Impact of Rock Physics Depth Trends and Markov Random. Fields on Hierarchical Bayesian Lithology/Fluid Prediction.

Transcription

1 Impact of Rock Physics Depth Trends and Markov Random Fields on Hierarchical Bayesian Lithology/Fluid Prediction. Kjartan Rimstad 1 & Henning Omre 1 1 Norwegian University of Science and Technology, Trondheim, Norway. rimstad@math.ntnu.no; omre@math.ntnu.no (August 30, 2009) Running head: Hierarchical Lithology/Fluid Prediction ABSTRACT Lithology/fluid prediction is phrased in a Bayesian setting, based on prestack seismic data and well observations. The likelihood model contains a convolved linearized Zoeppritz relation and rock physics models with depth trends caused by compaction and cementation. Well observations are assumed to be exact. The likelihood model contains several global parameters, like depth trend, wavelets, and error parameters; and inference of these are an integral part of the study. The prior model is based on a profile Markov random field parameterized to capture different continuity directions for lithologies and fluids. The posterior model captures both prediction and model parameter uncertainty, and is assessed by Markov chain Monte Carlo simulation based inference. The inversion model is evaluated on both a synthetic and a real data case. It is concluded that geologically plausible lithology/fluid predictions can be made. Rock physics depth trends have impact whenever cementation is present and/or predictions at depth outside the well range are made. Inclusion of model parameter uncertainty make the prediction uncertainties more realistic. 1

2 INTRODUCTION Lithology/fluid prediction based on seismic data and a few wells constitutes an important step in reservoir evaluation. Seismic data have large uncertainty but good spatial coverage, while well observations are precise but only available along a few well traces. The model used for prediction will contain several global parameters that must be determined. Hence assessing global model parameters from well observations and associated seismic data along the well traces, and predicting the regional lithology/fluid characteristics supported by seismic data appears as a feasible approach to lithology/fluid prediction. Seismic data contain information about contrasts only; hence information about slowly varying reservoir characteristics need to be included in the model. Rock physics models with depth trends may provide this information, Avseth et al. (2003). Lithology/fluid inversion based on prestack seismic data and well observations appears as an ill-posed inversion problem due to imprecise processing and observations errors. We choose to solve this inversion problem in a Bayesian framework, see Eidsvik et al. (2004). The likelihood model contains a rock physics model with a depth trend caused by depth dependent compaction and cementation effects, see Avseth et al. (2003). The seismic response is modeled by a convolutional linearized Zoeppritz likelihood, see Buland and Omre (2003a). Well observations are considered to be exact along the well trace. The prior lithology/fluid model is defined to be a profile Markov random field, see Ulvmoen and Omre (2009), although a new parameterization is specified here. This prior model captures both vertical and lateral couplings in the lithology/fluid characteristics. The variability of global model parameters are integrated in the study by using a hierarchical Bayes method, see Gelman et al. (2004). The posterior model, constituting the Bayesian inversion solution, 2

3 is uniquely defined by the likelihood and prior models. This posterior model is not analytically tractable; hence it must be assessed by simulation based inference and a efficient Markov-chain Monte Carlo algorithm is defined for this purpose. Key references for the current work are: Avseth et al. (2003) which defines the rock physics depth trend models; Ulvmoen and Omre (2009) which defines the lithology/fluid profile Markov random field models with associated simulation algorithm; and Buland and Omre (2003b) which presents an approach to include model parameter uncertainty into seismic inversion. Several new features are presented in the current paper: a rock physics likelihood model with depth trends is defined and the model parameters are estimated from observations and seismic data along a well trace; the prior profile Markov random field model is reparameterized and an improved simulation algorithm is defined; and the uncertainties in the lithology/fluid predictions include also model parameter uncertainties. The inversion model is presented with associated simulation algorithm. The properties of the model is evaluated on a synthetic reservoir. It is concluded that rock physics depth trends have large impact on the lithology/fluid prediction whenever cementation effects occur and/or predictions at depths outside the range of well observations are made. Moreover, more plausible uncertainty statements are made when model parameter uncertainties are included. Lastly, the inversion approach is demonstrated on a real data case. MODEL Consider a cross section of a sedimentary layered reservoir in two dimensions. The reservoir consists of different lithologies, and the lithologies are saturated with fluids. The reservoir D is discretized downward in the time axis by the regular lattice L t D of size 3

4 n t and in the horizontal direction by L x D of size n x with L D = L x D Lt D. The first objective in this study is to model lithology/fluid classes in the cross section π = {π x ;x L x D } = { πtx ;x L x D,t } Lt D. There are four classes: oil-, gas-, and brine-saturated sandstone, and shale, which define the sample space π tx {SandGas,SandOil,SandBrine,Shale}. Consider seismic data in all lattice nodes L D and a vertical well in one trace w L x D. The observations are denoted o = [d,m o w,πo w ], where the vector d is the prestack seismic AVO data and the vectors [m o w,π o w] are time-to-depth converted well observations. The seismic data in d contain seismic observations for n θ reflection angles. The well observations in [m o w,π o w] consist of the seismic elastic parameters: log-p-wave, log-s-wave and log-density denoted m o w, and the lithology/fluid classes denoted πo w. Let o w = [d w,m o w,πo w ] be the observations available in the well trace w. Another objective is to estimate the porosity/cementation depth trend parameters λ, the seismic wavelets s, the rock physics covariance matrix Σ m and the seismic covariance matrix Σ d. The depth trends, wavelets, and covariance matrices are treated as global parameters that do not vary spatially, and they are denoted τ = [λ,s,σ m,σ d ]. The inversion is solved in a Bayesian framework; hence the posterior model for the lithology/fluid configuration given the available observations p(π o) are the objective of the study. By extending the posterior model by the global model parameters it can be written as p(π o) = = p(π,τ o) dτ p(π o,τ) p(τ o) dτ p(π o,τ) p(τ o w ) dτ, (1) where p( ) is a generic term for probability mass function or probability density function 4

