U.S. patent application number 11/353405 was filed with the patent office on 2006-10-05 for hydraulic fracturing.
This patent application is currently assigned to COMMONWEALTH SCIENTIFIC AND INDUSTRIAL RESEARCH ORGANISATION. Invention is credited to Brice Tanguy Alphonse Lecampion.
Application Number | 20060219402 11/353405 |
Document ID | / |
Family ID | 37068935 |
Filed Date | 2006-10-05 |
United States Patent
Application |
20060219402 |
Kind Code |
A1 |
Lecampion; Brice Tanguy
Alphonse |
October 5, 2006 |
Hydraulic fracturing
Abstract
Method and apparatus for monitoring growth of fracture during
hydraulic fracturing treatment of a ground formation. A series of
tiltmeters are positioned at spaced apart tiltmeter stations at
which tilt changes due to hydraulic fracturing treatment are
measurable by those tiltmeters. Tilt measurements obtained from the
tiltmeters at progressive times during the fracture treatment are
compared with estimated tilt changes estimated from an assumed
hydraulic fracturing propagation. Useful values for initially
undetermined values of parameters of the model are derived using an
inverse procedure. The data analysis may be performed in real time
as the hydraulic treatment progresses using differing propagation
models to determine a most probable model.
Inventors: |
Lecampion; Brice Tanguy
Alphonse; (Richmond, AU) |
Correspondence
Address: |
MERCHANT & GOULD PC
P.O. BOX 2903
MINNEAPOLIS
MN
55402-0903
US
|
Assignee: |
COMMONWEALTH SCIENTIFIC AND
INDUSTRIAL RESEARCH ORGANISATION
Campbell
AU
|
Family ID: |
37068935 |
Appl. No.: |
11/353405 |
Filed: |
February 14, 2006 |
Current U.S.
Class: |
166/250.1 ;
166/308.1; 166/66 |
Current CPC
Class: |
E21B 47/02 20130101;
E21B 49/00 20130101; E21B 43/26 20130101 |
Class at
Publication: |
166/250.1 ;
166/066; 166/308.1 |
International
Class: |
E21B 47/02 20060101
E21B047/02; E21B 43/26 20060101 E21B043/26 |
Foreign Application Data
Date |
Code |
Application Number |
Feb 16, 2005 |
AU |
2005900724 |
Claims
1. A method of monitoring growth of a fracture during hydraulic
fracturing treatment of a ground formation comprising: measuring at
progressive times through an extended time interval during the
treatment values of physical variables dependent on propagation of
the fracture; estimating, from an assumed hydraulic fracturing
propagation model that is reliant on initially undetermined
parameters, values of the variables at said progressive times
through the extended time interval for differing values of said
parameters; and comparing the estimated values of said variables
with the measured values to derive useful values of said
parameters.
2. A method as claimed in claim 1, comprising the further step of
estimating from the assumed hydraulic fracturing propagation model
and the derived useful values of said parameters future propagation
dependent variables.
3. A method as claimed in claim 1, wherein the useful values of
said parameters are derived by an inverse procedure.
4. A method as claimed in claim 3, wherein the inverse procedure is
a Bayesian procedure.
5. A method as claimed in claim 1, wherein said physical variables
comprise tilt changes measurable by tiltmeters disposed at spaced
apart locations in the vicinity of the fracture.
6. A method as claimed in claim 1, wherein said physical variables
comprise fracturing fluid injection pressure values.
7. A method as claimed in claim 1, wherein said physical variables
comprise acoustic emissions data.
8. A method of monitoring growth of a fracture during hydraulic
fracturing treatment of a ground formation comprising: positioning
a series of tiltmeters at spaced apart tiltmeter stations at which
tilt changes caused by the hydraulic fracture growth are measurable
by those tiltmeters; estimating, from an assumed hydraulic
fracturing propagation model, tilt changes at the tiltmeter
stations at progressive times through an extended time interval
during the treatment; obtaining, from the tilmeters, tilt
measurements at said progressive times throughout the extended time
interval; and comparing the estimated tilt changes through said
time interval with the actual tilt measurements throughout that
time interval.
9. A method as claimed in claim 8, further comprising deriving from
the comparison of the estimated and actual tilt changes useful
values for initially undetermined values of parameters of the
model.
10. A method as claimed in claim 9, wherein the useful values of
said parameters are derived using an inverse procedure.
