Vol.2, No.9, 1030-1034 (2010) Natural Science
Copyright © 2010 SciRes. OPEN ACCESS
Orthogonal optimization of horizontal well fracturing
Chi Dong*, Kao-Ping Song, Xiao-Qi Chen
Key Laboratory of Ministry of China on Enhanced Oil and Gas Recovery, Da Qing, China;
*Corresponding Author: dongchidongchi@163.com
Received 18 May 2010; revised 29 June 2010; accepted 5 July 2010.
In many applications, streamline simulation sh-
ows particular advantages over finite-difference
simulation. By the advantages of streamline
simulation such as its ability to display paths of
fluid flow and acceleration factor in simulation,
the description of flooding process gets more
visual effect. The communication between wells
and flooding area has been represented appro-
priately. In low permeability reservoirs horizon-
tal wells have been applied to stimulate. And
some horizontal wells have been fractured. But
different fracturing methods are followed by
different effects. Through streamline simulation
this paper presents that the fractures improve
both the productivity of horizontal well and the
efficiency of the flooding unit. And the syntag-
matic relation of distance between artificial frac-
tures, fracture length and fracture conductivity
has been given by orthogonal experiment. This
method has been applied to the D160 reservoir
in Daqing Oilfield. The production history of this
reservoir is about 2 years. The reservoir is main-
tained above bubble point so the simulation
meets the slight compressibility assumption.
New horizontal wells are fractured followed from
this rule.
Keywords: Streamline Simulation; Horizontal Well;
Using horizontal wells to stimulate and enhance oil re-
covery, has been applied widely all over the world. It is
necessary to use horizontal wells to develop low perme-
ability reservoirs. In the long process of practice, it was
discovered that the advantages of horizontal well would
not be made full use of when only the horizontal well
drilling techniques not the combination with the actual
development conditions has be paid attention to. In low
permeability reservoirs the horizontal wells need to be
fractured to stimulate. As the increase of the length of
horizontal section, the number of artificial fractures
should be increased too. And the syntagmatic relation of
distance between fractures, fracture length and fracture
conductivity is necessary to be optimized.
As the application of high-resolution geologic models
in simulations, more attention had been paid to the use
of simulation in oil field development. Finite difference
method which applied widely cannot meet the needs
such as high computational efficiency and being visually
appealing. So the 3D streamline simulation approach has
been applied to simulation as complementary to finite
difference simulation techniques [1-4].
Streamline simulation is in the advanced and matured
stage in slightly compressible system. It calculates the
saturation via 1D streamline instead of saturation field
[5]. Both computational efficiency and matching high
resolution geologic models becomes possible. Not like
the unity of the saturation in finite difference grid cell,
streamline simulation provides several values of the
saturation in one grid cell, then the fineness of the solu-
tion gets elevated.
Taking advantage of streamline simulation’s particular
capability, the syntagmatic relation of distance between
artificial fractures, fracture length and fracture conduc-
tivity has been optimized by orthogonal experiment
2.1. Velocity Field Solution and Streamline
The streamline model is a model with the assumptions as
follows: 1) Considering the effects of gravity and capil-
lary; 2) There are oil phase and water phase in the res-
ervoir; 3) The fluid flow in the reservoir is incompressi-
C. Dong et al. / Natural Science 2 (2010) 1030-1034
Copyright © 2010 SciRes. OPEN ACCESS
ble; 4)The fluid flow obeys Darcy’s Law; 5)The flow is
under an isothermal condition. On the basis of these as-
sumptions, and utilizing the continuity equation and flow
equation, the pressure equation of streamline model can
be established as follows:
twv 0cow
 
 
 (1)
where λo, λw are, respectively, mobility of oil and water,
μm2/Pa·s; λt is total mobility, μm2/Pa·s; γo, γw are, re-
spectively, gravities of oil and water, dimensionless; qvo,
qvw are volume flow rate of producer and injector, m3/s;
qv is the total flow rate of injection or production wells,
m3/s; D is the depth from the reference level, its direc-
tion is the same to gravity acceleration, m; pcow is the
capillary force between oil phase and water phase, Pa.
Eq.1 can be solved by finite difference method. And
the pressure distribution, i.e. pressure field, can be de-
According to the pressure field, streamline tracing is
implemented by using Pollock method which is defined
with the following assumptions: the velocity field in a
grid cell as a linear interpolation, and the velocity in
each direction is only a function of the coordinate in that
In a 2D grid system, with the pressure at the face, the
potential at the face of the grid cell can be calculated.
Then the velocity in each direction is derived, the TOF
can be calculated also. The face at which the streamline
exits is the one corresponding to the minimum positive
exit time. The exit point will be the entry point of the
adjacent grid cell, repeating this calculation process until
the result converges to the grid cell which a production
well is in. Connecting the exit points in chronological,
an integrated streamline can be derived.
In a 3D system, the exit points can be obtained by
utilizing Pollock method to calculate the values in the
z-direction. By connecting these points, production and
injection wells, the streamline can be derived.
The advantages of this method: being analytical; and
it constructs a velocity field that satisfies the flux con-
servation condition.
2.2. Saturation Field
After the pressure filed is solved and the streamlines are
traced by utilizing finite difference method, the next step
is to solve the saturation along the streamlines by intro-
ducing the time-of-flight concept. The time-of-flight is
defined as the time required for a particle to travel a dis-
tance along the streamline,
 
