Technique predicts oil recovery from waterfloods

Sameh Macary
Egyptian Petroleum Research Institute (EPRI)
Cairo

Walid A. Al Hamid
Gulf of Suez Petroleum Co. (Gupco)
Cairo

• X-Plot Calculations [51,984 bytes]

The X-plot technique can predict oil recovery and obtain the permeability ratio and fractional-flow curves for determining displacement efficiency.

Two field examples show how this approach was applied in the El-Morgan South Kareem and South Belayim reservoirs in Egypt. Both reservoirs have produced with water cuts greater than 50% since 1987.

This X-plot technique is for reservoirs that produce with water cuts greater than 50%. It requires the linearization of the relationship between oil recovery and water cut. Once obtained, the straight line allows extrapolating the line to any desired water cut to obtain the corresponding oil recovery.

Before running a linear regression for the straight line, the production data may have to be filtered and examined to determine whether it is relevant.

The slope and the intercept of this straight line are used to generate a plot to determine a relative permeability ratio, which forms the basis for creating the fractional-flow curve.

In addition to recovery predictions, the created permeability ratio and fractional-flow curves, unlike the laboratory-derived ones, are composite curves that are affected not only by fluid displacement characteristics but also by reservoir geometry, heterogeneity, and field operations.

Therefore, the created fractional-flow curve can serve as a quality-check tool for lab measurements.

Uncertainty

The level of uncertainty controls the degree of success of any decision. The lower the uncertainty, the more likely one will realize the objectives.

Often, reducing uncertainty requires significant expenditures. But methods such as the X-plot only require an evaluator's time and data that are usually readily available.

Special core analysis for obtaining relative permeability data and determining fractional-flow curves, on the other hand, is costly and sometimes technically impossible to obtain. But relative permeability data and fractional-flow curves are very important for planning the development of oil fields and determining displacement efficiency.

The X-plot technique, introduced by Ershaghi and Omaoregie,1 can generate the main portion of the relative permeability curve and hence the fractional-flow curve.

The X-plot technique is a convenient method for representing production performance of waterflooded or natural water drive reservoirs. The technique is based on relating oil recovery in terms of original oil in place (Er) vs. water cut as a straight line that can be extrapolated.

The X-plot technique is based on fractional flow and the Buckley-Leverett calculations.

Reference 1 describes the derivation of the relationship.

X-plot technique

To tabulate the time series of both water cut (F _w) and E _r, one needs to know the production performance of an oil reservoir. The reservoir has to produce under either water injection or natural water drive and has to have water cuts greater than 50%. The original oil in place also has to be known.

For this case, a graph of E_r vs. X will result in a straight line. The X is a weighted function of F_w, and is derived from the linearization process. It is obtain from Equation 1 (see equation box).

Because X has a parabolic shape (Fig. 1 [61,815 bytes]), the recommendation is to restrict this technique to water cuts greater than 50%. Differentiating X with respect to Fw and equating the first derivative with zero can prove this restriction. Thus, the inflection point is at Fw = 0.5. Reference 2 provides a detailed explanation of this concept.

Although cumulative production could be used, it has been found that for better mathematical handling, an easier and more suitable method is to use the overall reservoir recovery as a fraction of original oil in place. The resulting straight line indicates that the performance is being controlled by the relative permeability ratio characteristics of the reservoir. The straight line may be extrapolated out to any desired water cut to obtain the corresponding recovery.

Equation 1, which is based on Ershaghi's technique, may be changed to the form shown as Equation 2.

In Equation 2, m and n are the slope and the intercept, respectively. After a fit has been found, calculations are started with m to obtain b, if initial water saturation (S_wi) is obtained from Equation 3.

In Equation 3, b is the slope of the main portion of the k_ro/k_rw-vs.-S_w curve plotted on semilog paper. Equation 4 is obtained by using n and b and having an estimate of the mobility ratio (µ_o/µ_w).