5 (pdf). The Gaussian pdf, in particular, is denoted N(µ, Σ) where µ is expectation vector and Σ is covariance matrix. The approximation p(τ o) p(τ o w ), implies that only observations in the well trace are used to estimate the global parameters τ. This approximation makes simulation from the posterior models easier. The approximation can be justified since almost all information about the parameter τ appears in the well trace where the lithology/fluids, elastic parameters, and seismic observations are observed. Hence [π o] can approximately be simulated by a sequential algorithm which first simulates τ from p(τ o w ) and then simulates π from p(π o,τ). as The posterior model of the global parameters p(τ o w ) can by Bayes formula be written p(τ o w ) = const p(o w τ) p(τ), (2) where const is a normalizing constant, p(o w τ) is a well likelihood model and p(τ) is a prior model. The lithology/fluid posterior model p(π o, τ) can also by Bayes formula be written as p(π o,τ) = const p(o π,τ) p(π τ), (3) where const is a normalizing constant, p(o π,τ) is a likelihood model and p(π τ) is a prior model for the lithology/fluid, and both given the global parameters. Likelihood models The likelihood models contain the forward models; hence the links between the observations and the variables of interest. The dependence structure in the model is illustrated in Figure 1. This graph illustrates for example that the seismic elastic parameters m and wavelets 5

6 s are prior independent. The likelihood p(o π,τ) can, by including the seismic elastic parameters m and using the dependence structure in Figure 1, be written as p(o π,τ) = p(d,m o w,π o w π,λ,s,σ m,σ d ) = p(d,m o w,π o w,m π,λ,s,σ m,σ d ) dm = p(d m,s,σ d ) p(m o w m) p(m π,λ,σ m) p(π o w π) dm, (4) where p(d m,s,σ d ) is the seismic likelihood, p(m π,λ,σ m ) is the rock physics likelihood and p(m o w m) and p(πo w π) are the well observation likelihoods. It is assumed that each trace of the likelihood model is conditionally independent and can be considered separately, which entails: p(o π,τ) = p(d w m w,s,σ d ) p(m o w m w ) p(m w π w,λ,σ m ) p(π o w π w ) dm w p(d x m x,s,σ d ) p(m x π x,λ,σ m ) dm x, (5) x =w with x w represent all traces except the well trace. The global parameters likelihood p(o w τ) can similarly be written as p(o w τ) = p(d w m w,s,σ d ) p(m o w m w) p(m w π w,λ,σ m ) p(π o w π w) p(π w τ) dm w. π w (6) Hence the likelihood expressions needed are p(d x m x,s,σ d ), p(m x π x,λ,σ m ), p(m o w m w ) and p(π o w π w). Rock physics likelihood model Consider first the rock physics likelihood p(m x π x,λ,σ m ). In the porosity and lithology/fluid space the classes are depth dependent, and the depth trends in the model are 6

7 porosity/cementation depth trends, see Figure 2. The porosity φ(t) is the porosity trend at depth t and it is parameterized similar to Ramm and Bjørlykke (1994): φ sh (λ,t) = φ 0 sh exp { α sh (t t 0 ) }, (7) φ 0 ss exp { α ss (t t 0 ) } if t t c φ ss (λ,t) =, (8) φ ss (t c ) κ ss (t t c ) if t > t c where sh indicates shale, ss indicates sand, t 0 is the depth to the top of the reservoir in the well trace, φ 0 is the porosity at depth t0, the cementation initiates at depth t c, and α and κ ss are regression coefficients. Denote the trend parameters λ = [ φ 0 sh,φ0 ss,α sh,α ss,κ ss,t c]. The conditional expected values for the elastic parameters E[m tx π tx,λ,σ m ] = h πtx (t,λ) are calculated by using a Hashin-Shtrikman Hertz-Mindlin model for unconsolidated sand, see Avseth et al. (2005), a Hashin-Shtrikman shale model for shale, see Holt and Fjær (2003), and Dvorkin-Nur constant/contact cement model for cemented sandstone, see Dvorkin and Nur (1996). Fluid effects are calculated by Gassmann s relations, see Gassmann (1951). The rock physics forward model at each location tx is assumed to have the form [m tx π tx,λ,σ m ] = h πtx (t,λ) + e m with e m N(0,Σ m ). Moreover, conditional independence between locations is assumed; hence the seismic elastic properties m x given the global parameters λ,σ m and the lithology/fluids π x can be written as p(m x π x,λ,σ m ) = t N(h πtx (t,λ),σ m ). (9) Seismic likelihood model The seismic likelihood model is defined to have a form similar form to Buland and Omre (2003a). The seismic forward model has the form [d x m x,s,σ d ] = WADm x + e d with e d N(0,Σ d ), where W is a convolution matrix based on the wavelets s, A is a weak 7

8 contrast approximation reflection matrix (Aki and Richards, 1980) and D is a differential matrix. Hence the likelihood has Gaussian form: p(d x m x,s,σ d ) = N(WADm x,σ d ). (10) Well likelihoods model We assume exact observations in the well; hence the well likelihoods are of Dirac form: p(m o w m w) = D(m w ), (11) p(π o w π w ) = D(π w ). (12) These assumptions are justified by the errors in well observations being ignorable relative to seismic errors. They also simplify Expression 5 and 6 because some of the integral and sum expressions vanish. Remember that we still have model errors in the rock physics likelihood and seismic likelihood. Prior models The lithology/fluids π and the global parameters τ are assumed to be prior independent; hence p(π,τ) = p(π)p(τ), where p(π) is the lithology/fluid prior and p(τ) is the global model parameter prior. Lithology/fluid prior model The prior model for the lithology/fluids p(π) is defined by a profile Markov random field by p(π x π x ) for all x L x D, along the lines of Ulvmoen and Omre (2009). The following 8

