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SPE 151960
Predicting Well Performance in Complex Fracture Systems by Slab Source Method Jiajing Lin, SPE, and Ding Zhu, SPE, Texas A&M University
Copyright 2012, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Hydraulic Fracturing Technology Conference held in The Woodlands, Texas, USA, 6–8 February 2012. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessar ily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.
Abstract Multiple hydraulic fracture treatments in reservoirs with natural fractures create complex fracture networks. Predicting well
performance in such a complex fracture network system is an extreme challenge. The statistical nature of natural fracture
networks changes the flow characteristics from that of a single linear fracture. Simply using single linear fracture models for
individual fractures, and then summing the flow from each fracture as the total flow rate for the network could introduce
significant error. In this paper we present a semi-analytical model by a source method to estimate well performance in a
complex fracture network system. The method simulates complex fracture systems in a more reasonable approach. We
statistically assigned a fracture network of natural fractures, based on the spacing between fractures and fracture geometry.
We then added multiple dominating hydraulic fractures to the natural fracture system. Each of the hydraulic fractures is
connected to the horizontal wellbore, and some of the natural fractures are connected to the hydraulic fractures through the
network description. Each fracture, natural or hydraulically induced, is treated as a series of slab sources. The analytical
solution of superposed slab sources provides the base of the approach, and the overall flow from each fracture and the effect
between the fractures are modeled by applying the superposition principle to all of the fractures. The fluid inside the natural
fractures flows into the hydraulic fractures, and the fluid of the hydraulic fracture from both the reservoir and the natural
fractures flows to the wellbore. The finite conductivity of hydraulic fractures is modeled by additional pressure drop inside
the fracture, but it is neglected in natural fractures. This paper also shows that non-Darcy flow effects have an impact on the
performance of fractured horizontal wells. In hydraulic fracture calculation, non-Darcy flow can be treated as the reduction of
permeability in the fracture to a considerably smaller effective permeability. The reduction is about 2% to 20%, due to non-
Darcy flow that can result in a low rate.
The semi-analytical solution presented can be used to efficiently calculate the flow rate of multistage-fractured wells.
Examples are used to illustrate the application of the model to evaluate well performance in reservoirs that contain complex
fracture networks.
Introduction North America has had a substantial growth in its unconventional oil and gas market over the last two decades. The primary
reason for that growth is because North America, being a mature market, is beyond the peak production from its conventional
hydrocarbon resources. New technology applications of multi-fractured horizontal wells allow us to produce at economical
rates from these low permeable oil and gas resources. Since commercial exploration and production of oil and gas reservoirs
began, there have been circumstances where the reservoir character or depositional model has caused difficulty in
assessment. Production assessment of unconventional reservoirs using standard methodology has been problematic. The
complexity of the fractured system posts the challenges to analytical models, and reservoir simulation of such a system is
extremely time-consuming.
Over the past decades, point source integrated over a line and/or a surface has been mostly used in solving single-phase
flow problems in porous media when fluid movement is from a complex fractured well system; horizontal well models with
point source solution have been presented in many literatures. Gringarten and Ramey’s work (1973) is the first application of
the Green’s source function to the problem of unsteady-state fluid flow in the reservoirs. They introduced Green’s functions
for a series of source and boundary conditions. They used the integration of the response to an instantaneous source solution
to get the response for a continuous source solution. The application of the Newman’s principle in breaking a problem of 3-D
into the product of three 1-D solutions is also discussed in their work. Gringarten et al. (1978) applied the Green’s function
2 SPE 151960
later to the unsteady state pressure distribution created by a vertical fractured well with infinite conductivity fracture. By
dividing the fracture into N segments, a series of equations had been solved to calculate the pressure distribution and
contribution of each segment to the total flow by assuming each segment as a uniform flux source. Cinco-Ley el al. (1981)
used the Green’s function with Laplace transform to develop a model of finite conductivity vertical fracture in an infinite
reservoir. They presented a new technique for performing pressure transient analysis for vertical finite-conductivity fractures
using a bilinear flow model. Ozkan (1995) developed a point source solution in Laplace domain in order to remove the
