U.S. patent application number 14/353432 was filed with the patent office on 2014-09-25 for method for determining the inflow profile of fluids of multilayer deposits.
The applicant listed for this patent is SCHLUMBERGER TECHNOLOGY CORPORATION. Invention is credited to Vyacheslav Pavlovich Pimenov, Valery Vasilievich Shako, Bertrand Theuveny.
Application Number | 20140288836 14/353432 |
Document ID | / |
Family ID | 48168147 |
Filed Date | 2014-09-25 |
United States Patent
Application |
20140288836 |
Kind Code |
A1 |
Shako; Valery Vasilievich ;
et al. |
September 25, 2014 |
METHOD FOR DETERMINING THE INFLOW PROFILE OF FLUIDS OF MULTILAYER
DEPOSITS
Abstract
A method for determining the profile of fluids inflowing into
multi-zone reservoirs provides for a temperature measurement in a
wellbore during the return of the wellbore to thermal equilibrium
after drilling and determining a temperature of the fluids
inflowing into the wellbore from each pay zone after perforation at
an initial stage of production. Specific flow rate for each pay
zone is determined by a rate of change of the measured
temperatures.
Inventors: |
Shako; Valery Vasilievich;
(Domodedovo, RU) ; Pimenov; Vyacheslav Pavlovich;
(Moscow, RU) ; Theuveny; Bertrand; (Moscow,
RU) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
SCHLUMBERGER TECHNOLOGY CORPORATION |
SUGAR LAND |
TX |
US |
|
|
Family ID: |
48168147 |
Appl. No.: |
14/353432 |
Filed: |
October 25, 2012 |
PCT Filed: |
October 25, 2012 |
PCT NO: |
PCT/RU2012/000872 |
371 Date: |
April 22, 2014 |
Current U.S.
Class: |
702/12 |
Current CPC
Class: |
E21B 49/0875 20200501;
E21B 47/103 20200501; E21B 43/14 20130101; E21B 49/08 20130101 |
Class at
Publication: |
702/12 |
International
Class: |
E21B 49/08 20060101
E21B049/08 |
Foreign Application Data
Date |
Code |
Application Number |
Oct 23, 2011 |
RU |
2011143218 |
Claims
1. A method for determining profile of fluid inflow from multi-zone
reservoirs into a wellbore comprising: measuring a temperature in
the wellbore during a wellbore-return-to-thermal-equilibrium time
after drilling, perforating the wellbore, determining a temperature
of the fluids inflowing into the wellbore from each pay zone at an
initial stage of production, and determining a specific flow rate
for each pay zone by a rate of change of the measured
temperatures.
2. The method of claim 1, wherein the temperature of the fluids
inflowing into the wellbore from the pay zones is determined by a
direct measurement of temperature of the fluids inflowing into the
wellbore from each pay zone, and a specific flow rate of each pay
zone is determined by the formula Q i = 4 .pi. .chi. a h 1 ( T . in
, i T . s - 1 ) , ##EQU00027## where Q.sub.i is a flow rate of the
ith pay zone, {dot over (T)}.sub.s is a rate of temperature
recovery in the wellbore before perforation, {dot over
(T)}.sub.in,i is a rate of temperature variation of the fluid
inflowing into the wellbore from the ith pay zone at the initial
stage of production, h.sub.i is a thickness of the ith pay zone, a
is a thermal diffusivity of the reservoir, .chi. = c f .rho. f
.rho. r c r , ##EQU00028## .rho..sub.fc.sub.f is a volumetric heat
capacity of the fluid,
.rho..sub.rc.sub.r=.phi..rho..sub.fc.sub.f+(1-.phi.).rho..sub.mc.sub.m
is a volumetric heat capacity of the rock saturated with the fluid,
.rho..sub.mc.sub.m is a volumetric heat capacity of a rock matrix,
.phi. is a porosity of the reservoir.
3. The method of claim 1, wherein the
wellbore-return-to-thermal-equilibrium time is 5-10 days.
4. The method of claim 1, wherein the temperature of the fluids
inflowing into the wellbore from each pay zone at the initial stage
of production is measured within 3-5 hours after start of
production.
