Using copulas for implementation of variable dependencies in petroleum resource assessment: Example from Beaufort-Mackenzie Basin, Canada Zhuoheng Chen, Kirk G. Osadetz, James Dixon, and James Dietrich
ABSTRACT Petroleum resource potential modeling seeks to characterize undiscovered petroleum resources. This information from the modeling can contribute to a reduction in corporate risk while characterizing the commercial potential of the undiscovered resources. Such models consider different types of variable dependencies arising from geologic risk evaluation, volumetric calculation, and resource aggregation to higher geographic levels. Commonly, the available data are not sufficient to specify such variable correlations or interdependencies, particularly in frontier regions. It is also a challenge to formulate variable correlations in resource calculations because geologic variables have to be fit to a multivariate lognormal distribution or other specific multivariate distributions with an appropriate correlation structure. However, variable correlations are common among the geologic variables, and ignoring the interdependencies may lead to a serious bias in the resource potential estimation and the uncertainty range. Recent methodological developments in statistics indicate that the use of copulas permits more flexibility for the consideration and incorporation of variable interdependency, thus analogs can be introduced to problems where estimating correlation structures are impossible
Copyright ©2012. The American Association of Petroleum Geologists. All rights reserved. Manuscript received December 13, 2010; provisional acceptance April 5, 2011; revised manuscript received June 20, 2011; final acceptance June 30, 2011. DOI:10.1306/06301110196
AAPG Bulletin, v. 96, no. 3 (March 2012), pp. 439–457
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AUTHORS Zhuoheng Chen Geological Survey of Canada, 3303-33rd Street, North West, Calgary, Alberta, Canada, T2L 2A7;
[email protected] Zhuoheng Chen obtained his Ph.D. from the Norwegian University of Science and Technology in 1993 and held a position as an associate professor at China University of Petroleum (Beijing) before joining the Geological Survey of Canada in 1998. His research interests include petroleum resource assessment (methods and applications), petroleum systems, and basin analysis. Kirk G. Osadetz Geological Survey of Canada, 3303-33rd Street, North West, Calgary, Alberta, Canada, T2L 2A7;
[email protected] Kirk Osadetz graduated from the University of Toronto (B.Sc. degree, 1978; M.Sc. degree, 1983). He manages the Earth Science Sector Gas Hydrates Fuel of the Future Program and is the head of the Laboratory Services Subdivision at the Geological Survey of Canada in Calgary. He is active regarding petroleum resource evaluation and has research interests in gas hydrates, tectonics, and thermochronology. He previously worked at Gulf Canada Resources, Inc., and PetroCanada Resources, Inc., in Calgary. James Dixon Geological Survey of Canada, 3303-33rd Street, North West, Calgary, Alberta, Canada, T2L 2A7;
[email protected] James (Jim) Dixon has a B.Sc. degree (Reading, United Kingdom) and a Ph.D. (Ottawa, Canada) in geology, with more than 37 yr of experience in the petroleum industry and as an officer of the Geological Survey of Canada. His expertise is in stratigraphic analysis, clastic sedimentology, basin analysis, and play definition. He recently retired from the Geological Survey of Canada. James Dietrich Geological Survey of Canada, 3303-33rd Street, North West, Calgary, Alberta, Canada, T2L 2A7;
[email protected] Jim Dietrich is a petroleum geophysicist at the Geological Survey of Canada (Calgary Division). He graduated from the University of Waterloo in 1977 with a B.Sc. degree in earth sciences. His research activities include studies of seismic reflection data and petroleum resource assessments of sedimentary basins in Canada.
ACKNOWLEDGEMENTS We thank Giles Morrell of Indian and Northern Affairs Canada for all the discussions on petroleum geology and resource assessment of Beaufort-Mackenzie Basin and helpful inputs on this study. The reservoir volumetric data in this study are from the National Energy Board. We also thank the AAPG Bulletin reviewer Kurt J. Steffen, an anonymous reviewer, and our internal reviewer D. Lepard of the Geological Survey of Canada for the helpful comments and suggestions. This is ESS contribution no. 20110031. The AAPG Editor thanks the following reviewers for their work on this paper: Kurt J. Steffen and an anonymous reviewer.
DATASHARE 42 Appendixes 1 and 2 are accessible in electronic version on the AAPG Website (www.aapg.org /datashare) as Datashare 42.
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and wider choices of statistical distributions become available. This article proposes the use of copulas for handling variable dependency in petroleum resource assessment. The methods and procedures are illustrated using examples from a hypothetical data set and the crude oil resource appraisal of Tertiary clastic plays in Beaufort-Mackenzie Basin in Arctic Canada. Comparisons of crude oil resource estimates obtained using different correlation scenarios for these plays suggest that when positive correlations are used, the mean value of the oil resource is increased about 1.6 times that estimated, assuming a complete independence among the input variables.
