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energies
Article
Numerical Simulation of the Propagation of
Hydraulic and Natural Fracture Using
Dijkstra’s Algorithm
Yanfang Wu * and Xiao Li *
Key Laboratory of Shale Gas and Geoengineering, Institute of Geology and Geophysics,
Chinese Academic of Sciences, Beijing 100029, China
*Correspondence: wuyanfang@mail.iggcas.ac.cn (Y.W.); lixiao@mail.iggcas.ac.cn (X.L.);
Tel.: +86-10-8299-8040 (Y.W.)
Academic Editor: Mofazzal Hossain
Received: 2 March 2016; Accepted: 27 June 2016; Published: 5 July 2016
Abstract:
Utilization of hydraulic-fracturing technology is dramatically increasing in exploitation
of natural gas extraction. However the prediction of the configuration of propagated hydraulic
fracture is extremely challenging. This paper presents a numerical method of obtaining the
configuration of the propagated hydraulic fracture into discrete natural fracture network system.
The method is developed on the basis of weighted fracture which is derived in combination of
Dijkstra’s algorithm energy theory and vector method. Numerical results along with experimental
data demonstrated that proposed method is capable of predicting the propagated hydraulic fracture
configuration reasonably with high computation efficiency. Sensitivity analysis reveals a number
of interesting observation results: the shortest path weight value decreases with increasing of
fracture density and length, and increases with increasing of the angle between fractures to the
maximum principal stress direction. Our method is helpful for evaluating the complexity of the
discrete fracture network, to obtain the extension direction of the fracture.
Keywords:
discrete fracture network; hydraulic fracturing; Dijkstra’s algorithm; hydraulic fracture
1. Introduction
Hydraulic fracturing is one of the key technologies in exploitation of natural gas, which plays a
major role in enhancing petroleum reserves and daily production especially from unconventional
reservoirs [
1
]. The hydraulic fracture and branch fractures could be generated during Hydraulic
fracturing, as shown in Figure 1[
2
]. The hydraulic fracture can provide a high-conductivity path for
methane migration in the low permeability reservoir. In recent years, to investigate propagation and
re-orientation of hydraulic fracture, lots of research has been conducted [2–5].
The Finite Element Method (FEM) and Displacement Discontinuity Method (DDM) are two
popular methods. Mohammadnejad and Khoei used a fully coupled extended finite element method
for hydraulic fracture propagation of porous media [
6
]. Ferté et al. combined extended the finite
element method and cohesive zone models to predict crack paths and to compute the crack deflection
angle [
7
]. Fu et al. simulated hydraulic fracturing in arbitrary discrete fracture networks by using an
Energies 2016,9, 519; doi:10.3390/en9070519 www.mdpi.com/journal/energies
Energies 2016,9, 519 2 of 15
explicit coupling simulation strategy [
8
]. However, due to the multiple dynamic physical processes
with characteristic length-scales across several orders of magnitude, the time steps of the Finite
Element Method (FEM) for simulating fracture propagation are very small. Furthermore, the matrix
in physical simulation is not only complex, but also ill-conditioned. Hence, FEM is not suitable
to be used to simulate hydraulic fracture propagation. Verde and Ghassemi implemented Fast
Multipole Displacement Discontinuity Method to accelerate the solution of large scale coupled fluid
flow-geomechanical problems [
9
]. Dong and De Pater used a general displacement discontinuity
formulation to deal with the problems of curved and straight-line cracks [
10
]. Behnia et al. studied
the boundary element method based on the displacement discontinuity formulation for mixed-mode
crack tip propagation of pressurized fractures [
11
]. However, the influence of the coefficient matrix
of the boundary element method is not sparse. In addition, with the increase of gird, the time
consumed increases rapidly. Hence, the application of DDM in simulating complex fracture network
is also limited. In this paper, due to the weaknesses of these two methods for simulating hydraulic
fracture propagation in a large scale, a new numerical model based on graph theory, energy theory
and vector methods was studied to obtain the final configuration of fracture propagation.
