An Improved Fractal Model for Characterizing Spatial Distribution of Undiscovered Petroleum Accumulations

Zhuoheng Chen, Kirk Osadetz, and Peter Hannigan

Geological Survey of Canada, 3303 � 33rd Street, NW, Calgary Alberta, T2L 2A7, Canada Email: ess-cal-zchen@x1.nrcan.gc.ca,kosadetz@nrcan.gc.ca, phanniga@nrcan.gc.ca

Abstract Study of a mature petroleum play in the Western Canada Sedimentary Basin (WCSB) indicates that the spatial distribution of petroleum accumulations exhibits a self-affinity characteristic. This characteristic motivated the examination of a fractal model for a quantitative description of petroleum resource spatial distribution. The proposed approach transforms the spatial information with respect to discovered petroleum accumulations into a frequency domain, represented by an amplitude map and a phase spectrum. The amplitude map is then calibrated using a fractal model, inferred from exploration data, to account for the sampling bias in exploration procedure. The information in the obtained phase spectrum provides no clue with respect to the locations of undiscovered accumulations, and cannot be enhanced by the established fractal model either. If the calibrated amplitude map and a random phase map are transformed back to the spatial domain and conditioned on the discovered petroleum accumulations, the resulting map is equivalent to one of equal-probable realisations from a conditional simulation. Improvement can be made by extracting information with respect to locations of undiscovered petroleum deposits from geological factors controlling the formation of petroleum accumulations in a petroleum system. Using additional quantitative models, such as a geological favorability map or a map of probability of petroleum occurrence, allows an improved characterisation of spatial distribution of petroleum accumulations by the fractal model. An example from the Western Canada Sedimentary Basin illustrates the application of the method. Key words: Fractal, Fourier integral, calibrated amplitude, improved phase spectrum, Rainbow gas play 1 Introduction During the past three decades method development for hydrocarbon resource assessment has focused mainly on assessing the aggregate properties of hydrocarbon resources, such as the total potential and number of pools/fields, or the distribution of pool/field sizes in a hydrocarbon play. Little attention was paid to the spatial characteristics of undiscovered resource. New demands for both a better natural resource management and improved exploration efficiency make it important to understand the spatial characteristics of undiscovered hydrocarbon resource. Quantification of the spatial distribution of petroleum resource must simultaneously consider two fundamental elements: the size of hydrocarbon accumulation and its location. It is well known that the data generation process with respect to the discovery of petroleum accumulations in an exploration program is a biased one, from which larger features are sampled with higher priority. This exploratory phenomenon precludes a consensus on a unique model for pool/field size distribution. The discussion of the shape of pool/field size distribution has taken place for

more than three decades (e.g., Mandelbrod, 1962, Kaufman, et al. 1965; Arps and Robert 1968; Coustau, 1980; Schuenemeyer and Drew, 1983; Drew, et al., 1988; Davis and Chang, 1989; Lee 1993; Crovelli and Barton, 1995). More evidence suggests that the lognormality of the field size distribution in a play could be due to a consequence of economic truncation and the limitations of exploration technology, and that the distribution of field size could be fractal (Schuenemeyer and Drew, 1983; Drew, 1990). The scaling of natural properties on the Earth surface has broad implications for a wide variety of problems varying from hydrocarbon reservoir, tectonics, and topographical surface characterizations, to seismic hazard assessment (e.g. Scholz and Cowie, 1991; Marrett, et al. 1999; Veneziano and Iacobellis, 1999). For assessing the aggregate properties of petroleum accumulations at the regional, basin and play levels, fractal model has been discussed and applied to estimate the pool/field size distribution by many authors (e.g., Mandelbrod; 1962, Coustau, 1980; Drew, et al., 1988; Houghton, 1988; Crovelli and Barton, 1995). Barton et al. (1991) and La Pointe (1995)