11. A method as claimed in claim 10, wherein the inverse procedure
is a Bayesian inversion procedure.
12. A method as claimed in claim 8, wherein the fracture
orientation during said time interval is estimated from the tilt
measurements.
13. A method as claimed in any one of claim 8, wherein measurements
of data other than tilt changes are used to constrain the
propagation model.
14. A method as claimed in claim 13, wherein said data include
hydraulic injection pressure data.
15. A method as claimed in claim 13, wherein said data include
acoustic emissions data.
16. A method as claimed in claim 8, wherein the comparison of
estimated tilt changes with measured tilt changes is performed in
real time as the hydraulic fracturing treatment progresses.
17. A method as claimed in claim 16, wherein comparisons are made
through a first time interval from a start time and sequentially
through successive longer time intervals from the same start
time.
18. A method as claimed in claim 8, and comprising the steps of:
estimating from one or more alternative hydraulic fracturing
propagation models tilt changes at the tiltmeter stations at said
progressive times throughout said time interval; comparing the
estimated tilt changes estimated from the assumed and alternative
propagation models with the actual tilt measurements to determine a
most probable propagation model.
19. Apparatus for monitoring growth of a fracture during hydraulic
fracturing treatment of a ground formation comprising: instruments
to measure physical variables dependent on propagation of the
fracture; and a signal processing unit to receive measurement
signals from the instruments at progressive times through an
extended time interval during the fracturing treatment and operable
to estimate, from an assumed fracturing propagation model that is
reliant on initially undetermined parameters, values of said
physical variables at said progressive times through the extended
time interval for differing values of said parameters and by
comparison of the estimated values with the measurement signals to
derive useful values of said parameters.
20. Apparatus as claimed in claim 19, wherein the signal processing
unit is operable to derive the values of said parameters by an
inverse procedure.
21. Apparatus for monitoring growth of a fracture during hydraulic
fracturing treatment of a ground formation comprising: a series of
tiltmeters positionable at spaced apart tiltmeter stations to
measure tilt changes due to hydraulic fracturing treatment; and a
signal processing unit to receive tilt measurement signals from the
tiltmeters at progressive times through an extended time interval
during the fracturing treatment and operable to estimate from an
assumed hydraulic fracturing propagation model tilt changes at the
tiltmeter stations at progressive times through an extended time
interval during the treatment and to compare the estimated tilt
changes through said time interval with said tilt measurement
signals.
22. Apparatus as claimed in claim 21, wherein the data processing
unit is operable to derive from the comparison of the estimated
tilt changes and the tilt measurement signals useful values for
initially undetermined values of parameters of the model.
23. Apparatus as claimed in claim 22, wherein the data processing
unit is operable to derive the useful values of said dependent
parameters by an inverse procedure.
24. Apparatus as claimed in claim 21, wherein the signal processing
unit is operable to derive from the tilt measurements estimates of
fracture orientation during said extended time interval.
25. Apparatus as claimed in claim 21, wherein the signal processing
unit is operable to compare the estimated tilt changes with the
tilt measurement signals in real time as the hydraulic fracturing
treatment progresses.
26. Apparatus as claimed in claim 21, wherein the signal processing
unit is operable to estimate from one or more alternative hydraulic
fracturing propagation models tilt changes at the tiltmeter
stations at progressive times throughout the extended time
interval, and to compare the estimated tilt changes estimated from
the assumed and alternative propagation models with the actual tilt
measurement signals to determine a most probable propagation model.
Description
TECHNICAL FIELD
[0001] This invention relates to hydraulic fracturing of natural
ground formations which may be on land or under a sea bed.
[0002] Hydraulic fracturing is a technique widely used in the oil
and gas industry in order to enhance the recovery of hydrocarbons.
A fracturing treatment consists of injecting a viscous fluid at
sufficient rate and pressure into a bore hole drilled in a rock
formation such that the propagation of a fracture results. In later
stages of the fracturing treatment, the fracturing fluid often
contains a proppant, typically sand, so that when the injecting
stops, the fracture closes on the proppant which then forms a
highly permeable channel (compared to the permeability of the
surrounding rock) which may thus enhance the production from the
bore hole or well.
[0003] In recent years, hydraulic fracturing has been applied for
inducing caving and for preconditioning of rock for caving in the
mining industry. In this application, the fractures are typically
not propped but are formed to modify the rock mass strength and
weaken the ore or country rock.