2 1 0 1 2
2
1.5
1
0.5
0
0.5
1
1.5
2
x/m
y/m
Fracture Network
Fluid Pressure
0MPa
0.2MPa
0.4MPa
0.6MPa
0.8MPa
1MPa
1.2MPa
1.4MPa
1.6MPa
1.8MPa
Hydraulic fracture
Figure 1. hydraulic fracture.
A graph is an important mathematical method to describe the special relationship between
variables [
12
]. A graph includes some vertices and arcs. The vertices represent the members,
and the arc represents the unique relationship between them [
12
]. The shortest path problem is
one of the most fundamental problems with widespread applications [
13
], which is to find the
shortest path between two vertices. The shortest path problem’s solution can be used in a number of
fields. Thus, more researchers have focused on this problem. Shirinivas et al. [
14
] have given an
overview of the graph theory in various fields to some extent and mainly focused on the computer
science applications. Klampfer et al. [
15
] have used mathematical analysis methodologies based
on circular graphs to solve a shortest path routing problem in Routing Information Protocol. Sarbu
and Valea [
16
] have proposed that an optimal solution could be obtained based on graph theory for
selection of source location and the main path of water transmission. All of the above applications
are given with complete graph information, that is, all vertices and arcs are given in this graph.
However, parts of vertices and arcs in graph are not given in some special fields. For example, new
vertices and arcs are generated with the fracture propagation in discrete fracture networks (DFN)
during hydraulic fracturing. This paper conducts exploratory research based on this special field.
In this paper, a new method is presented to obtain the final configuration of fracture propagation
with high computation efficiency, which combines Dijkstra’s algorithm, energy theory and vector
method. Changes in hydraulic fracture propagation are generally attributed to various distribution
Energies 2016,9, 519 3 of 15
properties of fractures (such as fracture length, fracture density and dip angle) and simulation
conditions (like elastic modulus, normal stress pressure, Poisson’s ratio and so on). Hence, different
parameters are analyzed, which can show how these parameters influence the propagation of
hydraulic fracture. Some preliminaries are given in Section 2. Section 3discusses how to obtain the
configuration of propagated hydraulic fracture using Dijkstra’s Algorithm. Parameter analysis is
done in Section 4. Section 5states a discussion and some conclusions are given in Section 6.
2. Preliminaries
Graph
G(V
,
E)
is a collection, which is composed of a vertex set
V(G)
and edge set
E(G)
. In
this paper, the element of E(G)denotes fracture.
Definition 1.
[
12
] Make
G= (V
,
E
,
W)
be a weighted graph, where
V
is the vertex set,
E
is the edge set,
and
W
is the weight matrix for
E
.
W
represents the cost from one vertex
u
to other vertex
v
(indicated by
ω(u,v)). If e = (u,v), replace ω(u,v)with ω(e).
Definition 2.
[
12
] The length of a path
p= (e0
,
e1
, ...,
ek)
is the sum of the weights of its constituent edges:
length(P) =
k
∑
i=1
ω(ei−1,ei). (1)
Definition 3. [12] The distance from u to v, shown as δ(u,v), is the shortest path with the minimum sum
of the weights on the edges in a u −v path. u,v denoted start vertex and sink vertex, respectively.
The following example is given to illustrate the meaning of
Definition 1–3
, which is shown in
Figure 2. Vertex
A
and
D
are the start vertex and terminal node, respectively. The weight value and
shortest path can be obtained:
V={A,B,C,D},
E={ω(A,B) = 3, ω(A,D) = 9, ω(A,C) = 5, ω(B,D) = 2, ω(C,D) = 1}.
Figure 2. The flowchart of a graph.
From
Definition 2
, it is easy to obtain that
length(A
,
D) = ω(A
,
D) =
9 or
length(A
,
D) =
ω(A,B) + ω(B,D) = 5. Hence, the distance from Ato Dis 5.
