1

Re-addressing the Missing Scale Using Edges

Lisa Stright and Jef Caers

Petroleum Engineering Department, Stanford University, Stanford, CA

lstright@pangea.stanford.edu

ABSTRACT

Current pixel-based techniques prove insufficient when attempting to model the missing

scale, in particular, when the missing scale is critical to reservoir flow, either as fine scale

barriers or flow paths. The classical geostatistical approach, i.e. working with calibrated

or effective properties, falls short of providing enough modeling flexibility. For example,

representation of continuous, fine scale, irregularly shaped flow barriers on a

geostatistical, or more importantly, simulation grid scale fails using current pixel-based

techniques. In order to solve this problem, we must start to think beyond pixels, yet not

resort to the alternative: object models, as they are too specific and difficult to condition,

and are usually converted to pixels before entering a flow simulator anyway.

This paper presents a proposal for using edge properties in combination with multiple-

point geostatistics to represent fine scale flow barriers, although the idea of using "edges"

can be extended to many other geological phenomena. An edge property is a continuous

or categorical value that is associated with the cell face, rather than the cell center. The

first challenge in using edge properties is to generate an edge training image. Once the

training image is generated, it is then coupled with the pixel-based property in SIMPAT

to generate multiple realizations of the reservoir that include the missing scale.

One of the great advantages of modeling with edges is their direct link with

transmissibility in flow simulators, hence their appeal in addressing the missing scale as

an added dimension to effective property modeling. As a matter of fact, any

geostatistical model of permeability needs to be converted to a transmissibility

(essentially an edge model) before flow simulation can be applied. The goal of this work

2

is to take advantage of the transmissibility modifier term that is included in the

transmissibility calculations to adapt the edge model and perform a coupled geostatistical

simulation with the edge and effective block values. This idea potentially opens a new

world of modeling complex geological features efficiently at the scale that they are

relevant.

1 Introduction

Reservoir stratigraphy is dominated by multi-scale structures that may impact the flow

and recovery of reservoir fluids. One of the major challenges in reservoir modeling and

simulation lies in our ability to capture the information at these various scales when it

impacts the recovery prediction of a reservoir model. Figure 1 illustrates the large

disparity of scale at which core, well-log, seismic, and production data inform the

reservoir. The scale of core and well-log data is easily five orders of magnitude smaller

than a geocellular model which is significantly more than the two orders of magnitude

change when a high resolution geocellular model is upscaled to a flow simulation model.

The core data is at a much finer scale than what can actually be captured in a single pixel

of a geocellular or simulation model.

Often the lithology, porosity or permeability properties of an individual core are ascribed

to the entire grid cell, thereby removing observed fine scale features. This implicit

assumption, whereby the heterogeneity within that grid cell is ignored and an explicit

upscaling is performed, is termed the missing scale problem.

3

Figure 1 The Missing Scale Problem

Challenges in Modeling the Missing Scale

It is often sufficient to represent these small scale heterogeneities as effective properties

at the grid block scale. However, when the small scale heterogeneity is a controlling

factor in the flow behavior of the reservoir, effective property representation is no longer

sufficient. This occurs when geological features are thin, irregularly shaped, and control

reservoir flow and recovery. Additionally, this can happen when features are not easily

described by the orthogonal gridding that is compatible with reservoir simulation.

An example geological environment where of the missing scale is likely to occur is

shown in the Figure 2. Figure 2 shows a detailed turbidite channel system containing

approximately 2 million cells. (Data courtesy of Shell). Channels are lined with

continuous shale drapes. These shale drapes are flow barriers that compartmentalize the

reservoir. If they are not included in the simulation model, predictions would severely

over estimate connectivity and subsequent recovery. Therefore, it is not only the

presence of the shale drape that is important but the continuity as a flow barrier that is

critical to describing the reservoir flow. However, the shale thickness could be on the

Scale of Property (m)

App

roxi

mat

e N

umbe

r of

Grid

Cel

ls

Vol

ume

Sup

port

/Res

olut

ion

109

108

107

106

105

10-2 10-1 100 101 102

100

101

106

107

108

GeocellularModel

SimulationModel

Cores

Logs

“Missing Scale”

Seismic

Production

Scale of Property (m)

App

roxi

mat

e N

umbe

r of

Grid

Cel

ls

Vol

ume

Sup

port

/Res

olut

ion

109

108

107

106

105

109

108

107

106

105

10-2 10-1 100 101 102

100

101

106

107

108

100

101

106

107

108

GeocellularModel

SimulationModel

Cores

Logs

“Missing Scale”

Seismic

Production

4

order of centimeters, which again, is orders of magnitude smaller than a gridblock that is

normally meters or 10s of meters, hence a problem presents itself.

