Note: Descriptions are shown in the official language in which they were submitted.
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SYSTEM AND METHOD FOR PERFORMING
WELLBORE FRACTURE OPERATIONS
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This application is a continuation in part of US Patent Application No.
12/479,335, filed
on June 5, 2009, the entire contents of which is hereby incorporated by
reference. This
Application also claims priority to US Provisional Application No. 61/574,130
filed on July 28,
2011, the entire contents of which is hereby incorporated by reference.
BACKGROUND
[0002] The present disclosure relates generally to methods and systems for
performing wellsite
operations. More particularly, this disclosure is directed to methods and
systems for performing
fracture and production operations, such as investigating subterranean
formations and
characterizing hydraulic fracture networks in a subterranean formation.
[0003] In order to facilitate the recovery of hydrocarbons from oil and gas
wells, the
subterranean formations surrounding such wells can be hydraulically fractured.
Hydraulic
fracturing may be used to create cracks in subsurface formations to allow oil
or gas to move
toward the well. A formation is fractured by introducing a specially
engineered fluid (referred to
as "fracturing fluid" or "fracturing slurry" herein) at high pressure and high
flow rates into the
formation through one or more wellbore. Hydraulic fractures may extend away
from the
wellbore hundreds of feet in two opposing directions according to the natural
stresses within the
formation. Under certain circumstances, they may form a complex fracture
network.
[0004] The fracturing fluids may be loaded with proppants, which are sized
particles that may be
mixed with the fracturing fluid to help provide an efficient conduit for
production of
hydrocarbons to flow from the formation/reservoir to the wellbore. Proppant
may comprise
naturally occurring sand grains or gravel, man-made or specially engineered
proppants, e.g.
fibers, resin-coated sand, or high-strength ceramic materials, e.g. sintered
bauxite. The proppant
collects heterogeneously or homogenously inside the fracture to "prop" open
the new cracks or
pores in the formation. The proppant creates a plane of permeable conduits
through which
production fluids can flow to the wellbore. The fracturing fluids are
preferably of high viscosity,
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and therefore capable of carrying effective volumes of proppant material.
[0005] The fracturing fluid may be realized by a viscous fluid, sometimes
referred to as "pad"
that is injected into the treatment well at a rate and pressure sufficient to
initiate and propagate a
fracture in hydrocarbon formation. Injection of the "pad" is continued until a
fracture of
sufficient geometry is obtained to permit placement of the proppant particles.
After the "pad," the
fracturing fluid may consist of a fracturing fluid and proppant material. The
fracturing fluid may
be gel, oil based, water based, brine, acid, emulsion, foam or any other
similar fluid. The
fracturing fluid can contain several additives, viscosity builders, drag
reducers, fluid-loss
additives, corrosion inhibitors and the like. In order to keep the proppant
suspended in the
fracturing fluid until such time as all intervals of the formation have been
fractured as desired,
the proppant may have a density close to the density of the fracturing fluid
utilized.
[0006] Proppants may be comprised of any of the various commercially available
fused
materials, such as silica or oxides. These fused materials can comprise any of
the various
commercially available glasses or high-strength ceramic products. Following
the placement of
the proppant, the well may be shut-in for a time sufficient to permit the
pressure to bleed off into
the formation. This causes the fracture to close and exert a closure stress on
the propping agent
particles. The shut-in period may vary from a few minutes to several days.
[0007] Current hydraulic fracture monitoring methods and systems may map where
the fractures
occur and the extent of the fractures. Some methods and systems of
microseismic monitoring
may process seismic event locations by mapping seismic arrival times and
polarization
information into three-dimensional space through the use of modeled travel
times and/or ray
paths. These methods and systems can be used to infer hydraulic fracture
propagation over time.
[0008] Conventional hydraulic fracture models may also assume a bi-wing type
induced fracture.
These bi-wing fractures may be short in representing the complex nature of
induced fractures in
some unconventional reservoirs with preexisting natural fractures. Published
models may map
the complex geometry of discrete hydraulic fractures based on monitoring
microseismic event
distribution.
[0009] In some cases, models may not be constrained by accounting for either
the amount of
pumped fluid or mechanical interactions both between fractures and injected
fluid and among the
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fractures. Some of the constrained models may provide a fundamental
understanding of involved
mechanisms, but may be complex in mathematical description and/or require
computer
processing resources and time in order to provide accurate simulations of
hydraulic fracture
propagation.
[00010] Unconventional formations, such as shales are being developed as
sources of
hydrocarbon production. Once considered only as source rocks and seals, shale
formations are
now considered as tight-porosity and low-permeability unconventional
reservoirs. Hydraulic
fracturing of shale formations may be used to stimulate and produce from the
reservoir.
[000iii Patterns of hydraulic fractures created by the fracturing stimulation
may be complex and
form a fracture network as indicated by the distribution of associated
microseismic events.
Complex hydraulic fracture networks (HFNs) have been developed to represent
the created
hydraulic fractures. Examples of fracture models are provided in US
Patent/Application Nos.
6101447, 7363162, 7788074, 20080133186, 20100138196, and 20100250215.
[00012] Due to the complexity of HFNs, production from a stimulated shale
reservoir may be
numerically simulated. Numerical simulation for stimulation job design and
post-job analysis
may be time-consuming, and it may be inconvenient to construct a numerical
model and carried
out runs for each of the numerous designs of a stimulation job. The
effectiveness and efficiency
of a fracturing job may ultimately be judged by production from the stimulated
reservoir.
SUMMARY
[00013] The present application discloses methods and systems for
characterizing hydraulic
fracturing of a subterranean formation based upon inputs from sensors
measuring field data in
conjunction with a hydraulic fracture network model. The fracture model
constrains geometric
properties of the hydraulic fractures of the subterranean formation using the
field data in a
manner that significantly reduces the complexity of the fracture model and
thus significantly
reduces the processing resources and time required to provide accurate
characterization of the
hydraulic fractures of the subterranean formation. Such characterization can
be generated in real-
time to manually or automatically manipulate surface and/or down-hole physical
components
supplying fracturing fluids to the subterranean formation to adjust the
hydraulic fracturing
process as desired, such as by optimizing fracturing plan for the site (or for
other similar
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fracturing sites).
[00014] In some embodiments, the methods and systems of the present disclosure
are used to
design wellbore placement and hydraulic fracturing stages during the planning
phase in order to
optimize hydrocarbon production. In some embodiments, the methods and systems
of the present
disclosure are used to adjust the hydraulic fracturing process in real-time by
controlling the flow
rates, compositions, and/or properties of the fracturing fluid supplied to the
subterranean
formation. In some embodiments, the methods and systems of the present
disclosure are used to
adjust the hydraulic fracturing process by modifying the fracture dimensions
in the subterranean
formation in real time.
[00015] The method and systems of the present disclosure may also be used to
facilitate
hydrocarbon production from a well, and subterranean fracturing (whereby the
resulting fracture
dimensions, directional positioning, orientation, and geometry, and the
placement of a proppant
within the fracture more closely resemble the desired results).
[00016] In another aspect, the disclosure relates to a method of performing an
oilfield operation
about a wellbore penetrating a subterranean formation. The method involves
performing a
fracture operation. The fracture operation involves generating a plurality of
fractures about the
wellbore and generating a fracture network about the wellbore. The fracture
network includes the
fractures and a plurality of matrix blocks positioned thereabout. The
fractures are intersecting
and hydraulically connected. The matrix blocks are positioned about the
fractures. The method
also involves generating flow rate through the fracture network, generating a
fluid distribution
based on the flow rate, and performing a production operation, the production
operation
comprising generating a production rate from the fluid distribution.
[00017] In another aspect, the disclosure relates to a method of performing an
oilfield operation
about a wellbore penetrating a subterranean formation. The method involves
performing a
fracture operation. The fracture operation involves stimulating the wellbore
and generating a
fracture network about the wellbore. The stimulating involves injecting fluid
into the
subterranean formation such that fractures are generated about the wellbore.
The fracture
network includes the fractures and a plurality of matrix blocks positioned
thereabout. The
fractures are intersecting and hydraulically connected. The plurality of
matrix blocks is
positioned about the fractures. The method also involves placing proppants in
the fracture
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network, generating flow rate through the fracture network, generating a fluid
distribution based
on the flow rate, and performing a production operation. The production
operation involves
generating a production rate from the fluid distribution.
[00018] Finally, in another aspect, the disclosure relates to a method of
performing an oilfield
operation about a wellbore penetrating a subterranean formation. The method
involves designing
a fracture operation based on job parameters and performing the fracture
operation. The fracture
operation involves generating a fracture network about the wellbore. The
fracture network
includes a plurality of fractures and a plurality of matrix blocks. The
fractures are intersecting
and hydraulically connected. The matrix blocks are positioned about the
fractures. The method
also involves optimizing the fracture operation by adjusting the fracture
operation based on a
comparison of a simulated production rate with actual data, generating flow
rate through the
fracture network, generating a fluid distribution based on the flow rate, and
performing a
production operation. The simulated production rate is based on the fracture
network. The
production operation involves generating a production rate from the fluid
distribution.
[00019] This summary is provided to introduce a selection of concepts that are
further described
below in the detailed description. This summary is not intended to identify
key or essential
features of the claimed subject matter, nor is it intended to be used as an
aid in limiting the scope
of the claimed subject matter.
BRIEF DESCRIPTION OF THE DRAWINGS
[00020] Embodiments of the system and method for characterizing wellbore
stresses are
described with reference to the following figures. The same numbers are used
throughout the
figures to reference like features and components.
[00021] Figs. 1.1-1.4 are schematic views illustrating various oilfield
operations at a wellsite;
[00022] Figs. 2.1-2.4 are schematic views of data collected by the operations
of Figures 1.1-1.4;
[00023] Fig. 3 is a pictorial illustration of geometric properties of an
exemplary hydraulic fracture
model in accordance with the present disclosure;
[00024] Fig. 4 is a schematic illustration of a hydraulic fracturing site that
embodies the present
disclosure;
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[00025] Figs. 5.1 and 5.2, collectively, is a flow chart illustrating
operations carried out by the
hydraulic fracturing site of Fig. 4 for fracturing treatment of the
illustrative treatment well in
accordance with the present disclosure.
[00026] Figs. 6.1-6.4 depict exemplary display screens for visualizing
properties of the treatment
well and fractured hydrocarbon reservoir during the fracturing treatment of
the illustrative
treatment well of Fig. 4 in accordance with the present disclosure;
[00027] Figs. 7.1-7.4 depict exemplary display screens for visualizing
properties of the treatment
well and fractured hydrocarbon reservoir during the fracturing treatment and
during a subsequent
shut-in period of the illustrative treatment well of Fig. 4 in accordance with
the present
disclosure;
[00028] Figs. 8.1 - 8.3 are schematic diagrams illustrating an elliptical
hydraulic fracture network
about a well;
[00029] Fig. 9 is a schematic diagram depicting proppant placement;
[00030] Fig. 10 is a schematic diagram illustrating a cross-sectional view of
the elliptical
hydraulic fracture network of Fig. 8.1 and a detailed view of a matrix block
therefrom,
respectively;
[00031i Fig. 11 is a schematic diagram illustrating fluid flow through a dual
porosity medium;
[00032] Figs 12 is schematic flow diagrams depicting methods of performing
production
operations;
[00033] Figs. 13.1 and 13.2 are various schematic diagrams for depicting fluid
flow through a
medium;
[00034] Fig. 14 is a flow chart depicting a fracture design and optimization;
[00035] Fig. 15 is a flow chart depicting a post-production operation; and
[00036] Fig. 16 is a flow chart depicting a method for performing a production
operation.
DETAILED DESCRIPTION
[00037] The description that follows includes exemplary systems, apparatuses,
methods, and
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instruction sequences that embody techniques of the subject matter herein.
However, it is
understood that the described embodiments may be practiced without these
specific details.