[0004] One of the most important issues in the practice of the
hydraulic fracturing technique is knowledge of the geometry
(orientation, extent, volume) of the created fracture. This is of
particular importance in order to estimate the quality of the
treatment performed. However, operators presently have only limited
direct measurement capability that allows them to verify the
quality and effectiveness of their operations. It is only
afterwards when production has restarted that the performance of
the created fracture can be assessed.
[0005] In order to map hydraulic fractures, several types of
indirect measurements can be carried out such as microseismic
acoustic monitoring and tiltmeter mapping, but until recently such
surface tiltmeter techniques have not been capable of producing
accurate real time information which can be used during the course
of a hydraulic fracturing treatment and have generally been used to
provide data for later analysis. However, our International
Application PCT/AU2004/001263 discloses a method and apparatus
whereby it is possible to obtain useful data on the effectiveness
of a hydraulic treatment as the treatment progresses.
[0006] In accordance with the invention disclosed in
PCT/AU2004/001263, tilt measurements may be obtained at spaced
apart tiltmeter stations at progressive times during the fracturing
treatment and a fluid driven fracture volume derived from those
measurements by performing an analysis at each of those times as
the treatment is in progress. The analysis may be based on
minimisation of misfit between the tilt measurements at each given
time and tilt predicted by a fracture model, for example a
displacement discontinuity model.
[0007] The present invention involves an alternative kind of
analysis which can be performed from tiltmeter or other data
recorded during an interval of the fracturing treatment. Rather
than analysing the data sequentially at different times
independently, analysis in accordance with the present invention is
carried out using data over an extended time interval, for example,
the complete treatment time. By the present invention, it is
possible to use a hydraulic fracturing propagation model to analyse
the growth of the fracture through the time interval during which
the relevant data is recorded and to use the recorded data to
calibrate the model. The recorded data may be tiltmeter data, but
it is possible to measure other data such as fracturing fluid
injection pressure and acoustic emissions to constrain the growth
model.
DISCLOSURE OF THE INVENTION
[0008] The invention broadly provides a method of monitoring growth
of a fracture during hydraulic fracturing treatment of a ground
formation comprising:
[0009] measuring at progressive times through an extended time
interval during the treatment values of physical variables
dependent on propagation of the fracture;
[0010] estimating, from an assumed hydraulic fracturing propagation
model that is reliant on initially undetermined parameters, values
of the variables at said progressive times through the extended
time interval for differing values of said parameters; and
[0011] comparing the estimated values of said variables with the
measured values to derive useful values of said parameters.
[0012] More specifically, the invention may provide a method of
monitoring growth of a fracture during hydraulic fracturing
treatment of a ground formation comprising:
[0013] positioning a series of tiltmeters at spaced apart tiltmeter
stations at which tilt changes caused by the hydraulic fracture
growth are measurable by those tiltmeters;
[0014] estimating, from an assumed hydraulic fracturing propagation
model, tilt changes at the tiltmeter stations at progressive times
through an extended time interval during the treatment;
[0015] obtaining, from the tiltmeters, tilt measurements at said
progressive times throughout the extended time interval; and
[0016] comparing the estimated tilt changes through said time
interval with the actual tilt measurements throughout that time
interval.
[0017] The method may further comprise deriving from the comparison
of the estimated and actual tilt changes useful values for
initially undetermined values of parameters of the model.
[0018] The useful values of said parameters may be estimated using
an inverse procedure.
[0019] Furthermore, other measurements such as recorded injection
pressure, or/and acoustic emissions may be used to constrain the
growth model if such measurements are available.
[0020] The method may also comprise the steps of:
[0021] estimating from one or more alternative hydraulic fracturing
propagation models tilt changes at the tiltmeter stations at said
progressive times throughout said time intervals;
[0022] comparing the estimated tilt changes estimated from the
assumed and alternative propagation models with the actual tilt
measurements to determine a most probable propagation model.
[0023] The comparison of estimated tilt changes with measured tilt
changes may be performed in real-time as the hydraulic fracturing
treatment progresses. For example, comparisons may be made from a
first time interval from a start time and sequentially through
successive longer time intervals from the same start time.
