Geostatistical Reservoir Modelling of an Indian Oilfield using Kriging and Simulated Annealing B. C. Sarkar 1 , Monosvita Chaliha 2 , and Kalyan Saikia 3 1 Indian School of Mines University, Dhanbad; 2 Cairn Energy India Pty Limited, Gurgaon; 3 Geopetrol International Inc., New Delhi. (Corresponding author E-mail: [email protected]) ABSTRACT Reservoir characterization involves accurate modelling of reservoir parameters, viz. porosity, permeability, reservoir thickness, average oil content, sand-shale ratio, acoustic impedance, among others. Geostatistical reservoir modelling of an Indian oilfield has been attempted in this paper employing kriging and simulated annealing techniques. Omni-directional semi-variography of selected reservoir parameters revealed that sample values are correlatable within 800 m distance apart adequately represented by a spherical model fit. Block kriging of sample values yielded block-wise estimate of reservoir parameters together with associated error of estimation. A comparison of kriged and simulated results reveal that kriging procedure provided maps with relatively smoothed variations while simulated annealing procedure generated variability that are close to sample distribution. Introduction Reservoir characterization involves adequate and accurate modelling of reservoir parameters, viz. porosity, permeability, reservoir thickness, average oil content, sand- shale ratio, acoustic impedance, among others. However, estimates obtained from conventional techniques such as isopach and isochore are less reliable as they under- predict the variability and over-predict the reservoir properties. Geostatistics based on Matheron’s (1971) theory of ‘Regionalized Variable (ReV)’ provide a means of describing spatial continuity together with associated error of estimation in building a reservoir model. Kriging based on semi-variogram model provides local estimate of the reservoir attributes under study. Stochastic simulation, on the other hand, provides a means of building alternative, equi-probable, high resolution models of spatial distribution of reservoir attributes (Deutch and Journel, 1997; Journel, 1990; Olea, 1999). In this paper, an attempt to reservoir modelling of an Indian oilfield has been made using kriging and geostatistical simulated annealing technique and to characterise reservoir parameters. Based on the availability of drill core sample measurements and the need for geostatistical modelling study, exploratory data in respect of porosity, reservoir thickness and average oil content (AOC) have been considered for reservoir modelling and characterization. Geology of the Oilfield The oilfield under reference constitutes an important part of the principal oil bearing onshore basin in India. For the sake of confidentiality, name and location of the oilfield have not been mentioned in the paper. Lithologically, the oilfield is comprised of thick 1
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Geostatistical Reservoir Modelling of an Indian Oilfield using Kriging and Simulated Annealing
B. C. Sarkar1, Monosvita Chaliha2, and Kalyan Saikia3
1Indian School of Mines University, Dhanbad; 2Cairn Energy India Pty Limited, Gurgaon; 3Geopetrol International Inc., New Delhi. (Corresponding author E-mail: [email protected])
ABSTRACT
Reservoir characterization involves accurate modelling of reservoir parameters, viz. porosity, permeability, reservoir thickness, average oil content, sand-shale ratio, acoustic impedance, among others. Geostatistical reservoir modelling of an Indian oilfield has been attempted in this paper employing kriging and simulated annealing techniques. Omni-directional semi-variography of selected reservoir parameters revealed that sample values are correlatable within 800 m distance apart adequately represented by a spherical model fit. Block kriging of sample values yielded block-wise estimate of reservoir parameters together with associated error of estimation. A comparison of kriged and simulated results reveal that kriging procedure provided maps with relatively smoothed variations while simulated annealing procedure generated variability that are close to sample distribution.
