Which Saturation Height Function?

SPWLA 57thAnnual Logging Symposium, June 25-29, 2016

WHICH SATURATION-HEIGHT FUNCTION? Stephen J. Adams, The Petrophysicist/WellEval.com

Copyright 2016, held jointly by the Society of Petrophysicists and Well Log Analysts (SPWLA) and the submitting authors. This paper was prepared for presentation at the SPWLA 57th Annual Logging Symposium held in Reykjavik, Iceland June 25-29, 2016.

INTRODUCTION When reservoir models are built, the hydrocarbon saturations within these models are initialized using a definition of water saturation (Sw) based on reservoir properties and height above a Free Water Level (FWL). Such descriptions are known as Saturation-height functions and are usually based on laboratory measurements of water saturation variation with capillary pressure.

ABSTRACT The industry currently uses over 20 different equations to describe the variation of water saturation with height and reservoir quality. Although papers comparing some common functions with new models have been published, no consistent review exists comparing quantitatively the performance of most of these equations over a selection of different reservoirs.

Over the years, many mathematical forms have been created and used to describe these variations of Sw with reservoir properties and height. But with so much choice, how does an Interpreter choose which equation is “best”.

This paper will present the results of quantitative comparisons between modelled and measured capillary pressure measurements over most common functions and through different reservoirs. The work was carried out in order to independently assess which equations should be tested first during saturation-height studies.

This paper examines many of these equations, testing them against three different capillary pressure datasets. Quantitative measures of how well the equations describe each dataset are made, allowing the best performing equations to be identified.

Sixteen common equations are fitted to three capillary pressure datasets. Each dataset exhibits different porosity to permeability characteristics. The differences between the water saturations estimated using the equations and those measured on the samples for each capillary pressure are examined both graphically and quantitatively. The results are then summarized and the equations ranked according to their performance.

Note that this work has been carried out in support of Adams (2016), which contains more detailed analysis and description than possible herein. PRIOR WORK The literature contains a number of reviews of saturation-height models. Some of these studies comparing different model equations are outlined below.

The results of this study show that the most commonly used Leverett-J model is one of the poorer performing saturation-height functions. Of the conventional “equation based” approaches, the Thomeer and Skjaeveland models appear to have the most utility and are recommended as first choice saturation-height models to test. However, the best results are achieved using interpolation, but this method is also the most difficult to execute in petrophysical and static modelling software.

Sondena (1992) present a saturation-height model based on Wright et. al. (1955) and compare its performance with Leverett (1942), Johnson (1987) and Alger et. al. (1989)(also known as CapLog). The new model and Johnson (1987) were deemed equivalent, giving matches to the raw capillary pressure data that were superior to both Leverett-J and Alger et. al..

The review presented could not encompass all possible equations, but does show which functions, of those most frequently cited, are likely to be of utility. Areas for future improvement are also highlighted.

Harrison et. al. (2001) compared four saturation-height models commonly used in the North Sea using two Fields from that Basin. The equations examined were those of Leverett (1942), Johnson (1987), Cuddy et. al. (1993) (also known as FOIL) and Skelt et. al. (1995).

1

SPWLA 57thAnnual Logging Symposium, June 25-29, 2016

All were found to give similarly ranging results, with no clear preference being identified. Sohrabi et. al. (2007) compare the performance of the Leverett-J, Alger et.al., Johnson, Cuddy et. al., Skelt et. al. and Sodena models in a North Sea Reservoir. They conclude that the Skelt et. al. model could be the best option. (Kamalyar et. al. (2012) continue the work of Sohrabi et. al. (2007) by investigating the performance of similar saturation-height equations using core and log data from a South Iranian reservoir. Here the models tested were: Leverett-J, Alger et.al. (and a modified version thereof), Johnson, Cuddy et. al. (and a modified version thereof), Skelt et. al. and Sodena. In this case, they conclude that Skelt et. al. gives the best match, while Leverett-J gives the poorest match.

Figure 1. Permeability variation with porosity in the study Fields. The light blue squares are from a carbonate reservoir, while the orange circles and black triangles are from two different sandstone reservoirs.

The author has not yet found a study providing a comparison of different saturation-height modelling equations across different reservoirs that approaches being as comprehensive as that reported herein. COMPARISON DATASETS In order to carry out this study, three capillary pressure datasets were selected, coming from three different continents, with one carbonate system and two different sandstone reservoirs. The range of reservoir properties represented in each dataset selected can be summarized using permeability vs. porosity plots of the capillary pressure samples. These data are shown in Figure 1 for all three Fields. Although there is significant overlay between the G and S Fields, one is a carbonate (S Field) and the other a sandstone (G Field), while T Field data behaves very differently from either of the other two datasets.

