CSIRO PUBLISHING
www.publish.csiro.au/journals/eg
Exploration Geophysics, 2009, 40, 257–264
Stochastic simulation of fracture strikes using seismic anisotropy induced velocity anomalies Samik Sil1,2,4 Sanjay Srinivasan3 1
University of Texas at Austin, Institute for Geophysics, J.J. Pickle Research Campus, Bldg. 196, 10100 Burnet Road (R2200), Austin, TX 78758-4445, USA. 2 Present address: Conoco Phillips, PO BOX 2197, Houston, TX 77252-2197, USA. 3 Department of Petroleum and Geosystems Engineering, University of Texas at Austin, 1 University Station C0300, Austin, TX 78712, USA. 4 Corresponding author. Email:
[email protected]
Abstract. Availability of a fracture map of a producing reservoir aids in increasing productivity. Generally, accurate information related to fracture orientation is only available at a few sparse well log locations. However, fractures introduce velocity anomalies in seismic data by making the medium azimuthally anisotropic. When multi-azimuth data is available then it is possible to map the fracture attributes in the entire reservoir zone by analysing the anisotropy induced velocity anomalies in the seismic data. In the absence of 3D data, seismic anisotropy induced velocity anomaly from 2D data (as fracture strikes are not constant and data contains multi-azimuthal effect even when it is 2D) can still be used as a secondary source of information for the purpose of fracture strike simulation. To validate the above hypothesis, fracture strike information in a reservoir from the Mexican part of the Gulf of Mexico is derived using Markov-Bayes stochastic simulation. In this simulation process, accurate well log derived fracture information is used as hard or primary data and seismic velocity anomaly/uncertainty based fracture information is used as soft or secondary data. The Markov-Bayes Stochastic simulation provides multiple realisations of the fracture patterns and thus helps to estimate the uncertainty associated with the fracture strikes of the reservoir. Accuracy of the simulation process is also estimated and the simulation result is compared with simple and ordinary kriging methods of fracture strike simulation. Key words: fracture, geostatistics, Markov-Bayes, seismic anisotropy, stochastic simulation. Introduction Economic importance of fractured reservoirs makes fracture mapping an attractive area of research. The main problem of fracture mapping lies in the fact that the information related to fracture orientations are limited to a few sparse locations (where well log data or shear-wave splitting data are available). So to fill the gap in the inter-well zones of the reservoir with fracture information, geostatistical analysis of seismic data plays an important role. Simple or ordinary kriging is a way to fill the inter-well zones of the model. For the purpose of fracture strike mapping, fracture orientation information from the well log data are generally used as primary data and a kriged estimate of fracture strike information is generated for the entire area (Viruete et al., 2001). This method works well when enough log information is available and distribution of primary data over the field area is not localised. However, in general, kriging (even cokriging and collocated cokriging) returns smooth estimates and do not ensure the true variance between two estimated nodes. Both of these facts lead the researcher to use geostatistical simulation for the purpose of estimating fracture information in the inter-well region (Deo and Morgan 1998; Gringarten, 1998; Angerer et al., 2003; Liu and Srinivasan, 2004; El Ouahed et al., 2005; Suzuki et al., 2005; Tran et al., 2006). The advantages of stochastic simulation are that it reproduces the true variance value between two estimated nodes, incorporates natural randomness in the estimated results and uses additional information (secondary data) for the purpose of generating the final estimated values. Multiple possible results are available from stochastic simulation ASEG 2009
