U.S. patent application number 11/145424 was filed with the patent office on 2005-12-08 for method for continuous interpretation of monitoring data.
This patent application is currently assigned to SCHLUMBERGER TECHNOLOGY CORPORATION. Invention is credited to Raghuraman, Bhavani, Ramakrishnan, Terizhandur S..
Application Number | 20050270903 11/145424 |
Document ID | / |
Family ID | 35448760 |
Filed Date | 2005-12-08 |
United States Patent
Application |
20050270903 |
Kind Code |
A1 |
Ramakrishnan, Terizhandur S. ;
et al. |
December 8, 2005 |
Method for continuous interpretation of monitoring data
Abstract
It is shown that a pressure pulse originating in a well is
correlated to a pulse observed at a distant well with a
characteristic time. The correlation time is directly related to
the diffusion time scale arising out of the pressure diffusion
equation. The relationship is affected by the source-observer or
observer-observer distance but the correction is small for large
distances. In practice, further corrections have to be included for
finite width pulses. For these pulses, a practical scheme for
continuous permeability monitoring is presented.
Inventors: |
Ramakrishnan, Terizhandur S.;
(Bethel, CT) ; Raghuraman, Bhavani; (Wilton,
CT) |
Correspondence
Address: |
SCHLUMBERGER-DOLL RESEARCH
36 OLD QUARRY ROAD
RIDGEFIELD
CT
06877-4108
US
|
Assignee: |
SCHLUMBERGER TECHNOLOGY
CORPORATION
Ridgefield
CT
|
Family ID: |
35448760 |
Appl. No.: |
11/145424 |
Filed: |
June 3, 2005 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
60577269 |
Jun 4, 2004 |
|
|
|
Current U.S.
Class: |
367/40 |
Current CPC
Class: |
G01V 1/40 20130101 |
Class at
Publication: |
367/040 |
International
Class: |
G01V 001/40 |
Claims
What is claimed is:
1. A method for monitoring characteristics of a region of earth
formation over time, comprising: a. obtaining source data at at
least one location in a source well; b. obtaining observer data at
at least one location in an observer well, wherein said observer
data may be correlated to said source data; c. repeating (a) and
(b) one or more times; d. developing a correlation between said
source data and said observer data as a function of time; and e.
analyzing said correlation to monitor one or more characteristics
of said earth formation.
2. The method of claim 1, wherein said correlation is based on a
propagation function.
3. The method of claim 1, further comprising analyzing the shape of
said correlation over time.
4. The method of claim 1, further comprising determining the time
location of the maximum value of said correlation and analyzing the
decay of said maximum value over time.
5. The method of claim 4, wherein said source data are pressure
pulses, said observer data is pressure data, wherein said
correlation is governed by diffusion, and wherein at least one of
said one or more characteristics is permeability.
6. The method of claim 1, wherein said source data and said
observer data is differentiated pressure data.
7. The method of claim 1, wherein said source data is flow data and
said observer data is pressure data.
8. The method of claim 1, further comprising repeating (a), (b) and
(c) for more than one observer well.
9. The method of claim 1, further comprising repeating (a), (b),
and (c) for more than one source well.
10. The method of claim 5, further comprising correcting said
correlation time for storage effects at said source well.
11. The method of claim 5, further comprising correcting said
correlation time for the distance between said observer well and
said source well.
12. The method of claim 5, further comprising correcting said
correlation time for the width of said pressure pulses.
13. A method of monitoring permeability of a region of earth
formation over time comprising: a. inducing one or more pressure
pulses at at least one location in at least one source well,
wherein data on said induced pressure pulses is gathered; b.
measuring pressure pulse data at at least one location in at least
one observer well, wherein said pressure pulse data includes
pressure pulses having propagated through at least a portion of
said earth formation; c. repeating (a) and (b) one or more times;
d. developing a correlation between said induced pressure pulses
and said propagated pressure pulses as a function of time; and e.
analyzing said correlation to monitor the permeability the region
of earth formation.
14. The method of claim 13, wherein said correlation is based on a
propagation function.
15. The method of claim 13, further comprising analyzing the shape
of said correlation over time.
16. The method of claim 13, further comprising determining the
determining the time location of the maximum value of said
correlation and analyzing the decay of said maximum value over
time.
17. The method of claim 16, wherein said correlation is governed by
diffusion, and wherein at least one of said one or more
characteristics is permeability.
18. The method of claim 13, wherein said induced pressure pulse
data gathered at the source well and said propagated pressure
pulsed data gathered at the observer well is differentiated
pressure data.
19. The method of claim 13, wherein said induced pressure pulse
data is flow data and said propagated pressure pulse is pressure
data.
20. The method of claim 13, further comprising repeating (a), (b)
and (c) for more than one observer well.
21. The method of claim 13, further comprising repeating (a), (b),
and (c) for more than one source well.
22. The method of claim 18, further comprising correcting said
correlation time for storage effects at said source well.
23. The method of claim 18, further comprising correcting said
correlation time for the distance between said observer well and
said source well.
24. The method of claim 18, further comprising correcting said
correlation time for the width of said induced pressure pulses.
Description
RELATED APPLICATIONS
[0001] This application claims priority from co-pending U.S.
provisional patent application Ser. No. 60/577,269 filed Jun. 4,
2004 (incorporated by reference herein in its entirety).
FIELD OF THE INVENTION
[0002] The present invention relates to the cross-correlation of
data to monitor formation properties, and more particularly, to the
cross-correlation of pressure data to monitor permeability in a
reservoir using permanent or semi-permanently installed
sensors.
BACKGROUND OF THE INVENTION
[0003] Permanent surface and downhole sensor technology is
increasingly being implemented to enable real time monitoring and
reservoir management. Acquiring vast quantities of data in real
time at high frequencies is useful only if data processing and
interpretation can be done at same time scales. Otherwise the value
of high frequency information is lost. U.S. patent application Ser.
No. 09/705,674 to Ramakrishnan et al. (the '674 Application)
(incorporated herein by reference in its entirety) teaches methods
for real time data acquisition and remote reservoir management
using information from permanent sensors. U.S. patent application
Ser. No. 10/442,216 to Raghuraman et al. (the '216 Application)
(incorporated herein by reference in its entirety) teaches methods
to efficiently process and interpret vast quantities of data
acquired from permanent sensors. The '216 Application outlines
algorithms for processing data at relevant time scales. The
subsequent interpretation is then done using increasingly detailed
levels of modeling so that the levels of modeling match the time
scale of data processing. For example a pressure-pressure
derivative correlation across two locations in a reservoir may be
done to estimate the pressure diffusion time between these two
locations and used as a tracker for formation properties
(permeability, fluid mobility, porosity) over time (see the '674
Application, and Raghuraman, et al., "Interference Analysis of
Cemented-Permanent-Sensor Data from a Field Experiment," (M019),
Jun. 11-15, 2001, EAGE 63rd Conference & Technical Exhibition,
Amsterdam, incorporated herein by reference in its entirety). This
is a quick look interpretation that can easily be done over a time
scale of hours/days as opposed to a full-scale reservoir simulation
that takes a time scale of weeks/months. Such quick look
interpretation methods, which match time scales of data collection,
are needed if one has to maximize the value of high frequency
information from permanent sensors. They can be useful for tracking
changes in formation properties over time as well as for
constraining more detailed reservoir models.