Figure 2. Capillary pressure curve end-point water saturations are plotted against permeability for the study Fields. Here the behaviour of the carbonate (S Field) clearly diverges from the sandstone reservoirs. SATURATION-HEIGHT MODELS TESTED As previously noted, there are numerous equations reported in the literature to model capillary pressure data and create equations describing water saturation as a function of height or pressure above the FWL.

To further illustrate the differences between the datasets, the end-points of the capillary pressure curves have been used to approximate irreducible water saturations and are plotted against permeability for each Field in Figure 2. In this case, all three Fields show different behaviour, with the carbonate “S Field” showing the greatest divergence.

Some equations, such as Leverett-J, are often used with the justification that there is a theoretical basis for its form. However, since no model is perfect, this author is of the opinion that the theoretical basis for an equation is unimportant. What is important is the ability of any approach to describe the measured capillary pressure data.

These Figures confirm that the three example datasets are sufficiently different that dissimilar saturationheight function parameters at least will be required to model each Field.

The most commonly used functions to fit to capillary pressure data are listed below. They are presented in

2

SPWLA 57thAnnual Logging Symposium, June 25-29, 2016

their order of importance i.e. the functions most commonly used are presented first.

In addition, a second set of statistics was calculated being the coefficient of determination (R2). Confirming that models obtained were reasonable was also important and carried out by plotting comparisons between the measured and modelled data along with some representative saturation-height curves for each model.

Leverett-J (Leverett 1942) Brooks-Corey (Brooks 1966) Lambda (Adams et. al. 1993, Adams 2016) Thomeer (Thomeer 1960) Skjaeveland (Skjaeveland 2000) Skelt-Harrison (Skelt 1995) Johnson (Johnson 1987) Heseldin (Heseldin 1974) FZI (Amaefule 1993) Entry-Height (Adams et. al. 1993, Adams 2016) Exponential (Adams et. al. 1993, Adams 2016) Hyperbola (Adams et. al. 1993, Adams 2016) Polynomial (Adams et. al. 1993, Adams 2016) Sigmoidal (Adams et. al. 1993, Adams 2016) Trig-Tangent (Adams et. al. 1993, Adams 2016)

DEMONSTRATION To demonstrate how the comparisons were made Figure 3 and Figure 4 show the comparison plots resulting from the Leverett-J model through the G Field dataset. While similar plots for the Thomeer model through the same data are shown as Figure 5 and Figure 6. Even cursory examinations of Figure 3 and Figure 5 shows that the Thomeer model is a better match to the data than the Leverett-J. The statistics bear out this observation showing R2 as 0.84 for Leverett-J and 0.93 for Thomeer.

Relationships between the different reservoir properties were also investigated using a Machine Learning approach (Schmidt, 2009, 2014). This method was tested rather than “neural networks” since the latter method does not yield equations for ready implementation in other software.

To illustrate that the other test Fields do not behave similarly to the G Field, the Leverett-J models developed for the S and T Fields are shown as Figure 7 and Figure 8.

Interpolation within the measured dataset using porosity-permeability classes was similarly tested – this approach being independent of any equations, but somewhat cumbersome to implement at present (Adams, 2016). COMPARISON STATISTICS With lots of different models to test, the “best” model should be that showing the smallest divergence from the measured dataset. There are a number of ways to quantify the differences between the modelled and measured data. These methods include looking at the sum of the squares of the differences and the average difference between the estimated and measured values. Moreover, since the oil and gas industry requires proven, probable and possible numbers for volumetric purposes, this author recommends a methodology that allows determination of these estimates in addition to the model uncertainty.

Figure 3. The measured water saturations are compared with those from the Leverett-J model through the G Field capillary pressure dataset.

The approach used was to calculate the differences between the modelled and measured estimates of water saturation. The mean difference should be zero for a good model, while the standard deviation of these differences can be used to compare different models and decide which has the lowest divergence from the measured data. 3

SPWLA 57thAnnual Logging Symposium, June 25-29, 2016

Figure 4. Some representative capillary pressure curves are shown from the Leverett-J model through the G Field capillary pressure dataset.

Figure 6. Some representative capillary pressure curves are shown from the Thomeer model through the G Field capillary pressure dataset.

Figure 5. The measured water saturations are compared with those from the Thomeer model through the G Field capillary pressure dataset.

Figure 7. The measured water saturations are compared with those from the Leverett-J model through the S Field capillary pressure dataset.

