U.S. patent application number 14/126209 was filed with the patent office on 2014-06-05 for system and method for performing wellbore fracture operations.
This patent application is currently assigned to SCHLUMBERGER TECHNOLOGY CORPORATION. The applicant listed for this patent is Wenyue Xu. Invention is credited to Wenyue Xu.
Application Number | 20140151033 14/126209 |
Document ID | / |
Family ID | 47601584 |
Filed Date | 2014-06-05 |
United States Patent
Application |
20140151033 |
Kind Code |
A1 |
Xu; Wenyue |
June 5, 2014 |
SYSTEM AND METHOD FOR PERFORMING WELLBORE FRACTURE OPERATIONS
Abstract
Methods for performing oilfield operations are provided. The
methods involve performing a fracture operation. The fracture
operation involves generating fractures and a fracture network
about the wellbore. The fracture network includes a plurality of
fractures and a plurality of matrix blocks positioned thereabout.
The fractures are intersecting and hydraulically connected. The
matrix blocks are positioned about the plurality of fractures. The
method also involves generating flow rate through the fracture
network, generating a fluid distribution based on the fracture
network, and performing a production operation comprising
generating a production rate from the fluid distribution.
Inventors: |
Xu; Wenyue; (Sugar Land,
TX) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Xu; Wenyue |
Sugar Land |
TX |
US |
|
|
Assignee: |
SCHLUMBERGER TECHNOLOGY
CORPORATION
Sugar Land
TX
|
Family ID: |
47601584 |
Appl. No.: |
14/126209 |
Filed: |
July 30, 2012 |
PCT Filed: |
July 30, 2012 |
PCT NO: |
PCT/US2012/048877 |
371 Date: |
February 7, 2014 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
61574130 |
Jul 28, 2011 |
|
|
|
Current U.S.
Class: |
166/250.01 ;
166/308.1 |
Current CPC
Class: |
E21B 43/26 20130101;
E21B 41/0092 20130101; E21B 43/267 20130101; E21B 43/00
20130101 |
Class at
Publication: |
166/250.01 ;
166/308.1 |
International
Class: |
E21B 43/26 20060101
E21B043/26; E21B 41/00 20060101 E21B041/00; E21B 43/00 20060101
E21B043/00; E21B 43/267 20060101 E21B043/267 |
Claims
1. A method of performing an oilfield operation about a wellbore
penetrating a subterranean formation, the method comprising:
performing a fracture operation, the fracture operation comprising
generating a plurality of fractures about the wellbore and
generating a fracture network about the wellbore, the fracture
network comprising the plurality of the fractures and a plurality
of matrix blocks positioned thereabout, the plurality of fractures
intersecting and hydraulically connected, the plurality of matrix
blocks positioned about the plurality of fractures; 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.
2. The method of claim 1, wherein the fracture network is
elliptical.
3. The method of claim 1, wherein the performing the fracture
operation comprises stimulating the subterranean formation by
injecting fluid into the subterranean formation.
4. The method of claim 1, wherein the performing the fracture
operation comprises simulating hydraulic fracturing about the
wellbore.
5. The method of claim 1, further comprising placing proppants in
the fracture network.
6. The method of claim 1, further comprising designing the fracture
operation based on job parameters.
7. The method of claim 6, wherein the job parameters comprise
formation parameters, fracture parameters, stimulation parameters,
fluid parameters, pumping parameters, proppant parameters,
microseismic parameters, reservoir parameters and combinations
thereof.
8. The method of claim 1, further comprising optimizing the
fracture operation by adjusting the fracture operation based on a
comparison of the production rate with actual data.
9. The method of claim 1, further comprising repeating the method
over time.
10. The method of claim 1, further comprising performing a post-job
analysis comprising generating a wiremesh hydraulic fracture
network based on job parameters, generating an elliptical fracture
model, generating fracture parameters, modeling the elliptical
fracture network based on the generated fracture parameters and
proppant parameters, and performing a production simulation.
11. The method of claim 10, wherein the fracture parameters
comprise spatial coordinates at an extremity of the plurality of
fractures, conductivity, averaged conductivity, height, averaged
height, reservoir pressure, averaged reservoir pressure at a
fracture location, permeability, averaged reservoir permeability at
the fracture location and combinations thereof.
12. The method of claim 10, further comprising modeling proppant
placement based on the proppant parameters.
13. The method of claim 1, wherein the generating flow rate
comprises passing fluid across the fracture network and through at
least one of the plurality of matrix blocks.
14. The method of claim 1, wherein the performing the production
operation comprises simulating a production using the fracture
network.
15. The method of claim 1, wherein the performing the production
operation comprises deploying tubing into the wellbore and
producing fluid from the wellbore therethrough.
16. The method of claim 1, wherein the fluid distribution comprises
one of a fluid pressure distribution, a fluid density distribution,
and combinations thereof.
17. A method of performing an oilfield operation about a wellbore
penetrating a subterranean formation, the method comprising:
performing a fracture operation, the fracture operation comprising
stimulating the wellbore and generating a fracture network about
the wellbore, the stimulating comprising injecting fluid into the
subterranean formation such that a plurality of fractures are
generated about the wellbore, the fracture network comprising the
plurality of the fractures and a plurality of matrix blocks
positioned thereabout, the plurality of fractures intersecting and
hydraulically connected, the plurality of matrix blocks positioned
about the plurality of fractures; placing proppants in the fracture
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
comprising generating a production rate from the fluid
distribution.
18. The method of claim 17, wherein the placing comprises
transporting the proppant one of horizontally and vertically
through the fracture network.
19. The method of claim 17, wherein the placing comprises
transporting the proppant in all directions through the fracture
network.
20. A method of performing an oilfield operation about a wellbore
penetrating a subterranean formation, the method comprising:
designing a fracture operation based on job parameters; performing
the fracture operation, the fracture operation comprising
generating a fracture network about the wellbore, the fracture
network comprising a plurality of fractures and a plurality of
matrix blocks, the plurality of fractures intersecting and
hydraulically connected, the plurality of matrix blocks positioned
about the plurality of fractures; and optimizing the fracture
operation by adjusting the fracture operation based on a comparison
of a simulated production rate with actual data, the simulated
production rate generated from the fracture 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 comprising generating a
production rate from the fluid distribution.
21. The method of claim 20, wherein the performing the fracture
operation comprises stimulating the wellbore by injecting fluid
into the subterranean formation such that fractures are generated
about the wellbore.
22. The method of claim 20, wherein the job parameters comprise at
least one of formation parameters, stimulation parameters, fracture
parameters, fluid parameters, pumping parameters, proppant
parameters, microseismic parameters, reservoir parameters and
combinations thereof.
23. The method of claim 20, wherein the designing comprises
generating proppant curves from the job parameters.
24. The method of claim 23, wherein the designing further comprises
generating a wiremesh fracture network and simulating proppant
placement based on the proppant curves and the job parameters.
25. The method of claim 24, further comprising visualizing the
fracture network.
26. The method of claim 25, further comprising comparing the
production rate with actual data.
27. The method of claim 26, wherein the performing a production
operation comprises producing fluid from the wellbore.
28. The method of claim 26, further comprising analyzing the
designed fracture operation.
29. The method of claim 27, further comprising adjusting the
fracture operation based on the analyzed, designed fracture
operation and repeating the fracture operation.
30. The method of claim 28, further comprising repeating the
operation.
31. The method of claim 20, further comprising placing proppants in
the fracture network.
32. The method of claim 31, further comprising determining proppant
placement from the job parameters and placing according to the
proppant placement.
33. The method of claim 20, further comprising performing a post
job analysis comprising generating a wiremesh hydraulic fracture
network based on the job parameters, forming an elliptical fracture
model, generating fracture parameters, modeling the elliptical
fracture network based on the generated fracture parameters and
proppant parameters, and performing a production simulation.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This application is a continuation in part of U.S. patent
application Ser. No. 12/479,335, filed on Jun. 5, 2009, the entire
contents of which is hereby incorporated by reference. This
application also claims priority to U.S. Provisional Application
No. 61/574,130 filed on Jul. 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, 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 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.
[0010] 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.
[0011] 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 U.S. Pat. Nos. 6,101,447, 7,363,162, 7,788,074,
20080133186, 20100138196, and 20100250215.
[0012] 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
[0013] 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 fracturing sites).
[0014] 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.
[0015] 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).
[0016] 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.
[0017] 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 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.
[0018] 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.