[0004] In principle, well-testing constitutes an inverse problem.
One starts with assumptions regarding the reservoir, and based on
the transient response attempts to estimate the relevant
properties. System identification techniques through deconvolution,
type-curve matching for the reservoir model, and variants of Newton
iterative techniques are common in well-test interpretation.
Conventional well testing implies a production or an injection
test, and monitoring the resulting pressure behavior at the
wellbore. (See Earlugher, "Advances in Well Test Analysis," Soc.
Pet. Eng. AIME, New York (1977); Matthews et al., "Pressure Buildup
and Flow Tests in Wells," Soc. Pet. Eng. AIME, New York (1967); and
Raghavan, Well Test Analysis, Prentice Hall (1993), incorporated by
reference herein in their entireties). Adequate shut-in prior to a
well test, or alternatively, inclusion of rate data, is important
in order to eliminate the influence of historical production in the
interpretation. Significant emphasis is placed on the determination
of skin-factor in such tests. In contrast, in interference testing,
pressure is measured in a shut-in well, termed the observation
well. Other wells continue to be active. Such tests can be useful
in estimating reservoir scale permeabilities between the observer
and the other wells, if one chooses to deploy a suitable testing
scheme. The physics of the method is substantively the same as
conventional testing schemes; but the methodology and the
procedures are different.
[0005] An enhancement to interference testing is the pulsed
interference testing, wherein a periodic pulsing of a well is
carried out, and the response at an observation well is analyzed.
(See Brigham, "Planning and Analysis of Pulse-Tests," J. Pet.
Technol. (1970), volume 22, pages 618-624; Johnson et al.,
"Pulse-testing: A new method for describing reservoir flow
properties between wells," J. Pet. Technol. (1966), volume 18,
pages 1599-1604; Kamal et al., "Pulse-testing response for unequal
pulse and shut-in periods," J. Pet. Technol. (1975) volume 27,
pages 399-410, incorporated by reference herein in their
entireties). An extensive set of "type-curves" is available to
translate the magnitude and the time-lag of the pulse responses at
the observation well. The lag is the time between the beginning of
the source pulse and the peak in the pressure response. While such
a method was an advance over conventional techniques for estimating
inter-well permeability, a certain degree of regularity and
ideality for the pulses is required. Calculations are based on
extensive table/type-curve look-up, and the applicability of these
techniques assume ideal conditions (e.g. periodicity, uniformity
etc.). A translation of these methods for continuous updating of
interwell permeability with nonsystematic or irregular pulsing is
difficult.
[0006] Accordingly, the present invention provides a quick look
interpretation methodology for cross-correlation of sensor data.
Such techniques may be applied to cross-correlation of data from
different sensor types at same or different locations.
SUMMARY OF THE INVENTION
[0007] It is shown that a pressure pulse originating in a well is
correlated to a pulse observed at a distant well with a
characteristic time. The correlation time is directly related to
the diffusion time scale arising out of the pressure diffusion
equation. The relationship is affected by the source-observer or
observer-observer distance but the correction is small for large
distances. In practice, further corrections have to be included for
finite width pulses. For these pulses, a practical scheme for
continuous permeability monitoring is presented.
[0008] Wellbore storage also has a strong influence on the
correlation time-scales. A simple forward calculation that removes
this effect and allows a direct estimation of reservoir scale
permeability is also provided. The techniques described herein are
simpler to practice than conventional interference testing and do
not require the same level of periodicity or uniformity of
pulses.
[0009] In addition, a technique is described that is based on
parameter estimation through forward calculations. No nonlinear
parameter estimation is needed. The method relies on the
correlation of signals at the source and the observer. The
advantage of the method is that it can be performed on a dynamic
basis over a window of data to give a continuous estimate of
permeability in the region between wells. It is insensitive to
nonperiodicity and the strength of the pulses, which makes the
approach practical. The method is robust when multiple wells
operate simultaneously in the region of the observer. Preferably,
the pulsing is carried out in a separable manner.
[0010] In a first embodiment, a method for monitoring
characteristics of a region of earth formation over time is
described comprising: (a) obtaining source data at at least one
location in a source well; (b) obtaining observer data at at least
one location in an observer well, wherein the observer data may be
correlated to the source data; (c) repeating (a) and (b) one or
more times; (d) developing a correlation between the source data
and the observer data as a function of time; and (e) analyzing the
correlation to monitor one or more characteristics of the earth
formation.
[0011] In a second embodiment, a method of monitoring permeability
of a region of earth formation over time is described, comprising:
(a) inducing one or more pressure pulses at at least one location
in at least one source well, wherein data on the induced pressure
pulses is gathered; (b) measuring pressure pulse data at at least
one location in at least one observer well, wherein the pressure
pulse data includes pressure pulses having propagated through at
least a portion of said earth formation; (c) repeating (a) and (b)
one or more times; (d) developing a correlation between said the
induced pressure pulses and the propagated pressure pulses as a
function of time; and (e) analyzing the correlation to monitor the
permeability the region of earth formation.
[0012] The correlations described above can be based on a
propagation function. In some instances, it may be preferable to
analyze the shape of the correlation over time. Furthermore, the
analysis may include determining the time location of the maximum
value of the correlation and analyzing the decay of the maximum
value over time. In some instances, the source data can be pressure
pulses and the observer data will be pressure data. In this
example, the correlation will governed by diffusion, and at least
one of the earth formation characteristics will be permeability. It
is noted that other sensor data may be used and appropriate
corresponding correlations developed.
[0013] In some instances, it may be preferable to use
differentiated pressure pulses from the source and differentiated
pressure data from the observer. Alternatively, source data may be
flow data and observer data may be pressure data.
[0014] Furthermore, (a), (b) and (c) of the above embodiments may
be repeated for more than one observer well or for more than one
source well.
[0015] Also provided are methods for correcting the propagation
function for errors due to storage effects at the source well, for
errors due to the distance between the source and the observer, and
for the width of the pressure pulse.
[0016] Additional advantages and novel features of the invention
will be set forth in the description which follows or may be
learned by those skilled in the art through reading these materials
or practicing the invention. The advantages of the invention may be
achieved through the means recited in the attached claims.
BRIEF DESCRIPTION OF THE FIGURES
[0017] FIG. 1 is a graph showing pressure-pressure correlation-peak
dimensionless time, suitably normalized, with respect to normalized
dimensionless pulse width time; this is applicable for short pulse
widths (.alpha.<.eta..sub.c).
[0018] FIG. 2 is a graph showing dimensionless correlation time for
differentiated pressures, as a function of dimensionless pulse
width time, both normalized to X.sup.2; this is applicable for
large pulse widths (.alpha.>.eta..sub.c).
[0019] FIG. 3 is a graph showing dimensionless correlation time for
differentiated pressures, as a function of dimensionless pulse
width time, normalized to {circumflex over (.eta.)}.sub.c and
X.sup.2, respectively; this is applicable for large pulse widths
(.alpha.>.eta..sub.c).
[0020] FIG. 4 is a graph showing short (.alpha.<.eta..sub.c) and
large (.alpha.>.eta..sub.c) pulse width results for
dimensionless correlation time for differentiated pressures, as a
function of dimensionless pulse width time, normalized to
{circumflex over (.eta.)}.sub.c (X) and X.sup.2 respectively.
[0021] FIGS. 5(a) and 5(b) are graphs showing flowrates in wells 1
(FIG. 5(a)) and 3 (FIG. 5(b)).
