Evaluating Data Conditioning Accuracy of MPS Algorithms ...pangea.stanford.edu/departments/ere/dropbox/scrf/... · complex geological heterogeneity within a high-resolution reservoir

Evaluating Data Conditioning Accuracy of MPS Algorithms and

the Impact on Flow Modeling

Indira Saripally↑ and Jef Caers↑

May 2008

ABSTRACT

This report aims at evaluating the modeling accuracy of Multi Point Statistics

(MPS) algorithms, namely, Snesim (Guardino and Srivastav, 1993 and Strebelle, 2002),

Real Time Post Processesing (RTPP) and Early Stage Resimulation (ESRS) (Suzuki and

Strebelle, 2006). In practice, Snesim is one of the most effective tools available to model

complex reservoir heterogeneity. Nevertheless, in some cases, it fails to produce

connected channels. The latter methods, RTPP and ESRS, were developed to overcome

the limitations of Snesim, thus, improving modeling accuracy.

In order to estimate the efficiency of the algorithms in terms of modeling

accuracy, the training patterns reproduction and hard data conditioning of each of the

algorithms has been compared. While training pattern reproduction is examined using

unconditional simulation, conditional simulations are employed to study hard data

conditioning.

It is observed that all the three algorithms introduce artifacts, such as lengthening

of channels and increased degree of uncertainty of finding sand, particularly when

↑ Stanford Center for Reservoir Forecasting

2

conditioned to well data. Flow simulation studies in the report quantitatively demonstrate

the impact of algorithm inaccuracies on flow response.

3

INTRODUCTION

Multi Point Statistics (MPS) is a practical and efficient geostatistical tool to model

complex geological heterogeneity within a high-resolution reservoir model. It simulates

the reservoir using patterns extracted from the training image, and conditioned to the well

data. The now state-of-the-art Snesim algorithm (Guardino and Srivastav, 1993 and

Strebelle, 2002) is an implementation of a pixel-based sequential simulation which

simulates the depositional facies on a Cartesian grid by visiting the grid node one at a

time along a random path. Due to the random path node visit in Snesim, the Markov

random field property (Daly 2004; Holden 2006) fails which results in anomalies

(disconnected channels) in the realizations. This limitation of producing discontinuous

channels is addressed with the so-called Real Time Post Processesing (RTPP) technique

(Suzuki and Strebelle, 2006). In contrast to Snesim, RTPP is based on the uniaxial fixed

path. Early Stage Resimulation (ESRS) is a further improvisation of the RTPP method

wherein the accuracy of pattern reproduction is enhanced through post-processesing

(Strebelle and Remy, 2004).

For the purpose of this study, modeling accuracy is primarily evaluated based on

reproduction of training patterns and well data conditioning. Unconditional simulations

are reasonable indicators of pattern reproducibility of an algorithm. Therefore,

unconstrained simulations are carried out with all three algorithms- Snesim, RTPP and

4

ESRS, and subjected to different training images. Patterns obtained in the simulated

realizations are compared with the training image patterns based on the estimate of the

number of geobodies reproduced, using a technique called geobody count.

However, besides reproduction of patterns from the training image, it is important

that the model is also constrained to well data since training images are quantitative

representation of the geological heterogeneity, not necessarily constrained to any well

data. Thus, conditional simulation is a crucial quantitative measure of modeling accuracy

for assessing the algorithms.

CHAPTER 1: BACKGROUND INFORMATION

PIXEL-BASED SEQUENTIAL SIMULATION ALGORITHMS

All the MPS algorithms discussed in this report are pixel-based sequential

simulations. In these simulations, the depositional facies are simulated on a Cartesian grid

by visiting the grid nodes one at a time. 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 MPS algorithms consist 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 conditional probability of occurrence of facies k at grid node ui is computed

as:

{ }( )

{ }

Prob facies at | cond. data in neighbourhood of

# facies at facies pattern cond. data event at # facies pattern cond. data event at

i i

i i

i

u k u

u k uu

=

= ∩ ≡=

≡ (1.1)

6

Training facies patterns that match the conditioning event are stored in a ‘search tree’.

The three algorithms: Snesim, RTPP and ESRS, differ in the sequence in which the

unsampled grid nodes are visited and in the way hard data conditioning is performed.

SNESIM

Snesim is the state-of-the-art MPS algorithm simulating complex geology as

described in the following steps1 (figure 1):

1. Scan the training image(s) to construct the search tree for the data template

nτ consisting of n vectors { }, 1,2,...,ih i n= such that the n locations iu h+ are

defined in n grid locations located closest to u .The n vectors are ordered as

increasing modulus.

2. Assign sample data to the closest grid node. Define a random path visiting every

unsampled grid node only once.

