bayesian monte carlo method for seismic predrill prospect assessment

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GEOPHYSICS, VOL. 75, NO. 2 �MARCH-APRIL 2010�; P. O9–O19, 9 FIGS., 4 TABLES.10.1190/1.3339678

ayesian Monte Carlo method for seismic predrill prospect assessment

eidi Kjønsberg1, Ragnar Hauge1, Odd Kolbjørnsen1, and Arild Buland2

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ABSTRACT

Predrill assessment of the probability that a potential drill-ing spot holds hydrocarbons �HC� is of vital importance toany oil company. Of equally great value is the assessment ofhydrocarbon volumes and distributions. We have developeda methodology that uses seismic data to find the probabilitythat a vertical earth profile contains hydrocarbons and theprobability distribution of hydrocarbon volumes. The meth-od combines linearized amplitude variation with offset�AVO� inversion and stochastic rock models and predicts thejoint probability distribution of the combined lithology andfluid for the entire profile. We use a Bayesian approach andfind the solution of the inverse problem by Markov chainMonte Carlo simulation. The stochastic simulation benefitsfrom a new and tailored simulation algorithm. The computa-tional cost of finding the full joint probability distribution isrelatively high and implies that the method is best suited tothe investigation of a few potential drilling spots. We appliedthe method to a case with well control and to two locations ina prospect: one in the center and one at the outskirts. At thewell location, we identify the two reservoir zones and obtainvolumes that fit the log data.At the prospect, we obtain signif-icant increases in HC probability and volume in the center,whereas there are decreases at the outskirts. Despite the largenoise components in the data, the risked volumes in the centerchanged by a factor of three. We have designed an algorithmfor computing the joint distribution of lithology, fluid, andelastic parameters for a full vertical profile. As opposed towhat can be done with pointwise approaches, this allows us tocalculate success probability and HC volumes.

INTRODUCTION

There has been a wide interest in lithology and fluid predictionLFP� from seismic amplitude variation with offset �AVO� response,

Manuscript received by the Editor 9 February 2009; revised manuscript rec1Norwegian Computing Center, Blindern, Oslo, Norway. E-mail: Heidi.Kj2Statoil, Stavanger, Norway. E-mail: [email protected] Society of Exploration Geophysicists.All rights reserved.

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ee e.g., Castagna �1993�, Mavko and Mukerji �1998�, and Avseth etl. �2005�. The uncertainty related to seismic LFP is generally largend several papers discuss the uncertainty aspect and describe meth-ds formulated in a statistical framework, see e.g., Mallick �1995�,idsvik et al. �2002�, Gunning and Glinsky �2004�, Gallop �2006�,arsen et al. �2006�, Malinverno and Parker �2006�, and Buland et al.

2008�. Houck �2002� shows how uncertainty in AVO interpretationas two layers of uncertainty because the link from the seismic datao the lithology and fluid goes via elastic parameters. The first uncer-ainty level is in the relation between seismic data and elastic param-ters, in which different elastic parameters can give very similareismic responses. The second is in the relation between elastic pa-ameters and lithology fluid classes. Different lithology and fluidombinations can have the same values for the elastic parameters.

Several papers relate seismic analysis to rock physical parametersuch as porosity, saturation, and clay content, see e.g., Doyen1988�, Mallick et al. �2002�, Bosch �2004�, Bachrach �2006�,oures and Moraes �2006�, and Spikes et al. �2007�. Of special com-ercial interest is the estimation of hydrocarbon volume or pay vol-

me, see e.g., Gastaldi et al. �1998�, Neff �1990�, Neff �1993�, Con-olly �2007�, Connolly and Kemper �2007�, and Sengupta andachrach �2007�.Most existing methods for LFP are pointwise methods. They pre-

ict the presence of hydrocarbon at one depth point at a time. Point-ise statistical methods make it possible to predict the probability ofydrocarbon presence independently at every point but cannot pre-ict the probability of whether a hydrocarbon reservoir is present be-ause this requires the joint probability of hydrocarbon presence atvery point. If hydrocarbon presence is highly correlated in depth,he probability of finding hydrocarbons is reduced but the expectedolume given hydrocarbon presence is increased. Lower correla-ions give a higher probability of finding hydrocarbons but less ex-ected column thickness. To handle this consistently, we need theoint posterior distribution. This is a demanding problem. The mainhallenge is that the number of possible lithology and fluid combina-ions is extremely large; therefore, searching through all combina-ions is not feasible. It is necessary to use smart algorithms to reduce

September 2009; published online 1April [email protected]; [email protected]; [email protected].

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he computational complexity. This paper presents a solution to thisroblem.

We use a Bayesian approach in which we define prior probabili-ies for lithology-fluid combinations in a vertical profile and a prior

odel for the rock properties. A likelihood model defines the linkrom the elastic properties of the rock to the seismic data. We workith prestack data as considered in Gouveia and Scales �1998� but

dopt the linearized model for seismic data described in Buland andmre �2003�. This model uses standard assumptions of the convolu-

ion model and weak contrast approximation. In contrast to thesewo Bayesian approaches that consider elastic parameters, our mainocus is on the joint probability distribution of lithology and fluid.

e construct an object-based lithology model and model the fluidontacts independently of this. This preserves the ordering of fluidsnd gives long-range dependencies in the resulting facies models.he elastic parameters within each lithology object have a verticalorrelation that makes the model less sensitive to the samplingnterval.