2.1. Dijkstra’s Algorithm
The shortest path problem can be divided into four situations in actual problems, and there are
always four kinds of the corresponding algorithms [
17
]: (1) Dijkstra’s algorithm for Single Source
Shortest Path, the weight of edge is non-negative [
18
]; (2) Bellman–Ford’s algorithm, it is also for the
Single Source Shortest Path, and the weight of edge can be negative. However, the negative weight
circuit cannot exist [
19
]; (3) Shortest Path Fastest Algorithm for Single Source Shortest Path—it is an
improved algorithm based on the Bellman–Ford algorithm [
20
]; (4) Floyd algorithm for Shortest
Path among all vertices—the weight of edge can be negative [
21
]. However, the negative weight
Energies 2016,9, 519 4 of 15
circuit cannot be existed. For hydraulic fracturing, the weight of fracture is non-negative and the
hydraulic fracture propagation belongs to the Single Source Shortest Path problem. Hence, Dijkstra’s
algorithm is the most suitable method to simulate the hydraulic fracture during propagation of
hydraulic fracturing.
Dijkstra’s algorithm can be used to find the shortest path between the node and every other [
18
].
Here,
V={v0
,
v1
,
v2
, ...,
vn}
is divided into two vertex sets:
S={v0}
and
T={v1
,
v2
, ...,
vn}
.
Assume
v0
is the initial vertex, and
dist[vi]
is used to store the upper bound of the shortest path
distance from
v0
to
vi
. If the final shortest path from
v0
to
vi
is determined, denoted
S=S∪vi
,
T=T\vi
. The algorithm repeatedly selects the node
vi
from
T
with the minimum
dist[vi]
, and puts
it in set S. The main steps for Dijkstra’s algorithm can be described as follows:
Step 1. Set v0is initial vertex,
S[vi] =
0, vi/∈S
1, vi∈S
,
and
dist[vi] =
ω0i,i6=0&(v0,vi)∈E
∞,(v0,vi)/∈E
0, i=0&(v0,vi)∈E
i=1, 2, ..., n.
Step 2.
Select the vertex
vi
in
T
, such that
dist[vi] = min
vj∈Tdist[vj]
. If
dist[vi] = ∞
, stop, else go to
Step 3.
Step 3. Set S=S∪vi,T=T\vi, and S[vi] = 1. If T=∅, stop, else go to Step 4.
Step 4. Update dist[vj] = min{ω0j,dist[vi] + ωij }. Go to Step 2.
An example is given to illustrate the working of the Dijsktra’s algorithm, which is shown in
Figure 3. To begin with, the weight matrix should be computed. Assume that
A
is the start vertex,
the weight of the minimum-weight paths from Ato every other vertex is +∞, and Ato Ais 0.
Figure 3. The solution procedure of Dijsktra’s algorithm.
Energies 2016,9, 519 5 of 15
3. Model Description
In this section, Dijsktra’s algorithm is used to simulate the hydraulic fracture propagation in the
DFN. DFN is given under the meter scale, which has two groups of fractures. During the hydraulic
fracture propagation, two kinds of energies are used to describe different fracture propagation:
mechanical energy and interface energy. The mechanical energy is used to keep natural fracture
open, and interface energy is used to create new fractures. Combined the mechanical energy and
interface energy, the fracture weighted formula is successfully deduced.
3.1. Discrete Fracture Network
An example of DFN is a two-dimensional horizontal section of a rock with 4 m length, 4 m
width, which is shown in Figure 4. DFN is composed of two groups of random fractures, and
the blue lines represent natural fractures. The fracture number is 250, the length of a fracture is
0.2 m. The angle between the two groups of fractures and the dip angle (which also means the angle
between one set fractures to the maximum principal stress direction) is 60
◦
. Figure 4also shows that
the DFN is not fully connected from one side to the other. During hydraulic fracturing, new fractures
are the only way to link natural fractures. Therefore, new fractures should be an important factor in
Dijsktra’s algorithm.