5. The method of claim 1, wherein the temperature of the fluids is
determined by sensors installed on a tubing string above each
perforated interval, a specific flow rate of a lower pay zone is
determined by the formula Q 1 = 4 .pi. .chi. a h 1 ( T . 1 T . s -
1 ) , ##EQU00029## where Q.sub.1 is a flow rate of the lower zone,
{dot over (T)}.sub.s is a rate of temperature recovery in the
wellbore before perforation, {dot over (T)}.sub.1 is a rate of
temperature change of the fluid inflowing into the wellbore from
the pay zone at the initial stage of production as measured above
the lower perforated interval, h.sub.1 is a thickness of the lower
pay zone, a is a thermal diffusivity of the reservoir, .chi. = c f
.rho. f .rho. r c r , ##EQU00030## .rho..sub.fc.sub.f is a
volumetric heat capacity of the fluid,
.rho..sub.rc.sub.r=.phi..rho..sub.fc.sub.f+(1-.phi.)
.rho..sub.mc.sub.m is a volumetric heat capacity of the rock
saturated by the fluid, .rho..sub.mc.sub.m is a volumetric heat
capacity of the rock matrix, .phi. is a porosity of the reservoir,
and a specific flow rate of overlying pay zones is determined by
temperatures measured by the sensors installed on the tubing
string, using the flow rates determined for the underlying pay
zones.
6. The method of claim 5, wherein the
wellbore-return-to-thermal-equilibrium time is 5-10 days.
7. The method of claim 5, wherein the temperature of the fluids
inflowing into the wellbore from each pay zone at the initial stage
of production is measured within 3-5 hours after start of
production.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This application is a U.S. National Stage Application under
35 U.S.C. .sctn.371 and claims priority to Patent Cooperation
Treaty Application No. PCT/RU2012/000872 filed Oct. 25, 2012, which
claims priority to Russian Patent Application No. RU2011143218
filed Oct. 26, 2011. Both of these applications are incorporated
herein by reference in their entireties.
FIELD OF THE DISCLOSURE
[0002] The disclosure relates to the field of geophysical studies
of oil and gas wells, in particular to determining the inflow
profile of fluids inflowing into the wellbore from multi-zone
reservoirs.
BACKGROUND OF THE DISCLOSURE
[0003] Usually when estimating flow rate of individual pay zones by
temperature data, temperature measurement along the entire wellbore
is conducted, while temperature of a reservoir near the wellbore is
assumed close to the temperature of the undisturbed reservoir.
[0004] In particular, a method for determining relative flow rates
of pay zones by quasi-stationary flow temperatures measured along a
wellbore is known. This method is, for example, described in
Cheremsky, G. A. Applied Geothermics, Nedra, 1977, p. 181. The main
assumption of the traditional approach is that an undisturbed
temperature of a reservoir near a wellbore is known prior to the
tests. This assumption is not performed if temperature is measured
at a first stage of production shortly after perforation of the
well. The influence of the perforation itself is not very
significant, but as a rule the temperature of the near-wellbore
part of formation is considerably lower than the temperature of the
undisturbed reservoir due to the cooling resulting from previous
technological operations: drilling, circulation and cementing.
SUMMARY OF THE DISCLOSURE
[0005] The method for determining a profile of fluid inflow from a
multi-zone reservoir provides the possibility to determine the
inflow profile at an initial stage of production, just after
perforating a well, and in enhancing the accuracy of inflow profile
determination due to the possibility of determining inflow profile
by transient temperature data.
[0006] The method comprises measuring temperature in a wellbore
during a wellbore-return-to-thermal-equilibrium time after drilling
and then perforating the wellbore. Temperature of fluids inflowing
into the wellbore from pay zones is determined at an initial stage
of production and a specific flow rate for each pay zone is
determined by rate of change of the measured temperatures.