[0019] 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
[0020] 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.
[0021] FIGS. 1.1-1.4 are schematic views illustrating various
oilfield operations at a wellsite;
[0022] FIGS. 2.1-2.4 are schematic views of data collected by the
operations of FIGS. 1.1-1.4;
[0023] FIG. 3 is a pictorial illustration of geometric properties
of an exemplary hydraulic fracture model in accordance with the
present disclosure;
[0024] FIG. 4 is a schematic illustration of a hydraulic fracturing
site that embodies the present disclosure;
[0025] 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.
[0026] 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;
[0027] 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;
[0028] FIGS. 8.1-8.3 are schematic diagrams illustrating an
elliptical hydraulic fracture network about a well;
[0029] FIG. 9 is a schematic diagram depicting proppant
placement;
[0030] 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;
[0031] FIG. 11 is a schematic diagram illustrating fluid flow
through a dual porosity medium;
[0032] FIG. 12 is schematic flow diagrams depicting methods of
performing production operations;
[0033] FIGS. 13.1 and 13.2 are various schematic diagrams for
depicting fluid flow through a medium;
[0034] FIG. 14 is a flow chart depicting a fracture design and
optimization;
[0035] FIG. 15 is a flow chart depicting a post-production
operation; and
[0036] FIG. 16 is a flow chart depicting a method for performing a
production operation.
DETAILED DESCRIPTION
[0037] The description that follows includes exemplary systems,
apparatuses, methods, and 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.
[0038] 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.
[0039] FIGS. 1.1-1.4 depict various oilfield operations that may be
performed at a wellsite, and FIGS. 2.1-2.4 depict various
information that may be collected at the wellsite. FIGS. 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.
[0040] 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.
[0041] 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 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.
[0042] 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.
[0043] 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.
[0044] 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.
[0045] 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 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.
[0046] 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.
[0047] 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.
[0048] 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.
[0049] 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 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.
[0050] 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.
[0051] 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.
[0052] 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.
[0053] 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).
[0054] 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 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.
[0055] 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.
[0056] 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.
[0057] 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.
[0058] 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 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
[0059] 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-.sigma.c, fluid injection
rate q and duration tp, matrix plane strain modulus E, fluid
viscosity .mu. (or other fluid flow parameter(s) for non-Newtonian
fluids), confining stress contrast .DELTA..sigma., fracture network
sizes h, a, e, and fracture spacing dx and dy.
[0060] 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 FIG. 3.
Consequently, the numbers of fractures, per unit length, parallel
to the x-axis and the y-axis, respectively, are
n x = 1 d y and n y = 1 d x . ( 1 ) ##EQU00001##
[0061] 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 = b a . ( 2 ) ##EQU00002##
[0062] The governing equation for mass conservation of the injected
fluid in the fractured subterranean formation is given by:
2 .pi. ex .differential. ( .phi..rho. ) .differential. t + 4
.differential. ( Bx.rho. v _ e ) .differential. x = 0 , or ( 3 a )
2 .pi. y e .differential. ( .phi..rho. ) .differential. t + 4
.differential. .differential. y ( By .rho. v _ e e ) = 0 , ( 3 b )
##EQU00003##
which for an incompressible fluid becomes respectively
2 .pi. ex .differential. .phi. .differential. t + 4 .differential.
( Bx v _ e ) .differential. x = 0 , or ( 3 c ) 2 .pi. y e
.differential. .phi. .differential. t + 4 .differential.
.differential. y ( By v _ e e ) = 0 , ( 3 d ) ##EQU00004## [0063]
where .phi. is the porosity of the formation, [0064] .rho. is the
density of injected fluid [0065] v.sub.e is an average fluid
velocity perpendicular to the elliptic boundary, and [0066] B is
the elliptical integral given by
[0066] B = .pi. 2 [ 1 - ( 1 2 ) 2 ( 1 - e 2 ) - ( 1 3 2 4 ) 2 ( 1 -
e 2 ) 2 3 - ( 1 3 5 2 4 6 ) 2 ( 1 - e 2 ) 3 5 - ] . ( 4 )
##EQU00005##
The average fluid velocity v.sub.e may be approximated as
v _ e .apprxeq. 1 2 [ v ex ( x , y = 0 ) + v ey ( x = 0 , y = ex )
] .apprxeq. 1 2 ( 1 + e ) v ex ( x , y = 0 ) .apprxeq. 1 2 ( 1 + 1
/ e ) v ey ( x = 0 , y = ex ) with ( 5 ) v ex ( x , y = 0 ) = - [ k
x .mu. .differential. p .differential. x ] ( x , y = 0 ) , ( 6 a )
v ey ( x = 0 , y = ex ) = - [ k y .mu. .differential. p
.differential. y ] ( x = 0 , y = ex ) , ( 6 b ) ##EQU00006## [0067]
where p is fluid pressure, [0068] .mu. is fluid viscosity, and
[0069] k.sub.x and k.sub.y 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.
[0070] Using equations (5) and (6), governing equation (3) can be
written as
2 .pi. ex .differential. .phi. .differential. t - 2 .differential.
.differential. x ( B ( 1 + e ) xk x .mu. .differential. p
.differential. x ) = 0 , or ( 7 a ) 2 .pi. y e .differential. .phi.
.differential. t - 2 .differential. .differential. y ( B ( 1 + e )
yk y e 2 .mu. .differential. p .differential. y ) = 0. ( 7 b )
##EQU00007##
[0071] The width w of a hydraulic fracture may be calculated as
w = 2 l E ( p - .sigma. c ) H ( p - .sigma. c ) , H ( p - .sigma. c
) = { 0 p .ltoreq. .sigma. c 1 p > .sigma. c ( 8 ) ##EQU00008##
[0072] where H is the Heaviside step function, [0073] .sigma..sub.c
is the confining stress perpendicular to the fracture, [0074] E is
the plane strain modulus of the formation, and [0075] l is the
characteristic length scale of the fracture segment and given by
the expression
[0075] l=d+(h-d)H(d-h) (9) [0076] where h and d are the height and
the length, respectively, of the fracture segment.
[0077] 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, respectively, can be expressed as
w x = 2 d x A Ex E ( p - .sigma. cy ) H ( p - .sigma. cy ) , ( 10 a
) w y = 2 d y A Ey E ( p - .sigma. cx ) H ( p - .sigma. cx ) ( 10 b
) ##EQU00009##
where .sigma..sub.cx and .sigma..sub.cy are the confining stresses,
respectively, along the x-direction and the y-direction,
respectively, and A.sub.Ex and A.sub.Ey are the coefficients for
defining the effective plane strain modulus along the x-axis and
y-axis, respectively.
[0078] For complex fracture networks the coefficients A.sub.Ex and
A.sub.Ey may be approximately represented by the following
expressions
A Ex = d x [ 2 l x + ( d y - 2 l x ) H ( d y - 2 l x ) ] d y l x ,
( 11 a ) A Ey = d y [ 2 l y + ( d x - 2 l y ) H ( d x - 2 l y ) ] d
x l y . ( 11 b ) ##EQU00010##
where l.sub.x and l.sub.y are the characteristic length scale along
the x-axis and the y-axis, respectively. The value of the
coefficient (A.sub.Ex) for the effective plane strain modulus along
the x-axis can be simplified for different cases of d.sub.x,
d.sub.y, and h by any one of Tables 1-2 listed below. The value of
the coefficient (A.sub.Ey) for the effective plane strain modulus
along the y-axis can be simplified for different cases of d.sub.x,
d.sub.y, and h by any one of Tables 3-5 listed below.