The posterior distribution represents the complete solution of theayesian inverse problem, including optimal predictions and the as-

ociated uncertainty. For many realistic inversion problems, the pos-erior distribution cannot be found analytically; therefore, we repre-ent it by a set of realizations generated from the posterior distribu-ion by stochastic simulation. We construct a Metropolis-Hastingslgorithm that draws realizations from the joint posterior probabilityistribution, see e.g., Green �1995�. The Metropolis algorithm is in-roduced in geophysical inverse problems by Mosegaard and Taran-ola �1995�.

Early approaches of stochastic simulation of lithology using seis-ic amplitude data is presented in Haas and Dubrule �1994� andorres-Verdin et al. �1999�. Larsen et al. �2006� compute the jointrobability of lithology and fluid combinations using a Markov-hain model to represent the prior model for lithology and fluid com-inations. Given the lithology and fluid combinations, they modelhe elastic properties with a general distribution depending on the li-hology and fluid class, but unlike us they assume that the elasticroperties at any two depth points are independent. To make an effi-ient algorithm, they also resort to an approximate likelihood, whichs based on pointwise approximations. The results in Larsen et al.2006� therefore represent an approximate solution and show a clearendency to underestimate the real uncertainty. The works of Contr-ras et al. �2005� and Merletti and Torres-Verdin �2006� apply Mar-ov chain Monte Carlo methods to draw from complex distributions.hese works are directed toward reservoir characterization when de-

ailed well information is available. The lithology model is less suit-d for an exploration setting because it is made to honor a predefinedlobal measure of facies proportion. In our approach, the probabilityor the presence of a reservoir is the main outcome. The approach ofosch et al. �2007, 2009� applies Markov-chain Monte Carlo meth-ds to do seismic amplitude inversion coupled with a petrophysicalnd geostatistical model. Their methods focus less on the lithology-uid model than we do and rely on conditional Gaussian assump-

ions. The benefit obtained by a simpler model is computationalpeed. Also, their methods are more targeted toward reservoir char-cterization.

In the next section, we describe the details of the prior and the like-ihood models and describe the algorithm we used to generate real-zations from the posterior distribution. A case study offshore Nor-ay is presented to illustrate the methodology.

METHODOLOGY

Our goal is to predict the properties of a vertical earth profile bytochastic inversion of seismic prestack data. We use a Bayesian ap-roach in which we explore the complete solution by drawing real-zations from the posterior distribution. The drawn realizations arehen used to make predictions about the profile such as the probabili-y for hydrocarbon presence and volume distributions.

We use the term “facies” for the combination of lithology and fluidn a cell. To deduce facies, we also need elastic parameters becausehese are the link between facies and seismic data. This means thate model the facies vector f for the trace and the vector of elastic pa-

ameters m. If the trace has nt cells, the facies vector and each of thelastic parameters is of length nt. We use three elastic parameters toescribe the seismic response of the earth, these being the P-waveelocity VP, S-wave velocity VS, and density �.

rior model

Our prior model for facies has two components: one for lithologynd one for fluid. The lithology is modeled by a marked point pro-ess in which objects of sand are placed on a background of shale sohat the sand objects erode into each other from above. Marked pointrocesses are stochastic models commonly used for representing anite number of events located in space and/or time. In our case,ach sand object has, in addition to its location in the vertical profile,parameter that describes its thickness. In statistical terms, the

hickness defines a mark �thus the name marked point process�. Weodel the location of the sand objects by a Poisson process so that

he locations of the objects are independent of one another. Thehicknesses of the sand objects are also mutually independent. Theocations of the fluid phases are modeled by fluid contacts. Here weave used two contacts separating gas, oil, and brine. The fluid con-acts can be located anywhere in the vertical profile but must be or-ered, i.e., the oil-water contact �OWC� is below the gas-oil contactGOC�. If the fluid contact is situated inside a sand object, this givesfluid transition in the object. The sand bodies are considered to beonnected away from the current vertical profile, meaning that for auid contact situated inside a shale region, sand objects above thehale and below the shale have different fluid content. This allowshree fluids — gas above the first contact, oil between them, andrine below the second. We use a discrete distribution for the loca-ion of the contacts. The stochastic models for the lithology and theuid contacts define the prior facies probability distribution p�f�.Rock physics models define the link from rock properties for each

acies to effective elastic properties �Avseth et al., 2005�. Typicalock properties are mineral composition, fluid saturation, porosity,nd texture characteristics. The rock physics relations are generallyonlinear and assumed local in the sense that the rock parameters inne location only affect the elastic properties in that specific loca-ion. Generally, different rock models can be used for differentithologies and different rock parameters ri can describe the differentock models at location i. For a specific facies f i, the probability den-ity function for the elastic properties for this facies can be modeledith a stochastic rock model as

p�mi�f i�� p�mi�ri,f i�p�ri�f i�dri, �1�

here p�ri � f i� is the prior PDF for the rock parameters ri for facies f i

nd p�m �r ,f � is the stochastic rock physics model. This model can
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Seismic predrill prospect assessment O11

e estimated from well-log data or set by expert knowledge. We willater use these relations to find the porosity from the bulk density.he stochastic rock models define the prior models for the elastic pa-

ameters for each facies f i at a single location.For each facies type, the distribution for the elastic parameters VP,

S, and � contained in m follows the specified rock physics model.e expand this local model to a model defining vertically correlated

lastic parameters. The vertical dependency within a lithology isiven by a correlation structure typically defined by a stationary cor-elation function. By using multidimensional quantile transforms,e extend the single-cell rock physics model to a correlated vector

seeAppendix A for details�. This defines the conditional probabilityistribution for the elastic parameters given the complete facies pro-le p�m � f�.The important feature of the rock physics model is that the separa-

ion of elastic parameters between the different facies is representedorrectly. If this is ensured, seismic reflection coefficients can beroperly reproduced. The absolute levels of the elastic parametersnly have a secondary effect in our approach.