[000381 The present disclosure relates to techniques for performing fracture
operations to estimate
and/or predict production. The fracture operations involve fracture modeling
that utilize elliptical
and wire mesh modeling to estimate production.
[00039j Figures 1.1-1.4 depict various oilfield operations that may be
performed at a wellsite, and
Figures 2.1-2.4 depict various information that may be collected at the
wellsite. Figures 1.1-1.4
depict simplified, schematic views of a representative oilfield or wellsite
100 having subsurface
formation 102 containing, for example, reservoir 104 therein and depicting
various oilfield
operations being performed on the wellsite 100. FIG. 1.1 depicts a survey
operation being
performed by a survey tool, such as seismic truck 106.1, to measure properties
of the subsurface
formation. The survey operation may be a seismic survey operation for
producing sound
vibrations. In FIG. 1.1, one such sound vibration 112 generated by a source
110 reflects off a
plurality of horizons 114 in an earth formation 116. The sound vibration(s)
112 may be received
in by sensors, such as geophone-receivers 118, situated on the earth's
surface, and the geophones
118 produce electrical output signals, referred to as data received 120 in
FIG. 1.1.
[00040j In response to the received sound vibration(s) 112 representative of
different parameters
(such as amplitude and/or frequency) of the sound vibration(s) 112, the
geophones 118 may
produce electrical output signals containing data concerning the subsurface
formation. The data
received 120 may be provided as input data to a computer 122.1 of the seismic
truck 106.1, and
responsive to the input data, the computer 122.1 may generate a seismic and
microseismic data
output 124. The seismic data output may be stored, transmitted or further
processed as desired,
for example by data reduction.
[00041j FIG. 1.2 depicts a drilling operation being performed by a drilling
tool 106.2 suspended
by a rig 128 and advanced into the subsurface formations 102 to form a
wellbore 136 or other
channel. A mud pit 130 may be used to draw drilling mud into the drilling
tools via flow line 132
for circulating drilling mud through the drilling tools, up the wellbore 136
and back to the
surface. The drilling mud may be filtered and returned to the mud pit. A
circulating system may
be used for storing, controlling or filtering the flowing drilling muds. In
this illustration, the
drilling tools are advanced into the subsurface formations to reach reservoir
104. Each well may
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target one or more reservoirs. The drilling tools may be adapted for measuring
downhole
properties using logging while drilling tools. The logging while drilling tool
may also be adapted
for taking a core sample 133 as shown, or removed so that a core sample may be
taken using
another tool.
[00042] A surface unit 134 may be used to communicate with the drilling tools
and/or offsite
operations. The surface unit may communicate with the drilling tools to send
commands to the
drilling tools, and to receive data therefrom. The surface unit may be
provided with computer
facilities for receiving, storing, processing, and/or analyzing data from the
operation. The surface
unit may collect data generated during the drilling operation and produce data
output 135 which
may be stored or transmitted. Computer facilities, such as those of the
surface unit, may be
positioned at various locations about the wellsite and/or at remote locations.
[00043] Sensors (S), such as gauges, may be positioned about the oilfield to
collect data relating
to various operations as described previously. As shown, the sensor (S) may be
positioned in one
or more locations in the drilling tools and/or at the rig to measure drilling
parameters, such as
weight on bit, torque on bit, pressures, temperatures, flow rates,
compositions, rotary speed
and/or other parameters of the operation. Sensors (S) may also be positioned
in one or more
locations in the circulating system.
[00044] The data gathered by the sensors may be collected by the surface unit
and/or other data
collection sources for analysis or other processing. The data collected by the
sensors may be used
alone or in combination with other data. The data may be collected in one or
more databases
and/or transmitted on or offsite. All or select portions of the data may be
selectively used for
analyzing and/or predicting operations of the current and/or other wellbores.
The data may be
may be historical data, real time data or combinations thereof. The real time
data may be used in
real time, or stored for later use. The data may also be combined with
historical data or other
inputs for further analysis. The data may be stored in separate databases, or
combined into a
single database.
[00045] The collected data may be used to perform analysis, such as modeling
operations. For
example, the seismic data output may be used to perform geological,
geophysical, and/or
reservoir engineering analysis. The reservoir, wellbore, surface and/or
processed data may be
used to perform reservoir, wellbore, geological, and geophysical or other
simulations. The data
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outputs from the operation may be generated directly from the sensors, or
after some
preprocessing or modeling. These data outputs may act as inputs for further
analysis.
[00046] The data may be collected and stored at the surface unit 134. One or
more surface units
may be located at the wellsite, or connected remotely thereto. The surface
unit may be a single
unit, or a complex network of units used to perform the necessary data
management functions
throughout the oilfield. The surface unit may be a manual or automatic system.
The surface unit
134 may be operated and/or adjusted by a user.
[00047] The surface unit may be provided with a transceiver 137 to allow
communications
between the surface unit and various portions of the current oilfield or other
locations. The
surface unit 134 may also be provided with or functionally connected to one or
more controllers
for actuating mechanisms at the wellsite 100. The surface unit 134 may then
send command
signals to the oilfield in response to data received. The surface unit 134 may
receive commands
via the transceiver or may itself execute commands to the controller. A
processor may be
provided to analyze the data (locally or remotely), make the decisions and/or
actuate the
controller. In this manner, operations may be selectively adjusted based on
the data collected.
Portions of the operation, such as controlling drilling, weight on bit, pump
rates or other
parameters, may be optimized based on the information. These adjustments may
be made
automatically based on computer protocol, and/or manually by an operator. In
some cases, well
plans may be adjusted to select optimum operating conditions, or to avoid
problems.
[00048] FIG. 1.3 depicts a wireline operation being performed by a wireline
tool 106.3 suspended
by the rig 128 and into the wellbore 136 of FIG. 1.2. The wireline tool 106.3
may be adapted for
deployment into a wellbore 136 for generating well logs, performing downhole
tests and/or
collecting samples. The wireline tool 106.3 may be used to provide another
method and
apparatus for performing a seismic survey operation. The wireline tool 106.3
of FIG. 1.3 may,
for example, have an explosive, radioactive, electrical, or acoustic energy
source 144 that sends
and/or receives electrical signals to the surrounding subsurface formations
102 and fluids
therein.
[00049] The wireline tool 106.3 may be operatively connected to, for example,
the geophones 118
and the computer 122.1 of the seismic truck 106.1 of FIG. 1.1. The wireline
tool 106.3 may also
provide data to the surface unit 134. The surface unit 134 may collect data
generated during the
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wireline operation and produce data output 135 which may be stored or
transmitted. The wireline
tool 106.3 may be positioned at various depths in the wellbore to provide a
survey or other
information relating to the subsurface formation.
[00050j Sensors (S), such as gauges, may be positioned about the wellsite 100
to collect data
relating to various operations as described previously. As shown, the sensor
(S) is positioned in
the wireline tool 106.3 to measure downhole parameters which relate to, for
example porosity,
permeability, fluid composition and/or other parameters of the operation.
[00051j FIG. 1.4 depicts a production operation being performed by a
production tool 106.4
deployed from a production unit or Christmas tree 129 and into the completed
wellbore 136 of
FIG. 1.3 for drawing fluid from the downhole reservoirs into surface
facilities 142. Fluid flows
from reservoir 104 through perforations in the casing (not shown) and into the
production tool
106.4 in the wellbore 136 and to the surface facilities 142 via a gathering
network 146.
[00052j Sensors (S), such as gauges, may be positioned about the oilfield to
collect data relating
to various operations as described previously. As shown, the sensor (S) may be
positioned in the
production tool 106.4 or associated equipment, such as the Christmas tree 129,
gathering
network, surface facilities and/or the production facility, to measure fluid
parameters, such as
fluid composition, flow rates, pressures, temperatures, and/or other
parameters of the production
operation.
[000531 While only simplified wellsite configurations are shown, it will be
appreciated that the
oilfield or wellsite 100 may cover a portion of land, sea and/or water
locations that hosts one or
more wellsites. Production may also include injection wells (not shown) for
added recovery or
for storage of hydrocarbons, carbon dioxide, or water, for example. One or
more gathering
facilities may be operatively connected to one or more of the wellsites for
selectively collecting
downhole fluids from the wellsite(s).
[00054j It should be appreciated that FIGS. 1.2-1.4 depict tools that can be
used to measure not
only properties of an oilfield, but also properties of non-oilfield
operations, such as mines,
aquifers, storage, and other subsurface facilities. Also, while certain data
acquisition tools are
depicted, it will be appreciated that various measurement tools (e.g.,
wireline, measurement
while drilling (MWD), logging while drilling (LWD), core sample, etc.) capable
of sensing
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parameters, such as seismic two-way travel time, density, resistivity,
production rate, etc., of the
subsurface formation and/or its geological formations may be used. Various
sensors (S) may be
located at various positions along the wellbore and/or the monitoring tools to
collect and/or
monitor the desired data. Other sources of data may also be provided from
offsite locations.
[00055] The oilfield configuration of FIGS. 1.1-1.4 depict examples of a
wellsite 100 and various
operations usable with the techniques provided herein. Part, or all, of the
oilfield may be on land,
water and/or sea. Also, while a single oilfield measured at a single location
is depicted, reservoir
engineering may be utilized with any combination of one or more oilfields, one
or more
processing facilities, and one or more wellsites.
[00056] FIGS. 2.1-2.4 are graphical depictions of examples of data collected
by the tools of FIGS.
1.1-1.4, respectively. FIG. 2.1 depicts a seismic trace 202 of the subsurface
formation of FIG. 1.1
taken by seismic truck 106.1. The seismic trace may be used to provide data,
such as a two-way
response over a period of time. FIG. 2.2 depicts a core sample 133 taken by
the drilling tools
106.2. The core sample may be used to provide data, such as a graph of the
density, porosity,
permeability or other physical property of the core sample over the length of
the core. Tests for
density and viscosity may be performed on the fluids in the core at varying
pressures and
temperatures. FIG. 2.3 depicts a well log 204 of the subsurface formation of
FIG. 1.3 taken by
the wireline tool 106.3. The wireline log may provide a resistivity or other
measurement of the
formation at various depts. FIG. 2.4 depicts a production decline curve or
graph 206 of fluid
flowing through the subsurface formation of FIG. 1.4 measured at the surface
facilities 142. The
production decline curve may provide the production rate Q as a function of
time t.
[00057] The respective graphs of FIGS. 2.1, 2.3, and 2.4 depict examples of
static measurements
that may describe or provide information about the physical characteristics of
the formation and
reservoirs contained therein. These measurements may be analyzed to define
properties of the
formation(s), to determine the accuracy of the measurements and/or to check
for errors. The plots
of each of the respective measurements may be aligned and scaled for
comparison and
verification of the properties.
[00058] FIG. 2.4 depicts an example of a dynamic measurement of the fluid
properties through
the wellbore. As the fluid flows through the wellbore, measurements are taken
of fluid
properties, such as flow rates, pressures, composition, etc. As described
below, the static and
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dynamic measurements may be analyzed and used to generate models of the
subsurface
formation to determine characteristics thereof. Similar measurements may also
be used to
measure changes in formation aspects over time.
FRACTURE OPERATIONS
[00059] In one aspect, these techniques employ a model for characterizing a
hydraulic fracture
network as described below. Such a model includes a set of equations that
quantify the complex
physical process of fracture propagation in a formation driven by fluid
injected through a
wellbore. In one embodiment, these equations are posed in terms of 12 model
parameters:
wellbore radius xw and wellbore net pressure pw¨cic, fluid injection rate q
and duration tp,
matrix plane strain modulus E, fluid viscosity 1.1 (or other fluid flow
parameter(s) for non-
Newtonian fluids), confining stress contrast Aki, fracture network sizes h, a,
e, and fracture
spacing dx and dy.
[00060] Various fracture networks as used herein may have natural and/or man-
made fractures.