Figure 2 Detailed geologic model containing fine scale heterogeneities (Data courtesy of Shell)

Current pixel-based techniques such as 2-point sequential indicator simulation or

multiple-point geostatistics could potentially predict the correct channel location and

large scale connectivity; however, they would prove insufficient when attempting to

model the missing scale, in this example, the shale barrier. These classical geostatistical

approaches work well with calibrated or effective properties, but they do not provide

enough modeling flexibility because they model the missing scale by directly assigning

properties to grid blocks. The following sections will review some of the current

approaches and discuss why they are not able to capture the effect of the missing scale in

a simulation model. Then a proposed approach for a coupled geostatistical modeling

method will be presented in an attempt to compensate where the other methods fall short.

Nx = 100

Nx = 100

Nz = 200

Shale Drapes

Nx = 100

Nx = 100

Nz = 200

Nx = 100

Nx = 100

Nz = 200

Shale Drapes

5

Pixel-Based Methods

The details of the channel geometry and fine scale flow barriers in the model presented in

Figure 2 can potentially be captured with a high resolution pixel-based model as shown in

the left image in Figure 3.

Figure 3 The challenge of pixel-based techniques in modeling fine scale flow barriers. High resolution model (left) that captures the character of the barriers

and upscaled model (right) wherein the barrier is compromised

The fluid flow in the channel (yellow) is isolated from the background facies (white) by

the continuous channel drape (black) based on a traditional 5-point scheme in a flow

simulator. Therefore, at this very high resolution, the pixels are able to resolve and

represent the missing scale because the barrier inhibits flow out of the channel in the

orthogonal grid directions. However, the final goal of most reservoir modeling is flow

prediction and high resolution models require upscaling to make flow simulation feasible.

For example, the right side of Figure 3 shows the same fine scale model once it has been

upscaled. In the upscaled model, the integrity of the shale barrier is compromised

because it no longer separated the channel facies from the background facies. In this

example, the resulting flow simulation would only be impacted if the background facies

is permeable. Another compromise made during upscaling is that the shale proportion

dramatically increased from 8.5% to 30% in attempting to maintain the sealing properties

of the barrier. This increase in shale proportion would coincide with a significant

decrease in pay volumes in place from the high resolution to the upscaled model.

6

Conformable Gridding

Another option would be to use a conformable grid to explicitly model the location and

the thickness of the channel and the channel drape. With this modeling approach the thin

shale layer could be represented with surfaces and the cells contained between the two

surfaces created as no flow barriers during simulation either with transmissibility

modifiers or zero vertical permeability values.

Figure 4 Conformably gridded model showing the distortion of cells in an attempt to capture definition of multiple channels and channel drapes

There are two problems with this approach. First, in addition to the subsurface geology

being complex, our knowledge of the geology is uncertain due to the limited sampling of

the reservoir. This approach models the uncertain location of the geology, i.e. channel

locations, with a fixed grid. During history matching of the reservoir model, including

the position of shale drapes needs to be perturbed, which would mean: changing the

conformable grid automatically. Currently, this automatic gridding is not yet a realistic

option. Secondly, as the number of channels and channel barriers increase the

conformable grid becomes increasingly complex with variable cell sizes and shapes

7

which may lead to convergence issues during simulation. While Cartesian grids may be

over simplistic, they are robust, a feature that is useful when automating history

matching.

Boolean (Object-Based Methods)

Boolean methods are able to model the connectivity and geometry of complex reservoir

features. However, models built with Boolean methods eventually required conversion to

a cell-based model for flow simulation with the risk of losing the fine scale detail.

Additionally, Boolean methods are limited by their inability to integrate primary (well

logs and cores as hard data) and secondary data (seismic information as soft data and

production data).

Current Workflow Practices

Other approaches applied to modeling the missing scale manifest during the history

matching process wherein the static geologic model is upscaled and then modified to

match the measured reservoir production and pressure data at the well over time. This

approach generally consists of altering relative permeability, Kv/Kh ratios, and/or utilizing

pore volume and transmissibility multipliers. These properties are perturbed manually

until a match is reached.