Introduction
Reservoir characterization involves adequate and accurate modelling of reservoir parameters, viz. porosity, permeability, reservoir thickness, average oil content, sand-shale ratio, acoustic impedance, among others. However, estimates obtained from conventional techniques such as isopach and isochore are less reliable as they under-predict the variability and over-predict the reservoir properties. Geostatistics based on Matheron’s (1971) theory of ‘Regionalized Variable (ReV)’ provide a means of describing spatial continuity together with associated error of estimation in building a reservoir model. Kriging based on semi-variogram model provides local estimate of the reservoir attributes under study. Stochastic simulation, on the other hand, provides a means of building alternative, equi-probable, high resolution models of spatial distribution of reservoir attributes (Deutch and Journel, 1997; Journel, 1990; Olea, 1999). In this paper, an attempt to reservoir modelling of an Indian oilfield has been made using kriging and geostatistical simulated annealing technique and to characterise reservoir parameters. Based on the availability of drill core sample measurements and the need for geostatistical modelling study, exploratory data in respect of porosity, reservoir thickness and average oil content (AOC) have been considered for reservoir modelling and characterization.
Geology of the Oilfield
The oilfield under reference constitutes an important part of the principal oil bearing onshore basin in India. For the sake of confidentiality, name and location of the oilfield have not been mentioned in the paper. Lithologically, the oilfield is comprised of thick
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sediments of shelf facies of Tertiary age. The study area represents two oil bearing horizons, viz. a Lower formation of Oligocene age and an Upper formation of Miocene age. These two horizons are referred to in this paper as Lower and Upper formations. Both the formations are comprised of arenaceous sandstone with minor shales. The rocks represent characteristics of clastic continental with meandering to flood plain environment. In the study area, hydrocarbon accumulations, confined within structural traps, are formed due to both faulting and folding. However, field studies revealed occurrence of some stratigraphic trap conditions also. A number of faults occurring within the reservoir rocks mainly serve as seals. While the reservoir rocks are constituted by sandstones, the cap rocks are argillaceous and clays.
Geostatistical Modelling Approach
Geostatistical modelling study of reservoir parameters of the oilfield performed for the study area constitutes the following steps:
i. Organization and generation of oil-well database; ii. Statistical data treatment of reservoir parameters;
iii. Computation of experimental semi-variograms; iv. Fit of mathematical functions to experimental semi-variograms; v. Segmentation of the lateral study stretch of the oilfield into small blocks;
vi. Estimation of reservoir parameters using block kriging technique; and vii. Generation of spatial distribution maps of kriged estimate and kriging variance of
reservoir parameters. The modelling study has been carried out using geostatistical softwares, viz. GEXSYS (Sarkar, 2001); GSLIB (Deutch and Journel, 1997) ISATIS (Bleiness et al., 2004) and other tailor-made specific computer programs developed for the present study. Oil-well data of 17 exploratory wells have been utilised pertaining to Lower and Upper oil bearing formations. These include exploratory data in respect of porosity, reservoir thickness and average oil content, formation thickness, depth of formation tops and water saturation data. The database consists of information pertaining to well identification number, coordinates of well location, viz. X (eastings), Y (northings) and Z (elevations), and values of parameters, viz. porosity (%), reservoir thickness (m) and average oil content (%). Statistical Data Treatment and Semi-variography
Frequency distribution analyses have been carried out for porosity, reservoir thickness and average oil content. The frequency distributions of respective reservoir parameters exhibit a skewed nature. Summary results of the statistical analyses are provided in Table 1. The coefficients of variations of respective variables for the two formations are found to be closely similar, which is a reflection of similar variability characteristics.
Omni-directional experimental semi-variograms for reservoir thickness, porosity and average oil content in respect of both Lower and Upper formations have been computed
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with a lag distance of 300 m based on average spacing of exploratory oil-wells. To detect the presence of spatial anisotropy, directional semi-variograms were also computed using an angle of regularization of 11.25º and lag distances of 300 m and a spread limit of twice the lag distance. However, the directional semi-variograms did not reveal any satisfactory regionalized phenomena. The omni-directional experimental semi-variograms in respect of reservoir thickness exhibit a low nugget variance for the Upper formation in comparison to the Lower formation. The experimental semi-variograms of porosity in respect of the two formations vary in terms of nugget variance, continuity, sill variance and range of influence. Similarly, the experimental semi-variograms of average oil content for the Lower and Upper Formations revealed spatial characteristics of different nature. Spherical functions to each of the reservoir parameters provided adequate fit with (i) ratio of estimation variance to kriging variance as 0.97, 0.99, and 1.02; and (ii) mean error of estimation as 0.002, 0.003, and 0.001 respectively for porosity, reservoir thickness and average oil content. The semi-variogram models (Table 2) in respect of the reservoir variables revealed a range of influence varying between 500 m and 800 m.