TABLE-US-00001 TABLE 1 Coefficient A.sub.Ex for different cases of
d.sub.x, d.sub.y, h A.sub.Ex d.sub.x .gtoreq. d.sub.y d.sub.x <
d.sub.y d.sub.x .ltoreq. h d.sub.x > h d.sub.x .ltoreq. h
d.sub.x > h 2 d x d y ##EQU00011## d.sub.y .ltoreq. 2h d.sub.y
> 2h d.sub.y .ltoreq. 2d.sub.x d.sub.y > 2d.sub.x d.sub.y
.ltoreq. 2h d.sub.y > 2h 2 d x d y ##EQU00012## d x h
##EQU00013## 2 d x d y ##EQU00014## 1 2 d x d y ##EQU00015## d x h
##EQU00016##
TABLE-US-00002 TABLE 2 Coefficient A.sub.Ex for different cases of
d.sub.x, d.sub.y, h A.sub.Ex d.sub.x .gtoreq. d.sub.y d.sub.x <
d.sub.y d.sub.x .ltoreq. h d.sub.x > h d.sub.y .ltoreq. h
d.sub.y > h 2 d x d y ##EQU00017## d.sub.y .ltoreq. 2h d.sub.y
> 2h d.sub.y .ltoreq. 2d.sub.x d.sub.y > 2d.sub.x d.sub.y
.ltoreq. 2h d.sub.y > 2h 2 d x d y ##EQU00018## d x h
##EQU00019## 2 d x d y ##EQU00020## 1 2 d x d y ##EQU00021## d x h
##EQU00022##
TABLE-US-00003 TABLE 3 Coefficient A.sub.Ey for different cases of
d.sub.x, d.sub.y, h A.sub.Ey d.sub.y .gtoreq. d.sub.x d.sub.y <
d.sub.x d.sub.y .ltoreq. h d.sub.y > h d.sub.y .ltoreq. h
d.sub.y > h 2 d y d x ##EQU00023## d.sub.x .ltoreq. 2h d.sub.x
> 2h d.sub.x .ltoreq. 2d.sub.y d.sub.x > 2d.sub.y d.sub.x
.ltoreq. 2h d.sub.x > 2h 2 d y d x ##EQU00024## d y h
##EQU00025## 2 d y d x ##EQU00026## 1 2 d y d x ##EQU00027## d y h
##EQU00028##
TABLE-US-00004 TABLE 4a Coefficient A.sub.Ey for different cases of
d.sub.x, d.sub.y, h A.sub.Ey d.sub.x .gtoreq. d.sub.y d.sub.x
.ltoreq. h d.sub.x > h d.sub.x .ltoreq. 2d.sub.y d.sub.x >
2d.sub.y d.sub.y .ltoreq. h d.sub.y > h 2 d y d x ##EQU00029## 1
d.sub.x .ltoreq. 2d.sub.y d.sub.x > 2d.sub.y d.sub.x .ltoreq. 2h
d.sub.x > 2h 2 d y d x ##EQU00030## 1 2 d y d x ##EQU00031## d y
h ##EQU00032##
TABLE-US-00005 TABLE 4b Coefficient A.sub.Ey for different cases of
d.sub.x, d.sub.y, h A.sub.Ey d.sub.x < d.sub.y d.sub.x .ltoreq.
h d.sub.x > h d.sub.y .ltoreq. h d.sub.y > h d.sub.x .ltoreq.
2h d.sub.x > 2h d.sub.x .ltoreq. 2d.sub.y d.sub.x > 2d.sub.y
d.sub.x .ltoreq. 2h d.sub.x > 2h 2 d y d x ##EQU00033## d y h
##EQU00034## 2 d y d x ##EQU00035## 1 2 d y d x ##EQU00036## d y h
##EQU00037##
TABLE-US-00006 TABLE 5 Coefficient A.sub.Ey for different cases of
d.sub.x, d.sub.y, h A.sub.Ey d.sub.x .gtoreq. d.sub.y d.sub.x <
d.sub.y d.sub.x .ltoreq. h d.sub.x > h d.sub.x .ltoreq. h
d.sub.x > h d.sub.x .ltoreq. 2d.sub.y d.sub.x > 2d.sub.y
d.sub.y .ltoreq. h d.sub.y > h 2 d y d x ##EQU00038## d.sub.x
.ltoreq. 2h d.sub.x > 2h 2 d y d x ##EQU00039## 1 d.sub.x
.ltoreq. 2d.sub.y d.sub.x > 2d.sub.y d.sub.x .ltoreq. 2h d.sub.x
> 2h 2 d y d x ##EQU00040## d y h ##EQU00041## 2 d y d x
##EQU00042## 1 2 d y d x ##EQU00043## d y h ##EQU00044##
[0079] The increase in porosity of the fractured formation
(.DELTA..phi.) can be calculated as
.DELTA. .phi. = n x w x + n y w y - n x n y w x w y .apprxeq. 2 d x
d y A Ex E ( p - .sigma. cy ) H ( p - .sigma. cy ) + 2 d y d x A Ey
E ( p - .sigma. cx ) H ( p - .sigma. cx ) ( 12 ) ##EQU00045##
The fracture permeability along the x-axis (k.sub.x) and the
fracture permeability along the y-axis (k.sub.y) can be determined
as
k x = n x w x 3 12 = 2 d x 3 3 E 3 d y A Ex 3 ( p - .sigma. cy ) 3
H ( p - .sigma. cy ) , ( 13 a ) and k y = n y w y 3 12 = 2 d y 3 3
E 2 d x A Ey 3 ( p - .sigma. cx ) 3 H ( p - .sigma. cx ) , ( 13 b )
##EQU00046##
along the x-axis and y-axis, respectively.
[0080] For p>.sigma.cy 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
4 ( p - .sigma. cy ) 3 p x = 3 A Ex 3 d y E 3 .mu. ( 1 + e ) Bd x 3
x ( 2 .pi. .intg. x w x .differential. .phi. .differential. t es s
- q ) . ( 14 a ) ##EQU00047##
Similarly for p>.sigma..sub.cx, the governing equation (7b) can
be integrated from x.sub.w to y using equation (12b) for the
permeability (k.sub.y) to yield
4 ( p - .sigma. cx ) 3 p y = 3 e 2 A Ey 3 d x E 3 .mu. ( 1 + e ) Bd
y 3 y ( 2 .pi. .intg. x w y .differential. .phi. .differential. t s
e s - q ) . ( 14 b ) ##EQU00048##
In equations (13a) and (13b), x.sub.w 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.
[0081] 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
p - .sigma. cy = [ 3 ( 1 + e ) B .intg. x a A Ex 3 d y E 3 .mu. d x
3 r ( q - 2 .pi. .intg. x w r .differential. .phi. .differential. t
es s ) r ] 1 / 4 . ( 15 a ) ##EQU00049##
Equation (14b) can be integrated from y to b yields a solution for
the net pressure inside the fractures along the y-axis as
p - .sigma. cx = [ 3 e 2 ( 1 + e ) B .intg. y b A Ey 3 d x E 3 .mu.
d y 3 r ( q - 2 .pi. .intg. x w r .differential. .phi.
.differential. t s ) r ] 1 / 4 . ( 15 b ) ##EQU00050##
[0082] For uniform .sigma.c, E, .mu., n and d, equation (15a)
reduces to
p - .sigma. cy = A px [ q ln ( a x ) - 2 .pi. e .intg. x a ( .intg.
x w r .differential. .phi. .differential. t s s ) 1 r r ] 1 / 4 A
px = ( 3 A Ex 3 d y E 3 .mu. ( 1 + e ) Bd x 3 ) 1 / 4 . ( 16 a )
##EQU00051##
Similarly, equation (15b) reduces to
p - .sigma. cx = - 1 / 2 A p y [ q ln ( b y ) - 2 .pi. e .intg. y b
( .intg. x w r .differential. .phi. .differential. t s s ) 1 r r ]
1 4 A py = ( 3 A Ey 3 d x E 3 .mu. ( 1 + e ) Bd y 3 ) 1 / 4 . ( 16
b ) ##EQU00052##
[0083] The wellbore pressure p.sub.w is given by the following
expressions:
p w = .sigma. cy + A px [ q ln ( a x w ) - 2 .pi. e .intg. x w a (
.intg. x w r .differential. .phi. .differential. t s s ) 1 r r ] 1
/ 4 , ( 17 a ) p w = .sigma. cx + 1 / 2 A py [ q ln ( b x w ) - 2
.pi. e .intg. x w b ( .intg. x w r .differential. .phi.
.differential. t s s ) 1 r r ] 1 / 4 . ( 17 b ) ##EQU00053##
By requiring the two expressions (17a, 17b) for the wellbore
pressure p.sub.w to be equal, one obtains the difference between
confining stresses (.DELTA..sigma..sub.c), which is also referred
herein to as stress contrast .DELTA..sigma..sub.c, as
.DELTA. .sigma. c = .sigma. cx - .sigma. cy = A px [ q ln ( a x w )
- 2 .pi. e .intg. x w a ( .intg. x w r .differential. .phi.
.differential. t s s ) 1 4 r ] 1 / 4 - 1 / 2 A py [ q ln ( ea x w )
- 2 .pi. e .intg. x w ea ( .intg. x w r .differential. .phi.
