Value of information of seismic amplitude and CSEM resistivity

Jo Eidsvik1, Debarun Bhattacharjya2 and Tapan Mukerji 3,

1 Department of Mathematical Sciences, NTNU, 7491 Trondheim, NORWAY. E-mail:

joeid@math.ntnu.no

2 Management Science and Engineering, Stanford University, California. E-mail:

debarunb@stanford.edu

3 Energy Resources Engineering, Stanford University, California. E-mail: mukerji@stanford.edu

ABSTRACT

We propose a method for computing the value of information inpetroleum exploration, where

decisions regarding seismic or electromagnetic data acquisition and processing are critical. We

estimate the monetary value that a seismic amplitude or electromagnetic resistivity dataset is worth,

in a certain context, before purchasing the data. Our methodis novel in the way we incorporate

spatial dependence to solve large-scale real-world problems by integrating the decision-theoretic

concept of value of information with rock physics and statistics.

The method is based on a statistical model for saturation andporosity on a lattice along the top

reservoir. Our model treats these variables as spatially correlated. The porosity and saturation are

tied to the seismic and electromagnetic data via non-linearrock physics relations. We efficiently

approximate the posterior distribution for the reservoir variables in a Bayesian model by fitting a

Gaussian at the posterior mode for transformed versions of saturation and porosity. The value of

information is estimated based on the prior and posterior distributions, the possible revenues from

the reservoir, and the cost of drilling wells. We illustrateour method with three different examples.

INTRODUCTION

The petroleum exploration and production industry has to contend with a great deal of uncer-

tainty, and information such as seismic and electromagnetic data reduces some of the uncertainty

in reservoir properties, at a price. In this paper, we present a computational approach for estimating

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how much a certain information gathering scheme is worth in the context of reservoir development.

Our method is novel in the way it integrates the decision-theoretic concept of value of information

(VOI) with rock physics and spatial statistics.

We illustrate our methodology with examples valuing seismic amplitude versus offset (AVO)

data and controlled source electromagnetic (CSEM) data. Consider, for instance, the typical situ-

ation where the decision maker has obtained seismic traveltime data at a reservoir, a-priori. One

must decide whether it is worthwhile to purchase seismic amplitude data, or CSEM data, or both.

The VOI for a certain information gathering scheme is the maximum that the decision maker should

pay to purchase that scheme.

We use a lattice model for reservoir properties along the topreservoir. When we consider

purchasing seismic AVO data, we are referring to processing, or simply re-processing seismic data

at top reservoir. In the case of CSEM data we consider purchasing a CSEM survey and subsequent

processing of resistivity data along the top reservoir zone. The CSEM resistivity data can be

processed from phase and amplitude data observed at sea bed logging sensors (Ellingsrud et al.,

2002), along with depth information and overburden resistivity properties. Using resistivity values

at the top reservoir is a different approach compared to other inversion methods for CSEM data

that use vertical models (Hoversten et al., 2006). For any kind of dataset, whether it is AVO or

CSEM, the important question is whether it is more valuable to the decision maker than its cost,

for a particular reservoir.

Our method is based on a Bayesian model for porosity and saturation along top reservoir, and

for the seismic AVO and CSEM resistivity data. We evaluate the prior and posterior probability

density functions (pdfs) for saturation and porosity, and from this estimate the prior and posterior

value, and finally the expected VOI. We now introduce some of the main notation: Brine (water)

saturation issi, oil saturation1 − si (assuming no gas), and porosity isφi. Here,i = 1, . . . , N

represents a spatial lattice location along top reservoir.One could alternatively use gas (or both

oil/gas) rather than oil as hydrocarbon variate. We use a regular lattice model, whereN = N1N2,

and (N1,N2) are the number of cells in the (north, east) directions. In general we letd denote the

experimental data. This can include seismic AVO or CSEM resistivity, or both. The particular data

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sources will be indicated with subscripts for seismic AVO andc for CSEM. Thus, the seismic AVO

data are denotedds,i = (ds,1,i, ds,2,i)′, i = 1, . . . , N , containing zero offset reflectivityds,1,i and

AVO gradient observationsds,2,i. We assume that both these seismic attributes will be available

at all N cells, if we decide to purchase AVO data. The CSEM resistivity data are denoteddc,j,

j = 1, . . . , M , where sizeM is typically much smaller thanN because CSEM data are relatively

low frequency signals inducing coarse sampling. Further, CSEM data, by current practice, are only

acquired along a few sail lines, and not exhaustively on a spatial lattice. The forward models for

seismic AVO and CSEM resistivity data are described using rock physics theory, tying porosity

and saturation to the elastic and electric properties, which are in turn linked with the data.

VOI is a popular notion in decision making under uncertainty, see e.g. Howard (1966) and

Matheson (1990). A good overview to VOI and decision theory in general is presented in Raiffa (1968).

There have been several recent approaches to estimating VOIfor high dimensional spatial prob-

lems, but very few of these incorporate spatial dependence.A key focus of this work is to de-

velop a method for estimating VOI with spatially dependent variables of interest. Polasky and

Solow (2001) compute the VOI for a spatial decision problem in conservation biology, assum-

ing spatial independence. They use a value criterion that isnot in monetary units. Houck and

Pavlov (2006) estimate the VOI for CSEM data, using a threshold method for a few outcomes

at every step and a sensitivity map for spatially distributed anomalies, but no spatial correlation

in the model. Bickel et al. (2006) study the VOI for seismic data using a model for net-to-gross,

with empirical relations between reservoir variables and value. They include no spatial correlation.

Bhattacharjya et al. (2006) compute the VOI for a discrete lattice model with spatial dependence,

showing that prior spatial dependence influences the VOI. However, the discrete model is limited

to small cases due to computational reasons, and reservoir properties like saturation and porosity

are only modeled indirectly. These properties may be more suitably integrated using continuous

variables at every cell. In this paper we present a method applicable for continuous variables with

spatial dependence, that can estimate the VOI for large problems.