3. At each unsampled location retain the n nodal locations i iu u h= + , 1,2,..., ,i n=

of the template; only ( )'n n≤ locations are inferred by the data. Let 'nd be the data

event constituted by those 'n conditioning data. Retrieve from the search tree the

numbers ( )'k nc d of training 'nd -replicates for which the central value at u is equal

to sk, k=1,2,…,K. Estimate the probability distribution conditioned to 'nd .

1 Strebelle, 2000

7

4. Draw a simulated s-value for node u from the previous conditional probability

distribution.

5. Move to the next grid along a random path and repeat steps 3 and 4.

6. Loop until all grid nodes are simulated.

Figure 1: Diagrammatic representation of conventional MPS algorithm, Snesim

LIMITATIONS:

Randomly visiting grid nodes leads to failure of the Markov random field

property resulting in anomalies such as disconnected channels. Additionally, channel

��

U

U U

U

Simulation Grid Training Image

Search for matching patterns

Drop data until matching pattern

found

Draw simulated

value

Go to next grid using

RANDOM PATH

Prob(u in sand)=2/3 Prob(u in shale)=1/3

Update Simulation

CHANNEL SAND BACKGROUND MUD

Simulation Grid

8

discontinuities are also caused due to dropping of conditioning data as a consequence of

computing facies probabilities from a limited size training image.

REAL TIME POST PROCESSESING (RTPP)2

RTPP is a technique addressing the limitations of Snesim that cause discontinuous

channels. It is based on the observation that anomalies can be observed as soon as the

coarsest-scale grid (stage 1) which are then carried over to finer-scale grids (stages 2, 3,

etc). As opposed to the Snesim algorithm, in this method, unsampled grids in the coarsest

grid are visited along a uniaxial path. For the remaining stages, multi-grid Snesim

(random path) is followed. Thus, RTPP is a two stage process-

• 1st stage: RTPP (only on coarsest grid)

• 2nd stage ~: MPS

The RTPP algorithm also includes an image post-processesing step at the very end. The

following steps briefly describe the RTPP algorithm:

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

Step 2: Step back along the uniaxial path, and select the best combination of facies for

locations 0u and 1u (figure 2) that maximizes the number of conditioning data that would

actually be used (i.e. not dropped) to simulate these two locations. In the example shown

in figure 2, sands both at location 1u and at location 0u renders the optimal combination of

2 Suzuki and Strebelle, 2006

9

indicators of the number of conditioning data used for each possible combination of

facies at locations 0u and 1u (22 conditioning data used in total: 10 for location 1u and 12

for location 0u ). As a consequence, the facies at location 1u will be changed from mud to

sand.

Step 3: Step back the path once again, and select the best combination of facies

indicators for locations 0 ,u 1,u and 2 ,u again in terms of total number of conditioning data

used. In this example, sand at location 2 ,u sand at location 1u and sand at location 0u is the

optimal combination (36 conditioning data used in total: 12 for location 2 ,u 12 for

location 1u and 12 for location 0u ), thus the facies indicator at location 2u 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.

10

Figure 2: Diagrammatic representation of RTPP algorithm

In this method, unlike Snesim, when a hard data is encountered the simulation

walks back along the uniaxial path and finds the best combination of facies indicators that

not only honors the well data, but also maximizes local data conditioning. The

conditioning neighborhood is expanded such that the hard data is included (figure 3).

There are no dropped nodes in this method which reduces discontinuous channels.

However, discontinuities are still observed, especially at the boundaries due to fewer

conditioning data at the boundaries, known as the boundary effect. Poor conditioning at

the boundaries propagates to other grid nodes.

Figure 3: Conditioning neighborhood expansion in RTPP algorithm

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EARLY STAGE RESIMULATION (ESRS)3

ESRS is an improvement over RTPP incorporating the resimulation method

(Strebelle and Remy, 2004) for improving model accuracy by resimulating those nodes

which were previously simulated with very limited conditioning. Similar to the RTPP

method, coarse grid nodes are simulated using a uniaxial path. Resimulation is done after

the 1st stage of simulation which is then followed by conventional Snesim for the

remaining grids. Thus, the following steps define the ESRS algorithm:

� 1st

stage: RTPP (only on coarsest grid)

� 2nd

stage: MPS + post-processesing

� 3rd

stage~: MPS

POST PROCESSESING

In ESRS, the image is post-processesed through resimulation to reduce boundary

effects due to fewer conditioning data at the grid boundaries. Resimulation is done as

follows:

� On simulating a grid, mark simulation nodes with very few replicating events

(‘bad nodes’)

� Resimulate nodes at marked locations till the number of ‘bad nodes’ decreases

sufficiently. 3 Suzuki and Strebelle, 2006

12

This process is repeated several times, until the number of the nodes that need to

be resimulated stops decreasing.

