Understanding Saudi Aramco’s "Rosetta Stone"

What lessons can you learn about saturation modelling from Aramco’s “Rosetta Stone”? This project uses saturation-height functions rather than dimensionally consistent J Functions. All data are fit exclusively with the Thomeer capillary pressure model. Despite these apparent drawbacks, Rosetta Stone publications report a high degree of satisfaction with the results. Isn’t this perplexing? This article reconciles this apparent conflict. Its message is that you can make up for ignoring physics and restricting your analysis to a single model by spending a lot of money on data collection. The Rosetta Stone database was acquired for Ghawar, the biggest oil field in the world. Can everyone else with smaller budgets afford to ignore physics as well? To diagnose what happened with Rosetta Stone, you must imagine how you would go about characterizing such a large field when you do not believe in J Functions. Once you go down this route, you will continually learn from your data what J Functions would have taught you in the first place. The coefficients of saturation-height functions end up depending on permeability when the role of permeability should have been clear from the beginning from dimensional analysis. Either you appreciate physics from the beginning or you will learn the implications of the physics as you collect more data. The phrase “saturation-height function” implies that saturation depends on height. A more neutral phase is “saturation function”. Why do modelers add “height” when dimensional analysis teaches you there is more to it than height? More confusedly, why do they use “height” when they know permeability is just as important?

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The Leverett J Function combines five variables in a dimensionally consistent function, where units balance. The five variables are capillary pressure, permeability, porosity, interfacial tension, and contact angle. This combination of five variables into a single function demonstrates the power of dimensional analysis. It is akin to the Reynolds Number of fluid mechanics combining four variables. Leverett’s dependence on contact angle stems from consideration of the Young-Laplace equation. His dependence on porosity stems from his consideration of the bundle of capillary tubes model. However, based on purely dimensional considerations, the J Function need not include porosity or contact angle. The usefulness of including porosity may be more apparent to those who are aware of the Reservoir Quality Index (RQI) which is defined as the square root of permeability divided by porosity, which has units of length (usually in micron).

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From this definition of J, we learn how permeability affects saturation. Without knowing this, you end up having to correlate the parameters of your chosen capillary pressure fitting function with permeability.? Then, you will discover dependencies on permeability that would have been apparent from dimensional analysis. For example, you will discover that the capillary pressure entry (or displacement) pressure correlates very well with permeability. The J Function also reveals the role of the product of interfacial tension and contact angle. Indeed, J reveals how to convert laboratory measurements (such as mercury capillary pressures) to reservoir conditions. Perhaps less appreciated is how the J Function shows how to model reservoirs where this product of interfacial tension and contact angle varies with depth.

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As mentioned, the dependencies of porosity and contact angle in the Leverett function are not required to make it dimensionless. To create a dimensionless function, you need to balance the units of capillary pressure, permeability, and interfacial tension. If you don’t buy into the usefulness of the Young-Laplace equation or Leverett’s interpretation of the bundle of capillary tubes model, you may treat porosity and contact angle as independent variables. Perhaps, this explains practitioners plotting capillary pressure versus bulk volume of oil or water instead of saturation. If you find better correlations with the product of saturation and porosity compared with saturation, you have introduced no dimensional inconsistency because both of these variables are dimensionless. Of course, you should then correlate J with the product of porosity and saturation. If you persist in using height or capillary pressure instead of J (as was done by “Rosetta Stone”), you continue to use a dimensionally inconsistent correlation.

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Please appreciate the remarkable contribution of dimensional analysis coupled with the extra reasoning about porosity and contact angle. The J equation collapses five independent variables into a single variable called “J”. This means that you do not have to vary all five of these variables independently in order to understand how they affect saturation. You need only vary J. This same benefit is what makes the Reynolds Number so important in fluid mechanics. Suppose you are trying to understand the onset of turbulence in a flow system (such as porous media). On the one hand, you could perform experiments which vary four variables independently: density, viscosity, superficial velocity, and a length scale (such as pore diameter). On the other hand, you could perform far fewer experiments by simply varying a single independent variable, the Reynolds Number.? If budget were no object, you might get away with the former. In this case, instead of understanding that turbulence occurs at a unique value of the Reynolds Number, you would construct a multivariate function (or look-up tables) to show which combinations of these four variables initiate turbulent flow.

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If you are trying to understand saturation modelling without J Functions, you will be forced into doing a lot of experiments to understand the effects that the five variables have on saturation. This is the Rosetta Stone method. However, it is worse than this. With J Functions, you see clearly how permeability affects saturation. With saturation-height functions, the effect of permeability on saturation is obscured. Have you noticed that saturation-height proponents use the phrase “rock type”? Not surprisingly, their rock types depend primarily on permeability. Why is this? It’s because they ignored what dimensional analysis teaches us about permeability. Physics wins in the end: they are forced into recognizing the importance of permeability in some other way, through defining rock types. Don’t be deceived by thinking these rock types align with what a geologist describes. When geologists describe rocks, they talk of lithology, texture, lithofacies, mineralogical composition, diagenesis, pore structure, etc.? For saturation-height modelers, rock type depends primarily on permeability. That’s because they fail to appreciate the power of dimensional analysis. Ironically, saturation-height modelers celebrate their ignorance by proclaiming that an advantage of saturation-height functions is they do not include permeability.

