Page 1
Impact of Rock Physics Depth Trends and Markov Random
Fields on Hierarchical Bayesian Lithology/Fluid Prediction.
Kjartan Rimstad1 & Henning Omre1
1Norwegian University of Science and Technology, Trondheim, Norway.
E-mail: [email protected] ; [email protected]
(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 re-
lation 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 pos-
terior model captures both prediction and model parameter uncertainty, and is assessed
by Markov chain Monte Carlo simulation based inference. The inversion model is evalu-
ated 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 when-
ever 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
Page 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 re-
sponse 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 lithol-
ogy/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
Page 3
is uniquely defined by the likelihood and prior models. This posterior model is not ana-
lytically 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 oc-
cur 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 LtD of size
3
Page 4
nt and in the horizontal direction by LxD of size nx with LD = Lx
D ×LtD. The first objective
in this study is to model lithology/fluid classes in the cross section π = πx;x ∈ LxD =
πtx;x ∈ LxD, t ∈ Lt
D
. There are four classes: oil-, gas-, and brine-saturated sandstone,
and shale, which define the sample space πtx ∈ SandGas,SandOil,SandBrine,Shale.
Consider seismic data in all lattice nodes LD and a vertical well in one trace w ∈ LxD. The
observations are denoted o = [d,mow,πo
w], where the vector d is the prestack seismic AVO
data and the vectors [mow,πo
w] are time-to-depth converted well observations. The seismic
data in d contain seismic observations for nθ reflection angles. The well observations in
[mow,πo
w] consist of the seismic elastic parameters: log-p-wave, log-s-wave and log-density
denoted mow, and the lithology/fluid classes denoted πo
w. Let ow = [dw,mow,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 Σm and the seismic covariance
matrix Σd. The depth trends, wavelets, and covariance matrices are treated as global
parameters that do not vary spatially, and they are denoted τ = [λ, s,Σm,Σd].
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 | ow) dτ , (1)
where p(·) is a generic term for probability mass function or probability density function
4
Page 5
(pdf). The Gaussian pdf, in particular, is denoted N(µ,Σ) where µ is expectation vec-
tor and Σ is covariance matrix. The approximation p(τ | o) ≈ p(τ | ow), 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 | ow) and then simulates π from p(π | o, τ ).
The posterior model of the global parameters p(τ | ow) can by Bayes formula be written
as
p(τ | ow) = const × p(ow | τ ) p(τ ), (2)
where const is a normalizing constant, p(ow | 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
Page 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,mow,πo
w | π,λ, s,Σm,Σd)
=
∫
p(d,mow,πo
w,m | π,λ, s,Σm,Σd) dm
=
∫
p(d | m, s,Σd) p(mow | m) p(m | π,λ,Σm) p(πo
w | π) dm, (4)
where p(d | m, s,Σd) is the seismic likelihood, p(m | π,λ,Σm) is the rock physics likelihood
and p(mow | 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(dw | mw, s,Σd) p(mow | mw) p(mw | πw,λ,Σm) p(πo
w | πw) dmw
×∏
x 6=w
∫
p(dx | mx, s,Σd) p(mx | πx,λ,Σm) dmx, (5)
with x 6= w represent all traces except the well trace. The global parameters likelihood
p(ow | τ ) can similarly be written as
p(ow | τ ) =
∫
∑
πw
p(dw | mw, s,Σd) p(mow | mw) p(mw | πw,λ,Σm) p(πo
w | πw) p(πw | τ ) dmw.
(6)
Hence the likelihood expressions needed are p(dx | mx, s,Σd), p(mx | πx,λ,Σm), p(mow |
mw) and p(πow | πw).
