1

Real-time Post-processing Method to Enhance

Multiple-point Statistics Simulation

Satomi Suzuki1 and Sebastien Strebelle2 1Department of Energy Resources Engineering, Stanford University 2Chevron Energy Technology Company

Abstract

Multiple-point statistics (MPS) simulation has gained increasing application to reservoir

modeling as an effective facies modeling tool, providing a way to simulate complex geological

features while honoring well and seismic data. The geological patterns to be simulated are provided

in the form of a training image. That training image should be large enough to display multiple

replicates of the desired geological patterns; otherwise, the simulation may fail to reproduce the

multiple-point statistics moments associated to those patterns, and anomalies may appear in the

simulated realizations (e.g. disconnection of sand channels). However, on the other side, the use of

large training images can make the simulation extremely computational time demanding.

This paper proposes a new method to improve the modeling accuracy of MPS simulation. That

method has been designed in the context of the multiple-grid sequential simulation approach

implemented in the MPS program snesim. The idea is that, if at any given node visited along the

simulation random path, no replicate of the local conditioning data event can be found in the training

image, then the nodes previously simulated are re-visited in order to maximize the conditioning of

both those previously simulated nodes and the currently visited node. That post-processing is applied

only during the first stage of the multiple-grid simulation approach, reducing computational cost, yet

improving pattern reproduction accuracy.

1. Introduction

For the last few years, multiple-point statistics (MPS) simulation (Guardiano and Srivastava, 1993;

Strebelle, 2002) has become one of the most favored facies modeling techniques in geostatistics

because of its ability to reproduce complex geological patterns (e.g. sinuous channels) that cannot be

modeled by two-point statistics moments (i.e. variograms). MPS simulation uses a “training image” to

describe the depositional facies patterns to be reproduced in the reservoir model conditional to well

2

data. The training image provides information about facies heterogeneity, i.e. facies geometries and

interactions among facies. Figure 1 gives an example of training image illustrating a fluvial channel

system. The training image is a purely conceptual geological model; it does not need to be constrained

to any specific reservoir data.

Unlike variogram-based simulation techniques such as Sequential Indicator Simulation, local

conditional probabilities in MPS simulation are directly borrowed from the training image without any

analytical formulation such as kriging. Because of the richness of the training image, the conditional

probabilities in MPS simulation are not limited anymore to two-point statistics, but they are estimated

from n-point statistics moments inferred from the training image, n being the number of available

conditioning data.

Figure 1: Example of training image for a fluvial channel system

One early implementation of MPS simulation, the snesim algorithm (Strebelle, 2002), uses a

pixel-based sequential simulation approach, which makes the conditioning to well and seismic data

much easier than in object-based modeling techniques (Viseur, 1999). The depositional facies are

simulated on a Cartesian grid by visiting the grid nodes one at a time along a random path. Sampled

grid nodes, i.e. nodes where depositional facies can be observed or interpreted from well data, are not

visited, but they are used as conditioning hard data for the simulation of the unsampled nodes. At each

unsampled grid node, the snesim algorithm consists of 1) looking for conditioning data in a local

neighborhood and retrieving from the training image all the facies patterns that match the data event

formed by those conditioning data, 2) computing the conditional probability of facies occurrence from

the central values of the retrieved training replicates, and 3) drawing a facies from the computed

probability.

The conditioning data include both hard data and previously simulated facies. The conditional

probability of occurrence of facies k at grid node ui is computed as:

Training ImageTraining Image

3

{ }{ }

{ }i

ii

ii

k

k

uuu

uu

atevent datacond.patternfacies#atevent datacond.patternfaciesatfacies#

ofodneighborhoindatacond.|atfaciesProb

≡≡∩=

=

= (1)

The set of training facies patterns that match the conditioning data event is denoted as {facies pattern

� cond. data event at ui}. In the actual implementation of the snesim algorithm, the training facies

patterns are stored before the simulation starts in a dynamic data structure called “search tree”, which

requires scanning the training image only once. Then during the simulation, the conditional

probabilities of type (1) are retrieved from the search tree in a very efficient way; they do not require

scanning the training image repeatedly (see Strebelle for details).