.differential. t s s ) 1 r r ] 1 4 . ( 18 ) ##EQU00054##
[0084] 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
q t p .pi. = e .intg. x w a .DELTA. .phi. x x x + 1 e .intg. x w b
.DELTA. .phi. y y y = e .intg. x w a 2 d x ( p x - .sigma. cy ) d y
A Ex E x x + e .intg. x w x .sigma. 2 d y ( p x - .sigma. cx ) d x
A Ey E x x + 1 e .intg. x .sigma. b [ 2 d x ( p y - .sigma. cy ) d
y A Ex E + 2 d y ( p y - .sigma. cx ) d x A Ey E ] y y , ( 19 a )
or q t p .pi. e = .intg. x w a [ .DELTA. .phi. x ( x ) + .DELTA.
.phi. y ( y = ex ) ] x x = 2 E [ .intg. x w x .sigma. ( d x d y A
Ex + d y d x A Ey ) ( p x - .sigma. cy ) x x + .intg. x .sigma. a d
x d y A Ex ( p x - .sigma. cy ) x x ] + 2 E .intg. x w a ( d x d y
A Ex + d y d x A Ey ) ( p y - .sigma. cx ) x x + 2 .DELTA. .sigma.
c E ( .intg. x w a d x d y A Ex x x - .intg. x w x .sigma. d y d x
A Ey x x ) , ( 19 b ) where x .sigma. is defined as x w .ltoreq. x
.sigma. < a where p .ltoreq. .sigma. cx if x .ltoreq. x .sigma.
p > .sigma. cx if x > x .sigma. p = .sigma. cx if x = x
.sigma. . ( 19 c ) ##EQU00055##
[0085] Equation (15a) can be rewritten for the case
p=.sigma..sub.cx at x=x.sub..sigma. as follows
.DELTA. .sigma. c = [ 3 ( 1 + e ) B .intg. x .sigma. a A Ex 3 d y E
3 .mu. d x 3 r ( q - 2 .pi. .intg. x w r .differential. .phi.
.differential. t es s ) r ] 1 / 4 . ( 20 ) ##EQU00056##
[0086] The surface area of the open fractures may be calculated as
follows
S .apprxeq. .pi. a b .times. 2 hn x + .pi. x .sigma. b .times. 2 hn
y , = 2 .pi. eah ( a d y + x .sigma. d x ) . ( 21 )
##EQU00057##
[0087] For a quasi-steady state, governing equations (7a) and (7b)
reduce to
- 2 B ( 1 + e ) xk x .mu. p x = q , ( 22 a ) - 2 B ( 1 + e ) e 2 yk
y .mu. p y = q . ( 22 b ) ##EQU00058##
Moreover, for the quasi-steady state, the pressure equations (15a)
and (15b) reduce to
p - .sigma. cy = [ 3 ( 1 + e ) B .intg. x a A Ex 3 d y E 3 q .mu. d
x 3 r r ] 1 / 4 , ( 23 a ) p - .sigma. cx = [ 3 e 2 ( 1 + e ) B
.intg. y b A Ey 3 d x E 3 q .mu. d y 3 r r ] 1 / 4 . ( 23 b )
##EQU00059##
For the quasi-steady state and uniform properties of .sigma..sub.c,
E, .mu., n and d, equations (16a) and (16b) reduce to
p - .sigma. cy = A px ( q ln a x ) 1 / 4 , ( 24 a ) p - .sigma. cx
= e 1 / 2 A py ( q ln b y ) 1 / 4 . ( 24 b ) ##EQU00060##
Correspondingly, for the quasi-steady state, the wellbore pressure
equations (17a) and (17b) reduce to
p w = .sigma. cy + A px ( q ln a x 2 ) 1 / 4 , ( 25 a ) p w =
.sigma. cx + e 1 / 2 A py ( q ln ea x w ) 1 / 4 . ( 25 b )
##EQU00061##
By requiring the two expressions (25a, 25b) for the wellbore
pressure p.sub.w to be equal, one obtains
[ 1 - e 1 / 2 A ea d x d y ( A Ey A Ex ) 3 / 4 ] ( p w - .sigma. cy
) = .DELTA. .sigma. c , A ea = [ ln ( ea / x w ) ln ( a / x w ) ] 1
/ 4 . ( 26 ) ##EQU00062##
[0088] For the quasi-steady state and uniform properties of
.sigma.c, E, .mu., n and d, equations (19a) and (19b),
respectively, reduce to
q t p .pi. = e A .phi. d y 1 / 4 A Ex 3 / 4 d x 3 / 4 [ ( d x d y A
Ex + d y d x A Ey ) .intg. x w x .sigma. ( ln a x ) 1 / 4 x x + d x
d y A Ex .intg. x .sigma. .alpha. ( ln a x ) 1 / 4 x x ] + A .phi.
d x 1 / 4 A Ey 3 / 4 e 1 / 2 d y 3 / 4 ( d x d y A Ex + d y d x A
Ey ) .intg. x w b ( ln b y ) 1 / 4 y y + .DELTA. .sigma. c E [ d x
e d y A Ex ( b 2 - x w 2 ) - e d y d x A Ey ( x .sigma. 2 - x w 2 )
] , A .phi. = [ 48 q .mu. ( 1 + e ) BE ] . and ( 27 a ) q t p .pi.
e = A .phi. ( d y A Ex 3 d x 3 ) 1 / 4 [ ( d x d y A Ex + d y d x A
Ey ) .intg. x w x .sigma. ( ln a x ) 1 / 4 x x + d x d y A Ex
.intg. x .sigma. .alpha. ( ln a x ) 1 / 4 x x ] + e 1 / 2 A .phi. (
d x A Ey 3 d y 3 ) 1 / 4 ( d x d y A Ex + d y d x A Ey ) .intg. x w
a ( ln a x ) 1 / 4 x x + .DELTA. .sigma. c E [ d x d y A Ex ( a 2 -
x w 2 ) - d y d x A Ey ( x .sigma. 2 - x w 2 ) ] , A .phi. = [ 48 q
.mu. ( 1 + e ) BE ] 1 / 4 . ( 27 b ) ##EQU00063##
Correspondingly, equation (20) can be solved to yield
x .sigma. = a exp [ - 1 q ( .DELTA. .sigma. c A px ) 4 ] . ( 28 )
##EQU00064##
The integrations in equation (27) can be numerically evaluated
rather easily for a given x.sub..sigma..
[0089] 1. Constraints on the Parameters of the Model Using Field
Data
[0090] 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 (.DELTA..sigma..sub.c) are constrained given wellbore
radius xw and wellbore net pressure pw-.sigma.c, fluid injection
rate q and duration tp, matrix plane strain modulus E, fluid
viscosity .mu., and fracture network sizes h, a, e, as follows.
Note that since x.sigma. in equation (27) is calculated using
equation (28) as a function of .DELTA..sigma.c, the solution
procedure is necessarily of an iterative nature.
[0091] Given these values, the value of
d.sub.x.sup.3/(A.sub.Ex.sup.3d.sub.y) is determined according to
equation (25a) by
d x 3 A Ex 3 d y = d 0 2 d 0 = [ 3 E 3 q .mu. ln ( a / x w ) ( p w
- .sigma. cy ) 4 ( 1 + e ) B ] 1 / 2 , ( 29 ) ##EQU00065##
[0092] If (2d.sub.y.gtoreq.d.sub.x.gtoreq.d.sub.y) and
(d.sub.x.ltoreq.h), equation (29) leads to
d.sub.y= {square root over (8)}d.sub.0. (30)
Equations (26) and (27) become, respectively,
[ 1 - A ea ( e d y d x ) 1 / 2 ] ( p w - .sigma. cy ) = .DELTA.