[0007] In case of direct measurement of temperature of the fluids
inflowing into the wellbore from each pay zone, specific flow rate
of each pay zone is determined by the formula
Q i = 4 .pi. .chi. a h i ( T in , i . T . s - 1 ) ,
##EQU00001##
[0008] where Q.sub.i is a flow rate of an ith pay zone,
[0009] {dot over (T)}s is a rate of temperature recovery in the
wellbore before perforation,
[0010] {dot over (T)}.sub.in, i is a rate of change of temperature
of the fluid inflowing into the wellbore from the ith pay zone at
an initial stage of production,
[0011] h.sub.i is a thickness of the ith pay zone,
[0012] a is a thermal diffusivity of a reservoir,
.chi. = c f .rho. f .rho. r c r , ##EQU00002##
[0013] .rho..sub.fc.sub.f is a volumetric heat capacity of the
fluid,
[0014]
.rho..sub.rc.sub.r=.phi..rho..sub.fc.sub.f+(1-.phi.).rho..sub.mc.su-
b.m is a volumetric heat capacity of the rock saturated with the
fluid,
[0015] .rho..sub.mc.sub.m is a volumetric heat capacity of a rock
matrix;
[0016] .phi. is a porosity of the reservoir.
[0017] In situations where it is not possible to directly measure
temperature of the fluids inflowing into the wellbore from each pay
zone, temperature of the fluids is determined with the use of
sensors installed on a tubing string, above each perforated
interval. A specific flow rate of a lower zone is determined by the
formula
Q 1 = 4 .pi. .chi. a h 1 ( T 1 . T . s - 1 ) , ##EQU00003##
[0018] where Q.sub.1 is a flow rate of a lower pay zone,
[0019] {dot over (T)}.sub.s is a rate of temperature recovery in
the wellbore before perforation,
[0020] {dot over (T)}.sub.1 is a rate of change of temperature of
the fluid inflowing into the wellbore from the pay zone at an
initial stage of production as measured above the lower perforated
interval,
[0021] h.sub.1 is a thickness of this pay zone,
[0022] a is a thermal diffusivity of a reservoir,
.chi. = c f .rho. f .rho. r c r , ##EQU00004##
[0023] .rho..sub.fc.sub.f is a volumetric heat capacity of the
fluid,
[0024]
.rho..sub.rc.sub.r=.phi..rho..sub.fc.sub.f+(1-.phi.).rho..sub.mc.su-
b.m is a volumetric heat capacity of the rock saturated with the
fluid,
[0025] .rho..sub.mc.sub.m is a volumetric heat capacity of a rock
matrix;
[0026] .phi. is a porosity of the reservoir.
[0027] Then with temperatures measured by the sensors installed on
the tubing string, specific flow rates of overlying zones are
determined, using values of flow rates determined for the
underlying zones.
[0028] The wellbore return-to-thermal-equilibrium time usually
lasts for 5-10 days.
[0029] Temperature of the fluids inflowing into the wellbore from
pay zones at the initial state of production is measured within 3-5
hours from start of production.
BRIEF DESCRIPTION OF THE FIGURES
[0030] The disclosure is illustrated by drawings where:
[0031] FIG. 1 shows a scheme with three perforated intervals and
three temperature sensors;
[0032] FIGS. 2a and 2b show results of calculation of inflow
profiles for two versions of formation permeabilities;
[0033] FIG. 3 shows temperatures of fluids inflowing into the
wellbore and temperatures of the corresponding sensors for the case
illustrated in FIG. 2a;
[0034] FIG. 4 shows temperatures of the fluids inflowing into the
wellbore and temperatures of the corresponding sensors for the case
illustrated in FIG. 2b;
[0035] FIG. 5 shows time derivatives of fluid temperature and
temperature of sensor 1 for the case illustrated in FIG. 2a;
[0036] FIG. 6 shows time derivatives of fluid temperature and
temperature of sensor 1 for the case illustrated in FIG. 2b;
f 21 = T . 2 T . 1 and f 32 = T . 3 T . 21 ##EQU00005##
[0037] FIG. 7 shows ratios of temperature growth rates for FIG.
5;
[0038] FIG. 8 shows the same ratios for FIG. 6; and
[0039] FIG. 9 shows correlation between the time derivative
T.sub.in and specific flow rate q.
DETAILED DESCRIPTION
[0040] The method may be used with a tubing-conveyed perforation.
It is based on the fact that a near-wellbore space, as a result of
drilling, usually has a lower temperature than the temperature of
surrounding rocks.
