limitations of statistical methods for estimating hydrocarbon reserves

8/8/2019 Limitations of Statistical Methods for Estimating Hydrocarbon Reserves

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tations of Statistical Methods For Estimating Hydrocarbon Reserves

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Limitations of Statistical Methods For Estimating Hydrocarbon Reserves

Abstract

Statistical (stochastic) methods are widely used to calculate hydrocarbon reserves. With modern software itis very quick and easy to generate probability distributions that describe uncertainty in the reservescalculations.

Without exercising extreme care, the statistical method can lead to serious errors in estimating reserves.Close examination of the data is essential to ensure appropriate use of parameter probability distributionfunctions (PDFs). Extensive documentation is also required that fully explains the derivation of these

PDFs. It is also essential that any parameter dependence is fully documented - parameter independencemust be demonstrated.

There should be no expectation that statistical methods provide either improved accuracy or betterassessment of uncertainty. The statistical method is unlikely to have a rigorous audit trail and themethodology gives no information about the spatial distribution of the hydrocarbon reserves.

Introduction

In this article we investigate some limitations of statistical methods of estimating hydrocarbon reserves.

Statistical reserves modelling has been promoted as a means to provide a more accurate calculation and tobetter characterize uncertainty. In the following sections we will show that the probability distribution ofcalculated hydrocarbon reserves is dramatically affected by the methods and assumptions used to analyseand process the available reservoir data. There should be no expectation that statistical methods provideeither improved accuracy or better assessment of uncertainty. The statistical method is unlikely to have arigorous audit trail and the methodology gives no information about the spatial distribution of thehydrocarbon reserves.

Only at the time of final field abandonment are the hydrocarbon reserves known with certainty. Until thattime there is always uncertainty in calculating the volumes of hydrocarbon that exist in reservoirs and thequantities that can be economically extracted.


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Uncertainty in reserves estimates results from limitations of the input parameter data where the data isincomplete either because of the high cost of obtaining the data or the inability to access existingproprietary reservoir data. A related source of uncertainty is bias, resulting from inappropriate use of theavailable data. Bias is a predisposition of the interpreter and is not necessarily conscious or intentional.

Hydrocarbon reserves can be estimated using either deterministic or statistical methods. The deterministicmethod uses a single value, or a very limited range of values, for each input parameter in the reservescalculation. The values for each parameter are those that are deemed to be the most appropriate.Limitations of the deterministic method are the logistics in revising the estimates and the very limited

scope for describing the uncertainty of the resulting reserves estimate. However, the deterministic methodis likely to have a robust audit trail and an excellent visualisation of the spatial hydrocarbon distribution.

Statistical methods are very widely used to model the uncertainty is estimating oil and gas reserves,especially in the exploration and early appraisal stages. Each parameter in the reserves calculation isassigned a range of possible values with their associated probability. Monte Carlo simulation is used torepeatedly sample random values from the parameter probability distributions. These values are then usedto calculate a reserves volume. The results of the Monte Carlo simulation are then sorted to yield aprobability distribution for the reserves. The expectation is that the process of incorporating uncertaintyfunctions will result in a correct analysis of the range of possible reserves.

Data for Stochastic Reserves Estimation

In this article we have used, for illustation and modelling purposes, data from;

Benmore, R., Cooper, M., Wells, B., "Stochastic Modelling for Reducing Risk in Prospect

Evaluation", AAPG Annual Convention, Houston, Tx., March 1995.

The Benmore et al. paper describes the statistical analysis of expected reserves in a North Sea satellitedevelopment prospect. Since the area is a mature basin, there is extensive reservoir data. Table 1 andfigure 1 show the reservoir parameters from the Benmore et al, 1995 paper.

Table 1: Drillable Prospect Reservoir Parameter Table

Parameter LowsideMostLikely

HighSide

Gross RockVolume(Acre-feet)

7600 70150 246400

Porosity (%) 14 18 24

Net to Gross(%) 26 40 58

HydrocarbonSaturation(%)

15 48 78

Formation VolumeFactor(RB/STB)

1.38 1.25 1.12

Recover Factor(%) 10 15 30


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Figure 1: Reservoir Parameter Triangular Probability

Distributions

Porosity Model

Figure 2 shows a histogram of the porosity data from the Benmore et al., 1995 paper on stochastic reservesmodelling. The histogram shows that the data is very negatively skewed - it has a long 'tail' towards lowporosity values. The average porosity value is 20.69 percent and the modal (most common) value is about22 percent.