The limitations and compromises made during a typical history matching approach can

be illustrated by a simple example. Consider a scenario where the simulation model

predicts too much communication between two wells. A standard approach would be to

use a transmissibility multiplier at a grid face between the two wells to reduce the

transmissibility, i.e. communication between wells. The transmissibility reduces the flow

between cells or a set of cells in a specified direction (x-, y- or z-directions). While this

may improve the match, the barrier is usually inserted rather arbitrarily without any

geologic consistency with the original high resolution reservoir model. Furthermore, a

model that is not geologically consistent is likely not to predict accurately future reservoir

performance.

8

Proposed Approach

In order to solve this problem, we must start to think beyond pixels, Boolean methods,

conformable gridding and post-geologic modeling history matching fixes. This paper

presents a proposal for using edge properties in combination with multiple-point

geostatistics to represent fine scale flow barriers, although the idea of using "edges" can

be extended to many other geological phenomena. This approach utilizes the fact that

block effective permeabilities are converted to transmissibilities in a reservoir simulator

to capture the flow between grid cells.

An example of such thinking is presented in Voelker (2005) who modeled thin, highly

conductive fracture networks that behaved like flow conduits, also called super-k

permeability, and spanned large portions of the reservoir. Voelker proposed using a well

model to include the dominating effects of the fracture networks. The well model is a set

of connections in hydrostatic equilibrium connected via transmissibilities. The

conventional well model is represented simply as a set, Sw, of connection

transmissibilities, Twj, in hydrostatic equilibrium,

{ }fluidwj

www TTTS ρ:,...,, 21=

where w is a fracture name, and j is the block being connected.

The condition that the connections are in hydrostatic equilibrium is represented by ρfluid,

the density of the well fluid. Voelker found that modeling the fracture networks in this

manner, as a set Sw of transmissibilities, decoupled the simulation grid from the geologic

description, allowed greater flexibility in fracture description and improved the ability to

model the impact of the fine scale, connected nature of the fractures all within the

framework of the general implementation in conventional flow simulators.

This is one of the great advantages of modeling with edges; their direct link with

transmissibility in conventional flow simulators, hence their appeal in addressing the

missing scale. As a matter of fact, any geostatistical model of permeability is converted

to a transmissibility, essentially an edge model, before flow simulation. By taking

advantage of this fact, each simulation cell can carry information about block or rock

matrix permeability, which contributes to the flow between two cells, and also edge

9

properties that can add a dimension to more accurately capture the influence of fine scale

flow barriers or baffles between two cells. This edge property, as applied to modeling

thin shale barriers, will be presented as a transmissibility multiplier. Using this approach,

it will be possible to surpass the limitations of the current approaches presented above,

and not only model the effect of fine scale flow barriers but also gain the flexibility to

utilize more powerful history matching techniques such as the probability perturbation

method. This idea potentially opens a new world of modeling complex geological

features efficiently at the scale that they are relevant.

2 Edges

Transmissibilities: An Application of Edges

As presented in the previous section, the missing scale can be captured in simulation via

transmissibility barriers. Transmissibility is a value that defines the magnitude of flow

between two grid nodes and is defined at the cell face between two nodes. For example,

review the simple 1-dimensional simulation model discretization scheme shown in Figure

5.

Figure 5 Example representation of two cells through which a channel barrier cuts

i - 1 ii - 1 i

10

The magnitude of flow of a given fluid, f, across the face from node i to node i-1 is

calculated as a function of transmissibility and the pressure drop between the two cells:

( )121 −− −= iiif ppq /γ (1)

Where the transmissibility at the face between the two nodes is defined as:

multiplierx

if

fi i

kxB

Aγ

λγ

21

21

21 /

/

/ −

−

−

∆= (2)

Where A is the cross sectional area across which the flow, qf, occurs, λf is the fluid

mobility, Bf is the fluid formation volume factor, and ∆x the distance between node i to

node i-1.

The block effective permeabilities, kxi-1/2, are transferred to a transmissibility value, γ,

along the face i-½ as a harmonic average of the two effective block permeabilities:

1

1

121

+

+

+

∆+∆∆+∆=

+

i

i

i

i

iix

kx

kx

xxk

i / (3)

γmultiplier in Eq. (2), is a transmissibility multiplier that can be altered during to control

flow between blocks, thereby overriding transmissibilities calculated from block

permeabilities generated during modeling and subsequent upscaling. By default γmultiplier

is equal to 1, which means that the flow in the reservoir is controlled by the effective

block permeabilities ki and ki-1.