INTEGRATION OF SEISMIC DATA

TAU MODEL:

In order to constrain the facies model to seismic data (secondary information),

Tau-model is used. For the events A, B and C, where A denotes the yet unobserved event

to be simulated, e.g. ,the true sand proportion; B denotes set of hard local data to be

reproduced exactly, e.g., the actual observed proportion of sand from wells; and C

denotes the covariate data, e.g., seismic data; increment in information, x, is measured

as:

,c

x ba

τ� �= � ��

where, x is the incremental information due to secondary information, expressed as:

( )( )

( )1 | ,

,| ,

P A B Cx

P A B C

−= (1.2)

a can be interpreted as the relative distance to the event A occurring, expressed as:

( )( )

( )1

,P A

aP A

−= ( )P A is the marginal probability,

b quantifies how much is not known about A knowing the information B, given by:

13

( )( )

( )1 |

,|

P A Bb

P A B

−= ( )|P A B is the simulated facies probability,

c quantifies how much is not known about A knowing the information C, given by:

( )( )

( )1 |

,|

P A Cc

P A C

−= ( )|P A C is the seismic data probability.

τ denotes the parameter of redundancy, i.e., dependency between the events B and C.

UPDATING THE ESRS ALGORITHM

In the ESRS algorithm, the cumulative probability distribution function is updated

to account for secondary information. The probability conditional to hard data at every

grid node is updated using eq (1.2) to account for the collocated soft datum value. This

updates the local probability distribution function conditional to hard data, before

drawing values in the simulation tree.

Figure 4 shows an example where seismic data is integrated in facies modeling.

The training image (250X250) is a two-facies fracture system. Synthetic seismic

probability map is generated by averaging the facies model obtained from the

conventional Snesim algorithm, using a moving window. Figure 4c shows the simulated

facies model (unconditional) without seismic data and figure 4d integrates seismic data

( 1τ = ). Some seismic features (encircled region in figure 4b) are absent in figure 4c.

14

Figure 4a) Training Image Figure 4b) Probability map from Snesim

Cosnesim, 4 multiple grids, 1τ =

Figure 4c) Facies model using ESRS (without seismic data)

Figure 4d) Facies model using ESRS (with seismic data)

Figure 4: Integration of seismic data in ESRS algorithm

CHAPTER 2: METHODOLOGY

The modeling accuracy of the MPS algorithms - Snesim, RTPP and ESRS, are

evaluated through a comparative study of performance of the algorithms in terms of the

pattern reproduction and well data conditioning. In this report, a study of the differences

in the algorithms due to difference in the sequence of node visits and hard data

conditioning is carried out. These dissimilarities are accounted for through comparison of

unconditional and conditional realizations obtained from each of the algorithms which

are also compared to the training image. Different training images are used for the study

in order to generalize the conclusions drawn about the modeling accuracy of the MPS

algorithms.

This project is divided into two phases:

• Unconditional simulation

This section assesses how well the training patterns are reproduced in the

simulated model. A reservoir is simulated with a given training image and all the three

algorithms, independently. The ‘best’ facies model from each of the algorithm is

compared to the training image based on geobody count wherein the number of

geobodies is estimated. For this purpose, all the objects in an image connected to each

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other are considered to be a single geobody. This definition, however, does not

distinguish between different facies. This study is done using different training images

that are generated using the object-based image generator-TIGENERATOR4 .

• Conditional Simulation

In order to capture the artifacts introduced by the algorithm generated due to

conditioning, two methods of conditioning are done:

a. Conditioning by rejection; and

b. Conditioning by hard data as implemented in sequential simulation.

E-types obtained from both methods are compared to assess the degree of certainty.

Study of flow simulation response quantitatively assesses the impact of conditioning at

and around the well location.

4 Maharaja, 2006

17

CHAPTER 3: UNCONDITIONAL FACIES MODELING

Unconditional facies simulation enables verification of the ability of MPS

algorithms to reproduce training patterns without any conditioning data which is indeed

the primary task of any MPS algorithm. This report presents unconditional facies models

generated with three different training images and simulated using Snesim, RTPP and

ESRS. As only categorical cases are considered for the study, conclusions are based on

comparison between numbers of geobodies reproduced (as defined in geobody count) in

the simulated realization and that in the training image. To do this, the training image and

simulation grids are taken to be the same size. Based on the sensitivity studies with

different modeling parameters, it is observed that simulations are most sensitive to the

number of multiple grids and template size.

EXAMPLES:

Example 1: Single Channels

A two facies channel-sand system with 30% sand and 70% mud training image

(250X250X1) is constructed (figure 5). Channels are oriented at 500 from North-South

18

axis and do not touch each other. These are moderately thick channels (13 grid cells) with

wavelength of 75 grids cells. These channels are not very sinuous with amplitude of only

8 grid cells.

Figure 5: Training Image of 2 facies single channel system

It is found that with a 48-node grid template and 6 multiple grids for Snesim and 5

multiple grids in case of RTPP and ESRS, the ‘best’ realizations are obtained (Figure 6).