Rock physics likelihood model
Consider first the rock physics likelihood p(mx | πx,λ,Σm). In the porosity and lithol-
ogy/fluid space the classes are depth dependent, and the depth trends in the model are
6
Page 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):
φsh(λ, t) = φ0sh exp
−αsh(t − t0)
, (7)
φss(λ, t) =
φ0ss exp
−αss(t − t0)
if t ≤ tc
φss(tc) − κss(t − tc) if t > tc
, (8)
where sh indicates shale, ss indicates sand, t0 is the depth to the top of the reservoir in the
well trace, φ0∗ is the porosity at depth t0, the cementation initiates at depth tc, and α∗ and
κss are regression coefficients. Denote the trend parameters λ =[
φ0sh, φ0
ss, αsh, αss, κss, tc]
.
The conditional expected values for the elastic parameters E[mtx | πtx,λ,Σm] = hπtx(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
[mtx | πtx,λ,Σm] = hπtx(t,λ) + em with em ∼ N(0,Σm). Moreover, conditional indepen-
dence between locations is assumed; hence the seismic elastic properties mx given the global
parameters λ,Σm and the lithology/fluids πx can be written as
p(mx | πx,λ,Σm) =∏
t
N(hπtx(t,λ),Σm). (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 [dx | mx, s,Σd] = WADmx + ed with
ed ∼ N(0,Σd), where W is a convolution matrix based on the wavelets s, A is a weak
7
Page 8
contrast approximation reflection matrix (Aki and Richards, 1980) and D is a differential
matrix. Hence the likelihood has Gaussian form:
p(dx | mx, s,Σd) = N(WADmx,Σd). (10)
Well likelihoods model
We assume exact observations in the well; hence the well likelihoods are of Dirac form:
p(mow|mw) = D(mw), (11)
p(πow|πw) = D(πw). (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(πx|π−x) for all x ∈ LxD, along the lines of Ulvmoen and Omre (2009). The following
8
Page 9
lateral and depositional first order Markov properties are assumed:
p(πx | π−x) = p(πx | πx−1,πx+1)
=∏
t
p(πtx | π(t−1)x,πx−1,πx+1) all x ∈ LxD, (13)
where π−x = [π1, . . . ,πx−1,πx+1, . . . ,πnx ]. The parameterization of Expression 13 is de-
scribed in Appendix A. The parameters in p(πx | π−x) are the vertical transition matrix
P, the lateral coupling parameters βl and βf . The transition matrix P controls the vertical
lithology/fluid sorting, βl is related to the dependence structure in a sedimentary direction
for the lithologies and βf 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(Σm) p(Σd); hence the prior models of respectively λ, s, Σm, and Σd are
needed. We define the prior models to be conjugate priors for s, Σm, and Σd, see Geman
and Geman (1984), similar to the prior models in Buland and Omre (2003b). For the
wavelet the prior is p(s) = N(µs,Σs), where µs is the expected value for the wavelet,
Σs = Σθs ⊗ Σ0
s is the prior covariance matrix and ⊗ is the Kronecker product. The matrix
Σθs represents the correlation between the different angles and Σ0
s represents the vertical
covariance in one wavelet. The matrix Σ0s imposes wavelet smoothness and decays towards
the ends of the wavelet. Both Σθs and Σ0
s are assumed known.
The prior for the error covariance matrices are inverse Wishart distributed; p(Σm) =
IW (Σ0m, n1) and p(Σd) = IW (Σ0
d, n2), where Σ0m,Σ0
d, n1, and n2 are assumed known.
9
Page 10
The prior of λ is defined by
p(λ) = const ×∏
i∈sh,ss
I(0 ≤ φi(λ) ≤ 1) ×
6∏
i=1
p(λi), (14)
where I(A) takes the value one whenever A is true and zero otherwise, and p(λi) ∼
U(lλi, uλi
) is a discrete uniform distribution in the range [lλi, uλi
]. 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 (πw,mw,dw).
Posterior model
All the components in the posterior model in Expression 1 are now defined, with
p(π | o, τ ) = const × δ(πow = πw)
∏
x 6=w
∫
p(dx | mx, s,Σd) p(mx | πx,λ,Σm) dmx p(π),
(15)
p(τ | ow) = const × p(dw | mow, s,Σd) p(mo
w | πow,λ,Σm) p(λ) p(s) p(Σm) p(Σd). (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 | ow) and p(π | o, τ ).