Figure 2: Histogram of porosity data

For their stochastic prospect evaluation, Benmore et al. modelled porosity using a nearly symmetricaltriangular probability distribution function (Low = 14%, ML = 18%, High = 24%). The green dashed linein figure 3 shows that the cumulative probability distribution function (CDF) for this triangulardistribution is a very poor match for the actual data (blue line in figure 3).

The orange line in figure 3, derived from a highly asymmetric triangular probability distribution (Low =14%, ML = 22.2%, High = 24%) , matches the available data quite well.


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Porosity data is very frequently misused in statistical reserves calculations. We are interested in modellingthe uncertainty of the average value in the Gross Rock Volume. It is absolutely incorrect to use the rangeof values seen in a data set. The porosity average uncertainty function is expected to have a normalprobability density function (bell-shaped curve) centered on the population mean and having a standarddeviation (Standard Error) that is far less than the sample standard deviation. The red curve in figure 3 is aCDF showing the uncertainty of the average porosity value. This CDF has a mean of 20.69 percent(average of all of the available porosity data) and a standard deviation of 0.576 percent (the sampleStandard Error).

Standard Error is calculated by dividing the sample standard deviation by the

square root of the number of samples. As the number of samples increases the

Standard Error (uncertainty of the average value) will decrease.

Figure 3: Porosity Cumulative Distribution Functions (CDF)

Hydrocarbon Saturation Model

The saturation-height functions in the upper portion of figure 4 show that there is a strong correlationbetween porosity and hydrocarbon saturation. The lower part of figure 4 shows two functions relatinghydrocarbon pore-volume to porosity. The blue curve is derived from the saturation-height curves. Thepurple curve is derived using the porosity and hydrocarbon saturation triangular distribution data in table 1and assuming that there is a correlation between porosity and hydrocarbon saturation. There is a very

significant difference between these two functions.


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Figure 4 : Saturation-height Functions Showing a Strong Porosity-ShDependence

Figure 5 shows four possible hydrocarbon saturation CDF curves. The black curve shows the hydrocarbonsaturation CDF calculated using the triangular probability density function data in table 1. This curveappears to be a very poor match with the saturation-height derived Sh data.

The blue, red and green curves incorporate the Sh-porosity dependence derived from the saturation-heightcurves in figure 4.

The blue curve uses the porosity CDF calculated from the individual porosity measurements (data is sortedby increasing value and plotted against the corresponding cumulative probability).

The red curve is a reasonable match for the sample data. It was derived using the porosity triangularprobability density data in table 1. The apparent good match with the data is misleading, since it wasshown previously that the porosity triangular probability distribution data in table 1 is a very poor matchfor the available porosity data (blue curve in figure 3).

The green curve in figure 5 is an estimate of the probability distribution of the Sh average value. Porositydata was first used to calculate hydrocarbon saturation. The average and Standard Error of these estimates


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Net To Gross Model

Figure 7 shows a crossplot of porosity and net to gross data. For this data the correlation between porosityand net to gross is very weak and is if no practical use for modelling hydrocarbon reserves.

Note: In some data sets there is a strong porosity-NTG correlation that should be modelled

for Monte Carlo simulation.

The Net to Gross data shown in figure 7 is clearly derived from Petrophysical zone summaries (coupledporosity and NTG values). This Net to Gross data is of only very limited use for statistical prospectreserves calculations;

- We have no way of knowing the location of the net reservoir within the stratigraphic interval ofinterest.

- We do not know if the data is for contiguous reservoir units. Petrophysics summary tables rarely listthe properties of non-net intervals.

- We do not know the relationship between unit thickness and net to gross. It is quite possible that thehigh NTG values correspond to relatively thin stratigraphic intervals.

All of the above factors are absolutely critical for a robust modelling of drillable prospect reserves

uncertainty. It is highly likely that non-reservoir (zero net-to-gross) intervals exist in the stratigraphicinterval of interest. These intervals are rarely included in Petrophysics summary tables. It is also essentialto know the vertical distribution of the reservoirs. Low (or high) NTG zones near the base of the intervalof interest will have a very dramatic effect on the expected hydrocarbon reserves (net sands have a largearea). Zone thickness information corresponding to the porosity and NTG data is also vital.