These are the realizations which best reproduces training patterns and have similar

geobody count estimates. For the simple training image under consideration (figure 5), all

the algorithms reproduce training patterns reasonably well, in terms of the channel

geometry. However, the following differences can be observed in the simulated

realizations:

Training image

Channel sand Background mud

19

• In terms of reproducibility of the training image, uniaxial search path algorithms

(RTPP and ESRS) do not produce discontinuous channels unlike conventional

Snesim.

• Straightening of channels: Simulated channels are less sinuous than the training

image which results in lesser variability along the channel (compare figures 5 and

6). The impact of the reduction in variability can be observed in the flow

simulation (see Chapter 5).

• All three algorithms result in higher geobody count than the training image (figure

7).

20

Figure 6a) Snesim (unconditional) # multiple grids: 6 Template: 7X7

Figure 6b) RTPP (unconditional) # multiple grids: 5 Template: 7X7

Figure 6c) ESRS (unconditional) # multiple grids: 5 Template: 7X7

Anomaly

STRAIGHTENING OF CHANNELS �

Figure 6: Simulated facies model showing straightening of channel

21

Figure 7: Comparison of geobody count of simulated facies models

Example 2: Branched channels

The training image (250X250X1) under consideration is a two-facies system with

horizontal but branched channels of 30% sand proportion and 70% mud proportion.

Training Image # Geobodies:8

Snesim # Geobodies:9

RTPP # Geobodies:9

ESRS # Geobodies:9

22

Figure 8: Comparison of simulated facies from Snesim, RTPP and ESRS algorithms for example 2

Channel sand

Background mud

Training Image Grid size: 250X250

Snesim (unconditional) Grid size: 250X250 Multiple grids: 6 Template size: 7X7 # Geobodies: 6

Training Image Grid size: 250X250 # Geobodies: 4

ESRS (unconditional) Grid size: 250X250 Multiple grids: 5 Template size: 7X7�# Geobodies: 5

RTPP (unconditional) Grid size: 250X250 Multiple grids: 5 Template size: 7X7 # Geobodies: 7

Boundary effect

23

It can be observed that the simulated realizations produce more disconnected

channels than the training image (figure 8). Training image patterns are best reproduced

using the ESRS method (number of geobodies is five) because of post processesing

which results in fewer disconnections at the boundary. However, straightening of

channels persists in all the methods.

Example 3: Fracture system

In figure 9, a three-facies training image (250X250X1) represents fractures

comprising 30% of facies proportion and rest is background mud. Unlike examples 1 and

2, in this case, the geobodies are oriented in two orthogonal directions (horizontal and

vertical fractures).

Training Image

Snesim (unconditional) Grid size: 250X250 # Grids: 5 Template: 9X9

24

It can be observed that pattern reproduction is not accurate in this case which can

be attributed to the presence of geobodies oriented in orthogonal directions (figure 9).

Yet, RTPP simulation reproduces better training patterns than the other methods. Snesim

simulates more discontinuous fractures in both directions.

RTPP (unconditional) Grid size: 250X250 # Grids: 3 Template: 7X7

ESRS (unconditional) Grid size: 250X250 # Grids: 3 Template: 7X7

Figure 9: Comparison of simulated facies from Snesim, RTPP and ESRS algorithms for example 3

CHAPTER 4: HARD DATA CONDITIONING

So far it is seen that in unconditional simulation, uniaxial path methods (RTPP

and ESRS), indeed improve pure pattern reproduction but the aim of MPS facies

modeling is not to reproduce the training image exactly but to reproduce training patterns

conditioned to well data. Therefore, in this chapter, a single hard data point is used to

accurately evaluate the impact of conditioning at and around the well location. Also in

order to minimize boundary effects, simulation grids (100X100) are constructed smaller

than the training image (250X250). Models are simulated using the following two

methods of conditioning:

1. Conditioning by rejection

To begin with, an unconditional simulation is carried out. Then, all those

realizations which do not satisfy given condition(s) are rejected. Through this method,

artifacts in the model introduced by the algorithm due to conditioning are not

encountered because the conditional model is in fact selected from the set of

unconditional realizations. The following flow diagram (figure 10) shows an example

where 300 unconditional realizations are conditioned using the rejection method. Here all

those realizations which do not have sand at the grid node (50, 50) are rejected.

26

Figure 10: Flow diagram showing an example of conditioning by rejection

2. Conditioning with sequential simulation

This method concerns the conventional conditioning method where the facies

simulation is constrained to well data. The hard data value is simply frozen at the hard

data location.

It is interesting to observe that the two methods for conditioning lead to different

facies models. In fact, comparing the two methods bring forward the differences due to

constraining the simulation to hard data. These differences are attributed to the following

approximations introduced in the algorithm when a well data is encountered:

• In case of uniaxial path methods (RTPP and ESRS), the conditioning neighborhood is

expanded when a well is encountered. In unconditional simulation, however, only a

raster template path is used.