limitations of the Gringarten and Ramey’s model in considering the wellbore storage and skin effects. By integrating the
point source to line source, Babu and Odeh (1988) developed a solution to predict horizontal well performance in a closed
reservoir. The model is under pseudo-steady state condition. One of the limitations of this method is that the well must be
parallel to the reservoir boundary. Goode and Kuchuk (1991) introduced a solution for productivity of a horizontal well in a
reservoir with no-flow boundary and constant pressure boundary. Their solution is expressed in the form of an infinite
condition. A simplified solution for a short well was developed in their study. Ouyang (1997) presented a 3D horizontal well
model to describe wellbore pressure and reservoir pressure change with time and location. The formula is in the Laplace
space. The transient pressure behaviors in physical space can be easily obtained by means of the Stehfest algorithm. Amini
and Valko (2007) developed a method with distributed volume sources to simulate fractured horizontal wells in a box-shaped
reservoir. A source term was added to the diffusivity equation to calculate the pressure distribution. Then the production rate
from a fracture is computed. Different from the other point source methods, the volume source approach is able to describe
the pressure behavior inside sources and its influence to the flow field. Magalhaes and Zhu (2007) showed applications of the
volumetric source model and field cases are presented in their work. Meyer and Bazan (2010) presented a comprehensive
methodology using the trilinear solution to predict the behavior of multiple transverse finite conductivity vertical fractures in
horizontal wellbores. Miskimins and Barree (2005) demonstrated that non-Darcy flow effects can influence well productivity
across the entire spectrum of flow rates. They showed that even in low velocity situations, non-Darcy effects can influence
the productivity. Non-Darcy flow can have a major impact on reduction of a propped half-length to a considerably shorter
effective half length, thus lowering the well’s productive capability and overall reserve recovery.
In this paper, we present a different approach to the problem of unsteady state flow of an uncompressible fluid in a
rectilinear reservoir. The model is based on the solution of a series of slab sources. It can be used to calculate well
performance for horizontal gas wells with or without fractures. Fractures can be longitudinal or transverse, single or multiple,
and can be infinite or finite conductive. Using the slab source approach, we assigned the sources (horizontal wells or
fractures) geometry in 3-dimensions and the effect of pressure behavior inside sources are considered by superposition
principles. This method is relatively easy to apply because flow rate could be calculated directly from pressure difference
between initial reservoir pressure and pressure in the fracture, which is the same as wellbore flow pressure for an infinite
conductivity fracture.
Methodology Method of slab sources. The slab source method solves the flow problem in a parallellpiped porous medium with a slab
source, s, placed in the domain, as shown in Fig. 1. The porous medium is assumed to be an anisotropic reservoir. Following
the same approach as the conventional point source solution to apply Newman’s principle, the three-dimensional pressure
response of the system to an instantaneous source can be obtained as the production of the solutions of three one-dimensional
problems from each principal direction. With slab source, it promises a smooth transition from transient flow regime to
pseudo-steady state regime.
For uniform flux condition, the solution for the flow problem of the system shown in Fig. 1 (Gringarten, 1973) is
dtsssc
Bqtzyxppp
t
zyx
t
0
int ,,,
(1)
where
2
22
1
coscos2
sin14
1
e
x
n ee
o
e
f
f
e
e
f
xx
knxpe
x
xn
x
xn
x
xn
nx
x
x
xs
Here, B is the formation volume factor, is the fluid viscosity, q is the flow rate at the point (x,y,z), and sx, sy, and sz are the
one-dimensional instantaneous infinite slab source functions (Green’s functions). We use a no-flux boundary condition, for
example, sy, sz has the same format as sx but in y and z directions. The details of the model are discussed by Lin (2010). Eq. 1
can be easily used to describe a different problem. For a single fracture as shown in Fig. 2, the solution is
dxdydzdssswhxc
Bqtzyxpp
z
z
y
y
x
x
t
zyx
fft
2
1
2
1
2
1 0
int2
,,, (2)
where xf is the half length of the fracture, hf is the height of the fracture, and w is the width. In this case, flow into the
horizontal wellbore is neglected.
SPE 151960 3
Fig. 1—Schematic of the slab source model. Fig. 2—Schematic of a single fracture.
Fracture system in Slab Source Solution. For fractures, we divided each fracture into N*N segments, each segment has a
flow rate from the reservoir. For example, we have a fracture with 25 segments as shown in Fig. 3a. The segments are in
contact with each other. Because of the flow rate in the segment 1, the pressure for the other 24 segments will change.
Similarly, the flow rate of the second segment would change the pressure distribution in the other 24 segments. With N*N
segments, we have a set of N*N linear equations for pressure response to the flow in the fracture. The horizontal well takes
flow rate from the fracture, but not directly from the reservoir.