2 1 0 1 2
2
1.5
1
0.5
0
0.5
1
1.5
2
Sample Fracture Length (meter)
Sample Fracture Width (meter)
Discrete Fracture Network
Figure 4. An example of discrete fracture network.
3.2. The Fracture Intersection Point
There are many fracture intersection points in Figure 4. During hydraulic fracturing, when
fracture fluid flows through a fracture intersection, fracture fluid will enter into two fractures.
Hence, the new vertices generated from fracture intersections have a significant influence on DFN
connection, and these new vertices must be found out. Many mathematic methods to detect
the fracture intersection have been studied, such as parameter equation and vector method [
22
].
Vector method is a great approach to this problem.
e1= (u1
,
v1)
and
e2= (u2
,
v2)
represent two
fractures, respectively. Defining the two-dimensional vector cross product:
e1×e2=u1v2−u2v1
.
Vector quantities
r= (u3
,
v3)
and
s= (u4
,
v4)
represent the other two fractures, respectively.
Supposing the two line segments run from
e1
to
e1+r
and from
e2
to
e2+s
. The main steps are as
follows [23]:
(1)
If
r×s=
0 and
(e2−e1)×r=
0, then the two lines are collinear. If either 0
≤(e2−e1)•r≤r•r
or 0 ≤(e2−e1)•s≤s•s, then the two lines are overlapping.
(2)
If
r×s=
0 and
(e2−e1)×r=
0, but neither 0
≤(e2−e1)•r≤r•r
nor 0
≤(e2−e1)•s≤s•s
,
then the two lines are collinear but disjoint.
Energies 2016,9, 519 6 of 15
(3) If r×s=0 and (e2−e1)×r6=0, then the two lines are parallel and non-intersecting.
(4) If r×s=0 and 0 ≤t,u≤1, the two line segments meet at the point e1+tr =e2+us.
(5) Otherwise, the two line segments are not parallel but intersect.
Using this method, all of the fracture intersection points can be found in DFN, which are shown
in Figure 5, and marked by the red points.
−2 −1 0 1 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Sample fracture length (meter)
Sample fracture width (meter)
Fracture intersection
Figure 5. The fracture intersection points of discrete fracture networks.
3.3. The Fracture Weighted Formula
During hydraulic fracturing, energy can well explain the fracture propagation, which includes
the elastic modulus, normal stress, Poisson’s ratio and so on [
1
]. Hence, the fracture weighted
formula with energy will be innovatively deduced in this section. Energy can be divided into
two categories: mechanical energy and interface energy. The mechanical energy is for keeping
natural fractures open, and it is related to the fracture length. Interface energy is for creating new
fractures, which is related to the fracture length [
24
]. The energies during fracture propagation are
described by weighted value in the shortest path problem. Considering fracture length, fracture
type and so on, the weighted values of fracture are different from each other. Mechanical energy
should be considered in natural fractures. However, for new fractures, both mechanical energy and
interface energy should be considered during computing the weight values of new fractures [
24
].
The derivation processes of the fracture weighted formulas with energy are described as follows:
Assuming that fracture propagation length is
c
and the normal stress is
P
. The expression is [
24
]:
P=σxx sin2β−2σxy sin βcos β+σyy cos2β, (2)
where S[σxx,σxy,σyy]is in situ stress, and σxx and σyy are the xand ydirection stress. σxy is shear stress
and βis the dip angle, and the schematic describing the fracture and stresses is shown in Figure 6.
Figure 6. Transformation of stress.
Energies 2016,9, 519 7 of 15
Mechanical energy
UM
is the sum of the bearing system potential energy
UA
and elastic strain
energy UE[25]:
dUM=dUA+dUE=−P2dλ(c) + 1
2P2dλ(c) = −1
2P2dλ(c), (3)
G=−dUM/dc =1
2P2dλ(c)/dc, (4)
λ(c) = 2(1−ν)
G·c, (5)
where
G
is the energy per unit area or length [
25
],
c
is the length of fracture,
λ(c)
is elastic compliance,
and νis Poisson’s ratio. Generating Equations (4) and (5) into Equation (3), it is easy to obtain
UM=P2c2(1−ν)
G. (6)
In another case, new fracture length is c, then, interface energy USis:
US=4·c·γ, (7)
where γis the surface energy per unit area or length.