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studied the data from well-explored petroleum basins and concluded that spatial distributions of hydrocarbon accumulations are fractal. La Pointe (1995) proposed the use of fractal geometry to estimate the total potential in a region. Barton and Scholz (1995) proposed the use of fractal dimension as an indicator for exploration planning. Although attempts have been made recently with respect to the spatial characteristics of undiscovered petroleum accumulations, the current available methods (e.g., Chen et al, 2000; Gao et al, 2000) do not allow a full integration of data to account for both, the accumulation size and its location. Our study of a mature play in the Western Canada Sedimentary Basin indicates that the distribution of petroleum resource exhibits a self-similar characteristic, both in pool size distribution and spatial geometry. This characteristic motivates the examination of a fractal model for a quantitative description of hydrocarbon resource spatial distribution. Figure 1 presents the results from a box counting method for all the gas discoveries in the Rainbow gas play. The box size and the number of boxes containing gas pools show a linear relationship when plotted on a log-log scale, indicating that the spatial distribution of the gas accumulations in this play is fractal. After fitting a straight line through points in the six largest boxes, a slight deviation from linearity is observed among the smaller box sizes. This deviation is interpreted to represent missing small pools, which are the remaining objects to be sought. This interpretation is backed by the exploration results in the region. Figure 2a is a discovery sequence showing gas pool size with respect to its order of discovery. It displays a clear decline of pool size with the number of discoveries. Figure 2b plots cumulative reserve vs. the order of discovery. More than 80% of the gas reserve was found in the first half of the discovery sequence. The conclusion of small undiscovered gas pools in this gas pay is also consistent with the results of a gas resource assessment from the Geological Survey of Canada (GSC) in 1993 (Reinson, et al., 1993). The scaling property of the spatial objects implies that spatial characteristics of the objects at a

larger scale are similar to spatial characteristics at smaller scales. If the spatial distribution of petroleum accumulation is fractal, the spatial characteristics of large petroleum accumulations could be used to infer the spatial characteristics at smaller scales. The proposed fractal approach transforms the spatial information of the discovered hydrocarbon accumulations into a frequency domain by Fourier transform, resulting in an amplitude map and a phase map. The amplitude map contains spatial correlation information and the phase map contains location-specific information. Both the spatial correlation in the amplitude map and the location-specific information in the phase map are incomplete due to biased and inadequate sampling. The amplitude map can be calibrated using a fractal model inferred from exploration results under the assumption of spatial scale invariant to account for the biased sampling. However the phase map derived from previous exploration results contains information on discovered accumulation locations only and provides no clue regarding the locations of undiscovered accumulations. An inverse FFT transformation of the calibrated amplitude map and a random phase map conditioned on discovered petroleum accumulations, result in an equal-probable realization of spatial characteristics of discovered and undiscovered hydrocarbons in the play area. Although the location-specific information can not be inferred from the established fractal model, it may be extracted from geological factors controlling the formation of petroleum accumulations. Improvement can therefore be made regarding the location of undiscovered accumulations, through a study of the necessary and sufficient geological conditions for forming petroleum accumulations in a petroleum system. In this paper we discuss the utility of genetic geological models, such as geological favorability maps or probabilities of petroleum occurrence, for additional information with respect to location of undiscovered petroleum deposits and make use of this information in characterizing the spatial distribution of undiscovered accumulations in a play. An application example from the Middle Devonian Rainbow gas play in the WCSB illustrates how the fractal model can be used to assess petroleum resources.