[0024] The invention also provides apparatus for monitoring growth
of a fracture during hydraulic fracturing treatment of a ground
formation comprising:
[0025] instruments to measure physical variables dependent on
propagation of the fracture; and
[0026] a signal processing unit to receive measurement signals from
the instruments at progressive times through an extended time
interval during the fracturing treatment are operable to estimate
from an assumed fracturing propagation model that is reliant on
initially undetermined parameters, values of said physical
variables at said progressive times through the extended time
interval for differing values of said parameters and by comparison
of the estimated values with the measurement signals to derive
useful values of said parameters.
[0027] The invention further provides apparatus for monitoring
growth of a fracture during hydraulic fracturing treatment of a
ground formation comprising:
[0028] a series of tiltmeters positionable at spaced apart
tiltmeter stations to measure tilt changes due to hydraulic
fracturing treatment; and
[0029] a signal processing unit to receive tilt measurement signals
from the tiltmeters at progressive times through an extended time
interval during the fracturing treatment and operable to estimate
from an assumed hydraulic fracturing propagation model tilt changes
at the tiltmeter stations at progressive times through an extended
time interval during the treatment and to compare the estimated
tilt changes through said time interval with said tilt measurement
signals.
BRIEF DESCRIPTION OF THE DRAWINGS
[0030] The invention and the manner in which it may be put into
effect will now be described in more detail with the aid of the
references listed at the end of this specification and the
accompanying drawings, in which:
[0031] FIG. 1 illustrates the principle of tiltmeter
measurement;
[0032] FIG. 2 illustrates diagrammatically a dipping fracture and
corresponding uplift at the ground surface;
[0033] FIG. 3 illustrates the evolution in time of the inclination
recorded at a tiltmeter station during a fracturing treatment;
[0034] FIG. 4 illustrates tilt vectors at an array of tiltmeter
stations at a particular instant of time during a fracturing
treatment; and
[0035] FIG. 5 shows the time evolution of the half-length of the
fracture and opening of the fracture center of a PKN fracture
growth model, both the correct (i.e simulated) value and the one
obtained from the inversion of tilt data using the same fracture
growth model are displayed,
[0036] FIG. 6 displays the fit for the recorded tilts obtained from
the inversion of a PKN fracture growth model;
[0037] FIG. 7 displays a typical set-up to carry out the analysis
in the field.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
[0038] In order to explain the operation of the preferred method
and apparatus and according to the invention, it will be necessary
to analyse in some detail the current state of the art in the
operation of hydraulic fracture monitoring, especially tiltmeters,
and the modelling and inverse techniques required to derive
meaningful data from tiltmeter measurements.
Hydraulic Fracture Monitoring State of the Art
[0039] Several techniques have been developed in order to image
hydraulic fracture. Three type of measurements are typically
encountered in practice: [0040] 1. well pressure monitoring [0041]
2. passive seismic system recording acoustic emission during the
fracturing process [0042] 3. tiltmeters remotely located from the
fracture measuring change of deformation as the fracture grows.
These methods have all their strength and weaknesses.
[0043] Well pressure monitoring is a simple measurement with a
relatively low cost. When no pressure sensor is lowered down the
injection well at the level of the injection interval, pressure is
usually recorded on the surface rig together with the injected
flow. In that case, corrections of the pressure drops due to
viscous flow along the well have to be performed. The
interpretation of well pressure with respect to hydraulic fracture
growth can be rendered difficult due to damage around the well bore
disturbing the fluid pressure gradients. Typical analysis of
pressure decline, shut-in and fall-off data (G-plot etc.) can help
estimating a number of characteristics of the fracture (treatment
efficiency, closure stress etc.), Economides and Nolte, 2000 [5]. A
number of simplifying assumptions are usually made in order to
analyse pressure records.
[0044] Acoustic emission monitoring (also sometimes call passive
seismic) consists of recording using precise accelerometers the
seismic signal produced by the growth of the fracture. An array of
geophones is used, and analysis of the signal enables to locate the
seismic source. The velocity of seismic waves in the rock mass must
be known. Usually, operators calibrate a velocity model by
recording the signal generated by a known source (typically during
the perforation of the casing). This method can provide good
results in homogeneous rock mass if the array of sensor covers well
the fracture area. In the case, where natural joints are present in
the vicinity of the Hydraulic fractures, multiple shear events
along the natural joints can occur. This can `blur` the overall
picture given by the analysis of the recorded signals.
