WITH SEISMIC INFORMATION - Stanford Earth · 2010-08-20 · Figure 2.1: Seismic grid and well locations-Typical seismic line. • Figure 2.2: p, V p,, and Impedances histograms

CONSTRAINING RESERVOIR MODELS

WITH SEISMIC INFORMATION

SUBMITTED TO THE DEPARTMENT

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

MASTER OF SCIENCE

By

Louis-Jerome Bortoli

June, 1992

n

Icertify that Ihave read this report and that in my

opinion it is fully adequate, in scope and in quality, as

partial fulfillment of the degree of Master of Science in

Applied Earth Sciences.

Andre G. Journel(Principal advisor)

Approved for the UniversityCommittee on Graduate Studies:

Ill

Acknowledgements

Many people have contributed to the research presented in this thesis. Iam

particularly indebted to my advisor, Andre G. Journel, who first discovered that the

Californian climate was so propitious to Geostatistics; he was always available for

discussion, guidance, and help.

Open discussions with Alberto Almeida, Jinchi Chu, Clayton Deutsch, Felipe

Guardiano, Chris Murray, and Wenlong Xv greatly contributed to my understanding

and to the solution of the many problems that have arisen.

I'd also like to express my gratitude to Claude Chambon, Professor at the Ecole

dcs Mines de Nancy (France), for his enthousiast in teaching Geostatistics.

Iam thankful to Francois Alabert, Andre Haas, and Francois Hindlet for their

suggestions during a learning and collaborative period in Pau (France, Fall 1990).

Last, Iwant to aknowledge the help of Elf Aquitaine that provided me with

adequate technical and material support for this research.

IV

Abstract

Selection of conditional simulations of acoustic variables is done by matching the

simulated values with a series of actual seismic vertical sections. This is done by for-

ward convolution of the simulated acoustic models. The proposed algorithm allows

an iterative adjustment of the convolved simulated vertical section to the original seis-

mic data: first impedances along a vertical trace are simulated, then this impedance

trace is convolved into a synthetic trace at the seismic scale. An acceptance/rejection

criterion based on a correlation function is applied to compare the synthetic with

the actual trace. Ifaccepted, the algorithm proceeds and builds the synthetic image

by simulating an impedance trace at the next node, otherwise another simulation of

the same trace is drawn. Particular attention is given to calibration of the synthetic

seismograms at well locations and calibration of acoustic variables to porosity at the

log scale. Once the adjustment of the vertical section is obtained, the stochastic

impedance image -i.e. the adjusted section before convolution- is easily converted

into a porosity image. This image can directly be used for interpretation or flow sim-

ulator processing. The method is developped and tested on a real field,a shaly/sand

formation covered by an 3D seismic survey.

V

Contents

Acknowledgements ill

Abstract iv

Table of Contents v

List of Figures vii

1 Introduction 1

2 Geostatistical Modelling 8

2.1 Exploratory data analysis: 8

2.2 Variogram analysis: 12

3 Stochastic Simulation of Synthetic Seismograms 16

3.1 Sequential Simulation Techniques 16

3.2 Application to the Data Set 19

3.3 Convolution of the Stochastic Images 21

4 Optimization 27

vi

4.1 Optimization algorithm: 27

4.2 Results 32

4.3 The final porosity image: 33

5 Conclusion 37

Appendix 39

A Optimization Program: optim 39

B Inputs and Outputs files 45

Bibliography 49

vii

List of Figures

• Figure 1.1: The posterior matching approach.

• Figure 1.2: Methodology flow chart.

• Figure 2.1: Seismic grid and well locations - Typical seismic line.

• Figure 2.2: p, Vp, <j>, and Impedances histograms.

• Figure 2.3: Infering an horizontal variogram model from seismic data.

• Figure 2.4: Three dimensional experimental variograms and models.

• Figure 3.1: Definition of the simulation grid: 30 CDP x 100 Lines x 108 ms.

• Figure 3.2: Model reproduction by the average variograms over 10 realizations

• Figure 3.3: Wavelet retained for convolution and synthetic well logs.

• Figure 3.4: Calibration at well locations: synthetic well logs.

• Figure 3.5: Line 32: Three synthetic realizations v. Actual seismic line.

• Figure 4.1: Local optimization algorithm.

• Figure 4.2: Impedance variogram reproduction after final optimization.

viii

• Figure 4.3: Correlations between the simulated synthetic and the actual seismic

Line 32 for different levels of optimization.

• Figure 4.4: 2D correlation maps.

• Figure 4.5: Superposition of the synthetic and actual Line 32.

• Figure 4.6: Final porosity image.

1

Section 1

Introduction

Imaging the subsurface, and the modelling ofreservoir properties such as lithology,

porosity, permeability or fluidsaturations, represents a major challenge for petroleum

geologists and geophysists. As a result of very high costs ofdrilling, only a minimum

exploration and appraisal wells can be justified. Therefore lateral variations of hetero-

geneous reservoir properties cannot be accurately predicted from only well data. In

that context, traditional deterministic interpolation methods limited to welldata usu-

ally give an oversmooth unrealistic reservoir description. Stochastic models provide

a solution to this problem ofpaucity of well data. Conditional simulation techniques

allow generating potentially an infinite set of equiprobable images honoring the few

The integration of 3D seismic data beyond its traditional use in mapping large

scale subsurface structures, may significantly improve the modelling of petrophysical

properties. Seismic data integration has been approached indifferent ways. One way

SECTION 1. INTRODUCTION 2

is to combine large scale "soft" seismic data and small scale "hard" well data in one

set of prior conditioning data through algorithms such as straight cokriging[7], krig-

ing with an external drift[l6],or indicator coding of seismic data[l, 14, 22]. Another

way is to derive an acoustic model from seismograms inversion through deconvolution

of seismic amplitudes in reflectivity sequences. These reflectivity sequences provide

acoustic data which can be used as soft information. However, acoustic parameters,

such as velocity, density, impedance or reflectivity, derived from band-limited and

noise-contaminated seismic data are non unique, they have only large scale resolution

(decametric versus decimetric for well data) and are difficult to relate to the petro-

physical parameters relevant to reservoir production.

This study investigates a third way: perform a stochastic imaging of an acoustic-

related variable based only on well data. From each equiprobable realization, a syn-

thetic seismogram is derived, and the synthetic seismogram that best matches the

actual seismic data indicates the most realistic image. This method can be seen as a

"posterior matching": the actual 3D seismic data are used as a reference data set to

select from various prior stochastic images, see Figure 1.1.

Under some approximations, any acoustic-related variable can be deterministi-

cally transformed in the unit of the seismic data, here relative amplitudes. Therefore,

the choice of the acoustic parameter to be simulated is made dependent on the final

petrophysical parameter to be mapped. Ifthe goal is a distribution of porosity, then

the simulated variable should be the acoustic parameter best related to porosity: for

example impedance. In the following, porosity-impedance is used as the typical pair

of petrophysical/acoustic variables.