[0022] FIG. 6 is a graph showing pressure response corresponding to
flowrates of FIGS. 5(a) and 5(b).
[0023] FIG. 7 is a graph showing pressure derivatives in wells 1,
2, and 3.
[0024] FIG. 8 is a graph showing correlation of pressure
derivatives in wells 1 and 2.
[0025] FIG. 9 is a graph of correlation pressure derivatives in
wells 3 and 2.
[0026] FIGS. 10(a) and 10(b) are diagrams of the hypothetic
reservoirs of Example 1 (FIG. 10(a)) and Example 2 (FIG.
10(b)).
[0027] FIGS. 11(a) and 11(b) are graphs showing pressure derivative
curves when zone 3 is opened for the two (2) reservoir model
scenarios of Example 2.
[0028] FIG. 12 is a graph showing the correlation functions for
zone 3 pressure derivative with zone 2, and zone 1 pressure
derivative for the two scenarios of Example 2.
[0029] FIGS. 13(a) and 13(b) are graphs showing annular pressure
derivatives (pressure differences over 5-minute intervals) for
zones 2 and 3 of Example 3.
[0030] FIGS. 14(a) and 14(b) are graphs showing: (a) the
correlation function of zone 2 and zone 3 pressure derivatives
(Example 3) when zones are opened individually after a shut-in and
(b) a detailed look at early times of FIG. 14(a).
[0031] FIG. 15 is a graph showing the correlation function of zone
2 and zone 3 pressure derivatives of Example 3.
[0032] FIG. 16 is a graph showing the effect of storage on peak
correlation time and its variation with source-observer
distance.
DETAILED DESCRIPTION OF THE INVENTION
[0033] Efficient data processing and interpretation of vast
quantities of data from permanent sensors is necessary to realize
the full value of permanent monitoring. The '674 Application
teaches a method of real-time interpretation using correlation time
from cross-correlation of sensor data. This concept proposes more
robust methods of cross-correlation of permanent sensor data
streams in real time. In addition to tracking correlation time in
such cross-correlations, it is proposed to also monitor the shape
of the correlation function. This shape is essentially
characteristic of the nature of the sensor response to a
disturbance and gives more information about the formation in real
time. The shape of the response together with correlation time also
offers better constraints for forward models. In some cases where
it is difficult to estimate correlation time, the shape parameters
may offer a more robust approach.
[0034] A source well in an infinite nondeformable formation is
assumed. Pressure for a slightly compressible fluid satisfies the
diffusion operator. Therefore, 1 p t = D p ( 1 )
[0035] The diffusion constant D is 2 k c
[0036] where c is the fluid compressibility, .mu. is the shear
coefficient of viscosity, .phi. is the porosity of the porous
medium, k is the permeability of the porous medium, p is pressure,
and t is time. The properties of the porous medium are assumed to
be representative homogenized constants.
[0037] As is known in the art, it can be assumed that the source of
the flux is a line. Here pressure is computed at the wellbore
radius of the source r.sub.w (see Raghavan (1993)). If one assumes
a fully-penetrating line-source in an infinite, cylindrical medium,
the governing equation for pressure in terms of t and the radial
distance r from the source reduces to 3 p t = D 1 r r [ r p r ] ( 2
)
[0038] At a distance r.sub.o from this source well, there is an
observation line, the pressure of which is labeled as p.sub.o.
Below, the case of an ideal pulse at the source and the resulting
correlation with the pressure at the observer is considered.
[0039] Ideal pulse. Given a flow pulse of magnitude Q at time t=0,
the boundary condition at the source is 4 lim r -> 0 - 2 rh k p
r = Q ( t ) , ( 3 )
[0040] where h is the formation layer thickness and .delta. has the
usual meaning of a Dirac delta functional. A positive flow rate is
used to denote injection.
[0041] With the restriction that the pressure must decay to its
initial value of zero when r approaches infinity, the solution at
the source, p.sub.s, is 5 p s ( t ) = Q 4 kh 1 t exp [ - r w 2 4 Dt
] H ( t ) ( 4 )
[0042] where H(t) is the Heaviside function, D is the pressure
diffusivity, the subscript s represents the source, and the
subscript w represents the wellbore. At the observation point, the
pressure is 6 p o ( t ) = Q 4 kh 1 t exp [ - r o 2 4 Dt ] H ( t ) (
5 )
[0043] where r.sub.o is the radius of the observer and p.sub.o is
the pressure of the observer (the subscript o represents the
observer). A correlation function between the source and the
observer is constructed. For the ideal case of an impulse source,
the correlation may be carried out between the two pressures. [It
is noted that while pressure measurements are used (with a
diffusivity propagation function), other sensors may be used with
different propagating functions. For example, acoustic measurements
may be used with an acoustic wave equation.] The correlation
function C of p.sub.s and p.sub.o (which depends on t) is defined
to be (see Press et al., Numerical Recipes in FORTRAN, Cambridge
University Press (1992), incorporated by reference in their
entireties.) 7 C ( p s , p o ) ( t ) = - .infin. .infin. p o ( t +
) p s ( ) t ( 6 )
[0044] where .tau. is a dummy variable. The peak in the correlation
function with respect to t gives the characteristic time scale
t.sub.c (where the subscript c represents the characteristic or
correlation) which in turn is related to interwell permeability.
For the purpose of identifying parameters, it is useful to define
dimensionless groups. First, the characteristic diffusion time
scale T is based on the length scale r.sub.w so that
T=.phi..mu.cr.sub.w.sup.2/k, where, T is the diffusion
(representative time), and c is the compressibility of the fluid.
The characteristic pressure scale is given by Q.mu./(2.pi.kh).
Differentiating the function C with respect to t, defining
4D.tau./r.sub.w.sup.2=4.tau./T as .xi., and
4Dt.sub.c/r.sub.w.sup.2=4t.su- b.c/T as .eta..sub.c, the value of
.eta. (=4Dt/r.sub.w.sup.2 ), for which the correlation function
becomes a maximum, is obtained by satisfying: 8 0 .infin. 1 ( c + )
exp [ - X 2 ( c + ) ] exp [ - 1 ] { X 2 ( c + ) 2 - 1 c + } = 0 ( 7
)
[0045] where X is the dimensionless ratio of distances r.sub.o to
r.sub.w (that is, the ratio of the observer to source radii) and
.xi. is a dimensionless dummy value. Eq. 7 enables the calculation
of .eta..sub.c. Here, {circumflex over (.eta.)}.sub.c (X) is the
function that gives .eta..sub.c when the source is an impulse;
accordingly, .eta..sub.c depends only on X. In practice, rather
than solving for .eta..sub.c with this integral equation, a maximum
of the function 9 ( ; X ) = 0 .infin. 1 ( + ) 1 exp [ - X 2 + ] exp
[ - 1 ] ( 8 )
[0046] is determined with respect to .eta., for a given X. This is
equivalent to maximizing the correlation function of Eq. 6 with
respect to t.
[0047] The results for .eta..sub.c for various values of X are
provided in Table 1 below.