4

SPWLA 57thAnnual Logging Symposium, June 25-29, 2016

Model Interpolation Thomeer Lambda Skelt-Harrison Sigmoidal Leverett-J

Mean Difference, ±P90/P10 -0.01, ±0.14 0.01, ±0.14 0.00, ±0.15 -0.01, ±0.14 -0.02, ±0.15 -0.01, ±0.21

R2 0.966 0.966 0.963 0.963 0.960 0.920

Table 3. The summary statistics for the best five models and the Leverett-J model (15th best) in the T Field are shown. DISCUSSION Table 2 illustrates an important point about statistics; once the Interpolation Method is excluded, the lowest remaining combination of mean difference and uncertainty is actually in the Machine Learning model. However, the coefficient of determination (R2) is poorer than the Sigmoidal, Skjaeveland and FZI approaches. Hence, which of these models is actually best is not immediately apparent. Statistics do not give the complete answer. It is necessary to plot some representative curves for each model to ensure that they actually behave in the appropriate fashion. Figure 9 shows some curves for the Machine Learning model, while Figure 10 shows curves with the same permeabilities from the Skjaeveland model. The latter model shows curve shapes more in keeping with capillary pressure curve behaviour and hence is preferred, despite having slightly poorer mean difference and uncertainty.

Figure 8. The measured water saturations are compared with those from the Leverett-J model through the T Field capillary pressure dataset. RESULTS When all the models have been fitted through each Fields capillary pressure dataset, a series of summary statistics are available for each model. For the G Field, the statistics shown in Table 1 are reported for the best five models plus the Leverett-J. The latter model is included to allow comparisons with the most commonly used technique. Similar data is reported for the S Field in Table 2 and the T Field in Table 3. Model Interpolation Machine Learning Thomeer Skjaeveland Brooks-Corey Leverett-J

Mean Difference, ±P90/P10 0.00, ±0.06 0.00, ±0.07 0.00, ±0.08 0.00, ±0.08 0.00, ±0.08 0.02, ±0.12

R2 0.970 0.953 0.933 0.931 0.930 0.842

Table 1. The summary statistics for the best five models and the Leverett-J model (12th best) in the G Field are shown. Model Interpolation Sigmoidal Skjaeveland FZI Machine Learning Leverett-J

Mean Difference, ±P90/P10 -0.01, ±0.09 0.01, ±0.18 0.01, ±0.18 0.00, ±0.19 0.00, ±0.17 0.07, ±0.26

R2 0.880 0.803 0.802 0.797 0.793 0.720

Figure 9. Some representative capillary pressure curves are shown from the Machine Learning model through the S Field capillary pressure dataset.

Table 2. The summary statistics for the best five models and the Leverett-J model (14th best) in the S Field are shown. 5

SPWLA 57thAnnual Logging Symposium, June 25-29, 2016

capillary pressure datasets from three Fields. These Fields have been shown to have different reservoir properties, allowing different saturation-height function forms to be tested. In no case did the most commonly used Leverett-J model approach giving the closest match to the measured data. Despite this equation’s supposed theoretical basis, it does not describe the observed variations of water saturation with reservoir properties and height (pressure) above the Free Water level as well as some other functions. The best results obtained using a conventional (formulaic) approach are most likely to be obtained using the Thomeer or Skjaeveland models. The Sigmoidal, Lambda, Brooks-Corey and Skelt-Harrison models have also been found to be useful. A Machine Learning approach that yields actual formulae (as opposed to a neural network) is also worth testing, should the capability be available. The existence of an actual equation means that the model can be readily implemented in all types of reservoir modelling software.

Figure 10. Some representative capillary pressure curves are shown from the Skjaeveland model through the S Field capillary pressure dataset. When the three study Fields are considered, the Interpolation approach gives the best match between the modelled and measured water saturations in all cases. In other words, were this approach readily implemented in Petrophysical, Geological and Reservoir Engineering models, this method would be recommended. However, implementation is not currently straightforward, although Adams (2016) does give some suggestions as to how to proceed. In the meantime, other methods will be preferred by most modellers.

However, the best match between modelled and measured water saturations will be obtained using an Interpolation method. Unfortunately, this approach cannot presently be readily implemented in all reservoir modelling software. Hence, further development in this area is recommended.

From the tests carried out in the three study Fields, the following conventional models are recommended to be those first tested for saturation-height modelling an any Field. The first model given is considered most likely to have a variant giving the “best” match, while each successive model is considered a little less likely to be “best”:

REFERENCES Adams, S.J. & van den Oord, R. 1993. Capillary Pressure and Saturation-Height Functions. Shell International Petroleum Maatschappij. Adams, S.J. 2016. Saturation-Height Modelling for Reservoir Description. 1. Auckland: The Petrophysicist Limited.