The pressure drop as a result of each fracture segment produces at a constant rate is
),(,,,int jiFqtzyxpp ji (3)
where F(i, j) represents the right-hand side of Eq. 2.
dxdydzdsssjiF
z
z
y
y
x
x
t
zyx 2
1
2
1
2
1 0
),( (4)
The pressure at segment i as a result of the production, qj, at segment j is evaluated by multiplying qj with F(i,j). For the
entire fracture (N segment), we obtain a set of linear equation shown as,
1321 ),1(...)3,1()2,1()1,1( pNFqFqFqFq N
2321 ),2(...)3,2()2,2()1,2( pNFqFqFqFq N
3321 ),3(...)3,3()2,3()1,3( pNFqFqFqFq N
.
.
.
NN pNNFqNFqNFqNFq ),(...)3,()2,()1,( 321
(5)
where, qj is a constant flow rate flow into segment j and ∆pi is the pressure drop calculated at segment i as a result of the
production into every segment. The total production from the fracture is
total
n
j
j qq 1
(6)
By using the above method, we can calculate the fractured horizontal well performance in uniform flux boundary conditions,
infinite boundary conditions, and finite boundary conditions. The well system is predicted by either a constant flow rate
constraint or constant wellbore pressure constraint. The inflow distribution along the wellbore or fracture depends on the
inner boundary conditions.
Solution for Infinite Conductivity Condition. For infinite conductivity fractures, we divided the source into N*N segments.
For a source under infinite conductivity, uniform pressure over the source is assumed. Figure 3a shows an example of how
the source is discretized into 25 segments. For the infinite conductivity inner boundary condition, the wellbore pressure is
constant along the fracture. The right-hand side in Eq. 7 has the same pressure (Eq. 8). The set of liner equations could be
solved.
125321 )25,1(...)3,1()2,1()1,1( ppFqFqFqFq i
225321 )25,2(...)3,2()2,2()1,2( ppFqFqFqFq i
325321 )25,3(...)3,3()2,3()1,3( ppFqFqFqFq i
.
.
.
2525321 )25,25(...)3,25()2,25()1,25( ppFqFqFqFq i (7)
s
a
b
h
x
z
y
(x,y,z)
a
b
h
x
z
y
4 SPE 151960
252 ... pppi (8)
Solution for finite conductivity condition. For the cases of fracture with finite conductivity the similar approach is used as
the infinite conductivity case except we have to introduce another term to account for the pressure drop inside fracture
between source segments because of the source conductivity.
We first define the inner boundary condition at the interface of the source and the domain (for example, the wellbore and
the reservoir). Then we divide the fracture into multiple segments. The segments are then connected to each other by super
position in the space. By using this technique, a set of linear equations is generated and solved to predict the fractured
horizontal well performance. Fig. 3a shows an example for a 25 sources fracture. We first allow source 1 to exist in the
reservoir and let it generate a flow rate of q1 at the location. The flow results in corresponding pressure changes at locations
of sources 2 through 25. Then if we only let source 2 exist, the pressure also changes at all source locations. We can apply
this procedure to all 25 sources in the system.
(a) (b)sdf
Fig. 3—Segment fracture for finite conductivity.
An additional term will be added to calculate pressure drop inside the fracture, which is showed in Fig. 3b. Three circles
are defined in this case. The fluid from the reservoir to the first circle (segs. 1-6, 10-11, 15-16, and 20-25) will flow to the
second circle (segs. 7-9, 12, 14, and 17-19), join the fluid from the reservoir to the second circle, then flow to the center circle
(seg. 13) and finally the total fluid flow to the wellbore. In such a way, we could approximate the flow inside the fracture as
radial flow through porous media (proppant pack). The pressure drop can be calculated by
qsrC
A
kh
Bpp
wA
wf
2
4ln
2
12.141
(9)
where, A is drainage area, CA is shape facture, and is Euler’s constant. The Earlougher’s shape factor is shown in Table 1.
Table 1—SHAPE FACTOR (Earlougher, 1977).