To facilitate production from a wellbore, the wellbore may be stimulated by
performing fracture
operations. For example, a hydraulic fracture network can be produced by
pumping fluid into a
formation. A hydraulic fracture network can be represented by two
perpendicular sets of parallel
planar fractures. The fractures parallel to the x-axis may be equally
separated by distance dy and
those parallel to the y-axis are separated by distance dx as illustrated in
Figure 3. Consequently,
the numbers of fractures, per unit length, parallel to the x-axis and the y-
axis, respectively, are
n=---and nY = ¨1. (1)
x d y d x
[000611 The pumping of fracturing fluid over time produces a propagating
fracture network that
can be represented by an expanding volume in the form of an ellipse with
height h, major axis a,
minor axis b or aspect ratio
e = ¨ . (2)
a
[000621 The governing equation for mass conservation of the injected fluid in
the fractured
subterranean formation is given by:
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27-cex a(0/0) + 4 a(BxioTe)
= 0 , (3a)
at ax
or
22-ty a(0p) + 4 a i BypTe = 0,
(3b)
e at ay e i
which for an incompressible fluid becomes respectively
27rex ¨a 0 + 4 a(BxTe)
= 0 , (3c)
at ax
or
22-ty a0 A a i yTe
-- + 9- B ___________________ = 0 , (3d)
e at ay e i
where 0 is the porosity of the formation,
pis the density of injected fluid
Te is an average fluid velocity perpendicular to the elliptic
boundary, and
B is the elliptical integral given by
B = ¨A- 1¨r ¨12(1¨ e2) ¨r1.3 2 (1¨ e2)2 r 1.3.5 2 (1¨ e2)3
. (4)
2 \,.2, 2 = 4, 3 2 = 4. 6, 5
The average fluid velocity Te may be approximated as
ii
Te .---, ¨ [vex (x, y = 0) + v ey (x = 0, y = ex)]
2
--:--, ¨1 (1+ e)vex(x, y = 0) (5)
2
--:--, ¨1 (1+1/e)vey(x = 0, y =ex)
2
with
kx ap
vex(x, y = 0) = ¨ --, (6a)
_ax_(x,y=0)
k ap
vey(x = 0, y = ex) = ¨,
-- (6b)
_ (x=0,y=ex)
where p is fluid pressure,
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ii is fluid viscosity, and
kx and ky are permeability factors for the formation along the x-direction
and the y-direction, respectively.
For the sake of mathematical simplicity, equations below are presented for an
incompressible
fluid as an example, with the understanding that fluid compressibility may be
accounted for by
using a corresponding equation of state for the injected fluid.
[000631 Using equations (5) and (6), governing equation (3) can be written as
27-rex a0 a 2 1 B (1+ e)xk x ap
= 0 , (7a)
at ax id ax I
or
27ty a0 a 1 B (1 + e)yk y ap
(7b)
e at ay e2 du ay ,
[000641 The width w of a hydraulic fracture may be calculated as
21
w = T,(19¨ (011(P ¨ u,),
0 p ific. (8)
H(p¨ 0¨,)={
1 p > 19¨,
where H is the Heaviside step function,
o-, is the confining stress perpendicular to the fracture,
E is the plane strain modulus of the formation, and
1 is the characteristic length scale of the fracture segment and given by the
expression
/ = d + (h ¨ d)H (d ¨ h) (9)
where h and d are the height and the length, respectively, of the fracture
segment.
[00065] When mechanical interaction between adjacent fractures is accounted
for, assuming that
the size of stimulated formation is much larger than either the height of the
ellipse or the
averaged length of fractures, the width of fractures parallel to the x-axis
and the y-axis,
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respectively, can be expressed as
2d
Wx = __________________
AExE eY cy , (10a)
2d
WY = _______________________________________ )
Y AEyE c cx (10b)
where acx and acy are the confining stresses, respectively, along the x-
direction and the y-
direction, respectively, and AEx and AEy are the coefficients for defining the
effective plane
strain modulus along the x-axis and y-axis, respectively.
[000661 For complex fracture networks the coefficients A
Ex and AEY may be approximately
represented by the following expressions
d[21 + (d - 21x)H (d -21x)]
A Ex _ x x Y Y
dl , (11a)
y x
d[21 + (dx - 21 )H (dx - 2l)]
Y Y Y Y
AEy =
dl . (1 lb)
x y
where lx and ly are the characteristic length scale along the x-axis and the y-
axis, respectively.
The value of the coefficient ( AEx ) for the effective plane strain modulus
along the x-axis can
be simplified for different cases of dx, dy, and h by any one of Tables 1-2
listed below. The value
of the coefficient ( AEy) for the effective plane strain modulus along the y-
axis can be
simplified for different cases of dx, dy, and h by any one of Tables 3-5
listed below.
Table 1 - Coefficient AEx for different cases of dx, dy, h
AEx
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dxcly dx<dy
dx<h dx>h dx<h dx>h
dy2h dy>2h dy2d, dy>2d, dy2h dy>2h
2d,
1
dy h dy dy h
Table 2 - Coefficient AE, for different cases of dx, dy, h
AEx
d,(1), dx<dy
dx<h dx>h dy<h dy>h
dy2h dy>2h dy2d, dy>2d, dy2h dy>2h
2d,
1
dy h dy dy h
Table 3 - Coefficient AEy for different cases of dx, dy, h
AEy
dyd, dy<d,
dy<h dy>h dy<h dy>h
dx2h dx>2h dx2dy dx>2dy dx2h dx>2h
2dy
2dy dy 2dy 2dy dy
d, 1
d, h d, d, h
Table 4a - Coefficient AEy for different cases of dx, dy, h
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AEy
dxd,
dx< h dx> h
dx2d, dx>2d, dy <h dy > h
dx2d, dx>2d, dx2h d, >2h
2dy
1 2dy 2dy dy
d, 1
d, d, h
Table 4b - Coefficient AEy for different cases of dx, dy, h
AEy
d, <dy
dx< h dx> h
dy <h dy > h dx2h dx>2h
dx2d, dx>2d, d, <2h dx>2h
2dy dy
2dy
1 2dy dy d, h
d, d, h
Table 5 - Coefficient AEy for different cases of dx, dy, h
AEy
dxdy d, <dy
x
dx< h dx> h d<h dx> h
dx2d, d, >2d, d<h d> h 2dy dx2h dx>2h
y y
d,
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dx2d y d, >2d y d, 2h d, >2h
2d y 2dy d y
1
d y 2d y 2d y dy d y
1
dy d y
[00067] The increase in porosity of the fractured formation (1\ ) can be
calculated as
AO = n w+n w ¨n n w w
xx yy xyxy
2d x
d yAExE(19¨ )1I(P acy)+
cy
(12)
2dY (P Cicx)11(P acx)
dxAEyE
The fracture permeability along the x-axis (kx) and the fracture permeability
along the y-axis (ky)
can be determined as
n w3
kx= x x
12
2d 3(13a)
(13a)
3E3 dxA3
y Ex
and
n w3
k = Y Y
12
2d3
(13b)
3E3 dYA3 __________________ (P Clcx)3 11(P cx)
x Ey
along the x-axis and y-axis, respectively.
[00068] For p>sacy and a negligible virgin formation permeability as compared
to the fracture
permeability along the x-axis, the governing equation (7a) can be integrated
from xw to x using
equation (13a) for the permeability (kx) to yield
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4(p ¨ )3 dp =3AEx3 d yE3 11 r 21-
1-fx ¨aesds ¨ . (14a)
Y dx (1+ e)Bd x3 x x, at
Similarly for p> o, the governing equation (7b) can be integrated from x to y
using equation
(12b) for the permeability (ky) to yield
dp 3e2 Ar3 yd xE3 Y ao s ,
4(p ¨ acx )3 = ______ 3 21-c ¨ ¨ as ¨ q . (14b)
dy (1+ e)Bd yy at e
In equations (13a) and (13b), x, is the radius of the wellbore and q is the
rate of fluid injection
into the formation via the wellbore. The inject rate q is treated as a
constant and quantified in
volume per unit time per unit length of the wellbore.
[00069] Equation (14a) can be integrated from x to a and yields a solution for
the net pressure
inside the fracture along the x-axis as
-1/4
fa ¨AE3xdy E3/1 I
3 r
P cy = q ¨ 27-cf ¨0 esds)dr . (15a)
(1+ e)B d x3 r x, at
Equation (14b) can be integrated from y to b yields a solution for the net
pressure inside the
fractures along the y-axis as
-1/4
3e2P AdxE
E3 3 it q ¨ 27r1ds)dr . acx = fb y
(15b)
(1+ e)B d3r x, at e
000701 For uniform ac, E, ji, n and d, equation (15a) reduces to
p ¨ o_cy = q lnr¨a ¨27rel rfr sds,\ ¨1 dr-1/4
Xi x r
(16a)
3A3 3 -\ 1/ 4
A = ExdyE
(1+ e)Bdx3
Similarly, equation (15b) reduces to
, I I ao ,
P ¨ = ell2 Apy qm ¨ 271- b r ¨
sas ¨ ar
e x. at r
/ 3 -\ 1/4 (16b)
3A3 d E ,u
A = Ey x
PY (1 e)Bd3
Y
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[00071] The wellbore pressure Pw is given by the following expressions:
r -\
-1/4
a r r a0 ' 1
pw = acy + Apx qln ¨ 2;ref f -- sds ¨dr , (17a)
xwi xõ, \,. xõ, at j r
r -\
, b r r ao , 1 , -1/4
ell 2 Apy q in ¨ ¨ ¨ i i ¨_, sas ¨ar . (17b)
x e xw xw dt j r
w i
By requiring the two expressions (17a, 17b) for the wellbore pressure Ay to be
equal, one obtains
the difference between confining stresses ( Au, ), which is also referred
herein to as stress
contrast Au, , as
Ao-, = o-, ¨ o-,
( a a ( f sds r a0 '1
= A q in ¨ ¨ 27-ref ¨ ¨ dr (18)
px
Xwi xõ, xõ, at j r
r-1/4
, ea 2g- sea (jr 0 1
¨e'12Apy q in ¨ ¨ ¨ ¨sds ¨dr .
x, õ j e x, x, at j r
[00072] Assuming negligible leakoff and incompressible fluid, the time tp for
the ellipse edge
propagating from xw to a along the x-axis and xw to b along the y-axis is
determined as
a b
¨qt p = e f A0xxdx + ¨1 f A,0 ydy
zx,v e xw Y
a 2dx(px¨Grcy) x,2dy(px ¨ Grcx)
= e _________________________
f xdx+ e ___________
f xdx (19a)
xw d y AEx E xw dxAEy E
1 fb 2dx(py¨o-cy) 2dy(py¨o-,x)
_________________________________________________ ydy ,
e Jx.- d y AEx E dxAEy E
or
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qtp ja r
¨ = LA0x(x) A0y(Y = ex)ixdx
ze xõ,
- i
d a d
= 2 fx, d x
__________________________ __ Y (px acy)xdx + f __ x (p x ¨ acy)xdx
E ix d AEx dx AEy j x.- d y AEx
\,. y
I d
(19b)
2 fa d x
__________________________ _______________ Y (Py ¨ 0- cx)xdx
E ix \,.dyAEx dxAEy j
2 I
6,o-c fa d
+ ________________________ x x, d
_____________________________ xdxxdx ,
E x,,, d yAEx ilw d AY
x Ey j
where xo-is defined as x,,,, xo- < a where
p 0- cx if x x,õ
p > c if x > x,
(19c)
p = acx if x = x, =
[00073] Equation (15a) can be rewritten for the case p = acx at X = Xa as
follows
- -1/4
3 a _
A3 dyii
E3 I
fEx r a 0
= ________________________________ q 2;z1 ¨esds dr . (20)
3 _________________________________
(1+ e)B ix, d dr xw at ,
[00074] The surface area of the open fractures may be calculated as follows
S==-=-= ;Tab x2hn x +2- i x õb x 2hn y ,
i
, a x, (21)
=2.Trean
d d
j
[000751 For a quasi-steady state, governing equations (7a) and (7b) reduce to
õ xkx dp
¨2B(1+ e )-- (22a)
du dx
2B(1 + e) Yk y dp
= q . (22b)
e2
du dy
Moreover, for the quasi-steady state, the pressure equations (15a) and (15b)
reduce to
- -1/4
3 fx da AEx3 dyE3q1-1
19 ¨ a y =
C (1+ e)B i 3 __ dr , (23a)
xr
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- -1/4
3e2 fb AE3 yd xE3 q,u
_____________________________________ dr . (23b)
(1+ e)B jY d3 r
Y
For the quasi-steady state and uniform properties of ac, E, ,u, n and d,
equations (16a) and (16b)
reduce to
i =\1/4
p ¨ au = Apx q in ¨a , (24a)
\,. X.'