processes. The standard deviation of all those models is a measure of uncertainty for the final fracture map (Xu et al., 1992). A common practice in fracture simulation is to use lithology and mechanical properties as soft or secondary information (Liu and Srinivasan, 2004). Gringarten (1998) used several constraints such as curvilinear fracture geometry, spacing histogram, and length histogram as secondary information for this purpose. Tran et al. (2006) used spatial distribution of fracture density (obtained independently from the other studies) as soft information. In a different type of work where fracture mapping was based on fuzzy logic, El Ouahed et al. (2005) used slope, curvature, distance to nearest fault, and bed thickness as secondary information for producing the final discrete fracture map. The basic data used to generate the secondary information are porosity, permeability and Vsh values. Suzuki et al. (2005) used elastic stress information as a source of secondary data. However all these sources of secondary information have only an empirical relationship with fracture orientation. Angerer et al. (2003) propose seismic anisotropy signatures (amplitude variation with offset (AVO) or velocity variation) as another possible secondary data for fracture strike simulation. Their proposal is based on the fact that the effect of anisotropy is a physical consequence of the presence of fractures (a mathematical relationship exits between anisotropic signature and orientation of the fractures) and is available for the entire horizon of interest as seismic data is generally exhaustive. So far, many different stochastic simulation algorithms have been developed starting from the LU decomposition (the first stochastic simulation method using decomposition of a matrix 10.1071/EG08129
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into lower (L) and upper (U) part). Gotway and Rutherford (1994) carried out comparison of performances between five different types of simulations (LU, turning bands, sequential Gaussian (SG), sequential indicator, and truncated Gaussian random function) and the result in their words is: ‘for imaging spatial uncertainty in a continuous variable, this study suggests that, for the variety of exhaustive data set/transfer function scenarios considered, Gaussian-based approaches, and the sequential Gaussian simulation algorithms in particular, are flexible and accurate methods for stochastic simulation of random fields’. Similarly, comparison between Markov-Bayes (MB) simulation and sequential indicator simulation was made by Zhu and Journel (1993) and the result shows superiority of MB simulation. Later another comparison of performance is made between MB simulation and SG simulation by Fredericks and Newman (1998). Their result shows superiority of MB simulation over SG simulation at least for certain important aspects like local accuracy and reproduction of local heterogeneity. In a different work, Deo and Morgan (1998) prove the superiority of MB simulation for fracture characterisation over other methods of simulation. The above discussion suggests that use of seismic anisotropy data in conjunction with well log based fracture attributes, in a MB simulation is a robust method of spatial mapping of fracture attributes. In this paper we report on a case study of fracture strike simulation in a 2D plane using a MB simulation technique. The study area is located in the Mexican part of the Gulf of Mexico, and the reservoir is densely fractured. We restricted our work to a small area of ~30 km2. Fracture strike information as primary (hard) data is used from more than 50 locations (taken from the work of Shen et al., 2008) in this area for the purpose of fracture strike simulation. An exhaustive anisotropy induced velocity uncertainty map from the residual normal moveout (NMO) analysis of the 2D prestack common midpoint (CMP) gathers is used as secondary (soft data) data in this simulation. The anisotropy in this field is related to the strikes of the fractures. As the strike is not constant at each CMP location, the velocity uncertainty data contains multi-azimuth information. The localised nature of the hard data locations affects our simulation results. Hence in regions away from wells, the constraint placed on stochastic simulation is only due to the anisotropic velocity uncertainty. Next, we describe the process of simulation of the fracture map and uncertainty estimation and the limitations associated with the procedure. It is impossible to obtain fracture strike information alone from 2D anisotropic seismic data analysis and in that case multi-azimuth data is required. But that does not restrict the use of 2D anisotropic data as soft information in the process of stochastic fracture strike simulation when it is supported by enough hard data. Our work shows that in the case of unavailability of 3D seismic, we can utilise 2D seismic data for the purpose of fracture mapping by exploiting the stochastic simulation process, when sufficient hard information related to fracture strike is available. Multi-azimuth information can be extracted from 2D data utilising the correlation between soft and hard data at the collocated locations.