1TABLE 1 10 Correction Factors , 1 X 2 ^ c ( X ) X
.eta..sub.c/X.sup.2 2 0.4423 4 0.6563 6 0.7310 8 0.7702 10 0.7948
20 0.8489 40 0.8820 100 0.9092 200 0.9229 400 0.9328 1000 0.9427
2000 0.9485 4000 0.9531
[0048] Ideally (and as described in the '674 Application), it is
expected that .eta. should equal X.sup.2, i.e.,
t.sub.c=r.sub.o.sup.2/(4D). For small X, {circumflex over
(.eta.)}.sub.c deviates from X.sup.2 and is less than unity; but
for practical values of X, .eta..sub.c/X.sup.2 approaches unity.
Under any case, the correction factor is known a priori. The result
for .eta..sub.c is the basis for interpretation via forward
calculation. An impulse flow is induced at a source well, observe
the pressure at both the source and a monitoring well. With a
suitable time window, the correlation function is computed,
preferably by FFT, and the result scanned. The location of the
function maximum gives t.sub.c, obtaining 11 D = 1 4 c r w 2 t c =
1 4 ^ c ( X ) r w 2 t c . ( 9 )
[0049] Thus, using a noniterative method, an estimate for the
diffusion constant may be made in real-time with occasional
pulsing. This is translated to permeability using
k=.phi..mu.cD, (10)
[0050] assuming that .phi..mu.c is known. If the pulse is
sufficiently ideal (although it may be of finite width), one may
look at the magnitude of the correlation function to estimate other
properties.
[0051] Since .eta..sub.c grows as X.sup.2, it is convenient to
consider an O (1) quantity, u.sub..eta..sub..sub.c, a normalized
quantity:
u.sub..eta..sub..sub.c=.eta..sub.c/X.sup.2. (11)
[0052] While the idea of an impulse flow rate is useful for
understanding the concept behind continuous interpretation,
practical constraints allow the implementation of only a finite
amplitude pulse for a nonzero time duration. Again, an ideal finite
pulse is considered, i.e., the reservoir is quiescent, except for
this pulse of a rectangular shape.
[0053] Finite width pulsing. For a pulse of magnitude Q, but with a
pulse width in time equal to pulse width (.nu.), the flow rate at
the source fixes the wellbore boundary condition may be written as
12 - 2 rh k p r = Q v [ H ( t ) - H ( t - v ) ] ( 12 )
[0054] where H(t) is the Heaviside function. Accordingly, the
pressure responses at the source well is 13 p s ( t ) = Q 4 kh 1 v
{ E 1 ( r w 2 4 Dt ) H ( t ) - E 1 ( r w 2 4 D ( t - v ) ) H ( t -
v ) } ( 13 )
[0055] where E.sub.1 is the exponential integral. The corresponding
observer pressure is 14 p o ( t ) = Q 4 kh 1 v { E 1 ( r o 2 4 Dt )
H ( t ) - E 1 ( r o 2 4 D ( t - v ) ) H ( t - v ) } ( 14 )
[0056] For the finite pulse, a correlation function C(p.sub.s,
p.sub.o)(t) as in Eq. 6 is constructed. Unlike the pulse problem,
the functional form here is different depending upon the location
of the maximum of C. If the wells are sufficiently far apart the
correlation function maximum occurs at a time larger than .nu.. For
t>.nu., the following correlation may be derived 15 C ( p s , p
o ) ( t ) = ( Q 4 kh ) 2 1 v 0 .infin. E 1 ( 1 ) [ 2 E 1 ( X 2 + )
- E 1 ( X 2 + + ) - E 1 ( X 2 + + ) ] ( 15 )
[0057] where .alpha., a dimensionless pulse width, is equal to 4D
.nu.v/r.sub.w.sup.2. Note that .eta. is a scaled form of t, and
since t>.nu., .eta.>.alpha.. Clearly, as .alpha..fwdarw.0,
the results should approach those of the impulse correlation
provided above.
[0058] When the wells are closely spaced, the characteristic
correlation time becomes shorter than any practical pulse width
time. Then, as per the above procedure, a different result given by
16 C ( p s , p o ) ( t ) = ( Q 4 kh ) 2 1 v 0 .infin. E 1 ( 1 ) { 2
E 1 ( X 2 + ) - E 1 ( X 2 + + ) - E 1 ( X 2 ) ( 1 + - ) } ( 16
)
[0059] is obtained. This is applicable for t<.nu. or
.eta.<.alpha..
[0060] Numerical evaluation of the analysis. The location
(.eta..sub.c or equivalently t.sub.c) of the peak of the
correlation function has been computed for the above results.
Unlike the pulse correlation, where .eta..sub.c depends only on X,
here .eta..sub.c for which C is a maximum will depend on both
.alpha. and X. Furthermore, computations of the quadrature with the
exponential integrals in the integrand is limited by the numerical
accuracy that may be achieved, especially for small values of
.alpha.. Therefore, for .alpha..fwdarw.0, the values generated by
the correlations obtained with the impulse responses are used.
[0061] An interesting feature of the computed results for
.eta..sub.c is that a normalized plot virtually collapses the
dependency of .eta..sub.c on two parameters .alpha. and X to a
single curve. In FIG. 1, the y-axis is the ratio of .eta..sub.c to
{circumflex over (.eta.)}.sub.c (X), i.e., the ratio of .eta..sub.c
when a finite pulse is used to that of an impulse result. The ratio
.alpha./X.sup.2 is on the x-axis. Thus, as the x-axis goes to zero
the disturbance is akin to an impulse and the ordinate should
approach unity. Regardless of the value of X, all of the curves
seem to lie within a narrow band, leading to an almost universal
correction for the correlation-peak time. Knowing the width of the
pulse, and the distance to the source, a simple algorithm for
calculating the diffusivity of the formation, and hence the
permeability, can be constructed.
[0062] The algorithm for the calculation is simple. Knowing the
width of the pulse, start with a value for D from Eq. 9 assuming
.eta..sub.c={circumflex over (.eta.)}.sub.c (X) and compute .alpha.
and .alpha./X.sup.2. From FIG. 1, or equivalently from an algebraic
form of the figure, .eta..sub.c can be evaluated. From the known
value of t.sub.c obtained by correlating the appropriate data
windows, .eta..sub.c is converted to D using
D=.eta..sub.cr.sub.w.sup.2/(4t.sub.c). The process is continued
until D is satisfactorily converged.
[0063] The numerical results above have been obtained by
considering the case of .alpha.<.eta..sub.c, i.e., the
correlation-peak time is larger than the width of the pulse. For
the second case of the large pulse widths, however, the
differentiated data is correlated as provided below.
[0064] Differentiated data correlation. The pressure at the source
and observer is differentiated with respect to time and a
correlation function is constructed. For this correlation, the
responses due to a finite pulse are used. The idea is that for a
finite pulse of a sufficiently large width, the correlation of
differentiated results will be the same as the one with impulse
responses and the correlation function peak should approach the
value of {circumflex over (.eta.)}.sub.c (X). The applicability of
this for pulses of finite width is examined here.