Drainage Area Earlougher Shaper Factor
30.9
21.8
5.38
2.36
The procedures of calculating the flow rate and pressure drop for a finite conductivity fracture has two parts. One is from
hf
2xf
1 5 2 4 3
11 15 12 14 8
16 20 17 19 18
6 10 7 9 8
21 25 22 24 23
hf
2xf w
Circle 1
Circle 2
Circle 3
2
1
1
4
1
5
SPE 151960 5
the reservoir to the fracture, the other is to calculate pressure drop inside fracture by Eq. 9. The two parts are connected to
calculate the flow rate and pressure along the fracture. The general equation for each circle based on Eq. 9 is
circleicircleiinnercirclei Mqpp ,int, (10)
where, circlei is the circle number, pint is the initial pressure at the middle point of each segment, pinner is the inner boundary
pressure of each segment, qcirclei is the flow rate at each segment and M is
2
4ln
2
1
wiA
circlei
rC
AM
(11)
The following equations describe the procedures discussed above. For circle 1
11,1, circlecircleinnercirclei Bqpp (12)
Then in the circle 2, the fluid flows from the out boundary of circle 2 to the middle of the segment, then to the inner boundary
of circle 2. The flow rate will be the fluid flow inside of the segment plus the fluid comes from the circle 1.
22,2, circlecircleinnerciclei Cqpp (13)
22,2, circlecircleicirlceout Eqpp (14)
where, pout is the out boundary pressure or each segment. Finally the fluid flows inside circle 3 from out boundary of circle 3
to the wellbore, it could be written as
33, circlewfcircleout Dqpp (15)
where, pwf is the wellbore pressure. To solve these equations, substituting Eqs. 12, 13, and 15 into Eq. 3 yields
icircleoutcircleinnercircleinner qDCBFpppp 3,2,1,int (16)
Add Eq. 13 and Eq. 14, we have
icircleinnercircleout qCEpp 2,2, (17)
where B, C, D, and E are from Eq. 11 with different rw, CA, and A. In additional, anther two equations are generated
2,3, circleinnercircleout pp (18)
and
1,2, circleinnercircleout pp (19)
The procedure of calculating the production and pressure distribution is showed in Fig. 4. First the values of pinner,circle1 and
pinner,circle2 are assumed to calculate the gas flow rate, and then pout,circle2. Comparing pout,circle2 with pinner,circle1, if they are not the
same, a new pinner,circle1 value is set to pout,2 to iterate until pout,cirlce2 is equal to pinner,circle1. Then we can move to the next step to
calculate pwf,cal. Comparing with the true pwf, if pwf,cal and pwf are same then the flow rate and pressure distribution are
calculated in the fracture.
6 SPE 151960
Fig. 4—Flow chart for pressure drop inside fracture.
Non-Darcy Flow. The existence of non-Darcy effects in the flow of fluids through porous media has been studied by
petroleum industry for many years. In 1901, Forchheimer observed that the deviation from linearity in Darcy’s law increases
with flow rate, where the second term on the right-hand side is the non-Darcy response.
2
fkL
p (20)
When the flow velocity is low, the second term in Eq. 20 can be neglected. However, for high velocities this term becomes
significant, especially for low viscosity fluids. If dividing Eq. 20 by g we obtain
f
f
k
kvL
p1
1 (21)
for non-Darcy flow. Using this form of the equation, with non-Darcy effect, the permeability will decrease as flow rate
increases. To count this effect, an effective permeability (determining the actual pressure drop) is defined as
f
f
efffk
kk
1
(22)
The second term in the denominator of the right-hand side is dimensionless and acts as a Reynolds Number for porous media
flow. The Reynolds number in a porous media can be defined as
fkN Re
(23)
where, kf is in cm-g/sec2-atm, is in g/cm
3, is in cm/s, is in g/cm-sec and is in atm-sec
2/gm. Suggested by Geertsma
(1974) , substituting Eq. 23 into Eq. 22, the final expression of kf-eff describing the non-Darcy flow effects is
Re1 N
kk
f
efff
(24)
Where velocity in Eq.23
A
q353.0 (25)
where, q is in Mscf/day and A is in ft2. The constant in Eq. 25 is a result of unit conversion from the IS unit to the field unit.
Non-Darcy effects can significantly decrease well production especially for high flow rate wells. Smith, et al (2004),
claimed that non-Darcy flow effects can decrease productivity up to 35% in a hydraulically fractured high-rate oil well. They
also showed a productivity index reduction of 20% in a high rate (120 MMscf/d) gas well. Cramer (2004) also concluded that
qg
pout,circle2
pout,circle2=pinner,circle1 N
pwf,cal
Y
pwf=pwf,ture N
Assume pinner,circle1
Assume pinner,circle2
qg and p
SPE 151960 7
Non-Darcy flow in the fracture exists to some extent in most gas well completions and it shows up as a rate dependent
pseudo-skin, reducing the calculated effective xf (half-length) in an extensive analysis of non-ideal cases.