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(Deutsch and Journel, 1998). Co-IK can generate posterior conditional cumulative distribution considering soft data and other previous information. Different previous information can come from: (i) local hard indicator data i(ua; z1) originating from local hard data z1(ua) (for our case fracture strike information at sparse locations from log and shear wave splitting studies); (ii) local soft indicator data 0 y(ua ; z2) originating from ancillary information providing prior probabilities0 about the primary attribute given the secondary value z2(ua ) (for this case observed velocity uncertainty at each CMP location due to fracture anisotropy). (iii) F(z1) global previous information common to all locations u within the study area A, where (a) z1(u) is the primary variable of interest at a particular location u over the field of interest A, (b) z1(ua) are sparse hard data at the location ua, a = 1, 2, . . ., n (n = number of 0 hard data, for our case the number of wells with data), (c) z2(ua ) a0 0 are the secondary data points at the location u , a = 1, 2. . . n0 (n0 = number of secondary data, for our case velocity uncertainty). Then the indicator kriging or Bayesian updating of0 the local prior cdf Fz1(z1) into a posterior cdf Fz1(z1|z1(ua), y(ua )) can be done using the following formula: ProbfzðuÞ zjðn þ n0 Þg ¼ l0 ðuÞFðzÞ þ
n X
la ðu; z1 Þiðua ; z1 Þ
a¼1
þ
n0 X
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ua0 ðu; z2 Þiðua ; z2 Þ;
where la and ua0 are the weights associated with the primary and secondary indicator data and l0 is the weight of the prior. These weights are obtained by solving the Co-IK system: n X
lb ðu; z1 ÞC I ðhab ; z1 Þ þ
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a0 ¼1
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¼ C I ðha0 ; z1 Þ a ¼ 1; 2; . . . ; n n X
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n X
MB simulation In this section we describe, in short, the theory of MB stochastic simulation in a general sense, for completeness. We start our discussion with a description of the theory of indicator cokriging and then extend it to the MB simulation process. In order to incorporate the effect of soft data in a spatial interpolation, indicator cokriging (Co-IK) can be performed
:ð2Þ
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There are (n + n0 ) equations for each set of thresholds (z1, z2). The lag hab refers to the lag between locations ua and ub. To perform the updating, we require indicator covariance of hard data CI(h; z1), covariance of soft data CY(h; z2), and crosscovariance between hard and soft indicator data CIY(h; z1, z2) (Kelkar and Perez, 2002). If enough data (hard and soft) are available then it is possible to infer the covariance. However, in reality data, (especially hard data) are always limited. To overcome this difficulty a Markov’s hypothesis is used to simplify the problem. According to this hypothesis, hard information, i.e. z(ua) screens 0 any collocated 0 soft information (z(ua ) at the places ua = ua ). Under such an assumption, the required covariances can be derived as (Deutsch and Journel, 1998): C IY ðh; zÞ ¼ BðzÞC I ðh; zÞ
Theory
ð1Þ
a0 ¼1
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ð3Þ
BðzÞ ¼ msec ðZ > zÞ msec ðZ zÞ
ð4Þ
C Y ðh; zÞ ¼ B2 ðzÞC I ðh; zÞ8 h > 0 Here the coefficients B(z) are defined by:
and msec is the mean of the secondary data calculated on the basis of the scattergram of collocated primary and secondary data. The coefficients are therefore a measure of how effective the
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secondary data is in discriminating between different outcomes of the primary variable. Thus from equation 4 we can see that the soft auto-covariance model for h > 0 and cross-covariance model are deduced by simple linear rescaling of the hard covariance model. The hard covariance model can be generated from the semivariogram of the hard data. Then using the value of the Beez (a geostatistical term for B(z)), one can determine the soft covariance model and cross-covariance model. The value of B(z) can be 1. The B(z) values (or ‘value regulatets’) regulate the influence of the soft data on the estimation results. When B(z) = 1, soft indicator data corresponding to the threshold z will be treated as hard data, and when B = 0, soft indicator data will be ignored. Under the Markov assumption, the indicator kriging equation 1 and the corresponding system (2) simplifies because only the soft data coincident at the simulation location u is used in the interpolation. Equations 1 and 2 together with the auxiliary conditions (3) for the auto and cross-covariance amount to a Bayesian updating of the prior distribution Fz1(z1) to the posterior conditional probability distribution employing a Markov hypothesis that facilitates the inference of the requisite auto and cross-covariance. Interpolation algorithms such as Co-IK suffer from the drawback that covariance reproduction between estimated nodes does not reproduce the specified (true) covariance. In addition, kriging yields a deterministic (unique) map that is locally accurate, but does not provide an assessment of global uncertainty. The MB estimation procedure can therefore be extended to a sequential stochastic simulation in which the simulation nodes are visited along a random path and at each node the Bayesian updating procedure is carried out including the simulated value at the previously visited node. Several realisations can be generated by changing the random path and the random draw. These realisations reproduce (in an ergodic sense), the specified covariance model for the hard data as well as the calibration relationship between the hard and