PROBALISTIC SEISMIC INVERSION USING … · seismic inversion method based on matching large numbers...

© 2017 SOCIETY OF EXPLORATION GEOPHYSICISTS (SEG)Link to SEG digital library | https://doi.org/10.1190/segam2017-17565685.1

PROBALISTIC SEISMIC INVERSION USINGPSEUDO-WELLSEXTENDED ABSTRACT

AUTHORS: PATRICK CONNOLLY*, PCA LTDAUTHORS: MARK O`BRIEN, CEGAL LTD

BY PATRICK CONNOLLY* AND MARK O`BRIEN

SUMMARYThere is generally a high degree of uncertainty in anyattempt to characterize the subsurface. Bayesian methods provide a framework to account for the uncertainty of the prior knowledge and the data and to estimate the uncertainty of the result.

Connolly and Hughes (2016) describe a probabilisticseismic inversion method based on matching large numbers of pseudo-wells. From a Bayesian perspective the pseudowells are samples from the prior distribution. The samples are uncorrelated so, in essence, this is a simple Monte Carlo method. High efficiency is achieved by restricting dimensionality.

The method allows for the inclusion of a wide range ofprior data types and for the uncertainty of these data to be specified. This optimizes the balance between the prior data and the seismic. Here we provide further commentary on the method, describe some recent improvements and show more results.

INTRODUCTIONBayesian seismic inversion algorithms emerged in the early 2000’s (Buland and Omre, 2003; Gunning and Glinsky, 2004) and progress has continued within academia and oil companies (e.g. Leguijt, 2009; Grana and Della Rose, 2010).Over the past decade or so a preference has developed for single step inversion methods, also referred to as

petrophysical inversion, in which reservoir properties are estimated directly (e.g. Bosch et al, 2009), rather than a two-step process of first estimating elastic properties and mapping these to reservoir properties.

These two approaches have been combined (Buland et al, 2008) and more recently have appeared in commercial applications (Riise et al, 2012; Kemper and Gunning, 2014).

Connolly and Hughes (2016) describe an inversion method based on matching of pseudo-wells to seismic data. They directly estimate facies probabilities and reservoir properties with associated uncertainties within what can be interpreted as a Bayesian framework. Here we describe the method from a Bayesian perspective:

ONE DIMENSIONAL PROBALISTIC INVERSIONThe application uses a pure Monte Carlo approach in which independent samples, selected from a prior distribution, are tested against the data and then accepted or rejected. This contrasts with Markov chain Monte Carlo methods in which samples are correlated such that the outcome of each match influences the next sample.

The potential difficulty with Monte Carlo methods is that, if the model space is large, they can be slow. The pseudowell approach achieves high efficiency by restricting the size and spatial dimensionality of the samples.


PROBALISTIC SEISMIC INVERSION USING PSEUDO-WELLS

Each sample is a fairly short (<100ms) 1D vertical stratigraphic profile. Multiple profiles are vertically concatenated to extend over the zone of interest. Profiles are initially modelled as lithofacies columns and, by accessing prior geological and rock physics models, are developed into pseudo-wells; profiles with a full suite of reservoir and elastic property curves. Synthetics are calculated from the elastic curves and matched to the seismic. From the accepted best-match pseudo-wells reservoir property estimates are derived from the corresponding reservoir property curves.

The lithofacies columns are part deterministic and partstochastic. Interpreted horizons provide a framework for the individual vertical components. Within these macrolayers, a stack of stochastic micro-layers is built as a continuous time Markov chain (CTMC). Each micro-layer is a single lithofacies. The CTMC uses transitionprobabilities, usually derived from local well control, for the stochastic ordering of lithofacies. The micro-layerthicknesses are randomly selected from exponentialdistributions. Each lithofacies is modelled independentlyusing an appropriate rock physics model that incorporates the required level of uncertainty from global or locally calibrated empirical models.

The use of pseudo-wells allows a high degree of controland flexibility when specifying prior expectations anduncertainties. This can be important. The posteriordistribution is essentially the product of the prior andlikelihood distributions. If the seismic is able to tightlyconstrain the solution, implying a narrow likelihooddistribution, the precise definition of the prior distribution is of less importance; it just needs to encompass the solution. For example the net-to-gross of a relatively

thin layer, say less than 50ms, with a simple binary sand-shale impedance model can be quite accurately estimated from the seismic. This implies a low variance likelihood which should dominate the posterior. As long as the prior is broad enough, all should be well.

For a thicker interval, or more complicated geology withmultiple facies having variable properties, the solution is less constrained by the seismic and the likelihood variance will be large. In such situations the definition of the prior will significantly impact the posterior so it becomes important to be able to carefully specify our expectations for the reservoir.

The prior is providing the low frequencies for the inversion. The low frequencies will generally be uncertain, dependent, for example, on depth trends and the proportions of the various lithofacies. The pseudo-well parameterization allows the user to specify the elastic properties of the lithofacies and the expected distributions of their proportions. By assigning variance to the facies properties and distributions to their proportions, a distribution of low frequency information is being provided consistent with the prior knowledge of the reservoir. This contrasts with a conventional fixed low frequency model; effectively a very narrow prior that will inevitably bias the result to some extent (Grant, 2013).