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2 Method Description A) Fractal Model of Petroleum Accumulation In the proposed fractal approach, hydrocarbon resource is described by an image map, on which the value of each pixel represents the average magnitude of hydrocarbon accumulation within that pixel. Figure 3 (top left image) illustrates a conceptual model of the spatial distribution of hydrocarbon accumulations. Since the location where petroleum is generated is not necessary the location where petroleum is trapped, the value at each pixel represents the net effect of hydrocarbon accumulation. Negative values signify the net quality resulting from hydrocarbons migrating away from the pixel; whereas positive values represents a net accumulation at that location. Because the primary objective of hydrocarbon exploration in an area of interest is to find economically recoverable petroleum accumulations, only those exceeding certain threshold values are of significance. The remaining images in Figure 3 show a series versions of the same image with different threshold values. The top right image depicts only those pixels with net accumulations. The lower left image has an arbitrary threshold value of >0.2; whereas the lower right one has a threshold value of >0.5. From this series of images, it is obvious that even though spatial pattern of hydrocarbon accumulation varies depending on the size range of interest, the spatial correlation structure for any one of the images in Figure 3 can be inferred from the original one. The major problem associated with construction of a fractal model for describing the spatial distribution of hydrocarbon accumulations is that available information is biased with respect to exploration results. The fact that large fields/pools are likely found early in an exploration program can be used in the proposed fractal approach to infer the spatial characteristics of the undiscovered small accumulations. In other words, the spatial scale invariant property of the fractal model allows inference of spatial characteristics of the undiscovered petroleum accumulations from biased observations. Using the fractal model, the exploration bias can be corrected by extrapolating the straight line defined by large-size boxes to the extent that data permits (Figure 1).

B) Power Spectrum Representation of the Fractal Model

Any second stationary discrete stochastic process, y(k), can be expressed as a series of Fourier coefficients aj and bj. In the 1-D case, the series is written as:

( ) ( ) [ cos( ) sin( )]1 2 2

0

1

y k a bjjk

N jjk

Nj

N

� �

�

�

�� �

for j=0,1,2,�N-1; or equivalently using a complex exponential Fourier series:

( ) ( ) ( ) /2 2

0

1

y k A j ei kj N

j

N

�

�

�

� �

where A(j)=aj - ibj=|A(j)|exp{-i�(j)} is the jth complex Fourier coefficient, |A(j)|=(aj

2+bj2)1/2 is

the amplitude spectrum, �(j) = tan-1(-bj/aj) is the phase spectrum, and S(j) =|A(j)|2, j=0,1,�,N-1, is the power spectrum. For a time series of a self-similarity fractal, the power spectrum density has a power law dependency on frequency (Turcotte, 1994): ( ) ( )3 S k f�

� �

where f is frequency, and � is exponential coefficient. The 1-D time series power spectrum model can be easily converted to a 2D model for characterizing the spatial distribution of petroleum accumulations. In the fractal model, spatial correlation of objects is fully specified by the spectrum density function. The Wiener-Khintchine theorem states that any stationary process has a covariance function C(h) of the form (Pardo-Iguzquiza and Chica-Olmo,1993):

( ) ( ) ( )4 C h S e di h�

�

� � ��

�

�

which means the power spectrum function is the Fourier transform of the covariance function: where � is the angular frequency and �

symbolizes the Fourier transform pairs. This indicates that both functions (power spectrum and covariance) contain the same spatial correlation information, but expressed in different forms. To illustrate that all information of the second moment statistics in the covariance function is retained in the power spectrum, we applied Fast Fourier Transform (FFT) to the

(5) C(h) |A(j)|2�

3

image shown in Figure 4 which is a map of discovered gas pools in the Rainbow gas play of the Western Canada Sedimentary Basin (WCSB). This resulted in an amplitude map that is the square root of the power spectrum and a phase map of the image. In Figure 4, the size of gas pools is represented by pixel values. The location of the pool is represented by the central pixel where the discovery well is located. Figure 5 is a reconstruction of the original image by inverse FFT using the original amplitude map and a random phase map. This map is equivalent to one of the equal-probable realizations by stochastic simulation using spatial correlation information in the amplitude map. We can see in the image that the general shape and size of the spatial objects are similar to that of the original image, which demonstrates that second moment spatial statistics of the original image is well-retained in the amplitude map. However, the objects in the image are randomly distributed, indicating that the amplitude map does not contain any information with respect to their locations. The phase map contains all information regarding locations. Without the original phase map, amplitude alone can not restore the original image. Retaining the original phase, the upper left image of Figure 6 shows a reconstructed image using a uniform random amplitude map. The upper right image is derived from a normal random amplitude map; the lower left one uses a lognormal random amplitude map; and the lower right image uses a fractal amplitude map fitted to the original amplitude map. The general outlines of the features in the original image are maintained in all reconstructed images, indicating the phase map is very important in spatial modeling. By using the random uniform amplitude map, the image exhibits uniform magnitude throughout. The lognormal random amplitude, however, has its dominant magnitude represented by small values. From these images, we can see that retention of the original phase map ensures accurate locations for all features, while the magnitude and spatial structure of the features are dependent on the spatial correlation information contained in the amplitude map. In addition to the computational advantage of using the Fourier Integral Method (FIM) (Pardo-Iguzquiza and Chica-Olmo, 1993; Yao, 1999),