This multiplier is defined as the edge property and can either be a categorical variable, 0

or 1, or a continuous variable from 0 to 1 to capture the effects of fine scale flow barriers

between nodes. In the case of the categorical edge variable, when γmultiplier is equal to 0,

then a fully sealing fine scale flow barrier exists between the two nodes. A barrier that

inhibits flow, but does not completely prohibit it, can be captured with a continuous

variable.

11

The challenge is in defining a method for capturing this edge information on a regular

grid as a numeric value. Additionally, this edge information must be coupled with the

block property to remain geologically consistent and meaningful.

Defining a Coupled Edge Property and Block Property

Figure 6 outlines a scheme wherein the edges are defined as vectors of binary values at a

cell, where each binary value represents the presence or absence of an edge property on

each side of the cell. This binary number can be converted to a numeric number for a

single value representation.

Figure 6 Representation of edge properties on a regular grid

The attractiveness of defining the edge property in this manner lies in the fact that each

cell is coupled with its neighbors; 4 neighbors in 2-dimensions and 6 neighbors in 3-

dimensions. For example, a cell that is defined with a left edge (1000 or 8) shares the

same edge as the cell to its left as a right edge (0001 or 1). Therefore, the vector or single

numeric value defined at each cell carries details about the edge properties of the cell and

the neighboring cells. Additionally, each cell carries the categorical or continuous

property information at the cell center. This property should be explicitly coupled with

Edge Property

1 1 0 1

left rightbottom top

1000 0100 0001 0010

1010 1100 0101 0011

1011 1110 1101 0111

1111 0110 1001 0000

8 4 1 2

10 12 5 3

11 14 13 7

15 6 9

= 13

0

8 4 1 2

10 12 5 3

11 14 13 7

15 6 9

= 13

8 4 1 2

10 12 5 3

11 14 13 7

15 6 9

= 13

0 0 2 2

3 12 4

12 0 0

0 2 2

3 12 4

12 0 0

12

the edge property is a vector to ensure that the edge definition is consistent with the

controlling geologic geometry in the subsequent geostatistical simulations.

The goal is to use this coupled definition of the block values and edge property in a

geostatistical simulation. Modeling this multi-scale geologic information can be

accomplished using multiple-point geostatistics with a multi-scale training image. The

choice of the multiple point geostatistical algorithm is motivated in the next section.

3 Modeling with Edge Properties

The Challenge: Modeling Multi-Scale Geology

Traditional 2-point geostatistical simulation approaches are unable to reproduce

connectivity and geometry of complex geologic structures; however, these techniques are

excellent at integrating multiple types of diverse data. Object-based methods are able to

reproduce connectivity and geometry of complex geologic structures, however, cannot

integrate data. Additionally, parameterization and programming of the object

relationships are often very difficult. Advances in multiple-point geostatistics (MPS), i.e.

using multi-scale and multiple-point correlations coupled with the use of a geologically

consistent training image, have made it possible to reproduce the connectivity and

geometry of complex geologic structures while conditioning to the diverse suite of soft

and hard data typically encountered in petroleum reservoirs. (Srivastava, 1992 and

Guardiano and Srivastava, 1993) The training image, which comes from prior geologic

concept, is often times qualitative, i.e. using geologic expertise, and contains relative

shape/dimensions and correct association between facies. The training image allows

added control in environments of sparse and low-quality data control and in generating

geologically consistent models. Introduction of the search tree (Strebelle, 2000) made

MPS a reality with the program SNESIM, and with speed came the ability to generate

increasingly complex training images and scales of geology. However, MPS has its

limitations, for example, capturing the missing scale.

Figure 7 shows the result of modeling a complex channel system and the ability of MPS

to predict the large scale connectivity of the channels. However, where the channels

13

cross, MPS is unable to delineate the separation of the two channels because the pixels

are unable to distinguish channel barriers contained in the multi-scale nature of the

geologic setting.

Figure 7 Example result from MPS. Where the two channels cross MPS cannot delineate the fine scale flow barrier that separates the two channel

The multi-scale nature of this type of geologic setting calls for a realistic multi-scale

training image and a different approach for reproduction of the multi-scale training image

patterns. Capturing the multi-scale patterns of the training image can be accomplished

using SIMPAT relying on the multi-band capabilities of SIMPAT to jointly simulate edge

and cell properties, which are reviewed next.