Criteria different from VOI are popular for designing spatial acquistion designs under depen-

dence. Two such criteria are entropy, see e.g. Zidek et al. (2000), and marginal prediction variance,

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see e.g. Diggle et al. (2006). An important challenge in these design specifications is to place the

measurement stations such that the correlation parameterscan also be specified accurately. In this

paper we treat the correlation as fixed from prior assumptions, and use VOI as a criterion since

it is directly related to the monetary cost of the experimental test. Further, we can have potential

measurements at every station, e.g seismic AVO data, so our problem is not primarily a question

of optimal placement of stations.

Note that VOI is a commonly used expression in the petroleum industry. In this work, only the

exploration challenge of deciding whether to purchase seismic AVO or CSEM data is considered.

We do not regard the larger challenge of reservoir appraisaland depletion which comprises multiple

sequential decisions and operational tasks.

This paper is organized as follows: We first present the basicassumptions and equations for

computing the VOI. Then we discuss the methodological aspects, and finally we illustrate our

methods with numerical examples. The computational details are presented in the Appendix.

VALUE OF INFORMATION

The value of a reservoir field is related to the porosity and hydrocarbon saturation of local

reservoir zones. At each celli, we assume that the value (potential revenue obtainable) isRφi(1−

si). Here,R is a scaling term which involves factors such as oil price, recovery factor, net-to-gross

and thickness. While all of these could in principle also be varied stochastically along the lattice,

like saturationsi and porosityφi, we have assumed a fixed scaling factor ofR = R0Aihi for our

computations. We assume the oil price to beR0 = $300 per m3 ≈ $50/bbl, cell area ofAi = 502

m2, and reservoir thickness ofhi = 20 m, resulting inR = $15 million, but of courseR will

vary for different reservoir scenarios. Note in particularthat we treat reservoir thickness fixed, say,

already determined from seismic traveltime data. In practice, we could also include thickness as a

random variable, like porosity and saturation.

At the level of celli we now consider drilling a well at a costC. We set a cost ofC = $2

million, but this is case specific. For instance, deviated wells are more expensive than vertical

ones, drilling in deep offshore or high temperature/pressure zones is expensive, and so on. In our

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case,$2 million refers to a ballpark cost of drilling one well in a setof wells from a platform

developed field. The value obtained from the lattice cell is the revenue minus the cost of drilling.

We do not know the potential revenue obtainable beforehand and use the expectation in its place.

In this way we assume that the decision maker is risk-neutral, i.e. indifferent between accepting

a lottery with uncertain monetary prospects and accepting the expected value of the lottery for

sure. Risk aversion is probably a more common type of risk behavior, implying that the decision

maker would prefer to accept the expected value rather than enter the lottery. Different types of

risk behavior can be accomodated by assigning non-linear utility functions for risk averse or risk

seeking people. (See Raiffa (1968) for example.) The utility function for a single decision maker

or an organization can be assessed by the analyst or estimated from previous investment actions.

The expected prior value of celli is

Prior Valuei = max{RE[φi(1 − si)] − C, 0}, i = 1, . . . , N, (1)

whereE[φi(1− si)] is the expected value of porosity times oil saturation underthe prior pdf. Note

that the prior value in equation 1 is0 if the drilling costC is larger than the expected revenues at

this cell. Clearly, we will decide not to drill if the expected profit is negative.

Assume next that we have the option of purchasing datad, referring generically to seismic

AVO data on the lattice and possibly a sail line with CSEM resistivity data. The expected posterior

value of celli is then

Posterior Valuei = Ed (max{RE[φi(1 − si)|d] − C, 0}) , i = 1, . . . , N, (2)

whereE[φi(1−si)|d] denotes the expected value of porosity times oil saturationunder the posterior

pdf, conditioning on all datad. In contrast the prior expectation in equation 1 is unconditional,

not considering the experiment at all. If there is no correlation in the prior model or likelihood, we

only need to condition on data at celli in the posterior expectation, and not the entire experimental

datad. Equation 2 refers to the more general and common case where the reservoir variables can

be spatially dependent, see also Bhattacharjya et al. (2006). We marginalize over the experimental

datad in equation 2. We needEd, i.e. the expectation with respect to the marginal pdf of data d

to integrate over possible experiments. Thus, we can make the computations before the data are

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actually purchased. In practise this integration over all possible experiments is done by drawing

several realizations of datad, computing the posterior value for each realization, and taking the

average value of these. Note that equation 2 gives the value of the free experiment since we have

not included the price of purchasing the datad in the expression.

The VOI is always non-negative. The expected posterior value is always larger or equal to the

prior value since we make better informed decisions. One particular experimental datad may not

support increased revenues, and the posterior value can be smaller for this dataset alone. Another

dataset will support increased revenues, and on average, integrating over all possible experiments,

the posterior value is larger than the prior value. The notion of VOI arises by comparing a decision

maker’s prior decision problem with the problem created by changing the time by which a certain

information gathering scheme is observed. (See Merkhofer (1977) and Shachter (1999).) We

define the VOI at celli as the difference in expected posterior and prior value:

VOIi = Ed (max{RE[φi(1 − si)|d] − C, 0}) − max{RE[φi(1 − si)] − C, 0}. (3)

This kind of computation for the VOI is exact for a risk-neutral decision maker and for decision

makers with exponential utility functions, and is often a good approximation for other utility func-

tions as well (Raiffa, 1968). We define the total VOI of the field as the sum of the VOI for the

cells:

VOI =N∑

i=1

VOIi. (4)

If the VOI is larger than the price of purchasing the experiment d, the decision maker should

acquire / process this data. If this is not the case, the experiment should not be carried out since its

value is less than the cost of data acquisition and processing. Decisions regarding experimentation

and information gathering clearly depend on all parametersused to compute the VOI as above, as

well as the best estimate for the price of the experiment. We discuss this further with the help of

numerical examples.

In addition to valuing one test, the notion of VOI can also be used to compare different tests.

There are often several alternatives to evaluate and to keepin mind before purchasing data. The

final decision should be a trade off between the cost and the potential gain in information. For

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instance, one can choose to purchase data for the entire reservoir domain, or just a smaller geo-

graphical region of more interest. In situations with several geological constraints, a partial test

might be more valuable than a full experiment (Bhattacharjya et al., 2006). Selecting partial re-

gions then becomes a combined problem of site selection (Zidek et al., 2000) as well as VOI over

the selected partial region.