Assessment of the posterior model
A Markov chain Monte Carlo (McMC) algorithm is used to explore the posterior distribu-
tion, 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
Page 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, . . . , nλ in random order
Generate λi from p(λi | ow, τ−λi)
End
Generate s from p(s | ow, τ−s)
Generate Σm from p(Σm | ow, τ−Σm)
Generate Σd from p(Σd | ow, τ−Σd)
Generate π:
For all x ∈ 1, 2, . . . , w − 1, w + 1, . . . , nx in random order
Draw π′x from q(πx | dx,π−x, τ )
Calculate α = min
1, p(π′
x|dx,π−x,τ) q(πx|dx,π−x,τ)p(πx|dx,π−x,τ) q(π′
x|dx,π−x,τ)
Set πx =
π′x with probability α
πx else
End
End
11
Page 12
A Metropolis-Hastings algorithm is used to update πx, 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(πx | dx,π−x, τ ) = const×[p∗(dx | πx, τ )]ν p(πx | π−x), by
the forward-backward algorithm, where p∗(dx | πx, τ ) is the approximation of p(dx | πx, 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 itera-
tions 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 πR, illustrated in Figure 3. The four lithology/fluid classes are SandGas, SandOil,
SandBrine, Shale. The vertical target zone is in the range 1800−2800 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
Page 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 vp/vs ratio versus
acoustic impedance vpρ. 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 signal-
to-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 parame-
ters p(τ | o) is displayed and discussed. The posterior model for the lithology/fluid vari-
ables p(π | o) is evaluated through their marginals
p(πtx | o); x ∈ LxD, t ∈ Lt
D
. From the
marginal posterior distribution a location-wise maximum posterior estimate can be calcu-
lated and it is used as lithology/fluid predictor:
π :
πtx = argmaxπtx
p(πtx | o); x ∈ LxD, t ∈ Lt
D
. (17)
The marginal distributions of p(τ | o) and p(π | o) and the predictor π 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
Page 14
similar; therefore the convergence is judged to be satisfactory.
Results and discussion
Consider first the posterior estimates of the global parameters p(τ | ow) 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 πR in (a), the location-wise maximum posterior prediction π 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(πtx = πRtx | o); and the lithology
error defined as the corresponding misclassification of the correct lithology. Recall that the
14
Page 15
uncertainties include both classification uncertainties and model parameter uncertainties.
The prediction π appear as similar to the reference reservoir πR. The lithology geom-
etry 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 conclu-
sion 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 uncer-
tainty 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 hy-
drocarbon classes, and an increase in the lateral fluid coupling parameters βf 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 πR, 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 pa-
rameters 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 differ-
ent model parameterizations in the prediction. Display (a) and (b) contain the reference
reservoir πR and prediction based on correct model formulation π, respectively. This so-
lution 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
Page 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 pre-
diction 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 proba-
bilities 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 hy-
drocarbon. 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
Page 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 predic-
tions 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 = (φ0sh, φ0
ss, αsh, αss).
The wavelets are given by the company providing the data, see Appendix B. Consequently,
the global model parameters to be estimated are: τ = [λ,Σm,Σd]. 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
Page 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 π 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
Page 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
2400 ms for example.
Figure 17 contains: the lithology/fluid predictions in (a), and marginal posterior prob-
abilities 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 understand-
ing 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
π is not conditioned directly on πw. 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 ob-
servations πw 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 propor-
tions 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 ap-
pear at larger depth which is not geologically plausible. Lastly, Figure 19(d) is based on a
19
Page 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 lithol-
ogy/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 cou-
pling parameters, increased prediction uncertainty would appear. In the likelihood model,
approximate forward function may cause biased prediction, but the error-bounds are ex-
pected 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
Page 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 uncer-
tainties 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
Page 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 πx 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:
p(πx | π−x) =
nt∏
t=1
p(πtx | π(t−1)x,π−x), (A-1)
where p(π1x | π0x,π−x) = p(π1x | 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: ∂hπtx is the
horizontal neighborhood, ∂sπtx is the sedimentary neighborhood and ∂vπtx is the vertical
neighborhood.