INTRODUCTION Petroleum resource potential modeling seeks to characterize undiscovered petroleum resources. This information can contribute to a reduction in corporate risk while characterizing the commercial undiscovered resource potentials. The undiscovered resource potential inference commonly requires calculations that consider material and physical geologic variables and risk characteristics. Direct indications that these variables may not be independent commonly exist, as reservoir characteristics such as porosity, cap rock strength, pressure, and temperature are clearly dependent on depth. Hence, variable dependency is a well-known problem for petroleum resource assessment. Three types of variable dependencies in resource assessment use the probabilistic volumetric approach (Lee and Wang, 1983): geologic risk evaluation, volumetric resource calculation, and resource aggregation from the assessment unit (AU) level to a higher political or geographic level. Lee et al. (1989) examined the dependence among play risk variables using a conditional probability approach. The study found that variable dependence has an impact on geologic risk estimation in the Huanghua depression case study from the Bohai Bay Basin, China. Chen et al. (1994) used data from well-explored basins to study the impacts of variable dependencies on reservoir volume calculations at the play level. Crovelli and Balay (1986), in their assessment package FASP (fast appraisal system for petroleum resource), formulated a perfect positive correlation between reservoir porosity and hydrocarbon saturation for volumetric calculation under a lognormal assumption while allowing for the full range of correlations in play resource aggregation. The U.S. Geological Survey (USGS), in their 2000 World Petroleum Resource Assessment, assumed a perfect positive correlation when aggregating undiscovered
Using Copulas for Implementation of Variable Dependencies in Resource Assessment
resource volumes from an AU level up to the regional level. In addition, they used a 0.5 positive correlation when aggregating the eight regions assessed into global potential estimate (Klett et al., 2000). However, no specific reasons were provided explaining why they used a perfect positive correlation at AU level and partial correlation (0.5) for global resource aggregation. In a recent CircumArctic Petroleum Resource Assessment, the USGS used a more complex scheme that used paired correlation coefficients for regional resource aggregation (Schuenemeyer and Gautier, 2009). This approach requires expert judgment of pairwise correlations (probabilistic dependencies) on charge, traps, and rocks among distinct but geologically correlated AUs, which may have difficulties in directly constructing a positive definite correlation matrix. Using the Alaska gas hydrate resource assessment as an example, Kaufman and Schuenemeyer (2010) demonstrate the use of hierarchical modeling for reducing data input subjectivity and solving the problem in having a positive definite correlation matrix. However, no variable dependence was considered for either the risk evaluation or volumetric resource calculation. Data in well-explored basins suggest that partial correlations are common among volumetric pool attributes (Chen et al., 1994, 2009; Kaufman, 1996) and that these are not restricted to reservoir porosity and oil saturation only. Although efforts have been made to address variable dependence in both methodology and tool development, the greatest emphasis and attention have been given to resource aggregation. Until now, the impact of interdependencies among variables in volumetric resource calculations has been mostly ignored, and the implementation of variable dependency remains a challenge to petroleum resource appraisal. In practice, inadequate data commonly exist to either specify a standard multivariate distribution with an appropriate correlation structure or to quantify the resource aggregation correlation matrices. However, variable correlations are so common among geologic variables that ignoring their interdependence may lead to serious bias, affecting both the resulting resource potential estimation and its uncertainty ranges.
Furthermore, recent worldwide field reserve growth studies indicate that, during the last two decades, crude oil reserve addition from the existing fields has added reserves comparable to the contribution from new discoveries (Klett et al., 2005). In the past, much field growth accompanied field development and delineation (Barrett, 2004). With improved geophysical tools and technologies, much current reserve growth comes now from improvements in recovery factor instead of from initial inplace volume revision. Thus, future reserve growth should be included in the estimated ultimate resource (EUR) in resource assessment. However, statistics from well-explored basins, such as the Western Canada sedimentary basin (WCSB), indicate a positive correlation between the ultimate crude oil recovery factors and the in-place crude oil reserve volume within a petroleum play. Thus, reserve growth can be treated as a variable dependency problem for the EUR at a play level, which becomes another variable-type dependency problem to be considered by petroleum resource assessment. Study and application suggest that copulas provide an alternative way to model joint distributions of random variables with greater flexibility both in terms of marginal distributions and the dependence structure. It has become increasingly popular both among academics and practitioners in many fields involving multivariate data simulation and nonlinear dependence structure, such as in finance (Embretchts et al., 2001), hydrological, and meteorological studies (Bárdossy and Li, 2008; AghaKouchak et al., 2010a, b). The use of copulas allows a more flexible way for the implementation of variable dependence in modeling. Copulas have no restrictions in the form of statistical models considered, and they permits the use of analogs, thus providing a convenient way to study and implement variable dependence. In this article, we discuss a variety of variable interdependencies and the use of copulas to incorporate variable dependencies in petroleum resource assessment. The impacts of variable dependency on crude oil resource assessments are illustrated using a hypothetical data set and four established petroleum plays from the Beaufort-Mackenzie Basin (BMB), Arctic Canada. Chen at al.