the transformation of discovered petroleum accumulations from a spatial domain into a frequency domain makes it possible to study the characteristics of size and location separately. Different geoscience data may contain unique information with respect to petroleum accumulation. For example, spatial data with respect to discovered pools, such as pool size, shape and location contain both the spatial correlation as well as location; whereas, the geological favorability map and geophysical survey base map may provide additional information of location rather than spatial correlation. A model for spatial correlation and a separate model for locations could be constructed using different types of geoscience data and constrained by different conditions in frequency domain. This information can then be transformed back using inverse FFT to a spatial domain which represents our understanding of the spatial distribution of petroleum accumulations. C) Modeling Procedure The procedure for modeling the spatial distribution of petroleum accumulations consists of the following steps: 1) Prepare a petroleum accumulation image

map from exploration results; 2) Estimate fractal parameters from the image

map; 3) FFT the image map to obtain amplitude and

phase maps; 4) Generate a fractal image of petroleum

accumulations using the inferred fractal amplitude and a phase map by inverse FFT;

5) Validate and adjust the fractal image against observations to produce a new petroleum accumulation map;

6) Replace the original petroleum accumulation map with the newly generated one in 5); and

7) Repeat steps 3) to 5) until a desired tolerance in an objective function in eq. (6) below is reached.

( ) | |( , ) ( , )( , )6

1obj Z i j Z i j

Z i j

nm o

m�

�

�

� � � � �

� �

�

where Zm(i�, j�) is the modeled value at grid node (i, j) corresponding to the closest node at the �th

4

observation, �=1,�,n, and Zo(i�,j�) is the value of �th observation at grid node (i,j). 3 Application Example To illustrate how the proposed fractal model can be used to assess the spatial characteristics of petroleum resources, exploration data from the Rainbow gas play was collected from the database at the Geological Survey of Canada, Calgary. The Rainbow gas play, located in northwestern Alberta, WCSB, is a mature exploration play with an areal extent of about 5000 km2. Up to 1994, 467 exploratory wells were drilled and led to the discoveries of 207gas pools with an estimated reserve of 45 x109 m3 in place. The gas is found in Devonian reefs and associated traps above the reefs. Figure 7 shows the play boundary and exploratory well locations with indications for gas discovery or dry well. The dry wells were used as areal constraints in the modeling of the spatial distribution of undiscovered gas resource. A rectangular area of 0.36 km2 is assumed to be exhausted by an exploratory well in this study. Using the Fourier transform of the discovered gas accumulations in Figure 3, amplitude and phase maps are derived. Figure 8 are amplitude profiles of the Original Amplitude Map (OAM) of the spatial distribution of discovered pools derived by FFT. As expected, due to the biased sampling procedure, small pools are under-represented on the spatial distribution map of discovered gas pools as indicated by the deviation from a linear relationship in high frequency regions in both directions on the profiles. The property of spatial scale invariant is used to infer the spatial correlation from completely mapped large gas pools to the unmapped smaller ones. The calibrated amplitude map, using estimated fractal parameters from the OAM, represents the spatial correlation for all gas accumulations in the size range indicated by the data. The phase map derived from the discovered gas accumulations contains no information regarding the locations of these undiscovered gas accumulations. The use of the calibrated amplitude map and the methods of phase identification proposed by Yao (1999) result in a random realization of a stochastic simulation conditioned on discovered gas accumulations. In such a realization, the spatial