Figure 7 : Crossplot of Net to Gross and Porosity Data

Figure 8 shows three possible NTG Cumulative Probability Distribution Functions. The blue CDF isderived by sorting the available data. The green dashed CDF shows that the available NTG data is quitewell described by the triangular distribution data in table 1. However, the probability distribution of theactual NTG values is of very limited use in reserves calculation. In fact it is both incorrect and misleading.We are most concerned with our ability to describe the uncertainty of the average value in the entire grossrock volume. The red curve in figure 8 describes the CDF for the sample average NTG. It is thecumulative probability distribution of a normal PDF with an average value of 42.5 percent and a standard


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error of 1.88 percent.

Figure 8 : Net to Gross Cumulative Distribution Functions

Recovery Factor

Rarely would it be appropriate to use a recovery factor PDF to estimate potential reserves in a multi-reservoir exploration prospect. Recovery factors are calculated using specific information on severalreservoir factors, for example; in-place reserves contained in a modelled drainage area, expected reservoircontinuity, drive mechanism, proximity of water or gas contacts, depletion strategy and individual welleconomics. In a following section we have used a very simple example to show the effects of modellingrecovery factor as a dependent function of gross rock volume. However, the recovery factor PDF must

also be linked to an actual or conceptual field development plan.

The Impact of Parameter Dependence on Stochastically Generated Reserves

The green curve in figure 9 shows the probability distribution of reserves calculated using a restrictedMonte-Carlo simulation. The Gross Rock Volume, Net to Gross, Formation Volume Factor and RecoveryFactor parameters were held constant at the most likely values in table 1. The Monte-Carlo simulationassumed that porosity and hydrocarbon saturation are independent variables.

The porosity and Sh CDF functions were sampled by finding the value corresponding to random numbers0


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Figure 9 : Stochastic Simulation of Reserves Showing Impact ofPorosity-Sh Dependence

The purple curve in figure 9 shows the results of modelling the Sh dependence on porosity. As with theprevious Monte-Carlo simulation, the Gross Rock Volume, Net to Gross, Formation Volume Factor andRecovery Factor parameters were held constant at the most likely values in table 1.

The porosity CDF function was sampled by finding the value corresponding to random numbers 0


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HPV = -0.4001 x Porosity 2 + 1.5275 x Porosity - 1.27Sh = HPV/Porosity

All of the other hydrocarbon reserves parameters in table 1 were treated as independent variables.

The reserves calculated using a porosity-dependent hydrocarbon saturation (green line) are far larger thanthe reserves calculated using independent variables (black line). For this data set, the difference resultsprimarily because the porosity and saturation triangular distribution data in table 1 is a very poor match

for the available data.

Figure 10 : Reserves Probability Distribution

Case 3: Porosity Probability Distributions with Modelled Porosity-Sh Dependence

The green curve in figure 11 shows the results of using the porosity CDF derived directly from theavailable data (blue curve in figure 3)

The purple curve in figure 11 shows the results of modelling the uncertainty of the average porosity in thegross rock volume. The available porosity data has an average of 20.69 percent and a standard deviation of2.57 percent. The distribution of averages is therefore modelled as having an average of 20.69 percent(assume average = population mean) and a Standard Error of 0.575 percent.

As with Case 2, hydrocarbon saturation is modelled as a dependent function of porosity.

As expected, the resulting reserves probability distribution has lower variance than the results from Case 2.


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Figure 11 : Monte Carlo Reserves Simulation

Case 4: Modelling the Results of Recover Factor Dependent On GRV

For the data described by Benmore et al. 1995 it is very likely that recovery factor is inversely related togross rock volume. The GRV vs depth plot from Benmore et al. 1995 is shown in the top left panel offigure 12. The top right panel shows Recovery Factor plotted against GRV. The recovery factor data isfrom the table 1 triangular probability distributions. The bottom panel in figure 12 shows the impact ofmodelling RF as a GRV-dependent variable. As expected, there is a significant reduction in the reservesbecause of the low recovery factor interpreted in the large area towards the lowest spill point.

Figure 12 : Modelling the effects of a relationship between GRV and

Recovery Factor


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Figure 13 and table 2 show the results of the above Monte Carlo simulation models. It is appealing toconclude that inclusion of parameter dependence into the Monte Carlo simulation resulted in a robustevaluation of the expected reserves probability distribution for this satellite development prospect. This isincorrect because we have shown that the available data is both incomplete and provides no information onthe spatial distribution of the reservoir parameters.