We assume the posterior distribution for each 1D window to be generally broad but unimodal, implying a large number of similar solutions rather than say a discrete number of distinct solutions. Our goal is therefore not to search for a particular solution having a residual error close to zero but rather to obtain


a representative sample of possible solutions.

The uncertainty of the output is derived from a set of bestmatch pseudo-wells; the thirty or so pseudo-wells having the lowest residual error from the seismic match. The pseudo-wells each comprise a full suite of elastic and reservoir property curves so, from the best-match set, facies probability volumes can be calculated or reservoir properties, both means and variances, can be estimated.

No lateral continuity is imposed on the solution; everyinput trace is inverted independently. High frequencies,above the top seismic frequency, are therefore unconstrained limiting the mean property estimates to the highest frequency of the seismic. We envisage that the output volumes could be used to constrain subsequent geostatistical modelling to produce higher frequency realization for use in reservoir simulation models.

An important requirement for any inversion algorithm is stability; similar input should result in similar output.Because of the absence of enforced lateral correlation,algorithmic stability can be judged by the continuity of the mean property estimates. Given an adequate number of pseudo-wells, the method is highly stable with trace-totrace continuity of the output similar to the input.

Between 2000-5000 pseudo-wells per trace are usuallyenough to produce good results. Even though the numbers are large the program is very efficiently. For example, a typical reservoir extending over one million traces with 2000 pseudo-wells per trace, giving a total of two billion pseudo-wells, would run in about 8 hours on a standard interpretation workstation and single test lines can be run almost interactively. This approach is therefore demonstrably practical.

Both performance and quality are improved by preconditioning the seismic data. Rather than inputting,

say, corrected gathers this algorithm works with seismic that has been optimized to correlate with the reservoir property of interest. This decreases the data volume and hence the internal calculation for the match and also increases signalto-noise. Optimization is achieved by coordinate rotating color inverted intercept and gradient data resulting in relative extended elastic impedance parameterized by the rotation angle chi (Lancaster and Whitcombe, 2000; Whitcombe et al, 2002).

Multiple chi volumes may be required if multiple properties are to be estimated or if the optimum angle changes at, say, a fluid contact. For multiple input volumes, multiple synthetics are calculated from each pseudo-well at appropriate chi angles. These are each matched to the corresponding seismic volume, match metrics are combined and the pseudo-wells that best match all volumes are selected. So, in other words, the process allows for simultaneous inversion.

The use of chi volumes provides a way to capture seismic uncertainty. Probably the largest error in seismic AVOdata is the scaling of the gradient. Scaling errors will arise from incorrect offset-angle relationships, incorrect prestack scaling, moveout error, ignored anisotropy and so on, all of which are likely to be present to some extent. Incorrectly scaled gradients will translate into incorrect chi angles. This uncertainty can be captured by generating pseudo-well synthetics with a range of chi angles consistent with the estimated uncertainty of the input data chi angle.

NORTH SEA EXAMPLEWe show results from a North Sea Tertiary turbidite oilfield. For this project we used the Petrel plug-in version of the application, Blueback ODiSI (one dimensionalstochastic inversion) from Cegal. Input seismic data(Figure 1) are two relative extended elastic impedancevolumes, relEEI(15) and relEEI(90), which are approximate fluid and lithology projections respectively.


Figure 1: Input data. Top; relative EEI(15). Bottom; EEI(90).

The fluid volume better images the oil sands while thelithology volume is better able to image the brine sands (the contact is below the main sand bodies in the sections shown in Figure 1).

The zone of interest is divided into four macro layers; anoverburden, two reservoir macro-layers and an underburden each tied to the interpreted horizons shown on Figure 1. Horizons do not need to be accurately

picked and for this example a single bulk-shifted horizon is used. The macro-layers are approximately 70ms thick.

Figure 2 shows the best-match synthetics and corresponding lithofacies columns at the left-hand welllocation. The bars on the left of each of the panels indicate the positions of the four macro-layers at this location. For the synthetics panels the first trace is the seismic, the second is the average of the best-match synthetics and the remaining traces are the thirty best-match synthetics. In the lithofacies panel the first track is the well lithofacies interpretation, track two is the estimated most likely lithofacies, with variable saturation indicating the degree of confidence, and the other tracks are the lithofacies columns for the best-match pseudo-wells.

Individual best-match synthetics show a fair degree ofvariability; for each individual chi projection there is adegree of compromise because the corresponding synthetic also has to match the other volume. The algorithm finds the best solutions consistent with all data. The average synthetics are a good match to the seismic traces and the predicted most-likely lithofacies are a good match to the well data.

A 4-10-48-68Hz trapezoidal wavelet was used for thesynthetics to match the parameterization of the coloredinversion. The half-cycle wavelength of the 10Hzcomponent is 50ms so any attempt to characterize layers greater than this, as is the case for this data, will suffer increased uncertainty due to the diminishing frequencies in the seismic. The 70ms macro-windows will be somewhat deficient in low frequencies but there will also be a low frequency component of the full,


Figure 2: Top; best-match synthetics for the relEEI(15) data. Middle; best-match synthetics for the relEEI(90) data, and bottom; the corresponding set of lithofacies columns, the facies probabilty column (track 2) and the well facies column (track 1). Refer to Figure 3 for lithofacies color codes.