9 lateral and depositional first order Markov properties are assumed: p(π x π x ) = p(π x π x 1,π x+1 ) = t p(π tx π (t 1)x,π x 1,π x+1 ) all x L x D, (13) where π x = [π 1,...,π x 1,π x+1,...,π nx ]. The parameterization of Expression 13 is described in Appendix A. The parameters in p(π x π x ) are the vertical transition matrix P, the lateral coupling parameters β l and β f. The transition matrix P controls the vertical lithology/fluid sorting, β l is related to the dependence structure in a sedimentary direction for the lithologies and β f is related to dependence in a horizontal direction for the fluids. Global parameters prior models The prior models for the global parameters τ are assumed to be independent; p(τ) = p(λ) p(s) p(σ m ) p(σ d ); hence the prior models of respectively λ, s, Σ m, and Σ d are needed. We define the prior models to be conjugate priors for s, Σ m, and Σ d, see Geman and Geman (1984), similar to the prior models in Buland and Omre (2003b). For the wavelet the prior is p(s) = N(µ s,σ s ), where µ s is the expected value for the wavelet, Σ s = Σ θ s Σ0 s is the prior covariance matrix and is the Kronecker product. The matrix Σ θ s represents the correlation between the different angles and Σ0 s represents the vertical covariance in one wavelet. The matrix Σ 0 s imposes wavelet smoothness and decays towards the ends of the wavelet. Both Σ θ s and Σ 0 s are assumed known. The prior for the error covariance matrices are inverse Wishart distributed; p(σ m ) = IW(Σ 0 m,n 1) and p(σ d ) = IW(Σ 0 d,n 2), where Σ 0 m,σ0 d,n 1, and n 2 are assumed known. 9

10 The prior of λ is defined by p(λ) = const 6 I(0 φ i (λ) 1) p(λ i ), (14) i {sh,ss} i=1 where I(A) takes the value one whenever A is true and zero otherwise, and p(λ i ) U(l λi,u λi ) is a discrete uniform distribution in the range [l λi,u λi ]. Hence the porosities are ensured to be in the range [0,1]. The stochastic relations in the well are similar to the graph in Figure 1 with the (π,m,d) replaced by (π w,m w,d w ). Posterior model All the components in the posterior model in Expression 1 are now defined, with p(π o,τ) = const δ(π o w = π w ) p(d x m x,s,σ d ) p(m x π x,λ,σ m ) dm x p(π), x =w (15) p(τ o w ) = const p(d w m o w,s,σ d ) p(m o w π o w,λ,σ m ) p(λ) p(s) p(σ m ) p(σ d ). (16) Note that the Dirac prior models in Expression 11 and 12 are simplifying Expression 5 and 6 to obtain Expression 15 and 16. The next step is to define algorithms to simulate from p(τ o w ) and p(π o,τ). Assessment of the posterior model A Markov chain Monte Carlo (McMC) algorithm is used to explore the posterior distribution, see Geman and Geman (1984). The parameters and variables are updated as specified in Algorithm 1. The global parameters τ are updated with a Gibbs sampler based on observations in the well only, either by discretized sample spaces or conjugate priors. 10

11 Algorithm 1: Initiate Initiate τ with p(τ) > 0 Initiate π with p(π) > 0 End Iterate Generate τ: For all i {1,2,...,n λ } in random order Generate λ i from p(λ i o w,τ λi ) End Generate s from p(s o w,τ s ) Generate Σ m from p(σ m o w,τ Σm ) Generate Σ d from p(σ d o w,τ Σd ) Generate π: For all x {1,2,...,w 1,w + 1,...,n x } in random order Draw π x from q(π x d x,π x,τ) { Calculate α = min 1, p(π x dx,π x,τ) q(π x d x,π x,τ) p(π x d x,π x,τ) q(π x d x,π x,τ) } Set π x = End π x π x with probability α else End 11

12 A Metropolis-Hastings algorithm is used to update π x, with an approximate Gibbs sampler as proposal distribution. The proposal distribution is the approximation from Larsen et al. (2006) with a tempering tuning parameter, and it is possible to simulate exactly from the proposal distribution q(π x d x,π x,τ) = const [ˆp (d x π x,τ)] ν p(π x π x ), by the forward-backward algorithm, where ˆp (d x π x,τ) is the approximation of p(d x π x,τ) in Larsen et al. (2006) and ν < 1 is a tempering tuning parameter. The variables [π, τ] in Algorithm 1 will have a probability distribution which converges towards the approximation of the posterior model p(π, τ o) in Expression 1 as the iterations approaches infinity. The rate of convergence is hard to evaluate in the general case, but will be evaluated for the specific cases presented later. SYNTHETIC DATA CASE The model is tested on a synthetic manually created 2D reservoir, termed the reference reservoir π R, illustrated in Figure 3. The four lithology/fluid classes are {SandGas, SandOil, SandBrine, Shale}. The vertical target zone is in the range ms on varying depth with constant thickness of 128 samples 4 ms apart. In the horizontal direction there are 101 traces. The actual model parameters are given in Appendix B. The reference reservoir has a complex lithological architecture with shale layers at varying angles and several fluid units containing separate fluid contacts. Prestack seismic data will be available at each grid node and well observations will be available along the well trace on top of the structure, see Figure 3. In Figure 4 the expected porosity and seismic elastic properties are plotted. The porosity trends in Figure 4 can be compared to Figure 2. The porosity trends look well separated 12