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The plots on the top of this article show capillary pressure (on the left) and J Function (on the right) for carbonate data released by Dr. Jorge Costa Gomes in his doctoral thesis at Heriot-Watt University. These data are from the Upper Kharaib Member (Early Cretaceous) from the northeastern part of the Rub Al Khali basin, a sub-basin of the wider Arabian Basin. As with every field I’ve ever seen, the J Function data are far better-behaved (more compact) than the capillary pressure plot (not both plots show five decades). This is not surprising: this is the power of dimensional analysis. The color-coding on both plots is by permeability over a range of 10 micro Darcy up to 1 Darcy. Notice the gradations of permeability on the capillary pressure plot. Saturation-height modelers would use such information to define, say, four rock types corresponding to colors blue, green, yellow, and red. The J Function plot shows that such rock-typing based on permeability is worthless because the J Function reveals the role of permeability.

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For J Functions, rock typing is less about permeability and more about pore structure (or pore topology). Let me use an imperfect metaphor that you might find helpful. Imagine you have a cast of a pore system. Imagine you “blow it up” by increasing the size of everything while keeping all of the pore interconnections intact. The permeability of the blown up pore system is going to be greater and its capillary entry pressure will be less. However, the J Function for both of these situations is likely to be (pretty much) the same. That’s because the J Function captures the effects of permeability. If however, you were to modify the connections between the pores, you would see differing J Functions. When saturation-height modelers talk about rock types, they conflate the effects of permeability and pore structure. When J Function modelers explain the variability of J Functions, they focus on pore structure because they have already accounted for the effects of permeability. J Function rock typing has more to do with how a geologist describes rocks.

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This disconnection of saturation and permeability has important implications. Think about history-matching. What’s the number one variable that reservoir engineers modify? Isn’t it permeability? In carbonates, we often see three or four orders of magnitude variations in permeability at a given porosity. However, J Functions – which include permeability – are far better behaved than capillary pressure curves. This means we can learn a lot about how to constrain permeability in 3D models by what we know from saturation logs (as measured through resistivity). J Functions show you to constrain permeability; saturation-height functions do not. Why introduce more uncertainty into permeability modelling by using dimensionally inconsistent saturation-height functions?

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Imagine now that you are forced into an experimental campaign to determine how capillary pressure curves depend on the five independent variables mentioned above rather than J. Naturally, you will want to use the simplest models possible to characterize capillary pressure curves. The Thomeer model has three parameters: entry (displacement) pressure, irreducible saturation, and the “pore geometric factor”. As they say in the Mission Impossible movies, “your mission, should you choose to accept it is to” figure out how these three parameters depend on the five key independent variables. You might well ask, “why the Thomeer model?” If you are looking for simple models, another three parameter model is the Brookes-Corey model, where capillary pressure is dependent on saturation raised to a power. Rosetta Stone papers refer to “Thomeer Hyperbolas”. A hyperbola takes the form of Y=A/X. For the Thomeer model, Y equals the logarithm of capillary pressure and X equals the logarithm of the non-wetting phase saturation (mercury in MICP experiments) or bulk volume of oil. Does the Thomeer model fit carbonates better than Brookes-Corey? Only Saudi Aramco can know the answer because – unlike the “Costa” dataset – the Rosetta Stone data are not public.

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In my experience in carbonates, I find four parameter models (such as Skelt-Harrison) have the extra degree of flexibility necessary to get good fits of J versus Sw. If you are a saturation-height modeler, you don’t want to hear this. First of all, this requires you to correlate four parameters rather than three. Second, it suggests it’s not a good idea to limit all fitting of data to a single model. If you are in the business of figuring out how model parameters are affected by the five key variables, you definitely do not want any flexibility in your models. You need to select one, simple model, as was done by the Rosetta Stone team. Of course, if you’re model fails to fit all of your data, you will conclude it’s “good enough” or is “fit for purpose”. By the way, you can easily change a capillary pressure model into a J Function model by substituting J for Pc. Furthermore, these models can easily be extended for use on bimodal and trimodal rocks. In this case, you have more parameters to correlate with the five independent variables.

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No doubt, the Rosetta Stone dataset is impressive. It’s simply a shame that it has been interpreted without J Functions. If Saudi Aramco were interested in having a look at this, I would be quite happy to assist in doing so. Do J Functions work for the Rosetta Stone dataset as well as they do for the Costa dataset? Certainly, it would be nice to have dataset to do the comparison. However, does anyone really think that dimensional analysis fails to apply on Saudi fields?

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If you want an introduction to the Rosetta Stone approach, please see GeoArabia, vol. 13, no. 4, 2008, p. 113-160 Arab D pore systems, Saudi Arabia, Gulf PetroLink, Bahrain, “Application of Thomeer Hyperbolas to decode the pore systems,

facies and reservoir properties of the Upper Jurassic Arab D Limestone, Ghawar field, Saudi Arabia: A Rosetta Stone approach” by Edward A. Clerke, Harry W. Mueller III, Eugene Craig Phillips, Ramsin Y. Eyvazzadeh, David H. Jones, Raghu Ramamoorthy and Ashok Srivastava.

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