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METHODS Variable Dependencies in Resource Assessment Three typical variable interdependencies are involved in a regional petroleum resource assessment. These include play geologic risk evaluation, resource volume potential calculation in an AU, and resource aggregation to higher geographic levels. Geologic Risk Calculation The traditional play analysis approach defines the dry-hole risk as two components: a shared risk common to all prospects in the play and a local risk specific to individual prospects (White, 1988). If all the regional geologic controls are present favorably in a play (e.g., a potential petroleum source rock is present), petroleum accumulations are likely to occur. If any one or more shared characteristics are missing or unfavorably developed (e.g., unfavorable timing: petroleum generation precedes trap formation significantly), it will be likely that all the prospects in such a play will be dry. The local risk is specific to an individual prospect, but independent of the others in the play. The play risk is the product of the shared risk and the average of the local specific risk. Let GRi denote the regional geologic risk factors and GPk the prospect-specific geologic risk factors. The probabilities of the presence of the shared and prospect specific geologic factors for hydrocarbon occurrence in a play are defined respectively by (Lee, 2008) q gr ðiÞ ¼ P½a regional geologic factor satisfied for HC occurrence; i ¼ 1; 2; . . . m ð1Þ q gp ðkÞ ¼ P½a prospect specific factor satisfied for HC occurrence; k ¼ 1; 2; . . . l ð2Þ If all the geologic factors are independent, the play hydrocarbon occurrence chance is given by the following expression: q¼
m Y i¼1
q gr ðiÞ
l Y
q gp ðkÞ
ð3Þ
k
Appendix 1 (Datashare 42, www.aapg.org /datashare) provides the geologic risk variable list 442
for the calculations of shared play and prospectspecific risks. Natural processes leading to petroleum accumulation are seldom independent events, but likely correlated in one way or the other, as illustrated above. Studies show that geologic risk factors are likely correlated (Lee et al., 1989). For example, if a source rock also serves as a top seal, the source rock probability and the top seal probability are likely correlated. Another example could be a regional fault serving simultaneously as both, a vertical hydrocarbon migration path as well as a part of the trap geometry. As a result, the migration and preservation probabilities could be correlated. In this case, the conditional probability formulation below can accommodate the correlation (Lee et al., 1989) q ¼ P½GR1 \ GR2 \ . . . \ GRm \ GP1 . . . \ GPl ð4Þ One of the major obstacles to geologic risk evaluation using equation 4 is that the available data are insufficient to define the correlations among the geologic factors. The play risk may vary playwide spatially, and it is commonly represented as a contour map (White, 1988; Hood et al., 2000; Hood and Stabell, 2006). Chen and Osadetz (2006) proposed a Bayesian multivariate approach for estimating play risk. Hu et al. (2009) provided an example that used a spatial statistical approach. These studies suggest that play risk can be treated as spatial random variable such that a distribution, instead of a single overall risk factor, would better characterize the true nature of the play risk. A major advantage of using a distribution is that it would result in a description of risk-variable dependence that would be similar to the description of reservoir attribute variations that are used in the reservoir volumetric resource potential calculation. Field Size Distribution The field size, zi, can be expressed as the product of volumetric variable, yi, using a reservoir engineering equation: p Y z¼C yi ð5Þ
Using Copulas for Implementation of Variable Dependencies in Resource Assessment
i¼1
where C is a unit of measure conversion and p denotes the number of volumetric variables. The exact expression and the meaning of the p volumetric attributes of petroleum accumulation from the reservoir engineering equation are discussed elsewhere in detail (Crovelli and Balay, 1986; Lee 2008). Appendix 1 (Datashare 42) provides a full formulation of the reservoir engineering equations for oil and gas resource calculations. Under a field size lognormal distribution assumption, the field size probability density function is given by 1 ðlnðzÞ mÞ2 f ðzÞ ¼ pffiffiffiffiffiffi exp ð6Þ 2s 2 2p sz with m ¼ lnðCÞ +
p X
mi
i¼1
and s2 ¼
n X
XX s 2ij s 2i + 2
i¼1
