**Title**: Rolling the Dice

**Author**: R. E. (Gene) Ballay, PhD, http://www.geoneurale.com

**Publication**: The Outcrop, April 2010, p. 10, 14-17

INTRODUCTION

*The only certainty in most of our formation evaluations is the presence of uncertainty and how that issue is (or is not) addressed. At the simplest level one may estimate the Best and Worst Case, for each input attribute, and then bound the evaluation with the resulting extreme values, even as we recognize that the simultaneous occurrence of multiple “best” or “worst” values is an unlikely event.*

It is, in fact, relatively simple to address the uncertainty question in a comprehensive, realistic and quantitative fashion, and to further identify where to focus time, and money, in search of an improved evaluation.

At the simplest level our Sw estimates are compromised by uncertainty in the various Archie equation attributes.

S_{w} n = a R_{w}/(Φ m Rt)

In an earlier article (Risky Business) we took the derivative of Archie’s equation (the same approach will suffice for a shaly sand equation), and calculated the individual impact of each term’s uncertainty upon Sw to identify where the biggest bang for the buck, in terms of a core analyses program or suite of potential logs, was to be found. At that time we noted the ‘link between parameters, in that the relative importance of a single attribute, can be dependent upon the magnitude of another attribute, so that the characterization must be done for locally specific conditions.

An alternative approach is Monte Carlo simulation, which can be implemented with routine Excel spreadsheet functions. The Monte Carlo method randomly assigns values, according to user specified probability distributions, to each of the input parameters and then calculates the result. When the simulation is repeated a statistically significant number of times (results herein are based upon 2000 passes, which Excel handles without a problem), one is able to determine the likely outcome within any specific probability band, and to further identify which parameter is dominating the uncertainty (and hence where time and money is most efficiently directed for an improved result).

As an example, with the specifications tabulated in Figure 1, there is a 95% probability (+/- two standard deviations) that 0.28 < Sw < 0.43, whereas the Best / Worst approach would bound the results with 0.24 < Sw < 0.50; the difference being the unlikely event of multiple, simultaneous Best or Worst events. Not only does Monte Carlo give us a more realistic summary, but by varying the input standard deviations (uncertainties), one is able to identify where to most efficiently concentrate time/money in an effort to improve results.

**Monte Carlo Technique**

The Monte Carlo method relies on repeated random sampling of user specified input probability distributions to model expected results. This approach is attractive when it is infeasible or impossible to compute an exact result with a deterministic algorithm.

An advantage of Monte Carlo is that any type of distribution can be used to characterize the uncertainty specification of input parameters, for example normal, log normal, etc; an issue since the phenomena governing frequency distributions in nature often favor log-normal (Limpert et al, 2001).

A limitation of Monte Carlo is that special software is typically utilized (commercial add-ons to Excel, etc), and is often not even an option in commercially available petrophysics s/w packages. Common oilfield distributions, however, such as Normal, Log Normal and Triangle are available in Excel and it is straight-forward to implement Monte Carlo within the Excel framework. In this approach, one remains in the familiar Excel environment, and actually leverages their Excel skill set via the additional hands-on experience within the platform.

Additional details may be found in Ballay, Rolling The Dice, July 2009, www.GeoNeurale.com and at http://www.spwla-abudhabi.com/p_start.htm (Ballay, In Search of the Biggest Bang for the Buck).

**Monte Carlo Modeling of Sw (Archie)**

With a basic understanding of what the Excel Distributions options are, one is able to model the Archie equation within that framework. For illustration purposes, we regard “a”, Rw and Rt to be well-known, and Φ, “m” and “n” subject to uncertainty as specified in Figure 2. Allowance for uncertainty in “a”, Rw and Rt may be addressed by a straight-forward extension of the techniques presented here. Also, while the focus here is on the simple Sw(Archie), any other model (shaly sand, for example) may be evaluated in a similar manner. Once the concepts are understood, locally specific models are readily developed.

Each of the uncertain attributes a random number is modeled as input to Normlnv, whose mean value and standard deviation are locally appropriate. For example (Exhibit 2), the first pass random estimate of porosity, with a distribution centered on 20 pu and having a standard deviation of 1 pu, results in an estimate of 21 pu. The random values of “m” and “n”, appropriate to the specified distributions, are independently and randomly determined, and Sw calculated per the Archie relation.