.sigma. c , and ( 31 ) q t a .pi. = e A .phi. 2 1 / 4 d y 1 / 2 [ 2
.intg. x w x .sigma. ( ln a x ) 1 / 4 x x + .intg. x .sigma. a ( ln
a x ) 1 / 4 x x ] + 2 3 / 4 A .phi. e 1 / 2 d x 1 / 2 .intg. x w b
( ln b y ) 1 / 4 y y + .DELTA. .sigma. c 2 E [ b 2 - x w 2 e - e (
x .sigma. 2 - x w 2 ) ] . ( 32 ) ##EQU00066##
Using solution (30), equations (31) and (32) can be solved to
obtain
.DELTA. .sigma. c = { q t a .pi. - e A .phi. 2 1 / 4 d y 1 / 2 [ 2
.intg. x w x .sigma. ( ln a x ) 1 / 4 x x + .intg. x .sigma. a ( ln
a x ) 1 / 4 x x ] - 2 3 / 4 A .phi. e 1 / 2 d x 1 / 2 .intg. x w b
( ln b y ) 1 / 4 y y } 2 eE b 2 - x w 2 - e 2 ( x .sigma. 2 - x w 2
) , and ( 33 ) d x = 8 d 0 e A ea 2 ( p w - .sigma. xy p w -
.sigma. cy - .DELTA. .sigma. c ) 2 . ( 34 ) ##EQU00067##
[0093] If (h.ltoreq.d.sub.x>2d.sub.y), equations (26) and (27)
become, respectively,
[ 1 - e 1 / 2 2 3 / 4 A ea ( d x d y ) 1 / 4 ] ( p w - .sigma. cy )
= .DELTA. .sigma. c , ( 35 ) q t a .pi. = 2 3 / 4 e A .phi. d y 1 /
2 [ ( 1 2 + d y d x ) .intg. x w x .sigma. ( ln a x ) 1 / 4 x x + 1
2 .intg. x .sigma. a ( ln a x ) 1 / 4 x x ] + A .phi. d x 1 / 4 e 1
/ 2 d y 3 / 4 ( 1 2 + d y d x ) .intg. x w b ( ln b y ) 1 / 4 y y +
.DELTA. .sigma. c E [ 1 2 e ( b 2 - x w 2 ) - e d y d x ( x .sigma.
2 - x w 2 ) ] . ( 36 ) ##EQU00068##
Combined with solution (30) and replacing .DELTA..sigma..sub.c with
equation (35), equation (36) can be solved for d.sub.x.
.DELTA..sigma..sub.c can then be calculated using equation
(35).
[0094] If (d.sub.x>h.gtoreq.d.sub.y), equation (29) leads to
solution (30). Furthermore, if (d.sub.x.ltoreq.2d.sub.y), equations
(26) and (27) lead to solutions (33) and (34). On the other hand,
if (d.sub.x>2d.sub.y), equations (26) and (27) lead to equations
(35) and (36).
[0095] If (d.sub.x.gtoreq.d.sub.y) and (h<d.sub.y.ltoreq.2h)
equation (29) leads to solution (30). Furthermore, if
(d.sub.x.ltoreq.2h), equations (26) and (27) lead to solutions (33)
and (34). On the other hand, if (d.sub.x>2h), equations (26) and
(27) become, respectively,
[ 1 - A ea ( 8 e 2 d 0 2 d x h 3 ) 4 ] ( p w - .sigma. cy ) =
.DELTA. .sigma. c , and ( 37 ) q t a 2 .pi. e = A .phi. 2 d 0 1 / 2
[ ( 1 + 2 h d x ) .intg. x w x .sigma. ( ln a x ) 1 / 4 x x +
.intg. x .sigma. a ( ln a x ) 1 / 4 x x ] - h ( x .sigma. 2 - x w 2
) ( p w - .sigma. cy ) .sigma. c E d x [ 1 ( 8 e 2 d 0 2 d x h 3 )
4 ] . ( 38 ) ##EQU00069##
Equation (38) can be solved for d.sub.x and then
.DELTA..sigma..sub.c can be calculated by equation (37).
[0096] If (d.sub.x.gtoreq.d.sub.y>2h), equation (29) leads
to
d y = h 3 d 0 2 . ( 39 ) ##EQU00070##
Equations (26) and (27) becomes, respectively,
[ 1 - e 1 / 2 A ea ( d 0 2 d x h 3 ) 7 / 4 ] ( p w - .sigma. cy ) =
.DELTA. .sigma. c , and ( 40 ) q t a 2 .pi. e = A .phi. d 0 3 / 2 h
2 [ ( 1 + h 3 d 0 2 d x ) .intg. x w x .sigma. ( ln a x ) 1 / 4 x x
+ .intg. x .sigma. a ( ln a x ) 1 / 4 x x ] - h ( x .sigma. 2 - x w
2 ) ( p w - .sigma. cy ) E d x [ 1 - e 1 / 2 ( d 0 2 d x h 3 ) 7 /
4 ] . ( 41 ) ##EQU00071##
Equation (41) can be solved for d.sub.x and then
.DELTA..sigma..sub.c can be calculated by equation (40).
[0097] If (d.sub.x<d.sub.y.ltoreq.2d.sub.x) and
(d.sub.x.ltoreq.h), equations (29), (26) and (27) lead to solutions
(30), (33) and (34).
[0098] If (d.sub.y>2d.sub.x) and (d.sub.x.ltoreq.h), equations
(29), (26) and (27) become, respectively,
d x 3 = d 0 2 d y , ( 42 ) [ 1 - 2 3 / 4 A ea ( e d 0 d x ) 1 / 2 ]
( p w - .sigma. cy ) = .DELTA. .sigma. c , and ( 43 ) q t a 2 .pi.
e = A .phi. d 0 3 / 2 d x 2 [ ( 1 + d x 2 2 d 0 2 ) .intg. x w x
.sigma. ( ln a x ) 1 / 4 x x + .intg. x .sigma. a ( ln a x ) 1 / 4
x x ] - ( x .sigma. 2 - x w 2 ) .DELTA. .sigma. c 2 E . ( 44 )
##EQU00072##
Equations (42), (43) and (44) can be solved for d.sub.x, d.sub.y
and .DELTA..sigma..sub.c.
[0099] If (h<d.sub.x<d.sub.y.ltoreq.2h), equations (29), (26)
and (27) lead to solutions (30), (33) and (34).
[0100] If (h<d.sub.x.ltoreq.2h<d.sub.y), equation (29) leads
to solution (39). Equations (26) and (27) become respectively
[ 1 - 2 3 / 4 e 1 / 2 A ea ( d 0 d x ) 1 / 2 ] ( p w - .sigma. cy )
= .DELTA. .sigma. c and ( 45 ) q t a 2 .pi. e = A .phi. d 0 3 / 2 h
2 [ ( 1 + h 2 2 d 0 2 ) .intg. x w x .sigma. ( ln a x ) 1 / 4 x x +
.intg. x .sigma. a ( ln a x ) 1 / 4 x x ] - 2 ( x .sigma. 2 - x w 2
) .DELTA. .sigma. c 2 E ( 46 ) ##EQU00073##
[0101] Equations (45) and (46) can be solved to obtain
.DELTA. .sigma. c = E 2 ( x .sigma. 2 - x w 2 ) { A .phi. d 0 3 / 2
h 2 [ ( 1 + h 2 2 d 0 2 ) .intg. x w x .sigma. ( ln a x ) 1 / 4 x x
+ .intg. x .sigma. a ( ln a x ) 1 / 4 x x ] - q t a 2 .pi. e } ( 47
) and d x = 2 3 / 2 e d 0 ( p w - .sigma. cy p w - .sigma. cy -
.DELTA. .sigma. c ) 2 ( 48 ) ##EQU00074##
[0102] If (2h<.sub.x<d.sub.y), equation (29) leads to
solution (39) while equations (26) and (27) become equations (40)
and (41), respectively.
[0103] 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.
[0104] 2. Simplification of Model for Parallel Equally-Spaced
Planar Fractures Whose Spacing DX and DY are Smaller than Fracture
Height H
[0105] The assumption that fracture spacing d is usually smaller
than fracture height h leads to
l.sub.x=d.sub.x
l.sub.y=d.sub.y. (49)
Consequently, equations (11a) and (11b) can be simplified as
A Ex = 1 d y [ 2 d x + ( d y - 2 d x ) H ( d y - 2 d x ) ] , ( 50 a
) A Ey = 1 d x [ 2 d y + ( d x - 2 d y ) H ( d x - 2 d y ) ] . ( 50
b ) ##EQU00075##
Equations (50a) and (50b) can be used to simplify equations (10a)
and (10b) as follows
w x = 2 d x d y ( p - .sigma. cy ) H ( p - .sigma. cy ) [ 2 d x + (
d y - 2 d x ) H ( d y - 2 d x ) ] E , ( 51 a ) w y = 2 d y d x ( p
- .sigma. cx ) H ( p - .sigma. cx ) [ 2 d y + ( d x - 2 d y ) H ( d
x - 2 d y ) ] E . ( 51 b ) ##EQU00076##
Equations (50a) and (50b) can also be used to simplify equation
(12) as follows
.DELTA..phi. = 2 d x ( p - .sigma. cy ) H ( p - .sigma. cy ) [ 2 d
x + ( d y - 2 d x ) H ( d y - 2 d x ) ] E + 2 d y ( p - .sigma. cx
) H ( p - .sigma. cx ) [ 2 d y + ( d x - 2 d y ) H ( d x - 2 d y )
] E . ( 52 ) ##EQU00077##
Equations (50a) and (50b) can be used to simplify equations (13a)
and (13b) as follows
k x = k x 0 + 2 d x 3 d y 2 3 [ 2 d x + ( d y - 2 d x ) H ( d y - 2
d x ) ] 3 E 3 ( p - .sigma. cy ) 3 H ( p - .sigma. cy ) , ( 53 a )
k y = k y 0 + 2 d y 3 d x 2 3 [ 2 d y + ( d x - 2 d y ) H ( d x - 2
d y ) ] 3 E 3 ( p - .sigma. cx ) 3 H ( p - .sigma. cx ) . ( 53 b )
##EQU00078##
These equations can be simplified in the following situations.