In streamline tracing, the time for streamlines across
each grid cell, Δte,i , can be calculated using Pollock
method. The time of flight can be expressed as follow:
where nblocks designates the number of grid cells when
a particle travel a distance as s; Δte,i is the time for a par-
ticle to travel through the i’th cell, s.
A. Distribution of volumetric flux of streamline.
Tracing streamline from the cell of injection well to the
cell of production well. For a grid cell with source or
sink, as the streamline is not linear in segments, the
streamline generalizes from the face of the grid cell in-
stead of generalizing from the centre of the grid cell.
For simplicity, distribution of flux is based on chang-
ing the number of streamlines with different flow rate of
injection well, and the volumetric flux along each
streamline is constant. The higher injection flow rate, the
more streamlines. Similarly, much more streamlines can
be traced in a high flow rate region.
In grid cell (i,j,k) with a source, the volumetric flux at
the interface (i ± 1/2,j,k) of grid cell (i,j,k) and grid cell (i
± 1,j,k) is Qi ± 1/2,j,k. The volumetric flux along each
streamline is qsli ± 1/2,j,k at the interface(i ± 1/2,j,k). The
number of streamlines generalized from this face is ni ±
1/2,j,k can be expressed as follow:
12, ,12, ,12, ,ijk ijkslijk
where Qi ± 1/2,j,k is given as:
12, ,12, ,, ,1, ,i jkijkijkijk
 
 (5)
where TXi ± 1/2,j,k represents the transmissibility in the
x-direction of oil phase and water phase between the
grid cell (i ± 1,j,k) and the grid cell (i,j,k); Pi,j,k is the
pressure in the grid cell (i,j,k).
Similarly, the number of streamlines on other faces
can be obtained.
B. Establishing the saturation equation in streamline
model. Considering capillary effect, the fluid flow equa-
tion can be expressed as follow:
itw 0cow
  (6)
Combining with saturation equation and mass con-
servation equation, the saturation equation of streamline
model for oil-water two phases can be derived:
ww cowvw
 
 
C. Dong et al. / Natural Science 2 (2010) 1030-1034
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Simplified as:
ww w
Sf q
 (8)
where w
G is defined as:
www cow
 
C. Numerical solution for saturation equation of
streamline model. Because the derived equation is a
complicated convective diffusion equation, for simplicity,
solve the convection term, gravity and capillary item in a
saturation equation by utilizing the technique of operator
separation. The convective diffusion equation can be
divided to two parts, one is a nonlinear hyperbolic equa-
tion describing the convection term, the other one is a
parabolic equation describing the effects of gravity and
As the streamlines generate from the grid cell face of
injection well and dissolved at the grid cell face of the
production well, the field along streamlines is source-
Decoupling (8) by utilizing operator separation tech-
nique, two equations are obtained as follows:
 
 
Eq.10 is a 1D saturation equation, and can be solved
by 1D numerical solution along streamlines (transforma-
tion of the 3D equation to 1D equation). The solution for
(10) is assumed as Sf(t). Eq.11 is a saturation equation
considering the effects of gravity and capillary, and can
be solved only by 3D numerical solution. The solution
for (11) is assumed as H(t). Finally, the solution at time
tn(tn = nΔt) is derived as:
 
