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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
11
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
16
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
(Conditioning data @ (32,32))
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
(Conditioning data @ (50,50))
Figure 16: ESRS Simulation at the coarsest grid
(Conditioning data @ (50,50))
a) Conditioning by rejection b) Conditioning by sequential simulation
Hard data
# Grids: 5
a) Conditioning by rejection b) Conditioning by sequential simulation
# 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
Realization 118 Realization 129
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).
Realization 0 Realization 12
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)
Conditioning by rejection (ESRS) Conditioning by sequential simulation (ESRS)
����������������������������
53
Figure 34: Comparison between Water Cuts of P90, P50 and P10 of 100 ESRS and Boolean Realizations
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)
Figure 36: Comparison between Water Cuts of P90, P50 and P10 of 100 Snesim and Boolean Realizations
I: 38,38; P: 50,50 (hard data)
P90
P10
P50
I: 38,38; P: 50,50 (hard data)
Conditioning by rejection (Snesim) Conditioning by sequential simulation (Snesim)
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)
Figure 39: Comparison between Water Cuts of P90, P50 and P10 of 100 Snesim and Boolean Realizations
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
�
Conditioning by rejection (Snesim) Conditioning by sequential simulation (Snesim)
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)
Figure 43: Comparison between Water Cuts of P90, P50 and P10 of 100 ESRS and Boolean Realizations
Injector is placed across the channel (for ESRS)
�
I: 63,50; P: 50,50 (hard data)
Conditioning by rejection (ESRS) Conditioning by sequential simulation (ESRS)
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.