13 and approximately parallel. The expected seismic velocities however are closer and cross each other due to the appearance of cementation at about 2100 ms where seismic velocities increase because the cementation stiff the frame of the rock. Figure 5 contains an illustration of realizations from the distribution of v p /v s ratio versus acoustic impedance v p ρ. The lines are the trends in Figure 4 and the dots are realizations containing trends and heterogeneity. It is not easy to separate the different lithology/fluid classes in this figure. Observations in the well are considered to be exact for lithology/fluid classes, see Figure 4, and for elastic properties, see Figure 5. The synthetic case the signalto-noise ratio is approximately equal 1.5, and the prestack seismic observations are plotted in Figure 6 for the three angles (10,21,36 ). In order to evaluate the inversion technique the posterior model of the global parameters p(τ o) is displayed and discussed. The posterior model for the lithology/fluid variables p(π o) is evaluated through their marginals { p(π tx o); x L x D,t D} Lt. From the marginal posterior distribution a location-wise maximum posterior estimate can be calculated and it is used as lithology/fluid predictor: ˆπ : { } ˆπ tx = argmax p(π tx o); x L x D,t L t D π tx. (17) The marginal distributions of p(τ o) and p(π o) and the predictor ˆπ are assessed from 5000 realizations of the posterior model p(π,τ o) based on the McMC simulation algorithm with tempering parameter ν = 0.66 resulting in average acceptance probability of about 0.5. The convergence of the McMC simulation algorithm is displayed in Figure 7. The trace plots of proportions of the four lithology/fluid classes are displayed for the first 500 sweeps, for four different extreme initiations. The burn-in period is defined to be 25 and the mixing appears to be satisfactory. The convergences for the other parameters look 13

14 similar; therefore the convergence is judged to be satisfactory. Results and discussion Consider first the posterior estimates of the global parameters p(τ o w ) displayed in Figure 8 and 9. Figure 8(a) contains the posterior distributions of the porosity/cementation depth trend parameters on the diagonal displays. Because well observations are used to estimate these parameters, variances are small as expected. Note that the prior distributions appear as almost uniform on the displays. When considering the cross plots on the off-diagonal there are positive correlations between φ and α. This is expected because a decrease in φ could partly be compensated by reducing α, see Expression 7 and 8. The posterior porosity trends in Figure 8(b) are reliably estimated, and the true trends are inside the 95% confidence bounds. Note that the uncertainty increases with depth since the well observations are only available at the top of the structure. The estimates of the diagonals of the error covariance matrices are presented in Figure 9 together with the estimated wavelets. The posterior variances of the diagonal components are smaller than in the prior model and the modes are closer the true value. The estimates of the wavelets appear as reliable. The lithology/fluid results are displayed in Figure 10 which contains eight displays: the reference reservoir π R in (a), the location-wise maximum posterior prediction ˆπ in (b), the marginal posterior probability for each of the four classes in p(π o) in (c)-(f), and two error plots in (g) and (h). The two error plots are: lithology/fluid error defined as the probability for misclassification of the correct class: 1 p(π tx = πtx R o); and the lithology error defined as the corresponding misclassification of the correct lithology. Recall that the 14

15 uncertainties include both classification uncertainties and model parameter uncertainties. The prediction ˆπ appear as similar to the reference reservoir π R. The lithology geometry is very well reproduced in spite the fairly complex architecture with variably dipping shale and pockets of sand. The locally defined Markov random field prior model appears as sufficiently flexible to capture this architecture. The marginal probabilities for shale are almost binary and the lithology error plot appear as frame-like, and this support the conclusion above. Note that the frame design of the lithology error plot indicates that transitions from sand-to-shale sometimes are shifted one grid node vertically, probably due to uncertainty caused by seismic convolution. The fluid filling is complex with several fluid contacts on varying depths. In the prediction the fluids are largely reproduced, but the marginal probability plots for gas and oil show that there is considerable ambiguity with respect to hydrocarbon type. This ambiguity makes it hard to reproduce lateral continuity in the hydrocarbon classes, and an increase in the lateral fluid coupling parameters β f would cause one of the classes to dominate. Lastly, the lithology/fluid error plot demonstrates that the posterior model p(π o) appears as very representative for the reference reservoir π R, with most uncertainty related to fluid filling. It can be concluded that lithology/fluid prediction can be made very reliably if the model parameterization is known. The actual model parameters can be estimated from well observations and reliable predictions of lithology/fluid classes can be provided. Figure 11 contains four displays which demonstrate the consequences of using different model parameterizations in the prediction. Display (a) and (b) contain the reference reservoir π R and prediction based on correct model formulation ˆπ, respectively. This solution is discussed above. Display (c) contains the prediction based on an empirical rock physics model inferred from well observations, similar to the approach in Ulvmoen et al. 15

16 (2009). The lithologies are misclassified, probably due to averaging over the cemented and uncemented regions. Some hydrocarbon pockets are identified. Display (d) shows the prediction based on vertical porosity/cementation trends but without spatial coupling through a prior Markov random field. The general outline of the reservoir is identifiable although with considerable high frequency noise. The impact of observation noise can be reduced by including a Markov random field prior model with lateral coupling. In the current study the global model parameters are inferred from observations along the well trace, and the uncertainty in these estimates are an integral part of the study. In Ulvmoen and Omre (2009) the model parameters are considered known and unrealistically precise predictions are made. Figure 12 contains results from a study of the impact of model parameters uncertainty on lithology/fluid prediction. In Figure 12(a) the marginal probabilities for SandGas are displayed both for a model including model parameter uncertainty and a model where the model parameters are assigned their true values. The former appears as less binary 0 1. Figure 12(b) contains histograms of the marginal probabilities in grid nodes where the reference reservoir contains either SandGas or SandOil; hence contains hydrocarbon. Figure 12(c) contains the corresponding cross-plot of marginal probabilities for the two cases. It is observed that inferring model parameters from the wells and including the uncertainty in the lithology/fluid predictions provides prediction uncertainties which appear as more realistic. Closing remarks The method manage to classify the lithology/fluid, estimate the porosity depth trends, wavelets, and covariance matrices quite precisely. The accuracy is generally good and vari- 16