i629 million bbl). Therefore, the reserve growth caused by enhanced recovery in a play can be accounted for by introducing a positive correlation between an inplace resource and the recovery factor like that shown in equation 7. Regional Resource Aggregation The resource potential Rtotal in a basin or a higher geographic unit is the sum of its resources Vi (g) in all AUs: Rtotal ¼
Using Copulas for Implementation of Variable Dependencies in Resource Assessment
m X i¼1
Vi ðgÞ
ð8Þ
where g indicates the correlation relationship among estimated resources. Inspections of geologic maps of variations within and among stratigraphic units that contain petroleum indicate that petroleum play parameters and petroleum pool variables are geographically dependent within plays and interdependent among plays. Regional tectonic processes establish spatially and temporally varying tectonic subsidence patterns (Beaumont, 1981; Bond and Kominz, 1984). The tectonic framework subsequently exerts influence on depositional and burial processes, which will persist through the depositional history and subsequent deformation of a petroleum basin, affecting everything from source rock facies and thermal maturity to reservoir thickness and quality, including the volume and composition of petroleum that enters secondary migration pathways and that might be subsequently entrapped for eventual discovery and production. The influence and interdependency of these processes are manifest as geographic controls and dependencies among petroleum accumulations and their characteristics such as reservoir thickness and quality variations that geographically follow reservoir depth and physical state (i.e., pressure and temperature) that may also be geographically correlated to petroleum source rock thermal maturity patterns. Likewise, when individual stratigraphically defined plays are geographically aggregated, the regional tectonic setting may contain persistent distal-proximal features such that specific pool characteristics, such as reservoir thickness, may depend strongly on geographic position regardless of stratigraphic level. As discussed above, the play resources in a basin are likely correlated because of shared petroleum system elements, such as source rocks, regional top seal, migration fairways, timing, regional tectonics for trap formation, and accumulation preservation factors. Thus, resource dependence must be considered in the calculation of resource aggregation. The USGS used an expert subjective judgment procedure to produce pairwise correlation coefficients for its circum-Arctic petroleum resource aggregation (Schuenemeyer and Gautier, 2009). Kaufman and Schuenemeyer (2010) proposed a hierarchical modeling for handling the dependence
of gas hydrate resource aggregation using a constant correlation value 0.9 for objects within an AU and 0.6 among the AUs. In other cases, constant correlation coefficients were applied during the resource aggregation calculation (Klett, 2004). No universally accepted method is publicly agreed to or available for quantifying the resource correlation during resource aggregation at different levels.
IMPLEMENTING VARIABLE DEPENDENCE USING COPULAS Copulas In this study, copulas are used as a general way to formulate a multivariate distribution of variables, in such a way that the various general types of dependence can be represented in the resource volume calculation and aggregation. The copula approach to formulating a multivariate distribution is based on the idea that a simple transformation can be made of each marginal variable in such a way that each of the transformed marginal variables has a uniform distribution. A copula is a multivariate probability distribution whose marginal distributions are uniform. Let x1,…, xn be random variables and H be an ndimensional joint distribution function with margins F 1,…, Fn . The Sklar’s theorem states that an n-copula C exists such that for all x in Rn (Embretchts et al., 2001) Hðx1 ; x2 ; . . . xn Þ ¼ CðF1 ðx1 Þ; F2 ðx2 Þ; . . . ; Fn ðxn ÞÞ ð9Þ From equation 9, it is clear that the univariate margins and the multivariate dependence structure can be separated for continuous multivariate distribution functions, that is, the dependence structure is represented by a copula function (Embretchts et al., 2001). This property allows the parameters of marginal distributions and the correlation structure to be estimated independently from different data sets, which allows a greater flexibility for the consideration of and information by interdependencies obtained from appropriately analogous Chen at al.