[0045] A Tiltmeter (which is tightly installed in the rock)
measures at its location changes of inclination in two orthogonal
directions with a precision down to nanoradians. The tilts recorded
are a direct measure of the small strain rotation at the apparatus
location. In the field, tiltmeter fracture monitoring involves the
deployment of a number of apparatus in the surroundings of the
future fracture. Typically, they are either located on the surface
of the earth in shallow boreholes (to minimize noise), or in an
off-set well (downhole tiltmeter array), even sometimes the
injection well itself (Davis, 1983 [4]; Evans, 1983 [7]; Branagan
et al., [2]; Warpinski et al., [11]; Wright et al. [13]).
[0046] During a hydraulic fracture treatment, a measure of the
different tiltmeters is recorded every 5 seconds or so, thus
providing the effect of the fracture propagation on the elastic
deformation field in the rock surrounding the fracture. FIG. 4
displays for a given tiltmeter station, the two inclinations
(north-south and east-west) recorded during a fracturing job. We
clearly see the evolution of the inclination during the injection
as well as the slow return toward their initial values after the
end of the injection. This return is associated with the hydraulic
fracture closing back on itself after injection stops. Another
representation of tiltmeter measurements is given in FIG. 5. In
case of a surface array, the so-called tilt vectors provide a
spatial view (plan view) of the recorded inclination at the
different tiltmeter station at a particular time during the
injection. The tilt vector is determined from a vector addition of
the two orthogonal components of the rotation tensor measured by
the two bubbles in the tiltmeter: v = ( .differential. u z
.differential. x - .differential. u x .differential. z ,
.differential. u z .differential. y - .differential. u y
.differential. z ) . ##EQU1##
[0047] Such tiltmeter measurements are extremely precise and can
furnish interesting data sets. In contrast to the relative
simplicity of the measurement, the modelling necessary to
quantitatively analyse tilt data pose difficult problems. Despite
the now common use of tiltmeters to map hydraulic fractures in the
petroleum industry, there is a widespread misunderstanding of what
information about the fracture can and cannot be obtained from such
measurements. Lecampion et al. [22] have shown that for measurement
located at a distance twice the fracture characteristics length,
tilt measurements only sense the effect of fracture volume and
orientation. Such situation occurs in the field when the tiltmeters
are too far from the fracture: we are in a case of so-called
far-field tiltmeter mapping. The specification of our International
Application PCT/AU2004/001263 discloses the use of a fundamental
Displacement Discontinuity model by which it is possible to carry
out a real-time analysis of the hydraulic fracture volume and the
treatment efficiency, particularly for far-field tiltmeter mapping.
The disclosure of that specification is imported here by
reference.
[0048] If the tiltmeter are sufficiently close to the fracture, the
fracture shape starts to affect the recorded tilts: this is a case
of near-field tiltmeter mapping. In near-field tiltmeter mapping,
the use of simple static fracture model (radial, elliptical,
rectangular DD . . . ) faces the problem that these models do not
have the correct boundary conditions on the fracture face, which
can results in doubtful conclusions.
New Analysis
[0049] By the present invention, an alternative method to analyse
data recorded around hydraulic fractures is proposed. The method
may use tiltmeter data but other data such as measurements of the
injection pressure and acoustic emission can also be used. The
method consists of coupling hydraulic fracture growth models (which
model the physical process of Hydraulic fracturing either
numerically or analytically depending on their assumptions) to the
recorded data, and to calibrate the parameters of the hydraulic
fracture growth models from the data. In contrast to existing
methods, where data are analysed sequentially at different times
independently, here the analysis is carried out using the data over
an entire time interval and uses physical models of the HF process,
not just static fracture models like the ones described in
Warpinski 2000; [15]; Evans, 1983 [7]. Essentially the data
recorded throughout the entire time interval are used in an inverse
procedure to derive useful values for initially unknown values of
parameters in the model. This will normally involve estimating data
at the relevant time with differing assumed values of the
parameters and comparing the various estimates with the recorded
data to derive values which can then be used in the model to
estimate evolution of the fracture.
