Lesson 6
Advanced Pressure Transient Analysis
Topic 1: Hydraulically Fractured Wells
1. Hydraulic Fractures
Hydraulic fracturing drastically increases the productivity and/or injecting. Every year,
thousands of production and/or injection wells are fractured hydraulically. Introduction of a
hydraulic fracture influences the pressure transient response of the system in a very pronounced
manner; therefore, well test analysts should be cognizant of the expected deviations from the
conventional pressure transient model. During the hydraulic fracturing process, it is possible to
create horizontal and vertical fractures. However, in formations located at depths greater than
2000 ft, a great majority of the fractures created are vertical fractures. Therefore, it is common to
base the discussion of hydraulically fractured wells on vertical fractures. A vertical fracture that
has been hydraulically induced will have the tendency to propagate in a direction perpendicular
to the minimum principal horizontal stress direction.
Once the fracture is created and the fracture pressure is released, the inside stresses will work
towards healing (closing) the fracture. This healing is prevented by packing the fracture space
with a relatively coarse proppant to create a conduit with high permeability. In well test analysis
models, hydraulic fractures are perceived as having a height equal to the formation thickness
(fracture is 100% contained within the formation of interest). Again, in mathematical treatments,
it is also assumed that propagation of a hydraulic fracture from the well in opposite directions is
symmetrical so that length of each wing of the fracture is the same. Figure 6.1.1.1 illustrates this
well test idealization and its characteristic dimensions. Note: xf = fracture length.
Figure 6.1.1.1 Well Test Idealization of a Vertical, Hydraulic Fracture
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For the study of hydraulic fractures, the following dimensionless groups are defined:

Dimensionless time based on fracture length:
t Dx f

rw2
 tD 2
xf
=>
tD 
k
t
cx 2f
(Equation 6.1.1.1)
Dimensionless fracture conductivity:
cD 
k f wf
(Equation 6.1.1.2)
km x f
where,
kf = fracture permeability,
km = matrix or reservoir permeability,
wf = fracture width,
wm = fracture length.
Fracture permeabilities are very high, with permeabilities ranging within the thousand
miliDarcies. There is almost no resistance to flow. In the limiting case, when zero resistance is
found, the fracture is assumed to be of infinite conductivity (no pressure drop along the fracture).
2. Infinite and Finite Conductivity Fractures
The mathematical models that are formulated to mimic the pressure transient behavior of
hydraulically fractured wells consider three types of fracture:



Finite conductivity fractures
Infinite conductivity fractures
Uniform flux fractures
In infinite conductivity fractures, no pressure drop occurs in the fracture. In other words, the
fracture represents the innermost equal contour and any particle entering the fracture is assumed
to be produced instantaneously at the wellbore. In general, when the dimensions of the fracture
conductivity is large (cD > 300), for all practical purposes the fracture is assumed to perform as if
it is an infinite conductivity fracture. In reality, there is always a pressure drop taking place along
the fracture length; that is, a finite conductivity fracture is the usual behavior to be considered. In
this case, it takes some time for the particle to travel through the fracture and move from the
external edges of the fracture into the wellbore. This allows the development of several different
flow patterns, which will be discussed in the next section.
A uniform flux fracture is a conceptual treatment of fractures where it is assumed that fracture
loading from the formation per unit length of fracture is uniform along its length. In this case, the
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flow from the formation to the fracture is linear. In reality, the flux into a real fracture is higher
at the tips because that zone is supported by the rest of the reservoir. The uniform flux
approximation, however, is much easier to tackle analytically.
Finite and infinite conductivity, and uniform flux fracture representations are illustrated in Figure
6.1.2.1.
Figure 6.1.2.1 Well Test Idealization of a Vertical, Hydraulic Fracture
3. Flow Periods in a Finite Conductivity Fracture
For the case of finite conductivity fractures, five different flow regimes can be experienced at
different times. This is shown in the schematic representation (Figure 6.1.3.1). These flow
regimes are described below:
(1) Linear Flow in Fracture: During very early times, flow will take place only in the
fracture and because of the linear nature of the fracture, flow regime, will be linear.
This period can be extremely short, and during this period the contents of the fracture
will be emptied. This type of flow is inexistent in infinite conductivity fractures.
(2) Bilinear Flow in Fracture Formation: As a result of the linear flow in the fracture,
pressure in the fracture will be much lower than fracture surface. This will generate a
linear flow from formation into fracture. This type of flow is inexistent in infinite
conductivity fractures.
(3) Linear Flow in Formation: This flow regime will be observed when fracture
conductivity is so high such that any fluid that enters into the fracture is immediately
produced at the wellbore (infinite conductivity fracture).
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(4) Elliptical Flow in Formation: This flow regime represents a transitional flow regime
between linear flow and radial flow regimes. Equal-pressure contours are confocal
ellipses and streamlines are confocal hyperbolas. It should be noted that both ellipses
and hyperbolas share the same set of foci located at the tips of the fracture.
(5) Radial Flow in Formation: At late times, pressure around the fracture will be lowered
over a relatively large area. This reduced pressure zone will serve as a sink and flow
lines will orient themselves towards this circular sink such that a full radial flow
regime is developed. At this point, we can use our exponential integral classical
solution for pressure transient analysis.
Figure 6.1.3.1 Well Test Idealization of a Vertical, Hydraulic Fracture
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4. Diagnostic and Horner Plot Analysis
The initial linear flow regime occurs in a really short period of time and thus has no impact on
the well test analysis. During the bilinear flow period, the pressure drop in the fracture can be
modeled as follows:
2.451
(Equation 6.1.4.1a)
p D 
t 1/ 4
1 / 2 Dx f
(k fD w fD )
where,
k fD 
w fD 
kf
(Equation 6.1.4.1b)
km
wf
(Equation 6.1.4.1c)
xf
This means that a log-log plot of pD vs. tDxf yields a straight line of slope 1/4 during the bilinear
flow period. However, this flow period is seldom predominant at early times and this effect may
not be seen in well test data.
The predominant behavior at early times is the second linear flow period (linear flow in the
formation). The pressure drop (true linear flow response in the presence of an infinite
conductivity fracture) is given by:
p  pi  p wf  16.26
 
