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GUSS14 - 21

Multiscale reservoir geological modeling

and advanced geostatistics

B. DOLIGEZ, M. LE RAVALEC, S. BOUQUET, M. ADELINET

IFP Energies Nouvelles

This paper has been selected for presentation for the 2014 Gussow Geosciences Conference. The authors of this material have been cleared by all interested

companies/employers/clients to authorize the Canadian Society of Petroleum Geologists (CSPG), to make this material available to the attendees of Gussow 2014

and online.

ABSTRACT

This presentation will discuss new methodologies and

workflows developed to generate geological models 1) that

look more realistic geologically speaking and 2) that respect

the well and seismic data characterizing the studied area.

Accounting simultaneously for these two constraints is

challenging as they behave the opposite way. The more

realistic the geological model, the more difficult the

integration of data.

A first powerful approach is based upon the non-

stationary plurigaussian simulation method. In this case, the

available seismic data make it possible to compute the 3D

probability distributions of facies proportions, which are then

used to truncate the Gaussian functions.

A second method is rooted in the Bayesian sequential

simulation. Recent developments have been proposed to

extend this method to media including distinct facies. We

suggest an improved variant to better account for the

resolution differences between sonic logs and seismic data.

This yields a more robust framework to integrate seismic

data.

A third innovative approach reconciles geostatistical

multipoint simulation with texture synthesis principles.

Geostatistical multipoint methods provide models, which

better reproduce complex geological features. However, they

still call for significant computation times. On the other hand,

texture synthesis has been developed for computer graphics:

it can help reduce computation time, but it does not account

for data. We then envision a hybrid multiscale algorithm with

improved computation performances and yet able to respect

data

INTRODUCTION

Numerical geological modeling classically uses different

geostatistical techniques, which face two conflicting

objectives: to make the model more geological from a

descriptive point of view and to make it consistent with all

available data. As is well known, the more realistic the model,

the more difficult the integration of data. Methods can be

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ranked from pixel-based (Sequential Gaussian, Truncated

Gaussian, Sequential Bayesian Simulations) to Object- or

Process-based models. The former make data conditioning

easy, but do not provide enough flexibility to reproduce

geological objects like channels. The latter result in models

that better reproduce geological concepts, but data

conditioning is then challenging.

Many works have been dealing with the impact of

geological heterogeneities on fluid flow. Various

methodologies have been developed to integrate seismic

data into geological models and to converge towards more

realistic images of the subsurface geological complexity

(Moulière et al., 1997; Yao and Journel, 1998; Doligez et al.,

2007; Emery, 2008; Le Ravalec and Da Veiga, 2011). Within

this framework, issues related to scale or resolution

differences between well log data and seismic data have to

be taken into account (Yao and Journel, 1998; Gilbert and

Joseph, 2000). Many approaches are based upon the

integration of 2D seismic map(s) and use the cross-

covariances computed between geological and seismic

properties (Behrens et al., 1996) .

Exploratory efforts are ongoing at IFPEN about the

development of geostatistical methods to simulate models

respecting constraints originating from seismic or from

genetic modeling to obtain more realistic geological

distributions of heterogeneities. This paper focuses on three

specific methods and workflows developed to generate

geological models with an improved geological flavor and

that respect the well and seismic data characterizing the

studied area.

The first approach is the plurigaussian simulation. It uses

a variogram and is thus restricted to the analysis of two-point

statistics. Despite this inherent limitation, the extension to a

non-stationary context through the computation of the 3D

probability distributions of facies proportions offers

numerous possibilities to use conceptual geological data and

seismic derived information, qualitatively or quantitatively,

depending on data.

The second method is based on the Bayesian sequential

simulation (Doyen et al., 1997), with a proposed improved

variant to better account for resolution differences between

sonic logs and seismic data.

The third innovative approach reconciles the well-known

geostatistical multipoint simulation concepts with texture

synthesis that has been developed in computer graphics. An

hybrid multiscale multipoint algorithm is then presented with

better computation performances and yet able to respect

data.

This exploratory project is still on development. Its final

objective aims to test and evaluate these alternative

approaches so that guidelines can be suggested for modeling.