In non-Darcy effect calculation, is one of the important parameters. factor is a property of the porous media. The
equations developed to estimate this factor are based on lab data. Cooke (1973) first developed equation to estimate factor
of proppants. Brady sand was used in the lab experiments. Based on the form of the Forchheimer equation presented in Eq.
16, Cooke plotted vL
p
vs
v to get the factor, which is the slope of the curve on the plot. Five sand sizes and various stress
levels were considered. The fluids used were brine, gas and oil. Cooke observed no difference of the results among fluids
evaluated. All curves followed a simple equation
b
fk
a
(26)
where, kf is in Darcy, is in atm-sec2/gm, a and b are dimensionless, correlation constants, as shown in Table 2.
Table 2—CONSTANTS A AND B OF COOKE’S EQUATION.
Sand Size
(mesh) a b
8/12 3.32 1.24
10/20 2.63 1.34
20/40 2.65 1.54
40/60 1.10 1.6
As previously discussed, non-Darcy flow in a gas reservoir causes a reduction of production. The effect on pressure drop
and production distribution inside fracture is estimated following the flow chart as shown in
Fig. 5. Using the slab source method, the flow rate is first calculated with the original proppant. Then, the effective proppant
permeability is compared with the proppant permeability at the flow rate to check whether they are the same. If it is not, the
effective proppant permeability is used to get the new gas production by the slab source method. The iteration stops until the
new effective proppant permeability is the same as the previous one. The final proppant permeability is then used to calculate
the pressure and production distribution along the fracture.
Fig. 5—Flow chart for non-Darcy flow.
Reservoir Properties
No
Yes
Velocity
𝑘𝑒𝑓𝑓 =𝑘𝑓
1 +𝛽𝑘𝑛𝑓𝜌𝜈𝜇𝑔
𝑘𝑒𝑓𝑓(𝑖) = 𝑘𝑒𝑓𝑓(𝑖−1) ?
qg
∆p, qg
Proppant Properties
Gas Properties
8 SPE 151960
Preliminary Solution for Complex Fracture System. The multiple hydraulic fractures combined with natural fractures
create very complex fracture networks. In this study, a simplified hydraulic fracture/natural fracture system is used to
illustrate the approach of using the slab source model to estimate flow rate in such a system.
Figure 6 shows the physical model used in the approach. We assumed that the natural fractures that are connected to the
hydraulically created fractures are all orthogonal to the hydraulic fractures, and the hydraulic fractures are the main fractures
and the natural fractures are branch fractures which only connect with the main fractures, but not the wellbore. The fluid
inside the branch fractures flows into the main fractures. For the main fractures, the fluid is from both the reservoir and the
connected branch fractures, and it flows to the wellbore. We considered each hydraulic fracture and natural fracture is an
individual source, and the sources are interacted by each other. The superposition method is applied to solve the pressure and
flow field.
(a) (b)
Fig. 6—Schematic of complex fracture system.
The natural fractures are local sources providing flow rate at the locations where they intercept with hydraulic fractures. If
a natural fracture intercepts more than one hydraulic fracture, such as NF3 and NF4 in Fig. 6a, then the natural fracture will
be treated as 2 fractures shown on Fig. 6b. The total flow rate from this natural fracture is the total of the two parts. For
example, the flow rate of NF4 is divided to NF5 and NF6.
The complex system is controlled by a constant wellbore pressure. With the system shown in Fig. 6, we have 4 hydraulic
fractures, HF1, HF2, HF3, and HF4, and 4 natural fractures NF1, NF2, NF3, and NF4. However, NF3 and NF4 are connected
with two hydraulic fractures, and then the total fracture number is 10.
We acquire a set of linear equations as shown in Eq. 27. This linear system can be solved by defining the inner boundary
condition as uniform flux, infinite conductivity, or finite conductivity.
1int11321 )10,1(...)3,1()2,1()1,1( ppFqFqFqFq
2int11321 )10,2(...)3,2()2,2()1,2( ppFqFqFqFq
3int11321 )10,3(...)3,3()2,3()1,3( ppFqFqFqFq
.