During fracture propagation, the weighted value of natural fracture is computed only by
considering mechanical energy
UM
, which can be better understood by the energy of keeping
the fractures open during gas extraction. The weighted value of new fractures is computed by
combining mechanical energy UMand interface energy US.
4. Result of Numerical Modeling
4.1. Determination of the Hydraulic Fracture
Assume that the direction of the maximum principal stress is horizontal. Predetermine that the
new fracture in DFN has connected its downstream vertex with its upstream vertex. For example,
vector
(u
,
v)
represents the start vertex (indicated by
u(ux
,
uy)
) and the final point (indicated by
v(vx
,
vy)
) of a branch. If
ux
is greater than
vx
, vertex
u
is marked as an upstream vertex. In addition,
vertex
v
is made as a downstream vertex. On the contrary, vertex
v
is marked as upstream vertex.
All the vertices in Figure 4and intersections in Figure 5are the elements of set V.
Select
N
vertices from the set
V
as source points. If the abscissa of these
N
vertices is smaller
than other vertices’, these
N
vertices are denoted as initial vertices. In addition, found
N
vertices are
also regarded as terminal points, if the abscissa of these
N
vertices are larger than other vertices’.
Assume
N=
12. The initial vertices and terminal points can be obtained in this DFN by comparing
their abscissa values, which are shown in Figure 7. The vertices marked by red ’*’ are the start
vertices (indicated by
v1
,
v2
, ...,
v12
), and the terminal points are marked by a blue ’*’ (indicated by
u1,u2, ..., u12).
Energies 2016,9, 519 8 of 15
−2 −1 0 1 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Sample fracture length (meter)
Sample fracture width (meter)
Source point and Terminal point
Figure 7. The start vertex and terminal vertex of discrete fracture network.
Basing on
Definition 3
of Section 2, the DFN is a collection of three sets, indicated by
G(V
,
E
,
W)
.
All endpoints of the fracture in Figure 3and intersection points in Figure 4are the element of set
V=v1,v2, .., vn
, where
vj+12 =uj(j=
1, 2, ..., 12
)
, and
V1=v25, v26, .., vn
describe the intersection
points. The element of set
E
is composed of the fracture in DFN. Set
W
is computed by Equations
(6)
and
(7)
in Section 3.3. Take in situ stress
S[σxx
,
σxy
,
σyy] = [
6.0
×
10
7Pa
, 1.0
×
10
7Pa
, 5.0
×
10
7Pa]
in Equation
(2)
, Poisson’s ratio
ν=
0.35 in Equations
(5)
and
(6)
,
γ=
1.0
×
10
6J/m
in Equation
(7)
. Based on
Definition 2.2–2.3
in Section 2,
pij = (vi
,
uj)
presents the weight from
vi
to
uj(i
,
j=
1, 2, ..., 12
)
, and
δ(u
,
v)
represents the shortest path with the minimum sum of the fracture weights.
Using Dijkstra’s algorithm with energy, the hydraulic fracture of DFN can be obtained. The main
steps of algorithm are as follows:
Step 1. Set x=1.
Step 2. For the initial vertex vx, initialize S={vx},T=V\S,dist[vi]and ver[x]:
S[vi] =
0, vi/∈S
1, vi∈S
,
dist[vi] =
UM,i6=x&(v0,vi)∈E&nat ure f rac ture
UM+US,i6=x&(v0,vi)∈E&new f r acture
∞,(v,vi)/∈E
0, i=x&(vx,vi)∈E
i=1, 2, ..., n,
and
ver[x] = 0, x=1, 2, ..., 12.
Step 3.