NON-STATIONARY PLURIGAUSSIAN WORKFLOW

The principles behind the truncated Gaussian simulation

method have been published in several reference papers

(Journel and Isaaks, 1984; Matheron et al., 1987; Galli and

Beucher, 1997). The extension to plurigaussian simulations

(Armstrong et al., 2011) provides greater flexibility to

simulate more realistic geological environments. Two three-

dimensional Gaussian fields of values (correlated or not) are

generated and truncated using a 3D distribution of facies

proportions to end up with a 3D distribution of geological

facies.

The truncated Plurigaussian model is defined 1/ by the

matrices of covariances and cross-covariances, which fully

define the model of the Gaussian functions (zero mean and

unit variance) and 2/ by the method used to transform the

set of Gaussian functions into a unique direct facies function.

This is done using a partition (truncation rule) of the

plane defined by the two Gaussian random functions,

depending 1/ on the geological characteristics of the field

(the truncation rule is used as facies substitution diagram to

reproduce the sequential and spatial organization of the

sedimentary facies) and 2/ on the facies proportions, which

may vary over the whole domain. The 3D distribution of

proportions (also called matrix of proportions), used in the

case of proportions varying vertically and laterally, is

estimated from local proportions known at wells. It can

integrate other sources of information, qualitative or

quantitative depending on the correlations between facies

proportions at wells and additional external constraint

(derived from seismic or geological data).

Thanks to its flexibility and the large panel of possibilities

to account for additional geological or seismic data, this

method is now widely applied in petroleum industry for

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building geological models. It has been used for handling

various applications and geological environments of which a

highly fractured Iranian carbonate reservoir (de Galard et al.,

2005), a cretaceous turbidite environment (Albertao et al.,

2005), the distribution of diagenetic properties in a

siliciclastic reservoir (Pontiggia et al., 2010). Plurigaussian

simulations have been also used in the mining industry for

modeling deposits (Fontaine and Beucher, 2006; Carrasco et

al., 2007; Rondon, 2009 among others), as well as in

hydrogeology and environmental sciences (Mariethoz et al.,

2009; Cherubini et al., 2009).

There are indeed numerous techniques to integrate

geological or seismic information into the distribution of

facies proportions (Doligez et al., 1999a, 1999b, 2007, 2013;

Dubrule 2003; Doyen 2007). The basic approach entails the

use of well data and local vertical proportion curves (VPCs)

calculated at wells to estimate facies proportions throughout

the entire field without any other constraint. This is

traditionally performed with ordinary kriging.

A bit more refined technique calls for additional

information. This may be a two-dimensional map of

paleogeographic environments, which delineates regions

with very specific VPCs (Figure 1-a). These ones are computed

from the wells included in the target regions. The final grid of

facies proportions is obtained by kriging the local VPCs within

each area, and with a possible smoothing between the areas

(Labourdette et al., 2005; Hamon et al., 2014, Figure 1-b).

Figure 1 : a/map of paleoenvironment for the studied unit; b/matrix of proportions constrained with the map 1-a as

background ; c - d/ two levels in the reservoir with simulated geological facies using the matrix of proportions.

The impact of the non-stationarity of facies proportions

is illustrated by the two horizontal cross-sections extracted

from the final resulting simulation in Figure 1-c & d (Hamon

et al., 2014).

a

b

c

d

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Lithoseismic interpretation may provide 2D or 3D grids of

seismic facies (or packages of reflectors with similar seismic

characteristics). In this case, a statistical pattern recognition

approach based on discriminant analysis techniques, and

supervised with training samples or not is performed on

selected seismic attributes data. A second possibility to get

the grid of geological facies proportions constrained by

seismic consists in using the map of seismic facies as a

background to identify the areas associated to a given seismic

facies. The following step is dedicated to the computation of

the VPCs from the wells belonging to the defined regions. At

this stage, each seismic facies is related to the vertical

sequence of geological facies within each region. The 3D grid

of proportions is then estimated by kriging. It is considered

for each level of the grid and each facies in each region

(Beucher et al., 1999, Doligez et al., 1999a).

Other techniques call for the definition of facies

proportions throughout the reservoir and the use of a 2D or

3D constraint given in terms of proportions. This additional

information is expressed as a 2D map or a 3D grid populated

with mean lithofacies thickness or proportions They can be

derived either from stratigraphic modeling (Doligez et al.,

1999b) or from statistical calibration using seismic attributes

when a correlation exists between some reservoir properties

(for instance between impedance and porosity). The resulting

2D map or 3D grid is used as a constraint to estimate VPCs.