17 ances are small. From the results in Figure 10 it appears as lithologies can be identified very reliably while distinguishing gas and oil in the hydrocarbon volumes is more complicated. From the results in Figure 11 it appears as depth trends are important for reliable predictions of lithology/fluid characteristics, particularly if cementation effects are present. The effect of estimating global parameters in the model and integrate the associated uncertainty in the lithology/fluid predictions make the prediction uncertainties more realistic. REAL DATA CASE The real reservoir data are from a gas reservoir offshore Norway, also evaluated in Ulvmoen et al. (2009). The geological setting is discussed in more detail in that paper. The prestack seismic data and well observations used in the current study are presented in Figure 13 and 14. The seismic data are collected in 724 traces of length 80 samples at inter-distance 4 ms. The amplitude data are preprocessed and three angel stacks at (10,21,36 ) are used. The well is located at trace 4990, and observations of porosity, elastic seismic properties, and lithology/fluid classes are available. Based on the well observations three lithology/fluid classes are defined: (SandGas, SandBrine, Shale). The objective of the study is to predict these lithology/fluid classes based on the seismic prestack data and the well observations. In the current study the rock physics likelihood model involves sand and shale porosity trends, but no cementation. Hence the depth trends are parameterized by λ = (φ 0 sh,φ0 ss,α sh,α ss ). The wavelets are given by the company providing the data, see Appendix B. Consequently, the global model parameters to be estimated are: τ = [λ,σ m,σ d ]. Both the likelihood and the prior models are very similar to the models used in the synthetic data case, and model parameter values that are changed are listed in Appendix B. 17

18 In Ulvmoen et al. (2009) a slightly different lithology/fluid inversion model, including a source rock class, is used. The important difference between the model in the current study and Ulvmoen et al. (2009) is the rock physics model formulation, however. In the current study rock physics depth trends are assessed from the observations in the well. Moreover, the uncertainty in the rock physics model is integrated into the lithology/fluid prediction uncertainty. In Ulvmoen et al. (2009) empirical rock physics relations averaging over all depth trends are used. No uncertainties are associated with these relations. To enforce depth trends in the lithology/fluid proportions, vertically changing proportions curves are introduced in Ulvmoen et al. (2009). In particular a global brine/gas contact level is defined. The marginal distributions of p(τ o) and p(π o) and the estimate ˆπ are assessed from 5000 realizations of the posterior model p(π,τ o) based on the McMC simulation algorithm, with ν = 0.7 resulting in average acceptance probability of about 0.6. The convergence of the McMC simulation algorithm is displayed in Figure 15. The trace plots of proportions of the three lithology/fluid classes are displayed for the first 500 sweeps, for three different extreme initiations. The burn-in period is considered to be 250 and the mixing appears to be satisfactory. The convergences for the other parameters are similar; therefore the convergence is judged to be satisfactory. Results and discussion The results from the lithology/fluid prediction study are presented in Figure 16 and 17. In Figure 16 the posterior models for the porosity depth trend parameters are displayed. From the diagonal displays in Figure 16(a), one observe that the posterior probability distributions are fairly compact relative to the uniform prior models. This indicates that the 18

19 well observations are informative for porosity trend estimation. In Figure 16(b) the resulting porosity trends with associated uncertainties are displayed. Note that the uncertainty is larger at depths without relevant information. Sand is only observed at depth 2275 ms to ms for example. Figure 17 contains: the lithology/fluid predictions in (a), and marginal posterior probabilities for the three lithology/fluid classes in (b)-(d). The lithology/fluid prediction is geologically plausible and compares very well with the results in Ulvmoen et al. (2009). Remember that the posterior probabilities capture the uncertainty in the model parameters as well, which is not the case in the corresponding results in Ulvmoen et al. (2009). Note also that the model in the current study integrate more available rock physics understanding than previous studies, and thereby justifies extrapolations into deeper sections of the reservoir. Figure 18 contains the results from cross-validation in the well trace, where the estimate ˆπ is not conditioned directly on π w. The well observations, lithology/fluid predictions and marginal posterior probabilities are presented. The match is relatively good, except for an erroneous prediction of a sand zone in the upper shale. Figure 19 contains lithology/fluid predictions based on four different models. Well observations π w are not used in these predictions. Figure 19(a) is based on the current model, Figure 19(b) is based on a model without spatial coupling in the prior lithology/fluid model but including porosity depth trends. The results are much more patchy, and the proportions of small classes like SandGas is severely reduced. Figure 19(c) is based on a spatially coupled prior model but no depth trends are included. Note that SandGas tends to appear at larger depth which is not geologically plausible. Lastly, Figure 19(d) is based on a 19

20 model without spatial coupling nor depth trends. The lithology/fluid units are very patchy with small pockets of SandGas distributed everywhere. The noise in the data causes this patchiness. Closing remarks The results from the real data from a reservoir offshore Norway demonstrate that reliable lithology/fluid predictions can be made. It is important to include both a prior lithology/fluid model with spatial coupling and rock physics depth trends to obtain reliable geologically plausible results. By integrating uncertainties in estimation of global model parameters more realistic uncertainties in the lithology/fluid predictions can be assessed. The uncertainty assessment is dependent on the model parameterization actually used. If uncertainties were assigned to fixed model parameters in the prior model, like spatial coupling parameters, increased prediction uncertainty would appear. In the likelihood model, approximate forward function may cause biased prediction, but the error-bounds are expected to be reliable since model parameters like error covariances are estimated from available seismic and well observations. If the seismic and well observations are combined during the pre-processing stage or the reservoir appears with large spatial heterogeneity, the prediction uncertainty may not be representative due over-fitting in the well. CONCLUSION A Bayesian lithology/fluid prediction approach based on prestack seismic data and well observations is presented. The model includes rock physics depth trends and a spatially coupled prior model for the lithology/fluid characteristics. The inversion approach is eval- 20