Situation I (2d.sub.x.gtoreq.d.sub.y.gtoreq.d.sub.x/2):
[0106] With (2d.sub.x.gtoreq.d.sub.y.gtoreq.d.sub.x/2), equations
(50a) and (50b) become
A Ex = 2 d x d y , ( 54 a ) A Ey = 2 d y d x . ( 54 b )
##EQU00079##
Furthermore, equations (51a) and (51b) become
w x = d y ( p - .sigma. cy ) H ( p - .sigma. cy ) E , ( 55 a ) w y
= d x ( p - .sigma. cx ) H ( p - .sigma. cx ) E . ( 55 b )
##EQU00080##
Furthermore, equation (52) becomes
.DELTA..phi. = 1 E ( p - .sigma. cy ) H ( p - .sigma. cy ) + 1 E (
p - .sigma. cx ) H ( p - .sigma. cx ) . ( 56 ) ##EQU00081##
[0107] Furthermore, equations (53a) and (53b) become
k x = k x 0 + d y 2 12 E 3 ( p - .sigma. cy ) 3 H ( p - .sigma. cy
) , ( 57 a ) k y = k y 0 + d x 2 12 E 3 ( p - .sigma. cx ) 3 H ( p
- .sigma. cx ) . ( 57 b ) ##EQU00082##
[0108] Furthermore, equations (24a) and (24b) become
p - .sigma. cy = A p d y 1 / 2 ( q ln a x ) 1 / 4 , ( 58 a ) p -
.sigma. cx = 1 / 2 A p d x 1 / 2 ( q ln b y ) 1 / 4 , where ( 58 b
) A p = [ 24 E 3 .mu. ( 1 + e ) B ] 1 / 4 . ( 59 ) ##EQU00083##
Furthermore, equations (25a) and (25b) become
p w - .sigma. cy = A p d y 1 / 2 ( q ln a x w ) 1 / 4 , ( 60 a ) p
w - .sigma. cx = e 1 / 2 A p d x 1 / 2 ( q ln ea x w ) 1 / 4 , ( 60
b ) ##EQU00084##
[0109] And furthermore, equation (26) becomes
[ 1 - ( ed y d x ) 1 / 2 A ea ] ( p w - .sigma. cy ) =
.DELTA..sigma. c . ( 61 ) ##EQU00085##
Equation (60a) can be solved for d.sub.y as follows
d y = A p 2 ( p w - .sigma. cy ) 2 ( q ln a x w ) 1 / 2 . ( 62 )
##EQU00086##
[0110] With (2d.sub.x.gtoreq.d.sub.y.gtoreq.d.sub.x/2), equations
(27) and (28) become
qt a .pi. = e A .phi. 2 1 / 4 d y 1 / 2 [ 2 .intg. x w x .sigma. (
ln a x ) 1 / 4 x x + .intg. x .sigma. a ( ln a x ) 1 / 4 x x ] + 2
3 / 4 A .phi. e 1 / 2 d x 1 / 2 .intg. x w b ( ln b y ) 1 / 4 y y +
.DELTA..sigma. c 2 E [ b 2 - x w 2 e - e ( x .sigma. 2 - x w 2 ) ]
, ( 63 a ) qt a .pi. e = 2 3 / 4 A .phi. d y 1 / 2 [ .intg. x w x
.sigma. ( ln a x ) 1 / 4 x x + 1 2 .intg. x .sigma. a ( ln a x ) 1
/ 4 x x ] + 2 3 / 4 A .phi. e 1 / 2 d x 1 / 2 .intg. x w a ( ln a x
) 1 / 4 x x + .DELTA..sigma. c ( a 2 - x .sigma. 2 ) 2 E , and ( 63
b ) x .sigma. = a exp [ - d y 2 q ( .DELTA..sigma. c A p ) 4 ] . (
64 ) ##EQU00087##
[0111] Equations (61), (63) and (64) can be solved iteratively for
d.sub.x and .DELTA..sigma..sub.c.
Situation II (2d.sub.y<d.sub.y)
[0112] With (2d.sub.x<d.sub.y), equations (50a) and (50b)
become
A Ex = 1 , ( 65 a ) A Ey = 2 d y d x . ( 65 b ) ##EQU00088##
Furthermore, equations (51a) and (51b) become
w x = 2 d x ( p - .sigma. cy ) H ( p - .sigma. cy ) E . ( 66 a ) w
y = d x ( p - .sigma. cx ) H ( p - .sigma. cx ) E . ( 66 b )
##EQU00089##
Furthermore, equation (52) becomes
.DELTA..phi. = 2 d x d y E ( p - .sigma. cy ) H ( p - .sigma. cy )
+ 1 E ( p - .sigma. cx ) H ( p - .sigma. cx ) . ( 67 )
##EQU00090##
Furthermore, equations (53a) and (53b) become
k x = k x 0 + 2 d x 3 3 d y E 3 ( p - .sigma. cy ) 3 H ( p -
.sigma. cy ) , ( 68 a ) k y = k y 0 + d x 2 12 E 3 ( p - .sigma. cx
) 3 H ( p - .sigma. cx ) . ( 68 b ) ##EQU00091##
Furthermore, equations (24a) and (24b) become
p - .sigma. cy = ( d y 8 d x 3 ) 1 / 4 A p ( q ln a x ) 1 / 4 , (
69 a ) p - .sigma. cx = e 1 / 2 A p d x 1 / 2 ( q ln b y ) 1 / 4 ,
( 69 b ) ##EQU00092##
Furthermore, equations (25a) and (25b) become
p w - .sigma. cy = ( d y 8 d x 3 ) 1 / 4 A p ( q ln a x w ) 1 / 4 ,
( 70 a ) p w - .sigma. cx = 1 / 2 A p d x 1 / 2 ( q ln ea x w ) 1 /
4 , ( 70 b ) ##EQU00093##
And furthermore, equation (26) becomes
[ 1 - ( 8 e 2 d x d y ) 1 / 4 A ea ] ( p w - .sigma. cy ) =
.DELTA..sigma. c . ( 71 ) ##EQU00094##
[0113] With (2d.sub.x<d.sub.y), equations (27) and (28) lead
to
q t a .pi. = eA .phi. d y 1 / 4 2 d x 3 / 4 [ ( 1 + 2 d x d y )
.intg. x w x .sigma. ( ln a x ) 1 / 4 x x + 2 d x d y .intg. x
.sigma. a ( ln a x ) 1 / 4 x x ] + A .phi. 2 1 / 4 e 1 / 2 d x 1 /
2 ( 1 + 2 d x d y ) .intg. x w b ( ln b y ) 1 / 4 y y +
.DELTA..sigma. c 2 E [ 2 d x ed y ( b 2 - x w 2 ) - e ( x .sigma. 2
- x w 2 ) ] , ( 72 a ) q t a .pi. e = A .phi. ( d y d x 3 ) 1 / 4 [
( d x d y + 1 2 ) .intg. x w x .sigma. ( ln a x ) 1 / 4 x x + d x d
y .intg. x .sigma. a ( ln a x ) 1 / 4 x x ] + e 1 / 2 A .phi. 2 3 /
4 d x 1 / 2 ( d x d y + 1 2 ) .intg. x w a ( ln a x ) 1 / 4 x x +
.DELTA..sigma. c E [ d x d y ( a 2 - x w 2 ) - 1 2 ( x .sigma. 2 -
x w 2 ) ] , and ( 72 b ) x .sigma. = a exp [ - 8 d x 3 qd y (
.DELTA..sigma. c A p ) 4 ] . ( 73 ) ##EQU00095##
Equations (70), (71), (72) and (73) can be combined and solved
iteratively for d.sub.x, d.sub.y and .DELTA..sigma..sub.c.