A realization of the combined facies and elastics model is definedy the combination of one specific facies configuration, f and onepecific profile of elastic parameters m. The joint prior probabilityistribution for the realization is

p�f,m��p�m�f�p�f� . �2�

eismic likelihood model

We use the linear seismic forward model for a time-angle gatherefined in Buland and Omre �2003�. Let the dimension nd denote theumber of seismic time samples per angle times the number of an-les. The seismic time-angle gather d is linked to the elastic parame-ers m by the expression

d�Gm�e . �3�

ere d is an nd-dimensional vector of the seismic time-angle gather,is the noise in the gather, and G is an nd �3nt-dimensional modelatrix. The model matrix is given by

G�WAD, �4�

here W is a block-diagonal matrix representing one wavelet forach angle, A is a matrix defining the weak contrast reflection coeffi-ients of Aki and Richards �1980�, and D is a discrete derivative ma-rix. For small to moderate reflection angles, which are typicallysed in prestack amplitude analysis, the three-term approximationefined by Aki and Richards �1980� is adequate and more accuratehan the commonly used two-term approximation.

The seismic likelihood model p�d �m� is completely defined byhe forward model in equation 3 and an error model p�e�. Note espe-ially that p�d �m,f��p�d �m� because the seismic forward models fully determined by the elastic model m.

enerating realizations from the posterior model

The posterior distribution is given by

p�f,m�d�� p�d�m�p�m�f�p�f�, �5�

here p�d �m� is the seismic likelihood model, p�m � f� is defined byhe stochastic rock model, and p�f� is the prior facies model. Real-

zations are drawn from the posterior distribution by using a Markovhain Monte Carlo method. We do this by constructing a Metropolis-astings algorithm. This is an iterative algorithm that in each Mar-ov chain step proposes a new facies configuration and a new set oflastic properties �m̃,f̃� and compares it to the properties obtained inhe previous step �m,f�. If the proposal is accepted, this becomes theealization of the current step; otherwise, the realization of the cur-ent step is set equal to the realization of the previous step. The deci-ion to accept or reject is random with a probability of acceptance

Paccept�min�1,p�f̃,m̃�d�q�m,f�m̃,f̃�

p�f,m�d�q�m̃,f̃�m,f�� . �6�

In this equation, the quantities p�f,m �d� and p� f̃,m̃ �d� are theosterior distribution from equation 5 evaluated at the previous real-zation and the current proposal, respectively. The term q�m̃,f̃ �m,f�s the probability rule that modifies the previous realization to makenew proposal. The term q�m,f �m̃,f̃� is the reverse proposal distri-ution, i.e., the probability of proposing the previous realization byodifying the proposed realization.By construction, the acceptance probability of equation 6 ensures

hat the limiting probability distribution for the realizations of thearkov chain is the posterior distribution of equation 5 �see e.g.,reen, 1995�. This holds for any choice of probability rule for gener-

ting the proposal �m̃,f̃� from �m,f�, also if the rule depends on theata. The important thing is to ensure that the forward probability�m̃,f̃ �m,f� and the reverse probability q�m,f �m̃,f̃� can be comput-d because this allows the calculation of the acceptance probability.he forward probability can be computed easily because it follows

he forward direction of the algorithm, i.e., the new proposal is con-tructed based on the previous realization. However, the reverserobability is not directly accessible from the construction of the al-orithm and care must be taken so it can be computed.

In our method, the task of generating a new realization from theosterior distribution consists of three main steps: In the first step, aacies configuration is proposed, then elastic parameters are pro-osed given this facies configuration, and finally the proposed real-zation is accepted with a certain probability. We use several mecha-isms to draw from the facies configuration space and the proposedacies configuration is generated based on the existing configuration.he proposed elastic parameters are generated using seismic inver-ion conditioned on the proposed facies configuration and the seis-ic data. The tasks included in one step of the Markov chain, assum-

ng there are L different mechanisms for generating a facies configu-ation, are

� Randomly draw a proposal facies configuration:

a� Randomly draw one of the L facies proposal mechanisms,k� �1,2, . . . ,L�,

k p�k� . �7�

b� Randomly draw a proposal facies configuration based onthe previous facies configuration using mechanism k,

f̃ qk�f̃�f� . �8�

� Randomly draw proposal elastic parameters conditioned on fa-cies configuration and seismic data,

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� Accept or reject proposed realization � f̃,m̃�:

a� Accept proposed state with probability

Paccept�min�1,p�f̃,m̃�d�qk�f�f̃�q�m�f,d�

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where p�f,m �d� and p� f̃,m̃ �d� are the posterior distribu-tion from equation 5, evaluated at the previous realizationand the current proposal, respectively.

b� If accepted, set current state to proposal state � f̃,m̃�; if re-jected, set current state to previous state �f,m�.