correlation in the amplitude map is retained, but the locations of the undiscovered gas pools could be anywhere. We realize that locations of petroleum accumulation are not randomly distributed, but follow some sort of pattern controlled by geological factors. In general, if there is any additional information available regarding the location of the undiscovered gas accumulations, the use of this additional information should improve the prediction of locations for undiscovered accumulations. In a study of the spatial characteristics of the hydrocarbon accumulations in the same area, a posterior probability map of hydrocarbon occurrence has been produced (Figure 9) (Chen et al, 2000). This probability map integrated information on hydrocarbon accumulation from both exploration results as well as available geological factors controlling the formation of hydrocarbon accumulations in this area. An initial phase map, therefore, was derived from the posterior probability map. This phase map, along with the modified fractal amplitude map, was transformed back to the spatial domain by inverse FFT to obtain an initial spatial distribution of the gas accumulations. The initial distribution is then checked against the discovered petroleum accumulations, dry well locations, and their exhaustion range. This iterative procedure is carried on until a desired tolerance value in the subjective function that minimizes the difference between modeled spatial distribution of petroleum accumulation and known observation conditions is met. Figure 10 is the Modified Fractal Amplitude Map (MFAM), which is a modified version of the OAM that considers the estimated fractal model. Figure 11 shows the result from modeling of the spatial distribution of gas accumulations using the MFAM and the initial phase from the posterior probability map with a cut-off value of 0.25 by inverse FFT and conditioned on dry wells and discoveries. This map represents the prediction of the spatial distribution of all gas accumulations in the play. Figure 12 shows the difference between the discovered gas pool distribution and the modeled spatial distribution

5

based on the MFAM, representing the remaining undiscovered gas accumulations in the play. 4 Discussion and conclusions In this paper, petroleum accumulations in a play are treated as a collection of spatially correlated objects and the Fourier Integral algorithms are used for generating images representing the spatial distribution of undiscovered gas accumulations in the defined play. We have shown through experiments that information concerning both spatial correlation in an amplitude map and locations in a phase map are equally important in modeling of characteristics of spatial objects. An appropriate phase map is the key in accurately locating the modeled spatial objects in space. We demonstrated that the use of additional geological and geophysical prospecting data will enhance spatial modeling by adding location-specific information in the phase map which has received little attention in the past. There are two major advantages in using the FIM method in studying the characteristics of spatial objects. One is the computational convenience as discussed by Pardo-Iguzquiza and Chica-Olmo (1993) and Yao (1999). The other advantage is the flexibility in allowing studies of spatial feature in a frequency domain, in which the spatial correlation structure and location-specific information can be studied separately by means of different methods and using different data sources. The proposed method uses a fractal model to infer the spatial correlation for undiscovered gas accumulation based on biased sampling in petroleum exploration and on power law scaling of spatial correlation of gas accumulations in this play. The use of the inferred fractal model provides exclusive information with respect to the spatial correlation in unmapped pools that are not included in the data. The calibrated amplitude map derived from the estimated fractal model is assumed to represent the true spatial correlation structure of petroleum accumulations in the range indicated by the exploration data. To overcome the problem of lack of information with respect to locations of undiscovered accumulations, a posterior probability map derived from