i =\1/4
p ¨ = ell 2Apy q ln¨b . (24b)
Y i
Correspondingly, for the quasi-steady state, the wellbore pressure equations
(17a) and (17b)
reduce to
/ ,\ 1/4
a
pw = au + Apx q in¨ , (25a)
= xw j
/ ,\ 1/4
ea
pw = a cx e1/2 AII q py 1
in¨ . (25b)
= xw j
By requiring the two expressions (25a, 25b) for the wellbore pressure p to be
equal, one obtains
clx /AE \314
1¨ ell2 Aea ¨ Y (P. ¨ a cy) = AU,. ,
d y Ex)
(26)
-1/4
A ln(ea I .x.)
ea = .
ln(a / .x.) _
[00076] For the quasi-steady state and uniform properties of ac, E, i.t, n and
d, equations (19a)
and (19b), respectively, reduce to
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1/ . 3/4 r _________________________ r =\ 1/4
qt = p Mod4 y AEx d d Y a
71-
¨ x .\ x,rln a =\ 1/4 xdx + dx i ln ¨a xdx
d31 4 dy AEx dx AEy j ix J
, dyAEx x, \, XJ
A d114 A314 r d d =\ r \ 1/4
'
0 x Ey Y
x ____ fb in!? ydy
e d
l/ 2 3 1 4
" y \,.dyAEx d x,
xA Ey j \. Y i (27a)
A o- d C x
+ ________________________ (b2 .Y2 ) edY (x6,2 ¨ .Y2 ) ,
E ed y AEx d xAEy
-1/4
A= _____________
48q,u
,
(1+ e)BE
and
r d A3 -N1/4 -r
r qtp -N1/4 = A y Ex d d a
ze
x Y 'N x,rln a-N1/ 4 xdx + dx f ln ¨a xdx 0 ,43 d A d A
fx, dyAEx x, /
\,. "x i \,. y Ex x Ey j /
r 3 -N1/ 4 r
'N r -N1/4
1/2 . d xAEy d d a a
+e ii x Y in¨ xdx
r ,43 d A d A fx
\,. " y i \ , . y Ex x Ey j ' \.. X i (27b)
A o- õ d x
+ ________________________ (a2 xT2y) dY (X26', ¨ XT2,) ,
E _dy AEx dxAEy
-1/4
A=
48q,u
(1+ e)BE
Correspondingly, equation (20) can be solved to yield
1 r Aa, '\ 4
Xo. = a exp (28)
a A =
. px j
The integrations in equation (27) can be numerically evaluated rather easily
for a given .x,-.
1. CONSTRAINTS ON THE PARAMETERS OF THE MODEL USING FIELD DATA
[00077j In general, given the rest of the equations, equations (25a), (26) and
(27) can be solved to
obtain any three of the model parameters. Certain geometric and geomechanical
parameters of
the model as described above can be constrained using field data from a
fracturing treatment and
associated microseismic events. In one embodiment, the geometric properties
(dx and dy) and
the stress contrast (Auc) are constrained given wellbore radius xw and
wellbore net pressure
pw¨ac, fluid injection rate q and duration tp, matrix plane strain modulus E,
fluid viscosity i.t,
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and fracture network sizes h, a, e, as follows. Note that since xa in equation
(27) is calculated
using equation (28) as a function of Ac, the solution procedure is necessarily
of an iterative
nature.
dx3 /(A3 d )
[000781 Given these values, the value of Ex Y is determined according to
equation (25a)
by
dx3 __ = d2
0
AE3xd y
¨1/2' (29)
d =
3E3 q,u ln(a 1 x,y)
0
(p Ty ¨ Gr cy)4(1+ e)B
[000791 If ( 2dY > cl, d)')
and ( dx < h), equation (29) leads to
dy =1Nd0. (30)
Equations (26) and (27) become, respectively,
r ed -N1/2
1¨ Ad (PT, ¨ (icy) = Au , , (31)
ea
x i
and
at a eA r ,N1/4 r -N1/4
0 __ 2f x' in ¨a xdx + fa in ¨a xdx
z 21/4 _41/2 x ,.
"Y w A/ K.- Xy
(32)
3/4 r -N
2
ydy + Au a b ¨ x 2 2 ,,, 2
+ 2 A fin¨ b
1/4
e(xo. ¨ x,y) .
1/2 a /1/2 2E e
e x ' A YJ
Using solution (30), equations (31) and (32) can be solved to obtain
A0_, =
Lit { eA r ,\1/4 r '\ 1/4
õA , 21x - ln ¨a xdx + c ln ¨a xdx
7.1. 2¨ , x, õ
ay _ Ai xo- \,. x,
(33)
2314A0 bib '\1/4
e
2eE
1/2 a /1/2 iln x ydy b2 _x2 _ e2(x,72 _
x w Y i
and
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-N2
.19 (I
= JdoeAe2, ,, cy (34)
Pw 47,
¨A o,
[000801 If ( h >2dY ), equations (26) and (27) become, respectively,
e1/2'1/4
1 ¨ Aõ ¨d, (pw ¨ acy) = Acrc , (35)
yl
2 d
and
qt 23/4eA d x r =\1/4
= ___________________ 1 r + ln xdx+ fa ln xdx
71- din 2d x
y x 2 xx j
ilocPx/4 dy b r b1/4
1/23/4 ¨ ln¨ ydy (36)
ed 2 d
Y x
Ao-c 12 edy 2
+ __________________
E 2e
Combined with solution (30) and replacing Ao-c with equation (35), equation
(36) can be solved
for dx. Ao-c can then be calculated using equation (35).
[000811 If ( >
d Y), equation (29) leads to solution (30). Furthermore, if ( d, 2(1)' ),
equations (26) and (27) lead to solutions (33) and (34). On the other hand, if
(d, >2d
equations (26) and (27) lead to equations (35) and (36).
[00082] If ( d, dY ) and ( h < dY 2h ), equation (29) leads to solution (30).
Furthermore, if (
d,
2h) equations (26) and (27) lead to solutions (33) and (34). On the other
hand, if (d, > 2h
), equations (26) and (27) become, respectively,
18eo
2d2dx
1¨ Aea ,3 (P w cy) = Ao-c (37)
"
and
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- x1 1 1/4 7 '\1/4
qt 2,11a _ A0 1+ I ' a in¨ xdx+ in-1
xdx
27-re 2d0u2 dy j x, x j x., x,
(38)
h(x,2 ¨ x.2)(P. ¨ a, ) 1 18e2do2dx 4
Ed y h3
I
Equation (38) can be solved for dx and then Au can be calculated by equation
(37).
[000831 If (d, dY > 2h ), equation (29) leads to
dy = l'3 . (39)
do2
Equations (26) and (27) becomes, respectively,
/ 2 V7/4
do d
1¨ el12 Aea ___________ y (pi, ¨ 0-,)= Au, , (40)
h3
)
and
A d312 1 h3 \ / a '\1/4 7 '\1/4
qta _ f10 u0 .x' in¨ xdx+ f ln 1
1+ _____________________________ i xdx
27-re h2 do2dx j x,
(41)
h(x,2 ¨ x.2)(p. ¨0-,) 7 d2d V7/4
__________________________________ 1 e1/2 __ x .
Edy h3,
Equation (41) can be solved for dx and then Au can be calculated by equation
(40).
[00084] If ( d, <d Y 2d X) and ( dx < h ), equations (29), (26) and (27) lead
to solutions (30), (33)
and (34).
[00085] If (dY > 2dx ) and ( dx < h) equations (29), (26) and (27) become,
respectively,
dx3 = do2dy, (42)
/ N1/2
edo
1-2314A -d (piy¨ 0-,)= Ao-, , (43)
x)
and
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qt
d2 a1/4 -1/4
a a
a = 0 A d3/2 (-1+ x in xdx+ in ¨ xdx
27-re dx2 2d02 x x, x,
(44)
(x,2 ¨ x.2)Acic
2E
Equations (42), (43) and (44) can be solved for dx, dy and Ao-c.
[ y ¨ 000861 If ( h < d <d <2h
X ), equations (29), (26) and (27) lead to solutions (30),
(33) and (34).
[00087] If ( h <dx2h<dY), equation (29) leads to solution (39). Equations (26)
and (27)
become respectively
N1/2
1 ¨ 23'4 e1/2 A"e d p w Crcy = Acrc (45)
a d
x
and
qt A d3/2 7- 2 r
a = ra
0 h x a -\1/4 7 -\1/4
in¨ xdx+ in ¨a
1+ __________________________________________________ xdx
27-re h2 2d02 Xi xo-\
(46)
¨ xw2 )A o-c
Equations (45) and (46) can be solved to obtain
A d a a a 3/2 I h2 N1/4 N1/4
Ao-c= 2E 2 1+ 2 in¨ xdx+ ln ¨ xdx (47)
2(x, ¨ xw) h2 2d 0 ) x) x 27-ce
and
crcy
dx= 23/2ed0 Pw (48)
Pw (3- cy c
[000881 If ( 2h< <dY), equation (29) leads to solution (39) while equations
(26) and (27)
become equations (40) and (41), respectively.
[00089] In many circumstances, such as where the formation is shale, the
fracture network may
consist of a number of parallel equally-spaced planar fractures whose spacing
d is usually
smaller than fracture height h. In other cases, the opposite is true. Both can
lead to significant
simplifications. An example is presented below.
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2. SIMPLIFICATION OF MODEL FOR PARALLEL EQUALLY-SPACED PLANAR
FRACTURES WHOSE SPACING DX AND DY ARE SMALLER THAN FRACTURE
HEIGHT H
[00090] The assumption that fracture spacing d is usually smaller than
fracture height h leads to
ly = = d
l y
(49)
y cly .
Consequently, equations (11a) and (11b) can be simplified as
Aõ = I [2d, +(d ¨ 2d y)H (d y ¨ 2d,)] , (50a)
dy
A,Y =-1[2d +(d ¨ 2d y)H(d y ¨ 2d y)] . (50b)
dy
Equations (50a) and (50b) can be used to simplify equations (10a) and (10b) as
follows
2dydy(p ¨ sacy)H(p ¨a)cy
wy = _________________________________________________________________ (51a)
[2d, + (dy ¨261,)H(dy ¨2d,)]E '
2d w ydx(p ¨sacx)H(p ¨o-cõ) =.
(51b)
Y [2dy +(d y ¨ 2d y)H(d y ¨ 2d OW
Equations (50a) and (50b) can also be used to simplify equation (12) as
follows
2d y(p ¨o-,x)H(p ¨ sacy)
AO= ________________________________ + ______________________________ .(52)
[2d, +(d y ¨ 2d x)H (d y ¨2d ,)]E [2d y +(d y ¨2d y)H (d y ¨ 2d y)]E
Equations (50a) and (50b) can be used to simplify equations (13a) and (13b) as
follows
2d,3d y2
ky = kyo + ________________________________________ (p¨a)3H(p¨a), (53a)
3[2d, +(d y ¨ 2d x)H (d y ¨ 2d ,)]3 E3 cy cy
2d y3 dy2 .