The sample space Ω of πtx can be decomposed as πtx = [πftx, πl
tx] ∈ Ωf × Ωl, where Ωf
represents the fluid and Ωl the lithologies. Then the transition probabilities in Expression
A-1 can be hierarchically defined by first considering the lithology and thereafter the fluid
22
Page 23
filling:
p(πtx | π(t−1)x,π−x) = p(πftx, πl
tx | π(t−1)x,π−x) (A-2)
= p(πftx | πl
tx,π(t−1)x,π−x) p(πltx | π(t−1)x,π−x), (A-3)
where p(πltx | π(t−1)x,π−x) is the lithology part and p(πf
tx | πltx,π(t−1)x,π−x) is the fluid
part. The lithology part in Expression A-3 is
p(πltx | π(−t)x,π−x) = const ×
∑
πftx∈Ωf
p(πftx, πl
tx | π(t−1)x) V l(πltx, ∂sπtx, βl), (A-4)
where const is a normalizing constant, p(πftx, πl
tx | π(t−1)x) is a vertical Markov chain tran-
sition matrix used for the vertical trends and V l(πltx, ∂sπtx, βl) a sedimentary lithology
correction term. The fluid part in Expression A-3 is
p(πftx | πl
tx,π(−t)x,π−x) = const × p(πftx, πl
tx | π(t−1)x) V f (πftx, ∂hπtx, βf ), (A-5)
where const is a normalizing constant and V f (πftx, ∂hπtx, βf ) a horizontal fluid correction
term. The correction terms are:
V l(πltx, ∂sπtx, βl) = exp
βl∑
∈∂sπtx
I(
l = πltx
)
, (A-6)
V f (πftx, ∂hπtx, βf ) = exp
βl∑
∈∂hπtx
I(
f = πftx
)
, (A-7)
where I(·) is an indicator function, and βl and βf are the lithology and fluid lateral coupling
parameter respectively.
The transition matrix P =
p(πftx, 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(πtx | π(t−1)x,π−x)
becomes non-stationary.
23
Page 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 φcss = 0.41 and φc
sh = 0.6, and the constant cement volume is 0.03.
The values for the depth trends λ in the synthetic case are listed in Table 2. The priors
are uniform:
λi ∼ U [lλi, uλi
]. (B-1)
The Markov chain in the vertical downwards direction has the transition matrix:
P =
0.72 0.04 0.04 0.20
0.00 0.72 0.08 0.20
0.00 0.00 0.80 0.20
0.07 0.05 0.08 0.80
, (B-2)
where the ordering of lithology/fluid classes is: (SandGas, SandOil, SandBrine, Shale).
The transition matrix has the stationary distribution [0.12 0.11 0.27 0.50], which represents
the proportion of each class in the prior model. The values used for the lateral coupling
parameters βf and βl, defined in Appendix A, are βf = 1.5, βl = 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 Σm = Σ0m ⊗ Int ,
24
Page 25
where Int is a nt × nt identity matrix and
Σ0m =
0.0202 0 0
0 0.0102 0
0 0 0.0172
, (B-3)
and the associated prior is
p([Σ0m]ii) = IG(2, 0.022), i ∈ 1, 2, 3, (B-4)
which is a special case of the inverse Wishart distribution. The expected value of [Σ0m]ii is
0.022 and the variance is infinite and undefined.