[0041] After drilling of a wellbore, circulation and cementing,
temperature of a reservoir in a near-borehole zone is (by 10-20 K
and more) lower than an original temperature of the surrounding
reservoir at a depth under consideration. After these stages, a
relatively long period of wellbore-returning-to-thermal-equilibrium
follows during which other working operations in the well are
carried out, including installation of a testing string with
perforator guns. In the process of
wellbore-returning-to-thermal-equilibrium after drilling resulting
cooling of near-wellbore formations, temperature measurements in
the wellbore are conducted.
[0042] After perforation, an initial stage of production
follows--cleanup of the near-borehole zone of the reservoir. At the
initial stage of production, when a change takes place in the
temperature of fluids inflowing into the wellbore (usually during
3-5 hours), temperature of the fluids inflowing into the wellbore
form each pay zone is measured.
[0043] In case of a homogeneous reservoir, radial profile of
temperature in the reservoir prior to start of the cleanup is
determined with the use of some general relationship that follows
from the equation of conductive heat transfer (1).
.differential. T .differential. t = a ( .differential. 2 T
.differential. r 2 + 1 r .differential. T .differential. r ) ( 1 )
##EQU00006##
[0044] where "a" is a heat diffusivity of the reservoir.
[0045] From the physical viewpoint, it will be justifiable to
suppose that with a long wellbore-returning-to-thermal-equilibrium
time, some near-wellbore zone (r<r.sub.c) exists within which
the rate of increase of temperature in the formation is constant,
i.e. it does not depend on distance from the wellbore:
.differential. T .differential. t .apprxeq. .PHI. ( t ) .ident. T .
s ( 2 ) ##EQU00007##
[0046] Equations (1) and (2) have the following boundary conditions
at the wellbore axis:
T ( r = 0 ) = T a ; T r | r = 0 = 0 ( 3 ) ##EQU00008##
[0047] where T.sub.a is temperature at the axis (r=0).
[0048] The solution of the problem (1), (2), (3) is
T(r).apprxeq.T.sub.a+br.sup.2 (4)
[0049] where
b = 1 4 a T . s ( 5 ) ##EQU00009##
[0050] Formulas (4), (5) give an approximate radial temperature
profile near the wellbore prior to start of production. A numerical
simulation demonstrates that after 50 hours of
borehole-return-to-thermal-equilibrium time, these formulas are
adequate for r<0.5-0.7 m (with accuracy of 1-5%) for an
arbitrary possible initial (before closure) temperature
profile.
[0051] Formulas (4), (5) do not take into consideration the
influence of heat emission in the course of perforation and radial
non-uniformity of thermal properties of the wellbore and the
reservoir, that is why after comparison with results of numerical
simulation, introduction of some correction coefficient might be
necessary.
[0052] After the start of production, the radial profile of the
temperature in the reservoir and transient temperatures of the
produced fluid is determined, mainly, by convective heat transfer
that is determined by the formula
.rho. r c r .differential. T .differential. t - .rho. f c f v
.differential. T .differential. r = 0 where ( 6 ) v = q 2 .pi. r (
7 ) ##EQU00010##
is a velocity of radial filtration of the fluid, q [m.sup.3/m/s] is
a specific flow rate, .rho..sub.fc.sub.f is a volumetric heat
capacity of the fluid,
.rho..sub.rc.sub.r=.phi..rho..sub.fc.sub.f+(1-.phi.).rho..sub.mc.sub.m
is a volumetric heat capacity of the rock saturated with the fluid,
.rho..sub.mc.sub.m is a volumetric heat capacity of the rock
matrix, .phi. is a porosity of the reservoir.
[0053] Equation (6) does not account for conductive heat transfer,
the Joule-Thomson effect and the adiabatic effect. The influence of
the conductive heat transfer will be accounted for below, while the
Joule-Thomson effect (.DELTA.T=.epsilon..sub.0.DELTA.P) and the
adiabatic effect are small due to a small pressure differential
.DELTA.P and a relatively big typical cooling of the near-wellbore
zone (5-10 K) before start of production.
