In Search of The Biggest Bang for the Buck

1. In Search of The Biggest Bang for the Buck R E (Gene) Ballay SPWLA UAE October 2009 Statistical Differential www.Entrac-Petroleum.com

2. In Search of The Biggest Bang for the Buck • Geoscientists are accustomed to dealing with uncertainty • While minimum and maximum variations (for example) are common, they may be incomplete, and even non-representative • There are two basic alternatives • Partial derivatives • Statistical simulation The Devil’s Promenade, SW Missouri

3. In Search of The Biggest Bang for the Buck • Why quantify the uncertainty? • Time and Budget • There is typically a limited timeframe and budget. Where do we focus? • Lack of data and/or previous experiences • Even with ‘focus’, we seldom have all the data we would like • A previous bad experience, by either our self or someone else on the Team, will compound the unease

4. In Search of The Biggest Bang for the Buck • Concepts (and spreadsheet) applicable to many common oilfield issues • The various, discrete measurements that make up routine and special core analyses • The various attributes that are required to convert lab capillary pressure to a reservoir conditions • Reservoir volumetrics • Once we have mastered one calculation, it is straight-forward to address another

5. In Search of The Biggest Bang for the Buck • As carbonate petrophysicists, we are typically faced with Sw(Archie) estimates, in the face of uncertainty • Mineralogy • Porosity • Formation brine resistivity • Formation resistivity • “m” and “n” exponents • Exhibit following

6. In Search of The Biggest Bang for the Buck • As carbonate petrophysicists, we are typically faced with …… • Mineralogy • In some (KSA Shuaiba, for example) reservoirs, mineralogy is nearly uniform, and hence well known. • In other (probably most) locales, uncertainty exists in this basic information • In West Texas one may be faced with three minerals (limestone – dolostone – anhydrite) , and only two logging tools (density-neutron) • Anhydrite may be present as obvious nodules, or as (more subtle) cement • Porosity • Porosity is often thought of as +/- “x” pu of uncertainty, but in actual fact the uncertainty may be a function of the amount of porosity present • Tool measurement techniques and statistical noise • Formation brine resistivity • We’ve all worked Fields for which we had high confidence in Rw, but that is not always the case • Exhibit following

7. In Search of The Biggest Bang for the Buck • Formation resistivity • The 6FF40, as an example, has a Skin Effect limitation at the low resistivity end, and a signal-to-noise issue at high resistivity • The mud resistivity (and hence borehole & invasion effects) may change from one well to the next, indeed one logging interval to the next • “m” and “n” exponents • How many of us have ever been “really sure” of our exponents? • In carbonates, wettability (and hence “n”) may vary with Sw, and present a significant challenge in the transition zone • Sw(Archie) involves the combination of all these attributes, and their respective uncertainties • Is it any wonder that “one size may not fit all feet”?

8. The Differential Approach Over-view • Chen & Fang have taken an In Depth Differential Look at Sw(Archie) Uncertainty, in terms of input attribute values and respective uncertainties Sensitivity Analysis of the Parameters in Archie‘s Water Saturation Equation. The Log Analyst. Sept – Oct 1986

9. The Differential Approach Over-view • Chen & Fang produced generic charts to facilitate locale-specific evaluation • Sw(Archie) attributes with associated Best Estimate and Uncertainties • More on this later • For attributes specified above, and in the case of f ~ 20 pu, “n” is a relatively minor issue • At F ~ 20 pu, for stated conditions, the saturation exponent is a relatively minor issue

10. The Differential Approach Over-view • Chen & Fang’s results have been coded to an Excel spreadsheet, to facilitate locale specific, digital evaluation

11. The Differential Approach Over-view • Chen & Fang’s results have been coded to an Excel spreadsheet, to facilitate locale specific, digital evaluation • Sw(Archie) attributes with associated Best Estimate and Uncertainties • For attributes specified above, and in the case of f ~ 30 pu, the priorities change. • Now ‘m’ & ‘n’ are of about equal importance

12. The Differential Approach Over-view • At F ~ 10 pu, tortuosity in the pore system is far more important than “n” variations These relations may be coded into nearly any petrophysical s/w package, to facilitate foot-by-foot evaluation

13. The Differential Approach Over-view • The differential approach is applicable to many common oilfield issues • ArchieUncertainty.pdffor derivation & additional considerations • How to calculate the derivatives • How to calculate the propagation of error Additional Details

14. The Statistical Approach Over-view • Many phenomena in nature are not Gaussian distributed. • The probabilistic method allows one to model non-Gaussian distributions • More difficult to set up • Notincluded in the petrophysical s/w that is in use

15. The Statistical Approach Over-view • An advantage of Monte Carlo is that many types of distributions can be used to characterize the uncertainty of a parameter • Normal, log normal, rectangular, triangular, etc

16. The Statistical Approach Over-view • The Monte Carlo method relies on repeated random sampling to model results • This approach is also attractive when it is infeasible or impossible to compute an exact result with a deterministic algorithm.