Select the vertex
vi
in
T
, such that
dist[vi] = min
vj∈Tdist[vj]
. If
dist[vi] = ∞
, stop, else go to
Step 4.
Step 4. Set S=S∪vi,T=T\vi, and S[vi] = 1. If T=∅, stop, else go to Step 5.
Step 5. Update dist[vj] = min{ω0j,dist[vi] + ωij }. Go to Step 3.
Step 6. Find px j , and let Px=min
j=1,2,...,12{pxj }.
Step 7. Set ver[x] = 1, x=x+1 and go to Step 2 until x=12.
Energies 2016,9, 519 9 of 15
Step 8. δ(u,v) = min
j=1,2,...,12{Pj}.
In addition, the algorithm flow chart is shown in Figure 8. From the above steps, it can be
shown that the potential distances are now recorded as updated edge weights so that no additional
memory is required. Through combining the vector method and geological parameter with Dijkstra’s
algorithm, the hydraulic fracture of DFN is obtained, which is marked by a red color in Figure 9.
Figure 8. The flow chart of the completed algorithm.
−2 −1 0 1 2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Sample fracture length (meter)
Sample fracture width (meter)
The shortest path
Figure 9. The hydraulic fracture of the discrete fracture network.
4.2. Experiment and Comparison
The sample is sampled in the Jiaoshiba shale gas field, Sichuan Basin, and the sample’s detail
is shown in Figure 10. We drilled a hole with diameter 10 mm and depth 50 mm in the center of
sample. During the hydraulic fracturing test, the axial stress and confining stress are 30 MPa and 20
MPa, respectively, and the injection rate is 30 ml/min.
Energies 2016,9, 519 10 of 15
200 mm
100 mm
50 mm
10 mmdrill hole
Figure 10. The dimensions of the sample.
For comparison, the parameters in Section 4.1 are equal to the experimental conditions.
The failure morphology of the sample after the experiment is shown in Figure 11a. It is worth
noting that the sample has obvious horizontal bedding faces. The direction of natural fractures in
the numerical model is along these horizontal bedding faces. As demonstrated in Figure 11a, it can
be shown that the hydraulic fracture propagates through the whole sample along the direction of
maximum horizontal in situ stress. Figure 11b shows the simulation results using the method above
in Section 4.1. It can be shown that the hydraulic fracture, obtained by the method in this paper, is
similar to the the failure morphology of the sample.
(a)(b)180
160
140
120
100
80
60
40
20
31 157 283
Length (mm)
Width (mm)
The Primary Frature
Figure 11. (a) the failure morphology of the sample; (b) the simulated fracture.
4.3. Parameter Sensitivity Analysis
During fracture propagation, several factors can influence the propagation of hydraulic fracture
in a discrete fracture network system, such as intensity of discrete fracture (fracture number, and
Energies 2016,9, 519 11 of 15
marked with
n
), fracture length (marked with
l
), and dip angle (marked with
α
). In this section,
the weighted value of hydraulic fracture is used to explore how these parameters influence the
hydraulic fracture propagation. Here, the DFN is a two-dimensional horizontal section of a rock
with 4 m length and 4 m width. Assuming that the DFN is static, partially connected and randomly
distributed. The maximum principal stress direction keeps in line with the
x
-axis, and many groups
of fractures are repeated computing for eliminating the influence of randomness. Figures 12–15
show quantitative comparison of the behavior of the models based on different parameters.
175 200 225 250 275 300 325 350
1
2
3
4
5
6
7x 106
The number of fracture
The hydraulic fracture weighted value
175 200 225 250 275 300 325 350
2.5
3
3.5
4
4.5
5x 106
The number of fracture
The hydraulic fracture weighted value
(a) (b)
Figure 12.