soft information. Anisotropic velocity uncertainty analysis Soft or secondary information related to fracture orientation (strike) for this study comes from near offset anisotropy velocity anomaly/uncertainty analysis of the seismic data. In this section we discuss how we obtain velocity uncertainty from residual NMO analysis of the near offset prestack seismic data. A common belief in industry is that anisotropy effects are restricted to far offset seismic data only. But the assumption that effects of anisotropy are limited to far offsets is incorrect. To the contrary, several researchers show that theoretically, fracture characterisation from a transversely isotropic medium with a horizontal axis of symmetry (HTI) or from an orthorhombic medium can be computed from the near offset NMO velocity information alone (Grechka and Tsvankin, 1999; Bakulin et al., 2000; Vasconcelos and Tsvankin, 2006). Below we are developing a mathematical framework for the analysis of the velocity uncertainty due to anisotropy from the near offset data analysis. For an isotropic earth model in seismic data processing, the relationship between two-way traveltime (TWT) and offset (x) is generally represented by the Dix (1955) equation. Partial differentiation of the Dix (1955) equation for a fixed offset, can relate the uncertainty in TWT with the uncertainty in estimated NMO velocity by the following equation: qV ¼ V 3NMO tqt=x2 ;
ð5Þ
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where qV is the velocity uncertainty related to the time residual (qt) at a particular offset x after NMO correction (residual NMO effect), using a velocity VNMO. So by estimating qt from the residual moveout for a particular offset x, and putting the NMO velocity in equation 5, we can determine the velocity uncertainty (qV) at each CMP location. The traveltime curve in an HTI medium can be described by the following equation (Tsvankin and Thomsen, 1994): t 2 ¼ t20 þ
x2 V 2NMO
þ
A 4 x4 1 þ Ax2
ð6Þ
where for weak anisotropy (Tsvankin, 1997): 2
V 2NMO ¼ V h0 ð1 þ 2dh cos2 Þ
ð7Þ
and V0h is the vertical velocity (obtained from well log data), dh is one of the anisotropic parameters (Tsvankin, 1997), is the azimuth angle between the principal axis and the seismic inline, and A4 and A are functions of the other anisotropic parameters and azimuth. A mismatch between isotropic and anisotropic traveltimes at far offsets is due to the third term of equation 7. However when far offsets are muted, we can still observe mismatch between the traveltime curves of isotropic and anisotropic media. This mismatch is due to the difference between isotropic and anisotropic NMO velocities. It is evident from equation 6, that when dh is zero (medium is isotropic), the NMO velocity of the HTI medium becomes the isotropic velocity. So near offset mismatch between anisotropic and isotropic media (using vertical velocity) is dependent upon the parameters dh and azimuth . To illustrate this, we first simplify equation 7 for a small value of dh (assuming weak HTI medium) and normalise it with the vertical velocity to obtain: V NMO V h0 qV ¼ h ¼ dh cos2 V h0 V0
ð8Þ
If we consider qV in equation 8 to be the same as qV in equation 5, (because vertical velocity and NMO velocity are the same when the medium is isotropic), then the velocity uncertainty normalised by the vertical velocity obtained from well logs is a function of anisotropic parameter dh and azimuth angle (and strike of the fracture, as strike = 90). This value of velocity uncertainty qV is used as soft information for our study (after indicator transform, in the 2nd part of the RHS of the equation 1). In the case that multiazimuth data is available, then equation 8 alone is sufficient enough for obtaining values (and thus strike information) and we can obtain a deterministic result. However, when 2D data is available along with hard data, we can utilise equation 1 for the fracture simulation. Our logic is, even though 2D data contains single azimuth information at each CMP location, for the entire field (where fracture strike is not constant) it contains multiazimuthal information. If one wants to perform simulation using 3D data (preference of probabilistic results over deterministic result), then covariance models derived from the 3D data can be used in equation 1. Implementation details In general, the process of simulation can be divided into the following steps: *
*
A calibration scattergram of the collocated primary values versus the secondary values is plotted. The primary variable is converted into indicators defined at different cutoffs. Cutoffs can be defined based on a cross-plot between (scattergram) collocated primary and secondary data.
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Indicator variograms are calculated from all the available primary data for each cutoff to calculate primary indicator covariance. The secondary variable is discretised in different classes guided by the same cross-plot of the collocated primary and secondary data. For each class of the secondary variable, the scattergram values are used to calculate the probability distributions.
Histogram of hard data 18 16 14 12
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The scattergram is also used to calculate the coefficients, B(z). *
*
These B coefficients and the primary variable covariances are used to calculate the covariances and cross-covariances for the secondary variables approximated by the Markov hypothesis. Once all the covariances are available, the posterior cdfs are calculated using the Co-IK formula (equation 1).
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Fig. 2. Histogram of hard strike data. The distribution pattern is multimodal with at least three distinct nodes.