Application of the iterative algorithm used for drawing realiza-ions from the posterior will be referred to as a stochastic simulation.he realizations obtained during simulation are used to make predic-

ions about the vertical earth profile in question. Generation of faciesonfigurations can be done for any choice of rock physics prior mod-l as long as it provides VP, VS, and �. Predictions on volumes requirehat porosity and saturation can also be extracted from the realiza-ions or are otherwise known.

Further details of the algorithm used to generate facies configura-ions, with eight facies generating mechanisms L�8, are given be-ow. Thereafter, the seismic inversion used to draw elastic parame-ers is explained.

enerating facies configurations

Generation of the facies configuration is the first job to be carriedut when a new state in the Markov chain is to be established: The fa-ies configuration f exists and the proposal f̃ is to be generated. Weo this by randomly picking and carrying out the tasks of one out ofight proposal functions. The eight functions are

1� keep the previous facies configuration2� draw new fluid contacts3� add a new sand object4� remove one existing sand object5� move and reshape one of the sand objects6� reshape one of the sand objects7� propose a completely new facies configuration8� create a new facies configuration based on a reinterpretation

of main reflection coefficients

Each of the functions 1 through 6 represents relatively small per-urbations of the old facies configuration, thus allowing the algo-ithm to explore the local neighborhood around the old state. This ismportant for the convergence of the distribution of realizations.owever, correct drawing from the posterior does require that thehole facies configuration space is explored properly; to achieve

his, one should allow also major changes to the facies configuration,ot just minor perturbations. Proposal function 7 does this by pro-iding the opportunity to simultaneously alter the facies of every celln the vertical profile. However, because no attention is paid to theeismic data, the acceptance probability for this proposal is oftenmall. Proposals 1 through 7 are standard within a point processramework, see e.g., Geyer and Møller �1994� and Green �1995�.

Proposal 8 consists of a reinterpretation of main reflection coeffi-ients. The main underlying idea is that seismic reflection ampli-udes depend mainly on relative changes in elastic parameters. Veryifferent facies configurations might have similar reflections. By in-isting that the main reflection coefficients are kept while changinghe facies configuration, we can move far in realization space but stillave high acceptance. Appendix B provides further details of the re-nterpretation algorithm.

enerating elastic parameters

For a proposed new lithology and fluid profile f, the next chal-enge is to propose elastic parameters m that give a reasonable fit tohe real seismic data. The simple approach of drawing proposalsrom the prior model p�m � f� will typically give a poor match withhe data and a very low acceptance ratio. A much better approach iso propose an elastic model based on seismic inversion. The posteri-r distribution for a Bayesian seismic inversion with p�m � f� as prioristribution is

p�m�d,f�� p�d�m�p�m�f� . �11�

The seismic inversion based on the linear seismic forward modeln equation 3 would be easy if the noise term e and the elastic param-ters m were Gaussian. It is often reasonable to model the noise e inerms of a Gaussian distribution but the elastic parameters m typical-y follow a prior distribution that is multimodal and non-Gaussian.

e approximate the elastic parameters with the Gaussian parame-ers m*. The approximation is linked to the distribution for elasticarameters given the facies configuration m � f by using the mean andovariance determined by the rock physics. In addition, we use a ver-ical correlation function. The construction thus takes into accounthe local correlations between elastic parameters and the verticalorrelation structure. For details, seeAppendix A.

Using equation 3 now with the Gaussian m* instead of m, the con-itional distribution for m* �d is Gaussian. Gaussian elastic parame-ers m* are randomly drawn from this posterior distribution. Subse-uently, we find the correct non-Gaussian parameters m by quantileapping. The details for this are also given inAppendix A. Note that

ven if the Gaussian approximation is used in the proposal distribu-ion, the algorithm actually draws from the true posterior due to thecceptance step.

CASE STUDIES

In the following, we present results for facies prediction and poreolume calculations for three vertical profiles offshore Norway.

The first vertical profile coincides with a well, which gives us thepportunity to compare simulation results directly to well logs. Weet this profile be labeled locationA. The data from this trace are alsosed for wavelet extraction. Seismic data and well-log data are pre-ented in Figure 1. The facies categories are determined from theell logs for shale content and brine saturation. The synthetic seis-ic is calculated from the well logs’ elastic parameters, using the

ame forward model as is used for inversion, and has a moderate tie.bove 2250 ms, the synthetic seismic has less energy than the seis-ic data, whereas below 2350 ms it gives a stronger response. The

eason for these amplitude errors might be factors that are not includ-d in the forward model, e.g., anisotropy, or they are caused by im-erfections in seismic processing.

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The two other locations are from a prospect in the vicinity of loca-ion A, here called locations B and C. The prospect might be a reser-oir holding mainly gas, possibly in several layers. Based on struc-ural information, location B is in the center of the prospect and loca-ion C is from the outskirts, where we believe there are no hydrocar-ons. This information was not used in our prior model because wenvestigate the impact of the seismic amplitudes. Figure 2 showseismic data for locations B and C. Seismic data for three offset an-les are displayed. We have carried out simulations for the two tracesingled out in the figure. The investigated time interval was choseno start at 2148 ms and end at 2544 ms.