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geologic settings when such data are insufficient to determine the correlation structure locally. Another beneficial property of copulas is their characteristic invariance or characteristically simple monotone transformations of the random variables (Embretchts et al., 2001). Two major types of copula function, elliptical and Archimedean, exist. Gaussian and student t copulas belong to the elliptical families that are easy to implement and that can be easily parameterized. Let F denote the standard univariate normal distribution function and let F nR denote the standard multivariate normal distribution function with a linear correlation matrix R and define ui the marginal probability distribution of a random variable. Then the Gaussian (normal) n-copula can be written in the following form: Cðu1 ; u2 ; . . . ; un Þ ¼ FnR ðF 1 ðu1 ; u2 ; . . . un ÞÞ ð10Þ In the bivariate case, the Gaussian copula can be written as (Nelsen, 2006) Z CGa R ðu; vÞ ¼
F
1
ðuÞ Z
F
1
ðvÞ
1
2pð1 R212 Þ1=2 s2 2R12 st + t 2 exp dsdt ð11Þ 2ð1 R212 Þ 1
1
where R12 is a positive definite linear correlation matrix of the corresponding bivariate normal distribution, u and v are marginal probabilities, and s and t are the two correlated random variables. The necessity of a positive definite matrix has been discussed in detail by Schuenemeyer and Gautier (2009). The other copula family is the Archimedean, which includes the Gumbel, Clayton, and Frank families. The Archimedean copulas are not generated by Sklar’s theorem, but from monotonically increasing functions of convex shape. They are widely used in actuarial science and finance for modeling risk dependencies. Procedure for Estimating Ultimate Resource Potential From equation 7, the ultimate recoverable crude oil resource in a play is the product of n correlated 446
random variables. Using an n-copula, one realization of the n-correlated variables in equation 7 can be generated using the following steps: 1. Generate a random variable u1 from U(0, 1) 2. Generate a random variable u2 from C2(·∣u1) … and so on, until 3. Generate a random variable un from Cn(·∣u1,…, un–1). where U(0, 1) is uniform distribution and Cj(·∣uj-1) is a copula function. The product of all n-correlated random variables generates a single play resource realization. Repeating the above steps M times, gives M resource estimate values: a distribution of the ultimate recoverable crude oil resource in a play. The distribution, so derived, captures the uncertainty by describing the uncertainty among possible ultimate recoverable resource estimates.
EXAMPLES We present two examples to demonstrate copula use for handling variable dependencies in resource assessment. The first example uses a hypothetical data set to show copula use for handling volumetric attributes with different shapes of probability distribution, and second one is an application of the copula to the resource estimation for some of the oil plays in the BMB. Example from a Hypothetical Data Set Suppose that an exploration program generates a data set by seismic interpretation, well-log calculation, and core analysis, resulting in four random variables xj(n), i = 1, 2, 3, 4, representing, for example, closure area, net pay thickness, porosity, and oil saturation in a play (Appendix 2) (AAPG Datashare 42, www.aapg.org.net). The product of the four random variables provides estimates of hydrocarbon pore volume of the play. The distribution characteristics (Figure 2) of the four variables are represented by the marginal distributions F1,…, F4. In this example, each of the random variables follows a distinct probabilistic distribution to
Using Copulas for Implementation of Variable Dependencies in Resource Assessment
Figure 2. Histograms of four hypothetical random variables show the distribution characteristics of the marginal distributions.
mimic possible variation in the shape of the reservoir parameters in different geologic settings. Under an independent assumption, the marginal distributions are independent from each other and no correlation structure or a joint distribution is needed for the estimation of hydrocarbon pore volume. A Monte Carlo procedure of multiplication of the four independent marginal distributions produces a mean of 6744.2 with P10 and P90 of 269 and 16,453, respectively, for the hydrocarbon pore volume. However, data analysis shows significant correlations among the variables. A Pearson’s linear correlation coefficient matrix (Table 1) estimated from the example data shows various dependencies
Table 1. Estimated Linear Correlation Coefficient Matrix from the Four Random Variables in Hypothetical Example*
Area Net pay Porosity Saturation
Area
Net Pay
Porosity
Oil Saturation
1.0000
0.5836 1.0000
0.5156 0.5377 1.0000
0.5835 0.5756 0.5194 1.0000
*See Appendix 2, AAPG Datashare 42, www.aapg.org/datashare.
among the random variables. Because of the correlations, a product of independent marginal distributions is no longer adequate for estimating hydrocarbon pore volume, and consideration of a joint distribution of all the four margins is necessary. As the four random variables follow different distributions, it is not possible to use a single multivariable distribution, such as a multidimensional lognormal model in equation 6 to represent the data with a correlation structure in Table 1. A copula is a multivariate probability distribution with uniform marginal distributions. It allows different statistical models for the margins and makes a joint distribution through a copula function (equation 9). To incorporate variable dependencies in mathematical manipulations using copulas, three components in the construction of a joint probability distribution exist: the marginal distributions representing the statistical characteristics of the random variables, a correlation matrix representing the dependencies among the random variables, and a copula function bringing the margins and correlation structure together to form a joint probability distribution. From the data analysis (Figure 3), it appears that F1 follows a normal distribution and F4 follows Chen at al.