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Crossline # Fig. 1. Fracture strikes used as primary or hard information for the MarkovBayes simulation. The hard data location lacks randomness which may introduce some bias in the simulation results.
Fig. 3. Typical Prestack Time Migrated (PSTM) gather used in this study. PSTM is performed using the isotropic assumption. Large residuals in the form of ‘hockey sticks’, may indicate the anisotropic nature of the medium. For residual two-way traveltime calculation, the overcorrected portion of the seismograms is fitted with a straight line (blue line) up to a fixed near offset (dx). dt is then estimated from the slope of the straight line. In this figure x-axis is offset and y-axis is time.
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Results
Hard Vs soft data
Data preparation
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Hard fracture strike information for the study area is available from 50 locations. These data are obtained from Formation MicroScanner logs and shear wave splitting measurements. The fastest polarization direction of split shear wave is considered to be the direction of fracture strike (Figure 1). We note that the data locations are not randomly distributed over the entire horizon and may cause problems in the final simulation result. The maximum value of the strike is 90 and the minimum value is 8. The resulting mean value of the hard fracture strike data is 60 with a standard deviation of 25. The frequency distribution is clearly multimodal for the hard data at least with three distinct modes (Figure 2). Soft data for the study is fracture anisotropy induced velocity uncertainty. This value is available for the entire study area (3146 points for this case). To obtain these data we fit the near offset prestack NMO corrected gather with a straight line. From the slope of the fitted line qt values are determined (Figure 3). Then using equation 5 qt is converted to velocity uncertainty qV (Figure 4). The minimum value of the velocity uncertainty is 0% and the maximum is 20%. The mean and standard deviation of velocity uncertainty is 6% and 4% respectively.
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Fig. 5. The cross-plot between primary (fracture strike with inline) and secondary variables (velocity uncertainty). We determined four thresholds for hard and soft data based on the cross-plot (space between the three straight lines). The increasing value of velocity uncertainty with increasing strike is an indication that we are observing fracture anisotropy effect.
Indicator Variogram-Threshold 1
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Fig. 6. Experimental indicator semivariogram corresponding to the four thresholds. Variograms showed anisotropic behaviour. Modelled variograms (red broken line – major axis of anisotropy and blue broken lines – minor axis of anisotropy) along with the lag distances (h) are used to determine indicator covariances.
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Case study
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We apply MB simulation to map the fracture patterns using available hard and soft data. First we prepare the scattergram to determine B(z) values. We generate a cross-plot between primary data and collocated velocity uncertainty values obtained from P-wave anisotropy analysis. The cross-plot is shown in Figure 5. We can see velocity uncertainty is increasing with strike of the fracture (measured from inline direction) indicating the effect of fracture anisotropy (HTI medium). A code BICALIB from GSLIB (Deutsch and Journel, 1998) is used to determine the value of B(z) (equation 3) for four thresholds of hard data corresponding to the 20, 40, 60, and 85 strike. Excellent B(z) values (0.89, 0.82, 0.69, and 0.92) are obtained for the four thresholds implying good influence of the soft data on the final simulations. BICALIB also performs indicator transform of soft data with the provided threshlods of 1%, 4%, 7%, and 12% of velocity uncertainty. For the final simulation, indicator covariances are determined from the experimental indicator semivariograms of the hard data corresponding to the four thresholds (Figure 6). The B(z) values are then used to generate soft indicator covariances and crosscovariances. Those covariances are then used to determine the required weights for equation 1. Preparation of hard and soft indicator data were done using the threshold values. A sequential indicator simulation code SISIM from GSLIB (Deutsch and Journel, 1998) is used to generate 20 simulation results. The simulation uses indicator covariances determined from the variograms of the hard data corresponding to the four thresholds. Some of the simulated results are plotted in Figure 7. A simple interpretation of the simulated results is that the ‘true’ fracture strike can be anywhere within the cloud of the simulated results. The standard deviation of all the simulated results can be considered as the uncertainty in the fracture strike at a particular location (Figure 8).
