Joseph M. HawkinsPetromation Corp. Houston
Donald L. LuffelResTech Houston Houston
Thomas G. HarrisResTech Inc. Houston
Gas/water and oil/water contacts can be predicted by a method based on a new capillary pressure model that relates four quantities: porosity, water saturation, air permeability, and capillary pressure.
This advanced capillary pressure method, Adcap, provides integration of laboratory-measured capillary pressure (P,) data with routine log and core analysis.
In the absence of laboratory P, data, the model depends on correlations from data obtained worldwide.
Water levels are predicted through regression analysis using porosity and water saturation from log and/or core analysis.
An important statistical result of the regression analysis is that an error range can be assigned to the predicted water level.
BACKGROUND
When exploring for or developing new oil and gas reservoirs, wells high on a structure may not penetrate the hydrocarbon/water contact. this contact needs to be determined as soon as possible to locate delineation wells, plan development drilling, and forecast reserves and economics, especially when operating in high-cost areas.
In principle, water levels can be predicted from a combination of capillary pressure data and log or core-derived o, Sw, and k. The J-curve correlation as presented by Leverett, 1 2 which was developed for unconsolidated sand packs, was the earliest method proposed to relate laboratory data of Pc, S,, and k.
Later, Alger 3 employed a multilinear regression approach, Caplog, to relate these same properties to both core and log data.
More recently, Smith' predicted the water level from logs with a method for generating synthetic P, curves if no actual P, data were available.
There are several reasons why these methods have had only limited success.
First, laboratory measured capillary pressure data specific to the reservoir are usually not available.
Second, to describe the entire reservoir, great care is required for integrating the laboratory P, data from a few core measurements with log data .
A final drawback is the inherent sensitivity of the method. If the wells in the reservoir are high above the transition zone, then a small error in water saturation produces a large error in the predicted water level.
The new method, Adcap, can:
- Be used in the absence of laboratory data
- Simplify the integration of capillary pressure data with log and core data
- Minimize the errors in estimating the water level.
The model is based on published studies involving large, diverse core data bases at Shell Oil Co. and Amoco Corp. As such, it is a worldwide correlation for sandstones, but also is useful for many types of carbonates, e.g., grain-dominated types.
In the model, regression analysis is used to find the water level that best honors all available log and core data in a reservoir.
Regression improves the resolution of the prediction by averaging down errors. An important byproduct is the identification of the error range bounding the predicted water level.
This error range is usually expressed with 95% confidence limits. This means that 95 times out of 100, the actual water level lies between the upper and lower limits.
MODEL
In 1960, Thomeer 5 proposed that mercury capillary pressure data could be described with hyperbolic curves employing the three parameters in Equation 1 (see equation and nomenclature boxes and Fig. 1).
If it is assumed that Sb is porosity and Sb is porosity times hydrocarbon saturation, Equation 1 can be rewritten in more familiar terms as Equation 2.
This simplified Thomeer equation adequately represents capillary pressure data, except for high pressures in complex rocks, where microporosity creates double-humped curves.
In 1981, Swanson 6 demonstrated that the coordinates from a special Point A on the capillary pressure curve could be related to air permeability (Fig. 1). This relationship, rewritten in more familiar terms, is Equation 3. SwA and PcA are values at Point A.
Fig. 2 illustrates Swanson's correlation with data from 319 core samples (carbonates and sandstones). Note that Point A in Fig. 1 is related to the parameters in Equation 2 by Equations 4 and 5.
Equations 4 and 5 show that a knowledge of 4), PcA, and SwA are enough to determine the entire capillary pressure curve through application of Equation 2.
The missing link, until recently, was some practical means of determining PcA and SwA.
Pittman 7 provided the link by demonstrating a relation between the pore-throat radius at Point A and air permeability in sandstones as Equation 6.
This relation was established using multiple regression on data from 202 samples of sandstone from 14 formations ranging in age from Ordovician to Tertiary.
The porosities and permeabilities ranged from 3.3 to 28% and 0.05 to 998 md, respectively. Pittman's correlation is shown in Fig. 3. Pc at Point A can be found by converting pore-throat radius to mercury capillary pressure with Equation 7, then substituting Equation 7 into Equation 6.
By substituting Equation 8 into Equation 3, Sw can be found for Point A in terms of porosity and permeability, Further algebraic manipulation with Equations 4 and 5 results in Equations 9 and 10.
This completes the derivation of Adcap, which is embodied in Equations 2, 9, and 10.
