SPE 7142 Estimation of Saturation Height Function Using Capillary Pressure by Different Approaches Mahmound Jamiolahmady, Mehran Sohrabi and Mohammed Tafat, Inst. of Petroleum Engineering, Heriot-Watt U. Copyright 7, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Europec/EAGE Annual Conference and Exhibition held in London, United Kingdom, 11 14 June 7. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, Texas 7583-3836 U.S.A., fax 1-972-952-9435. Abstract The saturation-height function greatly impacts reserve calculations and is used by geologists or reservoir engineers to predict the saturation in the reservoir for a given height above the free water level. If cores are preserved, the water saturation information can be obtained directly. However, cores are often subject to changes from their original state before brought to laboratories. Hence, capillary pressure (P c ) curves measured on different core samples of a heterogeneous formation are usually correlated to porosity, permeability, and/or rock type using various techniques to generate field wide saturation height function. This paper evaluates performance of six saturation-height methods (i.e., Leverret-J [1], Cap-Log [2], Johnson [3], Cuddy et al. [4], Skelt-Harrison [5] and Sodena [6] methods) employed in the oil industry. Measured capillary pressure data and core properties for a well in one of the North Sea reservoirs were used with rock permeability varying from less than 1 md to over md. We have reviewed advantages and disadvantages of each method while comparing the output of each method with saturation obtained from the log data. The results indicated that Skelt-Harson method with coefficient independent of core properties, a simple average P c function, could be the best option and complicating the functional form by including the porosity and/or permeability data might inversely impact the predictions. Furthermore, modifying Cap-Log Cuddy et al. and kelt-harison by including the core permeability data in the mutli-regressione exercise resulted in higher errors confirming that increasing the degree of freedom does not necessarily mean a better performance.. INTRODUCTION Original reservoir fluid saturations are often determined through a combination of well log and core analysis data. Key core-analysis information is capillary pressure measurement, which captures the static volumetric behavior of reservoir fluids in the rock. When capillary pressure is available and when the free water level (FWL) is known, original reservoir water saturation can be reconstructed. The water saturation distribution above the FWL in an oil reservoir is controlled by the balance of capillary and gravity forces. The pressure gradient between the nonwetting (oil) and wetting (water) phases, P c, is determined by the difference in fluid densities at the given height from FWL. Hence, the reservoir P c (in psi) is related to height (in meter) using Equation 1. h=.735p c / ρ (1) where ρ is the density difference between formation water and hydrocarbon in (gr/cc) and is equal to.29 for the data in this study. Therefore, by knowing the P c function, water saturation at any given height is known. This dictates that an accurate estimation of P c function of the reservoir rock is crucial and greatly affects reserves estimation. Capillary pressure reflects the interaction of rock and fluids and is affected by the pore geometry, pore size distribution, interfacial tension and wettability. Equation 2, which relates the capillary pressure for an immiscible fluid pair in a circular cross section pore with radius r, can be used to show how P c relates to rock and fluid properties. Pc=2σcos(θ)/r, (2) where σ is interfacial tension and θ is the contact angle. The capillary pressure measured in the laboratory can be adjusted for the effect of fluid properties and wettability compared to reservoir conditions using Equation 3. ( P ) ( σ cosθ ) ( σcosθ ) ( Pc) lab res c res =. (3) lab In this study the values of σ res Cosφ res and σ lab Cosφ lab are 26