Situation III (d.sub.y<d.sub.x/2)
[0114] With (d.sub.y<d.sub.x/2), equations (50a) and (50b)
become
A Ex = 2 d x d y , ( 74 a ) A Ey = 1. ( 74 b ) ##EQU00096##
Furthermore, equations (51a) and (51b) become
w x = d y ( p - .sigma. cy ) H ( p - .sigma. cy ) E , ( 75 a ) w y
= 2 d y ( p - .sigma. cx ) H ( p - .sigma. cx ) E . ( 75 b )
##EQU00097##
Furthermore, equation (52) becomes
.DELTA..phi. = 1 E ( p - .sigma. cy ) H ( p - .sigma. cy ) + 2 d y
d x E ( p - .sigma. cx ) H ( p - .sigma. cx ) . ( 76 )
##EQU00098##
Furthermore, equations (53a) and (53b) become
k x = k x 0 + d y 2 12 E 3 ( p - .sigma. cy ) 3 H ( p - .sigma. cy
) , ( 77 a ) k y = k y 0 + 2 d y 3 3 d x E 3 ( p - .sigma. cx ) 3 H
( p - .sigma. cx ) . ( 77 b ) ##EQU00099##
Furthermore, equations (24a) and (24b) become
p - .sigma. cy = A p d y 1 / 2 ( q ln a x ) 1 / 4 , ( 78 a ) p -
.sigma. cx = e 1 / 2 A p ( d x 8 d y 3 ) 1 / 4 ( q ln b y ) 1 / 4 ,
( 78 b ) ##EQU00100##
Furthermore, equations (25a) and (25b) become
p w - .sigma. cy = A p d y 1 / 2 ( q ln a x w ) 1 / 4 , ( 79 a ) p
w - .sigma. cx = e 1 / 2 A p ( d x 8 d y 3 ) 1 / 4 ( q ln ea x w )
1 / 4 , ( 79 b ) ##EQU00101##
And furthermore, equation (26) becomes
[ 1 - ( e 2 d x 8 d y ) 1 / 4 A ea ] ( p w - .sigma. cy ) =
.DELTA..sigma. c . ( 80 ) ##EQU00102##
[0115] With (d.sub.y<d.sub.x/2), equations (27) and (28)
become
q t a .pi. = eA .phi. 2 1 / 4 d y 1 / 2 [ ( 1 + 2 d y d x ) .intg.
x w x .sigma. ( ln a x ) 1 / 4 x x + .intg. x .sigma. a ( ln a x )
1 / 4 x x ] + A .phi. d x 1 / 4 2 e 1 / 2 d y 3 / 4 ( 1 + 2 d y d x
) .intg. x w b ( ln b y ) 1 / 4 y y + .DELTA..sigma. c 2 E [ 1 e (
b 2 - x w 2 ) - 2 ed y d x ( x .sigma. 2 - x w 2 ) ] , ( 81 a ) q t
a .pi. e = A .phi. 2 3 / 4 d y 1 / 2 [ ( 1 2 + d y d x ) .intg. x w
x .sigma. ( ln a x ) 1 / 4 x x + 1 2 .intg. x .sigma. a ( ln a x )
1 / 4 x x ] + e 1 / 2 A .phi. ( d x d y 3 ) 1 / 4 ( 1 2 + d y d x )
.intg. x w a ( ln a x ) 1 / 4 x x + .DELTA..sigma. c E [ 1 2 ( a 2
- x w 2 ) - d y d x ( x .sigma. 2 - x w 2 ) ] , and ( 81 b ) x
.sigma. = a exp [ - d y 2 q ( .DELTA..sigma. c A p ) 4 ] . ( 82 )
##EQU00103##
Equations (79), (80), (81) and (82) can be combined and solved
iteratively for d.sub.x, d.sub.y, and .DELTA..sigma..sub.c.
[0116] FIG. 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.
[0117] 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.
[0118] 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 FIG. 3.
[0119] 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. Pat. 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.
[0120] 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 FIG. 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.
[0121] 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 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.
[0122] 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 (.mu.) of the fracturing
fluid that is being supplied to the treatment well 401 and the
radius (xw) of the treatment wellbore.
[0123] In steps 503-511, the data processing system 409 is
controlled to operate for successive periods of time (each denoted
as .DELTA.t) that fracturing fluid is supplied to the treatment
well 401.
[0124] In step 505, the data processing system 409 processes the
acoustic signals captured by the receiver array 405 over the period
of time .DELTA.t 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 .DELTA.t. 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 FIG. 3.
[0125] 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 .DELTA.t, and derives the
net downhole pressure pw-.sigma.c of the fracturing fluid at the
end of the period of time .DELTA.t. The wellbore net pressure
pw-.sigma.c can be obtained from the injection pressure of the
fracturing fluid at the surface according to the following:
p.sub.w-.sigma..sub.c=p.sub.surface-BHTP-p.sub.pipe-p.sub.perf+p.sub.hyd-
rostatic (83)
where p.sub.surface is the injection pressure of the fracturing
fluid at the surface; BHTP is the bottom hole treating pressure;
p.sub.pipe 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;
[0126] p.sub.perf is the friction pressure through the perforations
of the treatment well that provide for injection of the fracturing
fluid into the reservoir; and p.sub.hydrostatic is the hydrostatic
pressure due to density of the fracturing fluid column in the
treatment well.
The wellbore net pressure p.sub.w-.sigma..sub.c can also be derived
from BHTP at the beginning of treatment and the injection pressure
p.sub.surface at the beginning of the shut-in period. The wellbore
net pressure p.sub.w-.sigma..sub.c at the end of treatment can be
calculated by plugging these values into equation (83) while
neglecting both friction pressures p.sub.pipe and p.sub.perf, which
are zero during the shut-in period.
[0127] In step 509, the data processing system 409 utilizes the
parameters (E, .mu., 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-.sigma.c 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 .DELTA.t, such as parameters dx and dy and
the stress contrast .DELTA..sigma.c as set forth above.
[0128] In step 511, the geometric and geomechanical properties
(e.g., dx, dy, .DELTA..sigma.c) 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
(.DELTA..phi. from equation 12), and change in fracture
permeability (kx and ky from equations (13a) and (13b)).
[0129] In optional step 513, the data generated in step 511 is used
for real-time 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 FIGS. 6.1-6.4.
[0130] 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.
[0131] 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 (.DELTA..phi. from equation 12), and
change in fracture permeability (kx and ky from equations (13a) and
(13b)).
[0132] 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. FIGS. 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, respectively. Note that the circles of FIG.
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.
[0133] 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: [0134] 1. assume
[0134] .differential. .phi. .differential. t ##EQU00104##
if t=t.sub.0 (i=0), otherwise [0135] 2. knowing
[0135] .differential. .phi. .differential. t ##EQU00105##
from t=t.sub.i-1, determine e using equation (18) [0136] 3.
knowing
[0136] .differential. .phi. .differential. t ##EQU00106##
and e, calculate p-.sigma..sub.cx and p-.sigma..sub.cy using
equations (15a) and (15b) or equations (16a) and (16b) [0137] 4.
knowing p-.sigma..sub.cx and p-.sigma..sub.cy, calculate
.DELTA..phi. using equation (12) [0138] 5. knowing e and
.DELTA..phi., calculate t=t.sub.i using equations (19), or (27) and
(28) [0139] 6. knowing .DELTA.t=t.sub.i-t.sub.i-1 and
A.DELTA..phi., calculate
[0139] .differential. .phi. .differential. t ##EQU00107##
as .DELTA..phi./.DELTA.t [0140] 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=a.sub.N.
Distributions of net pressure, fracture width, porosity and
permeability as functions of space and time for x<a.sub.N and
t<t.sub.N are obtained as well.
[0141] 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).
Production Operations
[0142] 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.
[0143] 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.
[0144] Examples of empirical forecasts are provided in U.S. Pat.
Nos. 7,788,074, 6,101,447 and 6,101,447, 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.