Because of the large core data bases involved, this model can be viewed as a worldwide correlation for capillary pressure curves. In the absence of measured laboratory data, the model can generate capillary pressure curves characterizing a reservoir.
Fig. 4 illustrates a typical fit between the worldwide model and measured capillary pressure data from a Miocene sandstone. Overall, the match is good, with the high permeability sample showing remarkable agreement.
To improve the model, however, capillary pressure data, when available, should be used to adjust the constants in Equations 9 and 10.
HEIGHT CONVERSION
To apply the method to well data, laboratory capillary pressures must be converted to height above the free-water level with Equation 11.
Reservoir fluid interfacial tension values are usually not measured, but may be obtained from correlations e.g., Schowalter, 8 and Firoo-zabadi and Ramey. 9
REGRESSION THEORY
In the regression approach applied here, all observed variables are considered to have associated errors. In this case, the observations are log derived or core analysis results.
At each depth level, there are four observed variables: depth, porosity, water saturation, and air permeability. Although some or all of the last three values may be calculated from logs rather than measured on cores, for the sake of simplicity these values are treated as observations subject to random error.
The sum of the weighted squares of these errors is minimized, constrained by the model equations relating the observed values.
Regression analysis solves for the water level and for improved values of the observations which minimize this sum. The solution is obtained through matrix algebra within which the model equation has been made linear by means of Taylor series approximations.
ERROR BARS
Important results of the regression are the error bars bounding the predicted water level. These are computed from the chi-square distribution using 95% confidence limits.
This means that for a large number of trials, the actual water level should lie between the upper and lower confidence limits 95 times out of 100.
Typical error bars are shown in Table 1. In the table, all values are measured in terms of distance below the lowest known hydrocarbon. As the table implies, the resolution of the method is better if the logged pay interval contains the transition zone rather than being high above the water level, on the steep part of the capillary pressure curve.
INDONESIA
Fig. 5 shows the log of a clean gas sand in an Indonesian well. A conventional core cut in the bottom half of the sand in oil-base mud shows good porosities, high permeabilities, and low water saturations.
Log-derived porosities and water saturations are in fair-to-good agreement with the core values. Permeability was calculated from log-derived porosity by a correlation designed to maximize agreement with core permeability.
Because laboratory measured centrifuge capillary pressure data were available on five plugs, coefficients were adjusted in the standard model, Equations 9 and 10, to fit the measured data.
A local model for the reservoir is given by Equations 12 and 13.
Fig. 6 shows excellent agreement between the water saturations calculated from the local Adcap model vs. the water saturations measured on the five plugs in the laboratory using the centrifuge.
From Equation 11, the conversion of capillary pressure to height is given by Equation 14.
With the local model consisting of Equations 2, 12, and 13 and the conversion Equation 14, the free-water level was predicted using two different data sets.
In the first set, core analysis values for porosity, permeability, and water saturation were entered into the regression at each core plug depth.
Based on this, a free-water level was predicted 409 ft below the sand base with very large error bars (the upper limit was 56 ft below sand base).
In the second data set, porosity, water saturation, and permeability, values were derived from log analysis. Based on this, a free-water level was predicted 294 ft below the sand base (9,469 ft on log in Fig. 5) with much smaller error bars (the upper limit was 134 ft below sand base).
On the basis of the smaller error bars, the water level as predicted from the log analysis data set is preferred.
A log from a downdip well in the reservoir shows a hydrocarbon-water contact 200 ft below the base of the sand (Fig. 5). Both water level predictions are in qualitative agreement with this.
When the free-water level is known, the model can be inverted to calculate water saturation at any height in the reservoir.
From the free-water level as found in the downdip well, water saturation was calculated within the well shown in Fig. 5.
The curve shows fair-to-good agreement with log-derived water saturations, except in intervals where the induction resistivity is reduced by thin bed or shoulder bed effects.
AFRICA
Fig. 7 is the log of an oil sand in a well offshore Ivory Coast, Africa. Resistivity and gamma ray logs show a clean sandstone at its base, with a fining upward sequence.
Conventional core data at the sand base shows excellent porosity and permeability. Pressure buildup analysis indicates a kh value of 13,272 md-ft from a drill stem test that flowed 5,000 bo/d.
Because no laboratory measured capillary pressure data are available, the standard model, Equations 2, 9, and 10, was used for water level predictions. From Equation 11, h equals 0.64 Pl.
Data entered into the regression were foot-by-foot log analysis values for porosity, water saturation, and permeability for the 52-ft sandstone interval. Permeability values were calculated from porosity and shaliness using an empirical equation calibrated to the core permeabilities.