The MPS simulation can honor any order of multiple-point statistics in theory. However, in

practice, because of the limited size of the training image, hence the limited variation of patterns

provided by the training image, only a small amount of multiple-point statistics moments can be

actually inferred from the training image. Thus computing conditional facies probabilities from the

training image very often requires dropping conditioning data until at least one training facies pattern

matching the conditioning data event can be found. Strebelle and Remy (2004) have shown that the

more dropped conditioning data, the more anomalies (e.g. channel disconnections) are generated in

the simulated realization, see Figure 2.

Figure 2: (left) Facies model simulated by the snesim algorithm (250*250) using the training image

(250*250) depicted in Figure 1. (right) The grid nodes where conditioning data are dropped during the

MPS simulation. The circles associated by the arrows highlight the strong correlation between poor

conditioning and anomalies. (Courtesy of Strebelle and Remy)

This paper proposes a new method to enhance the pattern reproduction accuracy of MPS

simulation, reducing the number of anomalies by improving the data conditioning during the

simulation.

MPS modelMPS modelLocations of dropped

conditioning dataLocations of dropped

conditioning dataMPS modelMPS modelLocations of dropped

conditioning dataLocations of dropped

conditioning data

4

2. Method

2.1 Multi-grid Simulation and Unilateral Model

Our method is designed in the context of the multiple-grid approach implemented in snesim. That

approach consists of simulating nested increasingly finer-scale grids, as illustrated in Figure 3 (Tran,

1994; Strebelle, 2002). Nodes simulated in coarser-scale grids are used as conditioning data to

simulate nodes in finer-scale grids.

Figure 3: Example of multi-grid simulation. The facies model (250 * 250) is simulated using the

training image (250 * 250) in Figure 1. Anomalies are highlighted by circles.

As highlighted in Figure 3, anomalies (channel disconnections) can be observed as soon as in the

coarsest-scale grid (stage 1). They are then carried over finer-scale grids (stages 2, 3, and 4). This

example suggests that the key step of the multiple-grid simulation approach is the first stage: if facies

patterns are successfully reproduced without anomalies at the coarsest-scale grid stage, the final

simulated realization is expected to achieve satisfactory pattern reproduction.

The primary reason for the generation of anomalies is attributed to the failure of the Markov

Random Field (MRF) property (Daly, 2004; Holden, 2006):

( ) ( )iikiik ProbProb uuuu of odneighborho| except nodes all| = (2)

Daly (2004) showed that Eq. 2 does not hold if a random path is used for the sequential simulation,

resulting in the inaccurate reproduction of input statistics. Yet, property (2) is a critical hypothesis of

sequential simulation algorithms, which rely on the decomposition of a joint probability into a series

of conditional probabilities:

Stage 1 (8x8) Stage 2 (4x4) Stage 3 (2x2) Stage 4 (1x1)Stage 1 (8x8) Stage 2 (4x4) Stage 3 (2x2) Stage 4 (1x1)

5

)(Pr)|(Pr),,,|(Pr),,,|(Pr

),,,(Pr),,,|(Pr),,,|(Pr

),,,(Pr),,,|(Pr),,,(Pr

1122211121

2212211121

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AobAAobAAAAobAAAAob

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NNNN

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−−−

−−−−

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���

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(3)

where Aj, j = 1, …, N, denote dependent events (e.g. facies occurrences at grid nodes) that the modeler

wants to simulate by conditioning them jointly to some input statistics (e.g. statistics inferred from a

given training image). In MPS simulation, the series of conditional probabilities of Eq. (3) are inferred

from the training image.

If a restricted conditioning data search neighborhood is used to infer the conditional probabilities of

Eq. (3), as it is the case in snesim, then failure in Eq. (2) directly results in failure of Eq. (3), meaning

that the final realization may not be, in the rigorous sense, a sample of the joint probability Prob(A1,

A2, ….., AN), i.e. it may not honored all input statistics.