Figure 1.1: The posterior matching approach:Seismic data are used to select from prior realizations conditioned to well data.

The approach proposed presents an important advantage. Seismic and welldata

are considered at different steps of the processing: hence, the approach does not call

for combining data taken at different scales. As opposed to inversion techniques, the

finalbest acoustic model is derived only from well data at a scale allowing a good cal-

ibration between acoustic and petrophysical properties. As compared to methods in-

tegrating seismic data as prior conditioning data, the approach presented does not re-

quire a prior probabilistic calibration step, such as a Markov-Bayes calibration[l4, 22].

Such calibration is critical, and often very difficult to establish 1. The calibration re-

quired by the method proposed here is limited to the determination of a wavelet

allowing a good match of synthetic and actual seismograms at well locations.

methods are usually not able to deal with 3D seismic amplitudes since there is often aweak correlation between amplitudes and any acoustic or petrophysical parameter.


The flow chart of Figure 1.2, details the method used to get stochastic seismic

synthetic images of the reservoir. It shows the acquisition order of the different im-

ages until the final comparison with the actual seismic data, and the production of

the final porosity image.

The following terminology willbe used in reference to the sequence of the different

images produced:

Conditioning data: density and velocity data from well-logs.

Stochastic image: simulated image over the field of an acoustic-related variable,

here impedance, conditional to well-log data.

Acoustic image: stochastic image transformed inreflectivity.

Synthetic image: acoustic image convolved trace by trace 2.

Actual seismic image: original 3D seismic data.

Final porosity image: selected best stochastic image transformed inporosity by

a relation porosity/impedance defined at the log scale.

The first phase of the methodology proposed includes an exploratory data analy-

sis and the geostatistical modelling of the acoustic impedance (Section 2). Key steps

are the choice of impedance as the best acoustic-related variable to simulate and the

modelling of its lateral spatial correlation using quantitative data from well-logs and

qualitative information from geologic and seismic interpretations.

2A trace is a vertical sequence of acoustic amplitudes.


Figure 1.2: Methodology flow chart: Critical calibration phases are emphasized in


Section 3 describes the sequential Gaussian simulation algorithm used to provide

equiprobable 3D stochastic impedance images of the reservoir, and the calibration

procedure. The synthetic seismic traces built at well locations are calibrated with

collocated actual seismic data; the wavelet derived from this calibration is then used

to convolve the stochastic acoustic images, as obtained from the sequential Gaussian

simulation (SGS) algorithm, into synthetic seismic images of the reservoir. Among

all equiprobable realizations, one selects the one that best matches the actual seismic

data. An algorithm to perform that selection has been developed in Section 4.

Section 4 focuses on the elaboration of the optimization algorithm for dynamic

integration of 3D seismic data. The principle of the algorithm is to apply the poste-

rior matching approach (Figure 1.1) trace by trace instead of seeking a global match

over the whole stochastic image. This allows an iterative adjustment of the convolved

simulated vertical section to the original seismic data: first impedances along a ver-

tical trace are geostatistically simulated, then this impedance log is convolved into

a synthetic trace at the seismic scale. An acceptance/rejection criterion based on

a correlation function is applied to compare the synthetic with the actual trace. If

accepted, the algorithm proceeds by simulating an impedance trace at the next node.

Ifnot accepted, another simulation of the same trace is drawn.

What is gained, through this optimization, is the selection of the "best" realiza-

tion. What is lost is the uncertainty measure provided by the original distribution of

stochastic images.

Last, through a calibration between acoustic and petrophysical properties (i.e.

relation between impedance and porosity at the log scale), the single optimized


impedance simulated image is transformed into the petrophysical image which can

be interpreted and used for further processing.

Section 2

Geostatistical Modelling

2.1 Exploratory data analysis

The data used originate from a shaly sandstone reservoir. The study area, named

Eldorado, is about 2 km EW x 7 km NS and informed by 7 wells, see Figure 2.1. The

reservoir interval is characterized by a low sand/shale ratio withunknown extension

and direction of sand deposition. The whole zone Eldorado has been surveyed by a

3D seismic of 100 Lines x3O CDP 1. The seismic horizon at the base reservoir has

been picked on a 3D workstation and transformed into a flat event so that the base

of the reservoir occurs at apparent constant seismic time of 100 ms. The window

retained for the study is about 100 ms thick, see Figure 2.1.

The quantitative data used to describe the reservoir are well-logs, deviation tables

for deviated wells and seismic time picks at the top and base of the reservoir. The

1Common Depth Point.

8

SECTION 2. GEOSTATISTICAL MODELLING 9

Eldorado* Well locations

Seismic Section (Line 32)

M(DC 50

J

Figure 2.1: Seismic grid and well locations - Typical seismic line.

following logs were retained to characterize the reservoir acoustic and petrophysical

properties, see Figure 2.2:

Sonic transit time, expressed in ms/ ft, equal to the inverse of the p-wave in-

Bulk density p, expressed in ton/m 3 ;

Acoustic impedance p Vp,calculated from sonic and density log data;

Water saturation (fraction);

Clay content Shale%, expressed in per cent of total volume;

Porosity <f>, expressed as a fraction: this is the petrophysical variable to model.


VanaMa

Figure 2.2: p, Vp, </>, and Impedances histograms (7 wells).

Particular care was devoted to the choice of the variable to be simulated. This

variable must be correlated to the acoustic properties and must be well calibrated to

the petrophysical variables (porosities) to characterize. The acoustic impedance pVp

was retained as the variable to simulate because it is used directly in the definition

of the synthetic amplitude:

A multivariate analysis should check that impedance is the acoustic variable best

related to petrophysical properties of interest, such as porosity or clay content (see

table below) .2lnmany cases, densities are very stable and the seismic response is mostly controlled by sonic

Amplitude = wavelet ? d\n(pVp)


Matrix ofcorrelation (7 wells):

To build a reservoir model consistent with the 3D seismic format, the following

appoximations were considered:

A depth-to-time transformation, based on arough calibration of the true vertical

depth Ztvd versus the two-way travel time Ztwt, is applied to the vertical

coordinate 3.

Wells are relocated to the nearest nodes on the seismic grid.

Acoustic well-logs (sonic and density logs) are resampled from the original 15

cm sampling to the simulation discretization interval 4.

The stratigraphic transformation applied to the vertical coordinate of the well-

log data,

data. This remark would amount to adopt the following approximation:

Amplitude = wavelet ?——-~

wavelet ? -r^-pvP vP

However, such approximation should be carefully checked during the calibration phase.3Working directly in time voids the need for a posterior depth-to-time transformation after the

simulation.4An initial resampling is important because the spatial model of correlation must be derived at

the simulation scale which may not be the initial scale of the data.