Kriging
Blocks of dimensions of 300m x 300m were selected for kriging based on semi-variogram lag distance, which led to a total of 48 blocks. Block kriging of the reservoir parameters have led to generate spatial distribution maps of kriged estimates and associated kriging variances (Figs. 1 and 2). From the kriged maps, it may be observed that for the Lower formation, most of the estimates range between 85m and 230m in the case of reservoir thickness while in the case of porosity, the block kriged estimated values range between 24% and 28% and that in the case of average oil content, the values range between 4% and 7.5%. In the case of Upper formation, the block estimates range between 50m and 100m for reservoir thickness. Porosity kriged estimate of Upper formation reveal a concentration of higher values in the north-eastern part of the demarcated area while that for average oil content, the blocks have estimates in the range of 6% to 10% with a concentration of higher values in the western part of the demarcated boundary. The summary results of block kriging are provided in Table 3. Reservoir Modelling using Simulated Annealing
Simulated annealing (SA) is a relatively more popular and often used method for reservoir modelling and characterization on a node-by-node basis (Olea, 1999; Lantuejoul, 2002. The first step in the simulated annealing technique is to divide the entire study area into regular grids on a node-by-node basis. In the present study area, total number of grid nodes generated is 48 with a spacing of 300 m. The grid spacing has been based on the average distance between the exploratory oil-wells in the study area. If some sampling points have not fallen on the grid nodes, those have been moved to the closest node. The nodes where the original sampling occurs have been assigned with their actual values while for remaining of the nodes, values have been assigned by drawing at random from a cumulative distribution function, which is typically that of the data sampling. These values have been assigned at random to the grid nodes that are without original sampling. Statistical analyses of the simulated values in respect of the variables carried out revealed histograms of the simulated values closely resembling that of the
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original values. The SA technique requires an objective function (G) to achieve a match between a function (typically the semi-variogram) of the original observations (sample points) and the one for simulated realizations through the annealing procedure (cooling schedule). In the simulation process, a match is reached by reducing the objective function below a small threshold (Olea, 1999) through swapping pairs of values chosen at random. The initial value of G is termed as Gold. After swapping pairs of values, the objective function, G is recalculated as Gnew. In the simulation procedure, a swap is accepted provided that:
i. Neither of the locations involved in the swapping coincides with a sampling site; ii. There is a decrease in the objective function; iii. Even if the objective function does not decrease, some of the swaps are still retained,
but the frequency with which these unfavourable swaps are retained decreases with
t
GG newold
e−
, where ‘t’ mimics the temperature parameters in the Boltzman distribution.