 
2.3. Streamline Update
For the actual development process in the oil field, well
pattern and production system is not fixed. Especially
for the oil filed with very long production periods, we
often need to establish some methods to stimulate, such
as pattern adjustment, shut-in, isolation of individual
zones and water shut off, fracturing and acidizing, in-
filling, which will change the streamline distribution. So
we have to update the streamlines immediately after
those conditions to accurately represent the displacing
dynamic information.
2.4. Compare with Finite Difference
Compare the water cut of streamline simulation and fi-
nite difference simulation, it is easy to know that the
streamline simulation’s result is more close to the his-
torical data at initial stage. This is because the improve-
ment on saturation calculation. So streamline simulation
is more suitable for study on flooding.
3.1. Simulation of Fractured Horizontal
The flow field of flooding unit in combined well pattern
of horizontal and vertical wells is drew basing on the
result of streamline simulations as Figure 1. Respec-
tively, the situation with fractured horizontal well and
the one with unfractured horizontal well are simulated.
As shown in Figure 1, the fractures inflect the flow field
around the middle of horizontal section. As the former
study, the inflow rate is lowest in the middle of horizon-
tal section. So the fractures improve both the productiv-
ity of horizontal well and the efficiency of the flooding
3.2. Orthogonal Experimental Design
In flooding unit contains fractured horizontal well, the
main factors affecting the sweep efficiency are distance
between fractures, fracture half length and fracture con-
ductivity. In orthogonal design experiments, influence
factors are called factors, and the data points of factors
Figure 1. Streamlines representing oil saturation.
C. Dong et al. / Natural Science 2 (2010) 1030-1034
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called levels. There are three factors which effect devel-
opment, so using L9 (33)-type orthogonal to arrange ex-
Basing on the orthogonal experimental design table,
orthogonal experiment is designed as Table 2.
3.3. Simulation of Orthogonal Experiment
Simulate projects and record the sweep efficiency when
water cut is 98%. The results of streamline simulation
show that NO.6 project has the highest sweep efficiency
with the number 88.57%. But the results of simulation
cannot distinguish the sequence of priority of factors. To
study the impact on sweep efficiency by factors the
range of factors should be analyzed.
3.4. Analysis of Factors
According to the size of range, the impact of factor on
the size of the experimental results can be determined. In
orthogonal experiment method, range is the difference
between the maximum and minimum values at different
levels of the same factor. The bigger range a factor has,
the more it influences the size of the experimental result.
Before the analysis of the sequence of priority of fac-
tors, the experimental result should be processed as fol-
lows: 1) Sum the values of results in same level of one
Table 1. Factors and levels.
Level Fracture Half
Length (m)
Distance between
Fractures (m)
1 75 25 100
2 100 35 150
3 125 45 200
Table 2. Orthogonal experiment.
No. Fracture Half
Length (m)
Conductivity (D·cm)
Distance between
1 75 25 100
2 100 25 150
3 125 25 200
4 75 35 150
5 100 35 200
6 125 35 100
7 75 45 200
8 100 45 100
9 125 45 150
factor. As there are three levels in one factor, K1, K2 and
K3 are used to indicate the sums respectively. 2) Average
the values of results in same level of one factor. Use k1,
k2 and k3 to indicate the averages respectively. 3)
Solve the size of range of each factor, and use R to in-
dicate it.
According to the above rulesthe ranges is calculated
as Table 4. It shows, the range of factor A is biggest with
Table 3. Results of simulation.
Level Sweep Efficiency
1 0.8837
2 0.8493
3 0.7684
4 0.8808
5 0.8076
6 0.8857
7 0.8680
8 0.8635
9 0.7823
Table 4. The range of experimental result.
A B C Measurement
No. Fracture Half
Length (m)
1 75 25 100 0.8837
2 100 25 150 0.8493
3 125 25 200 0.7684
4 75 35 150 0.8808
5 100 35 200 0.8076
6 125 35 100 0.8857
7 75 45 200 0.8680
8 100 45 100 0.8635
9 125 45 150 0.7823
K1 2.632 2.501 2.633
K2 2.521 2.574 2.512
K3 1.654 2.514 2.444
k1 0.877 0.834 0.878
k2 0.840 0.858 0.837
C. Dong et al. / Natural Science 2 (2010) 1030-1034
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k3 0.551 0.838 0.815
R 0.326 0.024 0.063
the value 0.326. In consequence, the main factor effect-
ing the sweep efficiency is the fracture half length, fol-
lowed by distance between fractures and the weakest
which is fracture conductivity. The sequence of priority
of factors is as follow: A > B > C.
In this paper, the streamline model considering fractured
horizontal well is derived. The streamlines provide us
with a clear picture of flow field of flooding unit in
combined well pattern of horizontal and vertical wells.
The streamline simulation results indicate that the
fracture plays an important role in enhancing sweep effi-
ciency and stimulation.
The sequence of priority of distance between fractures,
fracture length and fracture conductivity has been de-
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