The covariance matrix of seismic observations Σd is parameterized Σd = Σ0d ⊗ Υd,
where Σ0d is a 3 × 3 matrix. The correlation matrix Υd is a nt × nt matrix:
Υd =1
100Int +
99
100W1W1
′, (B-5)
where Int is a nt × nt identity matrix and W1 is a normalized convolution matrix based
on the wavelets. The first term in Υd 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 Σ0d is set such that a wanted signal-to-noise ratio is acquired. The
prior distribution of Σ0d is
p(Σ0d) = IW (0.012I3, 5). (B-6)
The expectation of Σ0d is 0.012I3. By using five degrees of freedom the prior will be very
vague.
The prior for the wavelet w is p(w) = N(0,Σw), where Σw = Σθw ⊗ Σ0
w. The angle
covariance matrix Σθw is constructed by a Gaussian correlation function with range 30. The
25
Page 26
wavelet covariance matrix Σ0w 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 =
0.30 0.20 0.50
0.00 0.50 0.50
0.10 0.20 0.70
, (B-7)
which gives the stationary distribution [0.09 0.29 0.62]. The values used for the lateral
coupling parameters βf and βl in the lateral part of the Markov random field are βf = 3
and βl = 2.
Wavelets are plotted in Figure B-1, and the prior distribution of Σ0d is:
p(Σ0d) = IW (5 0002I3, 5), (B-8)
where I3 is a 3 × 3 identity matrix.
26
Page 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, 1004–1011.
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, 185–
198.
——–, 2003b, Joint AVO inversion, wavelet estimation and noise-level estimation using a
spatially coupled hierarchical Bayesian model: Geophysical Prospecting, 51, 531–550(20).
Dvorkin, J., and A. Nur, 1996, Elasticity of high-porosity sandstones: Theory for two North
Sea data sets: Geophysics, 61, 1363–1370.
Eidsvik, J., P. Avseth, H. Omre, T. Mukerji, and G. Mavko, 2004, Stochastic reservoir
characterization using prestack seismic data: Geophysics, 69, 978–993.
Fjær, E., R. M. Holt, P. Horsrud, A. M. Raaen, and R. Risnes, 2008, Petroleum related
rock mechanics: Amsterdam : Elsevier.
Gassmann, F., 1951, Uber die Elastizitat poroser Medien: Vierteljschr. Naturforsch. Ges.
Zurich, 96, 1–23.
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, 721–741.
27
Page 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, 79–96.
Krumbein, W. C., and M. F. Dacey, 1969, Markov chains and embedded Markov chains in
geology: Mathematical Geology, 1, 79–96.
Larsen, A. L., M. Ulvmoen, H. Omre, and A. Buland, 2006, Bayesian lithology/fluid pre-
diction 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, 475–490.
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 lithol-
ogy/fluid inversion from seismic prestack data and well observations: Part II – Real case
study. Submitted for publication in Geophysics.
28
Page 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
Page 30
LIST OF FIGURES
1 Graph of stochastic model. The nodes represent stochastic variables and the ar-
rows represent probabilistic dependencies. The arrows from π, λ, and Σm to m represent
the rock physics likelihood, the arrow from m, s, and Σd to d the seismic likelihood, and
from π to πow and from m to mo
w the well likelihoods.
2 Schematic illustration of porosity φ depth trends for sand and shale, where t0 is a
reference depth and tc is the initiation of sand cementation.
3 Reference reservoir πR. The well is marked at trace 13.
4 Expected porosity and seismic elastic properties. Realizations of vp,vs 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, vp/vs ratio against acoustic impedance vpρ. 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 ex-
treme 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: pos-
terior 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).
9 Posterior model of covariance and wavelet in synthetic case. (a) Posterior distribu-
30
Page 31
tions of the diagonal components in Σ0m and Σ0
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 reser-
voir πR. (b) Maximum posterior estimate π. (c)-(f) Probability plots for the classes:
SandGas, SandOil, SandBrine, Shale. (g) Lithology error: 1−p(πtx = “the correct lithology class” |
o). (h) Lithology/fluid error: 1 − p(πtx = “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 (vp, vs, 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 φsh(z) and sand porosity trend φss(z). Posterior mean (solid black
31
Page 32
line) and 95% posterior confidence bounds (hatched black line).