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We also compare our results with the performance of kriging. Both simple and ordinary krigings are performed using the same hard data. Kriging results in a smooth fracture map which is not realistic in nature. In Figure 9 we plot the kriging derived fracture map for the same horizon. We also check the performance of the MB simulation by performing blind test (or Jack-Knife test in geostatistical terminology). For that test, hard data is eliminated from one set of simulation. We then compare all the simulated results for the eliminated hard data with its original value. Similarly, another set of simulation is run excluding another hard data and the values get compared. In this way we check the reproducibility of all the hard data using MB simulation. A result from a Jack-Knife test is shown in Figure 10. In this figure the x axis is the original values of the hard data and the y axis is the simulated values for them. Error bars are added from the standard deviation of all the simulations. A black line is representing here 1 : 1 relationship between the simulated values and the original values of the hard data. Clustering of all the data in the plot along that black line proves satisfactory performance of the MB simulation. Discussion
Fig. 7. Few simulation results. Markov-Bayes simulation returns nonsmooth results of fracture strikes which is realistic in nature.
The simulation technique described above has two potential pitfalls. In our analysis we assume that velocity uncertainty observed in the seismic data is due to the fractured layer. But the near offset velocity uncertainty could also be due to using the wrong velocity model while processing the data (even when the medium is isotropic) or could be due to the presence of a polar
anisotropic medium (VTI) above the fractured layer. In both cases simulation will result in a faulty fracture map. This problem can easily be avoided in the case of the presence of multi-azimuth data. But while using single azimuth data for stochastic simulation this problem can also be handled. For that
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Fracture strike simulation
Exploration Geophysics
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Fig. 9. A fracture strike estimation using ordinary kriging. The result shows extremely smooth pattern which may not be realistic in nature.
we suggest careful checking of B values analysed from the crossplot between collocated fracture strike and velocity uncertainty data (equations 3, 4). If the observed velocity uncertainty is due to a top VTI layer or faulty velocity model we can expect to observe low B values indicating no correlation between fracture strike and velocity uncertainty at the collocated locations. Whereas when B values are high (like the case presented here) we can expect the velocity uncertainty is due to the fractured medium only. In the case of a presence of a VTI medium over a fractured medium the velocity uncertainty signal will bear the combined signal from both the HTI and top VTI medium. Because the effect of the VTI medium will be constant (no azimuthal effect), we can still observe the effect of fracture orientation from the cross-plot analysis which will result in a high B values. Thus in that case,
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Hard data Fig. 10. Jack-Knife test is performed to check the quality of the simulation. Mean value of simulated results are plotted against the original hard data value which are excluded from that particular set of simulation. Standard deviations (previous slide) are used as error bar. Black line is 1 : 1 correlation. Cluster of data around the black line proves satisfactory performance of the simulation.
velocity uncertainty from single azimuth data can still be used for fracture strike simulation purposes. In our case we have additional independent information regarding the presence of fracture anisotropy. Other than that, nice correlation (high B values) exists between the collocated velocity uncertainty and fracture strike data. This validates the fact that the observed velocity uncertainty is from the target fractured medium. Another advantage of MB stochastic simulation is, unlike kriging, it does not require a Gaussian distribution of hard and soft data. Indicator transform of the data itself provides the measure of conditional probability for the hard and soft data.
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Conclusions We performed a fracture simulation using the MB hypothesis. The simulation used fracture strike information at sparse location as hard data and seismic anisotropy driven near offset velocity uncertainty as soft data. High correlation between collocated soft and hard data indicates that the observed velocity uncertainty is due to the presence of a fractured medium. Simulation results in several realisations of fracture strike maps. Maximum uncertainty in fracture strike estimated from the standard deviation of all the realizations is 35 at the locations away from the hard data. Little work on fracture simulation is done where seismic anisotropy is used as soft data when anisotropy is a physical measure of fracture orientation. In that way this work is unique. For this purpose we describe a technique of deriving near offset velocity uncertainty from the Residual Moveout analysis of the HTI medium. We compare our results with kriging, and simulation definitely results in a more realistic fracture orientation picture of the reservoir as it returns non-smooth results. Results from simulation also retain the true variance between two estimated nodes. Multiple realizations also help to estimate uncertainty associated with the fracture simulation. We use Jack-Knife test plots to judge the performance of our simulation. Clustering of points along a 45 black line proves the satisfactory performance of MB simulation for fracture mapping. Acknowledgments We are thankful to the editor Lindsay Thomas and Vinay Vaidya of Exploration Geophysics for their help. We are also thankful to Dr Ravi P. Srivastava and Dr Abhijit Gangopadhya for their constructive criticism and suggestions which made this manuscript better. This work is a part of the reservoir model uncertainty estimation project conducted by the petroleum engineering department of UT Austin with funding received from G&W Systems.
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