Our assessment of total VOI in equation 4, given by the sum over all cells, requires large cells

and exploits our assumption that drilling decisions are taken at the level of cells. If the cells are

small, this assumption may not be justified because there is astronger connection between the

wells. In fact, the volume that a particular well drains may depend on the reservoir properties

in a complicated joint problem. Another issue that requiresattention is the drilling strategy, for

instance regarding the use of vertical or deviated wells that cost more to plan, but are cheaper to

extend than drilling several new vertical wells from an installation. The VOI in such a complex

setting would still be defined as the difference between posterior and prior values, but these must

now be computed using an integral over the joint distribution and with a set of actions or constraints

included in the function to integrate over. These aspects require further research in specific case

studies. In this paper we have chosen to use total VOI as defined in equation 4, for the sake of

tractability.

BAYESIAN MODEL

The main modeling aspects are summarized in Figure 1. The prior pdf for porosity and satura-

tion are represented on the lattice, from which we estimate the prior value for each cell in equation

1. Likelihood models for seismic AVO or CSEM resistivity data are based on forward models

with additive noise. We combine prior and likelihood to obtain the posterior pdf, and estimate the

posterior value for each cell in equation 2. The VOI in equation 3 (and subsequently equation 4) is

obtained as the difference between posterior and prior value.

Prior model for saturation and porosity

Brine saturation for all the cells is denoted(s1, . . . , sN) with si ∈ (smin, smax). We obtain a

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variable on the real line, denotedms,i, by taking a logistic transform

ms,i = log(si − smin

smax − si), si =

1

1 + exp(ms,i)smin +

exp(ms,i)

1 + exp(ms,i)smax, i = 1, . . . , N. (5)

Similarly porosity is denoted(φ1, . . . , φN) with φi ∈ (φmin, φmax), and we have

mφ,i = log(φi − φmin

φmax − φi), φi =

1

1 + exp(mφ,i)φmin +

exp(mφ,i)

1 + exp(φs,i)φmax, i = 1, . . . , N. (6)

The maximum and minimum limits for saturation and porosity should be tuned for every applica-

tion. In this paper we consider the specified limits in Table 1.

At each lattice cell we definemi = (ms,i, mφ,i). We assign a multi-variate Gaussian prior pdf

for m = (m1, . . . , mN)′, i.e.

p(m) = Normal(m; µ,Σ), (7)

for prior meanµ and covariance matrixΣ. Here, meanµ is a fixed size2N vector based on prior

knowledge about the spatial distribution of saturation andporosity. In the simplest case with little

prior information we use only two mean parametersµs andµφ which are representative for all

lattice cells. In a case with much prior geologic information we would set different prior means to

different spatial domains. We discuss both situations in the numerical example.

The covariance matrixΣ is defined from the inverse covariance matrixQ = Σ−1, also called

precision, which is sparse in the assumed case of a Markov neighborhood structure on the lattice,

see Appendix. Precision matrixQ is composed of i) a spatial structure matrix with ones on the

diagonal and non-zero entries only between sitesi, j that are neighbors, in our case the eight

nearest cells for an internal cell, and ii) a global2 × 2 scale matrix for the precision at every

cell. For an exponential correlation function, a sparse precision matrix gives a very good fit to the

actual precision matrix, see Rue and Held (2005). We assume no prior correlation between logistic

saturation and porosity. (See also Bachrach (2006).) Hence, with the simplest prior mean we have

that the marginal prior pdf at each celli is p(mi) = Normal[m′i; (µs, µφ)

′,Σ0,i], with diagonal

2 × 2 matrixΣ0,i. We need the subscripti on the covariance matrix because the edge and corner

cells have fewer neighbors than the other cells, but this could be changed by wrapping the lattice

on a torus.

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The cellwise prior value in equation 1 is computed by approximatingRE{φi(mφ,i)[1−si(ms,i)]}

under the Gaussian prior. We do this by Monte Carlo integration; drawing several samplesmi from

the prior, and computing the average of the dollar values (which in turn depend on saturations and

porosities) over all Monte Carlo samples.

Likelihood for seismic AVO data

Saturation and porosity are linked to seismic amplitude data by rock physics relations. Like we

illustrate in Figure 1, the forward model uses elastic parameters to predict zero offset reflectivity

and AVO gradient observations at top reservoir (Mavko et al,1998). We assume that these relations

are valid across the lattice. Thus, the following equationshold for all cells. First, density is given

by

ρi = φisiρb + φi(1 − si)ρo + (1 − φi)ρq, (8)

whereρb, ρo andρq are the fixed density of brine, oil and quartz, respectively.All fixed parameters

in the forward model are specified in Table 1. Bulk and shear modulus for different porosities and

saturations are calibrated from wells and by Gassmann’s formula for fluid substitution. We assume

a linear relationship for brine saturated bulk modulusK and shear modulusG as a function of

porosity. This entails using a well log to specifyKmax andGmax for the smallest porosityφmin,

andKmin andGmin for the largest porosityφmax. Bachrach (2006) uses a non-linear function

which is a better fit for his dataset. Any appropriate modulus-porosity relation, calibrated to log

data, can be used. By Gassmann’s formula, the shear modulus remains constant for any saturation,

and hence

Gi = G(si, φi) = G(smax, φi), (9)

wheresmax represents brine saturation andG(smax, φi) is the shear modulus fitted from well data in

brine zone. For the bulk modulus calculation we first computethe fluid bulk modulus of saturation

si (assuming homogeneous fluid mix) using a Reuss average as:

Kf,i =

[

si

Kb+

(1 − si)

Ko

]−1

, (10)

whereKb andKo are the fixed bulk modulus of brine and oil. By Gassmann’s formula, the rock

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bulk modulusKi for saturationsi and porosityφi is

Ki =BiKq

1 + Bi

, Bi =K(smax, φi)

Kq − K(smax, φi)−

Kb

φi(Kq − Kb)+

Kf,i

φi(Kq − Kf,i), (11)

whereKq is the fixed quartz bulk modulus andK(smax, φi) the bulk modulus fitted from the well

data in brine sands. P- and S-wave velocity are given from bulk and shear modulus by

