FRAMEWORK OF FRACTURE NETWORK MODELING USING
CONDITIONED DATA WITH SEQUENTIAL GAUSSIAN
SIMULATION
A PREPRINT
Yerlan Amanbek1,∗, Timur Merembayev1,2,†, and Sanjay Srinivasan3
1Department of Mathematics, Nazarbayev University, Kabanbay Batyr Avenue 53, Nur-Sultan, Kazakhstan
2International Information Technology University, Almaty, Kazakhstan
3Department of Energy and Mineral Engineering, Pennsylvania State University, 110 Hosler Building, University Park,
State College, PA 16802-5000, USA
yerlan.amanbek@nu.edu.kz, timur.merembayev@gmail.com, szs27@psu.edu
March 4, 2020
ABSTRACT
The fracture characterization using a geostatistical tool with conditioning data is
a computationally efﬁcient tool for subsurface ﬂow and transport applications.The
main objective of the paper is to propose a framework of geostatistical method to
model the fracture network. In the method, we have chosen neighborhood area to
apply the Gaussian Sequential Simulation in order to generate the fracture network
in the unknown region. The angle was propagated from the seed where direction
is guided by the neighborhood data in simulation regime. Initial seeds can be
distributed by Poisson procedure. The method is applied for geological faults from
the Central Kazakhstan and for ﬁeld data from Scotland, UK. The simulation results
are compatible with the original fracture network in the ﬂow and transport modeling
setting. From the research that has been carried out, it is possible to conclude that
∗Corresponding author: yerlan.amanbek@nu.edu.kz
†Corresponding author: timur.merembayev@gmail.com
arXiv:2003.01327v1  [cs.CE]  3 Mar 2020

Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
the numerical simulation of fracture network is a valuable tool in the subsurface ﬂow
and transport applications.
Keywords fracture network model · Sequential Gaussian Simulation · geostatistical method
1
Introduction
Fractures characterization plays vital role in ﬂow and transport problems in the subsurface applica-
tions such as oil and gas production, groundwater remediation, CO2 sequestrations and etc. Due
to limited information about the subsurface property, the prediction of the fracture network in the
porous media is still a challenge.
A variety of approaches has been studied to predict behavior of fracture at particular regions.
For several years great effort has been devoted to numerical and experimental studies of fracture
propagation described using complicated geomechanical equations using various methods [1, 2, 3, 4,
5]. Such geomechanical models require more computational resources to model a fracture network
with provided subsurface parameters. However, the key reasons for fracture propagation can be
detected from such analysis. These reasons are necessary to adopt in the design of the fracture
behavior. Input parameters have a large uncertainty due to spatial variations of subsurface properties
such as stress ﬁeld, existing fracture dimensions, fracture toughness, physical properties and etc.
Permeability and porosity proﬁles are typically interpreted from the fracture geometric network,
which can be generated stochastically. This structure might not be exactly resemble a real fracture
structure, however, it provides a reasonable conﬁguration for the ﬂow and transport models. On
the basis of the available data, for example data collected near wellbore and other techniques,
a geostatistical technique can be trained based on known regions to model the fracture network
in unknown regions. In the literature, stochastic approaches have attracted much attention from
researchers in reduction of subsurface uncertainty.
In [6] the dfnWorks workﬂow was introduced to simulate single or multi-phase ﬂow and transport
problem coupled with geostatistically generated discrete fracture network (DFN). A combination of
different types of tools in single platform is necessary to validate modeling uncertainty.
In [7] the geostatistical modeling of fracture system was developed using pattern statistics from
pattern bar chart. The multiple point statistics(MPS) was used with pixel based data in the growth-
based fracture network method. As reported in [8], the model was trained from satellite images
2

Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
which were obtained by detecting faults in the surface. The training and testing were performed for
chosen surface samples.
Authors [9] have proposed the fracture network system simulation based on MPS [8, 10] using the
sophisticated relations among cracks and faults. In [10] the method was developed based on MPS
honoring surrounding data and multiple point histogram. The propagation orientation is utilized
in the MPS simulation. This method was applied for real-world datasets such as Teapot Dome,
Wyoming. Another application of discrete fracture network simulation work using a commercial
software MoFrac were presented in [11] for the Canadian Shield dataset.
An interesting approach to this issue has been proposed by [12]. This method is adaptable for
additional user deﬁned constraints which honors other types of subsurface data. Algorithms
including Sequential Gaussian Simulation (SGS) is used to model with simple kriging approach of
the fracture orientation. In the model, the process is similar to a propagation of fracture where in
each iteration the orientation and step length is computed honoring provided subsurface information.
However, there is a necessity on systematic workﬂow in the fracture pattern prediction.
In this paper, we present the fracture characterization model using geostatistical analysis with
conditioned data based on the approach in [12]. The workﬂow of algorithm is provided for
the natural fracture network simulation in the undiscovered zone. Numerical simulations were
conducted for different examples such as realistic faults from the Central Kazakhstan and the natural
fracture dataset from South Glasgow, Scotland, UK. To verify the realization, we have compared
major variables such as the orientation histogram and the concentration proﬁle at production well,
where the adaptive numerical homogenization method was applied to simulate the single-phase and
incompressible ﬂow and transport model in the fractured reservoir due to multiscale feature in this
setting, see [13, 14] for more details.
The remaining part of the paper proceeds as follows. Section 2 describes the numerical method.
The process of modeling is shown in the ﬂow-chart. Numerical results are presented in Section 3.
In Section 4, the results were discussed and the conclusion is reported in Section 5.
2
Numerical Methods
We ﬁrst illustrate the fracture network simulation in 2D with neighbor selection procedure, and
then the Sequential Gaussian Simulation (SGS) process for simulating fracture direction. In our
case, there is data from certain regions is provided, however, the rest of the region is unknown due
3

Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
to sparsity of data. By using the information of fracture network we model fractures in unknown
regions. In the simulation of fracture networks, key parameters are fracture direction (angles) and
length of propagation. The orientation or the angle of fracture was computed by SGS method using
the data from known regions. In each iteration of simulation, fracture grows in all domain honoring
the information of neighbourhood region.
It is common to observe the family of fractures, not only single fracture. Such family of fractures
has a common features including orientations. The changes of directions in the fracture family
are usually smooth and transition of certain patterns are soft [15]. Therefore, the average value of
fracture orientation, the intensity distribution, etc are necessary in the fracture simulation.
2.1
Fracture Network Simulation with Conditioned by Neighboring Data
From a geomechanical standpoint, a fracture direction is inﬂuenced the systematic organization
of neighboring subsurface property such as geomechanical features, fracture intersections and
others [15]. The stress ﬁeld nearby the fracture tip is a key factor in the simulation of fracture
propagation. This is measured by a stress intensity factor (KI), which depends on loading (stress)
and the fracture dimensions. The fracture keeps its propagation as long as KI > KIC, where KIC is
fracture toughness. In addition, previous studies indicate that the energy minimization is important
in the systematic fracture path simulation using the phase-ﬁeld model [1, 3]. Therefore, the effect
of neighbor fracture should be taken into account fracture direction simulations.
We describe the key ingredients of algorithm in the fracture network simulations including initiation,
propagation and termination [15]. We will address all steps below.
To illustrate the proposed method of fracture characterization, we show the major steps in 2D for
surface traces.
In our model, a fracture is a collection of fracture segments with corresponding length and orientation.
The initial fracture segment will be growing honoring the surrounding information. We ﬁrst deﬁne
the initial fracture segments (coordinates and azimuth angles) in the entire domain. In the unknown
region, we use the Poisson procedure to seed initial fracture segments, however, these locations can
be indicated by user. In known area, fractures will be propagated from the middle of the original
fracture length and repeat their original path in both directions from the middle. Below you can see
the key steps in the method for deﬁning the neighborhood region, as shown in Figure 1.
4

Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
Step 1. The initial points will be seeded in the domain. For each point, which is the middle of
fracture segment, there are corresponding parameters such as azimuth angles, coordinates
of points and the neighboring segments. The ﬁlled point is target point from where a new
fracture segment will grow.
Step 2. We deﬁne the neighborhood zone as sector of inﬂuencing points that was created from the
following parameters: the initial azimuth angle, a standard deviation of angle and radius.
From the initial point along the line, which follows provided azimuth, we draw sector
with radius r and angle of sector equals to two standard deviations. The line generated
from azimuth divides the sector angle equally. In here, we compute azimuth of angle from
fracture angle distribution of known region. A radius of the sector is deﬁned from the
variogram model of fracture segments in the known region. It is within this search sector
that conditioning data for fracture angles will be searched.
Step 3. There will be included a virtual point additionally to the selected points in the sector. To
consider the inﬂuence of the nearest fracture segment, we draw virtual fracture segment in
the center of the segment, which connects points between the current point and the nearest
fracture segment. The virtual fracture segment keeps the angle. The selected points in the
sector including the additional data will be used in the SGS. This approach was initially
proposed in [12].
Step 4. We compute important features of a new fracture segment such as the orientation and the
fracture segment length. We use the points in the neighbourhood sector as input data in the
SGS analysis to compute a newly simulated angle. The angle is chosen from the computed
distribution in random fashion. The length of simulated segment equals the mean of fracture
segment distribution of known region. We note that the length can be speciﬁed by user.
Combining the information of angle and length of segments we draw a new fracture segment
which is colored in blue.
By propagating the target or initial points and repeating step 2 - step 4 we can achieve a conﬁguration
shown in Figure 1 (Final result). Standard deviation and radius are constant for further iterations.
After several propagation iterations a newly simulated fracture will be terminated until crossing any
fractures or length of fracture reaches the constraints.
5

Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
Figure 1: Schema of a fracture simulation.
6

Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
2.2
Sequential Gaussian Simulation
We brieﬂy go through SGS algorithm in perspective of fracture segment orientation. In step 4 from
the previous section, we need to identify a new angle of the next fracture segment by considering
angles of neighboring fractures.
The method of SGS assumes the presence of a normal distribution of the simulated random variable.
For any point in the region, the local distribution function will be distributed according to the normal
law and will be determined by two parameters such as mean and standard deviation. Before to
make simulation, it is assumed that the stationary is exist for random function Z(x) and a random
function Y (x) is exist such that
Y (x) = φ[Z(x)]
(1)
where Y (x) is N(0, 1) and φ - the normal score of the transformation.
The ordinary kriging is the main method that helps to propagate the fracture segment with the
provided spatial distribution. The kriging is the linear unbiased predictor and can be explicit as the
linearly - weighted average function of observations in the unknown area of fractures x, location of
fractures . Y (x) is a set of random variables (azimuthal fault angles). The ordinary kriging estimate
Y ∗
sk(x) at x is calculated as
Y ∗
sk(x) =
n(x)
X
i=1
λsk
i (x)Y (xi)
(2)
The ordinary kriging weights λsk
i that deﬁned by by solving the system n(x) of ordinary kriging
equations:
n(x)
X
j=1
λsk
j Cij = Ci0, ∀i = 1, ..., n(x)
(3)
where n(x) is the total number of neighboring points xi used in estimating the point x. The ordinary
kriging score variation is given by
σ2
sk = C(0) −
n(x)
X
i=1
λsk
i C(xi −x)
(4)
As a result, we obtain the parameters of the local normal distribution function N(Y ∗
sk(x), σ2
sk(x)) at
the estimation point x. This distribution is required in the identiﬁcation of the fracture propagation
angle.
7

Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
2.3
Estimation of fracture orientation
In step 4, we have used the SGS for the modeling of fracture orientation in each iteration of
the fracture network simulation. The SGS is the most commonly used sequential simulation
algorithm for modeling continuous variables. A detailed introduction to this method can be found
in [16, 17]. For visual representation of the ﬂow-chart of the proposed method the reader is referred
to Figure 2. In addition, the pseudocode of the proposed method is depicted in Figure 3. In the
algorithm,simulation of azimuth angles in SGS proceeds as follows:
1. Validate whether the distribution of the provided areas dataset is normal distribution or not.
This dataset is a collection of the fracture segment azimuth;
2. If necessary, transform the distribution of the data into a normal distribution;
3. Seed the initial fracture segments randomly using the Poisson process in unknown region;
4. Identify the points in the neighbor zone as described in Step 2-3, Section 2.1;
5 & 6. If number of neighbors are less or equal 1 then the direction of the new fracture segment will
be chosen by the Monte-Carlo method for the normal distribution of the provided dataset, as
described in Step 1, Section 2.1. This produces a new azimuth angle that is combined with
the length of fracture segment in order to propagate a new fracture segment;
5 & 7. If number of the neighbor points are more than 1 then to add point of halway to the nearest
neighbor for the inﬂuence of the nearest fracture segment;
8 & 9. As subroutine of SGS procedure, the ordinary kriging approach is utilized to obtain kriged
estimate and the corresponding kriging variance;
10. Termination of the fracture segment occurs when it crosses another one. We do not consider
another option for termination a propagation of fractures, criteria of achievement limit length
of full length of fracture, the criteria can be integrated;
11. Transform from the normal distribution to the original univariate distribution.
8

Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
Sequential Gaussian Simulation
Start
1.Dataset
normal distr?
2.Transform to norm dataset
3.Seed unknown fracture
4.Find neighbors
5.Count
neighbors >=1
7.Add point halfway
to the nearest neighbor
6.Calculate new strike of
fracture randomn(µ, σ)
8.Calculate weights by
closest fracture segment
9.Simulate strike trace by invers
covariation matrix randomn(µ, σ)
10.Intersect
fracture traces
11.Back transform
from norm to original
Finish
no
yes
no
yes
no
yes
Figure 2: Flow-chart of the fracture algorithm.
9

Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
Algorithm 1: Sequential Gaussian Simulation
Input
:mean, standard deviation std, fracture length len, iteration iter, matrix azimuth A, count of seed fractures m
Output :A(m, n)
1 for i ←1 to m do
2
for j ←1 to iter do
3
[xidij, yidij] ←Calculate of a mid of fracture
4
[xidk, yidk] ←Find neighbours A(X, Y )
5
if count[xidk, yidk] ≤1 then
6
xidi,j+1 = len ∗cos(randomn(mean, std)) + xidij
7
yidi,j+1 = len ∗cos(randomn(mean, std)) + yidij
8
if A(xidij, yidij) ∩A(X, Y ) then
9
close loop j
10
end
11
else
12
λ, σ ←Calculate weights(idk, covar−1)
13
A(xidi,j+1, yidi,j+1) =Simulation(λ, σ, randomn(mean, std))
14
if A(xidij, yidij) ∩A(X, Y ) then
15
close loop j
16
end
17
end
18
end
19 end
Figure 3: Pseudo code of fracture characterization .
3
Numerical results
In this section, we show several numerical examples such as comparison with previous result from
[7], application for faults from Zhezkazgan, Kazakhstan and fractures of McDonald limestone
dataset from Scotland, UK. To illustrate applications of the proposed method, we show numerical
simulations for realistic datasets in two dimensions with veriﬁcations.
3.1
Example 1
In this example, we have applied our method to the scenario with initial conditions presented in
[12]. We have compared the results in here to depict the similarity of proposed method. To verify
the method, our result of simulation is compared with the result of fractures from [12], as shown in
Figure 4.
The example was digitized by using GRABIT [18]. Fractures from the gray region is given in
the right side. In the original work, fracture of the dark region was predicted. We have used the
following parameters as input data in our algorithm. The mean of orientations is 70 degrees and the
standard deviation is 12. The variogram model is a spherical. Nugget is 1, spherical is 2, range is
50, r = 50, fracture length is 10.
10

Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
In here, the azimuth angles were generated by Monte Carlo method in the SGS procedure. As can
be seen from Figure 4b, simulated fractures are presented in different colors such as green, blue, red
and black. It is important to note that the fracture path in the known region was reproduced and the
part of it was honored during the simulation regime.
(a) The result of simulation in [12].
(b) Our result of simulation.
Figure 4: The results of simulation by proposed algorithm.
Various numerical examples are performed with different initial seeds in the hidden region. Poisson
procedure is applied to determine coordinates of initial points for simulated fractures. Figure 5
presents different realizations where red squares are middles of initial seeds.
(a) Example 1.
(b) Example 2.
Figure 5: Simulation fractures from initial points generated by Poisson process.
In Figure 5a and 5b, blue fractures (initial azimuth: 45 and 25) are similar to the original fracture.
During the simulation, we have observed that there is a tendency of having traces in similar
directions for nearby fractures. This phenomena was discussed in Section 2.1 with perspective
of geomechanics. We observe similar pattern for a black fracture (initial azimuth 120), which
connected to the known fracture, see Figure 5a, Figure 5b gives an examples of similar direction
11

Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
pattern for a black and blue fractures(initial azimuth 125 and 25). After several more iterations the
resulting path of black and blue fractures become close to each other in the center of domain.
3.2
Example 2
The analysis of conditioned data is necessary for the fracture characterization simulation. We ﬁrst
provide the results obtained from the FracPaq toolbox in Matlab [19]. Second, we show the one of
realizations of simulated fractures for the region in Zhezkazgan, Kazakhstan.
3.2.1
Fracture Data analysis
Before applying SGS algorithm we need to make data analysis of fractures given in known area
using FracPaq [19]. The software allows to obtain the quantitative fracture patterns, digitalization
of fractures and their distribution in two dimensions for length, segment scales, and rock types.
Fracture is deﬁned as a collection of one or more segments which creates a continuous line in the
software. It is assumed that segments at any scale are formed because of the interaction and fusion
of small cracks, so analysis of these segments separately can be useful.
(a) Traces colored by length.
(b) Segments colored by length.
Figure 6: Result of data analysis.
The FracPaQ provides an analysis of the traces and the segment of fractures by length so it helps
to indicate major and minor fractures, analysis of segment traces provides visual conﬁrmation
12

Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
that fractures have one mean of fracture segments. The traces colored and the segment colored of
fractures by length are shown on Figure 6.
(a) Angles of fracture segments.
(b) Angles of fracture segments.
Figure 7: Fracture orientation.
(a) Frequency-size histogram for trace lengths.
(b) Frequency-size histogram for segment lengths.
Figure 8: Histograms of analyzed data. Red lines show the minimum and maximum lengths.
The distribution of fracture angles for the known region is shown in Figure 7 and is presented in a
rose plot and a histogram. The rose plots and the histograms show the main fracture population
(about 120 degrees) in the known region, the second peak of the fracture population is an opposite
direction (120 degrees) a fracture segment. Based on the angles shown in the rose plot, the mean of
the azimuth angle of fracture segments is 120.
13

Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
Base on the data there are two frequency distributions – size plots for trace and segment lengths
can help to estimate the mean length of segment. The length will be used for simulation of fracture
propagation. Figure 8 shows histograms of trace and segments lengths of geology faults.
FracPaQ generated the variogram (or semi-variogram) of segment lengths. Semivariogram had been
calculated as a function of the separation of two segments from the stored coordinate positions of
all the segment mid-points (centroids) for every pair of segments.
In Figure 9 the calculated variogram model showed the correlation range is achieved on 15000
meters. This range value is used as a radius of the sector for searching neighbors.
Figure 9: Variogram of segment lengths.
3.2.2
Result of simulation for structural measurements
We examined the SGS algorithm in 7011,9 km2 area near of Zhezkazgan city, Kazakhstan. The study
area is located in the central part of Kazakhstan. Red beds and volcanic suites are well captured.
The rocks wereformed in age from the Paleozoic, Carboniferous, Devonain and Ordovician. They
have a good visibility in deﬁned fold structures. This geological data is obtained from [20]. The
main purpose of such maps is the investigatigation of the potential mines in the area, see Figure
10. We take the digital data: geology and faults for the area of interest to make the test of fracture
characterization. More detail about the description and color of the legend is explained in source
[20].
14

Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
(a) The image has faults and geology.
(b) The image has only faults.
Figure 10: Simpliﬁed geological.
Based on data analysis from the tool we get next information that is necessary for SGS algorithm:
• Mean of segment length – 2289.27 meters, The value is used as constant length of simulated
fractures segment. As previously noted the length can be speciﬁed by an user;
• Exist bimodal normal distribution of angle faults, 120 and about 300 degrees. 300 degrees
direction is the opposite direction (120 degrees) so we choose 120 degrees as the mean of
the azimuth angle of distribution for the known domain;
• Base on the variogram model of segment length, we deﬁne lag of between segments - 15000
m. Variogram model is represented in spherical mode.
Propagations for each fracture started from the middle of fracture length and propagate to both sides
from a middle point. As shown in Figure 11a, a trace indicated in red was selected to be predicted
by the proposed method. After removal of that fracture from the conditing data set, the method
used fractures in blue as conditioned data and then a fracture marked in green is generated at the
location of the removed fracture. As can be seen from Figure 11b, there is a good match between
the original fracture (red) and the simulated fracture(green).
15

Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
(a) Geology faults with middle points.
(b) The simulated chosen fault.
Figure 11: Result of SGS simulation for one fracture.
To test the algorithm for realistic condition, we hide the fracture information in the center area
(cropped from the original domain). We have used the midpoints of the original fractures as initial
seeds for simulation of green fractures. In Figure 12a, red color is the boundary of crop domain
from the original fractures, the blue color is original fractures or fractures from the known domain.
In Figure 12b the original fractures are in the blue color. In Figure 12c there are the simulated
fractures by the proposed algorithm in the green color. The result of the algorithm is consistent with
the major trends of the original fracture traces.
(a) Hidden zone of fractures
(b) Original fractures
(c) Simulated fractures
Figure 12: Result of Proposed algorithm, simulation for KZ geology fractures.
Geological fractures are divided into primary and secondary features, The primary fractures are
long extensional features, the secondary fractures are of insigniﬁcant length, and their prevails over
the primary. These fractures have different structures and origins. If the midpoint of long fractures
are not inside the hidden square we keep such faults as conditioned data. This is to ensure that we
reproduce the length distribution accurately.
16

Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
3.3
Example 3
The realistic fracture obtained from [19] is used in the fracture characterization to illustrate the
proposed model. Collected data is a fractured bedding plane in the McDonald (or Hosie) Limestone
in the Spireslack opencast coal pit. Overall, in Midland Valley there are mining carboniferous
coal-bearing ﬂuvial-deltaic rocks and have been started since the 19th century, Figure 13. The count
of initial segment fractures is user-deﬁned for an unknown area. In this case, we choose the 50
initial segment fractures. 50 initial fracture segments randomly seed by the Poisson process in the
hidden center area.
As the previous ﬁgure, we got 3 images: hidden domain (cropped region), original fractures and
simulated in Figure 13. If we compare the result of fractures simulation and original fractures on
the ﬁgures, we can see that the trend of fractures direction is reproduced. To prove this statement
we will resort to a histogram of fracture angles distribution.
(a) Hidden zone of fractures
(b) Original fractures
(c) Simulated fractures
Figure 13: Result of proposed algorithm, simulation for McDonald Limestone dataset from the Spireslack open cast
coal pit, south of Glasgow in Scotland, UK.
The veriﬁcation was given by comparing the distribution of fracture angles for original fracture
networks (left ﬁgure) and simulated fracture networks (right ﬁgure) within the hidden area, see
Figure 14. We have observed similarity between histograms for original and simulated fractures. In
particular, shape and means of distribution are similar that was illustrated in Figure 14 (vertical red
lines in histogram).
17

Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
(a) For original fractures.
(b) For simulated fractures.
Figure 14: Histograms of veriﬁcation.
Analysis of the plotted histograms yielded a multimodal distribution (4 peaks), it is due to the
fact that we have two main directions of fractures and two conjugate directions for propagation of
fractures. Based on histogram we get means of trace segment angles for original fracture networks:
20; 110; 185; 290. The corresponding angles of the simulated fracture networks: 18; 85; 190; 275.
These histograms give us the opportunity to assert that the proposed method shows a good result
and conﬁrms its ability to simulate fracture networks.
Figure 15: Location of injection and production wells.
The additional veriﬁcation of the fracture network model is given by comparing the concentration
proﬁle at the production well in a tracer test setting. We have a single-phase , incompressible ﬂow
and transport model which is based on work presented in [13]. Here, the ﬂow problem solved initially
18

Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
and then the transport problem. For more details we refer to earlier work [13, 14, 21, 22, 23, 24].
We locate the production well in the top right corner of the domain and the injection well in the
bottom left of the domain, see Figure 15. Dimension of the reservoir is 6900 ft x 5700 ft. In the
adaptive regime, the local cell dimension is 10 x 10 grids for homogenization of permeability, i.e.
domain dimension is 69 x 57 grids in coarse scale. Porosity was taken as 0.2 and diffusion is 0.0001
ft2/day. No ﬂow boundary conditions in the domain. Permeability at the fracture was chosen 200
times more than non-fracture media. Time step is 5 days, ratio between coarse and ﬁne scale is 10
and dx=10 ft. The injection rate at the bottom left well is chosen 100 stb/day.
As shown in Figure 16, the ﬂow and transport model with proposed fracture network algorithm was
able to match concentration at production well for the model with original fracture network. There
is a good agreement in concentration proﬁle for both scenarios as well as on breakthrough time.
We limit in this work with provided realization of fracture network in Figure 14 to illustrate the
veriﬁcation.
Figure 16: Location of injection and production wells.
Our veriﬁcation was performed for the provided realization. Future work will involve more ﬂow
and transport simulations for various realizations in an efﬁcient way using the multicore machines
in parallel mode.
4
Discussion
One can look at the history of fracture generation in porous media. In other words, if model allows
to go back in time and afterwards to take into account main geological events from the past which
was reason for generation of younger fractures along with older ones. Thus, our method can be seen
19

Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
as similar as the fracture propagation scenario where it starts from seed nodes obtained from the
Poisson Distribution or user-speciﬁed locations. In addition, the algorithm can be conditioned by
data from available sources such as seismic, wellbore and others.
We assumed that the fracture orientation has a gaussian distribution in the simulation regime. The
understanding of behaviour of 2D fracture network is an important base for studying the fracture
network in 3D. The extension from two dimensional to three dimensional can be done in a similar
fashion by considering azimuth, length and polar angles. A combination of layers can be interpreted
as characterization of the subsurface volume [12]. During the generation of a sector zone, we have
used the unique and precomputed standard deviation and radius for each sector in the simulation
process. However, these parameters can be speciﬁed by user.
5
Conclusion
The purpose of this paper is to provide the framework of the characterization of fracture network in
different scales using the geostatistical method. We have applied the proposed algorithm for natural
fracture path from the Central Kazakhstan, the area includes Zhezkazgan city, and for McDonald
limestone from Scotland, UK. To our best knowledge, this is the ﬁrst study to deal with the fracture
network modeling of the Central Kazakhstan faults. In the growth-based fracture network model,
the Gaussian Sequential Simulation is used for orientation of fracture honoring the neighborhood
information. We currently believe that the computation time of the fracture network characterization
using the proposed algorithm is less than the CPU time of the geomechanical models that considers
the coupled partial differential equations in the subsurface. To verify the proposed fracture network
method, the distribution of the segment angles for the original fracture path is compared with the
same distribution for the simulated fracture path. The main features of both distributions are in
good agreement. In addition, we have compared the concentration proﬁles at the production well
for different fracture networks in the single-phase and incompressible ﬂow and transport models.
There is a good match between the concentration history used simulated fracture network and the
concentration history used original fracture network. In our future research we intend to concentrate
on machine learning algorithms for fracture characterization using image training approaches for
real-ﬁeld dataset.
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Framework of Fracture Network Modeling using Conditioned Data with Sequential Gaussian Simulation
Acknowledgement
Financial support from the Nazarbayev University Faculty Development Competitive Research
Grant, No 110119FD4502, is gratefully acknowledged.
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