One solution to avoid this problem is the unilateral model or Markov Mesh model (Pickard, 1980;

Daly, 2004). The unilateral model consists of using a raster scan path that proceeds along one of the

axes of the model grid instead of a random path (Figure 4). As shown in Figure 4, the facies at

location ui is simulated conditionally to the closest previously simulated facies, which are all located

behind the current position ui.

Figure 4: Raster scan path and conditioning neighborhood of the unilateral model

In theory, Eq. 2 holds for the unilateral model only if the size of the simulation grid is infinite; in

practice, the reproduction of input statistics is limited by grid boundary effects (Daly, 2004). Grid

boundary effects are attributed to the simple fact that previously simulated facies data do not exist

outside the simulation grid. Thus the number of data that can be used for simulating grid boundary

nodes is necessarily smaller than for non-boundary nodes, resulting in the reproduction of smaller

order input statistics that may conflict with higher order input statistics.

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Other practical limitations of the implementation of a unilateral model in MPS simulation would

be the following:

1) The unilateral model should be used only at the first stage of the multiple-grid simulation

approach.

2) The unilateral model should not allow conditioning the simulation to well data.

Indeed, to ensure the Markov Random Field (MRF) property (Eq. 2) in the unilateral model, the

simulation of any node ui should be conditioned only to nodes located behind ui, which excludes well

data and previously simulated facies carried from previous stages of the multiple grid simulation

approach, located ahead of ui.

However, remember that, as illustrated in Figure 3, the critical time of MPS simulation in terms of

pattern reproduction is the first stage of the multiple-grid simulation approach. The later stages can

use a random path without losing much of the pattern reproduction accuracy achieved at the first

stage. Another favorable property of the unilateral model is the structured simulation path. As

described in Figure 4, in the unilateral model, the facies simulated at location ui-1 is only used for

simulating facies at ui. Similarly, the facies simulated at location ui-2 is only used for simulating facies

at ui-1 and ui, etc. Thus if the facies simulation at location ui fails due to the poor conditioning, one

could change the facies previously simulated at ui-1 (or the combination of facies at ui-2 and ui-1) such

that the conditioning at location ui is improved. In other words, whenever the facies simulation at a

particular node fails, the few previously simulated facies used for conditioning the simulation of that

node can be “repaired” by simply walking back the raster path, without affecting the rest of the

simulation.

2.2 Real-time Post-processing Method (RTPP)

Our method for improving the MPS simulation is based on the use of the unilateral model in the

first stage of the multiple-grid simulation process. The idea is to detect any failure of conditioning

during the simulation, and as soon as a failure is detected at a particular location (i.e. as soon as

conditioning data need to be dropped), to walk back the raster path and “repair” previously simulated

facies so that the number of data conditioning the node currently visited is maximized. Figure 5

illustrates the schematic steps of the method:

7

Figure 5: Schematic steps of real-time post-processing method. Blue figures attached to facies

categories indicate the number of data actually used for conditioning if the facies category is placed on the

grid

Step 1: Suppose that the conditioning failed at location 0.

Step 2: Step back the raster path, and select the best combination of facies for locations 0 and 1

that maximizes the number of conditioning data that would be actually used (i.e. not

dropped) to simulate these two locations. Figure 5 indicates the number of conditioning

data used for each possible combination of facies at locations 0 and 1. In this example,

sand at location 1 and sand at location 0 is the optimal combination (22 conditioning data

used in total: 10 for location 1 and 12 for location 0), thus the facies at location 1 will be

changed from mud to sand.

Step 3: Step back the path once again, and select the best combination of facies for locations 0, 1

and 2, again in terms of total number of conditioning data used. In this example, sand at

location 2, sand at location 1 and sand at location 0 is the optimal combination (36

conditioning data used in total: 12 for location 2, 12 for location 1 and 12 for location 0),

thus the facies at location 2 will be changed from mud to sand.