[0065] The pressure derivative for the source, {dot over (p)}.sub.s
(t), is 17 p s ( t ) t = Q 4 kh 1 v [ H ( t ) t exp ( - r w 2 4 Dt
) - H ( t - v ) t - v exp ( - r w 2 4 D ( t - v ) ) ] ( 17 )
[0066] The pressure derivative for the observer {dot over
(p)}.sub.o (t) is 18 p o ( t ) t = Q 4 kh 1 v [ H ( t ) t exp ( - r
o 2 4 Dt ) - H ( t - v ) t - v exp ( - r o 2 4 D ( t - v ) ) ] ( 18
)
[0067] The correlation of differentiated pressures is 19 C ( p . s
, p . o ) ( t ) = - .infin. .infin. p . o ( t + ) p . s ( ) ( 19
)
[0068] With .alpha., .xi. and .eta. defined as before, substituting
Eq. 17 and 18 in Eq. 19 gives for .alpha.<.eta. (short pulse) 20
C = ( Q 4 kh ) 2 1 v [ 0 .infin. - 1 { 2 exp ( - X 2 + ) + - exp (
- X 2 + + ) + + - exp ( - X 2 + - ) + - } ] . ( 20 )
[0069] For .alpha.>.eta., with differentiated pressures, the
responses are close to a pure impulse response. Therefore, when
.eta..sub.c is expected to be smaller than .alpha. (approximately
X.sup.2 small compared to .alpha.), the differentiated pressure
correlation may be considered. Small .alpha. values are less
feasible for reasonable source-observer distance, and therefore
.alpha.>.eta. will be needed for practical applications. The
expression for the correlation function with .alpha.>.eta. is 21
C ( p . s , p . o ) ( t ) = ( Q 4 kh ) 2 1 v [ 0 .infin. - 1 { 2
exp ( - X 2 + ) + - exp ( - X 2 + + ) + + } - exp ( - X 2 - 1 + - )
+ - ] ( 21 )
[0070] Here for .alpha..fwdarw..infin., .eta..sub.c={circumflex
over (.eta.)}.sub.c(X).
[0071] To obtain .eta..sub.c, as before, a maximum in C with Eq. 20
or Eq. 21 is determined. One has to exercise caution that the peak
value satisfies the restriction of .alpha.>.eta..sub.c or
.eta..sub.c>.alpha., whatever the case may be. For the above
correlation integrals, the numerical results for the case of
.alpha.>.eta..sub.c are shown in FIG. 2. In this figure,
.alpha./X.sup.2 is shown on the abscissa; the ordinate is
.eta..sub.c/X.sup.2, which may be expected to approach {circumflex
over (.eta.)}.sub.c (X)/X.sup.2, for large values of .alpha.. This
indeed it does, except for the inevitable numerical errors. This
point is further amplified in FIG. 3 where all of the .eta..sub.c
curves collapse into essentially one curve when the ordinate is
normalized to {circumflex over (.eta.)}.sub.c (X). The collapse is
almost perfect for X>20, down to values of .alpha./X.sup.2=1
even with an exaggerated ordinate.
[0072] In general, for large values of .alpha., the ratio
.eta..sub.c/{circumflex over (.eta.)}.sub.c (X) is close to unity
for the entire range of .alpha.. This allows a fast computation of
the formation diffusivity. For practical purposes, it probably
suffices to use .eta..sub.c={circumflex over (.eta.)}.sub.c (X).
Also, .eta..sub.c is seen to be nonmonotonic, reaching a peak at
about an abscissa value of 2. If one wishes to be more accurate,
but nevertheless use a relatively fast computation algorithm, the
universal curve result of FIG. 3 may be used. Diffusivity may be
first computed using .eta..sub.c={circumflex over (.eta.)}.sub.c
(X), and then .alpha./X.sup.2 calculated. This will allow for a
correction to .eta..sub.c and, given the weak dependence of
.eta..sub.c/[{circumflex over (.eta.)}.sub.c (X)] on
.alpha./X.sup.2, a quick convergence is expected.
[0073] The differentiated correlation case provides useful results
for .alpha.>.eta..sub.c. Contrarily, for .alpha.<.eta..sub.c,
C({dot over (p)}.sub.s, {dot over (p)}.sub.o)(t), is not
particularly attractive to apply. To illustrate this, the numerical
results corresponding to X=40 and X=100 are shown in FIG. 4; large
pulse result starts approximately at .alpha./X.sup.2=0.819 for X=40
and 0.874 for X=100. One can see a fairly large dependency on
.alpha. when .alpha.<.eta..sub.c, but there is a fairly smooth
overlap between large (.alpha.>.eta..sub.c) and small
(.alpha.<.eta..sub.c) .alpha. results. The correlation peak when
normalized as .eta..sub.c/{circumflex over (.eta.)}.sub.c (X)
overlap well for different values of X, when plotted against
.alpha./X.sup.2, a conclusion similar to that discussed above. As
.alpha. becomes large, but still constrained by
.alpha.<.eta..sub.c, .eta..sub.c approaches the case for
.alpha.>.eta..sub.c. But as .alpha. becomes small, there is
rapid decrease in .eta..sub.c to roughly half the magnitude.
Although for this case C(p.sub.s, p.sub.o) is preferred, C({dot
over (p)}.sub.s, {dot over (p)}.sub.o) may also be used.
[0074] To summarize, for large pulse widths, it is preferable to
correlate the differentiated pressure signals and compute the
characteristic correlation time. It is preferred that this time be
smaller than the pulse width time for the correlation result to be
applicable. Further, when the characteristic pulse width is small,
a direct pressure correlation is preferable.
[0075] Flow-pressure correlation. The underlying concept for the
correlation is readily illustrated with a flow-pressure correlation
where analytical results are easier to derive. Consider an impulse
in flow rate of magnitude Q at the source. As before, the observer
pressure is given by Eq. 5. The correlation of the impulse flow
rate and the pressure is 22 C ( q , p o ) ( t ) = Q 2 4 k h 0
.infin. ( ) t + exp [ - r o 2 4 D ( t + ) ] H ( t + ) ( 22 )
[0076] C becomes a maximum for t=t.sub.c, where 23 t c = r o 2 4 D
( 23 )
[0077] While this result suggests that it alone is sufficient to
estimate the diffusivity of the formation, practical considerations
may make pressure based correlations preferable. Noise in flow rate
measurements may be large, and if the pulse width is also large,
one is forced to deal with differentiated flow/pressure
correlations. Differentiation of noisy data is difficult. Where
flow rate measurements are available and are of good quality, both
forms of correlations may be considered as an added consistency
check.
[0078] Numerical Illustrations. Thus far, the calculations have
been with a single source. Numerical simulations have been carried
out to verify that the correlation concept works when production
occurs from several wells that may interfere with the deliberate
pulsing. In particular, the influence of production noise in the
correlation function is addressed. The numerics compute pressures
in arbitrarily distributed observation/sink lines fully penetrating
an infinite reservoir. If the response function (G) for pressure in
well i due to a unit flow rate in well j is G.sub.ij, then for all
practical purposes the result may be superimposed to write 24 p i (
t ) = j 0 t G ij ( t - ) q j ( ) ( 24 )
[0079] The response function G.sub.ij is 25 G ij ( t ) = 4 k h E 1
( r ij 2 4 kt ) ( 25 )
[0080] where r.sub.ij is the distance from well i to j. (This
solution ignores the requirement of uniform pressure within the
distant wells, i.e., all wells are line sources/observers.)
Obviously r.sub.ij for i.noteq.j is much larger than r.sub.w. For
i=j, r.sub.ij=r.sub.w. In the computational algorithm, each source
accepts pulses of widths that may be specified and the magnitude of
the pulses may also be varied. Random fluctuations in flow rates
are allowed in addition to the imposed steps. In experimentation
with numerically generated data, it was found that the pressure
responses as a result of small random fluctuations show no
discernible peaks in the correlation function. Although
reinforcement of correlation through random fluctuations in flow
rates approach is appealing, if the transient time for diffusion is
much larger than the time scale of the small fluctuations, then the
time signature of the random fluctuations is essentially lost at
the remote points, and it is unlikely that no measurable
correlation results.