VP,i =

√

√

√

√

Ki + 43Gi

ρi

, VS,i =

√

Gi

ρi

. (12)

The seismic attributes, here represented by zero offset reflectivity and AVO gradient, can be

evaluated from the elastic contrasts in the cap rock and in the reservoir using Aki and Richards

formula (Mavko et al, 1998), as follows:

z1,i =1

2

(

δVP,i

V̄P,i+

δρi

ρ̄i

)

, (13)

z2,i =1

2

δVP,i

V̄P,i

− 2V̄ 2

S,i

V̄ 2P,i

(δρi

ρ̄i

+ 2δVS,i

V̄S,i

),

where, for general propertyξi, ξ̄i = ξi+ξcr

2is the average between cap rock and reservoir properties,

while δξi = ξi − ξcr is the difference between reservoir and cap rock property. For the cap rock

properties, we use laterally constant values specified in Table 1. When incorporating the seismic

AVO data, we assume that the elastic properties of the cap rock are available and that top and

bottom reservoir traveltimes have already been specified. Similar assumptions have previously

been used for seismic lithology prediction; see e.g. Avsethet al. (2005).

We treat the relations in equations 5 - 13 as a forward model giving the expected seismic AVO

responses for fixed values of logistic porosity and saturation. This holds for all lattice cells, and

henceE(ds,i|mi) = E[(ds,1,i, ds,2,i)′|mi] = (z1,i, z2,i)

′ = f s(mi), i = 1, . . . , N . The functionf s

consists of a series of function evaluations: First, equations 5 and 6 tie logistic porosity and satura-

tion to saturation and porosity. Next, equation 8 defines thedensity, while equations 9 - 11 assign

the shear and bulk modulus corresponding to these saturation and porosity values. The velocities

are calculated in equation 12, and finally the seismic AVO attributes at top reservoir are computed

from the density and velocity values along with the cap-rockproperties in equation 13. Altogether,

this defines a non-linear forward model for the seismic AVO data ds = (d′s,1, . . . , d

′s,N), but a

forward model that is easy to compute in a stepwise manner.

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We model the likelihood pdf as Gaussian with conditional independence between allN sites:

p(ds|m) =N∏

i=1

Normal[ds,i; f s(mi),Σs], (14)

whereΣs is a2×2 matrix indicating the conditional covariance in zero offset reflectivity and AVO

gradient. The entries in this matrix include the standard deviation for zero-offset reflectivity de-

notedσs,1, standard deviation for AVO gradient denotedσs,1 and cross correlation between the two

denotedσs,1,2, all specified in Table 1. Conditional independence in equation 14 is an assumption

indicating no added correlation in seismic data processing. Additional spatial dependence can be

modeled in the likelihoodp(ds|m), but we must then specify the correlation / smoothing parame-

ters and take these into account in the computations for the posterior below. Note that the Gaussian

likelihood indicates an additive noise term, a modeling aspect that might not always be realistic

since the tuning of linear curves for bulk and shear modulus also contains noise, and similarly the

density relation. Nevertheless, the additive nature can bejustified by forward modeling several

Monte Carlo runs fromf s with noise at every stage, and then fitting the best Gaussian pdf for the

seismic AVO data.

In Figure 2(left) we illustrate the sensitivity of seismic AVO data to saturation and porosity

changes for our choice of reservoir, fluid and caprock parameters. The shifts in mean level for zero

offset reflectivity and AVO gradient are rather small for changing saturation, especially when we

consider the large overlap indicated by the uncertainty ellipses. A change in porosity is easier to

detect with our parameter settings. Note that the zero-offset reflectivity and gradient are negatively

correlated, which explains the rotated ellipses. A change in saturation is somewhat easier to detect

based on AVO gradient, than on zero-offset reflectivity. This is natural as the saturation leaves

shear modulusG invariant, while bulk modulusK changes according to Gassmann. Hence, the

ratio betweenVP andVS is expected to change for different fluid saturations, and this ratio goes

into the relation for AVO gradient in equation 13.

In applications with real data it is useful to assess the expected levels and associated uncertainty

as indicated by the Gaussian ellipses in Figure 2(left). This is important for posterior evaluation

of reservoir properties and also for estimating the VOI. Further, such displays allow us to study

the reservoir potential, for instance by asking questions like “what if the sensitivity to saturation is

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twice as large?” or “what if the expected value is shifted tenpercent to the right?”, and so on.

Likelihood for CSEM resistivity data

CSEM data, see e.g. Ellingsrud et al. (2002) or Edwards (2005), are generally tied to reservoir

variables by complex forward models and computed from the specific acquisition setting and local

properties of the subsurface. An inversion method for a vertical section of a reservoir zone is

presented in Hoversten et al. (2006), while another Bayesian inversion method is presented in Chen

et al. (2007). We consider CSEM resistivity datadc = (dc,1, . . . , dc,N), given along a lateral line

at top reservoir. Here we mean inverted resistivity (not apparent resistivity). Resistivity inversion

from CSEM data is gradually becoming more common. Nevertheless, use of these inversions as

inputs should be done with care after careful quality control. We assume that datadc are tied to

the saturation and porosity values by Archie’s law (Mavko etal, 1998), stating that resistivity is

r = γsαφβ. The fixed parametersα, β andγ are specified in Table 1, and are identical to the

ones used in Hoversten et al. (2006) and Chen et al. (2007). Since CSEM data typically have much

lower spatial resolution than seismic AVO data, the CSEM resistivity data are treated as aggregated

values, averaged over many lattice cells. Hence, for each site j we take a neighborhood denoted

Dj and use Archie’s law for saturation and porosity in that neighborhood as

fc(mi; i ∈ Dj) =1

|Dj|

∑

i∈Dj

ri =γ

|Dj |

∑

i∈Dj

sαi (ms,i)φ

βi (mφ,i), j = 1, . . . , M. (15)

Here,|Dj| denotes the number of lattice cells in the neighborhoodDj. For our examples, we use

north-south sail lines for the CSEM data, andDj consists of either5 or 9 cells, i.e. including

two or four cells to each side of the data cellj. This aggregation is our way of ensuring that the

CSEM resistivity data are on a different resolution than theseismic AVO data. An alternative is to

tune physically derived non-uniform weight functions smoothing over several cells derived from

forward numerical modeling and inversion over synthetic earth models.