Step 4: Continue stepping back the path until changing the previously simulated facies does not

improve the total number of conditioning data anymore. Then, place the selected best

combination of facies on the model grid, and continue the simulation.

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In the actual implementation, the optimal combination of facies is evaluated based on the

“conditioning rate”, which is defined as the number of conditioning data used divided by the total

number of data in the neighborhood, in order to account for grid boundary effects.

The Real-time Post-processing (RTPP) technique can be used for conditioning the unilateral

model to well data. When the raster path arrives at a well location, the simulation walks back the

raster path and find the best combination of facies that not only honors the well data, but also

maximizes the local data conditioning. The conditioning neighborhood is expanded as depicted in

Figure 6 in case of the conditioning to well data. This neighborhood expansion results in the failure of

Markov Random Field (MRF) property (Eq. 2). However, the resulting inaccuracy of pattern

reproduction can be reduced by using the Real-time Post-processing (RTPP) method.

Figure 6: Neighborhood expansion for conditioning to well data

2.2 RTPP Combined with Re-simulation Method (RTPP+RS)

The proposed method (RTPP) is combined with the re-simulation (RS) method proposed by

Strebelle and Remy (2004) for further improvement. Strebelle and Remy identify the nodes simulated

with a very limited conditioning at the end of each stage of the multi-grid approach, and re-simulate

them. That process is repeated several times, until the number of the nodes that need to be re-

simulated stops decreasing. This method is coupled with the Real-time Post-processing (RTPP)

method, with some minor modifications, as shown in Figure 7.

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Figure 7: Flowchart of Real-time Post-processing

(RTPP) method combined with the re-

simulation (RS) method. MPS denotes the

regular MPS simulation by the snesim

algorithm. The ‘bad points’ denote the grid

nodes where the number of nearest facies

events honoring the training image is less

than a given criteria.

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As illustrated, the RTPP method is applied only to the first stage of the multiple-grid simulation

approach, the RS method to the second stage, and then regular MPS simulation is performed at the

later stages. At the end of the first stage, the simulated realization is scanned by a template as depicted

in Figure 8a to evaluate at each grid node the maximum number of data forming an event for which at

least one replicate can be found in the training image. The greater that maximum number of data, the

more accurate the reproduction of input (training) statistics is at the grid node. That evaluation is

performed with an almost negligible CPU cost using search trees in snesim (Strebelle, 2002).

If the maximum number of data is less than a given criteria, the corresponding grid node is marked as

a “bad node”. The same process is applied again to mark “bad nodes” at the end of the second stage of

the multiple-grid simulation approach. All the “bad nodes” are re-simulated at the end of that stage,

starting with the least bad nodes (in terms of reproduction accuracy of input statistics). That re-

simulation process is repeated several times, until the number of “bad points” stops decreasing (Figure

8b). Then, the simulation can proceed to the third stage.

Figure 8: (a) Scan of a realization by a template window, (b) Behavior of the number of “bad points”

during the repetitive re-simulation

3. Application Examples

The combined RTPP+RS technique is tested on the modeling of 2D and 3D channels. Figure 9

shows simulated channel realizations (2D) generated respectively by the original MPS simulation

technique, the unilateral model (unconditional simulation), and the proposed methods (RTPP and

RTPP+RS). Hereafter, “conditional simulation” and “unconditional simulation” respectively denote

the simulation with and without conditioning to well data. The same training image, displayed in

Figure 1, is used for all simulations. The well data (marked as dots on the realizations from

0

50

100

150

200

0 10 20 30 40

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# of

bad

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nts

(a) (b)

0

50

100

150

200

0 10 20 30 40

Iteration count

# of

bad

poi

nts

(a) (b)

10

conditional simulation) are synthetic; they were generated so that they conflict with the training

image.