17. The Statistical Approach Over-view • A limitation of Monte Carlo is that special software is often utilized • Not included with many commercially available petrophysics s/w packages • When available, the implementation of a MC Model over a large section of log data can be very time consuming

18. The Statistical Approach Over-view • The Monte Carlo method makes use of probability distributions to produce a cumulative probability result • The determination of a probability distribution is superior to the single deterministic value • Insight is gained into the “upside” and “downside” • + / - 1 s will encompass ~ 68% of the distribution • + / - 2 s ~ 95 % of the distribution http://en.wikipedia.org/wiki/Standard_deviation

19. The Statistical Approach Over-view • Many Oilfield analyses can be accomplished with Excel • This eliminates both the expense of, and need to learn, commercial programs • @Risk, Crystal Ball, etc • If you have the time, and budget, to use @Risk, etc…Great! • If not, and you want to leverage your Excel skill set, consider that option • Oilfield applications typically use “normal”, “log normal”, rectangular and “triangular” statistical distributions. • Relevant Excel functions • Rand, NormDist, NormInv, LogNormDist, LogInv, TriangleInv

20. The Statistical Approach Over-view • The statistical approach is applicable to many common oilfield issues • If a “reference” spreadsheet is available, Monte Carlo may be easier to implement than derivatives • MonteCarloModeling.pdf and MonteCarloIllustrations.pdf for additional details and considerations Additional Details Additional Details

21. The Statistical Approach Over-view • Monte Carlo simulation can be used to characterize the uncertainty in Sw, for the attributes (& associated uncertainties) tabulated at right • “a”, Rw and Rt are assumed to be well-known, reflected here by no STD specification Illustrative Monte Carlo Application • This simulation is approximating an Sw interpretation for which the porosity, “m” & “n” estimates are each subject to individually specified uncertainty • Porosity (for example) is described by a Gaussian distribution, centered on 20 puwith a standard deviation of 1 pu

22. The Statistical Approach Over-view Sw(Archie) Uncertainty • One issue of interest is the dependence of Sw upon individual attribute values / uncertainties • With the specifications at right • Sw(mean) = 0.357 • s (Sw) = 0.038 • There is a 95% likelihoodthat Sw is contained within + / - 2 s • (0.357 – 0.076) < Sw < (0.357 + 0.076) • 0.28 < Sw < 0.433 • Be aware of how Excel ‘bins’ data and ‘non-linear effects’ • Supplemental material for details

23. Statistical vs Best / Worst • There is a 95% likelihoodthat Sw is contained within + / - 2 s • 0.28 < Sw < 0.433 • The Best / Worst case would yield considerably more uncertainty • 0.239 < Sw < 0.50 • In practice, it’s unlikely (but not impossible) that Best / Worst of all attributes would occur simultaneously, so that Sw(Monte Carlo) provides a better representation

24. Derivatives vs Statistics Over-view • Chen & Fang identify the attribute resulting in the greatest Sw uncertainty, with a derivate approach • In the case at right, “m” is the dominant issue • This same issue can be addressed with Monte Carlo simulation (below) • The Base Case is at lower left, with each simulation towards the right reflecting an individual improvement in “Phi”, “m” and “n” precision by 10 %. • Exhibit following

25. Derivatives vs Statistics Over-view • Chen & Fang identify “m” as the dominant issue • The Monte Carlo Base Case is at lower left, with each simulation towards the right reflecting an improvement in “Phi”, “m” and “n” individual precision by 10 %. • Improved definition of cementation exponent results in the most efficient improvement in Monte Carlo Sw estimation(lowest standard deviation in Sw) per Monte Carlo simulation • Monte Carlo is consistent (as expected) with Derivatives

26. The Spreadsheet Spreadsheet may be modified for locally specific conditions • Set up to illustrate various distributions, and to model Phi(Rhob), “m” & Sw(Archie) • Be aware that illustrations are from Excel 2007, in the backwards compatible mode, and Excel 2003 screens may be different • Illustrative application follows

27. The Statistical Approach Additional Details as a Text Document • The Monte Carlomethod is attractive when it is infeasible or impossible to compute an exact result with a deterministic algorithm. • Excel can handle common probability distributions, and can thus serve as a Monte Carlo simulator. • Quantitative estimation of 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 is a linkage in that one parameter becomes more or less important as another parameter value is changed, so the simulation must be executed for locally specific conditions.

