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# Resources and reserves models

Estimating resource and reserves crosses the disciplines between geoscientists and petroleum engineers. While the geoscientist may well have primary responsibility, the engineer must carry the resource and reserve models forward for planning and economics. Volumetric estimates of reserves are among the most common examples of Monte Carlo simulation. They are calculated for:

• Known producing wells
• Reservoirs
• Fields
• Exploration wells
• Mapped prospects and plays
• Unmapped prospects.

The resource and reserve estimates are important in their own right, but in addition, these estimates are inputs and drivers for:

• Capital-expenditure
• Production
• Cash-flow models.

### Volumetric formulas

Consider the following typical volumetric formula to calculate the gas in place, G, in standard cubic feet.

G = 43,560Ahφ(1 - Sw) / Bg × E, ………………………………. (1)

where A = area, acres, h = net pay, ft, φ = porosity, Sw = water saturation, Bg = gas formation volume factor,

and

E = recovery efficiency.

In this formula, there is one component that identifies the prospect, A, while the other factors essentially modify this component. The variable h, for example, should represent the average net pay over the area, A. Similarly, φ represents the average porosity for the specified area, and Sw should represent average water saturation. The central limit theorem guarantees that distributions of average properties—net pay, porosity, and saturation—will tend to be normal. Another consequence of the theorem is that these distributions of averages are relatively narrow (i.e., they are less dispersed than the full distributions of net pays or porosities or saturations from the wells, which might have been log-normal or some other shape. The correct distributions for Monte Carlo analysis, however, are the narrower, normal-shaped ones.

### Input parameter estimation

While we often do not have ample information to estimate the average porosity or average saturation, we are able to imagine what kind of range of porosities might exist from the best to the worst portions of the structure. We do have ample information from many mature fields where material balance could provide estimates. We also have extensive databases with plenty of information, from which some ranges of average values could be calculated and compared to the broader ranges of well data.

Always remember that, as with all else in Monte Carlo simulation, one must be prepared to justify all realizations (i.e., combinations of parameters). Just as we must guard against unlikely combinations of input parameters by incorporating correlations in some models, we should ask ourselves if a given area or volume could conceivably have such an extreme value for average porosity or average saturation. If so, then there must be even more extreme values at certain points within the structure to produce those averages (unless the structure is uniformly endowed with that property).

Perhaps the contrast is even easier to see with net pays. Imagine a play where each drainage area tends to be of relatively uniform thickness, which might be the case for a faulted system. Thus, the average h for a structure is essentially the same as seen by any well within the structure. Then the two distributions would be similar. By contrast, imagine a play where each structure has sharp relief, with wells in the interior having several times the net sand as wells near the pinchouts. Although the various structures could have a fairly wide distribution of average thicknesses, the full distribution of h for all wells could easily be several times as broad.

The traditional manner of describing area and treating it as a log-normal distribution is based on prospects in a play. If we were to select at random some structure in a play, then the appropriate distribution would likely be a log-normal. Sometimes, not even the area parameter should be modeled by a log-normal distribution. The distribution for A could easily be log-normal if the drainage areas were natural. In a faulted system, however, where the drainage areas were defined by faults, the distribution need not be log-normal. Suppose a particular prospect is identified from 3D seismic. We have seen situations where the base case value of area or volume is regarded as a mode (most likely). When asked to reprocess and or reinterpret the data and provide relatively extreme upside (say P95) and downside (say P5) areas or volumes, the results are skewed left—there is more departure from the mode toward the downside than the upside. Because the conventional log-normal distribution is only skewed right, we must select another distribution type, such as the triangular, beta, or gamma distribution.

What if this is correct: that we should be using narrower and more symmetrical distributions for several of the factors in the volumetric formula? Does it matter in the final estimate for reserves or hydrocarbon in place? How much difference could we expect? The right way to judge whether the type of distribution matters for an input variable is to compare what happens to the output of the simulation when one type is substituted for another.

### Variations of the volumetric formula

Among the numerous variations of the volumetric formulas, there is usually only one factor that serves the role of area in the argument. For instance, another common formula estimates original oil in place (OOIP) by

OOIP = 7,758Vb(NTG)φSo / Bo, ……………………… (2)

Where

Vb = bulk rock volume, NTG = net to gross ratio, φ = porosity, So = oil saturation,

and

Bo = oil formation volume factor.

Here, Vb would be the dominant factor, which could be skewed right and modeled by a log-normal distribution, while the factors NTG, φ, So, and Bo would tend to be normally distributed because they represent average properties over the bulk volume.

### Recovery factors

Recovery factors, which convert hydrocarbon in place to reserves or recoverable hydrocarbon, are also average values over the hydrocarbon pore volume. The recovery efficiency may vary over the structure, but when we multiply the OOIP by a number to get recoverable oil, the assumption is that this value is an average over the OOIP volume. As such, the recovery factor distribution often should be normally distributed. Additional complications arise, however, because of uncertainty about the range of driving mechanisms. Will there be a waterdrive? Will gas injection or water injection be effective? These aspects of uncertainty can be modeled with discrete variables, much like the probability of stuck pipe in the drilling AFE example.