.
.
10int11321 )10,10(...)3,10()2,10()1,10( ppFqFqFqFq (27)
Validation of the Method For complex fracture system, we use a simple example to compare our results with commercial simulation result (ECLIPSE
100, Schlumberger). The input parameters are list in Table 3. The schematic of complex fracture system is show in Fig. 7.
Table 3— INPUT DATA FOR COMPLEX FRACTURE VALIDATION.
Parameter Value Unit
Reservoir length, b 2000 ft
Reservoir width, a 1000 ft
Reservoir thickness, h 100 ft
Part 1→NF5
Part 2→NF6
HF 3 HF 4
x
y
HF 1 HF 2 HF 3 HF 4
NF 1 NF 2
NF 3
NF 4
NF4
SPE 151960 9
Porosity, 0.09 ft
Reservoir initial pressure, pi 4000 psi
Reservoir temperature, T 146 F
Bottomhole pressure, pwf 3000 psi
Gas specific gravity, g 0.84
Horizontal permeability, kh 0.01 md
Vertical permeability, kv 0.001 md
Gas viscosity, g 0.0156 cp
Horizontal well length, L 1500 ft
Hydraulic Fractures
Fracture length, xf 250 ft
Fracture height, hf 100 ft
Fracture width, wf 0.033 ft
Natural Fractures
Fracture length, xf 125 ft
Fracture height, hf 50 ft
Fracture width, wf 0.008 ft
Figure 7 shows the schematic of a complex fracture system. Two hydraulic fractures are evenly distributed along the
horizontal well with a half-length of 250 ft. Two natural fractures are placed in the reservoir with a half-length of 125 ft.
Each of them is connect with one hydraulic fracture. The flow into wellbore is neglect in this case. Fig. 8 presents
comparison of the two methods; they show a good match. Both of the results show the flow rate drop quickly at first 50 days,
and reach the pseudosteady-state condition.
Fig. 7—Complex fracture system schematic. Fig. 8—Comparison results for complex fracture system.
Results and Discussions Complex Fracture System. The slab source method could also be used for complex fracture systems. We considered each
hydraulic fracture and natural fracture as a separate source, and the sources will be affected by the other sources in the
system. Superposition method is applied to capture the internal affection among the sources.
Figure 9 shows a randomly generate natural fracture network. There are 10 hydraulic fractures along the wellbore, evenly
placed. The horizontal well is 3000 ft long, and spacing between hydraulic fractures is 273 ft. The hydraulic fracture all have
the same dimension, 500 ft in half length, 100ft in height, and 0.033 ft in width. The natural fracture length is from 100 to
500 ft and the width is 0.08 to 0.1 in. The detail of the fracture system properties is shown in Table 4. In these 20 natural
fractures, only seven of them are connected with the hydraulic fractures. There is one natural fracture that is connected with
two hydraulic fractures. As mentioned before, in this case we divided the natural fracture into two parts and each of them is
connected with a hydraulic fracture near it.
0
50
100
150
200
250
300
350
400
0 100 200 300 400 500 600
Flo
w r
ate
, Msc
f/d
Time, days
Slab Source
Eclipse
b
a HW
HF1 HF2
NF2
NF1
10 SPE 151960
Fig. 9—Randomly generated complex fracture system.
Table 4—INPUT DATA FOR COMPLEX FRACTURE SYSTEM.
Parameter Value Unit
Reservoir length, b 4000 ft
Reservoir width, a 2000 ft
Reservoir thickness, h 200 ft
Porosity, 0.09 ft
Reservoir initial pressure, pi 4000 psi
Reservoir temperature, T 146 F
Bottomhole pressure, pwf 3000 psi
Gas specific gravity, g 0.836
Horizontal permeability, kh 0.05,0.001 md
Vertical permeability, kv 0.01,0.005 md
Gas viscosity, g 0.0156 cp
Horizontal well length, L 3000 ft
Hydraulic Fractures
Fracture length, xf 500 ft
Fracture height, hf 100 ft
Fracture width, wf 0.033 ft
Natural Fracture
No.