integration of geological information and exploration results was used to derive a phase map. A conditioning procedure was implemented so that exploration results, such as the location of dry wells and their exhaustion zones as well as discovered gas accumulations, were used to ensure that discovered pools on the modeled map are in the correct locations and dry wells indicates no undiscovered gas resource. By this method, we extracted information of both the spatial correlation and undiscovered pool location from different data sources separately in frequency domain and transformed back to spatial domain as the prediction of spatial distribution of gas accumulations. In many cases, additional information such as a geological favorability map or a probability map of petroleum occurrence are not readily available. The use of a random phase map, followed by iterative alternations of the phase map by conditioning on observations such as discovered petroleum accumulations, and dry well locations as well as their exhaustion range, will lead to a conditional simulation. However, the use of the calibrated amplitude map will improve the conditional simulation by providing a more complete representation of the spatial correlation structure. Further constraints, such as geographic information from geophysical prospecting can also be introduced to exclude the possible locations of remaining unmapped geophysical anomalies, thus further constraining the possible location of undiscovered gas accumulation in the area. Our study shows that spatial distribution of gas accumulations in the Rainbow gas play, WCSB, is probably fractal. There were suggestions that the spatial distribution of hydrocarbon accumulations might be multifractal (Barton and Scholz, 1995). We do not rule out the possibility of multifractality for the spatial distribution of hydrocarbon accumulation, but believe the assumption of mono-fractality is appropriate in this case. The gas resource assessment of the Rainbow gas play conducted by the Geological Survey of Canada in 1993 (Reinson et al, 1993) predicted that 256 small gas pools are to be discovered even under a lognormal assumption for the gas pool size. Among these 256 undiscovered pools, 233 are within the scale

6

range as indicated by the current data (1 � 1000 x106 m3) and should not be too far to fit the gap for a single fractal relationship in Figure 1. Further more, recent research in topography shows that �when more appropriate methods are used, topography shows little evidence of multifractality� and �this finding may be important beyond topography, since the same technique has been used to investigate multiscaling of a variety of geoscience records� (Veneziano and Iacobellis, 1999). This method predicts that most of the undiscovered remaining gas pools in this play are small (Figure 12). This is consistent with the independent gas assessment results from the Geological Survey of Canada (Reinson, et al., 1993). According to this assessment, the total undiscovered gas resource is about 11 x109 m3 in-place that is dispersed in 256 small undiscovered pools. In another independent study, a probability map, showing remaining hydrocarbons contained within pool closure areas greater than or equal to 0.1 km2 in size, was produced by means of an integrated approach using information such as exploration outcome, geological favorability, and seismic line location (Chen et al., 2000). When comparing this probability map with the spatial distribution of remaining gas accumulations, a general similarity with respect to geographical locations of possible remaining pools is apparent. The similarity in predictions of undiscovered gas accumulations by two different methods increases the credibility of the proposed method for spatial feature mapping. Modeling the spatial distribution can be conducted based on different assumptions by using different amplitude maps, such as the Fitted Fractal Amplitude Map. Comparison of spatial distribution maps derived from different amplitude maps shows that different assumptions in the amplitude model will result in different spatial patterns of gas accumulations. The use of FFAM, assuming a continuous random field, gives a rather smooth and continuous spatial pattern of gas accumulation. The use of MFAM, however, assuming a discrete spatial distribution, results in a discontinuous spatial pattern. The flexibility in allowing different spatial correlation models in this approach permits alternative non-

fractal models for the characterization of spatial distribution of petroleum resources to be used. 5 Acknowledgements This work was supported by GSC project#950003 and the Panel for Energy Research and Development, NRCAN. The authors thank Drs. Dale Issler, Maowen Li and Glen Stockmal at GSC Calgary for their helpful discussions, and Ms. Ping Tzeng and Dr. David Lepard at GSC Calgary for their help with data recovery and coordinates conversions 6 References Arps, J.J., and Robert, T.G., 1968, Economics of drilling for Cretacous oil on east flank of Denver-Julesburg Basin, AAPG Bulletin, v. 17, no.11, p.2549-2566. Barton, C. C. and Scholz, C. H., 1995, The fractal size and spatial distribution of hydrocarbon accumulations, in Fractal in Petroleum Geology and Earth Processes, edited by Barton and La Pointe, 1995, Plenum Press, New York, p13-34. Barton,C.C. and Scholz, C.H., Schutter, P.R.H., and Thomas,W.J., 1991, Fractal nature of hydrocarbon deposits, 2: Spatial distribution, abstract, AAPG, v.75, no.5, p.539. Chen, Z., Osadetz, K., Gao, H., Hannigan, P. and Watson, C., 2000, Characterizing the spatial distribution of an undiscovered hydrocarbon resource: the Keg River Reef play, Western Canada Sedimentary Basin, Bulletin of Canadian Petroleum Geology, v.48, no.2, p.150-163. Coustau, H., 1980, Habitat of hydrocarbons and field distribution, a first step towards ultimate reserve-assessment: in Assessment of undiscovered oil and gas, Issued by CCOP project office, UNDP Technique support for regional offshore projecting in East Asia, Kuala Lumpur, Malaysia, p.180-194. Crovelli, R. A., and Barton, C., 1995, Fractal and the Pareto distribution applied to petroleum accumulation-size distribution, in Fractals in Petroleum Geology and Earth Processes, edited