2 SPE 7142 and 42 dyne/cm respectively. The impact of pore geometry and pore size distribution reflected by the term (2/r) in Equation 2 should also be accounted for when extending the use of laboratory data to field data. This is not a straightforward task. That is, the reservoir rocks are often heterogeneous and there are not enough P c measurements to reflect all the variation of rocks properties. Normal procedure is that they are scaled to the petrophyscal properties of the rocks. This allows the extension of a limited number of core measurements to the entire reservoir. Levert in 1941 [1] proposed the use of (k/φ).5, which has the dimension of mean pore radius, in Equation 2 to reflect the change in petrophysical properties of rocks. Hence, he scaled the P c function of different rock types through the use of dimensionless J-Function defined by Equation 4. Pc J ( S w ) = k / φ. (4) σcosθ He reached to this equation by considering a sand pack as a bundle of capillary tubes with different pore radii. Therefore, Equation 4 accounts for changes of permeability, porosity and wettability of the reservoir as long as the general pore geometry remains constant. However, it is often unsatisfactory and different types of rocks exhibit different J function correlations. Since early work of Levert [1] numerous methods [1-8] have been proposed by researchers to overcome the shortcoming of this simple, but easy to use, method. In this work we evaluate the performance of some of the most common methods (i.e., Leverret-J [1], Johnson [2], Cap-Log [3], Cuddy et al. [4], Skelt- Harrison [5] and Sodena [6] methods) used in the oil industry. The measured capillary pressure data and core properties for a well in one of the North Sea reservoirs were used. The well 98/6-3 has a set of capillary pressure measurement from conventional core analysis and a number of logs, (GR, LLD, Sonic and CNL). The evaluation of the performance of each method was made in comparison with saturation information obtained from logs, which is assumed to be reliable. Reservoir Rock Capillary pressure data measured for 18 core plugs from this single well were used for this study, Figure 1. Table 1 contains the petrophysical properties of these core samples. These 1.5 '' diameter core samples had been drilled from the whole cores obtained from the well 98/6-3 between depth of -1573.m and -1638.7 m BRT. Core samples were cleaned using the mild miscible cleaning technique. Having cleaned the core samples, they were saturated with brine with no intermediate drying. Water /Oil permabilities were measured using high and low rate unsteady waterfloods. Air/brine capillary pressure characteristics were determined using the porous plate capillary. The test sequence used involved de-saturating the plugs at each seven pressures between 1 and 64 psi. The plugs were removed from the capillary pressure cells and weighted after a period of three weeks at each pressure stage. They were returned to the cell and the pressure was raised to the next desaturation pressure and the process repeated. Saturation Height Method Used 1- Leverret-J [1] In this method, initially the capillary pressure vs. saturation data for all core samples is converted to a single J function using Equation 4. Then a least square regression analysis is made using J value as the independent variable. The best correlation is often obtained using a power law equation of the form: J=A(S w ) B, (5) where A and B are constants. Figure 2 shows the corresponding dimensionless J-function versus saturation for the data of Figure 1 together with the best-fit power curve. The reservoir S w values can now be obtained by first converting the height above the free water level (h) to P c using Equation 1. Then J function is calculated by Equation 4 and the corresponding S w value is obtained from Equation 5 with A=.737 and B=-2.6558. 2- Cap-Log [2] Aufricht and Keopf [7] were first to relate P c and S w from core measurements to other rock properties by a family of parametric curves. Permeability was usually chosen as the rock parameter but porosity was sometimes preferable. Heseldin [8] modified Aufricht and Keopf method by relating the porosity to hydrocarbon bulk volume (V bh ) instead of S w with a family of parametric curves of constant P c. V bh is related to water saturation and porosity through the following relationship: V bh = φ (1-S w ) (6) Alger et al. [2] modified Heseldin s method by proposing a multi linear regression to relate P c to porosity and/or permeability and used the term Caplog method. They proposed the following general function to describe the dependency of V bh to P c and rock properties. V bh =A+B*Log(h)+C* φ +D*log(k), (7) where A, B, C and D are constants and h (height) is related to P c through Equation 1. However, they have stated that as k is not normally available from log data D is usually assumed zero. In this exercise, since we had log permeability data, Figure 3 both cases of with and without permeability data were studied. The values of the constants of Equation 7 for these two scenarios were A=-.2173, B=.568, C=1.2669,