Z— Zt\VT

—ZtWT at base reservoir + 100 7715,

Imp VP

0.86

VP 0.92 0.62

-0.63 -0.90 -0.35

Sh% 0.33 0.36 0.24 -0.81


amounts to take the base of the reservoir for reference level5, and to keep the

true thickness of the formation for each well6.

2.2 Variogram analysis

The purpose of this analysis is to model the prevailing spatial correlation (ranges,

anisotropics...) through a variogram model to be used as an input for the geostatis-

tical simulation. Since the random function model used is Gaussian, the model of

spatial correlation needed is that of the normal score transforms of the impedances.

The vertical density of well-log data after resampling is stillhigh. So the 7 wells over

the Eldorado zone allow building a reliable vertical variogram model.

Horizontal information density is typically sparse at early stages of production.

The average distance between two wells is one kilometer; hence, characterization of

horizontal correlations using only well-log data is very hazardous. Horizontal ranges

of correlation and anisotropy, were determined from information provided by seismic

data and geologic interpretation. Horizontal variograms were computed on the 3D

seismic data along several directions (see Figure 2.3). The main direction of conti-

nuity (NNW-SSE) shows a range of 30 units (i.e. 30 x 75m =2.25km), while in the

orthogonal direction the horizontal range is only 15 units 7. These findings confirm

STo5To be consistent with the seismic 3D gridof data, the reference level is put at the same arbitrarylevel: 100 ms.

6This transformation accounts for the main structural features of the field (dipping, erosion...).7Comparing seismic horizontal variograms for different planes (isochrones) does not reveal any

significant variability of the anisotropy directions and ratio.


the geological structural interpretation: a NNW-SSE rift was conjectured to control

the deposition history, and the continuity was expected to be higher along this strike

direction.

The analytical expression of the 3D variogram model retained is given below, and

graphed on Figure 2.4; the system of coordinates is based on the seismic grid (see

sketch below): the unit in x-East is a CDP interval (75m), the unit in y-North is a line

interval (75m) and z', as defined in the previous section, is the vertical coordinate.

This model is read as a single spherical structure with no nugget effect and respective

directional ranges 30, 15, and 25 distance units.

7(>4, XX)=Spherically + (§)*+ (g)*


E

5

0.000.

0.700.

0.000.

0.500.

0.400.

o.aoo.

o.aoo.0 lOC.

0.000.

Dlatanca Dlatanca

o.aoo.

0.700.

o.aoo.

o.soo.

0.400.

0.300.

0.200.

0.100.

0.000.

Dtatanca Otaanoa

Figure 2.3: Infering an horizontal variogram model from seismic data:The main direction NNW shows a range of 30 units, while in the orthogonal directionthe range is 15 units, revealing a significant anisotropy ratio of 0.5.


wriMMMkkWUJbM

y(Line)

Horizontal anisotropy ratio: 0.5Major direction: NNW

x(CDP)

Figure 2.4: Three dimensional experimental variograms and models. The models are

fitted on the experimental variograms of impedance normal scores for the vertical and

seismic data for the horizontal.

12000 I 1 I^ooo iawu

0.9600 / 05600' /"

~«• '

0.7200 / 0.7200 / 05000 /

0.4800 / 0.4800 / 0.6000 /

02400 / «« 7 0300° 7

OJOOO0 0i a 20.0 30.0 40.0 111 10.0 20.0 30.0 40.0 50.0 •»% 20.0 40.0 60.0 80.0 100.0

Section 3

Stochastic Simulation of Synthetic

Seismograms

3.1 Sequential Simulation Techniques

As stated in the introduction, conditional stochastic simulation was chosen to

produce realistic images which reproduce the spatial continuity of a prior model, in

addition to honoring the sparse well data.

General principle

Given n original data {z(xQ),a =1,...,n] i.c prior data, we want to generate L

joint realizations (z(*,-)w) of the N dependent random events Z{x{),...,Z(x^), say

at N nodes.

The sequential simulation requires the N-variate posterior distributions

16

P(Z(x J)<zj,j=1,...,N\(n)) =FN(xj,zj,j=1,...,N\(n))

SECTION 3. STOCHASTIC SIMULATIONOF SYNTHETIC SEISMOGRAMSI7

This multivariate conditional cumulative distribution function (ccdf ) can be decom-

posed into the product of (N-l) univariate conditional cdf's using Bayes' axiom for

conditional probability:

*

This relation using conditional cdf's is completely general and suffers no exceptions.

Hence an algorithm for sequential simulations can be deduced easily:

Sequential Simulation Algorithm:

0 - Define a random path through all nodes to be simulated Xj,j=1,...,N.

1 - Draw z{xy) from the marginal ccdf F1\n(x1,Zi\(n)).

2 - given the relization z(xi), derive the conditional cdf of Z(x2), F2\i+n, then

(Allpreviously simulated nodes become conditioning data)

N - Derive the conditional cdf of Z(xj^), ifyAT-i+n? anc^ draw the last value

The Gaussian Model

In theory, the algorithm requires knowledge of all univariate conditional distribu-

tions usually not directly accessible from actual data. The practical approach calls for

Fn = i7V|AT-l+n(^N,^Ar|(n + TV— 1)) * FN_i\N_

2+n ¦• • *

SECTION 3. STOCHASTIC SIMULATIONOF SYNTHETIC SEISMOGRAMSIB

some particular random function model. Among these models, Gaussian, Indicators,

Boolean, Markov, Fractal ..., the Gaussian model is the simplest and the easiest to

implement.

It consists in assuming for the N-variate cdf a multinormal-related distribution easy

to infer from prior data and such as all conditional cdf's required by the sequential

simulation algorithm are easily retrievable.

The parametric Gaussian approach is particularly congenial because all conditional

cdf's are Gaussian and fully determined by the mean and the variance[3, B].

Hence, at step 1, Fm_i+n(xi,zi\(n -f /— 1)), is fully characterized by:

From that ccdf we can draw z(xi) which becomes a conditioning data for the next

step /-f1. The only requirement is the single covariance C(h) of the normal score

transforms. This covariance is used in the general regression system or simple kriging

system (SK):

Although, we can not check the Gaussian hypothesis at the N-variate level, there

are various ways to check for bivariate normality. The most discriminatory check con-

sists of comparing the sample indicator semi-variogram and the theoretical bivariate

normal cdf P(Y(x) < yp,Y(x -f h) < yp) for different cut-offs.