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by Barton and La Pointe, 1995, Plenum Press, New York, p59-72. Davis, J. and Chang, T., 1989, Estimating potential of small fields in mature petroleum province, AAPG, Bulletin, v. 73, n. 8, p.967-976. Drew,L.J., 1990, Oil and gas forecasting, reflections of a petroleum geologist, New York, Oxford University press, International Association for Mathematical Geology, Studies in Mathematical Geology #2. Drew, Attanasi, and Schuemeneyer, 1988, Observed oil and gas field distributions: A consequence of discovery process and prices of oil and gas, Math. Geology, V. 20, No. 8, p.939-953. Gao, Haiyu, Chen, Zhuoheng, Osadetz, Kirk, and Hannigan, Peter, 2000, A pool-based model of the spatial distribution of undiscovered petroleum resources, Math. Geology, v.32, n.6, p.725-749 Houghton, J. C., 1988, Use of the truncated shifted Pareto distribution in assessing size distributions of oil and gas fields: Mathematical Geology, v. 20, no. 8, p. 907-938. Kaufman,G. M., Balcer, Y. and Kruyt, D., 1975, A probabilistic model of oil and gas discovery, in J. Haun, ed., Estimating the volume of undiscovered oil and gas resources: AAPG Studies in Geology Series no. 1, p.113-142. La Pointe, P. R., 1995, Estimation of undiscovered hydrocarbon potential through fractal geometry, in Fractal in Geology and Earth Sciences, edited by Christopher C. Barton and paul R. La Pointe, Plenume Press, New York, 1995, p35-57. Lee, P. J., 1993, Oil and gas size probability distributions, J-shaped, lognormal, or Pareto?: GSC Current Research 93-1. Mandelbrod, B.B., 1962, Statistics of natural resources and the law of Pareto, IBM Research Note NC-146, June 29, 1962, reprinted in Fractal in Petroleum Geology and Earth Processes, edited

by Barton and La Pointe, 1995, Plenum Press, New York. Marret, R. & Allmendinger, R.W., 1992, Amount of extension on "small faults": an example from the Viking Graben, Geology, 20, p.47-50. Marrett, R., Ortega, O. and Kelsey C., 1999, Extent of power-law scaling for natural fractures in rock, Geology, v. 27, n. 9, p.799-802. Pardo-Iguzquiza, E. and Chica-Olmo, M.,1993, The Fourier integral method: an efficient spectral method for simulation of random fields: Math. Geology, v.25, no. 4, p.177-217. Reinson, G.E., Lee, P.J.,, Warters, W., Osadetz, K.G., Bell, L.L., Price, P. R., Trollope, F., Campbell, R. I., and Barclay, J.E., 1993, Devonian gas resources of the Western Canada Sedimentary Basin, Geological Survey of Canada Bulletin 452. Schoiz, C.H. and Cowie, P.A., 1990, Determination of total strain from faulting using slip measurements, Nature, Vol. 346, p.837-839. Schuenemeyer, J. H. and Drew, L. J., 1983, A procedure to estimate the parent population of the size of oil and gas fields as revealed by a study of economic truncation: Mathematical Geology, v. 15, no 1, p. 145-162. Turcotte, D.L., 1994, Fractals and chaos in geology and geophysics, 2nd edition, Cambridge University Press, p.148. Veneziano and Iacobellis, 1999, Self-similarity and multifractality of topographic surfaces at basin and subbasin scales, Journal of Geophysical Research, v. 104, no. B6, p.12797-12812. Yao, T., 1999, Conditional spectral simulation with phase identification: Math. Geology, v. 30, no. 3, p. 285-308.