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Figure 3. Scatter-histogram diagrams of the random variables showing the distributions and correlation characteristics. Different statistical models are used to demonstrate possible variations of reservoir parameters. The types of distribution include normal (F1), gamma (F2), Weibull (F3), and uniform (F4) (shown on the figure as x1, x2, x3, and x4).
a uniform distribution, and the model parameters of these two random variables can be estimated directly from observations with appropriate mod448
els. However, it might be difficult to judge what statistical models are the best for representing F2 and F3. In this case, we used data to generate
Using Copulas for Implementation of Variable Dependencies in Resource Assessment
Figure 4. Comparison of the resultant distributions of multiplication of the four random variables under independent and correlated scenarios from the hypothetical example. CDF = cumulative distribution function.
empirical distributions for these two dependent marginal distributions. A dependency structure, such as a correlation matrix in Table 1, can be directly estimated from the data. In case of insufficient data, an analog can be applied to obtain a correlation structure. In this study, we used a Gaussian copula function to produce the joint distribution H(x1,…,x4) (equation 9). A Monte Carlo simulation of multiplication of the four random variables from the joint distribution links a random entry of one variable to the others, resulting in a distribution of the hydrocarbon pore volume, which is different from that under the assumption of complete independence among the marginal distributions (Figure 4). The dependent scenario has a mean of 11,534 and P10 and P90 of 101 and 30,700, respectively, about 1.7 times of the independent mean and with a much larger uncertainty range, as indicated by the P10 and P90 (Figure 4).
Application to the Regional Resource Appraisal of the Beaufort-Mackenzie Basin Four established crude oil plays in the central BMB are selected as examples in this study to illustrate
copula use in resource assessment and demonstrate variable dependence impacts on the resulting estimated resource potentials. Geologic Background The BMB is located in the Canadian Arctic region. It hosts large contingent petroleum resources that are strategically important potential North American petroleum supplies (Figure 5). Petroleum exploration in this basin has resulted in 48 significant conventional crude oil and gas discoveries containing 53 distinct accumulations (National Energy Board, 1998). The discovered conventional petroleum resources are estimated to be 1.744 × 109 bbl (277.3 × 106 m3) recoverable crude oil and 11.74 tcf (332.4 × 109 m3) recoverable natural gas (Dixon et al., 1994). Recent studies of unconventional petroleum resources indicate that an immense natural gas potential exists in the form of methane hydrate in the BMB (e.g., Osadetz and Chen, 2010). The BMB exhibits a complex basin evolution, from an early Paleozoic open-marine setting to a later Paleozoic contractional orogeny followed by a rift-drift system in the Jurassic–Early Cretaceous. Once the Amerasian Basin formed, a major Late Chen at al.
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Figure 5. Location map showing the case study area (black dashed square). The red square in the small window indicates the location of Beaufort-Mackenzie Basin in Arctic Canada.
Cretaceous–Tertiary to Holocene deltaic sedimentation system developed. It deformed by the Late Cretaceous to current tectonic impingement of the Cordilleran orogen. Tectonically, the BMB can be divided into four structural domains: a stable craton in the south and southeast, a rifted margin to the Canada Basin in the southeast, the impinging Cordilleran fold belt in the southwest, and the Canada Basin in the north. The Upper Cretaceous–Cenozoic succession represents a postrift passive-margin basin composed of more than 14 km (>9 mi) of deltaic sediments (Dixon et al., 1992), parts of which have been and are now being accreted to the Cordilleran thrust belt. Three major BMB petroleum systems are recognized based on available data and our current petroleum system models. They include a Jurassic– Lower Cretaceous composite petroleum system in the southeast basin margin, a Jurassic–Tertiary com450
posite petroleum system in the delta, and an Upper Cretaceous–Tertiary composite petroleum system for the rest of the BMB. The delta region may overlie multiple source rocks, such as the Upper Cretaceous shale in the Boundary Creek and Smoking Hills formations (Li et al., 2008), coal seams in lower Tertiary Aklak and Taglu sequences (Snowdon et al., 2004), Jurassic–Lower Cretaceous shales in the Husky Formation (Dixon et al., 1994), and other Lower Cretaceous shaly intervals (Li et al., 2008). The four example play types are predominantly structural. Stratigraphicaly, they are constrained to the Oligocene Kugmallit and Eocene Taglu sequences deposited in a deltaic environment. Among the 53 significant discoveries, 38 discoveries accounting for about 77 and 70% of discovered natural gas and crude oil contingent resources, respectively, occur within the Tertiary Taglu and Kugmallit sequence reservoirs (Chen et al., 2007).
Using Copulas for Implementation of Variable Dependencies in Resource Assessment
Figure 6. Crossplots of reservoir volumetric variables showing variable dependence among the reservoir volumetric variables in the Ivik play of the Taglu play group. HC = hydrocarbon; corr coef = correlation coefficient.