We use equation 15 as the forward model connecting the logistic saturation and porosity to the

resistivity data. The likelihood pdf is modeled as Gaussian, with this non-linear conditioning in

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the expectation giving

p(dc|m) =M∏

j=1

Normal(dc,j; fc(mi; i ∈ Dj), σ2c ), (16)

where we also assume conditional independence, just like wedid for the seismic data. This entails

no smoothing in the data processing, additional to the aggregation in the forward modelfc. Such

added smoothing could be accomodated in the model, but requires that the process of ’smearing’

be specified.

In Figure 2(right) resistivity is plotted for different values of saturation and porosity. The

standard deviationσc is tuned from a well log of resistivity, saturation and porosity, and specified

in Table 1. Figure 2(right) shows that resistivity is sensitive to saturation, which gives a shift to

higher resistivity values for large oil saturation, using our parameter settings. On the other hand,

resistivity is not very sensitive to changes in porosity, for fixed saturation. This can also be seen

from the small absolute value of theβ parameter in Archie’s relation, compared to theα exponent

for saturation.

Posterior model

The posterior pdf for logistic saturation and porosity is

p(m|d) =p(m)p(ds|m)p(dc|m)

p(d), d = (ds, dc), (17)

where the denominator does not depend onm. The logarithm of the posterior is

log p(m|d) = κ −1

2(m − µ)′Σ−1(m − µ) −

1

2

M∑

j=1

[dc,j − fc(mi; i ∈ Dj)]2

σ2c

(18)

−1

2

N∑

i=1

[ds,i − f s(mi)]′Σ

−1s [ds,i − f s(mi)],

whereκ denotes a normalizing constant that does not depend onm. The posterior is not ana-

lytically tractable because of the non-linear forward models. Our solution is to approximate the

posterior by matching the mode and the curvature at the mode.We do this by maximizing equation

18. The optimization is based on sequential linearization of the non-linear likelihood forward mod-

els, using numerical differentiation. The fitted Gaussian approximation to the posterior is further

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described in the Appendix and equals

p(m|d) ≈ Normal(m; µm|d,Σm|d), (19)

whereµm|d is at the mode oflog p(m|d) in equation 18, and the covariance matrixΣm|d is

computed to match the curvature at the mode. We have defined logistic porosity and saturation

independent in the prior, but they will be correlated in the posterior because they couple in forward

modelsf s andfc.

The cellwise posterior value in equation 2 requires that we estimateRE{φi(mφ,i)[1−si(ms,i)]|d}

under the approximate Gaussian posterior formi = (ms,i, mφ,i), given the data. We do this by

Monte Carlo integration; drawing several times from the approximation to posteriorp(m|d), and

computing the average of the dollar values over all Monte Carlo samples.

Similarly, we need an integral over all possible datad to obtain the posterior value in equation 2.

This is also done using Monte Carlo integration. We draw samples ofd andm from p(d|m)p(m).

Thed sample is fromp(d); now we can simply discard them samples. For each Monte Carlo

data sample we estimate the posterior value (as explained inthe previous paragraph), and finally

average these posterior values over all Monte Carlo data samples.

The method of posterior approximation and VOI computation requires some CPU time, but

efficient tools can be used to minimize this cost, see Appendix. We could also run the code in

parallel for different Monte Carlo datasets. It is worth noting that there are no memory limitations

with our model, and thus VOI can be computed for large lattices, in contrast to the discrete model

presented in Bhattacharjya et al. (2006).

Pseudoalgorithm

To summarize the methodology we present an algorithm for computing the VOI:

• Estimateprior cellwise value in equation 1. In this step we use samples fromthe prior pdf

for mi, i = 1, . . . , N , to estimateRE{φi(mφ,i)[1 − si(ms,i)]}.

• Sum the values for all cells to get theprior value of the lattice.

14

• Repeat the followingw = 1, . . . , W times (W ≈ 1000) to approximate theposterior value

using Monte Carlo sampling:

1. Draw seismic AVO and CSEM resistivity datadw based on the statistical prior and

likelihood model.

2. Find the posterior mode and the curvature at the mode to approximatep(m|dw).

3. Estimate theposterior cellwise value for this dataset using equation 2. In this step we

use samples from the approximate posterior pdf form to estimateRE{φi(mφ,i)[1 −

si(ms,i)]|dw}, i = 1, . . . , N .

4. Sum the values for all cells to get theposterior value of the lattice for this datasetdw.

• Approximate theposterior value as an average over allW runs:

N∑

i=1

Ed (max{RE[φi(1 − si)|d] − C, 0}) ≈1

W

W∑

w=1

N∑

i=1

max{RE[φi(1 − si)|dw] − C, 0}.

• TheVOI is the difference between the expected posterior and prior value.

NUMERICAL EXAMPLES

In this section, we illustrate our methods by computing the VOI for several cases. We first

examine the relationship between VOI and the accuracy of theseismic AVO data in an illustrative

single cell example. We then study the value of seismic AVO and CSEM on a lattice. For laterally

constant prior means, first we compute the value of seismic AVO data only, and then the value

with seismic AVO plus CSEM resistivity data. Finally, we compare the VOI for one sail line of

CSEM data in a situation where seismic data has already been processed. Our specified values and

parameters in these examples should not be taken in earnest,they are used for illustration purposes

only and should be tuned carefully in a real application. Extensive field study is beyond the scope

of this manuscript.

Example 1: VOI for seismic AVO data, for a single cell

We first look at a single cell. The prior means for logistic variables areµs = −1/4 andµφ = 0,

which by equation 5 and 6 implies a mean brine saturation of0.45 and porosity of0.275. We set

15

prior standard deviations equal to1, with no correlation between logistic saturation and porosity.

This simple single cell case could represent a simplified model of a reservoir zone where we con-

sider re-processing seismic AVO data. The seismic AVO data can be processed very accurately

(expensive) or in a simple manner (less expensive). We incorporate this accuracy in the seismic

data using a constant scaling factor for the likelihood noise covariance matrix, i.e.̃Σs = η2Σs,

whereΣs is our reference specified in Table 1, and where the standard deviation scaling factorη

ranges from0.5 (the most accurate processing) to2 (simple processing). All other model and value

parameters are specified in Table 1. The VOI is computed according to the steps of the pseudoal-

gorithm outlined above. In this single cell example we drop the sum over all lattice cells for prior

value in step 2 and for posterior values in number4.