### Output from the simulation, OOIP, or gas initially in place (GIIP)

The Monte Carlo simulation yields a skewed right output (loosely speaking, “products are log-normal”), such as shown in Fig. 7, regardless of the shapes of the inputs. The result follows from:

• The definition of log-normal: a distribution the logarithm of which is normal
• The central limit theorem (sums are normal)
• The log of a product is the sum of the logs

One notable example from Caldwell and Heather uses five triangular distributions, two of which are sharply skewed left, one symmetric, and two slightly skewed right, to obtain coalbed gas reserves as a sharply right-skewed output. Regardless of the shapes of the inputs to a volumetric model—be they skewed right, skewed left, or symmetric—the output will still be skewed right, thus approximately log-normal. The central limit theorem guarantees this, because the log of a product (of distributions) is a sum of the logs (of distributions), which tends to be normal. Thus, the product, the log of which is normal, satisfies the definition of a log-normal distribution.

## Considerations

### Handling correlation among inputs

In the discussion so far, the input parameters have been described, and handled, as if they were each independent of one another. In many geologic settings, however, these input parameters would have an interdependency. This can be incorporated in our models by using correlation between the appropriate parameters. Some of the correlations that we apply result from fundamental principles in petroleum engineering. One such correlation that should be included in many volumetric models is that in a clastic, water-wet rock, water saturation and porosity are negatively correlated. In the volumetric formula, that relationship leads to a positive correlation between hydrocarbon saturation and porosity. Other correlations may be necessary, depending on the geologic story that goes hand in hand with the resource and reserve estimates. Fig. 8 shows the typical impact of positive correlation in a volumetric product model—the resulting distribution has greater uncertainty (standard deviation, range) and a higher mean than the uncorrelated estimate.

### Probability of geologic success

The hydrocarbon-in-place (resource) estimates become reserve estimates by multiplying by recovery factors. Until we model the probability of success, the implication is that we have 100% chance of encountering those resources. In reality, we must also incorporate the probability of geologic success in both our resource and reserve estimates. If the P (S) for the volumetric example is assigned as 20%, we would use a binomial distribution to represent that parameter, and the resulting reserve distribution will have a spike at zero for the failure case (with 80% probability). Fig. 9 illustrates the risked reserve estimate. (Note that the success case is as illustrated in Fig. 7.)

### Layers and multiple prospects

Often a well or a prospect has multiple horizons, each with its chance of success and its volumetric estimate of reserves. A proper evaluation of these prospects acknowledges the multiplicity of possible outcomes, ranging from total failure to total success. If the layers are independent, it is simple to assign probabilities to these outcomes in the manner already discussed. Whether one seeks a simple mean value or a more sophisticated Monte Carlo simulation, the independence assumption gives a straightforward procedure for estimating aggregate reserves.

When the layers are dependent, however, the aggregation problem becomes subtler: the success or failure of one layer alters the chance of success of other layers, and the corresponding probabilities of the various combinations of successes are more difficult to calculate. Moreover, the rules of conditional probability, notably consequences of Bayes’ Theorem, provide challenges to those who assign estimates for the revised values. Even in the case of two layers, some estimators incorrectly assign these values by failing to correctly quantify their interdependence. These issues have been addressed by Murtha, Delfiner, and Stabell, who offer alternative procedures for handling dependence, be it between layers, reservoirs, or prospects.

### Summary

The main factor in the volumetric equation, area or bulk volume or net rock volume, can be skewed left, symmetric, or skewed right. The other factors in a volumetric formula for hydrocarbons in place will tend to have symmetric distributions and can be modeled as normal random variables. Regardless of the shapes of these input distributions, the outputs of volumetric formulas, namely hydrocarbons in place and reserves, tend to be skewed right or approximately log-normal. Many of the natural correlations among the volumetric equation input parameters are positive correlations, leading to reserve distributions that have higher means and larger ranges (more accurately, larger standard deviations) than the uncorrelated estimates. The probability of geologic success can be modeled using a binomial variable. Finally, modeling layers or multiple prospects is accomplished by aggregating the individual layer or prospect estimates within a Monte Carlo simulation. In those cases where there is geologic dependence between the success of the layers (or prospects), that dependence can be modeled using correlation between the binomial variables representing P (S).

## Nomenclature

 A = area, acres; one or a set of mutually exclusive and exhaustive Bayesian-type events Bg = gas formation volume factor, dimensionless ratio Bo = oil formation volume factor, dimensionless ratio E = recovery efficiency, % h = net pay, ft NTG = net to gross ratio So = oil saturation, dimensionless ratio Vb = bulk rock volume, vol φ = porosity, dimensionless ratio