qB
t Dx f
AF
1/ 2
(Equation 6.1.4.2)
where,
AF = total exposed area of the fracture = 4 h xF.
The above equation indicates that a ½ slope straight line on a log-log plot will be formed during
the linear flow period since:
log p 
1
1 
qB 

log t Dx f  log 16.26
2
2 
AF 
(Equation 6.1.4.3)
Therefore, on the diagnostic plot, the data points that fall on a ½-slope straight line should not be
considered in defining the semilog straight line of the Horner plot of the same data set.
Conversely, a plot of p vs. t in Cartesian coordinates should yield an straight line whose slope
is equal to:
16.26qB  

m
AF  kct



1/ 2
(Equation 6.1.4.4a)
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or,
qB
m  4.064
hx F
 

 kct



1/ 2
(Equation 6.1.4.4b)
From the slope of Cartesian coordinate plot fracture, length xf, can be estimated. This procedure,
however, requires that permeability of the formation be available (either from the Horner plot
analysis of the infinite acting/radial flow portion of the test or from a prefracture test on the same
well).
Unitary slope (45o)
103
(pws-pwf)
Bilinear
m=1/4
102
Radial flow
Storage
m=1
Linear
m=1/2
10
Transitional
1
0.1
1.5 Cycles
Initial time for middle region (MTR)
logtDxf
Figure 6.1.4.1 Diagnostic (log-log) plot for build-up test analysis of
a hydraulically fractured well
pws
Boundary
effects
(LTR)
Region of
increasing slope
m(radial)
MTR =
constant slope
(radial flow)
log(tp+t)/t
Figure 6.1.4.2 Horner’s plot for build-up test analysis of
a hydraulically fractured well
Figure 6.1.4.1 shows the typical diagnostic plot (log-log) that one may expect when a build-up
test is conducted in a hydraulically fractured well. The plot shows the different regions that may
show up and their characteristic slopes. Figure 6.1.4.2 shows the typical Horner’s plot (semiP N G 425: Principles of Well Testing and Evaluation
Lesson 6, Topic 1 ~ Page 6 of 7
log) for a build-up test conducted in a hydraulically fractured well. We must keep in mind that
Horner’s plots have been devised for the analysis of radial flow into a well. Nevertheless, we still
plot all the data points that we have available and analyze the later time data points when radial
flow has been developed.
When the pressure data from the radial flow portion of the test (buildup) is analyzed using the
Horner plot (Figure 6.1.4.2), it will be seen that the curve with a continuously increasing slop
will form. This region of increasing slope is a distinctive feature of hydraulic fractures. True
radial flow regime is developed when the increasing slope reaches a maximum value and stays at
that value. The slope of this straight line can be used in calculating the formation permeability.
Recall that the Horner’s plot slope is an inverse function of permeability. Therefore, at the
beginning of the plot we are to expect a smaller slope because the fracture, which has the highest
permeability, has total control of the flow. The transition between fracture-permeability
dominated flow and formation-permeability dominated flow causes the region of increasing
slope. By looking at Figure 6.1.4.2, we may also conclude that if the fracture ends very close to
the boundaries, we will have a “S” shape curve with no MTR or radial flow region (no constant
slope would be found).
Written by Turgay Ertekin, Professor of Petroleum and Natural Gas Engineering
Copyright © January 2006, The Pennsylvania State University
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