(53b)
ky = k0+ 3124 y +(d y ¨ 2d y)H (d y ¨ 2d y)13 E3 (13 cr")311(13 cr')
These equations can be simplified in the following situations.
SITUATION I ( 2d, dy cl, / 2 ):
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[00091] With ( 2d, dY ci, 12 ), equations (50a) and (50b) become
(54a)
dy
2d y
A,Y =¨ . (54b)
d,
Furthermore, equations (51a) and (51b) become
d y(p ¨ so-cy)H(p ¨ so-cy)
w, = _____________________________ , (55a)
E
dx(p ¨ so-c,)H(p ¨ aõ)
= ____________________________________________________ (55b)
w .
Y E
Furthermore, equation (52) becomes
1 1
(56)
E E
Furthermore, equations (53a) and (53b) become
d2y3
k, = k,c, + ____________ (p¨ crcy)3 H(p¨ crcy) , (57a)
12E
d ,2
ky = ky0+ ______________ (p ¨ aõ)3 H(p ¨ aõ) . (57b)
12E3
Furthermore, equations (24a) and (24b) become
A I a`1/4
p ¨ o-cy = qln¨ , (58a)
dy Xi
1/2A
.
e
= 12 ' qln¨ , (58b)
d1, )1)
where
-1/4
24E3,u
A, = ________________________________________________________________ (59)
(1+ e)B
Furthermore, equations (25a) and (25b) become
29
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A / a -N1/4
p ¨ 0- = q ln¨ , (60a)
w cy din
xw )
"'Y
ein A / -\ 1/4
ea
Pw ¨ cc = dinP q in ¨ , (60b)
xw )
x
And furthermore, equation (26) becomes
_
/ , -N1/2
_
ea
1¨ ilea (pw ¨ 0-ey ) = AO-e . (61)
d
x )
Equation (60a) can be solved for d y as follows
A2 i -N1/2
, a
P
d y = 2 q in¨ . (62)
(Pw ¨ ucy Y xw )
¨ y ¨ x
[00092] With ( 2dx > d > d I 2 ), equations (27) and (28) become
_
/ -N1/4 / -N1/4 _
qt a eA
_____________________ 21' in ¨a xdx+ fa in¨a xdx
z ,-,114 A1/2 x. _
L' " y _
(63a)
1;:0/4 -
2 2 -
ydy + Au,. b ¨ Xii, 2 2
+ 2314A0 b lin e(x, ¨ xw) ,
1/2 71/2 ix 2E e
e a x ' Y )
at 2314 A0 far
, a = _____________________ a,\ 1/4 1 a r a ,\1/4
in¨ xdx + ¨ i in¨ xdx
ze A1/2 x. \,. .x j
2 xo- \,. xj
LI Y
(63b)
2314A0e1/2 a r a,\ 1/4 Ao_c (az _ x0.2)
+ __________________________ in¨xdx+ _____________ ,
din fc , x,
2E
and
d2 r,6,0-24
Y __________________________
(64)
x, = a exp .
q AP I
Equations (61), (63) and (64) can be solved iteratively for dx and Ao-c.
SITUATION II ( 2dy < dy )
[00093] With ( 2dy < dY ), equations (50a) and (50b) become
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Aõ =1, (65a)
2d
AEY = . (65b)
dx
Furthermore, equations (51a) and (51b) become
2dx(p¨acy)H(p¨acy)
ivx= (66a)
=
14) =dx(P¨aõ)11(P¨aõ)
(66b)
Furthermore, equation (52) becomes
2dx 1
A = d E(P-0-,y)11(P¨a,E(P¨aõ)11(P¨aõ)= (67)
Furthermore, equations (53a) and (53b) become
2dx3
(68a)
3dY E3
d x2
k =k n + _______________ (p¨aõ)3H(p¨aõ). (68b)
Y Y - 12E3
Furthermore, equations (24a) and (24b) become
r d 1/4 r a"4
p¨o-cy= Ap qln¨ , (69a)
\,.8dx xj
e1/2 . / =\1/4
11
p¨o-cx= 112' ____________ qln¨b , (69b)
dx Y
Furthermore, equations (25a) and (25b) become
r 1/4 r 1/4
U a
pw¨Gr, = Ap qln¨ , (70a)
Y 8 dx3 Xw
1/2 A 1/4
e (70b)
pw¨o-cx= _____________ 1/2P q ln ¨ea ,
dx xwi
And furthermore, equation (26) becomes
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r 2 -\ 1/4
8e d x
1 ___________________ Aea (Pw cy) = Aa c = (71)
Y
[000941 With ( < dY ), equations (27) and (28) lead to
,
-\1/4 a r a,\ 1/4
qt eA d1/4 r
a = 0 Y 2d x fxof a 2d
xdx+ in¨ xdx
2d31 4 dy =K, x d x
x y
r '\1/4
A0 1+2d x fb
in ydy (72a)
21/4 _1/2 /1/2
d y
e x y
c x (b_ _
Ao- 2d ') ')) ¨ x,24,) ,
2E edY
1/4 \1/4
qt A r dy d x xo-r ,\1/4 r a n
¨ ¨ n¨ +
¨ = l xdx ln=r xdx
;re d, 2 x d x
x jy
23/4 1d x a r a,\1/4
+ e1/2 A,¨ ¨ + ¨ in¨ xdx (72b)
din d 2 x, x
x y
Auc d x 2 2 1 2 2
+ _____________________ (a ¨ xw)¨ ¨ (x, ¨ x,y) ,
E d 2
_ Y
and
3 r \ 4
8dx Ao-c
x, = a exp (73)
qd y Ap j
Equations (70), (71), (72) and (73) can be combined and solved iteratively for
dx, dy and Ao-c.
SITUATION III ( dy <d x /2)
[00095] With ( dY < I 2 ), equations (50a) and (50b) become
2d x
A =- (74a)
Ex d y
AEy =1 . (74b)
Furthermore, equations (51a) and (51b) become
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wx __________________________________________________________________ (75a)
w = __________________________________________________________________ (75b)
Furthermore, equation (52) becomes
1 2d y
A 0 =E - acy)11(P ¨ acy)+ cl,E(P aõ)11(P ¨ aõ) = (76)
Furthermore, equations (53a) and (53b) become
d2
= k + _________________ (p ¨ o-c03 H(p¨ o-cy) , (77a)
x 12E3
2d3y
ky = kyo+ 3d3 ___________ (p o-õ)3 H(p ¨ o-õ) . (77b)
,E
Furthermore, equations (24a) and (24b) become
A, r C/1/4
P¨=-j- qln¨ , (78a)
dy x j
p ¨ o-cõ = ell2A ¨d, qln¨b , (78b)
8d 3
Furthermore, equations (25a) and (25b) become
A r 1/4
a
pw ¨0-c = 1P/2 qln¨ , (79a)
dy x,õ
-\1/4
pw¨o-cõ= e12 Ap ¨d, q ln ¨ea , (79b)
8d 3
Y x
And furthermore, equation (26) becomes
e d,
18d Aea (P w cy) ACI c = (80)
Y
[000961 With (d <d 12), equations (27) and (28) become
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at a eA r 2d r ci1/4 a r 1/4
____ ,,, 1+ fx' in¨ xdx + f in¨a xdx
21/ 4 A1,2
rt- d x, x
u y x j , x.- \,. x,
A (1114 r 2d b
0 x
1+ f in¨ ydy (81a)
2e 1/2d3/4
uY dx j xw y i
2 2
+ _____________________ (b x) 2edy (x0.2 _ x2 ) ,
2E e d x
A
qt 314 r d a4 1 a a"4
a 2 1 xr r
+ f ' in¨ xdx +f
¨ in¨ xdx
ze 0 d 1/2 2 d x,, x
y xj , 2 x.- \,. x,
r ,\1/4 ,
=\ 1/4
a
+ el/ 2 AA , ¨ 1 + i ln¨a xdx (81b)
Y ,I3
X i
L'tY)2 d x,, xj
Au c !2 2 dY 2 2
+ ____________________ (a¨ x)¨ ¨ (x, ¨ x) ,
E 2 d x
and
'\ 4
d 2 rAac
Y __________________________
(82)
x, = a exp .
q AP i
Equations (79), (80), (81) and (82) can be combined and solved iteratively for
dx, dy and A ac.
[00097] Figure 3 illustrates an exemplary operational setting for hydraulic
fracturing of a
subterranean formation (referred to herein as a "fracture site") in accordance
with the present
disclosure. The fracture site 400 can be located on land or in a water
environment and includes a
treatment well 401 extending into a subterranean formation as well as a
monitoring well 403
extending into the subterranean formation and offset from the treatment well
401. The
monitoring well 403 includes an array of geophone receivers 405 (e.g., three-
component
geophones) spaced therein as shown.
[00098] During the fracturing operation, fracturing fluid is pumped from the
surface 411 into the
treatment 401 causing the surrounding formation in a hydrocarbon reservoir 407
to fracture and
form a hydraulic fracture network 408. Such fracturing produces microseismic
events 410, which
emit both compressional waves (also referred to as primary waves or P-waves)
and shear waves
(also referred to as secondary waves or S-waves) that propagate through the
earth and are
recorded by the geophone receiver array 405 of the monitoring well 403.
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[00099] The distance to the microseismic events 410 can be calculated by
measuring the
difference in arrival times between the P-waves and the S-waves. Also,
hodogram analysis,
which examines the particle motion of the P-waves, can be used to determine
azimuth angle to
the event. The depth of the event 410 is constrained by using the P- and S-
wave arrival delays
between receivers of the array 405. The distance, azimuth angle and depth
values of such
microseismic events 410 can be used to derive a geometric boundary or profile
of the fracturing
caused by the fracturing fluid over time, such as an elliptical boundary
defined by a height h,
elliptic aspect ratio e and major axis a as illustrated in Figure 3.
[0001001 The site 401 also includes a supply of fracturing fluid and
pumping means (not
shown) for supplying fracturing fluid under high pressure to the treatment
well 401. The
fracturing fluid can be stored with proppant (and possibly other special
ingredients) pre-mixed
therein. Alternatively, the fracturing fluid can be stored without pre-mixed
proppant or other
special ingredients, and the proppant (and/or other special ingredients) mixed
into the fracturing
fluid in a controlled manner by a process control system as described in U.S.
Patent No.
7,516,793, herein incorporated by reference in its entirety. The treatment
well 401 also includes a
flow sensor S as schematically depicted for measuring the pumping rate of the
fracturing fluid
supplied to the treatment well and a downhole pressure sensor for measuring
the downhole
pressure of the fracturing fluid in the treatment well 401.
[000101] A data processing system 409 is linked to the receivers of the
array 405 of the
monitoring well 403 and to the sensor S (e.g., flow sensor and downhole
pressure sensor) of the
treatment well 401. The data processing system 409 may be incorporated into
and/or work with
the surface unit 134. The data processing system 409 carries out the
processing set forth in
Figure 5 and described herein. As will be appreciated by those skilled in the
art, the data
processing system 409 includes data processing functionality (e.g., one or
more microprocessors,
associated memory, and other hardware and/or software) to implement the
disclosure as
described herein.
[000102] The data processing system 409 can be realized by a workstation or
other suitable
data processing system located at the site 401. Alternatively, the data
processing system 409 can
be realized by a distributed data processing system wherein data is
communicated (preferably in
real time) over a communication link (typically a satellite link) to a remote
location for data
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analysis as described herein. The data analysis can be carried out on a
workstation or other
suitable data processing system (such as a computer cluster or computing
grid). Moreover, the
data processing functionality of the present disclosure can be stored on a
program storage device
(e.g., one or more optical disks or other hand-holdable non-volatile storage
apparatus, or a server
accessible over a network) and loaded onto a suitable data processing system
as needed for
execution thereon as described herein.