[0081] Unlike conventional pulse testing where a periodic sequence
is imposed, the present technique is quite flexible. Furthermore,
the interpretation is not weakened by relying on discerning peaks
in pressure signals--often corrupted rather strongly by extraneous
noise or storage. In addition, production fluctuations in other
wells may break the periodicity required in standard pulse-based
interference testing. Because the correlation technique analyzes a
large window of data, it is not strongly affected by small amounts
of unintended flow fluctuations. In fact, these only tend to
reinforce the correlation peak, though marginally. Finally, the
computation presented herein is rapid, through the use of FFTs.
[0082] Preferably, the correlation methods are carried out with
respect to pulses in flow rates. Preferably, the pulses are short
with respect to overall production schedules, but larger than the
characteristic correlation times. With a pulse of sufficiently
large width, the correlation is carried out with the differentiated
data. These types of numerical illustrations are shown in the '674
Application.
[0083] Shape Correlation. To describe the concept in detail,
pressure data streams are used as an example. The '674 Application
describes a method for pressure-pressure derivative correlation.
Pressure transients are created by various means including flow
rate pulsing. Correlation of derivatives of these pressure signals
gives a time lag between the stimulus at one location and the
response at another location. For a radially infinite reservoir
with low compressibility fluid and negligible wellbore storage,
this time lag or pressure diffusion time is related to the porosity
(.phi.), fluid compressibility (c) and viscosity (.mu.) and the
distance (r) between the two measurement points (as seen above): 26
t c c r 2 4 k ( 26 )
[0084] If all other properties are known, then the correlation time
may be used to estimate the formation permeability. Such
correlations can be automated and used in real-time at time scales
matching data acquisition frequencies. A trend or change in this
time would indicate change in formation property with time.
[0085] This was implemented on pressure data streams from an
injection experiment (see Raghuraman et al. (2001)) by
cross-correlating pressure derivative signals from an injection
well and an observation well. The experiment satisfied the
assumptions made in derivation of Eq. 26 and was used to infer the
presence of a fracture in the interwell region. Further, changes in
the correlation time over the one-year course of the experiment
were an indicator of change in the fracture characteristic. This
parameter was also used to constrain forward models. For more
complex reservoir geometries where the equation is not strictly
valid, the correlation time cannot be used to directly get an
estimate of formation properties but is indicative of some average
values. However, this parameter can be tracked to monitor changes
in formation properties in that region and also used to constrain
more detailed reservoir models (see Bryant et al., "Real-Time
Monitoring and Control of Water Influx to a Horizontal Well Using
Advanced Completion Equipped With Permanent Sensors," SPE 77525,
Sept. 29-Oct. 2, 2002, SPE Annual Technical Conference and
Exhibition, San Antonio, incorporated herein by reference in its
entirety).
[0086] The '674 Application observes that under certain conditions
(for example, when the wells are far apart or when the pressure
diffusivity is low), the correlation function may be broad and
diffused and a peak may not be easily identifiable. Also, in some
instances, the reverse may happen when the wells are very close and
the formation pressure diffusivity is very high. In such cases, the
correlation time may be very small and subject to large errors
depending on data acquisition interval, time synchronization errors
between different sensors, and numerical processing
(differentiation, end point effects etc.) of data. In such
instances the correlation time may not be the most robust parameter
to track.
[0087] As provided here, when cross-correlating sensor (or sensor
derivative) data, in addition to tracking correlation times, are
gathered, the shape of the correlation function curve is also
monitored. The shape of the correlation function curve together
with correlation time can be used to track changes in the formation
with time as well as to constrain the full-scale reservoir model
more robustly. The shape of the correlation function curve also
yields information about formation heterogeneity that is not
obtainable if correlation time alone were to be used. The details
of the concept are further explained using the examples provided
below (see Examples 2 and 3 below).
[0088] Field Example: A procedure based on time-lag on pressure
derivatives for a practical field experiment is applied. Pressure
data analysis was a part of an electrical array installation
project, which was implemented in Mansfield sandstone reservoir in
Indiana, onshore U.S.A. as described by Bryant et al., "Utility and
reliability of cemented resistivity sensors to monitoring
waterflood of the Mansfield sandstone, Indiana," SPE paper 71710 at
the SPE Annual Technical Conference and Exhibition, New Orleans
(2001), incorporated herein by reference in its entirety. An
experiment in this project included continuous interference testing
between an injection well at the center of an inverted five-spot,
surrounded by four production wells. A monitoring well was located
midway along one of the diagonals, i.e., between the injector and a
producer. The distance from the injector to the monitoring well was
71 m. Although the primary purpose of the experiment was to monitor
electrode responses in cemented arrays, the monitoring well also
had a cemented pressure gauge connected to the formation. The
injection pressure was also continuously recorded. This facilitated
the pressure derivative correlation, discussed by Raghuraman et al.
(2001). Start-up and shut-in pressure signal derivatives were
correlated which resulted in values of 28.2 ks and 21.6 ks,
yielding a permeability of a few orders of magnitude larger than
the expected matrix permeability. A fracture(s) connecting the
injection and monitoring well was hypothesized and was
independently corroborated by resistivity data. The changing
correlation times also indicate the nonlinearity of the fracture
with respect to fluid pressures (see Raghuraman et al. (2001)). At
the conclusion of the experiment, a lapse of eight months, the
correlation time further dropped to 3 ks, indicative of progressive
transmissibility enhancement. The correlation method proved to be
particularly powerful for identifying such formation property
changes.
EXAMPLE 1
[0089] In this example, there are three wells that are colinearly
located as shown in the '671 Patent and in FIG. 10(a). One of them,
well 2, is an observation well. Wells 1 and 3 are active wells, 40
m apart, and well 2 is a passive or observation well. Wells 1 and 2
are 40 m apart, whereas well 2 and 3 are 80 m apart. The
permeability of the formation is 0.1 .mu.m.sup.2 (.apprxeq.100 mD).
The porosity is 0.25, the viscosity is 0.001 kgm.sup.-1s.sup.-1,
and the compressibility is 4.times.10.sup.-9 m.sup.2N.sup.-1. All
the trial calculations included an initial step on which were
superimposed finite width pulses. For this example, the
interpretation steps are the following.
[0090] Pulse active wells 1 and 3 with respect to a background
rate.
[0091] Ensure that the pulses of each active well are sufficiently
separated from those of others and the pulses are wide enough to be
larger than the correlation time expected.
[0092] Create a window of data, preferably containing 2N (where N
is a natural number) points, surrounding the pulse(s) in the wells
of interest, with the time synchronized for all of the wells.
[0093] Differentiate the pressure data with respect to time.
[0094] Correlate the differentiated data between wells 1 and 2. A
similar exercise is carried out between wells 3 and 2, when well 3
is pulsed. Avoid correlation of data between the active wells, 1
and 3.
[0095] Locate the peak in correlation function. Convert this
according to Table 1 above for very large pulse width. When .alpha.
influences .eta..sub.c (only for small .alpha., an unlikely
possibility), the computation is iterative and FIG. 2 results are
necessary.