Because Excel recalculates equations each time the spreadsheet is opened, or specifications are changed, the various results will change (your line item spreadsheet values will change, each time you make a modification).

As a quality control device, we determine and display the distribution of random numbers, between zero and one, for the number of Monte Carlo passes being used in a specific simulation (2000, in this example). In a perfect world there would be 200 observations in each of the ten blue bins displayed in Figure 2.

The approach taken here is intended to parallel that of the LSIJ results (Must Read supplemental material), which also includes Log Normal and Triangle distributions, and so can be directly referenced if either of those distributions are required: http://www.enrg.lsu.edu/pttc/.

As an additional QC device, the statistical attributes of the simulated quantities (Φ, ‘m’ and ‘n’ in this example) are tabulated directly from the simulation population, compared to the input and displayed graphically. With 2000 simulations, we find the model population nicely replicates the input numerical specifications, and the porosity distribution takes on the expected appearance.

**Summary**

There are two basic ways in which the issue of uncertainty can be characterized; partial derivatives of the expression of interest (Sw in this situation) and Monte Carlo simulation. At the simplest level, they complement one another, and since each is easily coded into an Excel spreadsheet, we routinely perform both, as a QC cross-check.

The deterministic derivative approach yields an equation, which may be easily coded into foot-by-foot petrophysical analyses, in those cases for which the commercial petrophysics s/w does not include an uncertainty characterization option. One is then able to ‘bound’ the calculated results, foot-by-foot, which is an improvement over a ‘generic envelope,’ given the inter-dependence of the result and specific reservoir values. An illustration of this method may be found in Ballay, Risky Business, March 2009, http://www.GeoNeurale.com.

On the other hand, an attribute specific distribution may not be Gaussian. Focke and Munn, for example, investigated the dependence of the cementation exponent upon pore geometry. Suppose across a given interval we are unable to distinguish between interparticle and vuggy porosity; either is a possibility. The associated “m” distribution could then be rectangular, not Gaussian, an issue that the Monte Carlo approach can easily address (each input attribute can have its specific distribution, independent of the others).

In any case, it’s important to recognize the following.

- The uncertainty in Sw(Archie), and other common oilfield calculations, can be quantitatively addressed by both differential and statistical modeling approaches.
- Excel can handle common probability distributions, and can then serve as the Monte Carlo simulator. The derivative method will yield equations which may be easily coded into Excel, thereby facilitating a cross-check.
- Quantitative estimation of the uncertainty allows one to determine where time/money is most effectively spent, and to further avoid the trap of being misled as a result of a previous bad experience with a poorly defined parameter.
- The importance of the various input parameters will change, according to the various magnitudes. There may be a linkage in that one parameter becomes more or less important as another parameter value is changing. One size does not fit all feet.
- The equations resulting from the derivative approach may be coded, foot-by-foot, into the petrophysics s/w package, thereby providing a live-linked uncertainty estimate to the actual, local reservoir properties.

**Acknowledgements**

We appreciate the unidentified LSU faculty who posted his or her material (located via Google) to http://www.enrg.lsu.edu/pttc/.

George Eden (BP Canada) and Larry Maple (ConocoPhilIips) generously shared their thoughts, and suggested relevant reference material, as ideas and material on this question were gathered over the past few years.

Stefan Calvert (BG India, E&P) has kindly shared his thoughts and spreadsheet examples, as this overview was put together.

Omissions, typos etc remain, of course, my responsibility.

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R. E. (Gene) Ballay’s 32 years in petrophysics include research and operations assignments in Houston (Shell Research), Texas; Anchorage (ARCO), Alaska; Dallas (Arco Research), Texas; Jakarta (Huffco), Indonesia; Bakersfield (ARCO), California; and Dhahran, Saudi Arabia. His carbonate experience ranges from individual Niagaran reefs in Michigan to the Lisburne in Alaska to Ghawar, Saudi Arabia (the largest oilfield in the world).

He holds a PhD in Theoretical Physics with double minors in Electrical Engineering & Mathematics, has taught physics in two universities, mentored Nationals in Indonesia and Saudi Arabia, published numerous technical articles and been designated co-inventor on both American and European patents.

Subsequent to retirement from Saudi Aramco Gene established Robert E Ballay LLC, which provides physics-petrophysics consulting services.