Figure 13 : Summary of Monte Carlo Reserves Simulation

Table 2: Summary of Stochastic Modelling

ModelP90

Lowside

P50

Most_Likely

P10

Upside

Spread

(P90-P50)

Independent Triangular Distributions 1.40 3.84 8.32 6.92

Porosity CDF from datasetSh as a porosity-dependent function usingSaturation-height functions

2.90 7.54 15.91 13.01

Model as above but with Recovery Factor modelledas a function of GRV

2.77 5.61 9.53 6.76

Model as above but using the porosity averageuncertainty function

3.32 5.94 9.72 5.95

Summary and Conclusions

With modern software it is both quick and easy to implement statistical (stochastic) methods for estimatinghydrocarbon reserves. However, the method is not a robust and reliable method for modelling theuncertainty in the hydrocarbon reserves calculations.

Humans have a natural inclination to;- Underestimate the complexity of systems (in this case the geological parameters required to calculatereserves),


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- Be over confident in their ability to characterize uncertainty, and,- Propose that extreme values have a possibility of being representative of the entire system (porosity, forexample).

The vertical distribution of hydrocarbons in a reservoir is a complex function of porosity, grain size,permeability, fluid density/viscosity and distance from fluid contacts. The saturation-height functionsshown in figure 4 are gross simplifications. Statistical (stochastic) methods are typically inadequate formodelling these complexities. For example, it would be extremely rare to find an example where astochastic model adequately characterized the hydrocarbon-water transition zone. This is part of the

hydrocarbon system with the largest gross rock volume. In low porosity/permeability reservoirs, where thetransition zone can be extremely thick, failure to correctly model hydrocarbon saturation can lead to grossoverestimation of reserves.

Without exercising extreme care the statistical method can lead to serious errors in estimating reserves.Close examination of the data is essential for a robust reserves assessment. Extensive documentation isrequired that fully explains the derivation of the input parameter probability distribution functions.

It is imperative that any parameter dependence is adequately modelled. It also must be proven thatparameters assumed to be independent are not correlated with any other input parameter.

The Statistical method should address only the uncertainty of the expected (mean) value of the inputparameter in the entire gross rock volume of interest. The CDF must define the uncertainty in the estimateof the average. This is not the same as the actual range of values observed in the reservoir.

It is crucial that the parameter probability distributions are appropriate for the rock volumes beingevaluated. For example, it is completely incorrect to use the highest measured porosity value as having arealistic possibility in the gross rock volume being evaluated. An example would be the decision to use thehighest fracture porosity measured in a hand or core sample as having a possibility of being realised in afracture reservoir play.

The spatial distribution of reservoir parameters must be modelled and documented. Using tabulatedreservoir summaries of porosity, net to gross and hydrocarbon saturation is absolutely useless without

knowing the stratigraphic (vertical and areal) distribution of these properties.

Recovery factor probability distributions must be used with extreme caution in Monte Carlo reservessimulations. Simulator derived recovery factors that ignore economic flow rates or capital investmentreturns can be very misleading.

Appendix 1: Calculation of a CDF From Triangular Probability Distribution Data

Figure 14 shows how to calculate the cumulative probability distribution function (CDF) from thetriangular probability density function (PDF).

a is the minimum value,b is the maximum value,c is the most likely value, and,x is any value between a and b.CDF is the cumulative probability distribution function (CDF)


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Figure 14 : Triangular Probability Density Function

Appendix 2: Porosity Population Modelling

Reservoir porosity data sets are often composed of sub-populations that each typically have normalprobability density functions (the PDF's have a characteristic bell shape). The histogram in figure 2indicates that the porosity data is multi-modal (the data consists of a composite of sub populations).Examination of porosity data and the derived histogram indicates that there are at least 3 overlapping sub-populations with the following properties;

Sub-PopulationFraction of

Total Population(%)

Average

(%)

Standard Deviation

(%)

1 15 15.9 1.9

2 20 19.0 1.2

3 65 22.1 0.94

Figure 15 shows the normal probability distributions for these 3 sub-populations in the porosity data.


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Figure 15 : Stochastic Simulation of Porosity CDF

The porosity population CDF can be can be stochastically simulated if the following parameters areknown, or can be estimated;

Proportion of each sub-population and,The porosity average and standard deviation for each of the sub-populations,

The stochastic simulation program steps are;

1. Generate a random number from 0


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Porosity = WorksheetFunction.NormInv(u, 0.159, 0.019)

'Sub-population 2

ElseIf u


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Figure 16 : Porosity Sample Probability Density Functions

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