14

Review of SIMPAT

Arpat and Caers (2004) and Arpat (2005) introduced an MPS method using patterns to

better reproduce the geologic features found in increasingly complex training images and

directly bypass the reliance on probability theory. SIMPAT (SIMulation with PATterns)

is a sequential simulation algorithm that scans the training image for patterns that are

most similar to the data event. Arpat described this approach as one wherein reservoir

models are built by assembling training images patterns in a jigsaw puzzle fashion,

interlocking pieces while honoring local data. SIMPAT’s power lies in its ability to

reproduce complex patterns from the training image directly into the reservoir model as

opposed to multiple-point statistics and it’s flexibility lies in the fact that it can easily

integrate a vectorized training image; a property that we will rely on to jointly simulated

edge and block property. Additionally, Arpat’s implementation of distance transforms in

SIMPAT made it possible to more accurately reproduce the large scale geologic

continuity contained in a binary training images. The limitation of using the distance

transform on binary training images only was overcome with the introduction of “bands”

which made it possible to distance transform and simulate multiple-category training

images.

Simulation of multiple-category training images: Bands

A single band describes the occurrence of a category in a multiple-category training

image, ti, where the number of bands is equal to nb and each category is identified by an

index, bi, i.e. ti(u) = bi where i = 1,…,nb. The multiple-bands of the training image can

be represented as a vectorial variable ( )u→ti such that for a specific category index bi, the

vectorial variable has entities ( ) 0=bti ,u if ibb ≠ and ( )ibti ,u = 1 otherwise. A “training

image band” is then defined as the vector ( )ib→ti of all vectorial values ( )ibti ,u , tiG∈∀u

where tiG is the training image grid. (Arpat, 2005)

A data event ( )uved T

r is now comprised of several single data events each belonging to a

particular band:

( ) ( ) ( ) ( ) ( ){ }bi nbbb ,,....,,,...,,,, udevudevudevudevuved TTTTT 21=r

15

With,

( ) ( ) ( ) ( ) ( ){ }iniiii bdevbdevbdevbdevb ,,...,,,...,,,,,T

huhuhuhuudev TTTTT ++++= α21

A pattern,

( ) ( ) ( ) ( ) ( ){ }bi nbbb ,,...,,,...,,,, hpathpathpathpathtap TTTTT 21=r

is scanned from the pattern database, Ttdbapr

and the Manhattan distance to measure the

pattern dissimilarity is defined as the sum of all absolute differences of all bands:

( ) ( ) ( )

( ) ( )∑

∑

=

=

−+=

+=

bn

ii

ki

kn

k

bpatbdev,yxd

patveddd

1

TT0

TT hhupatudevT

,,

,,

ααααα

ααα

α

hhu TT

r

Where α = 1,…,nT the number of nodes in the template and the least dissimilar pattern is

the most similar.

The simulation is now performed based on multiple-category training image ( )ib→ti .

For an unconditional simulation the process is as follows:

Step 1. A random path is defined on the simulation grid.

Step 2. At the first node of the random path a vectorial pattern kTtap

r is randomly

selected from the pattern database and pasted into the grid.

Step 3. Step 2 is repeated until some of the previously simulated nodes start to appear within the bounds of the template at which point the pattern database is searched to match to most similar ( )uved T

r based on the Manhattan

distance ααα yxd , .

Step 4. The most similar vectorial pattern kTtap

r is pasted into the simulation grid

and the simulation proceeds to the next node.

The final result is vectorial since the patterns, kTtap

r are vectorial.

The key to this MPS, as with any MPS algorithm, is in obtaining a training image that

represents the field geology.

Arpat original implementation of banded simulation was so that distance transforms

could be calculated on a training image that contained more than one category (or facies)

since the distance transform was applicable to binary images.

16

Here, Arpat’s implementation is extended to edge and block properties as part of a

vectorized training image where the categories need not be binary so long as they do not

need to be distance transformed, i.e. numeric edge property. Distance transforms will be

addressed in future work.

Generating the Training Image

The above discussion presented an approach for using the dual bands option within

SIMPAT to generate a coupled geostatistical realization of block properties and block

faces. One of the requirements for this approach is to generate a vectorized training

image, ( )ib→ti . Within the current context of coupled pixel and edge definition, there are

two bands to define. b1 for the facies indicator of the pixel and b2 for the edge depiction

based on the numeric definition of edges (see Section 2 Edges).