300

Unconditioned Realizations

SELECT Is node (50, 50) sand?

REJECT

Calculate E-type

YES

NO

27

• Moving well data to the closest grid node: This approximation has significant impact

when the number of multiple grids used is large. In the coarsest simulation grid, when

conditioning is done by rejection method, well data is not required to move, as

opposed to the case where conditioning is done using sequential simulation where any

well data not located on a coarse grid node is moved to the closest grid node (figure

11a,b). This approximation causes over constraining to well data which can be seen

as an elongated smearing around the well location in the E-type (figure 12).

Figure 11: Snesim Simulation at the coarsest grid

(Conditioning data @ (50,50))

Hard data

Conditioning data

Snesim coarse grid (6 grids)

a) Conditioning by rejection b) Conditioning by sequential simulation

HARD DATA ≠ GRID NODE

28

Figure 12: Effect of moving well data to closest grid node

As expected, when the well data falls on a coarse grid node, e.g., at node (32, 32),

a conditioning artifact does not occur (figure 14).

Figure 13: Snesim Simulation at the coarsest grid


Snesim conditioning by rejection (100X100)

Snesim hard data conditioning (100X100)

Hard data

Nearest node

HARD DATA == GRID NODE

b) Conditioned by sequential simulation a) Conditioning by rejection

29

Figure 14: E-type showing no artifacts when hard data matches grid node

Similar effects can be observed in the RTPP and ESRS methods due to moving of

well node but with smaller influence since only 5 multi-grids were used (figures 15 a,b

and figures 16 a,b, respectively). Generally, it is observed that the number of multiple

grids used in conventional Snesim is greater than (or equal to) the number used in

uniaxial methods in order to get similar pattern reproduction accuracy.

Snesim conditioning by rejection (100X100)

Snesim sequential simulation conditioning (100X100)

30

Figure 15: RTPP Simulation at the coarsest grid


Figure 16: ESRS Simulation at the coarsest grid



Hard data

# Grids: 5


# Grids: 5

31

COMPARISON OF E-TYPES

To quantify the differences observed due to various conditioning methods, a

reference case is constructed. The reference is a realization obtained from Boolean

method (using TIGENERATOR) which is then conditioned by rejection. For the cases

considered in this report, the conditioning hard data is at grid node (50, 50). Thus, for

conditioning by rejection, all the realizations which do not have sand at (50, 50) are

rejected. Below is a qualitative comparison by means of the E-types.

32

Figure 17: Reference E-type (Boolean)- Conditioning by rejection

Realization # 129 Realization # 11

Conditioning data: sand @ (50, 50) Total # realizations: 300 # Realizations accepted: 115 Grid: 100 X 100 X 1 Channel width: 13 grids Length: mean 1000

Conditioning data

33

Figure 18: E-type by Snesim conditioning by rejection

Realization 0

Total # realizations: 300 # Realizations accepted:113

Grid size: 100X100X1

Conditioning data: 50,50

Straightening of channels

Realization 11

34

Figure 19: E-type by Snesim conditioning by sequential simulation

Straightening of channels is observed when the E-types of the reference Boolean

model and the Snesim model conditioned by rejection are compared (figures 17, 18).

Relatively less sinuous channels are also observed in facies models conditioned by

sequential simulation (figures 17, 19).

Realization 40

Grid size: 100X100

Hard data @ 50,50

Realization 11

Straightening of channels��

35

Figure 20: E-type by RTPP conditioning by rejection

Realization 0 Realization 106

Conditioning data @ (50,50)

Total # realizations: 300 # Realizations accepted:115 Grid size: 100X100

36

Figure 21: E-type by RTPP conditioning by sequential simulation

The following observations are made from the E-types of the models generated

using RTPP method and the reference Boolean model (compare figures 17, 20 and 21):

• Straightening of channels is observed in the realizations conditioned using both

rejection and sequential simulations methods.

• Smearing is observed around the well due to the artifacts introduced in the RTPP

algorithm by moving the well data to the closest node in order to condition to the

hard data by sequential simulation.

Realization# 0 Realization# 11

Grid size: 100X100

Hard data@ (50,50)

37

Figure 22: E-type by ESRS conditioning by rejection


Conditioning data @ (50,50)

# Realizations: 300 # Realizations accepted: 107 Grid size: 100X100

38

Figure 23: E-type by ESRS conditioning by sequential simulation

Similar to the observations for the Snesim and RTPP models, smearing around the

well and relatively less sinuous channels are observed, when the reservoir is simulated

using ESRS method (figure 23).


Hard data @(50,50)

Grid size: 100X100

Straightening of channels

39

DEGREE OF CERTAINTY

A more quantitative comparison of the E-types can be done through a measure

termed “Degree of Certainty”. Figure 24 shows the reference and the Snesim conditioned

E-types. Encircled region in the figures show the decreasing degree of certainty of

finding sand in the region or increasing certainty of finding mud, from left to right.

