Applications of Rate Normalization for Well Performance Analysis (I)

Theory

Oil and gas wells usually produce under a variable pressure/variable rate schedule. Such performance is rigorously analyzed by means of Duhamel's principle, in discretized form (Van Everdingen and Hurst, 1949)

where Pwd is the dimensionless pressure response, qD is the dimensionless rate, and PD,cr is the constant rate dimensionless pressure. PD,cr represents the solution of the diffusivity equation for a given set of initial and boundary conditions. Substituting Muskat's solution for an oil well producing at the center of a circle leads to, in field units (Blasingame and Lee 1986)

Note that this is a Pseudo-Steady State (PSS) approximation. To obtain a plotting function for graphical analysis, the concept of rate normalization for transient conditions is used (Odeh & Jones, 1965; Winestock & Colpitts, 1965); thus, for a smoothly varying rate profile,

where

Here, tmb, known as the material balance time, is given by Np/qn. Then, during PSS conditions, a plot of the left-hand side (ΔP/q) vs tmb results in a straight line (Figure 1). The slope of the line gives the In-place, and the intercept, bpss, yields the inverse of the productivity index.

Application

An oil well produces a black-oil fluid from a volumetric reservoir. The production history exhibits a variable rate/variable pressure schedule as shown in Figure 2 (Unneland, et. al, 1998)

The plot of the rate-normalized pressure (y-axis) against the material balance time (Figure 3) depicts a straight line from which slope N is determined to be 2.9 MMstb, and the PI is 2.9 stb/d/psi. Assuming a zero skin factor, the kh results in 3450 md-ft

Related data: Pi = 3200 psi. Bo = 2.3 stb/rb. Boi = 2.00 stb/rb. ct = 10⁻⁵psi⁻¹. Sw = 0.3. visc = 0.5 cps, h = 30 ft.

Limitations

• Low shrinkage fluid (black-oil fluid)
• Volumetric Reservoir
• No interference from offset wells
• Smooth rate changesCoclussions

Conclussions

• Normalization offers an effective way to approximate the Duhamel's integral; it can be stated that normalization is equivalent to rigorous superposition (Fetkovich, 1996)
• A Cartesian analysis yields a simple way to estimate relevant well and reservoir parameters, such as N and PI

The next article in the series will cover other practical applications of the PSS model

References

• Van Everdingen, A F, and Williams Hurst. 1949. The Application of the Laplace Transformation to Flow Problems in Reservoirs. AIME 186: 305-324.
• Winestock, A G, and G P Colpitts. 1965. Advances in Estimating Gas Well Deliverability J Can Pet Technol 4: 111-119. doi: https://doi.org/10.2118/65-03-01
• Odeh, A.S., and L.G. Jones. 1965. Pressure Drawdown Analysis, Variable-Rate Case. Journal of Petroleum Technology 960-964.
• Unneland, Trond, Yves Manin, and Fikri Kuchuk. 1998. Permanent Gauge Pressure and Rate Measurements for Reservoir Description and Well Monitoring: Field Cases. SPE Reservoir Evaluation & Engineering 1 (3): 224-230. doi:https://doi.org/10.2118/38658-PA.
• Fetkovich. 1996. Useful Concepts for Decline-Curve Forecasting, Reserve Estimation, and Analysis. SPE Res Eng 13-22. doi:https://doi.org/10.2118/28628-PA