The idea behind is to write the kriging system with an

aggregation constraint relative to the sum of facies

proportions in each cell of the proportion grid (Moulière et

al., 1997).

Another original example of integration and specific

workflow to compute the 3D grid of facies proportions for

plurigaussian simulation was published by Nivlet et al. (2007)

and Lerat et al. (2007) to take the most from seismic data of

exceptional quality. The 3D grid of facies proportions was

directly computed from the 3D high resolution seismic data

accounting for scale differences between seismic and well

data in several steps: 1/ an electrofacies analysis led to the

definition of seven geological facies from well logs data; 2/ a

second supervised electrofacies analysis based upon well

impedance logs permitted to correctly discriminate six

geological facies. Therefore, two of the original facies were

merged into an “heterolothic facies”. This emphasized that

geological facies could be discriminated from Ip and Is well

logs; 3/ upscaling of the available well logs informed with the

six geological facies to go to the seismic scale, keeping the

most probable electrofacies; 4/ definition of the seismic

facies grid, supervised by a training database of 5x5 traces

extracted from areas surrounding well positions; 5/

geological calibration of the seismic facies grid as a

proportion matrix based upon the computation of the

geological facies proportions within each seismic facies. Last,

the truncated Gaussian method was applied to generate

realizations constrained to well data and geological facies

proportions. The simulation of multiple realizations made it

possible to evaluate the uncertainty in the spatial distribution

of facies.

Bayesian sequential simulation

The Bayesian sequential simulation (BSS) method was

originally introduced by Doyen and Boer (1996) for the

interpolation and extrapolation of data. Its purpose is the

simulation of several realizations of a given primary variable

conditionally to intensive measurements of a secondary

variable throughout the model space. The main BSS

components are recapped below. A first step consists in

building the joint probability density function (pdf) between

the two variables of interest given collocated measurements.

This joint probability is assumed to be spatially invariant. It

can be estimated from the cloud plot using the

nonparametric kernel density estimation method. The second

step focuses on the simulation process. A grid block is

randomly selected in the discretized model space for which

the value of the primary variable is unknown. Then, the prior

pdf is derived from simple kriging given the measured values

or the values simulated previously for the primary variable.

The value of the secondary variable attributed to the current

grid block is used to identify a 1-D slice through the joint pdf.

This yields the likelihood function. The product of the prior

pdf and the likelihood function then gives the posterior pdf.

Last, a value is drawn from this pdf and attributed to the

current grid block. Repeating this simulation step until

populating the entire grid yields a realization of the primary

variable. This algorithm was improved by Dubreuil-Boiclair et

al. (2011) who proposed to use a Gaussian Mixture Model

(GMM) to approximate the likelihood function at each

iteration. The combination of several normal distributions

makes it possible to reproduce multimodal behaviors.

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Figure 2 : a/ Area studied: block square centered at well 1. b/ Part of the stratigraphic section of the Fort Worth Basin,

Texas. (from Adelinet et al., 2013)

Figure 3: a/ P-wave impedances and porosities against time at well 1 (red: derived from logs, blue: derived from

seismic). The red data area collected in the Marble Falls. b/ seismic P-wave impedances for time 670 ms.

We investigate the potential of BSS to model spatial

porosity variations in a sub-domain of the Marble Falls in the

south of the Fort Worth Basin, Texas. The early Pennsylvanian

Marble Falls formation is a fractured carbonate reservoir. It is

about 100m thick and lies right above the Barnett shales. The

studied area is roughly 1.13 km2 and centered at a well,

called well 1 (Figure 2)

The data used include high-resolution log and low-

resolution seismic data (Figure 3): the porosity and P-wave

impedances estimated from logs at well 1, and a cube of

seismic P-wave impedances derived from the inversion of

seismic data acquired in 2005 and 2006 (Adelinet et al.,

2013). For simplicity, we focus on the slice extracted from the

impedance cube for time 670 ms.

In a first study, we applied BSS following the workflow

described by Dubreuil-Boiclair et al. (2011). The log data

provided the joint pdf between P impedances and porosities.