17 Posterior model for lithology/fluid variables in synthetic case. (a) Maximum pos-
terior estimate π. (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 πw.
A-1 System of axis in prior Markov random field model. Fluid neighborhood (left) and
lithology neighborhood (right). Horizontal neighborhood ∂hπtx, sedimentary neighborhood
∂sπtx and vertical neighborhood ∂vπtx.
B-1 Wavelets, for angle (10, 21, 36) (left to right).
32
Page 33
d
Σdm
Σm
s
π λ
mo
w
πo
w
τ
o
Figure 1: Graph of stochastic model. The nodes represent stochastic variables and the
arrows represent probabilistic dependencies. The arrows from π, λ, and Σm to m represent
the rock physics likelihood, the arrow from m, s, and Σd to d the seismic likelihood, and
from π to πow and from m to mo
w the well likelihoods.
Rimstad & Omre –
33
Page 34
φ
t
Shale Sand
φ0
shφ0
ss
t0
tc
Figure 2: Schematic illustration of porosity φ depth trends for sand and shale, where t0 is
a reference depth and tc is the initiation of sand cementation.
Rimstad & Omre –
34
Page 35
CMP
Tim
e(m
s)
20 40 60 80 100
2000
2200
2400
2600SandGas
SandOil
SandBrine
Shale
Figure 3: Reference reservoir πR. The well is marked at trace 13. Rimstad & Omre –
35
Page 36
Sand
Shale
φ
Tim
e(m
s)
vp (m/s) vs (m/s) ρ (kg/m3)
2000 25001500 20002000 30000 0.2 0.4
2000
2200
2400
2600
Figure 4: Expected porosity and seismic elastic properties. Realizations of vp,vs and ρ
from the well trace in black. Gas-saturated sand (red), oil-saturated sand (green), brine-
saturated sand (blue) and shale (black).
Rimstad & Omre –
36
Page 37
vpρ (m/s×kg/m3)
vp/v
s
SandGasSandOilSandBrineShale
4 6 8 10×106
1.4
1.6
1.8
2
Figure 5: Rock-properties, vp/vs ratio against acoustic impedance vpρ. The lines are the
trends and the points are trends plus noise.
Rimstad & Omre –
37
Page 38
CMP
Tim
e(m
s)
20 40 60 80 100
2000
2200
2400
2600
(a)
CMP
Tim
e(m
s)
20 40 60 80 100
2000
2200
2400
2600
(b)
CMP
Tim
e(m
s)
20 40 60 80 100
2000
2200
2400
2600
(c)
-0.2
0.2
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
Page 39
Sweeps
Pro
port
ion
s
0 100 200 300 400 5000
0.2
0.4
0.6
0.8
1
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
Page 40
φ0 sh
φ0 ss
αsh
αss
κss
tc
φ0
shφ0
ss αsh αss κss tc
2095 21050.05 0.150 0.20.5 0.70.28 0.290.17 0.182095
2100
2105
0
0.1
0.2
-0.5
0
0.5
0.4
0.6
0.8
0.26
0.28
0.3
0
100
200
(a)
Tim
e(m
s)
φsh φss
0.16 0.22 0.280.1 0.15
2000
2200
2400
2600
(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
Page 41
[Σ0m]1,1 [Σ0
m]2,2 [Σ0m]3,3
[Σ0
d]1,1 [Σ0
d]2,2 [Σ0
d]3,3
0 5x10−30 5x10−30 5x10−3
0 5x10−40 2x10−40 1x10−3
(a)
Tim
e(m
s)
Amplitude Amplitude Amplitude
-2 0 2-2 0 2-2 0 2-100
0
100
(b)
Figure 9: Posterior model of covariance and wavelet in synthetic case. (a) Posterior dis-
tributions of the diagonal components in Σ0m and Σ0
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).