The mean E{Z(xt )\(n + /-!)}= [Z{xi)]*SK

The variance Var{Z(xi)\(n-f / — 1)} = cr\K{xi)

n+l-1

a- xp) — C(xi-xp) a = 1,...,n + /-l

o=l

SECTION 3. STOCHASTIC SIMULATIONOF SYNTHETIC SEISMOGRAMSI9

3.2 Application to the Data Set

Simulation grid:

To allow comparison with the seismic data set, the horizontal simulation grid is

identified to the regular grid of the 3D seismic survey which includes 100 Lines x 30

CDP, see Figure 3.1. Choosing a vertical grid amounts to decide on the simulation

vertical resolution. The sampling rate of the seismic data is 4 ms which gives, for a

window of 108 ms, 28 samples. The original sampling rate of the well-data is about

15 cm. For simplicity, the vertical simulation lag retained is the seismic lag, i.e. an

interval of discretization of 4 ms 1.

Line 32 (2

Reference level100 ms

120 ms

Figure 3.1: Definition of the simulation grid: 30 CDP x 100 Lines x 108 ms.

!The choice of a vertical simulation lag must be related to the particular use of the final stochasticimage, a flow simulator for instance.

SECTION 3. STOCHASTIC SIMULATIONOF SYNTHETIC SEISMOGRAMS2O

3D Simulations:

The algorithm selected to generate the geostatistical simulations of impedance

based on well data, is the Sequential Gaussian Simulation algorithm as implemented

in sgsim (GSLIB Library[6]). Gaussian simulations are based on a single covariance

model and the conditioning data. The conditioning data, normal scores of acoustic

impedances at seven well locations, are relocated at the nearest nodes of the sim-

ulation grid to allow a much faster simulation process. Simple kriging is used to

determine the conditional cumulative distribution function at every location. Fig-

ure 3.2 presents the variogram model reproduction by the average variogram over the

ten 3D impedance stochastic realizations.

Figure 3.2: Model reproduction by the average variograms over 10 realizationsVariograms of simulated values (dotted line) v. input models (continuous line).

SECTION 3. STOCHASTIC SIMULATIONOF SYNTHETIC SEISMOGRAMS2I

3.3 Convolution of the Stochastic Images

The simulation phase provides equiprobable 3D realizations of impedance values

on a regular grid. The next step consists of convolving these simulated impedance

values into a synthetic image of the reservoir that can be compared with the actual

3D seismic data. First the parameters of the convolution are determined through a

calibration of seismic traces at well locations. Then the calibrated wavelet is applied

to the simulated impedances transformed in reflectivities.

Calibration phase:

The calibration consists of comparing the synthetic trace along the conditioning

wells (derived by convolution of the reflectivity sequences from well data) with the

actual collocated trace from the 3D seismic survey.

This is a critical phase, a fail/pass step, for the feasibility of the study. In the

eventuality that the calibration at well locations is poor, comparison of actual versus

simulated seismic traces cannot be expected to be good between the wells. The

calibration is expected to solve the following equation for each well:

— **'*Reflectivity T"

where SeiSTnie is the actual seismic trace and Reficctivity ls the reflectivity sequence from

well-log data. At every well location, a wavelet w is estimated, then an "average" of

these w is used to convolve the whole reservoir.

There are several ways to evaluate the wavelet that provides the best fit between

synthetic and actual seismograms. A traditional approach consists of building the

SECTION 3. STOCHASTIC SIMULATIONOF SYNTHETIC SEISMOGRAMS22

synthetic seismograms at welllocations with different a priori wavelets until a maxi-

mum correlation is obtained at each location. This method was used for the 7 wells of

the Eldorado zone. Different correlation measures were tested to calibrate synthetic

and actual seismograms. Other factors such as well location or the existence of static

shifts can significantly affect the calibration. The best relocation and the best shift for

each well were automatically determined. The average wavelet retained is a ricker2,

zero phase, 30Hz, normal polarity plus an average static shift of the simulated traces

of +4 ms over the entire zone, see Figure 3.3.

RickerS, 30Hz

Figure 3.3: Wavelet retained for convolution.

The following table provides the Pearson's linear correlation coefficients attained

for the seven wells using the above wavelet withor without static shift and relocation,

see Figure 3.4 for the synthetic and actual seismograms of the 7 wells. The calibration

is considered as satisfactory whenever all coefficients of correlation are higher than

0.65.


Calibration at well locations: correlation coefficients between synthetic and actual

seismograms (7 wells):

Convolution of the stochastic impedance images

Seismic reflection records the contrasts of impedances at all interfaces (reflectivity)

filtered by the signal sent. Therefore, before convolution, the impedances Z, must be

transformed into reflection coefficients:

This expression represents a discrete approximation for the logarithm of the impedance

derivative.

The entire simulated 3D grid of reflectivity values is convolved using the wavelet

and shift determined in the previous calibration phase. The convolution is made

trace by trace , i.e. with ID convolution, in the time domain. Figure 3.5 presents

Zi+i+ Z,

WELL w/ shift w/ shift w/o shift w/o shift

w/ reloc. w/o reloc. w/o reloc. w/ reloc.

#1 0.70 0.63 0.76 0.79

#2 0.73 0.47 0.43 0.66

#3 0.80 0.63 0.50 0.81

#4 0.90 0.78 0.39 0.57

#5 0.68 0.55 0.55 0.70

#6 0.74 0.74 0.62 0.64

#7 0.67 0.35 0.39 0.57


three realizations (synthetic seismograms) of a particular section of the reservoir. The

comparison withthe collocated actual seismic section leads to the following remarks:

Main features like the base of the reservoir (Time 100 ms) are well reproduced.

The variability of the realizations reflects the uncertainty model resulting from

the geostatistical simulation. The actual seismic section appears as a possible

The different realizations show different degree of similarity with the actual

image, but it would be difficult to select one realization that is globally better.

These findings call for an algorithm to automatically select the realization that

matches best the actual seismic line. Instead of a tedious selection between multiple

simulated 3D realizations, the approach proposed consists of selecting the simulated

traces one by one, retaining at each horizontal node the best simulated trace to condi-

tion further neighboring simulated traces. Such algorithm would drive the simulated

image to converge towards the original seismic data.

SECTION 3. STOCHASTIC SIMULATIONOF SYNTHETIC SEISMOGRAMS2S

_inn I... I.... I¦ ¦¦¦ I.... I.... I ¦¦ .^-4-T

'

20 30 40 50 60 70 80 90 100 110 120

Well #310050

0-50_inn i... i.... i.... i.... i.... i.... i i i

20 30 40 50 60 70 80 90 100 110 120

10050

0-50_inn i... i.... i.... i i i ¦ i i i i i

120 30 40 50 60 70 80 90 100 110 120

Well #6100

500

-50_Inn I... I.... I.... I.... I.... I. I . ¦ I I I I I

,I I I I I I

20 30 40 50 60 70 80 90 100 110 120

Wen #710050

0-50

_« nn i ,. , i , . . . ivr ¦ , i i i . . iIUU 20 30 40 50 60 70 80 90 100 110 120

Figure 3.4: Calibration at well locations: actual (dotted line) and synthetic seismo-grams (7 wells).

ell10050 V0

---50 A A

-100 20 30 40 50 GO 70 80 90 100 110 120

Well #5

SECTION 3. STOCHASTIC SIMULATIONOF SYNTHETIC SEISMOGRAMS26Actual Seismic Line 32

Figure 3.5: Line 32: Three synthetic realizations v. Actual seismic fine.