For all three traces, the sampling period is 4 ms and the length ofhe investigated vertical segments is 0.4 s. Seismic processing wasarried out using standard procedures to preserve all amplitudes asell as possible. Upscaling for the well logs of location A was alsoerformed using standard methods and commercial software.

rior model

The prior model for facies probabilities is a combination of a Pois-on process for the sand objects and a simple discrete probability dis-ribution for the fluid contacts. The latter distribution was designedo give a higher probability for gas in the upper part of the trace withrine dominating toward the trace’s bottom. This corresponds to theatural layering of fluids, given there is only one fluid sequence. Theacies prior model is illustrated in Figure 3, left figure pane. Thehale probability of 0.84 is uniform in depth. The cellwise sand prob-bility of 0.16 is divided nonuniformly among the three fluids, withas dominating in the top of the trace and brine in the bottom of therace. The same facies prior model was used for all three locations.

The rock physics model is a parametric stochastic model with pa-ameters tuned to a nearby well. It consists of two lithologies —

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ters. Location A. Color code for facies categories: shale — black,rine sand — blue, hydrocarbon sand — magenta.

hale and sand — and three fluids — brine, oil, and gas. The modelombines parametric distributions specific to the lithologies withxed fluid parameters thereby providing probability distributions ofP, VS, and � for shale, brine sand, oil sand, and gas sand. Our modelarameters for the fluids are listed in Table 1.

The hydrocarbon pore volume V for each realization of the Mar-ov chain can be calculated according to the formula

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igure 2. Seismic data from the prospect that contains locations Bnd C. Vertical lines indicate the two locations, with location B onhe left.

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igure 3. Left: prior facies probabilities. Right: rock physics model.

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s the P-wave velocity, and �t /2�2 ms is the one-way traveltime.he resulting volumes are in units of meters because only a verticalrofile is considered. The porosity is calculated from the bulk densi-y, defined as a volume average

� ��0�1�� fl�, �13�

here �0 is the density of the solid rock and � fl is the fluid density.ere we have used 2650 kg /m3 for �0. For a given fluid, the porositycan be calculated from the bulk density of the rock. The effect of

et to gross was included by summing cells with hydrocarbon sand,hus excluding shale regions embedded between the sand bodiesequation 12�. Relative to the level of uncertainties from the inver-ion, this is considered sufficiently accurate for the current case. If aigher level of accuracy is considered necessary, one could decreasehe sampling interval or include the effect of net to gross on the sub-ampling scale in the rock physics model.

The rock physics model is illustrated in Figure 3, right figureanes. The figure shows elastic parameters drawn from the rockhysics model. There is significant overlap of the elastic parametersf the three sands. In particular, oil sand and gas sand have similarcoustic properties. Shale is clearly distinguished from the sands al-hough it has some overlap with brine sand.

Vertical correlations for the elastic parameters are modeled by anxponentially decreasing correlation function with a range of80 ms. The range is selected to be larger than the average sand bodyo create a stochastic level in each sand body.

esults

Figure 4 displays results for location A, i.e., the well location.eismic data are displayed to the left, followed by the computed re-ults for cellwise facies probability. The third and fourth figure panesisplay well-log data for brine saturation and shale content, respec-ively. The well-log data in Figure 4 show the two reservoir intervalslled with hydrocarbons located at roughly �2270,2320� ms and2340,2390� ms. Even though the well tie is moderate �Figure 1�,he seismic data has a clear impact on the predicted facies probabili-ies.

The predicted facies probabilities have a good match with the wellogs although a significant hydrocarbon probability above 2270 mss predicted. This region corresponds to the region in which the seis-

ic data has more energy than the synthetic thus it is not unnaturalhat this region has a result of an increased probability of discovery.

Figure 5 compares the contrast of elastic parameters as obtainedrom simulations to well-log contrasts. Comparison is done for

able 1. Fluid input parameters for the rock physics model.

Fluiddensity�kg /m3�

Fluid bulkmodulus

��kg m /s2� /m2� Fluid saturation

rine 1010 2.99 109 1.0

il 630 4.47 108 0.8

as 254 8.10 107 0.8

coustic impedance �AI� and VP /VS ratio. The figure illustrates thencertainty of the parameters and we see that the well-log data fallithin this uncertainty.Figure 6 provides a comparison between predicted sand porosity

nd well-log porosity. For each vertical position, we display the his-ogram of the sand porosity obtained from simulation. A dark colorndicates a high probability of sand porosity. This figure illustrateshe vertical positioning of sand and porosity values. The solid curveegments in the figure show the well-log porosity for sand and are in-luded for comparison. Figure 6 also shows the facies categories ob-ained from the well log. The simulation results show that there is

ostly shale in the top part of the trace and then an increasingmount of sand at the reservoir levels.

Figure 7 displays the results from locations B and C indicated inigure 2. Comparing the results at location B with the prior model inigure 3, the results show significant differences between prior andosterior facies probabilities. The combined hydrocarbon probabili-y is greatly enhanced relative to the prior model in two separate lay-rs. At location C, the hydrocarbon probability is quite low through-ut the trace with some minor increase of brine sand probability athe reservoir level.

The overall probabilities for finding oil and/or gas in all three trac-s, along with the analogous prior probabilities, are listed in Table 2.or location A, we find a significant increase in hydrocarbon proba-ility relative to the prior, with gas and oil having almost identicalrobabilities. In location B, the evidence for hydrocarbon presences very high. In location C, the probability for finding hydrocarbonss reduced relative to the prior.