Rimstad & Omre –
41
Page 42
πR: reference
Tim
e(m
s)
CMP20 40 60 80 100
2000
2200
2400
2600
(a)
π: full model
Tim
e(m
s)
CMP20 40 60 80 100
2000
2200
2400
2600
(b)
p(πtx =SandGas | o)
Tim
e(m
s)
CMP20 40 60 80 100
2000
2200
2400
2600
(c)
p(πtx =SandOil | o)
Tim
e(m
s)
CMP20 40 60 80 100
2000
2200
2400
2600
(d)
p(πtx =SandBrine | o)
Tim
e(m
s)
CMP20 40 60 80 100
2000
2200
2400
2600
(e)
p(πtx =Shale | o)
Tim
e(m
s)
CMP20 40 60 80 100
2000
2200
2400
2600
(f)
Lithology error
Tim
e(m
s)
CMP20 40 60 80 100
2000
2200
2400
2600
(g)
Lithology/fluid error
Tim
e(m
s)
CMP20 40 60 80 100
2000
2200
2400
2600
(h)
SandGas
SandOil
SandBrine
Shale
0.0
0.5
1.0
Figure 10: Posterior model for lithology/fluid variables in synthetic case. (a) Refer-
ence reservoir πR. (b) Maximum posterior estimate π. (c)-(f) Probability plots for
the classes: SandGas, SandOil, SandBrine, Shale. (g) Lithology error: 1 − p(πtx =
“the correct lithology class” | o). (h) Lithology/fluid error: 1−p(πtx = “the correct class” |
o).
Rimstad & Omre –
42
Page 43
πR: reference
Tim
e(m
s)
CMP20 40 60 80 100
2000
2200
2400
2600
(a)
π: full model
Tim
e(m
s)
CMP20 40 60 80 100
2000
2200
2400
2600
(b)
π: without depth trends
Tim
e(m
s)
CMP20 40 60 80 100
2000
2200
2400
2600
(c)
π: without Markov field
Tim
e(m
s)
CMP20 40 60 80 100
2000
2200
2400
2600
(d)
SandGas
SandOil
SandBrine
Shale
Figure 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.
Rimstad & Omre –
43
Page 44
Tim
e(m
s)
CMP
τ estimated
Tim
e(m
s)CMP
τ fixed
20 40 60 80 10020 40 60 80 100
2000
2200
2400
2600
2000
2200
2400
2600
(a)
0.0
0.5
1.0
τ estimated τ fixed
0 0.5 10 0.5 10
500
1000
0
500
1000
(b)τ estimated
τfi
xed
0 0.5 10
0.5
1
(c)
Figure 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.
Rimstad & Omre –
44
Page 45
Tim
e(m
s)
CMP4600 4700 4800 4900 5000 5100 5200 5300
2200
2300
2400
2500
2600
(a)
Tim
e(m
s)
CMP4600 4700 4800 4900 5000 5100 5200 5300
2200
2300
2400
2500
2600
(b)
Tim
e(m
s)
CMP4600 4700 4800 4900 5000 5100 5200 5300
2200
2300
2400
2500
2600
(c)
−2 × 104
2 × 104
Figure 13: The seismic data d in real case represented by angle stacks (10, 21, 36) in
display (a), (b) and (c) respectively.
Rimstad & Omre –
45
Page 46
φ
Tim
e(m
s)
log vp
(m/s)log vs
(m/s)log ρ
kg/m3
7.5 7.97 7.57.8 8.20 0.2 0.4
2200
2300
2400 SandGas
SandBrine
Shale
Figure 14: Well observations. Posterior (far left), elastic material properties (vp, vs, ρ) (left,
middle right) and lithology/fluid classes (far right).
Rimstad & Omre –
46
Page 47
Sweeps
Pro
port
ion
s
0 100 200 300 400 5000
0.2
0.4
0.6
0.8
1
Figure 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).