[000103] In step 501, the data processing system 409 stores (or inputs from
suitable
measurement means) parameters used in subsequent processing, including the
plain strain
modulus E (Young's modulus) of the hydrocarbon reservoir 407 that is being
fractured as well as
the fluid viscosity (Ix) of the fracturing fluid that is being supplied to the
treatment well 401 and
the radius (xw) of the treatment wellbore.
[000104] In steps 503-511, the data processing system 409 is controlled to
operate for
successive periods of time (each denoted as At) that fracturing fluid is
supplied to the treatment
well 401.
[000105] In step 505, the data processing system 409 processes the acoustic
signals
captured by the receiver array 405 over the period of time At to derive the
distance, azimuth
angle and depth for the microseismic events produced by fracturing of the
hydrocarbon reservoir
407 over the period of time At. The distance, azimuth and depth values of the
microseismic
events are processed to derive an elliptical boundary characterizing the
profile of the fracturing
caused by the fracturing fluid over time. In the preferred embodiment, the
elliptical boundary is
defined by a height h, elliptic aspect ratio e and major axis a as illustrated
in Figure 3.
[000106] In step 507, the data processing system 409 obtains the flow rate
q, which is the
pumping rate divided by the height of the elliptic fractured formation, of the
fracturing fluid
supplied to the treatment well for the period of time At, and derives the net
downhole pressure
pw¨ac of the fracturing fluid at the end of the period of time At. The
wellbore net pressure
pw¨ac can be obtained from the injection pressure of the fracturing fluid at
the surface according
to the following:
Pw (lc = P nuface BHTP ¨ pipe ¨
pelf P hydrostatic (83)
wheresteace is the injection pressure of the fracturing fluid at the surface;
BHTP is the bottom
v
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hole treating pressure; pp,pe is the friction pressure of the tubing or casing
of the treatment well
while the fracturing fluid is being injected into the treatment well; this
friction pressure depends
on the type and viscosity of the fracturing fluid, the size of the pipe and
the injection rate;
ppe,f is the friction pressure through the perforations of the treatment well
that
provide for injection of the fracturing fluid into the reservoir; and
Phydrostahc is the hydrostatic
pressure due to density of the fracturing fluid column in the treatment well.
The wellbore net pressure pw¨ac can also be derived from BHTP at the beginning
of treatment
and the injection pressure Pstaface at the beginning of the shut-in period.
The wellbore net pressure
pw¨ac at the end of treatment can be calculated by plugging these values into
equation (83) while
neglecting both friction pressures pinpe and ppe,f , which are zero during the
shut-in period.
[000107] In step 509, the data processing system 409 utilizes the
parameters (E, t, xw)
stored in 501, the parameters (h, e and a) defining the elliptical boundary of
the fracturing as
generated in step 505, and the flow rate q, the pumping period tp and the net
downhole pressure
pw¨ac as generated in step 507 in conjunction with a model for characterizing
a hydraulic
fracture network as described herein to solve for relevant geometric
properties that characterize
the hydraulic fracture network at the end of the time period At, such as
parameters dx and dy and
the stress contrast Acyc as set forth above.
[000108] In step 511, the geometric and geomechanical properties (e.g., dx,
dy, Acyc) that
characterize the hydraulic fracture network as generated in step 509 are used
in conjunction with
a model as described herein to generate data that quantifies and simulates
propagation of the
fracture network as a function of time and space, such as width w of the
hydraulic fractures from
equations (10a) and (10b) and the times needed for the front and tail of the
fracturing formation,
as indicated by the distribution of induced microseismic events, to reach
certain distances from
equation (19). The geometric and geomechanical properties generated in step
509 can also be
used in conjunction with the model to derive data characterizing the fractured
hydrocarbon
reservoir at the time period tp, such as net pressure of fracturing fluid in
the treatment well (from
equations (17a) and (17b), or (25a) and (25b)), net pressure inside the
fractures (from equations
=
(16a) and (16b), or (24a) and (24b)), change in fracture porosity (AOfrom
equation 12), and
change in fracture permeability (kx and ky from equations (13a) and (13b)).
[000109] In optional step 513, the data generated in step 511 is used for
real-time
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visualization of the fracturing process and/or optimization of the fracturing
plan. Various
treatment scenarios may be examined using the forward modeling procedure
described below. In
general, once certain parameters such as the fracture spacing and the stress
difference have been
determined, one can adjust the other parameters to optimize a treatment. For
instance, the
injection rate and the viscosity or other properties of fracturing fluid may
be adjusted to
accommodate desired results. Exemplary display screens for real-time
visualization of net
pressure change of fracturing fluid in the treatment well along the x-axis,
fracture width w along
the x-axis, changes in porosity and permeability along the x-axis are
illustrated in Figures 6.1 ¨
6.4.
[000110i In step 515, it is determined if the processing has been completed
for the last
fracturing time period. If not, the operations return to step 503 to repeat
the operations of step
505-513 for the next fracturing time period. If so, the operations continue to
step 517.
[0001iii In step 517, the model as described herein is used to generate
data that quantifies
and simulates propagation of the fracture network as a function of time and
space during the
shut-in period, such as width w of hydraulic fractures and the distance of the
front and tail of the
fracturing formation over time. The model can also be used to derive data
characterizing the
fractured hydrocarbon reservoir during the shut-in period, such as net
pressure of fracturing fluid
in the treatment well (from equations (17a) and (17b), or (25a) and (25b)),
net pressure inside the
fractures (from equations (16a) and (16b), or (24a) and (24b)), change in
fracture porosity (A
from equation 12), and change in fracture permeability (kx and ky from
equations (13a) and
(13b)).
[000112] Finally, in optional step 519, the data generated in step 511
and/or the data
generated in step 517 is used for real-time visualization of the fracturing
process and/or shut-in
period after fracturing and/or optimization of the fracture plan. Figures 7.1-
7.4 illustrate
exemplary display screens for real-time visualization of net pressure of
fracturing fluid in the
treatment well as a function of time during the fracturing process and then
during shut-in (which
begins at the time of 4 hours), net pressure inside the fractures as a
function of distance at a time
at the end of fracturing and at a time during shut-in, the distance of the
front and tail of the
fracturing formation over time during the fracturing process and then during
shut-in, fracture
width as a function of distance at a time at the end of fracturing and at a
time during shut-in,
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respectively. Note that the circles of Figure 7.3 represent locations of
microseismic events as a
function of time and distance away from the treatment well during the
fracturing process and
then during shut-in.
[000113] The hydraulic fracture model as described herein can be used as
part of forward
calculations to help in the design and planning stage of a hydraulic
fracturing treatment. More
particularly, for a given the major axis a = ai at time t = ti, calculations
can be done according to
the following procedure:
1. assume ¨ if t = to (i = 0), otherwise
at
2. knowing ¨ from t = t1, determine e using equation (18)
at
3. knowing ¨ and e, calculate p - acx and p - au using equations (15a) and
(15b) or
at
equations (16a) and (16b)
4. knowing p - o-cx and p - au, calculate AO using equation (12)
5. knowing e and AO, calculate t = t, using equations (19), or (27) and (28)
6. knowing At = t, ¨ti and AO, calculate ¨ as AwAt
at
7. repeat steps 2 to 6 till the whole calculation process converges
Carrying out the procedure described above for i = 1 to N simulates the
propagation of an
induced fracture network till front location a = aN. Distributions of net
pressure, fracture width,
porosity and permeability as functions of space and time for x < aN and t < tN
are obtained as
well.
[000114] Advantageously, the hydraulic fracture model and fracturing
process based
thereon constrains geometric and geomechanical properties of the hydraulic
fractures of the
subterranean formation using the field data to reduce the complexity of the
fracture model and
the processing resources and time required to provide characterization of the
hydraulic fractures
of the subterranean formation. Such characterization can be generated in real-
time to manually or
automatically manipulate surface and/or down-hole physical components
supplying fracturing
fluids to the subterranean formation to adjust the hydraulic fracturing
process as desired, such as
by optimizing fracturing plan for the site (or for other similar fracturing
sites).
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PRODUCTION OPERATIONS
[000115] In another aspect, these techniques employ fracture models for
determining
production estimates. Such estimations may be made, for example, by applying
the HFN
modeling techniques, such as those using a wiremesh HFN model with an
elliptical structure, to
production modeling. These techniques may be used in cases with multiple or
complex fractures,
such as shale or tight-sand gas reservoirs. Such models may use, for example,
an arbitrarily time-
dependent fluid pressure along hydraulic fractures. Corresponding analytical
solutions may be
expressed in the time-space domain. Such solutions may be used in high speed
applications for
hydraulic fracturing stimulation job design, optimization or post-job
analysis.
[000116] These techniques employ an analytical approach that provides a
means to forecast
production from reservoirs, such as shale reservoirs, using an HFN of elliptic
form. Such
forecasts may involve the use of analytical models for forecasting or
analyzing production from
oil and gas reservoirs with imbedded hydraulic fractures. The forecasting
models may be
empirical or analytical in nature.
[000117] Examples of empirical forecasts are provided in US Patent Nos.
7788074,
6101447 and 6101447, and disclosed in Arps, "Analysis of Decline Curves", SPE
Journal Paper,
Chapt. 2, pp. 128-247 (1944). Empirical forecasts may involve an estimate of
well production
using various types of curves with adjustable parameters for different flow
regimes separately
during a reservoir's lifespan.
[000118] Examples of analytical forecasts are provided in Van Everdingen et
al., "The
Application of the Laplace Transformation to Flow Problems in Reservoirs",
Petroleum
Transactions AIME, Dec. 1949, pp. 305-324; van Kruysdijk et al.,
"Semianalytical Modeling of
Pressure Transients in Fractured Reservoirs," SPE 18169, SPE Tech. Conf. and
Exhibition, 2-5
Oct. 1988, Houston, TX; Ozkan et al., "New Solutions for Well-Test-Analysis
Problems: Part 1 -
Analytical Considerations", SPE 18615, SPE Formation Evaluation, Vol. 6, No.
3, SPE, Sept.
1991; and Kikani, "Pressure-Transient Analysis of Arbitrarily Shaped
Reservoirs With the
Boundary-Element Method", SPE 18159 SPE Formation Evaluation March 1992.
Additional
analytical approaches have later been applied by de Swaan et al., "Analytic
Solutions for
Determining Naturally Fractured Reservoir Properties by Well Testing," SPE
Jrnl., pp. 117-22,
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Jun. 1976; van Kruysdij et al., "A Boundary Element Solution of the Transient
Pressure
Response of Multiple Fractured Horizontal Wells", presented at the 2nd
European Conf. on the
Mathematics of Oil Recovery, Cambridge, UK, 1989; Larsen, "Pressure-Transient
Behavior of
Horizontal Wells With Finite-Conductivity Vertical Fractures", SPE 22076, Soc.
of Petroleum
Engr., Intl. Arctic Tech. Conf., 29-31 May 1991, Anchorage, AL; Kuchuk et al.,
"Pressure
Behavior of Horizontal Wells with Multiple Fractures', 1994, Soc. of Petroleum
Engrs., Inc.,
Univ. of Tulsa Centennial Petroleum Engr. Symp., 29-31 Aug. 1994, Tulsa, OK;
Chen et al., "A
Multiple-fractured Horizontal Well in a Rectangular Drainage Region", SPE
Jrnl. 37072, Vol. 2,
No. 4, Dec. 1997. pp. 455-465; Brown et al., "Practical Solutions for Pressure
Transient
Responses of Fractured Horizontal Wells in Unconventional Reservoirs", SPE
Tech. Conf. and
Exhibition in New Orleans, LA, 2009; Bello,"Rate Transient Analysis in Shale
Gas Reservoirs
with Transient Linear Behavior", PhD Thesis, 2009; Bello et al., "Multi-stage
Hydraulically
Fractured Horizontal Shale Gas Well Rate Transient Analysis", North Africa
Tech. Conf. and
Exhibition, 14-17 Feb. 2010, Cairo, Egypt; Meyer et al, "Optimization of
Multiple Transverse
Hydraulic Fractures in Horizontal Wellbores", 2010, SPE 131732, SPE
Unconventional Gas
Conf., 23-25 Feb. 2010, Pittsburgh, PA, USA; and Thompson et al.,
"Advancements in Shale
Gas Production Forecasting ¨ A Marcellus Case Study," SPE 144436, North
American
Unconventional Gas Conf. and Exhibition, 14-16 Jun. 2011, The Woodlands, TX,
USA.