[0050] The propagation of a hydraulic fracture is a highly
non-linear process. Recent advances in the understanding of the key
parameters controlling the growth of such fracture have shown that
different regimes of propagation can be isolated (see Detournay,
2004 [17]; Adachi, 2001 [1]; Detournay et al. [18] for extensive
discussions). The propagation of a hydraulic fracture is governed
by two competing dissipative processes associated with fracturing
fluid viscosity and rock toughness, respectively and two competing
components of the fluid balance associated with fluid storage in
the fracture and fluid storage in the surrounding rock (leak-off).
Scaling laws as well as complete solutions for the propagation of
hydraulic fractures have been recently obtained for simple fracture
geometries such as a radial fracture and a plane strain fracture.
This approach is useful for the design of HF treatment and has been
described in reference [17]. Also, complex and computationally
efficient numerical models are now becoming available (Siebrits, E.
and Peirce, A. P.; 2001 [9]). We have also to note that empirical
growth models are sometimes used in the industry for practical
purposes [5]. All these hydraulic fracture growth models furnish a
solution at all times during the injection period for the fracture
shape, fracture aperture and internal fluid pressure (therefore
well pressure) etc. everywhere in the fracture. Moreover, for a
given fracture evolution, the corresponding tiltmeters at the
different observation point and at the different time can be simply
obtained from the solutions of an elasto-static problem (Eshelby,
1957 [6]) for each fracture configuration (i.e. each time). The
location of seismic events obtained from acoustic emission can be
qualitatively associated with the fracture front and the front of
fluid leaking into the rock formation.
[0051] Typically, a particular fracture growth model depends on a
number of parameters lumping physical properties of the rock, the
layering structure, state of initial stress, the properties of the
injected fluid driving the fracture as well as the pumping
schedule. Some, if not all, of these parameters are often poorly
known. Data recorded during the actual treatment can be used to
back calculate these parameters. Moreover, several growth models
with different type of assumptions can be tested against the data
recorded.
Fracture Models
[0052] For example, we can suppose a simple PKN model for the
fracture growth (see Economides and Nolte [5] for details). This is
a typical model use in the industry for a vertical fracture growing
in a confined layer. According to this model, the half-length of
the fracture and fracture aperture at the fracture center evolves
as: l .function. ( t ) = 0.68 .times. ( E ' .times. Q o 3 2 .times.
.mu. .times. .times. H 4 ) 1 / 5 .times. t 4 / 5 = 0.68 .times.
.times. .alpha. .times. t 4 / 5 ( 1 ) w .function. ( 0 , t ) = 2.5
.times. ( 2 .times. .mu. .times. .times. Q o 2 E ' .times. H ) 1 /
5 .times. t 1 / 5 = 2.5 .times. .times. .beta. .times. t 1 / 5 ( 2
) ##EQU2## where E' is the plane strain, .mu. the fluid viscosity,
H the layer in which the fracture is developing and Q.sub.o the
fluid injection rate (supposed constant). t denotes the time from
the beginning of the injection. For that particular growth model,
the unknown parameters will be .alpha. = ( E ' .times. Q o 3 2
.times. .mu. .times. .times. H 4 ) 1 / 5 .times. .times. and
.times. .times. .beta. = ( 2 .times. .mu. .times. .times. Q o 2 E '
.times. H ) 1 / 5 . ##EQU3##
[0053] Alternatively, we can assume that the created hydraulic
fracture is radial and grows in the so-called toughness dominated
regime (Detournay, 2004 [17]). When a Newtonian fracturing fluid is
injected at a constant rate, the radius and net pressure inside the
fracture evolves as power law of time: R .function. ( t ) = L k
.times. .gamma. ko .function. ( t t c ) 2 / 5 ( 3 ) p .function. (
t ) = k .times. E ' .times. .PI. o .function. ( t t c ) - 1 / 5 ( 4
) ##EQU4## where .gamma. ko = ( 2 .pi. .times. .times. 2 1 / 2 ) 2
/ 5 , .times. .PI. ko = .pi. 8 .times. ( .pi. 12 ) - 1 / 5 ##EQU5##
and the characteristic time t.sub.c is related to the injection
rate, plane strain Young's Modulus, fracture toughness and
fracturing fluid viscosity: t c = ( .mu. '1 / 5 .times. Q o 3
.times. E ' 13 K '18 ) 1 / 2 ##EQU6##
[0054] The characteristic lengthscale L.sub.k and dimensionless
number .epsilon..sub.k are given by (see Detournay, 2004 [17]): L k
_ = E '3 .times. Q o .times. .mu. ' K '4 ##EQU7## k _ = ( K '6 E '5
.times. Q o .times. .mu. ' ) 1 / 2 ##EQU7.2##
[0055] For that radial fracture toughness dominated growth model,
the unknown parameters to be identified during the inverse analysis
of the tilt series data are .alpha.={overscore
(L.sub.k)}(t.sub.c).sup.- and .beta.={overscore
(.epsilon..sub.k)}(t.sub.c).sup.1/5.