[0096] If the pulse width happens to be small compared to the
maximum in correlation function, correlate the pressures as opposed
to the derivatives, and use FIG. 1.
[0097] The calculations shown below are carried out with a 2% noise
in rates. The flow rates in wells 1 and 3 are shown in FIGS. 5(a)
and 5(b), respectively. There are 2048 discrete data points in this
figure, with the time axis in units of 100 s. The corresponding
pressure response for all three wells is shown in FIG. 6, where
well 1 is shown as line A, well 2 is shown as line C and well 3 is
shown as line B. The derivatives of pressures are in FIG. 7.
[0098] From the pressure plots, and as expected, the behavior of
the observation well is sluggish compared to the active ones. Also,
well 3 pulses are a larger step than well 1 because of the distance
between the observation point and well 3 (the larger the distance,
the less likely the peak in the propagating diffused pressure
pulse). The response characteristics of all three wells are more
readily seen in FIG. 7. In this figure, well 1 (peaks) are
designated by letter A; well 3 (peaks) are designated by letter B.
(It is noted that one peak represents overlapping data from both
wells 1 and 3.) Horizontal line C represents overlapping data from
both wells 1 and 3. Data from well 2 is represented by line D.
While the responses in the observation well to a disturbance is
clearly seen, the nature of the response is more sluggish when the
disturbance is made in the more distant well. The smearing of the
response, the characteristic width of which is also related to the
diffusion time may ultimately be comparable to the noise, at which
point the correlation technique fails to be robust, unless repeated
pulses are made to overcome the random noise. Also, the so-called
lag time is not clearly discernible with respect to well 3.
Therefore, methods based on lag time detection may be
unsatisfactory. But the correlation technique is reliable and gives
answers within close proximity of the true values, even when other
wells influence the pressure in the observation wells. The main
requirement is that discernible (and preferably) isolated pulses be
created at the source, when the property between the source and the
observer is desired.
[0099] The results of the correlation calculation between pressure
derivatives in wells 1 and 2 are shown in FIG. 8. The location of
the peak in this plot is at 3200 s (ignoring local fluctuations).
This is the correlation time t.sub.c. In carrying out this
calculation, it is assumed that the pulsing at well 3 would have a
negligible effect because well 1 dominates the well 2 response.
Because the pulse widths are sufficiently large compared to the
calculated t.sub.c, it is assumed that .eta..sub.c should be close
to {circumflex over (.eta.)}.sub.c (X). Then, an estimate of the
permeability between the two wells may be obtained from (X=400) 27
k = 0.9328 c X 2 r w 2 4 t c ( 27 )
[0100] which gives 0.093 .mu.m.sup.2, sufficiently close to the
original value. When the absolute peak is chosen in this example, a
permeability of 85 mD is obtained, a good extimate. Interestingly,
if .eta..sub.c=X.sup.2 was used in this example, the original
permeability would have been recovered with a t.sub.c=3200 s. The
nature of the pulsing and the effect of the noise and the
variations induced by well 3 probably caused this discrepancy.
Also, note that the resolution on the time axis is limited to 100
s.
[0101] For wells 2 and 3, the analysis is not as straight forward.
If the entire data is considered, the peak in the correlation
function is no longer reflective of the formation property. The
distance between these two wells is 80 m and the interaction signal
is dwarfed compared to the one between wells 1 and 2. For example,
consider a pulse imposed by well 1 with no change in well 3. Then,
both wells 2 and 3 experience a response corresponding to a
distance of 40 m. These two signal changes are essentially the same
in both the wells and will therefore have zero time displacement.
In turn, a correlation of pressure derivatives will indicate
infinite permeability between wells 3 and 2. Thus, interaction
between distant wells will be adversely affected by more dominant
intermediate wells. This could be circumvented by looking at a
targeted window correlation function calculation. For example, in
this case, a window around 10.sup.5 s where well 1 has no pulse
could be chosen. A correlation function based on a window between
68000 s and 119100 s is shown in FIG. 9. The correlation function
peak at 12300 s is in agreement with the diffusion time scale of
12800 s, yielding k=0.098 .mu.m.sup.2, quite close to the true
permeability. The small effect of finite .alpha. is ignored here in
estimating k. Thus, any automated pulsing sequence and windowing
should be implemented so that the observation-active well
interaction is the dominant one.
EXAMPLE 2
[0102] FIG. 10(b) shows the XZ cross-section of the hypothetical
reservoir used in this example. The reservoir is 3800 ft long, 2250
feet wide and 60 ft thick. The pressure support comes from an
aquifer connected to the XZ plane on the end of the reservoir, 1350
ft away from the well. The formation porosity is 15%. The viscosity
of the water and oil are 0.5 cP and 20 cP, respectively. The 600 ft
long horizontal producer is divided into 3 hydraulically isolated
zones, 160 ft, 180 ft and 180 ft long respectively by packers.
Within each zone, the flow can be independently controlled by a
variable choke valve. The valve controls the flow from each annular
zone into the common tubing. All zones are initially shut. A
numerical simulator (Eclipse) was used to simulate a pressure
transient created by opening zone 3 valve.
[0103] FIGS. 11(a) and 11(b) show pressure derivative responses of
zone 1 (line A) and zone 2 (line B) when zone 3 (line C) is opened
for two different reservoir model scenarios. The vertical
permeability is 50 mD in both cases, however the horizontal
permeability varies by a factor of 10. The response times and shape
of the curve vary both with formation permeability and the distance
from the pressure transient source. For zone 1 which is further
away, the response is smaller and more diffused. For zone 3, the
response is sharper and bigger. For both zones, the response is
faster and sharper when the horizontal permeability is increased by
a factor of 10. Thus, both the response time and the shape of the
response curve give information on the formation region between the
source and response locations.
[0104] FIG. 12 plots the correlation function for the pressure
derivative of zone 3 with zones 1 and 2 for the two model scenarios
(scenarios A and B). The shapes of the correlation function curves
capture the different responses shown in FIGS. 11(a) and 11(b). For
the zone 3-zone 2 pressure derivative correlation with scenario 2
(line 2B), the correlation time is determined to be between zero to
0.25 hours. The pressure data is computed every 0.25 hours. This,
together with any numerical errors in data processing of discrete
pressure data sets, can cause large error bars for the correlation
time. For the pressure derivative correlation between zones 1 and 3
for scenario 1 (line 1A), the correlation function is diffused with
no sharp peak identifiable. The correlation time is estimated to be
in the range of 13.6-15.1 hours. These problems become more severe
if there is random noise in the data. However, in such cases one
could track the formation more robustly, if one were to track the
shape of the curves as well. One way to parameterize the shape is
to track the time at which the correlation function falls to a
certain fraction of its peak value.
[0105] Table 2 below summarizes the correlation time and in
addition the times at which the correlation function falls to 75%,
50% and 25% of its maximum peak height. For the zone 2 and 3
correlation for scenario 2, it is seen that the percentage error
becomes lesser when looking for a time where the peak drops to 75%
or 50% of its value (see Table 2). Note, that the assumptions used
in derivation of Eq. 26 are no longer valid here because of the
complex geometry as well as different mobilities of the oil and
water. However, the correlation times and the other time parameters
derived are still a function of oil and water mobility,
compressibility and horizontal and vertical permeability. Hence,
they will still afford a means of tracking changes in these
quantities over time. They can also be used to constrain history
matching with full-scale models.