Rimstad & Omre –
47
Page 48
φ0 sh
φ0 ss
αsh
αss
φ0
shφ0
ss αsh αss
0 10.6 1 1.40.24 0.280.16 0.2-2
0
20
1
20.2
0.3
0.40.2
0.25
(a)
Tim
e(m
s)
φsh φss
0.2 0.30.1 0.15 0.2
2200
2300
2400
2500
2600
(b)
Figure 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 φsh(z) and sand porosity trend φss(z). Posterior mean (solid black
line) and 95% posterior confidence bounds (hatched black line).
Rimstad & Omre –
48
Page 49
π
Tim
e(m
s)
CMP4600 4700 4800 4900 5000 5100 5200 5300
2200
2300
2400
2500
2600
(a)
p(πtx =Gas | o)
Tim
e(m
s)
CMP4600 4700 4800 4900 5000 5100 5200 5300
2200
2300
2400
2500
2600
(b)
p(πtx =Brine | o)
Tim
e(m
s)
CMP4600 4700 4800 4900 5000 5100 5200 5300
2200
2300
2400
2500
2600
(c)
p(πtx =Shale | o)
Tim
e(m
s)
CMP4600 4700 4800 4900 5000 5100 5200 5300
2200
2300
2400
2500
2600
(d)
SandGas
SandBrine
Shale
0.0
0.5
1.0
Figure 17: Posterior model for lithology/fluid variables in synthetic case. (a) Maximum
posterior estimate π. (b)-(d) Probability plots for the classes: SandGas, SandBrine, Shale.
Rimstad & Omre –
49
Page 50
πRw πw
p(πtx =Gas|o)p(πtx =Brine|o)
p(πtx =Shale|o)
0 0.5 10 0.5 10 0.5 1
Figure 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).
Rimstad & Omre –
50
Page 51
π: full model
CMP
Tim
e(m
s)
4600 4700 4800 4900 5000 5100 5200 5300
2200
2300
2400
2500
2600
(a)
π: without Markov field
CMP
Tim
e(m
s)
4600 4700 4800 4900 5000 5100 5200 5300
2200
2300
2400
2500
2600
(b)
π: without depth trends
CMP
Tim
e(m
s)
4600 4700 4800 4900 5000 5100 5200 5300
2200
2300
2400
2500
2600
(c)
π: without depth trends and Markov field
CMP
Tim
e(m
s)
4600 4700 4800 4900 5000 5100 5200 5300
2200
2300
2400
2500
2600
(d)
SandGas
SandBrine
Shale
Figure 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 πw.
Rimstad & Omre –
51
Page 52
Current system of axis Sedimentary system of axis
∂hπtx
∂vπtx
∂sπtx
πtx
sedimentarysedimentaryhorizontal
vertic
al
vertic
al
Figure A-1: System of axis in prior Markov random field model. Fluid neighborhood
(left) and lithology neighborhood (right). Horizontal neighborhood ∂hπtx, sedimentary
neighborhood ∂sπtx and vertical neighborhood ∂vπtx.
Rimstad & Omre –
52
Page 53
Bulk Shear
modulus modulus Density
k (GPa) g (GPa) ρ (g/cm3)
Sand 37.0 44.0 2.65
Clay 15.0 10.0 2.65
Water (Free) 2.4 0.0 1.03
Bounded Water 4.0 6.0 1.03
Oil 0.76 0.0 0.71
Gas 0.133 0.0 0.29
Table 1: Rock physics model parameters.
53
Page 54
true lλiuλi
φ0sh 0.18 0.0 0.9
φ0ss 0.35 0.0 0.4
αsh 0.64 −∞ ∞ m−1
αss 0.1 −∞ ∞ m−1
kc 0.1 −∞ ∞ m−1
zc 2100 0 ∞ m
Table 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.
54
Page 55
Amplitude
Tim
e(m
s)
Amplitude Amplitude
−105 0 105−105 0 105
−105 0 105
-100
0
100
Figure B-1: Wavelets, for angle (10, 21, 36) (left to right). Rimstad & Omre –
55