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1 00

1 01

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B o x s iz e1 0

01 0

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umbe

r of

cou

ntin

gsFigure 1 Box counting results of the Rainbow gas play, Western Canada Sedimentary Basin. The linear relation between box size and the number of boxes containing gas pools indicates a fractal geometry of the spatial distribution for gas accumulations in this play.

0 5 0 10 0 15 0 2 0 0 2 5 01 0

0

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O rd e r o f d is c o very

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e of

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ry x

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1

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O rd e r o f d is c o very

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Figure 2 Discovery history of the Rainbow gas play. The discovery sequence (a), and a cumulative reserve of gas pools (b). Units in 109 m3.

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5 0 1 0 0 1 5 0 2 0 0

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Figure 5 Reconstruction of the image of discovered gas pools in Figure 4 using the original amplitude map and a random uniform phase map.

Figure 6 Reconstruction of the image of discovered gas pools in Figure 4 using the original phase map and different random amplitude maps. A) using a uniform random amplitude, B) using a normal random amplitude map, C) using a lognormal random amplitude map, and D) using a fractal amplitude map fitted from the original amplitude map. 11

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A B

C D

4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2 5.3

x 105

-10

-9

-8

-7

-6

-5

-4

-3

-2x 104

Figure 7 A play boundary map showing the locations of discovery and dry wells in the Rainbow gas play, WCSB.

1 0- 3

1 0- 2

1 0- 1

1 00

1 00

1 05

P o w e r S p e c t r u m , x - d i r e c t i o n

1 0- 3

1 0- 2

1 0- 1

1 00

1 00

1 05

P o w e r S p e c t r u m , y - d i r e c t i o n

Figure 8 Amplitude profiles for easting and northing directions. The amplitudes show the under-sampling of the smaller gas accumulations indicated by the deviation of the linear relationships inhigher frequency regions (horizontal axis: frequency and vertical axis: amplitude).

12

0

0 . 0 5

0 . 1

0 . 1 5

0 . 2

0 . 2 5

0 . 3

0 . 3 5

0 . 4

0 . 4 5

2 0 4 0 6 0 8 0 1 0 0 1 2 0

1 0

2 0

3 0

4 0

5 0

6 0

7 0

8 0

9 0

1 0 0

Figure 9 A posterior probability map of petroleum occurrence independently derived from an integrated approach by combining geological information with exploration results using Baysian theory. This probability map is used to derive a phase map.

Figure 10 Modified Fractal Amplitude Map (MFAM) derived from calibrating the original amplitude map by a fitted fractal amplitude map.

13

0

5

1 0

1 5

2 0

2 5

3 0

3 5

4 0

4 5

5 0T o t a l re s o u rc e

4 . 5 4 . 6 4 . 7 4 . 8 4 . 9 5 5 . 1 5 . 2

x 1 05

-9

-8

-7

-6

-5

-4

-3

x 1 04

Figure 11 Construction of spatial distribution of gas accumulations using the MFAM and the phase map from the probability map in Figure 10.

0

5

1 0

1 5

2 0

2 5

3 0

3 5

4 0

4 5

5 0R e m a in in g re s o u rc e

4 . 5 4 .6 4 .7 4 . 8 4 . 9 5 5 .1 5 . 2

x 1 05

-9

-8

-7

-6

-5

-4

-3

x 1 04

Figure 12 The spatial distribution of undiscovered remaining gas accumulations from the proposed fractal model. The geographical locations of the gas accumulations are in those areas with larger values of the estimated probability of petroleum occurrence

14