In Figure 3 we have plotted the prior value, posterior value and the VOI for this range of

accuracies in the seismic AVO data. Fluctuations are due to Monte Carlo error in the estimates.

The prior value is about $230, 000 for all likelihood accuracies. The posterior value is largest for

very accurate seismic data, and declines as the standard deviation increases fromη = 0.5 to η = 2.

The VOI inherits the shape of the posterior value. The VOI curve is almost linear for our range

of scaling factors. Recall that we should decide to purchasethe seismic AVO data if the VOI is

larger than the price of the seismic AVO data. Forη = 1 the VOI is about $80, 000, and therefore a

seismic AVO processing scheme of this quality that is cheaper than $80, 000 should be purchased.

Assume that a company that processes such data adjusts theirprice according to the quality

level of their product. The quality is determined by the number of days they work on this processing

project. Suppose that the price of processing is $2000 a day, and that the most accurate processing

(η = 0.5) takes40 work days (i.e. $80, 000), while a simpler method (η = 2) takes30 work days

(i.e. $60, 000). In Figure 3(top, dashed) we have marked a straight line between these two prices

for purchasing seismic AVO data. This line can be used to compare their sales strategy with our

estimated VOIs. In particular, we see that the two lines cross at aboutη = 1.2. Thus, if we process

data more accurately than this, the VOI is larger than the cost and we should buy the data. If we

can afford to pay $80, 000 for 40 days work, we should choose this strategy because the difference

between the VOI and the cost is the largest.

16

The CPU cost to make the plot in Figure 3 is some minutes, when the Monte Carlo integration

over the possible experimental data is done in the simplest manner on our desktop computers.

Example 2: VOI for seismic AVO and CSEM data, for a 45 × 45 lattice with constant mean

We now consider a lattice of size45×45. Prior means areµs = −1/4 andµφ = 0, for all cells,

indicating that oil is quite likely. The prior standard deviations for logistic porosity and saturation

are about1, with no correlation between the two, while the spatial correlation range is five lattice

cells. We consider the option of drilling a well at every lattice site. Since we have assumed high

oil saturation a priori, the prior value is very large, about$470 million.

We compute the VOI for the case of seismic AVO datads only, and with both seismic AVO and

CSEM resitivity datadc. The seismic AVO data is acquired at every cell, whereas the CSEM data

is acquired only along one sail line along the center north-south column. CSEM resistivity data is

acquired with resolution|Dj | = 9 and|Dj| = 5, which givesM = 5 andM = 9, respectively.

The VOI is about $112 million for seismic AVO alone with our specification of parameters. It

is hard to give a ballpark number for the price of acquisitionand processing of seismic AVO data.

Careful processing of top reservoir amplitudes from a seismic survey may cost about $2.5 million

if a team of five works on this survey for a year, for a daily price of $2000 per person. This would

definitely be worth the processing cost in our case, and we should decide to purchase the seismic

processing, since the VOI is much higher than the price of thedataset.

The additional VOI when including CSEM data to the seismic AVO data is $12 million in the

case of moderate resolution CSEM data, i.e.|Dj | = 9 while it is $16 million for higher resolution

CSEM, i.e. |Dj| = 5. The price of CSEM acquisition and processing should be lessthan this for

us to purchase the data. Let us assume, as a ballpark estimate, that the CSEM acquisition cost

is $1 million for a week of sailing and that the time spent for one sail line can be two weeks.

Additionally, we have the processing price, but this is usually small compared with the price of

acquisition for CSEM, and even if it makes a significant proportion, the price of a CSEM dataset

is smaller than the estimated VOIs of $12 and $16 million. In this case, the CSEM data would also

be valuable, and if the price difference between high and lowresolution CSEM data is less than $4

million, we decide to purchase the high resolution data.

17

Example 3: VOI for CSEM data, for a 55 × 55 lattice with spatially irregular mean

We next consider a spatially varying prior mean, where we useas current prior a Gaussian pdf

for logistic saturation and porosity that is conditioned onseismic AVO data. The decision to be

made next is whether to acquire and process a sail line of CSEMdata.

The prior is constructed from applying our model for logistic saturation and porosity to the

seismic AVO data at the top-Heimdal formation at the Glitne reservoir, see Avseth et al. (2005).

In this way we getp(m|ds) = Normal(m; µm|ds,Σm|ds

) as our prior distribution, and next

consider purchasing CSEM datadc which allows us to evaluate the posteriorp(m|ds, dc). Figure

4 shows the seismic AVO data (top), the oil saturation and porosity estimates derived from the

mode ofp(m|ds) (middle), and the approximate standard deviations of theseestimates at each

cell (below). The55 × 55 grid displayed in Figure 4 is a part of the reservoir that is quite likely

to contain oil. The AVO data is upscaled to50 × 50m2 cells. The prior pdf used in the inversion

of the seismic AVO data has constant meanµφ = 0 for logistic porosity, while the saturation

variable has a prior which depends on the traveltimes to top reservoir. We used prior standard

deviations of about1 and spatial correlation range of five lattice cells. The remaining model and

value parameters are taken from Table 1.

In our case the seismic AVO data and the associated estimatesof oil saturation and porosity in

Figure 4 illustrate the available information before CSEM data is acquired. We see that the north

and south central areas are quite likely to be oil saturated (middle, right plot), but the uncertainty

based on seismic AVO alone is quite large (below, right plot). The total prior value for this lattice

is about $516 million. This is estimated from the results of Figure 4 (middle and below).

We now compute the VOI for a line of CSEM at east coordinate27 (see Figure 4), i.e. along

the parts expected to have quite high oil saturation. We use aneighborhood size of|Dj| = 5, which

means that there areM = 11 aggregated resistivity observations in the processed CSEMdata along

the sail line. The VOI for this dataset is $3 million. Let us again assume that CSEM acquisition

costs about $2 million. Additionally, we would have to pay the processing price. If the processing

and planning cost less than $1 million, we should purchase the CSEM sail line, otherwise itis not

worth the cost.