~140ms, reservoir interval down to about 4Hz; almost completely undefined by the seismic. These low frequency components will, in part, depend on the net-to-gross of the reservoir and hence will be laterally highly variable. It is therefore important to be able to capture this uncertainty by specifying our prior expectations of the low frequencies, in the form of distributions for rock property trends and facies proportions, as they may have significant influence on the results.

Figure 3 shows the prior distributions of facies proportions for macro-layer three. The solid lines are the parameterizations and the histograms are the measured proportions from a set of test pseudo-wells. The prior expectation is that the average net-to-gross of this interval is about 65% but with fairly large uncertainty as expressed by the resultant net-to-gross histogram. This uncertainty is translated into variations in the low frequency component of the pseudo-wells. In this situation, a single low frequency model, recognizing no uncertainty, would clearly bias the result.

The estimated most-likely lithofacies and mean net-to-gross are displayed in Figure 4. These show good agreement with the well lithofacies columns. Individual lithofacies probabilities and property variances and high and low percentile volumes can also be generated.


Figure 3: Prior distributions of lithofacies proportions for macrolayer 3 and, bottom, implied prior net-to-gross expectation.

Figure 4: Most likely lithofacies (top) and mean estimated net-togross (bottom) and interesecting well lithofacies columns. (cs; cemented sand, sh-s; shaley-sand, sh; shale, ss;clean sand.)


CONCLUSIONSThe method of inversion by matching pseudo-wells allows a wide range of prior data to be utilized and its uncertainty to be specified. This is important to ensure the correct balance of contribution between the prior and the seismic data in the final results. The method is stable and practical with manageable run-times.

ACKNOWLEDGEMENTSWe wish to thank BP for providing the test dataset.

EDITED REFERENCESNote: This reference list is a copyedited version of the reference list submitted by the author. Reference lists for the 2017 SEG Technical Program Expanded Abstracts have been copyedited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web.

REFERENCESBosch, M., C. Campos, and E. Fernández, 2009, Seismic inversion using a geostatistical, petrophysical, and acoustic model: The Leading Edge, 28, no. 6, 690–696, http://dx.doi.org/10.1190/1.3148410. Buland, A., O. Kolbjørnsen, R. Hauge, Ø. Skjæveland, and K. Duffaut, 2008, Bayesian lithology and fluid prediction from seismic prestack data: Geophysics, 73, no. 3, C13–C21, http://dx.doi.org/10.1190/1.2842150.

Buland, A., and H. Omre, 2003, Bayesian linearized AVO inversion: Geophysics, 68, 185–198, http://dx.doi.org/10.1190/1.1543206.

Connolly, P. A., and M. J. Hughes, 2016, Stochastic inversion by matching to large numbers of pseudowells: Geophysics, 81, no. 2, M7-M22, http://dx.doi.org/10.1190/geo2015-0348.1.

Grana, D., and E. Della Rossa, 2010, Probabilistic petrophysical-properties estimation integrating statistical rock physics with seismic inversion: Geophysics, 75, no. 3, O21–O37, http://dx.doi.org/10.1190/1.3386676.

Grant, S. R., 2013, The impact of low frequency models on reservoir property predictions: 75th Annual International Conference and Exhibition, EAGE, Extended Abstracts, http://dx.doi.org/10.3997/2214-4609.20130335. Gunning, J., and M. Glinsky, 2004, Delivery: An open-source model-based Bayesian seismic inversion program: Computers and

Geosciences, 30, 619–636,http://dx.doi.org/10.1016/j.cageo.2003.10.013.

Kemper, M., and J. Gunning, 2014, Joint Impedance and Facies Inversion– Seismic inversion redefined: First Break, 32, no. 9, 89–95.

Lancaster, S., and D. Whitcombe, 2000, Fast-track “coloured” inversion: 70th Annual International Meeting, SEG Expanded Abstracts, 1572–1575, http://dx.doi.org/10.1190/1.1815711.

Leguijt, J., 2009, Seismically constrained probabilistic reservoir modeling: The Leading Edge, 28, 1478–1484, http://dx.doi.org/10.1190/1.3272703.

Riise, O., J. Elgenes, J. M. Frey-Martinez, Ø. Kjøsnes, and A. Buland, 2012, Detailed lithology and fluid mapping of the asterix gas discovery using bayesian inversion methodology: 74th EAGE Conference & Exhibition, Extended Abstracts, http://dx.doi.org/10.3997/2214-4609.20148160.

Whitcombe, D. N., P. A. Connolly, R. L. Reagan, and T. C. Redshaw, 2002, Extended elastic impedance for fluid and lithology prediction: Geophysics, 67, 63–67, http://dx.doi.org/10.1190/1.1451337.


PROBALISTIC SEISMIC INVERSION USING … · seismic inversion method based on matching large numbers...

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