In Equation 4, a is the intercept of the kro/krw-vs.-Sw plot. Equations 1-4 show the basis of the X-plot technique. The constants a and b, in turn, are used to derive an effective field relative-permeability ratio curve (Equation 5), assuming different values of S_w.

Equation 5 is equivalent to that proposed by Chierci,³ and considers the normalization of relative permeability due to the end points. The fractional-flow equation, neglecting the capillary pressure and gravity terms, because the dip angle is about 6° in the examples, may be written as Equation 6.

From this equation the fractional-flow curve can be created.

Technique applications

The X-plot approach was successfully applied in two field cases: El Morgan South Kareem and South Belayim reservoirs. Both are waterfloods operated by Gupco.

Since 1987, both reservoirs have had water cuts greater than 50%. Waterflooding has been ongoing in the South Kareem and South Belayim reservoirs since 1974 and 1983, respectively.

Pressure performance in both projects indicates that reservoir pressure has been maintained without an apparent active aquifer.

Table 1 [51,984 bytes] contains actual production data and the calculated X and Er for the South Kareem reservoir, based on an original oil in place of 1,960 million bbl. Table 2 [26,270 bytes] lists properties of both reservoirs.

Fig. 1 illustrates the scatter diagram of E_r vs. X on Cartesian paper (Fig. 2a [84,098 bytes] ). As seen from the scatter, the points are in need of filtering and smoothing. Filtering assumptions include:

• The starting point is the last recorded 50% water cut.
• For any point (production data), the coordinates of X and E_r should be greater than the previous point (continuous evolution).
• The error of the calculated value of E_r at 50% water cut by the regression line should not exceed 0.1%.

Fig. 2b [84,098 byutes] represents the same case but after filtering and linearizing the data. The R ² value (the goodness of fit) is high enough to meet the method's requirements.

The resulting plot can be used for predicting the reservoir's ultimate recovery at any water cut.

This is considered the most important application of the X-plot technique. In general for waterfloods, linearity is maintained when oil saturations are close to the residual saturation.⁴

For El Morgan South Kareem reservoir, S_wi was obtained from open-hole logs and the water/oil viscosity ratio was calculated from a pressure-volume-temperature (PVT) analysis. With these values, the two constants a and b were obtained.

Consequently the permeability ratio k_ro/k_rw at different Sw's was computed and plotted on a semilog paper (Fig. 3 [80,951 bytes]). The fractional-flow curve (Fig. 4 [80,504 bytes]) was created with Equation 6.

Figs. 5 [90,331 bytes], Fig.6 [80,213 bytes], Fig.7 [78,552 bytes] represent the same procedures for the South Belayim reservoir.

Predictions

Prediction methods based entirely on laboratory-derived data and with inadequately defined reservoir heterogeneity may be unsuccessful. Successful prediction techniques require input from real reservoir performance.

The X-plot technique for extrapolation of water-cut vs. recovery allows one to generate a field composite relative permeability ratio and fractional-flow curves that include reservoir properties as well as operational problems.

The main assumptions required are that:

• The plot of k_rw/k_ro vs. S_w is a straight line
• The leaky-piston displacement concept of Buckley and Leverett is applicable.

The linearity of E _r vs. X is a function in both the relative permeability ratio and the field operation conditions.

The deviation from the linearity of the X-plot indicates a definite new trend because of changes in the field conditions. The new trend must be used for extrapolation.

As seen from Figs. 2b and 5b, the overall recovery factor of Kareem and Belayim reservoirs at 90% water cut are 0.55 and 0.35, respectively. The calculated recovery factor for Kareem reservoir from the lab fractional-flow curve equals 0.55, while the recovery factor of Belayim was found by the decline curve analysis to be 0.34.

These values reflect the high accuracy of the X-plot technique.

The upward shift of this straight line or slope increases indicate good reservoir management and an operational success. In Fig. 2b, it is evident that the overall straight line has two jumps (upward shifts) while having the same slope at about 70-80% water cut, respectively.

These two periods are characterized by effective and successful operations that decreased the water production (Table 1).