[000119] The analytical approach may involve obtaining pressure or
production rate
solutions by solving partial differential equations describing gas flow in the
reservoir formation
and through the fractures. By way of example, Laplace transform and numerical
inversion may
be used. In another example, Laplace transformation may be used to obtain
asymptotic solutions
for early and late production periods, respectively, from a horizontally
radial reservoir subject to
either a constant pressure drop or a constant production rate at the wellbore.
The ordinary
differential equations in the Laplace domain may be solved using Green's and
point source
functions, and then transforming the solutions back to the time-space domain
through a
numerical inversion to study production from horizontal wells with multiple
transverse fractures.
[000120] The analytical approach may also involve using the time-space
domain.
Additional examples of the analytical approach are provided by Gringarten et
al., "The Use of
Source and Green's Functions in Solving Unsteady-Flow Problems in Reservoirs",
Society of
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Petroleum Engineers Journal 3818, October 1973, Vol. 13, No. 5, pp. 285-96;
Cinco et al.,
"Transient Pressure Behavior for a Well With a Finite-Conductivity Vertical
Fracture", SPE
6014, Society of Petroleum Engineers Journal, August 15, 1976; and in US
Patent No. 7363162.
Green's and point source functions may be corresponded to simplified cases.
Some of the
functions may be used to study production from a vertical well intersected by
a vertical fracture.
Time-space domain analytical solutions may also provide fluid pressure in a
semi-infinite
reservoir with a specified fluid source/sink.
MODEL AND SOLUTIONS FOR WIREMESH HFN
[000121] Figures 8.1 - 8.3 depict alternate views of HFN models 800.1,
800.2 and 800.3,
respectively, usable for hydraulic fracture modeling. The HFN models may be
created using the
HFN techniques described above. Application of the disclosed models to
hydraulic fracturing
stimulation job design and post-job analysis is described using wiremesh HFN
models
800.1,800.2,800.3 as an example. These figures each depict a wellbore 820 with
a hydraulic
fracture network (HFN) 822 thereabout.
[000122] The HFN 822 is an elliptical structure with a plurality of
vertical fractures 824
perpendicular to another a plurality of vertical fractures 826 forming a
wiremesh configuration.
The plurality of vertical fractures define a plurality of matrix blocks 828 of
the HFN 822. The
HFN 822 is a complex fracture network having a plurality of intersecting
fractures 824 and 826
that are hydraulically connected for fluid flow therebetween. The intersecting
fractures may be
generated by fracturing of the formation. Fractures as used herein may be
natural and/or man
made.
[000123] As shown in Figure 8.1, the HFN 822 has a height h along a minor
diameter, a
radius b along its minor axis and aligned with the wellbore 820, and a radius
a along its major
axis. Some of the dimensions of the HFN are also shown in Figure 3.
[000124] While Figures 8.1-8.3 depict complex HFN models 800.1, 800.2,
800.3, the
models may also be used with reservoirs having single or parallel hydraulic
fractures. Also,
while the wellbore 820 is depicted as passing through the HFN 822 parallel to
the vertical lines,
the HFN 822 may be oriented as desired about the wellbore 820. Application of
the disclosed
models to hydraulic fracturing stimulation job design and post-job analysis is
described using a
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wiremesh HFN 822 as an example. Application to reservoirs with single or
parallel hydraulic
fractures or a fracture network of non-elliptic shape can be done in a similar
manner, but
adjusted as needed to a comparably simpler or more complicated configuration.
PROPPANT PLACEMENT
[000125] Information about proppant placement in an HFN, such as the HFN
822 of
Figures 8.1-8.3, may be used to quantify production from the HFN. One or more
types of
proppant may be injected with an injection or treatment fluid during
stimulation to keep the
hydraulic fractures open after a fracturing job is done.
[000126] Figures 9 and 10 depict views of proppant placement about an HFN
and fractures
of an HFN, respectively. Figure 9 shows a cross-sectional view of the HFN 822
of Figure 8.2
taken along line 9-9. As shown in this view, proppant 823 is positioned in
wellbore 820, and
extends horizontally through the wellbore 820 along a major fracture and into
the surrounding
formation. As also shown in Figure 9, the proppant 823 may transport in
different transport
patterns 827, 829.
[000127] Fig. 10 is picture of a fracture 827 with proppant extending
therein. Fluid flows
through the fracture 827 from the left to the right. The proppant 823 is
carried by the fluid 827,
but settles on the left side of the fracture as it travels from left to right.
The proppant 827 as
depicted entering a left portion of the fracture 827 as indicated by the
lighter shaded regions.
[000128] The flow of proppant through an HFN may be defined by an analysis
of transport
of the proppant. For N types of proppant particles each with volume fraction
V1,,, the total
proppant volume fraction is
V =L17
VP p
1=1 (84)
[000129] The placement of proppant along the fractures of an HFN involves
horizontal
transport, vertical settling and possible bridging of the proppant. As shown
in Figure 9, proppant
type i is transported in all directions by the transport pattern 825. This can
be mathematically
described by the following:
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a(017 i) a
127-1-Akx ap
27-1-A v =0
at ax 11/ ax P'11
(85)
This equation also describes the horizontal flow of fluid in Figure 10.
[000130] If the proppant remains in the primary fracture along the x-axis
as shown in
transport pattern 829 of Fig. 9, then the proppant transport can be described
by
a(wxvp,i) ( wx3 ap v =
at ax 02,u OX "
(86)
[000131] For a uniform horizontal volume flow rate q, the above equations
reduce to,
respectively,
( V)i a(qv _________________ i)
27-t-yx P' + =0
(87)
For transport along a fairway only, the following equation applies:
a(wxvp,i) q
at _______________ + 2,2tyx P'',
(88)
When fluid leakoff qi is taken into consideration, the above equations become,
respectively,
a(017 i) akg¨q )v
27-t-A 1 " =0
Ot (89)
and
a(wxv i) 1q¨q1 __ V =0
Ot OX 27-cyx P,,
(90)
[000132] As shown in Figure 10, vertical settling may also occur as the
proppant 823 is
carried through the fracture 827. Proppant settling may be quantified by the
Stokes particle
terminal velocity
2
= _____________________ p f)d
vps,i
18,uf
(91)
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where pf and pf are the density and viscosity of the suspension fluid, pp,i
and dp,1 are the density
and mean particle diameter of proppant type i. When the size or concentration
of the proppant is
too large, bridging of proppant may occur. This is described by a modification
to the settling
velocity
vps,i = vst,if (V1,, dp,i,w)
(92)
where
'N 0.25
Wcr
if werj
f (VP d w) =
'
0 if w<wcr,i
w =min B,,l+VpB Cr ______________ dp,i Bcr = 2'5
0.17) (93)
Hindering factors may account for effects of fracture width, proppant size &
concentration, fiber,
flow regime, etc. Proppant movement may be further hindered by other factors
such as fluid flow
regime and the presence of fiber.
PRODUCTION
[000133] Figure 11 shows the HFN 822 taken along line 9-9. As shown in this
view, the
HFN 822 is depicted as having a plurality of concentric ellipses 930 and a
plurality of radial flow
lines 932. The radial flow lines 932 initiate from a central location about
the wellbore 820 and
extend radially therefrom. The radial flow lines 932 represent a flow path of
fluid from the
formation surrounding the wellbore 820 and to the wellbore 820 as indicated by
the arrows. The
HFN 822 may also be represented in the format as shown in Fig. 3.
[000134] Due to an assumed contrast between the permeability of the matrix
and that of the
HFN 822, global gas flow through the reservoir consisting of both the HFN 822
and the
formation matrix can be separated into the gas flow through the HFN 822 and
that inside of the
matrix blocks 828. The pattern of gas flow through the HFN 822 may be
described
approximately as elliptical as shown in Figure 11.
[000135] The HFN 822 uses an elliptical configuration to provide a coupling
between the
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matrix and HFN flows that is treated explicitly. A partial differential
equation is used to describe
fluid flow inside a matrix block that is solved analytically. Three-
dimensional gas flow through
an elliptic wiremesh HFN can be approximately described by:
r
aPfapf qg
at
x 1C - = (94) xax f ax apf
i , _______________________________
(/'f
aP
where t is time, x is the coordinate aligned with the major axis of the
ellipse, pf and g are fluid
pressure and density of fluid, 4 and xy are the porosity and the x-component
of the pressure
diffusivity of the HFN, and qg is the rate of gas flow from the matrix into
the HFN. All involved
properties may be a function of either t or x or both.
[000136] For each time t, calculations of fluid pressure using equation
(94) may begin from
the outmost ring of the elliptical reservoir domain and end at the center of
the HFN 822 at
wellbore 820, or in the reverse order. Fluid pressure along the elliptical
domain's boundary is
taken as that of the reservoir before production. It may be assumed that no
production takes place
outside of the domain.
[000137] Outside of the HFN, equation (94) still applies nominally, but
with qg = 0, 4 = m
and xy = 'em, where Om and Km are the porosity and the pressure diffusivity of
the reservoir matrix.
Given qg there are various ways available to solve equation (94), either
analytically or
numerically. Due to the complex nature of the HFN and fluid properties,
numerical approaches
may be used for the sake of accuracy. An example of numerical solution is
given below.
[000138] Dividing the elliptic reservoir domain containing the HFN into N
rings, the rate of
gas production from a reservoir matrix into the HFN contained by the inner and
outer boundaries
of the k-th ring is
qgk = qgxkAxk ggykAyk (95)
where Axk and Ayk are the total surface area of the fractures inside of the
ring, parallel to the major
axis (the x-axis) and the minor axis (the y-axis), respectively, and qgxk and
qgyk are the
corresponding rates of fluid flow per unit fracture surface area from the
matrix into the fractures
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parallel to the x- and y-axis, respectively. Fluid pressure pf and the rate of
gas production at the
wellbore can be obtained by numerically (either finite difference, finite
volume or a similar
method) solving equation (94) for any user specified initial and boundary
conditions and by
coupling the model with a wellbore fluid flow model.
[000139] Total surface area of fractures contained inside of the k-th ring
can be calculated
by
AXk = 4hk L,Ixk2 - 4( j1õny r)2 Vxk2 401õny /y)2]
(96)
Ny, ________________________________________________
Y, _______________________________
Ayk = 4hk Lvxk2 _4(iLmx)2 _ Lvxk2 _ 4(iLiwc)2
Y,
where y is the aspect ratio of the elliptical HFN, xk and hk are the location
and the height of the k¨
th ring, L and Lmy are the distances between neighboring fractures parallel to
the x-axis and the
y-axis, respectively, as shown in Figure 12. The Nx, and Nx, are the number of
fractures parallel
to and at either side of the x-axis inside the outer and the inner boundaries,
respectively, of the k¨
th ring, and Ny, and Ny, are the number of fractures parallel to and at either
side of the y-axis
inside the outer and the inner boundaries, respectively, of the k-th ring.
[000140] The pattern of gas flow through the HFN 822 may also be described
based on
fluid flow through individual matrix blocks 828 as shown in Figure 12. Figure
12 is a detailed
view of one of the blocks 828 of HFN 822 of Figure 11. As shown in this view,
the direction of
gas flow inside of a matrix block 828 can be approximated as perpendicular to
the edges of the
matrix block 828. Fluid flow is assumed to be linear flow toward outer
boundaries 1240 of the
block 828 as indicated by the arrows, with no flow boundaries 1242 positioned
within the block
828.