Figure 9: Simulated channel realizations, 2D case, (a) conditional and (b) unconditional simulations

Figure 10: Comparison of (a) number of realizations without anomalies out of 100 simulations and (b) average CPU time per simulation, 2D case

Training Image (250 x 250)

MPS Unilateral Model RTPP RTPP+RS

(a) Unconditional Simulation (250x250)

Well data (32 wells)

(b) Conditional Simulation (250x250)

MPS RTPP RTPP+RS

Training Image (250 x 250)

MPS Unilateral Model RTPP RTPP+RS

(a) Unconditional Simulation (250x250)

Well data (32 wells)

(b) Conditional Simulation (250x250)

MPS RTPP RTPP+RS

Proposed methods

Unconditional Simulation

2

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MPS

Unilateral

RTPP

RTPP+RS

Conditional Simulation

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MPS

RTPP

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(a) # of Realizations w/o Anomalies (out of 100)

Conditional Simulation

15.9 17.2 18.0

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(b) Average CPU time (per realization)

Proposed methods

Proposed methods Proposed methods

MPS Uni-lateral

RTPP RTPP+RS

RTPP+RS

RTPP RTPP+RS

MPSMPS RTPP RTPP+RS

MPS RTPP

Proposed methods

Unconditional Simulation

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(a) # of Realizations w/o Anomalies (out of 100)

Conditional Simulation

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(b) Average CPU time (per realization)

Proposed methods

Proposed methods Proposed methods

MPS Uni-lateral

RTPP RTPP+RS

RTPP+RS

RTPP RTPP+RS

MPSMPS RTPP RTPP+RS

MPS RTPP

11

Figure 10 compares a) the number of realizations without anomalies obtained out of 100

simulations and b) the average CPU time per simulated realization, for the different techniques. The

RTPP or RTPP+RS methods help decrease significantly the number of anomalies. Enhanced MPS

modeling accuracy is achieved without CPU time increase despite the additional simulations required

by the RTPP and RS methods. Indeed, when using RTPP or RTPP+RS, geological patterns are better

reproduced at the early stages of the multiple-grid approach, thus less conditional data need to be

dropped at the later stages, saving considerable CPU time. In other words, the cost of optimization/re-

simulation at early stages is compensated by the reduction of CPU cost at later stages.

Figure 11: 3D Channel simulated realizations: (a) conditional and (b) unconditional. The connected channels are labeled by the same color. Note that MPS tends to simulate overly-connected channels.

(a) Unconditional Simulation (80x80x20)

RTPPMPS RTPP+RS

Training Image (80x80x20)

(b) Conditional Simulation (80x80x20)

RTPPMPS RTPP+RS

Volume of connected sand (TI)

Well Data (5 wells)

Vol. of connected sand Vol. of connected sand Vol. of connected sand

Vol. of connected sand Vol. of connected sand Vol. of connected sand

(a) Unconditional Simulation (80x80x20)

RTPPMPS RTPP+RS

Training Image (80x80x20)

(b) Conditional Simulation (80x80x20)

RTPPMPS RTPP+RS

Volume of connected sand (TI)

Well Data (5 wells)

Vol. of connected sand Vol. of connected sand Vol. of connected sand

Vol. of connected sand Vol. of connected sand Vol. of connected sand

12

Figure 12: Comparison of average CPU time per simulation, 3D case

The results of the 3D case are presented in Figure 11 along with the training image used for the

simulation. Connected channel sands are labeled by the same color. Again the well data used for the

conditional simulation case were generated so that they conflict with the training image, by sampling

data from a model quite different from the training image (Figure 11). The histograms attached to the

realizations and to the training image compare the volume of the connected sand bodies. The regular

MPS simulation is prone to overestimate the connectivity of the channel bodies compared to the

training image, in both the conditional and unconditional cases. In the unconditional case, both the

RTPP method and the RTPP+RS method improve pattern reproduction accuracy. In the conditional

case, the RTPP method does not achieve much improvement compared to regular MPS simulation,

except when combined with the re-simulation method (RTPP+RS). Figure 12 compares the average

CPU time for simulating one realization. Again, both methods allow enhancing modeling accuracy

without increasing CPU time.

4. Acknowledgement

The authors would like to thank Chevron Energy Technology Company for their financial support

for this work and the permission for publication.

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in.) MPS

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