21 uated on a synthetic and a real case. The major conclusions from the study are: A McMC algorithm that converges reasonably fast can be defined for the posterior lithology/fluid model given the model parameters. The model parameters of the depth dependent rock physics model can be reliably assessed from observations and seismic data along the well trace. A reparameterization of the profile Markov random field makes it possible to capture complex structures in the lithology/fluid characteristics. The lithology/fluid predictions are improved by introducing a depth dependent rock physics model whenever cementation is present and/or predictions are made outside the depth range of the well observations. More plausible uncertainty quantifications can be made when model parameter uncertainties are included. The error-bounds are considered to be reliable for the current model parameterization since crucial model parameters are estimated from available seismic and well observations, but approximate forward functions in the likelihood model may cause somewhat biased predictions. Lithology/fluid predictions in the real data case based on a spatially coupled prior model and a depth dependent rock physics model appear as geologically plausible, but if the seismic and well observations are combined during the pre-processing stage or the reservoir appears with large spatial heterogeneity, the prediction uncertainty may not be representative. 21

22 ACKNOWLEDGMENTS The research is a part of the Uncertainty in Reservoir Evaluation (URE) activity at the Norwegian University of Science and Technology (NTNU). Discussions with P. Avseth and R. Holt were important for the study. APPENDIX A PRIOR MODEL FOR LITHOLOGY/FLUID The deposition process may be modeled by a Markov chain upwards, see Krumbein and Dacey (1969); hence the lithology/fluids in one trace π x given the lithology/fluids in all the other traces π x may be modeled as a Markov chain upwards. The reversed chain will then also be a Markov chain such that: n t p(π x π x ) = p(π tx π (t 1)x,π x ), t=1 (A-1) where p(π 1x π 0x,π x ) = p(π 1x π x ). In construction of the transition probabilities in Expression A-1 three trends are taken into account: (i) sedimentary trend that affects only lithology, (ii) horizontal trend that affects only fluid, (iii) vertical trend that affects both lithology and fluid. The three trends are illustrated in Figure A-1 and one neighborhood for each trend is assigned: h π tx is the horizontal neighborhood, s π tx is the sedimentary neighborhood and v π tx is the vertical neighborhood. The sample space Ω of π tx can be decomposed as π tx = [π f tx,πl tx] Ω f Ω l, where Ω f represents the fluid and Ω l the lithologies. Then the transition probabilities in Expression A-1 can be hierarchically defined by first considering the lithology and thereafter the fluid 22

23 filling: p(π tx π (t 1)x,π x ) = p(π f tx,πl tx π (t 1)x,π x ) (A-2) = p(π f tx πl tx,π (t 1)x,π x ) p(π l tx π (t 1)x,π x ), (A-3) where p(π l tx π (t 1)x,π x ) is the lithology part and p(π f tx πl tx,π (t 1)x,π x ) is the fluid part. The lithology part in Expression A-3 is p(π l tx π ( t)x,π x ) = const π f tx Ωf p(π f tx,πl tx π (t 1)x ) V l (π l tx, s π tx,β l ), (A-4) where const is a normalizing constant, p(π f tx,πl tx π (t 1)x) is a vertical Markov chain transition matrix used for the vertical trends and V l (π l tx, s π tx,β l ) a sedimentary lithology correction term. The fluid part in Expression A-3 is p(π f tx πl tx,π ( t)x,π x ) = const p(π f tx,πl tx π (t 1)x ) V f (π f tx, h π tx,β f ), (A-5) where const is a normalizing constant and V f (π f tx, h π tx,β f ) a horizontal fluid correction term. The correction terms are: V l (πtx, l s π tx,β l ) = exp βl V f (π f tx, h π tx,β f ) = exp βl s π tx I h π tx I ( ) l = πtx l, ( ) f = π f tx, (A-6) (A-7) where I( ) is an indicator function, and β l and β f are the lithology and fluid lateral coupling parameter respectively. The transition matrix P = { } p(π f tx,πl tx π (t 1)x ) defines a stationary Markov chain. When the correction terms V l and V f are included the Markov chain p(π tx π (t 1)x,π x ) becomes non-stationary. 23

24 APPENDIX B MODEL PARAMETER SPECIFICATION Synthetic Data Case The rock physics parameters are listed in Table 1 and are from Holt and Fjær (2003); Mavko et al. (2003); Avseth et al. (2005); Fjær et al. (2008). The critical porosities are set to φ c ss = 0.41 and φc sh = 0.6, and the constant cement volume is The values for the depth trends λ in the synthetic case are listed in Table 2. The priors are uniform: λ i U[l λi,u λi ]. (B-1) The Markov chain in the vertical downwards direction has the transition matrix: P =, (B-2) where the ordering of lithology/fluid classes is: (SandGas, SandOil, SandBrine, Shale). The transition matrix has the stationary distribution [ ], which represents the proportion of each class in the prior model. The values used for the lateral coupling parameters β f and β l, defined in Appendix A, are β f = 1.5,β l = 1.5. This means that if both the neighbors in the sedimentary direction are shale, then shale receive a multiplicative weight of 20 relative to sand. The covariance matrix for the seismic properties is parameterized Σ m = Σ 0 m I n t, 24