Then, we generated a porosity realization conditionally to the

seismic P impedances. The log P impedances used to

establish the joint pdf and the seismic P impedances that

provide the spatial constraint were considered the same way

despite the resolution difference. The joint pdf, as well as the

subsequent likelihoods, were approximated by a GMM with

two Gaussian densities (Figure 4-a). An example of porosity

realization simulated with this process is displayed in Figure

5-a. Its distribution is compared to the distribution of porosity

data in Figure 5-c. A divergence is pointed out: the simulated

realization does not properly reflect the significant

occurrence of small porosity values. This is actually related to

the fact that we equally treated seismic and log impedances.

The simulation process is driven by the seismic impedances

and the joint pdf between porosity and impedances is based

upon logs. There are less large impedance values for seismic

than for logs. On the other hand, large impedances and low

porosities characterize the first family of the GMM.

Therefore, the contribution of this first family is lessened,

which results in less low porosity values.

a

b

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Figure 4 : a/ scatter plot of porosity and log impedances superposed to the GMM imprint (with 2 kernels)

b/ scatter plot of log and seismic impedances superposed to the Gaussian density imprint.

Figure 5 : a/ porosity realizations simulated from basic BSS, b/ porosity realizations simulated from double BSS, c/ distribution of porosity data (red) compared with

distribution of the simulated porosity values 5-a d/ distribution of porosity data (red) compared with distribution of the simulated porosity values for 5-b.

a b

a

b

d

c

7

To avoid this pitfall and to account for resolution differences

between seismic and logs, we developed an improved

modeling workflow calling twice for BSS as suggested by

Ruggeri et al. (2013). The first call is used to simulate a log

impedance field conditionally to seismic impedances. At this

stage, the simulation grid is a sub gridded version of the

original seismic. Each grid block was split into 5×5 grid blocks.

The second call to BSS aims to simulate a porosity field

conditionally to the log impedance field obtained right

before. A second joint pdf is then required to relate log and

seismic impedances (Figure 4-b). This one was modeled by a

single Gaussian density. We performed a few tests to

investigate the potential of the double BSS approach. A

randomly drawn porosity realization is depicted in Figure 5-b

with its distribution in Figure 5-d. Clearly, there is now a

better agreement between this distribution and the one

calculated from the porosity data. The proposed workflow

leads to more reliable porosity realizations.

Multiscale multipoint simulation

Multiple-Point Statistics (MPS) simulation was

introduced as an alternate answer to the quest of more

realism into geological models (Guardiano and Srivastava,

1993). It belongs to the class of sequential non-parametric

pixel-based methods, but departs from the techniques

presented in the above sections as spatial variability is

inferred from multiple-point statistics instead of two-point

statistics. Heterogeneity is no longer characterized by a

variogram, but by a training image that is viewed as a

conceptual model of the expected heterogeneity. Multiple-

point statistics are then inferred from the training image and

integrated into the simulation process. MPS simulation yields

realizations that reflect the knowledge of the objects present

in the geological formation while still respecting

measurements at wells and auxiliary information like seismic.

On the other hand, the last decade also saw the emergence

of texture synthesis techniques (Efros and Leung, 1999; Wei

and Levoy, 2000) in computer graphics. These techniques aim

to produce large digital images from small digital sample

images by taking advantage of its structural content.

Although designed for different applications, MPS simulation

and texture synthesis techniques clearly share common

ideas. To date, MPS simulation is still tackling performance

issues. Referring to texture synthesis may help design

adequate strategies (Arpat and Caers, 2007; Straubhaar et al.,

2011; Tahmasebi et al., 2014): the training image can be

considered as a database of patterns instead of being used

from estimating probabilities, grid blocks can be populated

patch by patch instead of one by one, the path followed to

visit the entire grid can be regular instead of random, the

database can be organized to simplify its exploration, it can

be also partially and not fully investigated when looking for

an appropriate pattern. The interested reader can refer to Hu

and Chugunova (2008) and Mariethoz and Lefebvre (2014) for

comprehensive reviews. Following the same ideas, Gardet

and Le Ravalec (2014) developed a multiscale multipoint

algorithm. For simplicity, we restrict our attention to two

scales: the fine scale given by the training image and an

intermediate coarse scale. A preliminary step consists in

constructing a coarse training image by coarsening the

original training image. Then, a multiscale database is created

from the concatenation of the patterns extracted from both

the fine and coarse training images. The simulation process

involves two successive steps. First, a realization is simulated

at the coarse scale using the information provided by the

coarse training image. This is extremely fast and the resulting

coarse realization is viewed as secondary information in the

following step. Second, a realization is simulated at the fine

scale conditionally to the realization already simulated at the

coarse scale. This is performed using the information saved in

the multiscale database. The multiscale capability makes it

possible to capture large-scale objects with smaller

templates, which induces a significant decrease in the

computational overburden.