In the present study, the objective function G is calculated as (Eq. 1):
[ ]
( ) ( )∑ −=
h
G
h
hhG
2
2
)]([γγγ
…….…. (1)
where, is the semi-variogram model and γG(h) is the semi-variogram of the realization. If the value of G is near or equal to zero, then the model semi-variogram and the semi-variogram for the simulated values would be identical. Initially, generated values are assigned to nodes at random and G is calculated. This value is expected to be much higher than zero and hence the next step is to lower the G value in gradual stages leading to a value very close to zero. Attempts have been made in the present study to find a suitable annealing schedule as proposed by various researchers (Olea, 1999; Deutsch and Cocherham, 1994; Bhattacharya et al., 2003). After several attempts, the following annealing schedule has been found to be suitable in the present simulation study:
õ(h)
Initial temperature (T0) = 1 Cooling Schedule (Tx) = T0 (0.88) X, (where, X= 1, 2, 3, 4 …………n); and Minimum tolerance limit of objective function (G) = 0.001
The entire exercise was carried out using a computer program, SIMANN developed in C++ by the present authors. Fig. 3 shows the final experimental semi-variograms of simulated values together with model semi-variograms for respective reservoir variables of the two formations. In the simulation process, it is observed that in the initial stage, when the temperature is high, the variation in G value is highly fluctuating. As the temperature drops with increasing number of swaps, the objective function tends to stabilise. However, the global minimum G value is obtained at very low temperature (i.e. approximately 0.0001) which indicates the annealing is closer to near perfect. Discussion
While the kriging procedure provided maps with relatively smoothed variations, the simulated annealing procedure resulted in variations showing local heterogeneity that are
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apparently close to the actual variations. The stochastic images reproduced by simulated annealing simulation in the present modelling study indicate that there are apparent local fluctuations within the reservoirs. The images produced through geostatistical conditional simulation provide an improved means of realization of the uncertainties about the reservoir and thus can be of aid as ‘WHAT-IF’ tool for planning and for risk assessment.
Acknowledgement
The authors are thankful to the company engaged in exploration and development of the oilfield under study for necessary data support and information required for the modelling study. The authors also thankfully acknowledge Dr. Shalivahan of Department of Applied Geophysics for necessary help and the Department of Applied Geology, Indian School of Mines University, Dhanbad for providing necessary infrastructure facilities.
References 1. Bhattacharya, B. B., Shalivahan, and Sen, M. K. (2003) Use of VFSA for resolution and uncertainty
analysis in 1D DC resistivity and IP inversion, Geophysical Prospecting, Vol. 51, pp. 393 – 408. 2. Bleiness, C., Deraisme, J., Geffroy, F., Jeannee, N., Perseval, S., Rambert, F., Renard, D., Torres, O.,
and Touffait, Y. (2004) ISATIS, Isatis Software Mannual, 5th Edition, Geovariances and Ecole des Mines de Paris, France, 709 p.
3. Deutch, C. V. and Journel, A. G. (1997) GSLIB: Geostatistical Software Library and User’s Guide, Oxford Univ. Press, New York 380 p.
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5. Journel, A.G. (1990) Geostatistics for Reservoir Characterization Society for Petroleum Engg paper 20750prepared for presentation at the 65th Annual Technical Conference and Exhibition of the SPE, New Orleans, Sept.23-26.
6. Matheron, G. (1971) The Theory of Regionalised Variables and its Application, Booklet No. 5, Les Chiers de Centre de Morphologie Mathematique, Fontainebleau, 211p.
7. Olea, R. A. (1999) Geostatistics for Engineers and Earth Scientists, Kluwer Academic Publication, Boston, 303 p.
8. Sarkar, B. C. (2001) Computer based geostatistical modelling process for ore evaluation: a step by step case study on a bauxite deposit. In Computer Applications in Mineral Industry, eds. C. Bandyopadhyay and P. R. Sheorey, Oxford IBH Pub. Co., New Delhi, pp.107-115.
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Table 1 Summary results of statistical analyses of the reservoir parameters.
Upper Formation Lower Formation
Statistical Parameters
Porosity (%)
Reservoir thickness
(m)
Average oil content
(%)
Porosity (%)
Reservoir thickness
(m)
Average oil content
(%) Mean 24.38 90.62 7.92 26.02 164.15 5.36
Standard deviation 2.09 36.71 2.84 3.00 59.65 2.11
Variance 4.35 1347.83 8.05 9.02 3557.87 4.44
Degree of skewness -1.15 0.70 -.04 -0.96 -0.14 1.09
Degree of kurtosis 3.83 5.75 1.96 3.57 3.66 3.53
Coefficient of variation
0.08 0.40 0.35 0.12 0.36 0.39
Table 2 Summary of semi-variogram model parameters.