Based on these data, the predicted water level is at 8,500 ft, Fig. 7, or 152 ft below the lowest known oil. Error bars range from 127 to 184 ft below the lowest known oil.
Although this water level has not yet been confirmed by a well, there are supporting data. A structure map from existing well control and 3-D seismic shows a spill point at the predicted water level.
What appears to be shale below the sand base is actually low-quality sand with oil shows to 8,410 ft.
A second DST taken at 8,430 ft produced 6 bo/d, 230 Mcfd, and 1 bw/d. An RFT, repeatable formation tester, at 8,405 ft recovered 7,000 cc of oil.
As in the Indonesian case, the model was inverted and used to calculate water saturation. This time the predicted free-water level was used to determine Pc.
Fig. 7 shows excellent agreement between Adcap-based and resistivity-based water saturations. This kind of overlay provides a visual check on how well the method interrelates the petrophysical data. The better the harmony, the more certain the water level prediction.
The operator, UMC Petroleum Corp., is proceeding with field development.
OTHER TEST CASES
Table 2 compiles water level predictions made in seven different reservoirs. One field, Boomerang, is a limestone while the other six are sandstone reservoirs.
Permeability ranges from a low at Lake Creek and Stratton to high in the other four sandstone reservoirs.
In the three cases with P, data, local models were used. The standard model was used in the other four cases. Fair-to-good agreement is shown between water levels predicted from Adcap and limits known from other information.
LIMITATIONS
Since Pittman's equation was derived for sandstones, the standard model is currently restricted to clastic reservoirs.
Our experience in applying the method to carbonates, however, has been encouraging, especially for intercrystalline, intergranular, or interparticle-type textures.
In the sandstone cases tested, porosities have generally exceeded 12%. Limited application to low-porosity sandstones (4-10%) has shown that the model provides satisfactory results when calibrated with capillary pressure data from the specific reservoir rock.
For best results in low porosities, capillary pressures should be measured with the cores at reservoir stress.
A useful technique for determining the validity of method is the saturation overlay discussed in the Indonesian and African cases.
If a reasonably close match is not obtained between the Adcap and resistivity-based water saturations using the standard model, then the model has to be adjusted to find the best match.
OTHER APPLICATIONS
Although the locations of water levels has been emphasized, the method has multiple uses.
If any three of the four basic quantities (porosity,
permeability, water saturation, and capillary pressure) are known, then the fourth can be calculated. For example, if the water level is known and porosity and permeability are available, then a water saturation profile can be derived as illustrated in the case histories of Indonesia and Africa.
Saturations independent of the induction log can be particularly useful where sands are thinly laminated with shales.
Another application is the calculation of permeability values when the water level, porosity, and water saturation values are known with confidence.
In a Gas Research Institute project in the Lake Creek field, Montgomery County, Tex., effective gas permeability was calculated from the model. This permeability compared favorably with permeability derives from well tests (Table 3).
REFERENCES
1. Leverett, M.C., "Capillary Behavior in Porous Solids," Trans AIME, Vol. 142,1941, pp. 151-69.
2. Leverett, M.C., Lewis, W.B., and True, M.E., "Dimensional Studies of Oil-Field Behavior," Trans AIME, Vol. 146, 1942, pp. 175-93.
3. Alger, R.P., Luffel, D.L., and Truman, R.B., "New Unified Method of Integrating Core Capillary Pressure Data with Well Logs," SPEFE, June 1989, pp. 145-52.
4. Smith, D., "How to Predict Down-Dip Water Level," World Oil, May 1992, pp. 85-88.
5. Thomeer, J.H.M., "Introduction of a Pore Geometrical Factor Defined by the Capillary Pressure Curve," JPT, March 1960, pp. 73-77.
6. Swanson, B.F., "A Simple Correlation Between Permeabilities and Mercury Capillary Pressures," JPT, December 1981, pp. 2,498-2,504.
7. Pittman, E.D., "Relationship of Porosity and Permeability to Various Parameters Derived from Mercury Injection Capillary Pressure Curves for Sandstone," AAPG Bulletin, Vol. 76, No. 2, February 1992, pp. 191-98.
8. Schowalter, T.T., "Mechanics of Secondary Hydrocarbon Migration and Entrapment," AAPG Bulletin, Vol. 63, No. 5 May 1979, pp. 723-60.
9. Firoozabadi, A., and Ramey, H.J. Jr., "Surface Tension of Water-Hydrocarbon Systems at Reservoir Conditions," journal of Canadian Petroleum Technology, May-June 1988, pp. 41-48.
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