Three examples are presented hereafter. Two-

dimensional cases were preferred for illustrative purposes

only, but the above multiscale multipoint algorithm is also

able to handle three-dimensional simulation. The training

image of the first example (Figure 6-a) was extracted from a

satellite image. It shows a fraction of the Ganges delta with a

network of large and small channels. The coarsening process

based upon the arithmetic mean provided the coarse training

image in Figure 6-b. An interesting feature is the

disappearance of the smallest channels. We then moved to

the first simulation step and got the coarse realization in

Figure 6-c. This one reproduces pretty well the channels

described by the coarse training image. As expected, there

are only large channels. Then, this coarse realization is used

to constrain the simulation of the realization at the fine scale

(Figure 6-d).

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Figure 6 : a/ Fine scale training image, b/ coarse scale training image,

c/ Coarse realization. Coarse grids for both the training image and the realization: 66×66 pixels,

d/ Fine realization. Fine grids for both the training image and the realization: 200×200 pixels.

The large channels of the coarse realization are still

visible on the fine realization, but details were added all

around. We finally obtain a network of small and large

channels that looks like the one represented in the fine

training image.

The two following examples focus on fractured media.

The first one involves a training image created by mapping

fracture traces in a marble formation from the Germencik

field, Turkey (Jafari and Babadagli, 2010). The resulting

network is very complex and well connected. It comprises

large fractures with a prevailing diagonal orientation. There

are also many smaller fractures characterized by T

intersections (Figure 7-a). A realization simulated at the fine

scale is displayed in Figure 7-b. It shows a fracture network

very similar in appearance to the one portrayed by the

training image. The last example corresponds to systematic

joints. The training image, inspired by the Bloemendaal

reservoir model (Verscheure et al., 2012), exhibits 2 families

of small and large sub parallel joints (Figure 8-a). Again, the

resemblance between the simulated realizations (Figure 8-b)

and the training image is good.

a

b

d

c

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Figure 7 : a/ Training image with 100×107 pixels (from Jafari and Babadagli, 2010)

b/ Realization simulated at the fine scale with 200×200 pixels.

Figure 8 : a/ Training image with 200×200 pixels (from Verscheure et al., 2012),

b/ Realization simulated at the fine scale with 200×200 pixels.

Conclusion

The three methods and algorithms presented in this

paper can be used to generate geological models respecting

high resolution well data and low resolution seismic data. The

techniques considered to handle the required constraints are

different, but the final objective is the same: to obtain a

b

a

a

b

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result realistic enough in terms of properties and

heterogeneity distribution for fluid flow simulations. Each

approach has its own strong and weak points.

The non-stationary Plurigaussian method yields great

flexibility and a large panel of possibilities to account for

various geological or seismic data in the computation of the

matrix of facies proportions. On the other hand, the use of

this method and workflow implies the construction of a

geological facies model before focusing on the simulation of

petrophysical properties distribution. This can be viewed as a

benefice since it provides an additional control on the result,

or as a supplementary contribution to the global

uncertainties.

The Bayesian Sequential Simulation and proposed

variations have also the great advantage of flexibility and

simplicity in their implementations. The link between well

and seismic data is only introduced through probability laws.

However, the lack of geological control in the final results can

be considered as a weakness of this family of approaches

The proposed multiscale multipoint method is promising

in terms of geological realism: the simulated realizations can

reproduce very complex structures. However, the various

tests performed still stress the need for improved

computation performances. In addition, the simulation of

large objects remains a challenge.

The examples presented above illustrated that it is

possible to generate realistic geological 3D models while

integrating multiscale data. However, we may expect that in

real field studies, the quality of the results will strongly

depend on the available data, their quality, and the possible

links between the primary fine scale and secondary coarse

scale set of data. These relationships between data can be

qualitative or quantitative, with more or less uncertainty to

be taken into account.

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