2 TABLE 2 Time (hrs) Scenario Peak 75% 50% 25% 1. kx = ky = 50 mD;
3.25 8.24 12.98 27.82 kz = 50 mD 2. kx = ky = 500 mD; 0-0.25 1.37
2.67 4.87 kz = 50 mD
EXAMPLE 3
[0106] This example illustrates the utility of this approach using
real pressure data from a producing well. Details of the well and
the reservoir are given in Bryant et al. (2002). FIGS. 13(a) and
13(b) show that the pressure derivative response of zone 2 to zone
3 opening is very different from the reverse case of zone 3
response to zone 2 opening. FIG. 13(a) shows the zone 2 response
when zone 3 was opened after a shut-in. This is smaller and
diffused as compared to the zone 3 response to opening after a
shut-in shown in FIG. 13(b).
[0107] FIG. 14(a) shows the correlation function for the pressure
derivatives in these two zones for the two tests. FIG. 14(b) shows
a more detailed view of FIG. 14(a) at early times. (The arrows
indicate correlation times for the two tests.)
[0108] Both plots show a peak at time 0 min which corresponds to
relaxation of a hydraulic fluid filled rubber packer that is
isolating the two zones. The data acquisition interval here is 1
minute. The correlation peak for the zone 3 response (line A) is at
10 min. It is harder to identify the peak for the zone 2 response
(line B). The first big peak however is at 9 minutes very close in
value to the zone 3 response peak. This is to be expected as the
path connecting the two zones is the same in either test and the
pressure travel time should be equal. However, it is very evident
that the response of zone 3 to zone 2 opening is very sharp, while
that of zone 2 is very broad and diffuse.
[0109] The difference in the shape indicates that the formation is
changing in some way depending on the source of the pressure pulse.
The correlation time alone would not reveal this information. The
shape together with the correlation time is a much better
constraint for the forward model. FIG. 15 shows the correlation
plots when the frequency of data points is reduced to 1 in 5
minutes. In this figure, line A represents zone 3 opened and line B
represents zone 2 opened. The correlation plot is much less noisier
but correlation time accuracy is lost because of lower time
resolution. The correlation peak is at 10 minutes for both (note
that error in estimating correlation time is of the order of +/-5
min here as compared to +/-1 min in FIG. 14). Note also that the
peak at zero time corresponding to packer relaxation is not there
in FIG. 15 again due to loss in data frequency.
[0110] Effect of Storage. Much of the above discussion assumes an
ideal pulse shape, i.e., a rapid rise to a fixed flow rate and a
reduction at a later time. When such a flow rate is imposed at the
surface, wellbore storage will slow the rise to the final rate. To
take this into account, the conservation equation within the well
is combined with that of the formation. The prescription for the
correlation time involves an additional parameter related to the
storage equations.
[0111] For the discussion below, it is assume that a cemented
observation well is used so no storage at the observation point
needs to be considered.
[0112] The pressure equation for the formation remains the same as
Eq. 2. Conservation of fluid mass within the wellbore at the source
gives (see Raghavan (1970) and van Everdingen et al., "The
application of the laplace transformation to flow problems in
reservoirs," Trans. AIME (1949), volume 186, pages 305-324,
incorporated herein by reference in its entirety) 28 - lim r ->
0 2 r h k p r = Q ( t ) - V w c w p s t ( 28 )
[0113] here the variables with subscript w denote wellbore values
of the source. Here, the wellbore fluid may have a different
compressibility c.sub.w than that of the formation fluid. The
coefficient 29 V w c w 2 r w 2 h c
[0114] is the dimensionless storage constant (.beta.). This may be
arrived at by choosing a characteristic distance of r.sub.w and a
time-scale of 30 c r w 2 k .
[0115] Whenever the dimensionless time is large compared to .beta.,
storage effects become unimportant for the flow rate establishment
at the source well. Given that the characteristic dimensionless
time for correlation between the wells is X.sup.2 (note that
t.sub.c/T is actually .apprxeq.X.sup.2/4), in the absence of
storage, for .beta.<<X.sup.2, the influence of storage in
altering the correlation time is likely to be small. For practical
situations, this is satisfied even for an observer at a distance of
20 m. In terms of dimensionless quantities such as .eta..sub.c, the
problem is simply governed by two parameters, X and .beta.. The
effect of .alpha. is assumed to be negligible for .alpha.
sufficiently large compared to .eta..sub.c.
[0116] The forward problem for computing pressure transients is
easy to solve: one combines the wellbore boundary condition of Eq.
28 with the diffusion equation. But a scheme for providing a
real-time interpretation of permanent monitoring results is
developed. A simple prescription for correcting for the effect that
storage on the correlation time is therefore desired.
[0117] Real-time interpretation with storage. In its most
rudimentary form, the present approach assumes that the effect of
wellbore storage may be adequately represented by an exponential
approach to the surface rates. That is, given an imposed rate of
q(t)=H(t) at the surface, it is assumed that a good approximation
to the ratio of the bottom rate to the surface rate is (see
Kuchuck, "A new method for determination of reservoir pressure,"
SPE paper 56418 at the SPE Annual Technical Conference and
Exhibition, Houston, Tex. (1999), incorporated herein by reference
in its entirety) 31 q b ( t ) q ( t ) = ( 1 - exp [ - t T s ] ) (
29 )
[0118] Obviously, the true bottom rate is not accurately
represented by this form because the solutions to the differential
equation Eq. 2 and 28 is not given by Eq. 29. The intention here is
simply to have an adequate numerical representation of the effect
of storage. The interpretation procedure would therefore estimate
T.sub.s by knowing bottom and surface rates through standard least
square procedures, which may be performed rather rapidly. Once
T.sub.s is estimated, a direct correlation of differentiated
pressure data (for pulse width sufficiently large) should give a
value for the actual correlation time (t.sub.ca).
[0119] Assuming that the true correlation time is t.sub.ct, (i.e.,
the correlation time that would have been obtained if T.sub.s=0,
with an impulse rate i.e., .eta..sub.c={circumflex over
(.eta.)}.sub.c (X)), then two dimensionless variables
t.sub.ct/t.sub.ca and T.sub.s/t.sub.ca can be constructed. Because
these are dimensionless time scales, the dependency between them
should change only with X, the other dimensionless parameter.
[0120] To establish this relationship, several numerical exercises
in which three finite width pulses were imposed at a source well
with observations made at a distant well were carried out. The
observer distance was changed to establish dependency on X as was
T.sub.s. The final result of the entire numerical exercise is shown
in FIG. 16.
[0121] FIG. 16 illustrates how storage may be removed in the
forward calculations. A notable feature is that the source-observer
dependency is virtually absent. This result is expected, given the
flattening of {circumflex over (.eta.)}.sub.c (X) for practical
values of X. In the actual interpretation of this figure, one
computes T.sub.s and t.sub.ca from the measured data for a pulse
experiment, and reads the ordinate value. The ordinate when
multiplied by tca gives the correlation peak time in the absence of
storage, and from which diffusivity may be estimated as per
previous sections. Thus, the inverse problem is solved without
extensive iteration based optimization.
[0122] While the invention has been described herein with reference
to certain examples and embodiments, it will be evident that
various modifications and changes may be made to the embodiments
described above without departing from the scope and spirit of the
invention as set forth in the claims.
* * * * *