Section 4

Optimization

The proposed algorithm performs an iterative adjustment of the convolved sim-

ulated vertical sequences of impedances to the actual 3D seismic traces. Instead of

simulating at once the whole 3D grid of acoustic impedance, the impedances are

simulated trace by trace. Each simulated trace is convolved into a synthetic seis-

mogram and compared to the collocated actual trace. In case of an "acceptable"

match, that simulated trace is retained and the simulation proceeds to another node

on the horizontal seismic grid. Otherwise another impedance trace is drawn at that

same location. The final result is one single 3D "optimal realization" of impedance

according.

4.1 Optimization algorithm

A vertical sequence of nz simulated impedances is defined as a random vector Z con-

stituted of nz random variables. Z is simulated at every node (x, y) of the horizontal

seismic grid (CDP, Lines) according to the following algorithm:

27

SECTION 4. OPTIMIZATION 28

Node to be simulated

Best simulated trace

Maximum correlation wltn

Actual collocatedseismic trace

5 local realizations(Simulated impedances converted into syntnetlc amplitudes)

Figure 4.1: Local optimization process.

1. Define a random path through all nodes (x, y) to be simulated.

2. For node iat location (xt,y,-) perform a local optimization (see Figure 4.1):

Determine the vector of conditional cumulative distribution functions (ccdf)

by solving nz simple kriging systems sequentially along the vertical.

Draw Isim realizations (Z/a,m, jsim =1,...,Isim) from that ccdf.

After convolution, compare all Isim simulated traces to the collocated

actual seismic trace (matching function).

Retain the best trace which then becomes a conditioning data.

3. Go to the next node of the horizontal seismic grid.

The resulting realization is fine tuned by a "global optimization" process. A new

random path through the grid is defined and new simulated traces are retained only


iftheir fitis better than that achieved during the previous run.

The inputs of the algorithm are:

the conditioning wells relocated on the simulation grid (i.e. seismic grid).

the parameters of the convolution (wavelet and static shift) to build the syn-

thetic seismograms;

the 3D actual seismic amplitudes.

The main output is an optimal impedance image (3D grid of simulated values) to

be transformed into an image of the petrophysical variable of interest. Other out-

puts allow assessing the quality of the optimization achieved. Section by section, the

simulated synthetic images can be compared visually to the reference seismic data.

In addition, a 2D correlation map provides a quantitative measure of the correlation

attained at every location between the best synthetic and the actual seismic trace,

see Results: Figure 4.3 and Figure 4.4.

Aside from the parameters of the sequential Gaussian simulation algorithm (essen-

tiallya covariance model), the optimization program calls for specific parameters such

Isim: number of traces drawn at any location to select the best one (local

nsim: number of times all the nodes should be revisited along a new 2D random

path (global optimization).


The global optimization process protects the optimal image from biases that may be

associated to particular simulation paths. Note that ifIsim and nsim are set to 1,

the optimization is equivalent to a traditional Gaussian simulation with every trace

being simulated once, without any local selection process.

Validation:

The results shown hereafter indicate that the optimization algorithm succesfully

integrate the seismic data. However, to be valid it should also insures reproduction of

the input spatial correlation model of impedance normal scores, and consistency of the

optimal acoustic image with the original impedance well-data. Since deconvolution

does not give a unique solution, a matching synthetic trace could come from many

different impedance sequences. Therefore a selection based on similarity between syn-

thetic and actual seismic data does not necessarily ensure a realistic impedance image.

In the approach proposed, each impedance trace generated honors a prior model of

spatial covariance based on well data. Despite the built-in selection, the optimization

algorithm remains a sequential simulation process. Therefore it also ensures that ev-

ery 3D realization of impedance is consistent with the calibrated well-data. Another

faster optimization method, which could consists of applying independent small per-

turbations on the synthetic traces untilperfect match with the actual seismic traces,

would provide no control on the global covariance, thus may give spatially inconsis-

tent results (incompatible impedance sequences at close locations).

The global reproduction of the covariance model (see Figure 4.2) is expected to

31SECTION 4. OPTIMIZATION

Figure 4.2: Impedance variogram reproduction after final optimization (nsim=\o,

lsim=so).

vertically: each trace is simulated at once, therefore the optimization (selection)

process does not perturb the vertical variogram reproduction (the variogram of

the simulated values and the experimental variogram of the welldata (Figure

2.4) show the same departure from the model);

horizontally: because the 2D simulation path is random and because the impedance

horizontal variogram was modelled from the seismic data, the reproduction

should be reasonable.

With regard to program efficiency, the algorithm speed is of the same order as that

of the sequential Gaussian simulation algorithm (SGS, program sgsim of GSLIB). For

each trace (horizontal location), the vertical sequence of simple kriging systems is

solved only once, and then any number Isim of local realizations can be drawn from

the same set of ccdfs. As for the operations involved to build the synthetic amplitudes

and to compare them to the actual seismic data, their CPU requirement is relatively

unimportant.


4.2 Results

To analyse the performances of the optimization algorithm, a section (Line 32) was

extracted from the simulated 3D grid. With the parameters nsim and Isim set to 1,

the impedance simulated image is a straight realization from SGS. As Isim increases,

the image significantly converges towards the actual seismic reference image. For

each level of optimization the correlations between synthetic and seismic traces are

computed and presented below the sections, see Figure 4.3.

The following table provides a measure of the performance of the optimization

algorithm for that section: the average of the Pearson's linear correlation coefficients

between synthetic and actual seismic traces of Line 32 increases from 0.28 with no

optimization to 0.83 with lsim=loo local realizations.

Means and variances of correlation coefficients (Line 32, 30CDP):

To assess the quality of the optimization for a whole 3D realization, a 2D correlation

map is provided, see Figure 4.4.

nsim lsim Mean Variance 10 2 nsim lsim Mean Variance 10 2

0.28 13.5

10 0.68 1.50 0.70 1.86

50 0.78 0.86 20 0.86 0.67

100 0.83 0.70 100 0.86 0.46

150 0.84 0.50


4.3 The final porosity image

A single 3D acoustic image of the reservoir, constrained a posterioriby 3D seismic data

witha satisfactory correlation level, has been obtained. This optimal acoustic image

is transformed into the petrophysical parameter of interest, say porosity, through an

acoustic-pet rophysical calibration performed on well-data. The relation <f> = f(pVp )

proposed in this paper was empirically derived from simple multilinear regression

from well data at the log scale.