Of course, more complex numerical models with a larger number of
parameters can be used. Some parameters of the numerical simulation
can also be isolated in order to be estimated in the inverse
analysis.
Inversion
[0056] All the data from tiltmeters recorded during a time interval
of the treatment can be used to perform an inverse calculations to
estimates the parameters of a given growth models. Eventually, the
parameters of several competing models can be estimated and the
different models quantitatively ranked using a Bayesian model
selection procedure as discussed in Raftery, 1995 [21]. It is
therefore possible to detect the regime of propagation in which the
hydraulic fracture is growing during that time interval.
[0057] Let us describe the inverse procedure. Consider a fracture
growth model M, with parameters m=(.alpha., .beta., etc . . . ).
For a given value of its parameter, this model furnishes the
fracture shape S, aperture w and pressure p at all time. It is
therefore possible to compute, for this value of the parameters,
the tilts at all the tiltmeter location for all time where the
tilts are sampled. We shall denote the predicted tilts (or
eventually well pressure, or acoustic event location) as a function
of the unknown parameters of the model: g(m).
[0058] If we have N tiltmeter stations, each recording two
components of tilts, and p samples during the time interval for
each stations, we have a data vector d of length 2*N*p.
[0059] The inverse procedure seeks to minimize the error between
the predictions and the observation in order to obtain the best fit
(Vogel, 2000 [10]; Tarantola and Valette, 1982 [19]): J .function.
( m ) = 1 2 .times. g .function. ( m ) - d ( 5 ) ##EQU8## Prior
knowledge on the parameters can be input in the procedure (see
Tarantola and Valette, 1982 [19]; Ulrych T. J. et al. 2001 [20]).
The minimization of the error can be done by using several methods
(random search, Quasi-Newton Methods etc.). The best fit furnishes
the best estimates for the model parameters. The knowledge of the
best estimates of the model parameters allows to obtain the
evolution of the fracture shape, aperture etc. during the injection
period. The growth model is then calibrated. An Example
[0060] As an illustrating example, let consider the following
synthetic case. The tiltmeter data corresponding to the growth of a
Hydraulic Fracture of so-called PKN geometry (with coefficients
.alpha.=2, .beta.=2e-3, see eqns (1)-(2)) . . . are simulated and
noise is added to this synthetic data set. The data are logged by a
borehole array located 200 meters from the fracture (a far-field
array). Tilts are sampled every minute. The synthetic data are then
inverted using the same PKN model for the fracture growth model
(see eqns. (1)-(2)). The parameters (.alpha.,.beta.) are supposed
unknowns and are retrieved via an inverse calculation using the
simulated tilt data. The values of both parameters are retrieved
accurately, we obtain .alpha.=1.94, .beta.=1.98e-3. The fit of the
tilt data at a particular tilt station can be seen in FIG. 6. FIG.
5 shows the time evolution fracture half-length and aperture at the
fracture center obtained from eq. and the estimated values of the
coefficients .alpha. and .beta.. The estimated values of the model
parameters allow to obtain fracture length and aperture.
Steps of the Analysis
[0061] As a guideline, the practical steps of the analysis are
typically the following:
[0062] Collect data at all the tiltmeter stations for all sampling
times during the interval of interest t.di-elect
cons.[t.sub.a,t.sub.b], correct the tiltmeter data (drift, earth
tides, rotation to the geometrical coordinate system);
[0063] Perform the identification procedure for a particular
fracture propagation model using the tilt data at the different
instrument locations over the time interval of interest in order to
estimate the model parameters; and
[0064] Eventually repeat the preceding step with a different
propagation model and compare the different fracture propagation
models.
[0065] We have to note that the computational speed of the models
make it possible to carry out a real-time analysis, as described
above, during the treatment. For example, the analysis can be
performed on the first five minutes of the treatment, then on the
first ten minutes of treatment and so on, eventually allowing
design changes to be performed according to the calibrated
model.
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