[0145] 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,
December 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,
Tex.; Ozkan et al., "New Solutions for Well-Test-Analysis Problems:
Part 1-Analytical Considerations", SPE 18615, SPE Formation
Evaluation, Vol. 6, No. 3, SPE, September 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, June 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, Ala.; 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, Okla.; Chen et al., "A Multiple-fractured Horizontal
Well in a Rectangular Drainage Region", SPE Jrnl. 37072, Vol. 2,
No. 4, December 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, Tex., USA.
[0146] 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.
[0147] 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 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, Aug. 15, 1976; and in U.S. Pat. No.
7,363,162. 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
[0148] FIGS. 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.
[0149] 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.
[0150] As shown in FIG. 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 FIG. 3.
[0151] While FIGS. 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
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
[0152] Information about proppant placement in an HFN, such as the
HFN 822 of FIGS. 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.
[0153] FIGS. 9 and 10 depict views of proppant placement about an
HFN and fractures of an HFN, respectively. FIG. 9 shows a
cross-sectional view of the HFN 822 of FIG. 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
FIG. 9, the proppant 823 may transport in different transport
patterns 827, 829.
[0154] 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.
[0155] 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 V.sub.p,i, the total proppant
volume fraction is
V p = i = 1 N V p , i ( 84 ) ##EQU00108##
[0156] The placement of proppant along the fractures of an HFN
involves horizontal transport, vertical settling and possible
bridging of the proppant. As shown in FIG. 9, proppant type i is
transported in all directions by the transport pattern 825. This
can be mathematically described by the following:
2 .pi..gamma. x .differential. ( .phi. V p , i ) .differential. t -
.differential. .differential. x ( 2 .pi..gamma. xk x .mu.
.differential. p .differential. x V p , i ) = 0 ( 85 )
##EQU00109##
This equation also describes the horizontal flow of fluid in FIG.
10.
[0157] 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
.differential. ( w x V p , i ) .differential. t - .differential.
.differential. x ( w x 3 12 .mu. .differential. p .differential. x
V p , i ) = 0 ( 86 ) ##EQU00110##
[0158] For a uniform horizontal volume flow rate q, the above
equations reduce to, respectively,
2 .pi..gamma. x .differential. ( .phi. V p , i ) .differential. t +
.differential. ( q V p , i ) .differential. x = 0 ( 87 )
##EQU00111##
For transport along a fairway only, the following equation
applies:
.differential. ( w x V p , i ) .differential. t + .differential.
.differential. x ( q 2 .pi..gamma. x V p , i ) = 0 ( 88 )
##EQU00112##
When fluid leakoff q.sub.i is taken into consideration, the above
equations become, respectively,
2 .pi..gamma. x .differential. ( .phi. V p , i ) .differential. t +
.differential. ( q - q l ) V p , i .differential. x = 0 and ( 89 )
.differential. ( w x V p , i ) .differential. t + .differential.
.differential. x ( q - q l 2 .pi..gamma. x V p , i ) = 0 ( 90 )
##EQU00113##
[0159] As shown in FIG. 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
v ps , i = g ( .rho. p , i - .rho. f ) d p , i 2 18 .mu. f ( 91 )
##EQU00114##
where .rho..sub.f and .mu..sub.f are the density and viscosity of
the suspension fluid, .rho..sub.p,i and d.sub.p,i 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
v ps , i = v st , i f ( V p , d p , i , w ) where ( 92 ) f ( V p ,
d p , i w ) = { ( 1 - w cr , i w ) 0.25 if w .gtoreq. w cr , i 0 if
w < w cr , i w cr , i = min ( B cr , 1 + V p B cr - 1 0.17 ) d p
, i B cr = 2.5 ( 93 ) ##EQU00115##
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
[0160] FIG. 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.
[0161] 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 FIG. 11.
[0162] The HFN 822 uses an elliptical configuration to provide a
coupling between the 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:
.differential. p f .differential. t - 1 x .differential.
.differential. x ( x .kappa. f .differential. p f .differential. x
) = q g .phi. f .differential. .rho. f .differential. p ( 94 )
##EQU00116##
where t is time, x is the coordinate aligned with the major axis of
the ellipse, p.sub.f and .rho..sub.f are fluid pressure and density
of fluid, .phi..sub.f and .kappa..sub.f are the porosity and the
x-component of the pressure diffusivity of the HFN, and q.sub.g 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.
[0163] 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.
[0164] Outside of the HFN, equation (94) still applies nominally,
but with q.sub.g=0, .phi..sub.f=.phi..sub.m and
.kappa..sub.f=.kappa..sub.m, where .phi..sub.m and .kappa..sub.m
are the porosity and the pressure diffusivity of the reservoir
matrix. Given q.sub.g 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.
[0165] 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
q.sub.gk=q.sub.gxkA.sub.xk+q.sub.gykA.sub.yk (95)
where A.sub.xk and A.sub.yk 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
q.sub.gxk and q.sub.gyk are the corresponding rates of fluid flow
per unit fracture surface area from the matrix into the fractures
parallel to the x- and y-axis, respectively. Fluid pressure p.sub.f
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.
[0166] Total surface area of fractures contained inside of the k-th
ring can be calculated by
A xk = 4 h k [ j = - N xo N xo x k 2 - 4 ( j L my / .gamma. ) 2 - j
= - N xi N xi x k - 1 2 - 4 ( j L my / .gamma. ) 2 ] A yk = 4 h k
.gamma. [ i = - N yo N yo x k 2 - 4 ( i L mx ) 2 - i = - N yi N yi
x k - 1 2 - 4 ( i L mx ) 2 ] ( 96 ) ##EQU00117##
where .gamma. is the aspect ratio of the elliptical HFN, x.sub.k
and h.sub.k are the location and the height of the k-th ring,
L.sub.mx and L.sub.my are the distances between neighboring
fractures parallel to the x-axis and the y-axis, respectively, as
shown in FIG. 12. The N.sub.xo and N.sub.xi 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
N.sub.yo and N.sub.yi 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.
[0167] 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 FIG. 12. FIG. 12 is a detailed view of one of the
blocks 828 of HFN 822 of FIG. 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.
[0168] Fluid flow inside a rectangular matrix block 828 can be
approximately described by
.differential. p m .differential. t - .kappa. m .differential. 2 p
m .differential. s 2 = 0 p m ( t , s ) = p r p m ( t , L s ) = p f
( t ) .differential. p m .differential. s s = 0 = 0 ( 97 )
##EQU00118##
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, p.sub.m is fluid pressure and p.sub.r 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
q qxk = .phi. m .differential. .rho. m .differential. p
.differential. .differential. t .intg. 0 t p fk u [ L y 2 erfc ( L
y 4 .kappa. m ( t - u ) ) + 2 .kappa. m ( t - u ) .pi. ( 1 - L y 2
16 .kappa. m ( t - u ) ) ] u q qyk = .phi. m .differential. .rho. m
.differential. p .differential. .differential. t .intg. 0 t p fk u
[ L x 2 erfc ( L x 4 .kappa. m ( t - u ) ) + 2 .kappa. m ( t - u )
.pi. ( 1 - L x 2 16 .kappa. m ( t - u ) ) ] u ( 98 )
##EQU00119##
where p.sub.fk is the pressure of the fluid residing in fractures
in the k-th ring and .rho..sub.m is the density of the fluid
residing in the matrix. The coupling of p.sub.fk and q.sub.gk
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.
[0169] 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 FIG.
13.1, or depicting the reservoir using a sugar cube representation
as shown in FIG. 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,
September 1963.
[0170] 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, Colo., 30 October-2 November, 2011,
the entire content of which is hereby incorporated by
reference.
Hydraulic Fracturing Design and Optimization
[0171] 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.
[0172] For example, for design of a particular stage of a planned
job, hydraulic fracturing software, such as MANGROVE.TM. software
commercially available from Schlumberger Technology Corporation
(see:www.slb.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.
[0173] FIG. 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.
[0174] 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 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
[0175] Reservoir properties and hydraulic fracturing treatment data
can be used to obtain information about the created HFN, such as
fracture spacing d.sub.x and d.sub.y and stress anisotropy
.DELTA..sigma., 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 FIGS. 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, June
2009, San Antonio, Tex., 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 Oct. 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 Sep. 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 Jun. 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.
[0176] FIG. 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
(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.
[0177] 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.
[0178] FIG. 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.
[0179] 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.
[0180] 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.
[0181] 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.
[0182] 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.
[0183] 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. .sctn.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.
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