<t> = 0.6546 -0.0636/>Vp , with R2 = 60%.

This relation, established at the log scale, is used to transform the final simulated

image (see synthetic section of Figure 4.5) at the simulation scale into a porosity

image at the same scale (see Figure 4.6). A scatterplot of these simulated porosity

values versus the original porosity well-data is given in the same figure. The final

porosity section corresponding to the best synthetic image on the Line 32 example

appears very realistic. Such 3D porosity-image, produced at a scale defined by the

user, can directly be used for reservoir modelling and flow simulation.


cg3

10.75

0.5

LO

0.25

co3

!0.75

0.5L.L.O

0.25

0 t 4 0 0 10 1! U 10 10 00 «I 04 M SO SOear

0 I' '

III'

I'I'III

' ' 'I I I I I I I IIII L_<

12345678 0 10 11 12 13 14 15 16 17 18 IS 20 2122 3324252627 2828 30CDP

0 t 4 0 0 10 It 14 10 10 M 66 14 tt U SOCDP

q IIII III IIII I I'Ii I III1 IIII

—I—I—I—I—

l

12345678 0 10 1112 13 14 15 16 17 18 18 20 21222324 25262728 2030CDP

4 0 I 10 11 14 10 10 M It 14 10 10 SOcap

Actual Seismic Line 32

a

0 8 4 0 0 10 If 14 10 10 tO tt 14 10 tS 00CDP

Figure 4.3: Correlations between the simulated synthetic and the actual seismic Line32 for different levels of optimization.

nsim=l, lsim=l(Line 32) nsim=l, lsim=so (Line 32)

c.fi 1

5 0.75C 0.5

nsim=l, lsim=lso (Line 32) nsim=s, lsim=lo (Line 32)

c.fi 1

0 O«'SO 0.5


Eldorado* Wen locations

1.0

0.8

0.7

0.6

0.5

0.3

0.2

0.1

0.0

Figure 4.4: 2D correlation maps: nsim and Isim set to 1 (no optimization) resultin a poor level of similarity between synthetic and actual traces. The darker zonesindicate better correlations around the wells. With Isim = 10 and nsim = 1, thecorrelation map becomes darker indicating a better match, nsim = 10 and lsim=bOyield to very good correlations over the field.

Correlations(nsimm*l, lsim=l)

0.9

0 10 30 30CDP

Correlations(nsim=l, lsim=lo)

Correlations(nsim^lo, lsim»so)0.4


Optimal Image v. Actual Line 32

Figure 4.5: Superposition of the synthetic and actual Line 32 (note the excellentmatch).

0.30

02S

0.20

0.15

010

0.05

Origins!oomponQ

Figure 4.6: Final porosity image (Line 32 extracted from the 3D optimal realization):the boundaries of the reservoir are indicated by low-porosity zones. The lower westpart of the reservoir, featuring very high porosities, confirms to be richer.

Porosity Section (Line 32: nsim=lo,bim=so)

37

Section 5

Conclusion

This research has presented an approach to integrate seismic data into porosity

mapping: the acoustic image generated can be considered as an inversion of the 3D

seismic amplitudes honoring both well data and a prior model of spatial continuity.

The approach includes two steps: simulation of reservoir properties conditional to

hard well data using the sequential Gaussian algorithm and integration of3D seismic

amplitudes through an iterative selection of simulated impedance traces.

In practice, the results are found to be sensitive to the quality of the calibra-

tion at well locations as well as to the acoustic-petrophysical relation used to derive

the final porosity image. The results could be improved by making the convolution

and the acceptance/rejection criterion of a simulated trace dependent on the location

(by using the specific results of the calibration at the closest well).

The method does not aim at providing any measure of uncertainty, instead it

SECTION 5. CONCLUSION 38

yields a unique optimal image. However, the correlation map of synthetic and actual

seismic traces provides an indication of uncertainty: areas where high correlations

between simulated values and corresponding seismic data are obtained after few iter-

ations give confidence in the optimal image and in the prior calibration; areas where

correlations remain poor indicate seismic characters not fullyaccounted for,or locally

inconsistent with the well data.

39

Appendix A

Optimization Program: optim

The Optimization algorithm is implemented as amodification of the Sequential Gaussian

Simulation (SGS) program, sgsim, from GSLIB1 [6].

Program subroutines:

optimm.f an example driver program for optim

optim. inc an include file with maximum array dimensions

optim.f the subroutine optim

optim.par an example parameter file for optim.

The optimization algorithm is presented in section 4.1; however there are many imple-

mentation decisions that were not given. The first group relates to implementations specific

to the sequential Gaussian algorithm (SGS) as search strategy, relocation of data on grid

nodes, and type of kriging. For more information, the reader should consult the GSLIB

User's manual.

The second group is specific to the optimization algorithm and concerns primarily the

convolution of the simulated impedance traces into synthetic amplitudes traces, and the

definition of the seismic and simulation grids. In the present version, the simulation grid

1GSLIB is the name of a directory containing the geostatistics software developed at Stanford.

APPENDIX A. OPTIMIZATIONPROGRAM: OPTIM 40

is exactly the seismic grid, and, in particular, the vertical simulation lag is the seismic

sampling interval: 4ms. If, for any application, this simulation lag is considered not to

carry enough resolution, then the program can be easily customized. The comparison with

the actual seismic data after convolution of the impedance traces would, however, remain

limited by the resolution of the seismic data.

The convolution is a ID convolution performed in the time domain; the convolved

synthetic trace is truncated and shifted according to the calibration parameters. These

implementation decisions are described with the wavelet file description in Appendix B.


Parameter file:

The following parameters are required for optim (see also included file).

Optimization of 3DGaussian Sequential Simulationusing 3D seismic information

***a*»»»****a»»******aaa»*»»*»

START OF PARAMETERS:

veils. data

trans. out1 0.01 0.0

seismic. datavavelet.data1

calib.out1debug .outimpedance. out

synthampli.outcorrelation .out

0

15.00.0 0.0 0.0 1.0 1.020

\minimum value (missing values)

\o*transform the data, l*don't\Transformation table\Lover tail option and parameter\Upper tail option and parameter\lnput file: actual seismic data\lnput fiter file (with nfilt, shift)

\l*>perform calibration, o=don't\output File for Calibration\Debugging level: 1,2,3\output File for Debugging\output File for simulated impedances\output File for synthetic amplitudes\output File for 2D correlation map\number of local realizations (Isim)

\number of global simulations (nsim)

\Random number seed

\ny,ymn,ysiz\nz,zmn,zsiz\x,y and z block discretization\min, max data for simulation\o=two part search, l=da.ta-nodes\max per octant (0 -> not used)

\maximum search radius\sangl,sang2,sang3,sanisl ,sanis2\number simulated nodes to use\nst , nugget effect

datafl: the input data is a simplified Geo-EAS formatted file (see Appendix B).

icolx, icoly, icolz, icolvr, icolwt: the columns for x, y, and z coordinates, the

variable to simulate (acoustic impedance), and possibly, the declustering weights.

tmin: values less than tmin are ignored.

igauss: if set to 1 then the variable is already standard normal; otherwise transfor-

mation is required.