Thomeer, Skjaeveland, Sigmoidal, Lambda, Brooks-Corey, Skelt-Harrison and FZI. In addition, a Leverett-J model should also be tested. It is not likely that this model will be “best”, but the comparisons with such a model can be used to illustrate which model is preferred and why.

Alger, R.P., Luffel, DL., Truman, R.B. 1989. “New Unified Method of Integrating Core Capillary Pressure Data with Well Logs.” SPE Formation Evaluation, June: 145-152.

In all cases quantitative comparisons should be made, along with “check” plots, to ensure that the “best” model really is identified.

Amaefule, J.O., Altunbay, M., Tiab, D., Kersey D.G., Keelan, D.K. 1993. “Enhanced Reservoir Description: Using Core and Log Data to Identify Hydraulic (Flow) Units and Predict Permeability in Uncored Intervals/Wells.” Annual Technical Conference & Exhibition. Houston: Society of Petroleum Engineers.

CONCLUSIONS AND RECOMMENDATIONS A detailed review of commonly used saturation-height functions has been made using reasonably large 6

SPWLA 57thAnnual Logging Symposium, June 25-29, 2016

Brooks, R.H. and Corey, A.T. 1966. “Properties of Porous Media Affecting Fluid Flow.” Journal Irrigation and Drainage Division (ASCE) 92: 61-68.

Capillary Pressure By Different Approaches.” EUROPEC/EAGE Conferenec and Exhibition. London: Society of Petroleum Engineers.

Cuddy, S., Allinson, G, Steele, R. 1993. “A Simple, Convincing Model for Calculating Water Saturations in Southern North Sea Gas Fields.” SPWLA 28th Annual Logging Symposium. Society of Petrophysicist and Well Log Analysts.

Sondena, E. 1992. “An Empirical Method for Evaluation of Capillary Pressure Data.” Society of Core Analysts. Thomeer, J.H.M. 1960. “Introduction of a Pore Geometrical Factor Defined by the Capillary Pressure Curve.” Journal of Petroleum technology (JPT) (Society of Petroleum Engineers) 73-77.

Harrison, B., Jing, X.D. 2001. “Saturation-Height Methods and Their Impact on Volumetric Hydrocarbon in Place Estimates.” SPE Annual Technical Conferenec and Exhibition. New Orleans: Society of Petroleum Engineers.

Wright, H.D., Wooddy, L.D. 1955. “Formation Evaluation of the Borregas and Seeligson Field, Brooks and Jim Wells Country, Texas.” Symposium of Formation Evaluation. AIME.

Heseldin, G.M. 1974. “A Method of Averaging Capillary Pressure Curves.” Transactions of the Society of Professional Well Log Analysts, 15th Annual Logging Symposium.

ABOUT THE AUTHOR Stephen J. Adams is an independent Petrophysical Consultant, operating his own international consultancy since 1994. Prior to that, he worked for Shell International Petroleum in the Netherlands, the Sultanate of Oman and Australia. During his career as a Petrophysicist, Steve has had a great deal of exposure to some challenging problems involving capillary pressure and saturation-height modelling from Fields all over the world. Much of this work has been “leading edge” in that similar cases have not been described in the literature previously.

Johnson, A. 1987. “Permeability Averaged Capillary Data: A Supplement to Log Analysis in Field Studies.” 28th Annual Logging Symposium. Society of Petroleum Well Log Analysts (SPWLA). Kamalyar, K., Sheikhi, Y., Jamialahmadi, M. 2012. “An Investigation of Using Different Saturation Height Functions in an Iranian Oil Reservoir.” Petroleum Science and Technology 30 (4): 412-424. doi:10.1080/10916461003752512. Leverett, M.C., Lewis, W.B., True, M.E. 1942. “Dimensional Studies of Oil-field Behaviour.” Transactions. American Institue of Mining Engineers (AIME). 175-193. Schmidt, M., Lipson, H. 2014. Eureqa software. http://www.eureqa.com/products/eureqa-desktop/. —. 2009. “Distilling Free-Form Natural Laws from Experimental Data.” Science, 81-85. Skelt, C., Harrison, B. 1995. “An Integrated Approach to Saturation-Height Analysis.” 36th Annual Logging Symposium. Society of Petroleum Engineers. Skjaeveland, S.M., Siqveland, L.M., Hammervold Thomas, W.L. & Virnovsky, G.A. 2000. “Capillary Pressure Correlation for Mixed-Wet Reservoirs.” SPE Reservoir Evaluation & Engineering (SPE) 3 (1). Sohrabi, M., Jamiolahmady, M., Tafat, M. 2007. “Estimation of Satiration-Height Function Using 7