Ershaghi's technique considers the case of a reservoir under water injection to be analogous to a water drive reservoir. Regardless of the injection rate or pressure, the cumulative water and oil production is controlled by the composite relative permeability curve describing oil and water flow in the reservoir.

Numerous proposed methods for obtaining relative permeability data from reservoir core samples are tedious and too time consuming for practical use or have yielded questionable and inconsistent results.⁵

The X-plot technique completely relies on production data for creating the relative permeability ratio curve, so that its output is representative of the actual reservoir.

Another use of the X-plot is to obtain the relative permeability ratio used in material-balance calculations. All material balance variables except this ratio are functions of reservoir pressure. The k_ro/k_rw is a function of the saturation of the displacing phase.

The objective in the examples was to plot a fractional-flow curve. The main portion, between S_wi and water breakthrough, is a straight-line segment with a slope equal to the reciprocal of pore volumes injected at water breakthrough. This is also true after the breakthrough.

Furthermore, a line segment that is drawn tangent to the curve from S_wi and extended to F_w = 1.0 will terminate at the average water saturation S_w.

Jones, et al.,⁷ stated that because relative permeability and fractional-flow curves depend on history, waterflood performance is described more accurately by unsteady-state, waterflood-derived, relative permeability curves than by the composite curves obtained from simultaneous injection of water and oil (core lab analysis).

This is the case in the examples. Table 3 [26,981 bytes] compares the fractional-flow curves generated by both the lab and X-plot technique. As seen in this table, the output of the X-plot matches well with lab measurements.

To complete the fractional-flow curve at both ends S_wi and S_or need to be obtained. S_wi can be estimated from open hole logs or other production logs. However, there is no need to draw the upward concave part of the S-shape because it gives no additional information for displacement efficiency calculations and we are only concerned about the S_wi value for constructing the tangent.

The other end of the fractional-flow curve (S_or) could be obtained from TDT logs. The µ_o/µ_w is available from PVT analysis.

The generated fractional-flow curves (Figs. 4 and 7), unlike the laboratory-derived curves, are composite curves that include not only the displacement characteristics of the fluids but also the reservoir geometry, heterogeneity, and operational conditions of the field.

The height of the point of tangency to the S-curve gives the water saturation at the outflow face just after the time of water breakthrough. The slope of the tangent can be used to obtain the flowing water/oil ratio just after breakthrough.

After breakthrough, the calculations are based primarily on the S-curve above the point of tangency.⁶ The resulting fractional-flow curve, in contrast with the laboratory one, could be updated at any time after obtaining more production data.

Acknowledgment

The authors would like to express their gratitude to Gupco's administration for permission to publish this article and also to the operation's engineering staff for the valuable comments and help in preparing this article.

References

1. Ershaghi, I., and Omoregie, O., "A Method for Extrapolation of Cut vs. Recovery Curves," JPT, February 1978, pp. 203-04.
2. Ershaghi, I., and Abdassah, D., "A Prediction Technique for Immiscible Processes Using Field Performance Data," JPT, April 1984, pp. 664-70.
3. Chierici, G.L., "Novel Relations for Drainage and Imbibition Relative Permeability," JPT, June 1984, pp. 275-76.
4. Ershaghi, I., Handy, L.L., and Hamdi, M., "Application of The X-Plot Technique to The Study of Water Influx in the Sidi El-Itayem Reservoir, Tunisia," JPT, September 1987, pp. 1127-36.
5. Johnson, E.F., Bossler, D.P., and Naumann, V.O., "Calculation of Relative Permeability from Displacement Experiments," Petroleum Transactions, AIME, Vol. 216, 1959, pp. 370-72.
6. Henry, J., "A Simplified Method for Computing Oil Recovery by Gas or Water Drive," Petroleum Transactions, Vol. 195, 1952, pp. 91-98.
7. Jones, S.C., and Roszelle, W.O., "Graphical Techniques for Determining Relative Permeability from Displacement Experiments," JPT, May 1978, pp. 807-17.

The Author