applicability because of the compressible nature of the fluid. Strictly speaking, the only time the pressure-squared formulation is correct is when the μz product is constant as a function of pressure. This usually occurs only at low pressures (less than approximately 2,000 psia). As a result, it generally is recommended that the pseudopressure solutions be used in the analysis of gas well performance.

Single-phase analytical solutions

Radial diffusivity equations can be solved for numerous initial and boundary conditions to describe the rate-pressure behavior for single-phase flow. Eqs. 1 through 3 have similar forms, which lends themselves to similar solutions in terms of pressure, pressure-squared, and pseudopressure. Of primary interest to the petroleum engineer is the constant terminal-rate solution for which the initial condition is an equilibrium reservoir pressure at some fixed time while the well is produced at a constant rate. The steady-state and semisteady-state flow conditions are the most common, though not exclusive, conditions for which solutions are desired in describing well performance.

The steady-state condition is for a well in which the outer boundary pressure remains constant. This implies an open outer boundary such that fluid entry will balance fluid withdrawals exactly. This condition may be appropriate when the pressure is being maintained because of active natural water influx or under active injection of fluid into the reservoir. The steady-state solution for single-phase liquid flow in terms of the average reservoir pressure can be written as ....................(5)

The semisteady-state condition is for a well that has produced long enough that the outer boundary has been felt. The well is considered to be producing with closed boundaries; therefore, there is no flow across the outer boundaries. In this manner, the reservoir pressure will decline with production and, at a constant production rate, pressure decline will be constant for all radii and times. This solution for single-phase liquid flow in terms of the average reservoir pressure is ....................(6)

The stabilized flow equations also can be developed for a real gas and are presented in pressure-squared and pseudopressure forms. For steady state, the solutions are ....................(7)

and ....................(8)

The semisteady-state solutions for gas are ....................(9)

and ....................(10)

Steady-state or semisteady-state conditions may never be achieved in actual operations. However, these stabilized conditions are often approximated in the reservoir and yield an acceptable estimate of well performance for single-phase flow. In addition, these solutions provide a means to compare production rates for various estimates of rock and fluid properties and well completion options. These relationships are useful as they allow the petroleum engineer the opportunity to estimate production rates before any well completion operations or testing.

Little difference is obtained in estimates of production rates or pressure drops when using the steady-state or semisteady-state solutions and, in practice, many engineers use the semisteady-state solutions. While each solution represents a distinctly different physical system, the numerical difference is minor when compared with the quality of the estimates used for rock and fluid properties, drainage area, and skin factor, as well as accounting for the heterogeneous nature of a reservoir. Dake, Craft, Hawkins, and Terry, and Lee and Wattenbarger provide complete details regarding the development of the diffusivity equations and the associated stabilized-flow solutions.

Nomenclature

 B = formation volume factor, dimensionless, RB/STB ct = total compressibility, Lt2/m, psia–1 h = formation thickness, L, ft k = permeability, L2, md p = pressure, m/Lt2, psia pp = gas pseudopressure, m/Lt3, psia2/cp = average reservoir pressure, m/Lt2, psia pwf = bottomhole pressure, m/Lt2, psia re = external drainage radius, L, ft rw = wellbore radius, L, ft s = skin factor, dimensionless T = temperature, T, °R z = gas compressibility factor, dimensionless μ = viscosity, m/Lt, cp ϕ = porosity, fraction