12 3 0 6-0.00010

232519710

30 1.0 1.0100 1.0 1.028 12.0 4.01 1 10 01

1 0.00

1 30.0 1.00

157.5 0.0 0.0 0.5 0.8

\Data File in GEOEAS format\column: x,y,z,vt,vr

\Kriging type (O=SK, 1*01)

\nx,xmn,xsiz

\it,aa,cc: STRUCTURE 1\angl ,ang2,ang3 ,anisl,anis2


transfl: output file for transformation table iftransformation is required (igauss=o).

ltail«ltpar: specify the backtransformation implementation in the lower tail of the

distribution. ltail=limplements linear interpolation to the minimum data; Itail=2

implements linear interpolation to the lower limit ltpar.

utail,utpar: specify the backtransformation implementation in the upper tail of the

distribution. utail=l implements linear interpolation to the minimum data; utail=2

implements linear interpolation to the lower limit utpar; utail=3 implements an

exponential interpolation; utail=4 implements hyperbolic model interpolation with

uj= utpar.

seismicfl: 3D seismic data set defined on the regular simulation grid (Geo-EAS

formatted file: see Appendix B).

waveletfl: wavelet for convolution. The three first lines are: a title, the length of

the filter, and the phase shift (see Appendix B).

calibflag: ifset to 1 the program perform a calibration at well locations (see section

0.0 anu Appenaix d).

calibfl: calibration output file when calibflag is set to 1.

idbg: an integer debugging level between 0 and 4 (informations from the SGS pro-

dbgfl: the file for the debugging output.

imped .outfl: the output 3D grid of simulated impedances is written on this file,

cycling fastest on z(Time), then x(CDPs), then y(Lines).

synthtl: the corresponding 3D grid of synthetic amplitudes is written on this file.

corrfl: the linear correlation coefficients between each synthetic and and actual seis-

mic trace is written on this file, cycling fastest x(CDPs), then y(Lines).


Isim: number of local realization (impedance traces simulated at each location (x,

y))-

nsim: number of global iterations.

seed: random seed number.

ktype: the kriging type (0 = simple kriging, 1 =ordinary kriging).

Nx, xmn, xsiz: definition of the grid system (x axis).

Ny, ymn, ysiz: definition of the grid system (y axis).

nz, zmn, zsiz: definition of the grid system (z axis).

xblock,y block, zblock: x,y, and z block discretization.

ndmin, ndmax: minimum and maximum of original data to consider to simulate a

grid node.

sstrat: ifset to 0, the data and previously simulated grid nodes are searched sep-

arately. Ifset to 1, the data are relocated to grid nodes and the parameters ndmin

and ndmax are not considered.

noct: the number of original data to use per octant.

radius: the search radius. This radius can be made anisotropic by the following

parameters:

sangl, sang 2, sang 3, sanisl, and sanis2: parameters defining the 3D anisotropy

of the search ellipsoid (see GSLIB User's manual [6] section II.3).

ncnode: the maximum number of previously simulated nodes to use for the simula-

tion of another node.

nst, cO: the number of variogram structures; the isotropic nugget effect.


For each nested structure, one must define the type of structure, the range parameter,

the variance contribution, and the 3D anisotropy (see GSLIB User's manual[6] section

II.3).

45

Appendix B

Inputs and Outputs files

Input file: well dataThe following file presents the acoustic impedances in a file ready to condition the

optimization:

the horizontal coordinates x, y, are common to the seismic grid (CDP, Line);

the vertical coordinate has been submitted to a depth-to-time transformation, and to

a stratigraphic transformation so that the base reservoir occurs at a constant time:

density and sonic logs were used to derive the acoustic impedances;

the acoustic impedances have been resampled at the lag of the vertical simulationgrid (4 ms);

This file has been used to calculate the experimental vertical variogram.

Conditioning Data (Eldorado Zone :7 sells)

6

X-EastY-lorthZ (Two-way Travel Time)Density logVelocity VpAcoustic Impedance

20.00 33.00 12.00 2.38 2.95 7.0220.00 33.00 16.00 2.37 3.02 7.16

1.00 99.00 52.00 2.31 3.12 7.231.00 99.00 56.00 2.28 3.13 7.211.00 99.00 60.00 2.30 3.11 7.231.00 99.00 64.00 2.27 3.15 7.30

APPENDIX B. INPUTS AND OUTPUTS FILES 46

Input file: seismic 3D data setA cube corresponding exactly to the simulation grid (see section 3.2), has been extracted

from the 3D seismic data set. As for the well data, the base reservoir horizon has been pickedon a workstation and the data transformed so that it occurs at a constant time: 100 ms.The 3D seismic grid is regular, the amplitudes data can then be presented in a one-columnfile with z cycling fastest, then x, and then y.

Line [1-100] , CDP[I-30] , Z(TMT) [12ms-120ms]1

Input file: waveletTo get the synthetic amplitudes from the simulated impedances, one must use a con-

volution filter v. This wavelet can be provided by the seismic survey history, or by adeconvolution performed on one well within the study area or close to, or we can simplyinput a first try training signal. To actually perform the calibration or to test a calibratedsignal on every wells, the calibration flag must be set to 1in the parameter file by the user.

Ricker2, 30Hz, formal Polarity20

\comments should not be included:\number of samples

-1 \shift:-4ms-7.62e-05-0.00065-0.00432-0.0214

Given an input wavelet, the calibration subroutine in the program:

convolves the impedance well logs;


matches each synthetic well trace with the actual trace from the seismic data set at

the exact welllocation, but also with the surrounding traces (the seismic coordinates

could be inexact, and a better match could be achieved at a neighboring location);

by moving up and down the synthetic seismograms, the subroutine accounts for static

shifts or uncertainty of the depth-to-time calibration;

the matching criterion used to compare synthetic and actual traces is a linear corre-

lation coefficients.

The results of this calibration phase (see output file below) should be used to modify thewavelet ifnecessary, or to relocate some wells on the 3D simulation grid. In any case, thecalibration step allows the user to check that his calibrated signal perform a fair correlationfor all the wells and according to the criterion used by the optimization program.