SPE 7142 3 where height and porosity are in meter (m) and fraction of pore volume, respectively. The corresponding values, when permeability data were included, are: A=-.1736, B=.565, C=.9311, D=.17, where permeability is in md (milli-darcy). 3- Johnson [3] This technique relates water saturation derived from standard laboratory capillary pressure measurements to permeability and capillary pressure as described by Equation 8. Log(S w )=A*log(k)+ B*P c C, (8) where A, B and C are constants. This averaging technique involves the generation of a number of cross plots. First on the log-log plot a series of straight lines is fitted to the permeabilities and water saturation data for each of the laboratory capillary pressures (at the same P c ). That is, it is assumed that for a given capillary pressure, the saturation will vary from sample to sample depending on the petrophysical properties of the rock with better quality reservoir rock having typically lower wetting phase saturation. The slope of the lines, A in Equation 8, is then averaged to give a constant value. Then lines of constant slope were then fitted through each data set to generate a new intercept B`. On the log-log scale a linear relationship between B` and the capillary pressure P c is fitted, which is then converted to a power law functional form to give the first term of the right hand side of Equation 8. Following the above procedure for our data we obtained the following values for the constants of Equation 8: A=-.1669, B=2.92, C=-.682, where water saturation is expressed as percentage and capillary pressure is in psi. 4 Cuddy et al. [4] A simple function was developed by Cuddy that relates the product of porosity and water saturation to the height above the free water level as expressed by Equation 9. -log(φ.s w ) = A*log(h)+B. (9) This correlation has been developed based on the data of gas reservoirs in southern North Sea whereby, above transition zone, one observes an increase in porosity as water saturation decreases and vice versa. Hence, it takes no account of lithology and is biased towards fitting the water saturation data in the better quality sand. In our study we also included the effect of permeability in the above equation in the following form: -log(φ.sw) = A*log(h)+B+C*log(k), () where A, B and C are constants. For the present study, the values of the constants of Equations 9 and were obtained as: A=.2798, B=.7676, C=, and A=.292, B=.6113, C=.75, respectively. It should be noted that when using these values h (height), and k (permeability) are in (m) and (md), respectively, and porosity, water saturation are in fraction of pore volume. 5- Skelt-Harrison [5] Skelt Harrison proposed the following function relating water saturation to the height (h) above the free water level is: S w =1-A exp(-[b/(h+d)] C ) (11) where A, B, C, D are constants. The corresponding values for our data set were obtained by curve fitting as: A=.8491, B=2.5378, C=.64, D=1.7121, where h in (m) and s w is in fraction of pore volume. We also considered relating the coefficients of Equation 11 to permeability. Here it was noted that D merely shift the curves in the vertical direction, hence were kept constant. In a multiregression exercise it was observed that B is the most sensitive parameter to variation permeability for the core samples under study. A power laws function described the dependency of B to core permeabilities as follows: B=.86(k) -.586, (12) where k is in (md). The mean values of the A and C were.947 and.5482, respectively. 6- Sodena [6] In this method water saturation is related to capillary pressure and permeability through the following Equation: S w = A*P c C +B* P c D *log(k) (13) This averaging technique, similarly to Johnston method, involves the generation of a number of cross plots. Water saturation is first plotted versus the logarithm of permeability at constant values of capillary pressure. A straight line is fitted to the data for each value of capillary pressure. Then the slope and the intercept values are plotted against capillary pressure