E-type of reference E-type of Snesim facies (Conditioned by rejection)

Figure 24: Comparison of E-types showing decreasing certainty (from left to right)

The “Degree of Certainty”, ‘a’, is defined for each location in the grid as:

( )( )

( )( )

if E-type

1 ,

if E-type

.

P A

a P A B

P A

a P A B

<

= −

≥

=

( )P A is the marginal probability,

( )P A B is the simulated facies probability.

E-type of Snesim facies (Conditioned by sequential simulation)

40

b) Snesim conditioning by rejection

a) Reference: conditioning by rejection 100X100

41

c) Snesim conditioning by sequential simulation

d) RTPP conditioning by rejection

42

e) RTPP conditioning by sequential simulation

f) ESRS conditioning by rejection

43

It can be observed that the degree of certainty histogram of the reference Boolean

model (figure 25a) is skewed to the left which implies that the certainty of finding sand is

higher. On the contrary, histograms of the degree of certainty of the E-types of the

simulated models (figure 25b-g) are symmetrically bimodal which shows increase in

g) ESRS conditioning by sequential simulation

Figure 25: a) Certainty map of Reference (left); Histogram of reference certainty (right)

b) Certainty map of Snesim conditioned by rejection (left); Certainty histogram of Snesim conditioned by rejection (right)

c) Certainty map of Snesim conditioning by sequential simulation (left); Certainty histogram of Snesim conditioning by sequential simulation (right)

d) Certainty map of RTPP conditioned by rejection (left); Certainty histogram of RTPP conditioned by rejection (right)

e) Certainty map of RTPP conditioning by sequential simulation (left); Certainty histogram of RTPP conditioning by sequential simulation (right)

f) Certainty map of ESRS conditioned by rejection (left); Certainty histogram of ESRS conditioned by rejection (right)

g) Certainty map of ESRS conditioning by sequential simulation (left); Certainty histogram of ESRS conditioning by sequential simulation (right)

44

certainty of finding mud in the model, thus indicating decrease in degree of certainty of

finding sand.

45

CHAPTER 5: FLOW SIMULATION

The purpose of a flow simulation study is to assess what impact the method of

conditioning has on the flow response (water cut). The effect of different methods of

conditioning on the E-type is already seen in Chapter 4 and in this chapter the impact on

each facies realization is studied. For the flow simulation model, a 2D simulation grid of

size 100X100 is used. For simplicity, the reservoir to be simulated is assumed to have

constant porosity and permeability per facies. The training image used is a two facies

single channel system described in Chapter 3 (figure 5). For each of the 100 realizations

generated using the three algorithms and conditioning methods, a P10, P50 and P90 water

cuts are calculated and compared with the reference that is generated using the Boolean

method. Thus, the following 7 flow simulation models are compared:

1. Boolean (reference)

2. Snesim conditioning by rejection

3. Snesim conditioning by sequential simulation

4. RTPP conditioning by rejection

5. RTPP conditioning by sequential simulation

6. ESRS conditioning by rejection

7. ESRS conditioning by sequential simulation

There are two wells: the producer is located at the hard data location (node (50,50)) and

the injector is varied to study the following aspects:

46

ASPECT 1: DIFFERENCE BETWEEN CONDITIONING METHODS

Conditioning by rejection and conditioning by sequential simulation is different

largely because in the latter method hard data is shifted to the closest grid node and in

case of uniaxial path methods, conditioning neighborhood is expanded when a hard data

is encountered.

CASE 1: Moving well data to closest grid node

Shifting well node to the closest grid node in the coarse grid causes increased

radius of influence observed as an elongated smearing around the well in the E-type.

Thus, an injector is placed at the grid node (66, 65) such that it lies near the closest node

where the well data is shifted in the coarse grid, so that the impact of the shift may be

studied in the flow responses. Figures (26) and (27) show the E-types of the reservoirs

simulated using Snesim and the reference, respectively.

47

Figure 26: E-type of Snesim reservoir with Injector at (66,65) and Producer at (50,50)

Figure 27: E-type of reference reservoir with Injector at (66,65) and Producer at (50,50)

Injector (I): 66,65 Producer (P):50,50 (hard data)

Conditioning by rejection (Snesim) Conditioning by sequential simulation (Snesim)

Reference: Conditioning by rejection (Boolean)

48

Figure 28: Comparison between Water Cuts of P90, P50 and P10 of 100 Snesim and Boolean Realizations

Two important observations showing the impact of the inaccuracies in the Snesim

algorithm upon conditioning are as follows:

1. P10 and P50 of the simulated models are different from the reference.

2. Flow response of conditioning by sequential simulation is different from the

rejection method. In fact, the rejection method gives results closer to the

reference.