Fracture length
xf
Fracture width
wf
1 237 0.007
2 530 0.007
3 260 0.008
4 250 0.009
5 257 0.008
6 528 0.005
7 371 0.008
The production rate of the well is calculated by the slab source method, and the results are shown in Fig.10 and 11 with
NF1
NF2
NF3
NF4 NF5
NF6
NF7
HW
HF1 HF2 HF3 HF4 HF5 HF6 HF7 HF8 HF9 HF10
SPE 151960 11
different reservoir permeability. Fig. 10 shows the flow rate for reservoir permeability of 0.05 md. At the early time, the
hydraulic fracture produced 13 times more than the natural fractures (76.5 MMscf/day vs 5.8 MMscf/day), and the
production from natural fracture is only 7% of the total production. But the rate from hydraulic fractures drops to 4 times
more than the natural fractures at 500 days. Fig. 11 also plots the production rate from the complex fracture system with the
reservoir permeability of 0.001 md. The production from the natural fractures increased to 15% of the total production for the
same fracture system. When reservoir permeability is low, the production from natural fractures is increased.
Fig. 10—Flow rate for complex fracture system (kh=0.05md). Fig. 11—Flow rate for complex fracture system (kh=0.001md).
Non-Darcy Flow. The Slab source solution model was used to study the effect of non-Darcy flow in the fracture. To
understand the impact of the loss in fracture conductivity on well performance and cumulative gas production, we run cases
under transient conditions. The results are compared with Barree’s results (2005).
Three cases were run. The specifics of these three cases are shown in Table 5. A single transverse fracture is placed along
a horizontal well. The reservoir pressure is 4000 psia, bottomhole pressure is 1000 psia, fracture height is 100 ft, and the
proppant concentration is 1.0. The reservoir permeability for the first two cases is 0.01 md and changed to 0.1 md in case 3.
Each case was run with an effectively propped fracture half-length of 1000 ft, except for Case 2 which used xf of 100 ft. No
damage from filter-cake or gel residual in the pack was considered.
Table 5—CASES FOR NON-DARCY FLOW STUDY.
Case Reservoir
Perm. (md)
xf
(ft)
1 0.01 1000
2 0.01 100
3 0.1 1000
The results of the three cases are presented in Fig. 12 through 14. The results show that even for 0.01 md reservoirs the
impact of non-Darcy flow can be significant if the created fracture length is long. In case 1, the slab source result shows a
slightly higher production rate than Barree’s results. The non-Darcy effects result decrease in cumulative production after 10
years by 11%.
Case 2 is similar to case 1 except that the fracture length is decreased from 1000 ft to 100 ft. when the fracture is short,
Fig. 13 shows that non-Darcy effects are not significant.
Figure 14 shows the results for Case 3. This case the reservoir permeability is increase to 0.1 md. In this case the result
shows that the non-Darcy flow effects are magnified. The difference in production after 10 years is 20%. The difference is
caused by the high production which would introduce higher velocity, resulting in significant non-Darcy flow effect.
0
10
20
30
40
50
60
70
80
90
0 200 400 600
Flo
w r
ate
, MM
scf/
d
Time, days
HF
NF
Total
0
5
10
15
20
25
30
35
0 200 400 600
Flo
w r
ate
, MM
scf/
d
Time, days
HFNFTotal
12 SPE 151960
Fig. 12—Cumulative production for case 1. Fig. 13—Cumulative production for Case 2.
Fig. 14—Cumulative production for Case 3.
A summary of the 10-year cumulative production with and without non-Darcy flow effects for the three cases is shown in
Table 6. For the three cases studied, non-Darcy effect ranges from 2% to 20% in cumulative production over ten years.
Table 6—CUMULATIVE PRODUCTION RESULTS.
Case Barree’s model Slab source Model
No. Darcy Flow (MMscf)
Non-Darcy (MMscf)
Percentage Change
Darcy Flow (MMscf)
Non-Darcy (MMscf)
Percentage Change
1 3160 2840 11.3 3250 2870 11.6
2 1835 1825 0.5 1850 1830 2.0
3 15000 12700 18.1 16100 12900 19.8
Field Case Study (Marcellus, shale). The slab source was used to evaluate a gas well performance in Marcellus Shale. The
production history was presented by Meyer and Bazan (2010). They also presented history matching by their trilinear
solution. A 2100-ft long horizontal well was stimulated by seven stages of fracture treatment. Fig. 15 shows the original
published data. The production log showed a total gas flow rate of 3.166 MMscf/d and a water rate of 2541 bpd. Stage two
showed a minimal contribution of about 3% of the total production. Therefore, only six multiple transverse fractures were
used to history match. The reservoir and fracture properties for Marcellus Shale is given in Table 7. The Marcellus Shale gas
well history matched parameters are given in Table 8.