4 SPE 7142 and the coefficients of Equation 13 are found by curve fitting procedure. Following the above procedure for our data we obtained the following values for the constants of Equation 13: A=.2832, B=1.3473, C=-.2431, D=-.271, where water saturation is in fraction of pore volume, permeability in (md) and capillary pressure is in (psi). RESULTS AND DISCUSSION OF THE DIFFERENT METHODS The saturation height function from a single well 98/6-3 has been determined by six methods, each model was compared with the water saturation profile obtained from logging as a base case. The results are shown in Figures 4-9 for these six methods, respectively. Here we elaborate on their accuracy by looking at some statistical error indicator to supplement the eye balling error base analysis observed in these Figures. Two criteria have been made to quantify the accuracy of each method compared to saturation obtained from logging. i) Average absolute percentage deviation, AAD%, which is calculated using the following equation: n j= ( Sw ) ( S ) core w ( S ) w AAD% = 1 * (14) n ii) The Standard error of estimate calculated by SEE= n j= 1 [( S ) ( S ) ] w core n 1 2 w (15) Subscripts (logging) and (core) denote the values obtained from log and cores analysis. The results of this error exercise are found in Table 2. The Levert method, Figure 4, which is based on the bundle of capillary tube models, gives the highest error. Eyeballing the data of Figures 5-9 shows that, with the exception of Skelt- Harrison, there are not significant differences between the trends of saturation data predicted by the remaining five different methods. In the case of Johnston (Figure 7) and Sodena (Figure 9) methods there is a significant deviation for the first meters above the free water level. Excluding their predictions for this part has significantly reduced AAD% from 43.8 and 42. to 34.8 and 37.% respectively, Table 2. This is mainly due to the presence of a power law function in Equations 8 and 13, respectively. That is, for these functions towards the low P c values, a small deviation in P c values would results in a significant variation in the predicted water saturation. In the case of Sodena the presence of two power law functions exaggerates this effect. The difference between Caplog and Cuddy et al. methods is that the former is based on ratio of hydrocarbon volume to the bulk volume whilst the latter is based on the volume of water to the bulk volume. As it was mentioned earlier Cuddy et al. method was proposed based on the data of the North Sea reservoirs, which favour our North Sea well data giving lower error values compared to Cap-Log method, Table 2. It should be noted that similarly to Cuddy et al. method, Cap-Log method is biased towards fitting the water saturation data in the better quality sand. This has resulted in significantly overestimating the water saturation for the low permeability zones in both methods. This observation can be made by looking at the permeability data from logging, Figure 3, and the corresponding predictions by Cap-Log and Cuddy et al. methods, Figures 5 and 6, respectively. Skelt-Harson method with constant coefficients (with AAD% of around ) has the best performance. This suggests that a simple average P c function, Figure 8, independent of core properties could be the best option for the case under study and complicating the functional form by including the porosity and/or permeability data might inversely impact the quality of predictions. It was also noted that including the core permeability data resulted in a lower error for the functions defined by Equations 7 (Cap-log method), 9 (Cuddy et al.) and 12 (Skelt-Harrison). However, the predicted water saturation values resulted in higher error values for all these three methods, Table 2. This indicates that including more parameters to the multiregression exercise increase the degree of freedom but does not necessarily mean a better performance. CONCLUSIONS Several saturation height functions models were investigated in the generation of the saturation height function algorithm. These functions calculate water saturation based on one or more parameters like, porosity, height above free water level, and permeability. Based on the limited data available we have noted that: 1) The Levert method, which is based on the bundle of capillary tube models, gives the highest error. 2) The performance of Johnston and Sodena methods is poor at low Pc values (small distance form FWL) due to the presence of power law function, which dictates a significant variation in waster saturation for a small variation in Pc in this domain. 3) Cuddy et al. and Cap-Log methods, which have similar structure, are biased towards fitting the water saturation data in the better quality sand. Furthermore, as Cuddy et al. method, which has been proposed based on the data of the North Sea reservoirs, favour our North Sea well data it gives lower error values compared to Cap-Log method.