The influence of fracture number (
n
) on the hydraulic fracture weighted value. Here,
α=
30
◦
,
l=
0.2 m, and the angle between two groups fracture remains orthogonal. Eight kinds of
fracture numbers (175, 200, 225, 250, 275, 300, 325 and 350) have been analyzed, with 150 groups of
randomly generated fractures for each kind of fracture number. (
a
) scatter distribution of the the
hydraulic fracture weighted value with different values of
n
; (
b
) average of the hydraulic fracture
weighted value with different values of n.
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.5
0
1
2
3
4
5
6
7x 106
The length of fracture (meter)
The hydraulic fracture weighted value
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
1
2
3
4
5
6x 106
The length of fracture (meter)
The hydraulic fracture weighted value
(a) (b)
Figure 13.
The influence of fracture length (
l
) on the hydraulic fracture weighted value. Here,
α=
35
◦
,
n=
250, and the angle between two groups fracture keeps orthogonal. In addition, eight
kinds of fracture lengths (0.15 m, 0.2 m, 0.25 m, 0.3 m, 0.35 m, 0.4 m, 0.45 m and 0.5 m) have been
analyzed, with 150 groups of randomly generated fractures for each kind of fracture length. (
a
)
scatter distribution of the hydraulic fracture weighted value with different values of
l
; (
b
) average of
the hydraulic fracture weighted value with different values of l.
Energies 2016,9, 519 12 of 15
0 10 20 30 40 50 60 70 80
2
2.5
3
3.5
4
4.5
5
5.5 x 106
The angle (degree)
The hydraulic fracture weighted value
0 10 20 30 40 50 60 70 80
2
2.5
3
3.5
4
4.5
5
5.5 x 106
The angle (degree)
The hydraulic fracture weighted value
(a) (b)
Figure 14.
The influence of dip angle (
α
) on the hydraulic fracture weighted value when the angle
between two groups fracture remains orthogonal. Here,
n=
250 and
l=
0.2 m. The realizations
belong to fifteen kinds of
α
(0
◦
, 6
◦
, 11
◦
, 17
◦
, 22
◦
, 28
◦
, 34
◦
, 40
◦
, 46
◦
, 52
◦
, 57
◦
, 63
◦
, 69
◦
, 74
◦
and 80
◦
),
with 150 groups of randomly generated fractures for each kind of
α
. (
a
) scatter distribution of the
hydraulic fracture weighted value with different values of
α
; (
b
) average of the hydraulic fracture
weighted value with different values of α.
Firstly, the influence of fracture number on the weight value of hydraulic fracture is shown in
Figure 12. The trends of scatter distribution and average distribution are consistent with each other,
and the weight value of hydraulic fracture decreases with the increasing of fracture number. That
also means that the greater the fracture, the better for the hydraulic fracture propagation. Then,
Figure 13 shows that the weight value of hydraulic fracture decreases with increasing of fracture
length. It also indicates that as the fracture length is growing, the weight value of the hydraulic
fracture is reducing. Longer fracture length will benefit the hydraulic fracture propagation. Finally,
as demonstrated in Figure 14, it has little influence on the weight value of hydraulic fracture for
different values of
α
when the angle between two groups of fracture keeps orthogonal. Hence, the
influence of different values of
α
on the weight value of hydraulic fracture will also be studied under
two groups of fractures are not orthogonal. Figure 15 shows the weight value of the hydraulic
fracture firstly decreases and then increases with the increase of
α
when the angle between two
groups of fractures are 20
◦
. That being said, the farther away from the maximum stress direction,
the larger the weight value of the hydraulic fracture.
0 9 18 27 36 45 54 63 72 81
3
4
5
6
7
8
9
10
11
12 x 106
The angle (degree)
The hydraulic fracture weighted value
0 9 18 27 36 45 54 63 72 81
4
5
6
7
8
9
10
11 x 106
The angle (degree)
The hydraulic fracture weighted value
(a) (b)
Figure 15.