CALIBRATIOI FILE

Shift: -1Well f 1

Coord :( 19, 31), Corr: -0.3691Coord: ( 20, 31), Corr: -0.0689Coord :( 21, 31), Corr: -0.2221Coord :( 19, 32), Corr: 0.2917Coord: ( 20, 32), Corr: 0.3693Coord: ( 21, 32), Corr: 0.3318Coord :( 19, 33), Corr: -0.1931Coord: ( 20, 33), Corr: 0.2232Coord: ( 21, 33), Corr: -0.0434

Shift: 0Well t 1

Coord: ( 19, 31), Corr: -0.0926Coord: ( 20, 31), Corr: 0.3359Coord :( 21, 31), Corr: 0.0674Coord :( 19, 32), Corr: 0.6696Coord: ( 20, 32), Corr: 0.7048Coord: ( 21, 32), Corr: 0.6749Coord: ( 19, 33), Corr: 0.2642Coord: ( 20, 33), Corr: 0.6315Coord: ( 21, 33), Corr: 0.3995

Shift: 1

Well f 1

Coord: ( 19, 31), Corr: 0.2667Coord: ( 20, 31), Corr: 0.6405Coord: ( 21, 31), Corr: 0.4006Coord: ( 19, 32), Corr: 0.7737Coord: ( 20, 32), Corr: 0.7316Coord: ( 21, 32), Corr: 0.7498Coord :( 19, 33), Corr: 0.6295Coord: ( 20, 33), Corr: 0.7601Coord: ( 21, 33), Corr: 0.7021


Output file: 2D correlationsThe 2D correlation map provides a quantitative measure of the correlation attained at

every location between the best synthetic and the actual seismic trace (see section 4.3). Bycomparing this correlations with those attained with Isim set to 1 (no optimization), onecan quantify the effective integration of the seismic information. Futhermore, their spatialdistribution are important to control the performance of the optimization algorithm.

2D Correlation map1

Coefficients0.87880.88820.3434

0.61130.6451

0.69130.8249

Output file: synthetic amplitudesThe 3D grid of synthetic amplitudes is provided as an output file. It allows the user to

compare visually synthetic images with the corresponding actual images.

0PTIMIZATI0I:

1Synthetic Amplitudes

-1.6699-1.4370-3.340619.1648

63.677371.473013.2123

Output file: impedancesLast, the most important output is the simulated 3D grid of acoustic impedance (best

trace from Isim realizations at each horizontal location). This realization can be consideredas an inversion of the 3D seismic amplitudes honoring both well data and a prior model ofspatial continuity.

OPTIHIZATIOI1

Simulated Impedances7.04487.88796.8216

7.49008.56008.57257.8900

49

Bibliography

[1] F. Alabert, S. Thadani and A.G. Journel. An integrated geostatistical/pattern recogni-

tion technique for the characterization of reservoir variability. SEG Annual Meeting,

New Orleans 1987.

[2] F. Alabert. Stochastic Imaging ofSpatial Distributions Using Hard and Soft Informa-tion. Master's thesis, Stanford University, Stanford, CA, 1987.

[3] T. Anderson. Anintroduction to multivariate statistical analysis. John Wiley and Sons,

New York,NY, 1958.

[4] L.J. Bortoli, F. Alabert, A.Haas, and A.G.Journel. Constraining stochastic images to

seismic information. In Proc. of 4th International Geostatistical Congress, Portugal,m • *r\r\nTroia, 1992.

[5] M. deßuyl, S. Ullah, and T. Guidish. Seismic reservoir description: substantiation by

reservoir simulation. SPE paper 16781, presented at the 62nd Ann. SPE Tech. Conf.,

[6] C. Deutsch. and A.G. Journel. GSLIB: Geostatistical Software Library and User's

guide. Oxford University Press, New York, 1992, inpress.

[7] P. Doyen. Porosity from seismic data: A geostatistical approach. Geophysics, 53 (10),

pp 1263-1275, October 1988.

BIBLIOGRAPHY 50

[8] J. Gomez- Hernandez. A stochastic approach to the simulation of block conductivity

fields conditioned upon data measured at the short scale. PhD thesis, Stanford Univer-

sity, Stanford, CA, 1991.

[9] T.A. Hewett, U. Araktingi, W. Bashore, and T. Tran. Integration of seismic data

and well log data in reservoir modeling. In Report 5, Stanford Center for Reservoir

Forecasting, Stanford, CA, 1992.

[10] E. Isaaks and R. Srivastava. An introduction to applied geostatistics. Oxford University

Press, New York, 1989.

[11] A.G. Journel. Fundamentals of Geostatistics in Five Lessons. Short course in Geology,

vol.B, AGU publ.,Washington, D.C., 1989, 40p.

[12] A.G. Journel and C.J. Huijbegts. Mining Geostatistics. Academic Press, New York,

NY,1978.

[13] A.G. Journel, W. Xv, and T. Tran. Integrating seismic data in reservoir modeling. The

collocated cokriging approach. In Report 5, Stanford Center for Reservoir Forecasting,

Stanford, CA, 1992.

[14] A.G. Journel and H. Zhu. Integrating soft seismic data: Markov-Bayes updating, an

alternative to cokriging and traditional regression. In Report 3, Stanford Center for

Reservoir Forecasting, Stanford, CA, 1990.

[15] L. Lavergne and C. Willm. Inversion of seismograms and pseudo-velocity logs. Geo-

physical prospecting, v.ll,pages 231-250, 1977.

[16] A. Marechal. Kriging seismic data in presence of faults. In G. Verly et al., editors,

Geostatistics for natural resources characterization, pages 271-294, Reidel, Dordrecht,

Holland, 1984.

[17] K.Mosegaard and P.D. Vestergaard. A simulated annealing approach to seismic model

optimization with sparse prior information. Geophysical prospecting, v. 39, pages 599-

-611, 1991.

BIBLIOGRAPHY 51

[18] D.W. Oldenburg, T.Scheuer, and S. Levy. Recovery of acoustic impedance from reflec-tion seismograms. Geophysics, Volume 48, pp 1318-1337, 1983.

[19] W.J. Ostrander. Plane-wave reflection coefficients for gas sands at non-normal inci-

dence. Geophysics, Volume 49, Number 10, pp 1637-1648, October 1984.

[20] S.C. Key and S.C. Smithson. New approach to seismic-reflection event detection and

velocity determination. Geophysics, Volume 55 (8), pp 1057-1069, August 1990.

[21] R.E. Sheriff. Seismic Stratigraphy. IHRDC, Boston, 1980.

[22] H. Zhu. Modeling Mixture of Spatial Distributions with Integration ofSoft Data. PhD

thesis, Stanford University, Stanford, CA, 1991.

Documents

WITH SEISMIC INFORMATION - Stanford Earth · 2010-08-20 · Figure 2.1: Seismic grid and well locations-Typical seismic line. • Figure 2.2: p, V p,, and Impedances histograms