Results and Discussion Available data allow us to study the variable dependence among the reservoir volumetric variables, but the data are not adequate for an examination of the correlations among geologic risk factors. Thus, independence is assumed for all geologic risk factors, and this current study is focused on depen-
dency effects on the reservoir volumetric calculation and the total regional resource only. Data analysis suggests that correlation is common among the volumetric variables, although the degree of dependency varies from play to play. Figures 6 and 7 are crossplots, showing examples of variable correlations from the Kugmallit and Ivik Figure 7. Crossplots of reservoir volumetric variables showing variable dependence among the reservoir volumetric variables in the Kugmallit (East) play of the Kugmallit play group. HC = hydrocarbon; corr coef = correlation coefficient.
Chen at al.
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Table 2. Example Variable Correlation Matrices from Crude Oil Plays in the Beaufort-Mackenzie Basin, Arctic Canada, Kugmallit (East) Oil Play and Irik Oil Play Area
Net Pay
Porosity
Oil Saturation
Kugmallit (East) Oil Play Area 1.0000 0.2837 Net pay 1.0000 Porosity Saturation
0.2162 0.2041 1.0000
0.3907 0.3057 0.7712 1.0000
Ivik Oil Play Area 1.0000 Net pay Porosity Saturation
0.2112 0.3114 1.0000
0.3264 0.2245 0.8646 1.0000
0.5318 1.0000
oil plays, respectively. Table 2 shows correlation matrices of reservoir volumetric variables calculated from the same data sets shown in Figures 6 and 7, suggesting that a wide correlation range is characteristic of the paired variables among the example plays. The four strongly correlated reservoir volumetric variables include pool area, net pay, porosity, and oil saturation. Because no oil production in this basin exists, it is not possible to establish a relationship between the ultimate crude oil recovery factor and field size, nor the correlation structure using available data. Instead, we used an analog from a well-established mature play in the Western Canada sedimentary basin (Figure 1). The correlation coefficient between the recovery factor and in-place crude oil resource is assumed to be fixed at 0.6 for all four plays in this basin. A truncated multivariate lognormal distribution is used for the four volumetric variables listed in Table 2, and a multinormal distribution is used for the remaining reservoir variables, such as the formation volumetric factor, trap oil fill, and recovery factor. In the Monte Carlo simulation of resource potential, Gaussian copulas are used to formulate correlated multivariate distributions. Figure 8 shows the estimated pool size distributions and compares scenarios with and without consideration of variable dependences in the example 452
four plays. When variable dependence is considered in the computation, the pool size distributions have a greater uncertainty range, but the median pool sizes remain almost unchanged. Figure 9 displays the estimated play crude oil resources for the four plays. The estimated play resources are higher when variable dependencies are considered for all four example plays. The uncertainty shows a greater range than that for the independent cases in the four plays. Clearly, variable independence impacts the field size distribution variance estimate significantly. Incorporation of positive correlations among variables in pool size calculation increases the variance significantly, thus resulting in more dispersed pool size distributions. The size of large fields becomes even larger and the size of small fields becomes even smaller compared with the result when the input variables are considered to be mutually independent. Implementing positive correlations in play resource calculation leads to not only a wider range of uncertainty, but also greater value of the mean estimate. The difference depends on the degree of correlation. The larger the correlation coefficients are, the greater the differences between the dependent and the independent model results could be. A complete dependence among variables yields the maximum difference in comparison with the result obtained for the independent variable scenario. No quantitative method is available that permits us to quantify the degree of dependency between play resources in resource aggregation. The determination of such dependencies is commonly both difficult and subjective (e.g., Klett et al., 2000; Schuenemeyer and Gautier, 2009). The justification for assuming that play resources are positively correlated in the four example plays in this study is as follows. All four plays share the same petroleum system and are sourced from the same petroleum source rock strata. The late Miocene tectonics postdate the deposition of the regional top seals on reservoirs, and these geologic processes and event are inferred to have had similar impacts on such variables as trap formation, petroleum generation timing and migration, and preservation in the reservoir. In addition, the depositional settings
Using Copulas for Implementation of Variable Dependencies in Resource Assessment
Figure 8. Estimated pool size distributions with variable dependencies (solid line) and assuming independent (dashed line) of four established plays in this case study. The pool size distribution shows a greater uncertainty range when variable correlations are implemented than that of cases assuming a complete independency among the reservoir volumetric variables in the modeling.