25 where I nt is a n t n t identity matrix and Σ 0 m = , (B-3) and the associated prior is p([σ 0 m] ii ) = IG(2, ), i {1,2,3}, (B-4) which is a special case of the inverse Wishart distribution. The expected value of [Σ 0 m] ii is and the variance is infinite and undefined. The covariance matrix of seismic observations Σ d is parameterized Σ d = Σ 0 d Υ d, where Σ 0 d is a 3 3 matrix. The correlation matrix Υ d is a n t n t matrix: Υ d = I n t W 1W 1, (B-5) where I nt is a n t n t identity matrix and W 1 is a normalized convolution matrix based on the wavelets. The first term in Υ d is assumed to be measurement error and the second term represent source-generated noise. The variance is divided between the terms such that the variance in the second term are 100 times larger than the variance in the first term. The covariance matrix Σ 0 d is set such that a wanted signal-to-noise ratio is acquired. The prior distribution of Σ 0 d is p(σ 0 d ) = IW(0.012 I 3,5). (B-6) The expectation of Σ 0 d is I 3. By using five degrees of freedom the prior will be very vague. The prior for the wavelet w is p(w) = N(0,Σ w ), where Σ w = Σ θ w Σ 0 w. The angle covariance matrix Σ θ w is constructed by a Gaussian correlation function with range 30. The 25

26 wavelet covariance matrix Σ 0 w is defined such that the expected wavelet amplitude is of the order of one and based on a Gaussian correlation function with a range that corresponds to the Ricker wavelets in w and provides that the amplitude decay towards the end of the wavelet. Real Data Case The parameters that are different from the synthetic case are listed here. The new bulk and shear moduli are 25 GPa and 7 GPa. The transition matrix used is P = , (B-7) which gives the stationary distribution [ ]. The values used for the lateral coupling parameters β f and β l in the lateral part of the Markov random field are β f = 3 and β l = 2. Wavelets are plotted in Figure B-1, and the prior distribution of Σ 0 d is: p(σ 0 d ) = IW( I 3,5), (B-8) where I 3 is a 3 3 identity matrix. 26

27 REFERENCES Aki, K., and P. G. Richards, 1980, Quantitative seismology: Theory and methods: W. H. Freeman and Co., New York. Avseth, P., H. Flesche, and A.-J. V. Wijngaarden, 2003, AVO classification of lithology and pore fluids constrained by rock physics depth trends: The Leading Edge, 22, Avseth, P., T. Mukerji, and G. Mavko, 2005, Quantitative seismic interprestation - applying rock physics tools to reduce interpretation risk: Cambridge University Press. Buland, A., and H. Omre, 2003a, Bayesian linearized AVO inversion: Geophysics, 68, , 2003b, Joint AVO inversion, wavelet estimation and noise-level estimation using a spatially coupled hierarchical Bayesian model: Geophysical Prospecting, 51, (20). Dvorkin, J., and A. Nur, 1996, Elasticity of high-porosity sandstones: Theory for two North Sea data sets: Geophysics, 61, Eidsvik, J., P. Avseth, H. Omre, T. Mukerji, and G. Mavko, 2004, Stochastic reservoir characterization using prestack seismic data: Geophysics, 69, Fjær, E., R. M. Holt, P. Horsrud, A. M. Raaen, and R. Risnes, 2008, Petroleum related rock mechanics: Amsterdam : Elsevier. Gassmann, F., 1951, Über die Elastizität poröser Medien: Vierteljschr. Naturforsch. Ges. Zürich, 96, Gelman, A., J. B. Carlin, H. S. Stern, and D. B. Rubin, 2004, Bayesian data analysis, second edition: Chapman & Hall/CRC. Geman, S., and D. Geman, 1984, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images: IEEE Transactions on Pattern Analysis and Machine Intelligence, 6,

28 Holt, R. M., and E. Fjær, 2003, Wave velocities in shales - a rock physics model: EAGE 65th Conference & Exhibition, Stavanger, Norway 2-5 June, Krumbein, W. C., and M. F. Dacey, 1969, Markov chains and embedded Markov chains in geology: Mathematical Geology, 1, Larsen, A. L., M. Ulvmoen, H. Omre, and A. Buland, 2006, Bayesian lithology/fluid prediction and simulation on the basis of a Markov-chain prior model: Geophysics, 71, R69 R78. Mavko, G., T. Mukerji, and J. Dvorkin, 2003, The rock physics handbook: Cambridge University Press. Ramm, M., and K. Bjørlykke, 1994, Porosity/depth trends in reservoir sandstones; assessing the quantitative effects of varying pore-pressure, temperature history and mineralogy, Norwegian Shelf data: Clay Minerals, 29, Ulvmoen, M., and H. Omre, 2009, Improved resolution in Bayesian lithology/fluid inversion from seismic prestack data and well observations: Part I Methodology. Submitted for publication in Geophysics. Ulvmoen, M., H. Omre, and A. Buland, 2009, Improved resolution in Bayesian lithology/fluid inversion from seismic prestack data and well observations: Part II Real case study. Submitted for publication in Geophysics. 28

29 LIST OF TABLES 1 Rock physics model parameters. 2 Porosity/cementation depth trends model parameters. The second column contains the true values, and the third and fourth contain the parameters in the prior distributions. 29

30 LIST OF FIGURES 1 Graph of stochastic model. The nodes represent stochastic variables and the arrows represent probabilistic dependencies. The arrows from π, λ, and Σ m to m represent the rock physics likelihood, the arrow from m, s, and Σ d to d the seismic likelihood, and from π to π o w and from m to mo w the well likelihoods. 2 Schematic illustration of porosity φ depth trends for sand and shale, where t 0 is a reference depth and t c is the initiation of sand cementation. 3 Reference reservoir π R. The well is marked at trace Expected porosity and seismic elastic properties. Realizations of v p,v s and ρ from the well trace in black. Gas-saturated sand (red), oil-saturated sand (green), brine-saturated sand (blue) and shale (black). 5 Rock-properties, v p /v s ratio against acoustic impedance v p ρ. The lines are the trends and the points are trends plus noise. 6 The seismic data d in synthetic case represented by angle stacks (10,21,36 ) in display (a), (b) and (c) respectively. 7 Convergence of McMC algorithm for the 500 first sweeps, starting at the four extreme configurations. Proportions classified as gas (red), oil (green), brine (blue), and shale (black). 8 Posterior model of depth trend parameters in synthetic case. (a) Diagonal: posterior distributions (line) and true values (cross). Off-diagonal: cross-plot of realizations.. (b) Posterior distributions of shale porosity trend φ sh (z) and sand porosity trend φ ss (z). Posterior mean (solid black line), 95% posterior confidence bounds (hatched black line) and true trends (hatched gray line). 9 Posterior model of covariance and wavelet in synthetic case. (a) Posterior distribu- 30