28. The Differential Approach Additional Details as a Text Document • An alternative, deterministic approach to error analysis is accomplished by taking the derivative of Sw(Archie) with respect to each attribute • Swn = a Rw / (Fm Rt) • The same approach will suffice for a shaly sand equation • The various terms in the derivative expression quantify the individual impact of uncertainty in each term, upon the result • The relative magnitude then allows one to recognize where the biggest bang for the buck, in terms of a core analyses program, suite of potential logs, etc is to be found (Figure 1).

29. Monte Carlo Simulation of Sw(Archie) • Monte Carlo simulation can be used to characterize the uncertainty in Sw, for the attributes (and associated uncertainties) tabulated at right • “a”, Rw and Rt are assumed to be well-known, reflected here by no STD specification • This simulation is approximating a foot-by-foot Sw interpretation, for which the porosity, “m” & “n” estimates are subject to uncertainty • Porosity (for example) is described by a Gaussian distribution, centered on 20 puwith a standard deviation of 1 pu • Exhibit following Monte Carlo Simulation of Sw(Archie)

30. Monte Carlo Simulation of Sw(Archie) • Porosity is described by a Gaussian distribution, centered on 20 puwith a standard deviation of 1 pu • Two Thousand calculations are performed and the results are accumulated • Exhibit following

31. Monte Carlo Simulation of Sw(Archie) • Monte Carlo simulation can be used to characterize the uncertainty in Sw, for the attributes (and associated uncertainties) tabulated at right • “a”, Rw and Rt are assumed to be well-known, reflected here by no STD specification Monte Carlo Simulation of Sw(Archie) • “m” and “n” are also Gaussian distributed, and centered on 2.0, with standard deviations of 0.1 • “m” & “n” may be varied independently by simply modifying the above Table • Calculations are live-linked • “a”, Rw and Rt may also be varied in the MC model by a straight-forward extension • Exhibit following

32. Monte Carlo Simulation of Sw(Archie) Excel Details • Porosity, “m” & “n” are determined with individual Random Number inputs to NormInv, and are not driven by a single, common, random number source • Exhibit following

33. Monte Carlo Simulation of Sw(Archie) • Calculations are in the Sw_Random worksheet, live-linked to the Sw_Results worksheet Be Careful to enter values appropriately, and not over-write live links

34. Monte Carlo Simulation of Sw(Archie) • To modify spreadsheet to accommodate a different three parameter simulation, simply re-title the various attributes and adjust the individual calculations as required

35. Monte Carlo Simulation of Phi(Rhob) • To modify spreadsheet to accommodate a different three parameter simulation, simply re-title the various attributes and adjust the individual calculations as required. This worksheet models Phi(Rhob).

36. Monte Carlo Simulation of “m(Dual Porosity)” • To modify spreadsheet to accommodate a different three parameter simulation, simply re-title the various attributes and adjust the individual calculations as required. This worksheet models the Wang & Lucia Dual Porosity “m” Exponent

37. Monte Carlo Simulation of Sw(Archie) Cross Check • Porosity is specified as a Gaussian distribution, centered on 20 puwith a standard deviation of 1 pu • 2,000 calculations are done, and the result “checked’ by means of histograming the resulting porosity distribution and calculating the resulting statistics • Exhibit following

38. Monte Carlo Simulation of Sw(Archie) Cross Check • Porosity is specified as a Gaussian distribution, centered on 20 puwith a standard deviation of 1 pu • 2,000 calculations are done, and the result “checked’ by means of histograming the resulting porosity distribution and calculating the resulting statistics • The MC simulation is observed to reproduce the specified inputs, digitally and graphically • Random number “mean” & “Std” converge to input values • Exhibit following

39. Monte Carlo Simulation of Sw(Archie) • As a QC device, the distribution of Excel random numbers used to drive the Monte Carlo simulation, are ‘binned’ from zero to one • With 2000 simulation performed, we expect to find Frequency ~ 200 in each of the ten bins • The Excel Random() function has approached the ideal distribution QC the Monte Carlo

40. Monte Carlo Simulation of Sw(Archie) Sw(Archie) Uncertainty • One issue of interest is the dependence of Sw upon individual attribute values / uncertainties • With the specifications at right • Sw(mean) = 0.357 • s (Sw) = 0.038 • There is a 95% likelihoodthat Sw is contained within + / - 2 s • (0.357 – 0.076) < Sw < (0.357 + 0.076) • 0.28 < Sw < 0.433 • Be aware of how Excel ‘bins’ data • Exhibit following

41. Monte Carlo Simulation of Sw(Archie) Excel Bins • Be aware of how Excel ‘bins’ data • The Freq of a specific Bin is not a centered value • Sw_Bin=0.25  0.225 .LT. Sw .LE. .25 • Sw_Bin=0.275  0.25 .LT. Sw .LE. .275 • Sw_Bin=0.30  0.275 .LT. Sw .LE. .30 • Sw_Bin=0.325  0.30 .LT. Sw .LE. .325 • Sw(Mean)=0.357 but Bin Freq, and associated graphic, peaks to the high side of this value • Bin Values are shifted with respect to actual distribution