SPE 7142 5 4) Skelt-Harrison method with constant coefficients, which is a simple average P c function independent of core properties is the most suitable method. However, it should be added that all these methods are empirical and the robustness of each algorithm makes it a feasible alternative in a special area. REFERENCES 1. Leverett, M.C.: Capillary Behavior in Porous Solids, Transactions of AIME 142, pp. 151-69, 1941. 2. Johnson, A.: Permeability averaged capillary data: a supplement to log analysis in field studies, Proc. of 1987 SPWLA, 28th Annual Symposium, 29 June - 2 July. 3. Alger, R.P., Luffel, D.L. and Truman, R.B.: New Unified method of integrating core capillary pressure data with well logs, SPE Formation Evaluation, June, 1989. 4. Cuddy, S., Allinson, G. and Steele, R.: A simple, convincing model for calculating water saturations in Southern North Sea gas fields, Proc. of 1993 SPWLA, 34th Annual Symposium, 13-16 June. 5. Skelt C. and Harrison, R.: An integrated approach to saturation height analysis, Proc. of 1995 SPWLA, 36 th Annual Symposium, 26-29 June. 6. Forthinggton and Chardaire-Riviere C.: Advances in core evaluation III. 7. Aufricht, W.R. and Koepf, E.H.,: Interpretation of capillary pressure from carbonated reservoirs, Trans. AIME 2, 1957, pp. 2-5. 8. Heseldine, G.M.: A Method of averaging capillary pressure curves, Proc. of 1974 SPWLA Annual Symposium 2-5 June. 9. Harisson, B., Jing, X.D.: Saturation height function methods and their impact on hydrocarbon in place estimates SPE71326, Proc. of SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, September - 3 October, 1. Table 1-Depth, permeability and porosity of cores samples from well 98/6-3. Sample No. Depth /m k /md Φ % 12 1573. 22 15.9 17 1582.23 9.3.6 9 1582.87 21. 4 1585.93 83.6 14 1586.9 64 22.4 2 159. --- 23.5 35 1594.5 468 26.5 22 163.7 361 27.5 21 169.9 395 27.3 28 16.58 289 23.6 32 16.91 1421 27.2 26 1612.93 922 26.8 39 1619.68 781 29.6 51 1619.89.5 15.3 53 1624.81 51 26.8 42 1625.5 24 16.8 45 1631.25 --- 22.1 56 1631.36 966 24.4 Table 2-Average Absolute Deviation and Standard Error of Estimates between the calculated water saturations from the core data, using different methods, with those obtained from Log. Method AAD% SEE AAD% (*) SEE (*) Leverrett 48.9.224 43.4.195 Cap-Log 39..178 37.6.171 Cap-Log- k 41..182.5.181 Cuddy 36.6.168 32.9.148 Cuddy-k 39.3.177 36.3.162 Johnson 42..198 34.8.151 Skelt-Harison 32.2.15 29.4.137 Skelt-Harison-k.7.178 39.9.176 Sodina 43.8.19 37..156 (*) refers to the values exclusing the first m from the FWL contact.
6 SPE 7142 Capillary Pressure /psi 7 6 5.2.4.6.8 1 Wetting Phase Saturation /PV Core No. 12 17 9 4 14 2 35 22 21 28 32 26 39 51 53 42 45 56 Fig. 1-Capillary pressure curves for core samples of well 98/6-3 obtained from laboratory measuremnts. 35 y =.737x -2.6558 R 2 =.8179 7 6 5 25 J (S w ) 15 5.2.4.6.8 1 Wetting Phase Saturation /PV Fig. 2-J function vs. saturation for core samples of well 98/6-3 obtained from laboratory measuremnts..1.1 1 Permeability (md) Fig. 3- Permeability vs. height from FWL obtained from logging.
SPE 7142 7 7 Levert 7 Cap-Log Cap-Log-k 6 6 5 5.2.4.6.8 1.2.4.6.8 1 Fig. 4-Water saturation vs. height from FWL obtained from logging and J function method. Fig. 5-Water saturation vs. height from FWL obtained from logging and Cap-Log method and mdofied Caplog method including permeability (k).
8 SPE 7142 7 7 Cuddy et al. Cuddy et al.-k Johnson 6 6 5 5.2.4.6.8 1.2.4.6.8 1 Fig. 6-Water saturation vs. height from FWL obtained from logging and Cuddy et al. method and mdofied Cuddy et al. method including permeability (k). Fig. 7-Water saturation vs. height from FWL obtained from logging and Johnson method.
SPE 7142 9 7 Skelt-Harrison Skelt-Harrison-k 7 Sodena 6 6 5 5.2.4.6.8 1.2.4.6.8 1 Fig. 8-Water saturation vs. height from FWL obtained from logging and Skelt-Harison method. Fig. 9-Water saturation vs. height from FWL obtained from logging and Sodena method.