�P90

Rejection (Snesim)

Hard data (Snesim)

Reference (Boolean)

P10

I: 66,65 P:50,50 (hard data)

��

��

49

Figure 29: E-type of ESRS reservoir with Injector at (66,65) and Producer at (50,50)

Figure 29 shows the E-types of the simulated reservoir, using the ESRS

algorithm. As before the injector is placed at the grid node (66, 65) and the producer is

placed at the node (50, 50). Flow response in the ESRS simulation case is much more

consistent with the reference (figure 30) because the shift in the well location is much

smaller than that in the Snesim simulation. In the coarsest grid, the hard data node at (50,

50) is moved to the closest node at (48, 48). This small shift, in turn, results in smaller

difference in the two water cuts (figure 30).

I: 66,65 P:50,50 (hard data)

Conditioning by rejection (ESRS) Conditioning by sequential simulation (ESRS)

50

Figure 30: Comparison between Water Cuts of P90, P50 and P10 of 100 ESRS and Boolean Realizations

CASE 2: Effect of using a uniaxial path

Although using a uniaxial path improves reproduction of training patterns in

unconditional case, it results in deviation in flow response for some cases. It can be

observed that the smearing of the E-type around the well is elongated towards the left

corner of the grid which is essentially towards the direction in which the uniaxial path

moves (figure 31). To assess this situation, an injector is placed at the node (38, 38)

where the smearing effect around the well can be observed. The P50 water cuts from both

the conditioning methods are similar but much greater than the reference (figure 32) as a

result of using a uniaxial path, both with and without any constraining data.

I: 66,65 P:50,50 (hard data)

TIME

��

�P90

Rejection (ESRS)�

Hard Data (ESRS)

Reference (Boolean)

P10

51

Figure 31: E-type of RTPP reservoir with Injector at (38,38) and Producer at (50,50)

Figure 32: Comparison between Water Cuts of P90, P50 and P10 of 100 RTPP and Boolean Realizations

I: 38,38; P: 50,50 (hard data)

P90

�

�

�

�

P50

P10

Injector is placed at a previously simulated node (Uniaxial path)�

I: 38,38; P: 50,50 (hard data)

Conditioning by rejection (RTPP) Conditioning by sequential simulation (RTPP)

��

��

52

Similarly, in case of ESRS simulation, when the injector is placed at the node (38,

38) the P50 water cut of the simulated realizations is greater than the reference water cut

(figure 33).

On the contrary, with the injector placed at the node (38, 38), reservoirs simulated

using the Snesim method (figure 35) show flow responses similar to the reference water

cut (figure 36) as there are no uniaxial paths in the conventional MPS simulation causing

skewed elongated smearing around the well data location.

Figure 33: E-type of ESRS reservoir with Injector at (38,38) and Producer at (50,50)

I: 38,38; P: 50,50 (hard data)


��

53


I: 38,38; P: 50,50 (hard data)

P90

P50

P10

54

Figure 35: E-type of Snesim reservoir with Injector at (38,38) and Producer at (50,50)


I: 38,38; P: 50,50 (hard data)

P90

P10

P50

I: 38,38; P: 50,50 (hard data)


55

Figure 37: Comparison between P50 Water Cuts of 100 Snesim, RTPP, ESRS and Boolean Realizations

Thus, Case 2 is an example demonstrating the situation where flow response of

Snesim compares best to the reference Figure (37), despite RTPP and ESRS reproducing

better training patterns (Chapter 3). This case, in fact, shows the effect of the uniaxial

path on flow response.

ASPECT 2: STRAIGHTENING OF CHANNELS

Often, the simulated channels are less sinuous than the training image which

reduces the variability of properties along the ‘true’ (reference) channel orientation

resulting in reduced range of uncertainty (P50 – P10) in the simulated flow models.

Besides differences in the P50 water cuts due to the conditioning method employed,

figure (28) also shows higher P50 values for the simulated reservoir. Next, the injector is

Reject (ESRS)

Reference (Boolean) Reject (Snesim)

Reject (RTPP)

Sequential simulation (RTPP)

Sequential simulation (Snesim)

Reject (ESRS)

I: 38,38; P: 50,50 (hard data)

56

placed at a node (63, 50) across the channel where the variability is not affected by the

straightening of channels (figure 39).

Figure 38: E-type of Snesim reservoir with Injector at (63, 50) and Producer at (50,50)


P90

P50

P10

TIME

WATER CUT

Reference (Boolean)�

Sequential simulation Conditioning�(Snesim)�

Conditioning by rejection�

I: 63,50; P: 50,50 (hard data)

Injector is placed across the channel

�


57

Similarly, in case of reservoirs simulated using RTPP, the injector lies at a

location outside the channel when placed at a node (63, 50) (figure 40). Although the

range of uncertainty is similar to the reference case, the P50 water cut of the model

conditioned by sequential simulation deviates considerably after 5000 days (figure 41).