The influence of dip angle (
α
) on the hydraulic fracture weighted value when the
angle between two groups of fracture are 20
◦
. Here,
n=
250 and
l=
0.2 m. Ten kinds of
α
(0
◦
, 9
◦
, 18
◦
, 27
◦
, 36
◦
, 45
◦
, 54
◦
, 63
◦
, 72
◦
and 81
◦
) were simulated, with 150 groups of fractures randomly
generated for each kind of
α
. (
a
) scatter distribution of the hydraulic fracture weighted value with
different values of
α
; (
b
) average of the hydraulic fracture weighted value with different values of
α
.
Energies 2016,9, 519 13 of 15
5. Discussion
The operating parameters are effective only when the geological parameters are favorable
for fracturing treatment. Therefore, the evaluation of the geological parameters before hydraulic
fracturing is crucial to hydraulic fracturing treatment in shale formation. The effects of geological
parameters are investigated, including fracture density, fracture length, and dip angle (the angle
between one group of fracture and the maximum principal stress direction). By analyzing the
sensitivity of DFN with various parameters, it can be concluded that the hydraulic fracture weight
value decreases with increase in the intensity of fracture and fracture length, and increases with
increasing of the angle between fractures to the maximum principal stress direction. It means that
the fracture density and fracture length are significant to the DFN connectivity.
6. Conclusions
Increases in natural gas extraction are being driven by rising energy demands, mandates for
cleaner burning fuels, and the economics of energy use [
26
]. Directional drilling and hydraulic
fracturing technologies are allowing expanded natural gas extraction from organic-rich shales [
27
],
while the propagation of the hydraulic fracture is the crux of hydraulic fracturing, and can provide a
high-conductivity path for methane migration in the low permeability reservoir. Accompanying the
benefits of such extraction [
28
] are public concerns about hydraulic fracturing that are ubiquitous
but lack a strong scientific foundation. In this paper, Dijkstra’s algorithm, energy theory and vector
methods were combined to simulate the hydraulic fracture propagation in DFN during hydraulic
fracturing. We successfully give the shortest path weighted value formula of fracture propagation,
and obtained the hydraulic fracture in DFN during hydraulic fracturing. The direction of the
hydraulic fracture is also simulated, which plays a key role in identifying the spatial position of
the well bore during hydraulic fracturing. Numerical results with experimental data show that the
hydraulic fracture obtained by our method is similar to the failure morphology of the sample (see
Figure 11).
This study has played a very important role in the hydraulic fracturing field, including the
following aspects: (1) obtaining the extension direction of the fracture; through comparing the
energies of fracture propagation in different directions, the direction of the minimum energy fracture
propagation is the most likely direction of the hydraulic fracture; (2) evaluating the complexity of
the DFN. Through Dijkstra’s algorithm, several paths can be obtained whose energies are smaller. If
the positions of these paths are very near, it is shown that the hydraulic fracture is a great advantage
and it is harder to get a complex fracture network. On the contrary, if the positions of these paths are
very far away, it is shown that the hydraulic fracture lacks strength and it is easy to get a complex
fracture network; and (3) through computing the weight value of hydraulic fracture, the injected
energy during hydraulic fracturing can be obtained, which is strongly related with pore pressure
and fluid viscosity. During hydraulic fracturing treatment, the pore pressure decreases with the
distance far away from the well bore, and this factor has been considered while computing fracture
weight value.
Acknowledgments:
We thank Yu Wang, Zhaobin Zhang, Bo Zhang and Mingtao Li for their help with the
experiment. This work was supported by the National Natural Science Foundation of China (Grants Nos.
41227901, 41502294, 41502306) and the Strategic Priority Research Program of the Chinese Academy of Sciences
(Grants Nos. XDB10030300 and XDB10050400).
Author Contributions:
Yanfang Wu and Xiao Li conceived and designed the experiments; Yanfang Wu
performed the experiments; Yanfang Wu analyzed the data; Yanfang Wu and Xiao Li contributed
reagents/materials/analysis tools; Yanfang Wu and Xiao Li wrote the paper.
Energies 2016,9, 519 14 of 15
Conflicts of Interest: The authors declare no conflict of interest.
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