18

The CPU time for the lattice examples is overnight on our desktop computers, when the Monte

Carlo integration is done in the simplest manner. Large speed-up can be achieved by running this

in parallel.

CONCLUSION

We propose a method for computing the VOI of seismic AVO and CSEM resistivity for a

reservoir field under spatially correlated reservoir variables. The method relies on computing both

the prior and posterior value in a Bayesian model. Our model consists of a prior for saturation

and porosity on a lattice along top-reservoir, and likelihood functions exploiting the rock physics

relations known for seismic amplitude and CSEM data. The methods could be generalized within

our framework to include other reservoir variables and different types of data.

Our formulation allows for spatially correlated porosity and saturation, and the computation of

VOI should account for this coupling of reservoir variables. To compute the posterior for a dataset,

we use an efficient solution based on an approximate Gaussianpdf at the posterior mode. This

posterior approximation is very fast compared with simulation-based methods. Yet, the posterior

value is computed for several simulated datasets in a Monte Carlo integration approach, and this

takes a fair amount of CPU time. In a larger field example this could be run in parallel.

In the examples with seismic AVO and CSEM resistivity, we have chosen some plausible nu-

merical values for the different variables for computationand illustration purposes only. These

numbers will vary from scenario to scenario and the actual absolute values used in the examples

should not be taken in earnest. One challenge in more realistic examples is to set the revenues,

which could depend on multi-variate probabilities and possibly complex interactions of the spatial

reservoir variables. Our methods can treat multi-cell probabilities on the lattice, which remain

approximate joint Gaussian. Any linear combination of lattice variables is also Gaussian, but

complex forward models such as fluid flow simulators working on the spatial variables cannot be

handled directly within the current framework. Careful specification of revenues must then include

reservoir engineers and drilling engineers.

The method presented in this paper only studies a small part of the reservoir management

challenge. In practice, the VOI should be computed for a larger workflow including the different

19

stages of a reservoir’s lifetime. We believe that the described methodology, by integrating statistics

and decision theory with geophysics, can aid decision makers with data acquisition and processing

decisions for real-world problems in reservoir development.

ACKNOWLEDGEMENTS

We thank the Stanford Rock Physics project, Stanford University, California, for assisting JE

Nov.-Dec. 2006, and for supporting TM, and the Department ofMathematical Sciences, NTNU,

Trondheim, Norway, for assisting DB June-Sept. 2006.

20

APPENDIX - COMPUTATIONAL ASPECTS

Expressions for efficient Gaussian computation

Let m = (m1, . . . , mN)′ be a Gaussian random variable with pdf

p(m) =|Q|1/2

(2π)(N/2)exp[−

1

2(m − µ)′Q(m − µ)], (20)

with fixed sizeN × 1 mean vectorµ andN × N inverse covariance matrixQ = Σ−1, also called

precision matrix. For validity the precision matrix must bepositive definite. Expression 20 can be

written in quadratic and linear parts ofm as

p(m) ∝ exp(−1

2m′Qm + m′λ), λ = Qµ. (21)

Hence, the inverse covariance is in the quadratic part, and the inverse covariance matrix times the

mean is in the linear part.

We use the precision matrix since this is sparse for a Gaussian Markov random field, see Besag

and Kooperberg (1995) and Rue and Held (2005), making fast computations tractable. The only

non-zero elements are the diagonalQii, andQij for cells i andj that are neighbors on the lattice.

For a second order neighborhood, i.e. a3 × 3 neighborhood around each cell that we use here,

there are nine non-zero elements per row. For edge and cornercells this number reduces. For

an exponential covariance on the latticeQ can fitΣ−1 very accurately: If we scale entries in the

matrices such that the diagonal ofQ is 1, the fitted off-diagonal entries inQ areq1 = −0.36 for

north, east, south and west neighbors,q2 = 0.12 for diagonal neighbors, and0 otherwise, when the

exponential correlation range is five cells. The scale can bea scalar valid for all cells, or a matrix

that enters via a Kronecker product like we have here for multi-variate cell variables.

If we know the linear partλ and the quadratic partQ in equation 21, the mean valueµ can be

obtained by solving the linear system

Qµ = λ. (22)

We solve equation 22 using a Cholesky factorization of the sparseQ matrix. This entails finding a

lower triangular matrixL such thatQ = LL′, andµ is obtained by solving

Lw = λ, L′µ = w. (23)

21

The Cholesky matrixL is also sparse, though not as sparse asQ. Efficient computer code exists

for Cholesky factorization of large sparse matrices (Davisand Hager, 2005).

We sample a Gaussian variable fromp(m) using the Cholesky matrix as follows:

1. DrawN independent standard Normal variables usingp(w) = Normal(w; 0, IN).

2. Solve form in L′m = w.

3. Setm = m + µ.

Approximate posterior pdf

The exact posterior pdf is represented by equation 17 and 18.The Gaussian approximation is

fitted by matching the posterior mode and the curvature at themode. We maximize equation 18

using Newton–Raphson optimization. This entails iterative linearization of the non-linear forward

models for seismic AVO and CSEM resistivity data. Letm⋆ be a linearization point. For the

seismic AVO part a first order Taylor expansion entails

f s(mi) = f s(m⋆i ) +

df s

dmi

∣

∣

∣

∣

∣

m⋆i

(mi − m⋆i ), i = 1, . . . , N. (24)

Similarly for the CSEM resisitivity part at sitej we get

fc(mi; i ∈ Dj) = fc(m⋆i ; i ∈ Dj) +

∑

i∈Dj

dfc

dmi

∣

∣

∣

∣

∣

m⋆

(mi − m⋆i ), j = 1, . . . , M. (25)

To use a compact notation we collect all data in a length2N + M vectord = (ds, dc)′. We

further collect the two non-linear forward models as one, i.e. E(d|m) = f (m). Now, let F

denote the(2N + M) × 2N matrix of derivatives off , evaluated at linearization pointm⋆. Let

furtherΣd be the measurement covariance matrix of size(2N + M)× (2N + M) with entriesΣs

on the block diagonal for the2N upper diagonal entries, andσ2c on the lowerM diagonal elements.