Below is the workflow that describes a process for building this training image.

Step 1: Construct a very high resolution model at any scale or format that is able to capture the continuous nature of the edge property. An example is an image where the flow barrier is captured in the large number of pixels. Additionally, the input could be a deterministic or unconditional Boolean model where the edge is represented as a surface.

Step 2: Select a training image grid size ( tiG defined by number of cells in x- and y-directions, nx, ny and the cell size in the x- and y-directions, dx, dy). The selection of the training image grid size is related to the model or simulation grid size, depending on the goal of the modeling process. Superimpose tiG over the image from Step 1 and place grid nodes at the center of each grid cell.

Step 3: Start a search in the x-direction of the grid stopping to check for the presence of the barrier between grid nodes. The barrier is either a surface or pixel depending on the representation of the image. The assumption is that if a barrier is a continuous surface and if the barrier is detected along the line between the two grid points then there is no flow between the two grid nodes. If the barrier is encountered then the face of the cell is flagged as an edge. At this point, this value is not a transmissibility multiplier value; it is simply a flag to denote the presence or absence of the continuous fine scale feature between the nodes. Repeat for y- and z-directions

17

Step 4: Finally search all grid nodes to get categorical block properties or use an upscaling technique to obtain and effective continuous block property

4 RESULTS

The following section presents the results of coupling edges and block properties together

in a single SIMPAT simulation using bands with a vectorized training image, ( )ib→ti .

First, the training image is presented for a channel system containing continuous shale

drapes at the base of each channel, then, the modeling results presented. Training Images

A fine scale, 2-dimensional model of multiple, stacked, meandering channels was

generated using SBED, Figure 8. This model was then captured as a high resolution

bitmap with approximately 1200x800 pixel, (Step 1).

Figure 8 Bitmap image of 2-Dimensional SBED model with three facies: Channel Drape (blue), Channel (dark brown), Background (light brown)

18

A 60x40 cell training image grid was generated, Figure 9, where the edge property in this

example is defined as the presence or absence of a shale barrier (blue) between two

nodes, (Step 2). This is directly transferable to a transmissibility multiplier as described

in Section 2. At each node of the training image is a numeric value describing the

categorical edge configuration and a pixel-value describing channel or non-channel

facies.

An important point to note about this training image is that the multi-scale channel model

is essentially discretized onto a regular Cartesian grid through the training image building

process. Although the explicit description of the channel shape is compromised, the

training image should capture the impact of the missing scale on the simulation model.

Therefore it is acceptable to use this coarse discretization. Additionally, the coarse

discretization makes it possible to transfer the simulated results directly to a simulation

model, bypassing the upscaling process. This has significant implications for using

automatic history matching techniques. However, the modeling could also be performed

on a geocellular grid.

After applying Step 3, the resulting training image, Figure 9, shows some artifacts that

were generated when searching the bitmap image for the edge definition. This is

apparent in the visual representation of the shale barriers. Although the training image

does not exactly match what would visually be generated, it possible to proceed with this

nearly correct training image in SIMPAT to demonstrate the methodology. In the future

work, surfaces will be used instead of bitmap images to generate the training image due

to the more exact nature of the surface.

19

Figure 9 Vectorized training image, ( )ib→ti (top) with edge property,

presence or absence of shale barrier between nodes, (left) and block property, channel or non-channel, (right)

Artifacts created while generating the training image are shown circled in red.

This vectorized training image ( )ib→ti is the input into conditional and unconditional

simulation results presented in the next section.

20

Coupled (Dual) SIMPAT Simulation

Once an edge property training image is generated, it can be used with SIMPAT to

generate coupled simulation results. Two unconditional realizations with varying

simulation grid and template sizes are presented herein.

The results of multiple realizations for an unconditional simulation on a 30x20 grid with

an 11x11 template size, Figure 10, show a successful coupling of the block property and

the edge property in the simulation. The training image (60x40) is twice the size of the

simulation grid and the template size is more than half the size of the simulation grid in

the vertical direction. For the unconditional simulation, at the first node of the random

path a random 11x11 pattern is selected from the pattern database and placed in the

simulation grid. The resulting simulation is a function of that first pattern and the full

simulation model is subsequently a 30x20 cookie-cut out of the training image. This is

because of the large template size and small simulation grid.

However simplistic these results appear, they are quite significant because they prove the

ability to simulate a vector of information while preserving the edge continuity. The

edge continuity is, however, rather delicate and is sensitive to simulation grid and

template sizes, as shown in the next example.