Table 3 lists pore volumes predicted for the three hydrocarboncenarios of our model; the scenarios refer to the location containingnly oil, only gas, or both. We list the posterior expectation value andtandard deviation, the latter in parentheses. LocationsAand B haveore volumes that are significantly increased relative to the prior. ForocationA, the hydrocarbon pore volume extracted directly from theell-log data is 28.7 m. The realizations drawn from the posterior

0 20 40

2100

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Seismic

Tw

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aytr

avel

time

(ms)

Angle (°)0 0.5 1

Facies

Probability0 0.5 1

Brine

Saturation0 0.5 1

Shale

Vsh

igure 4. First pane: seismic data; second pane: facies probabilitiesrom simulation; third and fourth pane: well-log data for brine satu-ation and shale content, respectively. LocationA.

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re in good agreement with this value. For location C, pore volumesor all hydrocarbon scenarios are reduced relative to the prior.

Table 4 lists the corresponding risked volumes for the three cases,.e., the volumes in Table 3 multiplied with the probabilities of Table.

Figures 8 and 9 show hydrocarbon pore-volume distributions forocation B, conditioned on two hydrocarbon scenarios. Figure 8hows the pore-volume distribution, given that only gas is present inhe reservoir. The distribution is quite symmetric around the meanalue. Figure 9 is a scatterplot of oil and gas pore volume, given thatoth fluids are present. There is a clear correlation between the flu-ds: less gas means more oil and vice versa. The analogous distribu-ion for the case of oil but no gas is not displayed because the proba-ility for this scenario is quite low �6%�.

AI contrast

Tw

o−w

aytr

avel

time

(ms)

−0.4 0 0.4

2100

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2400 0.02

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2.00

VP

/ VS

contrast

−0.4 0 0.4

0.02

0.05

0.10

0.20

0.50

1.00

2.00

igure 5. Comparison of elastic parameters from simulation �histo-ram displayed with grayscale color code� to elastic parametersrom well log �red curve�. Left pane:AI. Right pane: ratio of VP to VS.ocationA.

Sand porosity

0 0.1 0.2 0.3

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2.00

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igure 6. Left: facies categories obtained from the well logs. Right:redicted sand porosity �histogram displayed with grayscale colorode� and well-log sand porosity �red curve�. LocationA.

Similar results for the hydrocarbon pore volume distributionsonditioned on hydrocarbon scenarios are also found for location A.hat is, we find bell-shaped histograms for hydrocarbon pore vol-mes, conditioned that only one type of hydrocarbon is present andlearly correlated pore volumes if both fluids are present.

For each of the locationsA, B, and C, the results presented here arebtained from realizations generated by running several indepen-

0 20 40

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Seismic B

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Probability0 20 40

Seismic C

Angle (°)0 0.5 1

Posterior C

Probability

igure 7. Seismic data and posterior facies probabilities for locationsand C.

able 2. Probabilities for finding oil and/or gas.

Wet Oil Gas Gas and Oil

rior 0.47 0.18 0.20 0.15

ocation A 0.24 0.17 0.17 0.41

ocation B 0.03 0.06 0.45 0.46

ocation C 0.56 0.18 0.13 0.13

able 3. Predicted pore volumes (m) for differentydrocarbon scenarios, standard deviation in parenthesis.

Oil Gas Gas and Oil

rior 10.1 �7.7� 14.3 �9.1� 21.8 �10.1�

ocation A 20.9 �11.0� 22.0 �11.1� 29.2 �11.0�

ocation B 21.6 �9.3� 24.3 �7.9� 28.1 �8.4�

ocation C 9.7 �7.0� 11.8 �9.2� 18.5 �8.3�

able 4. Risked hydrocarbon pore volumes (m).

Prior 7.9

Location A 19.4

Location B 25.2

Location C 5.7

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ent simulations. For each simulation, we used an initial burn-inhase of 15,000 iterations. After that, we kept each 5000th iteration,conservatively chosen sampling interval to ensure independencef the realizations. The results for location A are based on a total of40 independent realizations, 840 for location B, and 1100 for loca-ion C. Standard methods were used to assess convergence. For allhree locations, the number of iterations needed to reach conver-ence was approximately the same and not very sensitive to initialalues or seismic data.

DISCUSSION

The trace from location A coincides with a well, making the as-essment of the simulation results readily available. As judged fromigure 4, the hydrocarbon layers that are seen in the well-log data are

0 10 20 30 40 50 600.00

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Gas pore volume (m)

Pro

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lity

igure 8. Hydrocarbon pore volume distribution in location B whennly gas is present.

0 10 20 30 40 500

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45

Oil pore volume (m)

Gas

pore

volu

me

(m)

igure 9. Hydrocarbon pore-volume distribution in location B whenas and oil are present.

etected easily by our method. The method also suggests an addi-ional false hydrocarbon interval as a plausible explanation for themplitude anomalies just above the two hydrocarbon intervals thatre present. Note, however, that the three intervals are predicted withncertainty, i.e., they need not be present. Such uncertainty is alsoeen in the contrasts of the elastic parameters in Figure 5. Our meth-d identifies all major reflections present in the data and proposes ad-itional positions in which there could be large reflections. Due tooise in the data, we are not able to precisely determine the presencer position of the contrasts but only the possibility of their presence.he porosity in the sand region, as shown in Figure 6, is in good cor-

espondence with the well log. Again, the uncertainty is reflected inur approach. There is a possibility that the shale region at about325 ms is a sand layer but if this is the case it has lower quality thanhe sand layers directly above and below.

Recall from Figure 7 that the simulation results for location Bhowed indications of a gas reservoir. This prospect has been recent-y drilled and gas was indeed found in two separate layers.