[000141] Fluid flow inside a rectangular matrix block 828 can be
approximately described
by
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aPm --,2
u P
at 1( m 2 =o
m
as
pm(t,$)= pr
(97)
pm (t, Ls) = p f (t)
al, m
¨ =0
as s=0
where s is the coordinate, aligned with the x-axis or y-axis, L is the
distance between the fracture
surface and the effective no-flow boundary, pm is fluid pressure and p, is the
reservoir pressure.
Equation (97) can be solved to obtain the rate of fluid flow from the matrix
into the fractures
inside the k-th ring
( __________________________________________________ ( 4
a pin a ftdpfl, Ly ________ LY + 2 lc' (1' ¨ ") 1 e 161C (t-u) du
qgxk = Om ________________ erfc ,
ap atio du 2 41/ x-m (t ¨ u) V 7-t-
I I _
(98)
( __________________________________________________ ( ex
a pin a ftdp fl, Lx Lx
q gyk = O erfc __ , ___ +2 11 x.'n (t ¨ ") 1 e 161cm(t-u)
du
m _____________
ap &Jo du 2 411 x-m (t ¨u)I 7-t-
I_
where pfk is the pressure of the fluid residing in fractures in the k-th ring
and pm is the density of
the fluid residing in the matrix. The coupling of pfk and qgk calculations can
be either explicit or
implicit. It may be implicit for the first time step even if the rest of the
time is explicit.
[000142] Conventional techniques may also be used to describe the concept
of fluid flow
through a dual porosity medium. Some such techniques may involve a 1D pressure
solution with
constant fracture fluid pressure, and depict an actual reservoir by
identifying the matrix, fracture
and vugs therein as shown in Figure 13.1, or depicting the reservoir using a
sugar cube
representation as shown in Figure 13.2. Examples of conventional fluid flow
techniques are
described in Warren et al., "The Behavior of Naturally Fractured Reservoirs",
SPE Journal, Vol.
3, No. 3, Sep. 1963.
[000143] Examples of fracture modeling that may be used in the modeling
described herein
are provided in Wenyue Xu et al., "Quick Estimate of Initial Production from
Stimulated
Reservoirs with Complex Hydraulic Fracture Network," SPE 146753, SPE Annual
Tech. Conf.
and Exhibition, Denver, CO., 30 Oct.-2 Nov., 2011, the entire content of which
is hereby
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incorporated by reference.
HYDRAULIC FRACTURING DESIGN AND OPTIMIZATION
[000144] For each design of a particular stage of a planned hydraulic
fracturing job, the
wiremesh fracturing model may be applied to generate an HFN and associated
proppant
placement using reservoir formation properties and fracturing job parameters
as the input. The
result, including the geometry of the fracture network and individual
fractures and proppant
distribution along the fractures can be used as part of the input for
production simulation using
the wiremesh production model described above.
[000145] For example, for design of a particular stage of a planned job,
hydraulic fracturing
software, such as MANGROVETM software commercially available from Schlumberger
Technology Corporation (see:www.s1b.com), may be used to produce an HFN with
the
information needed for production calculations. Production from the HFN can be
calculated
using the models described above. Production rates calculated for various
designs may then be
compared and analyzed in combination with other economic, environmental and
logistic
considerations. The job parameters can then be adjusted accordingly for a
better design. The
best design for each of the stages may be chosen for the job.
[000146] Figure 14 depicts an example fracture operation 1400 involving
fracture design
and optimization. The fracture operation 1400 includes 1430 ¨ obtaining job
parameters relating
to formation parameters (e.g., dimensions, stresses, etc.) and 1432 ¨
obtaining job parameters
relating to stimulation parameters, such as pumping (e.g., flow rate, time),
fluid (e.g., viscosity,
density) and proppant parameters (e.g., dimension, material). The fracture
operation 1400 also
includes 1434 - generating plots of formation parameters 1436 (e.g, slurry
rate and proppant
concentration over time) from the obtained parameters.
[000147] A wiremesh HFN and proppant placement simulation 1438 may be
performed to
model the HFN based on the plots 1436 and obtained parameters 1430, 1432.
Visualization
1440.1 of an HFN 822 and its proppant placement 1440.2 may be generated. A
wiremesh
production simulation 1442 may then be performed. An analysis 1444 of the
simulation, for
example, by comparison of actual with simulated results to evaluate the
fracture operation 1400.
If satisfied, a production operation may be executed 1446. If not, job design
may be analyzed
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1448, and adjustments to one or more of the job parameters may be made 1450.
The fracture
operation may then be repeated.
POST FRACTURE OPERATION
[000148] Reservoir properties and hydraulic fracturing treatment data can
be used to obtain
information about the created HFN, such as fracture spacing dõ and dy and
stress anisotropy A o-,
by matching the modeled HFN with a cloud of microseismic events recorded
during the job. The
techniques for hydraulic fracture modeling as described with respect to
Figures 3-7 may be used
to simulate the growth and proppant placement of the HFN. Examples of
hydraulic fracture
modeling that may be used are provided in Wenyue Xu, et al., "Characterization
of
Hydraulically-Induced Fracture Network Using Treatment and Microseismic Data
in a Tight-Gas
Sand Formation: A Geomechanical Approach", SPE 125237, SPE Tight Gas
Completions Conf.,
15-17, Jun. 2009, San Antonio, TX, USA; Wenyue Xu, et al., "Characterization
of
Hydraulically-Induced Shale Fracture Network Using An Analytical/Semi-
Analytical Model",
SPE 124697, SPE Annual Tech. Conf. and Exh., 4-7 October 2009, New Orleans,
LA; Wenyue
Xu et al., "Fracture Network Development and Proppant Placement During
Slickwater
Fracturing Treatment of Barnett Shale Laterals", SPE 135484, SPE Tech. Conf.
and Exhibition,
19-22 Sept. 2010, Florence, Italy; and Wenyue Xu, et al., "Wiremesh: A Novel
Shale Fracturing
Simulator", SPE 1322188, Intl. Oil and Gas Conf. and Exh. in China, 10 June
2010, Beijing,
China, the entire contents of which are hereby incorporated by reference.
Production from the
HFN model 800 can be calculated using the models described above to help in
understanding the
effectiveness and efficiency of the job done.
[000149] Figure 15 depicts an example of a post-fracture operation 1500.
The post-fracture
operation involves 1550 ¨ obtaining job parameters, such a formation,
microseismic,
fluid/proppant and other data. From this information, wellsite parameters,
such as formation, job,
microseismic and other data, may be determined 1552. Proppant data may also be
determined
1554 from the job parameters. The wellsite parameters may be used to
characterize a wiremesh
HFN 1556. The wiremesh HFN can be configured in an elliptical configuration
1558. The HFN
parameters (e.g., matrix and ellipse dimensions) may then be defined 1560. The
HFN parameters
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(e.g., dimensions, stresses) and the proppant parameters may be used to define
the HFN model as
shown in visualization 1562.1, and proppant placement as shown in the
visualization 1562.2.
[000150] A wiremesh production simulation 1564 may then be performed based
on the
HFN model. An analysis 1566 of the simulation may be performed, for example,
by comparison
of actual with simulated results to evaluate the fracture operation 1400. If
satisfied, a production
operation may be executed 1446. If not, job design may be analyzed 1448, and
adjustments to
one or more of the job parameters may be made 1450. The fracture operation may
then be
repeated.
[000151] Figure 16 illustrates a method 1600 of performing a production
operation. This
method 1600 depicts how the models and solutions are applied to a wiremesh HFN
obtained by
hydraulic fracturing modeling. The method involves performing a fracture
operation 1660. The
fracture operation involves 1662 - designing a fracture operation, 1664 -
optimizing a fracture
operation, 1667 ¨ generating fractures by injecting fluid into the formation,
1668 - measuring job
parameters, and 1670 - performing a post-fracture operation. The method also
involves 1672 -
generating a fracture network about the wellbore. The fracture network
includes a plurality of the
fractures and a plurality of matrix blocks. The fractures are intersecting and
hydraulically
connected, and the plurality of matrix blocks are positioned about the
intersecting fractures.
[000152] The method also involves 1674 - placing proppants in the
elliptical hydraulic
fracture network, 1676 ¨generating a fluid distribution through the hydraulic
fracture network,
1678 ¨ performing a production operation, the production operation comprising
generating a
production rate from the fluid pressure distribution, and 1680 - repeating
over time. Part or all of
the method may be performed in any order and repeated as desired.
[000153] The preceding description has been presented with reference to
some
embodiments. Persons skilled in the art and technology to which this
disclosure pertains will
appreciate that alterations and changes in the described structures and
methods of operation can
be practiced without meaningfully departing from the principle, and scope of
this application.
Accordingly, the foregoing description should not be read as pertaining only
to the precise
structures described and shown in the accompanying drawings, but rather should
be read as
consistent with and as support for the following claims, which are to have
their fullest and fairest
scope.
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[000154] There have been described and illustrated herein a methodology and
systems for
monitoring hydraulic fracturing of a subterranean hydrocarbon formation and
extension thereon.
While particular embodiments of the disclosure have been described, it is not
intended that the
disclosure be limited thereto, as it is intended that the disclosure be as
broad in scope as the art
will allow and that the specification be read likewise. Thus, while a specific
method of
performing fracture and production operations is provided, various
combinations of portions of
the methods can be combined as desired. Also, while particular hydraulic
fracture models and
assumptions for deriving such models have been disclosed, it will be
appreciated that other
hydraulic fracture models and assumptions could be utilized. It will therefore
be appreciated by
those skilled in the art that yet other modifications could be made to the
provided disclosure
without deviating from its spirit and scope as claimed.
[000155] It should be noted that in the development of any actual
embodiment, numerous
implementation¨specific decisions must be made to achieve the developer's
specific goals, such
as compliance with system related and business related constraints, which will
vary from one
implementation to another. Moreover, it will be appreciated that such a
development effort might
be complex and time consuming but would nevertheless be a routine undertaking
for those of
ordinary skill in the art having the benefit of this disclosure. In addition,
the composition
used/disclosed herein can also comprise some components other than those
cited. In the
summary of the disclosure and this detailed description, each numerical value
should be read
once as modified by the term "about" (unless already expressly so modified),
and then read again
as not so modified unless otherwise indicated in context. Also, in the summary
of the disclosure
and this detailed description, it should be understood that a concentration
range listed or
described as being useful, suitable, or the like, is intended that any and
every concentration
within the range, including the end points, is to be considered as having been
stated. For
example, "a range of from 1 to 10" is to be read as indicating each and every
possible number
along the continuum between about 1 and about 10. Thus, even if specific data
points within the
range, or even no data points within the range, are explicitly identified or
refer to only a few
specific items, it is to be understood that inventors appreciate and
understand that any and all
data points within the range are to be considered to have been specified, and
that inventors
possessed knowledge of the entire range and all points within the range.
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[000156] Although only a few example embodiments have been described in
detail above,
those skilled in the art will readily appreciate that many modifications are
possible in the
example embodiments without materially departing from the system and method
for performing
wellbore stimulation operations. Accordingly, all such modifications are
intended to be included
within the scope of this disclosure as defined in the following claims. In the
claims, means-plus-
function clauses are intended to cover the structures described herein as
performing the recited
function and not only structural equivalents, but also equivalent structures.
Thus, although a nail
and a screw may not be structural equivalents in that a nail employs a
cylindrical surface to
secure wooden parts together, whereas a screw employs a helical surface, in
the environment of
fastening wooden parts, a nail and a screw may be equivalent structures. It is
the express
intention of the applicant not to invoke 35 U.S.C. 112, paragraph 6 for any
limitations of any of
the claims herein, except for those in which the claim expressly uses the
words 'means for'
together with an associated function.
53