Figure 11 shows the results of a simulation on a 40x30 simulation grid with a 5x5

template using the same training image as in the previous simulation. At first

observation, the integrity of the channel barrier has been compromised and channel facies

are no longer separated by the continuous no-flow barrier. However, the same pattern

still exists wherein the edge and block properties are closely coupled in the simulation

results.

21

Unconditional Realization: 11x11 Template, 30x20 Simulation Grid

Figure 10 5 Realizations of an unconditional SIMPAT simulation with a 30x20 simulation grid and an 11x11 template

22

Unconditional Realization: 5x5 Template, 40x30 Simulation Grid

Figure 11 5 Realizations of an unconditional SIMPAT simulation with a 40x30 simulation grid and an 5x5 template

23

5 CONCLUSIONS

In this paper a new property, called and edge property, was introduced in an attempt to

model the missing scale. The edge property was defined in terms of transmissibility

multipliers for direct use in conventional simulators. The challenge of the method was in

generating a training image that could be used in SIMPAT that would represent the multi-

scale and coupled nature of the edge property and the block property. The edge property

and block property were then used as a vectorized training image in SIMPAT to generate

multiple unconditional and conditional realizations of a meandering channel system with

multiple stacked channels separated by continuous shale drapes at the base of each

channel. The coupled simulation of a channelized reservoir was successful, in that

simulations of block property (channel/non-channel indicator) and edge property (shale

drape at the base of each channel) were consistent, however, there is a great deal of work

that can be done to improve the results to better model the continuity of the channel

bodies and drapes.

Although the example presented herein was for an edge property defined as

transmissibility, the concept can be extended to other geologic features such as faults and

fractures.

6 FUTURE WORK

This work is in its infancy and there are many exciting directions that it can be taken.

One of the first areas that will be explored will be in the distance transform to better

reproduce the connected nature of the edge properties. It will be a challenge develop a

method that distance transforms this irregularly shaped property that is defined in a

binary or numeric fashion. All of the modeling presented in this paper focused on using

the numeric representation of the edge. The distance transform may require a coupled

approach for all edges using the binary representation. Additionally, adapting SIMPAT’s

conditional simulation to multiple bands will be critical to making this process robust.

24

After solving the continuity issues in 2-dimensions, the next step will be to move to a 3-

dimensional form. 3-dimensions present many challenges, such as joint edge and pixel-

based property visualization for quality control purposes while building the training

image and viewing the simulation results. The goal then is to start to model increasingly

complex channel systems, such as the system shown in Figure 2. One significant change

from the current approach for building training, to accommodate the complexity of such

models, will be to transfer the training image building process from bitmap inputs to

working directly with model grids and surfaces. For example, channels and channel

locations can be explicitly described by a triangulated surface and the barrier can be

delineated by the intersection of that surface with the 3-dimensional Cartesian grid of the

training image.

Finally, further investigation is warranted in the modeling process to add more facies

when coupling pixel-based properties with edge properties and using block effective

properties of porosity and permeability directly as opposed to the categorical facies

description presented herein. This would all low for inclusion of the next level of

heterogeneity with the pixel-based properties.

REFERENCES

1. G. Arpat and J. Caers. A multiple-scale, pattern-based approach to sequential simulation. In GEOSTAT 2004 Proceedings, Banff, Canada, October 2004. 7th International Geostatistics Congress.

2. G. Arpat. Sequential Simulation With Patterns. PhD thesis, Stanford University,

Stanford, CA, USA, 2005. 3. F. Guardiano and M. Srivastava. Multivariate geostatistics: Beyond bivariate

moments. In A. Soares, editor, Geostatistics-Troia, pages 133–144. Kluwer Academic Publications, Dordrecht, 1993.

4. M. Srivastava. Iterative methods for spatial simulation. SCRF Annual Meeting

Report 5, Stanford Center for Reservoir Forecasting, Stanford, CA 94305-2220, May 1992.

25

5. S. Strebelle. Sequential Simulation Drawing Structures from Training Images. PhD thesis, Stanford University, 2000.

6. J. Voelker. The Use of the Conventional Well Model to Predict the Effect of Discrete

Fracture Network Flow on Reservoir Flow Performance. SCRF Annual Meeting Report 5, Stanford Center for Reservoir Forecasting, Stanford, CA 94305-2220, May 2004.