The method in this paper is based on seismic prestack amplitudes,mplying that the effect of all the different angle stacks is included inhe inversion. Seismic data should be properly processed so as toreserve the amplitudes of each angle as well as possible. The work-ow can also be applied with low-quality data but the conclusionsill be less clear. If, for instance, the inverted SI is less determined

han the AI, this comes out as a reduced ability to discriminate be-ween facies that differ in SI and not in AI. The rock physics modeletermines which facies this is.

The method requires a wavelet and a rock physics model to be pre-efined. For proper estimation of parameters, application of theethod is thus limited to prospects in the same region as existingells.If the wavelet amplitude fails to be properly estimated and is too

arge, the algorithm will interpret reflections as being weaker thanhey should have been. Then reflections that should have been con-idered as changes of facies can be misinterpreted as intrafacies re-ections. If the wavelet amplitude is too small, the simulation resultsill typically give facies changes that are too frequent. It is oftenossible to detect such biases by studying single realizations fromhe Markov chain. If the frequency of facies changes in each realiza-ion tends to be dramatically different from what is expected fromhe prior facies probability p�f�, this can indicate that the wavelet es-imation should be reconsidered.

If rock physics parameters are set so that rock physics probabilityistributions are too narrow, the simulation results will tend to pro-ide very rigid and possibly wrong conclusions. The latter is in ourpinion the most dangerous pitfall of our method. Simulation resultsiving probabilities that tend to be either close to zero or close to oneight indicate that rock physics distributions are too narrow. If on

he other hand rock physics distributions are too wide, the simulationesults will be close to the prior and, hence, uninformative.

We would like to stress that estimation of input parameters shouldlways aim to reflect the best available physical understanding of therospect at hand. The tests for detecting wrongly estimated input pa-ameters as indicated in the two previous paragraphs should never besed as a substitute for such understanding.

It is a general quality of our method that elastic contrasts are ro-ustly predicted even if the absolute levels of the prior rock physicsodel should happen to be affected by errors. The important point is

hat the prior rock physics covers the relative differences in elasticarameters. In our method, robust predictions for facies and elastic

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ontrasts are the main point, not the absolute elastic levels.The computation of the full joint distribution for a vertical profile

oes have a computational cost. When run on a standard laptopwithout any specific optimization effort�, our code usually takeseveral hours to perform the simulation of one trace. This makes theool most powerful in a predrill setting when the assessment of a fewandidate well locations is of great importance.

CONCLUSION

The method presented in this paper provides the joint lithologynd fluid distribution for an entire vertical earth profile. The methodses seismic prestack data and requires a wavelet and a rock physicsodel. The tool is formulated as a Markov chain Monte Carlo meth-

d that draws realizations from the combined space of facies config-rations and elastic parameters. For a rock physics model that in-ludes porosity, the method can be used to establish the probabilityistributions for hydrocarbon volumes.

The case results presented provide good evidence that our methodives reliable predictions for lithology and fluid and for the pore-vol-me distributions in a vertical profile. This does not mean that theredictions are perfect but that the results fall within the uncertain-ies of the predictions. It is well known that uncertainties in seismicredictions are large and our method takes this into account. Whenhe well tie of seismic data is good, the simulation results coincideery well with the well logs. In a predrill situation, it is of major im-ortance to assess a few possible well locations carefully. We believehe presented method can be of great value in this situation, provid-ng a cost-efficient way to increase the probability of successfullyhoosing the drilling locations.

ACKNOWLEDGMENTS

We would like to thank Statoil for the permission to publish theata included in this paper. We also want to thank the editors andnonymous reviewers for their constructive comments and helpfuluggestions.

APPENDIX A

CORRELATED ELASTIC PARAMETERS

Our goal here is to define a vertically correlated prior model forlastic parameters, conditioned on a given facies configuration. De-ote the number of cells in the trace by nt and let f be thet-dimensional facies vector.Assume the correlation structure

Corr�mi,mj��i� j��, �A-1�

ith i,j being cell indices, is given a priori. Also assume the exis-ence of the stochastic rock physics model, which is non-Gaussian.

e let the triplet of elastic parameters in cell i be denoted by the 31 column vector mi. Let �� f i��E�mi � f i� be the joint expectation

f the elastic parameters for facies f i and �� f i��Cov�mi � f i� be theorresponding covariance matrix. Both quantities refer to the rockhysics model.

We now consider Gaussian elastic parameters m*. Define a prioraussian distribution as the 3nt-dimensional Gaussian using the first

wo moments of m:

E�mi*�f i��f i�, �A-2�

nd

ov�mi*,m

j*�f i,f j�

� ��f i��i� j�� if i and j are in the same facies interval;

0 otherwise�A-3�

xcept that we also include analogous nonzero correlations acrossertically separated shale intervals. Equation A-3 defines how weombine the time-independent correlations inherent in the rockhysics model with the vertical correlation structure ��i� j��. Bysing the method in Buland and Omre �2003�, a Gaussian posterior

p�m* �d� can be obtained. Here d denotes the seismic data.An elasticonfiguration m* is drawn from the latter distribution.

Non-Gaussian elastic parameters are in turn found through a de-erministic quantile transformation carried out separately for eachell i. Let L� f i� be the Cholesky matrix for the rock physics covari-nce matrix �� f i��L� f i�L� f i�T. For each cell i do the following:

• Find the three-dimensional vector s*�L� f i��1�mi*

��*� f i��, giving s*N3�0,1� �for simpler notation we sup-press the i label of s�.