Figure 40: E-type of RTPP reservoir with Injector at (63, 50) and Producer at (50,50)

Injector is placed across the channel (for RTPP)

�

I: 63,50; P: 50,50 (hard data)

Conditioning by rejection (RTPP) Conditioning by sequential simulation (RTPP)

58

Figure 41: Comparison between Water Cuts of P90, P50 and P10 of 100 RTPP and Boolean Realizations

With the injector located at the node (63, 50) and producer at the node (50, 50), it

is observed that in the ESRS simulated reservoirs, difference between the P50 and P90

water cuts increases indicating increased variability in reservoir properties across the

channels which is actually in good agreement with the reference (figure 43). However,

sequential simulation conditioning shows larger deviation from reference.

P90

P50

P10

TIME

WATER CUT

Reference (Boolean)

Rejection (RTPP)

Sequential simulation (RTPP)

I: 63,50; P: 50,50 (hard data)

59

Figure 42: E-type of ESRS reservoir with Injector at (63, 50) and Producer at (50,50)


Injector is placed across the channel (for ESRS)

�

I: 63,50; P: 50,50 (hard data)


P90

P50

P10

TIME

Reference (Boolean)

Sequential simulation (ESRS)�Rejection (ESRS)

I: 63,50; P: 50,50 (hard data)

WATER CUT

60

Figure 44: Comparison between P50 Water Cuts of 100 Snesim, RTPP, ESRS and Boolean Realizations�

Thus, when the injector is placed at the node (63, 50) and the producer placed at

the node (50, 50), uncertainty in the water cut calculations increases. However, it can also

be observed that the P50 water cuts of the reservoir simulated using Snesim show better

agreement to the reference case than the RTPP and ESRS models, using sequential

conditioning method (figure 44).

I: 63,50; P: 50,50 (hard data)

Reject (Snesim)

Reference (Boolean)

Reject (ESRS) Reject (RTPP)

Sequential simulation (RTPP)�

Sequential simulation (ESRS)�

Sequential simulation (Snesim)�

61

CASE WHERE CONDITIONING METHOD HAS NO IMPACT

The simple two-facies training image (250X250X1) showing single horizontal

channels (figure 45) show no difference in the E-types of the simulated reservoirs when

conditioned using rejection method and using hard data as implemented in sequential

simulation method (figures 46 and 47). For RTPP and ESRS methods, as the channel

orientation is same in the direction as the uniaxial path, the effect of uniaxial path is not

observed. However, the impact of moving a well location to the closest grid node can still

be observed.

62

Figure 45: Training image with single horizontal channels

Figure 46: E-type by Snesim Simulation

Training Image: 250X250 Facies:2 Facies proportion: sand-40% : mud-60%

Facies modeling grid: 100X100 Hard data: (50,50)

Sand

Snesim conditioned by rejection

Conditioning Data @ (50,50)

Snesim conditioned by sequential simulation

63

Figure 47: E-type by RTPP simulation

Figure 48: E-type by ESRS Simulation

ESRS conditioned by rejection

RTPP conditioned by rejection

ESRS conditioned by sequential simulation

RTPP conditioned by sequential simulation

64

CHAPTER 6: CONCLUSION

In this report, we have addressed the issue of conditioning in reservoir modeling

using MPS algorithms. The conventional MPS algorithm, Snesim, is an efficient tool to

model complex reservoirs. However, it suffers from the drawback of introducing undue

discontinuities in the simulated reservoir. In order to overcome this disadvantage, two

other algorithms, Real Time Post Processesing and Early Stage Resimulation, methods

have been developed. As shown in this report, the latter methods indeed improve the

modeling accuracy, but only under unconditional simulation. It is demonstrated with

different training images that RTPP and ESRS methods reproduce the training image

better than conventional Snesim. Nevertheless, all the algorithms simulate straighter

channels than the reference model.

However, when the simulation is conditioned to well data, additional artifacts are

seen in simulated models. These artifacts result from the approximations/alterations that

were made in the algorithms to accommodate hard data. One such inaccuracy occurs

from moving the well data to the closest grid node when the well data does not coincide

with any grid node. This inaccuracy has a significant impact when the number of multiple

grids is large (e.g.in the Snesim simulation performed for this investigation, in which six

multiple grids are used). The second inaccuracy is pertinent to the uniaxial algorithms in

which the conditioning neighborhood is expanded when well data is encountered. This

65

effect is measured through the differences in the E-types of realizations obtained from

conditioning by rejection and sequential simulation conditioning.

Finally, a quantitative assessment of the impact of data conditioning is done

through a flow response study of the simulated reservoir models. It is seen that in some of

the cases, the Snesim algorithm generates models which have flow responses that are in

better agreement with the reference case; however, in some examples, the RTPP and

ESRS methods show better flow responses. These seemingly ambiguous findings do not

undermine the importance of any of the methods but, instead, emphasize the importance

of selecting the relevant algorithm for a particular scenario. It also emphasizes the need

of a more robust algorithm to improve model accuracy.

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