are deltaic and the reservoir facies in all four plays have similar dispersion patterns. The differences in trap size, structural configurations, and reservoir features are primarily characterized by pool size distributions in each play; however, the resource richness of each play is basically a function of both the oil charge and the preservation of accumulations that are mostly controlled by common petroleum system elements. In this regard, we infer that the resources in the four plays are highly correlated, although the pool size distributions among the four plays vary considerably (Figure 8). A Gaussian copula is used in resource aggregation of the four example plays, and two correlation
scenarios are used, assuming either correlation coefficients 0.6 or 1.0, respectively, to investigate the resource correlation impact on aggregation. Figure 10 is the aggregated crude oil resource from the four plays. It compares the model constructed using the different correlation scenarios, assuming either a complete correlation (solid line) or a partial dependence (dotted line). Both are compared with an independent scenario (broken line). It is obvious from Figure 10 that variable dependence affects the resource aggregation uncertainty range, and that the higher the variable correlation is, the larger the total resource variation could be. However, objective methods for quantifying resource Chen at al.
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Figure 9. Estimated play resources with variable dependencies (solid line) and assuming independent (dashed line) of four established plays in this case study. The estimated play resources are shifted toward the right, showing greater resource potentials when variable correlation is formulated than that from assuming a complete independency of the volumetric variables in the calculation. BMB = Beaufort-Mackenzie Basin.
Figure 10. Comparison of play aggregations of the four plays under different correlation scenarios in this case study. The solid line is the aggregated total crude oil resource when a complete dependence among the play resources is incorporated. The broken line is the completely independent case, and the dotted line is the partial correlated case (0.5).
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Using Copulas for Implementation of Variable Dependencies in Resource Assessment
Figure 11. Comparison of resource assessment results for the scenarios with and without formulating variable dependence in the assessment.
correlation in aggregation are needed to reduce current subjectivity. Figure 11 compares the estimated total crude oil resources in the four crude oil plays with and without implementing variable dependences on the reservoir volumetric calculation and the subsequent resource aggregation of the four example plays into a single regional resource potential. When positive variable dependences are considered, the resource estimate is not only larger in total crude oil resource, but it is also characterized by a wider uncertainty range compared with the resource estimate obtained without considering variable dependences. The total mean crude oil resource resulting from the consideration of variable dependence is
approximately 1.6 times the mean estimate resulting from the completely independent variable scenario (Table 3). In the BMB crude oil resource assessment, the incorporation of a positive correlation structure in volumetric play resource modeling and the subsequent aggregation of resources into a single potential increased the mean and widened the total resource potential uncertainty. The difference between the resulting total median basin crude oil resources differs by a factor of 1.6 (Chen et al., 2009). This suggests that the consideration of variable dependencies has an impact on resource estimation. We infer that, had suitable data been available, an analogous impact would be observed on risk factors.
Table 3. Comparison of Ultimate Recoverable Crude Oil Resource Estimates Under Different Scenarios for Variable Dependencies Case A B C D
Approach Partial (0.6) in aggregation Completely correlated in aggregation Uncorrelated in aggregation Completely independent Ratio (A/D)
Mean 864.6 870 867.1 533.2 1.6
(5.4) (5.5) (5.5) (3.4)
Median 854.4 (5.4) 849 (5.3) 863.4 (5.4) 531.6 (3.3) 1.6
Minimum 228.2 198.5 351.3 296.1 0.8
(1.4) (1.2) (2.2) (1.9)
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Maximum 1673.3 (10.5) 1911.4 (12.0) 1481.2 (9.3) 820.6 (5.2) 2.0 455
CONCLUSIONS Copulas can be used as a general way to formulate a joint probability distribution so that various variable dependencies can be represented easily in resource volume calculations and aggregations when Monte Carlo simulations are used. The applications of copulas in this study indicated that copulas provide a convenient way to consider variable dependency and to include information from analogous settings into petroleum resource assessment. The BMB petroleum play data suggest that partial positive correlations are common among reservoir volumetric variables and that data obtained from well-explored basins elsewhere support the concept that a positive correlation between the oil recovery factor and original oil in place volume in a play. Variable dependency consideration has significant impacts on resulting resource estimations. A positive correlation in pool size calculation increases pool size distribution variance such that large pools become larger and small pools become smaller in comparison with resource estimates obtained, assuming that the variables are independent. As shown in the examples, the incorporation of positive correlations in play resource calculations yields a larger mean and wider uncertainty range for the ultimate recoverable crude oil resource. In contrast, a negative correlation would shrink pool size distribution variance. Currently, it is not possible to objectively estimate the variable dependency on resource aggregation, and sound geologic judgments are needed to justify and determine such correlations. However, consideration of variable dependence in play resource aggregation increases the estimated variance of ultimate recoverable resource distributions.
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