42. Nonlinear relationship affects probability distributions. Nonlinear Relation Is An Additional Issue • The Archie relationship for a given formation, with constant “a”, “m”, “n” and Rt results in a hyperbolic relationship of Sw to porosity (blue) • This relationship distorts the frequency distributions, which are shown along the axes • A normal uncertainty distribution about a given porosity (green) becomes a log-normal distribution for the resulting Sw uncertainty (red) • The mean value (dashed yellow) and three sigma points (dashed purple) show the skewed Sw distribution • Sw distributions determined with Monte Carlo modeling are distorted at high values, since Sw cannot exceed 100%. Oilfield Review. Autumn 2002. Ian Bryant, Alberto Malinverno, Michael Prange, Mauro Gonfalini, James Moffat, Dennis Swager, Philippe Theys, Francesca Verga.

43. Monte Carlo vs Best / Worse Case • There is a 95% likelihoodthat Sw is contained within + / - 2 s • 0.28 < Sw < 0.433 • The Best / Worst case corresponding to + / - 2 s, would yield considerably more uncertainty • 0.239 < Sw < 0.50 • In practice, it’s unlikely (but not impossible) that Best / Worst of all attributes would occur simultaneously, so that Sw(Monte Carlo) provides a better representation

44. In Search of The Biggest Bang for the Buck • One issue of interest is the dependence of Sw upon individual attribute values / uncertainties • With the values specified at right • Sw(mean) = 0.357 • s (Sw) = 0.038 Where Is The Biggest Bang For The Buck? • Monte Carlo simulation also allows one to determine s for any possible (improved) input distribution, and to therefore identify where the ‘biggest bang for the buck” in reducing Sw uncertainty is at • Exhibit following

45. Uncertainty in Archie’s Equation Where to spend time, and money, in search of an improved Sw estimate Identifying the Biggest Bang for the Buck • In addition to quantifying the uncertainty in a single Sw(Archie) estimate, one will likely want to prioritize time and budget, in a manner that most efficiently reduces net Sw uncertainty • This can be done with either derivatives or MC simulation • Both options have attractive features • The analytical approach can be easily coded into a foot-by-foot petrophysical evaluation • Monte Carlo simulation can address non-Gaussian distributions. • Both options are adaptable to the spreadsheet environment Derivatives vs Monte Carlo

46. In Search of The Biggest Bang for the Buck • Chen & Fang’s results have been coded to an Excel spreadsheet, to facilitate locale specific, digital evaluation The Derivative Approach • As porosity increases, “m” or “pore system tortuosity” becomes less important, and the tortuosity of the conductive phase, represented by “n”, deserves increased attention, relative to “m” • Exhibit following • At F ~ 30 pu, for stated conditions, the three issues approach one another, in importance

47. In Search of The Biggest Bang for the Buck • Chen & Fang’s results have been coded to an Excel spreadsheet, to facilitate locale specific, digital evaluation Spreadsheet may be modified for locally specific conditions

48. In Search of The Biggest Bang for the Buck • As porosity decreases, “m” or “pore system tortuosity” becomes the dominant issue • In these circumstances the tortuosity of the conductive phase, represented by “n”, can be nearly neglected (in a relative sense) • At F ~ 10 pu, tortuosity in the pore system is far more important than “n” variations The Derivative Approach

49. In Search of The Biggest Bang for the Buck • Chen & Fang’s results should be apparent in a similarly specified Monte Carlo simulation • Consider the below three situations, with corresponding Chen & Fang results at right • “a” = 1: Uncertain(a) = 0% • Rw = 0.02 : Uncertain (Rw) = 0% • Rt = 40 : Uncertain (Rt) = 0% • Phi = 10, 20 & 30 pu : Uncertain (Phi) = 15% • “m” = 2.0 : Uncertain (m) = 10% • “n” = 2 : Uncertain (n) = 5% • In each case, the dominant parameter is identified with the red arrow (at right), per derivatives • Exhibit following Derivatives vs Monte Carlo

50. In Search of The Biggest Bang for the Buck • Derivatives identify the attribute resulting in the greatest Sw uncertainty • In the case at right (f ~10 pu), “m” is the dominant issue • This same issue can be addressed with Monte Carlo simulation • Recalling that + / - 1 s will encompass ~ 68% of the distribution, and 2 s ~ 95 % of the distribution, we approximate the various relative uncertainties per the following exhibit http://en.wikipedia.org/wiki/Standard_deviation