Table 7—MARCELLUS SHALE RESERVOIR AND FRACRTURE PROPERTIES.
Formation Marcellus
Formation height, ft 162
Porosity, % 4.2
Pore Pressure, psi 4726
Specific Gravity 0.58
0
500
1000
1500
2000
2500
3000
3500
0 1000 2000 3000 4000
Cu
mu
lati
ve p
rod
uct
ion
, MM
scf
Time, days
Barree Darcy
Barree Non-Darcy
Slab source Darcy
Slab source Non-Darcy
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 1000 2000 3000 4000
Cu
mu
lati
ve p
rod
uct
ion
, MM
scf
Time, Days
Barree Darcy
Barree Non-Darcy
Slab source Darcy
Slab source Non-Darcy
0
2000
4000
6000
8000
10000
12000
14000
16000
0 1000 2000 3000 4000
Cu
mu
lati
ve p
rod
uct
ion
, MM
scf
Time, days
Barree Darcy
Barree Non-Darcy
Slab source Darcy
Slab source Non-Darcy
SPE 151960 13
Temperature, °F 175
Drainage Area, acres 80
Reservoir Size, ft 933,3733
Bottomhole pressure, psia 1450-530
Formation Permeability, md 0.000546
Wellbore Radius, ft 0.36
Well Length, ft 2100
Number of Stages 7
Table 8—HISTORY MATHCED PARAMETERS.
Matched Parameters Slab Source
Model
Meyer’s
Model
Fracture Permeability, md 526 478
Propped Length, ft 353 363
Fracture Width, ft 0.0065 0.0063
Number Equivalent Fractures 6 6
The production data was matched with the single phase, multiple transverse fractures in horizontal wellbores, and the
result is shown in Fig. 16. The history match analysis of measured production was based on the parameters as shown in the
study done by Meyer and Bazan in Table 8. The calculated result shown in Fig. 16 agrees with the field observation. Because
fractured well performance depends on the assumption made in the model, it is not surprising that a perfect match was not
obtained at the first 10 days. The slab source model used transient flow conditions, which yields higher rate at first few days
of production.
An average permeability of 377 nanodarcy over a formation thickness of 162 ft that included the upper and lower
Marcellus was reported by a petrophysical analysis (Meyer, 2010). The resulting reservoir capacity (kh) was calculated to be
0.061 md-ft to match the production history, an average reservoir permeability of 546 nanodarcy was used in the slab source
approach. The matched parameters from slab source model show well agreement with Meyer’s model.
Fig. 15—Field Data (Meyer and Bazan, 2010). Fig. 16—History match of a gas well in Marcellus Shale.
Conclusion The slab source method is successfully developed to predict the performance in a closed rectangular reservoir under different
boundary conditions. The model can be used in complex well/fracture configurations such as horizontal wells, multiple
fractures, and complex fracture systems for both transient and pseudosteady-state conditions. The conclusions can be
summarized as following:
1. Non-Darcy flow effects have an influence on flow rates, these decrease can range from 2% to 20% under a given set
of conditions. Long-term production is significantly affected by non-Darcy conductivity losses, even in relatively
low permeability reservoirs where long fractures are needed.
2. For a complex fracture system, the production from natural fractures depends on natural fracture location,
dimension, and connection point with the hydraulic fractures. For low permeability formation, the production from
the natural factures is more important compared to moderate permeability formation.
1
1.5
2
2.5
3
3.5
4
0 50 100 150 200
pro
du
ctio
n r
ate
, MM
scf/
d
time, days
Field Data
Slab source model
14 SPE 151960
Nomenclature
a = reservoir width, ft
B = formation volume factor
b = reservoir length, ft
ct = compressibility, psi-1
h = reservoir height, ft
kx = permeability in x-direction, md
ky = permeability in y-direction, md
kz = permeability in z-direction, md
L = wellbore length, ft
pint = initial reservoir pressure, psi
p = average reservoir pressure, psi
pwf = bottomhole pressure, psi
Δp = pressure drawdown, psi
q = flow rate, Mscf/day
T = temperature, °R
xf = fracture width, ftyD
g specific gas gravity, dimensionless
= density, lb/ft3
= porosity, fraction
= viscosity, cp or g/100-cm-sec
= time
atm-sec2/gm
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