• Find the quantiles s�FN3�0,1��s*�. Due to the nature of theCholesky decomposition, these are now the quantiles ofp�m1�, p�m2 �m1�, and p�m3 �m1,m2�, with m1, m2, and m3 be-ing the components of the elastic vector mi� �m1,m2,m3�T

�again suppressing the i label�.• Find the three components of mi sequentially by

– m1�Ffi,m1

�1 �s1�,– m2 �m1�Ffi,m2�m1

�1 �s2�– m3 �m2,m1�Ffi,m3�m2,m1

�1 �s3�,

here Ffi,m1� · � is the marginal cumulative distribution function for

1 and Ffi,m2�m1� · � and Ffi,m3�m2,m1

� · � are conditional cumulatives for2 �m1 and m3 �m2, m1, respectively.All three cumulative distribution

unctions refer to the rock physics model for facies f i.When quantile mapping has been done for all cells i, the resulting

ector

m� �m1T,m2

T, . . . ,mnt

T �T �A-4�

ontains the sought correlated elastic parameters conditioned on theacies vector f.

For the Metropolis-Hastings acceptance step, we need the proba-ility distribution p�m � f� for the elastic parameters. This is found byalculating the joint probability for the Gaussian variables m* fromhe Gaussian prior distribution because m is uniquely given by m*.

APPENDIX B

REINTERPRETING MAIN REFLECTIONS

In this appendix, we provide a detailed explanation of the reinter-retation algorithm.

This proposal function is in itself a three-step procedure. In therst step of the algorithm, we simply establish the basis for the rein-

erpretation:

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Identify main reflection points of the old realization. We use thedefinition that facies changes occur at and only at the main reflec-tion points.For each interval, an interval here defined as the cells betweentwo adjacent main reflections, calculate the mean over the inter-val for each of the three elastic parameters of the old realization.Calculate reflection coefficients for three arbitrary angles 1, 2, 3. The angles should be well separated but need not be anyof those that we have data for. The reason for using three angles isexplained shortly. For each angle , the reflection coefficients arefound from the elastic mean values of the intervals �see previousbullet point�, using the weak contrast formula by Aki and Rich-ards �1980�,

ri� ��a� ��VP

V̄P

�b� ��VS

V̄S

�c� ��

�̄. �B-1�

his provides a set of reflections �ri�� ri� 1�,ri� 2�,i� 3��T�, � being the angle vector. The reflections are nonzero at theain reflection points and zero elsewhere. The set of nonzero reflec-

ions form the basis for the reinterpretation.For three given angles, the reflection vector ri can be viewed as a

unction of elastic parameters mi in cell i and cell i�1, ri

F�mi,mi�1�, where the details of the function F� · � are given byquation B-1. This function can be solved with respect to three un-nown elastic parameters on one side of the reflection point, givenhe elastic parameters in the cell on the other side. The result is twoonlinear relations

mi�G�ri,mi�1�, mi�1�H�ri,mi� . �B-2�

e refer to either of these solutions as the inverted weak contrast for-ula. The solutions satisfy ri�F�G�ri,mi�1�,mi�1� and ri

F�mi,H�ri,mi��.In the second step of the reinterpretation algorithm the inverted

eak contrast formula is used to create temporary elastic levels:

Pick a cell i0 at random and draw a new three-dimensional elasticvector mi0

in this cell from

p�mi0��

f

p�mi0�f i0

�p�f i0�, �B-3�

where p�mi0� f i0

� is the rock physics model and p� f i0� the prior fa-

cies model. This initializes the task of finding temporary elasticlevels from the main reflections.

For all cells i� i0�1,i0�2, . . . ,1

– Calculate mi from ri�1 and mi�1 using the inverted weak con-trast formula H of equation B-2.

For all cells i� i0�1,i0�2, . . . ,nt

– Calculate mi from ri and mi�1 using the inverted weak contrastformula G of equation B-2.

This gives a new temporary set of elastic levels that accuratelyreserves the main reflection coefficients.

In the last step of the reinterpretation procedure, a new faciesonfiguration is drawn conditioned on these temporary elastic lev-ls:

• Pick a cell i0 at random and draw a new facies from the distri-bution p� f i0

�mi0�� p�mi0

� f i0�p� f i0

�.

• For all cells i� i0�1,i0�2, . . . ,1:

– If ri�1�0, draw the facies f i from p� f i �mi,f i�1, . . . ,f i0�

� p�mi � f i�p� f i � f i�1, . . . ,f i0�,

– Else, set f i� f i�1.

• For all cells i� i0�1,i0�2, . . . ,nt:

– If ri�0, draw the facies f i from p� f i �mi,f1, . . . ,f i�1�� p�mi � f i�p� f i � f1, . . . ,f i�1�,

– Else, set f i� f i�1.

Here p�mi � f i� is the rock physics model, p� f i� is the facies priorrobability distribution, and p� f i � f i�1, . . . ,f i0

� and p� f i � f1, . . . ,f i�1�re facies probability distributions obtained from the prior but con-itioned on the restriction that facies changes occur at and only at theain reflection points and on the restriction that the fluids are lay-

red. The resulting facies configuration is adopted as the new pro-osal’s facies configuration.

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bayesian monte carlo method for seismic predrill prospect assessment

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