31 tions of the diagonal components in Σ 0 m and Σ 0 d (black line), prior distribution (gray line), and true values (cross). (b) Posterior models of the wavelets. Posterior means (solid black line), 95% confidence bounds (hatched black line) and true wavelets (hatched gray line) for angle (10,21,36 ) (left to right). 10 Posterior model for lithology/fluid variables in synthetic case. (a) Reference reservoir π R. (b) Maximum posterior estimate ˆπ. (c)-(f) Probability plots for the classes: SandGas, SandOil, SandBrine, Shale. (g) Lithology error: 1 p(π tx = the correct lithology class o). (h) Lithology/fluid error: 1 p(π tx = the correct class o). 11 Lithology/fluid prediction for different models in synthetic case. (a) Reference reservoir. (b) Full model. (c) Model without depth trends. (d) Model without Markov random field. 12 Posterior model for SandGas in hierarchical model (τ estimated) and model with given model parameter values (τ fixed). (a) Marginal posterior probabilities. (b) Histogram of marginal posterior probabilities in hydrocarbon cells. (c) Cross-plot of marginal posterior probabilities in hydrocarbon cells. 13 The seismic data d in real case represented by angle stacks (10,21,36 ) in display (a), (b) and (c) respectively. 14 Well observations. Posterior (far left), elastic material properties (v p, v s, ρ) (left, middle right) and lithology/fluid classes (far right). 15 Convergence of McMC-algorithm for the 500 first sweeps, starting at the three extreme configurations. Proportions classified as gas (red), brine (blue), and shale (black). 16 Posterior model of depth trend parameters in real case. (a) Diagonal: posterior distributions (line). Off-diagonal: cross-plot of realizations.. (b) Posterior distributions of shale porosity trend φ sh (z) and sand porosity trend φ ss (z). Posterior mean (solid black 31

32 line) and 95% posterior confidence bounds (hatched black line). 17 Posterior model for lithology/fluid variables in synthetic case. (a) Maximum posterior estimate ˆπ. (b)-(d) Probability plots for the classes: SandGas, SandBrine, Shale. 18 Cross-validation of well observations. Lithology/fluid in reference reservoir (far left), lithology/fluid prediction (left), marginal posterior probabilities SandGas (middle), SandBrine (right), and Shale (far left). 19 Lithology/fluid prediction for different models in real case. (a) Full model. (b) Model without Markov field. (c) Model without depth trends. (d) Model without depth trends and Markov random field. All the models are with no well data π w. A-1 System of axis in prior Markov random field model. Fluid neighborhood (left) and lithology neighborhood (right). Horizontal neighborhood h π tx, sedimentary neighborhood s π tx and vertical neighborhood v π tx. B-1 Wavelets, for angle (10,21,36 ) (left to right). 32

33 π λ Σ m τ π o w m s Σ d m o w o d Figure 1: Graph of stochastic model. The nodes represent stochastic variables and the arrows represent probabilistic dependencies. The arrows from π, λ, and Σ m to m represent the rock physics likelihood, the arrow from m, s, and Σ d to d the seismic likelihood, and from π to π o w and from m to m o w the well likelihoods. Rimstad & Omre 33

34 φ 0 sh φ0 ss φ t 0 Shale Sand t c t Figure 2: Schematic illustration of porosity φ depth trends for sand and shale, where t 0 is a reference depth and t c is the initiation of sand cementation. Rimstad & Omre 34

35 Shale SandBrine SandOil SandGas Figure 3: Reference reservoir π R. The well is marked at trace 13. Rimstad & Omre 35

36 2000 Shale Sand φ v p (m/s) v s (m/s) ρ (kg/m 3 ) Figure 4: Expected porosity and seismic elastic properties. Realizations of v p,v s and ρ from the well trace in black. Gas-saturated sand (red), oil-saturated sand (green), brinesaturated sand (blue) and shale (black). Rimstad & Omre 36

37 2 1.8 vp/vs 1.6 SandGas SandOil SandBrine Shale v pρ (m/s kg/m 3 ) 10 6 Figure 5: Rock-properties, v p /v s ratio against acoustic impedance v p ρ. The lines are the trends and the points are trends plus noise. Rimstad & Omre 37

38 (a) (b) (c) Figure 6: The seismic data d in synthetic case represented by angle stacks (10,21,36 ) in display (a), (b) and (c) respectively. Rimstad & Omre 38

39 Proportions Sweeps Figure 7: Convergence of McMC algorithm for the 500 first sweeps, starting at the four extreme configurations. Proportions classified as gas (red), oil (green), brine (blue), and shale (black). Rimstad & Omre 39

40 φ0sh φ0ss αsh αss κss tc φ0sh φ0ss αsh κss αss tc (a) φss φsh (b) Figure 8: Posterior model of depth trend parameters in synthetic case. (a) Diagonal: posterior distributions (line) and true values (cross). Off-diagonal: cross-plot of realizations.. (b) Posterior distributions of shale porosity trend φsh (z) and sand porosity trend φss (z). Posterior mean (solid black line), 95% posterior confidence bounds (hatched black line) and true trends (hatched gray line). Rimstad & Omre 40