Equation 18 is now approximated by

log p(m|d) ≈ κ −1

2(m − µ)′Σ−1(m − µ) (26)

−1

2[d − f(m⋆) − F (m − m⋆)]′Σ−1

d [d − f(m⋆) − F (m − m⋆)].

22

We collect linear and quadratic terms inm to get

log p(m|d) ≈ κ′ −1

2m′Qm + m′λ, (27)

Q = Σ−1 + F ′

Σ−1d F

λ = Σ−1µ + F ′

Σ−1d [d − f (m⋆) + Fm⋆],

whereκ′ is a normalizing constant. The quadratic and linear terms are used to fit the approximate

posterior mean under this linearization point as in equation 22. The posterior meanE(m|d) is

then represented by

QE(m|d) = λ. (28)

Equation 28 can be solved efficiently by a Cholesky factorization LL′ = Q, and nowQ = Σ−1 +

F ′Σ

−1d F . Prior precision matrixΣ−1 is sparse from the Markov assumption, whileF ′

Σ−1d F is

cellwise, except for the aggregated CSEM part in equation 25. To utilize the sparse structure we

split the lattice in three parts, one for the CSEM sail line, one to the east and one to the west of the

sail line. The east and west parts can be solved for separately in the Cholesky solution to equation

28, after the lattice column with CSEM data has been solved first.

The process above is iterated by setting the approximate mean as the new linearization point,

i.e. m⋆ = E(m|d). This continues until the method converges, typically around five steps. The

mean at the last step is close to the posterior mode and denoted µm|d. The fitted covariance

(curvature) at the mode is

Σm|d = (Σ−1 + F ′Σ

−1d F )−1, F =

df

dm

∣

∣

∣

∣

∣

µm|d

. (29)

We sample a Gaussian variable from the approximatep(m|d) by steps 1-3 from the previous

section, except now we draw2N standard Normal variables in step 1, solve the linear equation

using the new Cholesky matrixL′ from equation 28 in step 2, and useµm|d instead ofµ in step

3.

23

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and D. Vasco, 2006, Direct reservoir parameter estimation using joint inversion of marine

seismic AVA and CSEM data: Geophysics,71, c1–c13.

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25

Figure 1:

26

−0.1 0 0.1 0.2−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

s=0.75, φ=0.2

s=0.25,φ=0.2

s=0.25, φ=0.35

Zero offset reflectivity

AV

O g

radi

ent

Seismic AVO data.

−2 0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

pdf

Resistivity

CSEM resisitvity data.

s=0.25,φ=0.35

s=0.25,φ=0.2

s=0.75,φ=0.2

Figure 2:

27

0.5 1 1.5 220

40

60

80

100

120

VO

I

VOI curve

Price curve

0.5 1 1.5 2200

250

300

350

Prio

r an

d po

ster

ior

valu

e

Likelihood standard deviation scaling factor

Prior value

Posterior value

Better data quality

Figure 3:

28

Nor

th−

Sou

th

Zero offset reflectivitity

10 20 30 40 50

10

20

30

40

50−0.05

0

0.05

0.1

AVO gradient

10 20 30 40 50

10

20

30

40

50 −0.4

−0.2

0

Nor

th−

Sou

th

Estimate of porosity

10 20 30 40 50

10

20

30

40

500.2

0.25

0.3

0.35

Estimate of oil saturation

10 20 30 40 50

10

20

30

40

50 0.2

0.4

0.6

0.8

East−West

Nor

th−

Sou

th

Standard deviation of porosity estimate

10 20 30 40 50

10

20

30

40

50

0.04

0.05

0.06

East−West

Standard deviation of oil saturation estimate

10 20 30 40 50

10

20

30

40

50 0.16

0.18

0.2

0.22

CSEM line

Figure 4:

29

FIGURE CAPTIONS

Figure 1: Graphical display of reservoir variables, data and value. Decisions are connected to the

valueV , deduced from porosity and oil saturation along with revenue scaling factorR and the cost

of drilling C. We decide to drill a well only if this value is positive.

Figure 2: Illustration of likelihood models for seismic AVO (left) and CSEM resistivity values

(right) conditional on saturation and porosity. The ellipses in the left display show 95 percent

regions. For the seismic data, a change in porosity can be detected because the upper left ellipse

overlaps the others only slightly. A change in saturation isharder to detect. For the resistivity data,

a brine saturated rock has small resisitivity, while oil saturated rocks have large resisitivity values.

Resistivity is not very sensitive to a change in porosity, for fixed saturation.

Figure 3: Single cell case: Top plot shows VOI (solid) in $1000 and the price curve for processing.

Bottom plot shows prior value and expected posterior value in $1000. We evaluate the values for a

range of noise levels in the seismic AVO data.

Figure 4: Irregular mean: Top plots show the seismic AVO data for a selected domain in a seismic

dataset from the Glitne field. Middle plots show the irregular spatial estimates of porosity and

oil saturation based on the seismic AVO data. Bottom plots show the standard deviation of these

estimates in each cell across the lattice.

30

Table 1: Fixed parameters in the prior and likelihood model,and value computation.

Saturation and porosity limits:

Saturation smin = 0.1 smax = 0.9

Porosity φmin = 0.15 φmax = 0.4

AVO forward model:

Cap rock vP,c = 2.4km/sec vS,c = 1.0km/sec ρc = 2.25g/cm3

Bulk modulus Kq = 38GPa Kb = 2.8GPa Ko = 1.0GPa

Density ρq = 2.6g/cm3 ρb = 1.1g/cm3 ρo = 0.8g/cm3

Fitted bulk modulus in brine zone Kmin = 9GPa Kmax = 11GPa

Fitted shear modulus in brine zoneGmin = 2GPa Gmax = 5GPa

Noise covariance σs,1 = 0.025 σs,1,2 = −0.7 σs,2 = 0.075

CSEM forward model:

Archie’s law γ = 0.78 α = −1.31 β = −0.14

Noise covariance σc = 1

Other fixed parameters:

Monetary amounts R0 = $300/ m3 ≈ $50/bbl C = $2 million

Volumetric sizes Ai = 502 m2, all i hi = 20m, all i

31