[0054] Equation (6) has the following solution
T ( r , t ) = T 0 ( r 2 + .chi. .pi. q t ) , ( 8 ) ##EQU00011##
where T.sub.0(r) is an initial temperature profile in the reservoir
(4),
.chi. = c f .rho. f .rho. r c r . ##EQU00012##
[0055] Temperature of the fluid inflowing into the wellbore is (4),
(8):
T in ( t ) = T 0 ( r w 2 + .chi. .pi. q t ) or T in ( t ) .apprxeq.
T a + b ( r w 2 + .chi. .pi. q t ) = .alpha. + .beta. q t where ( 9
) .alpha. = T a + T . s r w 2 4 .pi. a , ( 10 ) .beta. = T . s
.chi. 4 .pi. a . ( 11 ) ##EQU00013##
[0056] In accordance with (9), rate of fluid temperature increase
at the inlet is
T in t = .beta. q . ##EQU00014##
[0057] This formula for rate of temperature increase of the
produced fluid is not fully correct because Equation (6) does not
take into consideration the conductive heat transfer. Even in cases
of very small production rates (q.fwdarw.0), temperature of the
inflow increases due to the conductive heat transfer and the
approximate formula accounting for this effect can be written in
the following way
T in t = .beta. q + T . s ( 12 ) ##EQU00015##
[0058] Thus, with direct measurement of temperature of the fluid
inflowing into the well, specific flow rate of each pay zone
Q.sub.i can be determined by the formula
Q i = 4 .pi. .chi. a h 1 ( T . in , i T . s - 1 ) , ( 13 )
##EQU00016##
[0059] For cases where no possibility exists to directly measure
temperature of the fluids inflowing into the wellbore from the pay
zones, it is suggested to use results of temperature measurements
above each perforated interval, for example, with the use of
sensors installed on a tubing string utilized for perforating. In
accordance with the numerical simulation, in 20-30 minutes after
start of production, the difference between temperature of the
fluid inflowing into the wellbore T.sub.in,1 and temperature
T.sub.1 measured in the wellbore above a first perforated interval
is practically constant: T.sub.in,1-T=.DELTA.T.sub.1.apprxeq.const
, and
T 1 t = T in , 1 t . ##EQU00017##
In accordance with Formula (12), this means that a flow rate of the
lower pay zone Q.sub.1 can be determined (Q.sub.1=h.sub.1q.sub.1)
(h.sub.1 is a thickness of this pay zone) by temperature measured
above the first perforated interval:
T . 1 = .beta. Q 1 h 1 + T . s ( 14 ) ##EQU00018##
or, taking into consideration Formula (11), we find
Q 1 = 4 .pi. .chi. a h 1 ( T . 1 T . s - 1 ) ( 15 )
##EQU00019##
[0060] The parameters in this formula can be approximately
estimated ("a" and .chi.) or measured. The value of {dot over
(T)}.sub.s is measured with the use of temperature sensors after
installing the tubing string before the perforation. The value of
{dot over (T)}.sub.1 is measured above the first perforation
interval at the initial stage of production.
[0061] In case of three or more perforated zones, numerical
simulation can be used for determining the inflow profile. For any
set of values of flow rate {Q.sub.i} (i=1, 2 . . . n, where n is
the number of perforated zones), transient temperatures of produced
fluids can be calculated in the following way (9):
T in , i = .alpha. i + ( .beta. Q i h i + T . s ) t ( 16 ) .alpha.
i = T a , i + T . s r w 2 4 .pi. a , ( 17 ) ##EQU00020##
[0062] The parameter .beta. (11) is one and the same for the zones;
the parameters .alpha..sub.i are different because they depend on
the temperature of the reservoir T.sub.a,i recorded in the wellbore
before start of production.
[0063] For this set of flow rate values, the numeric model of the
producing wellbore should calculate transient temperatures of the
flow at each depth of placement of the sensor with consideration of
heat losses into the surrounding reservoir, the calorimetric law
for the fluids being mixed in the wellbore, and the thermal
influence of the wellbore which is understood here as the influence
of the fluid initially filling the wellbore. The flow rate is
determined with the use of the procedure of model fitting that
minimizes differences between the recorded and calculated
temperatures of the sensors.
[0064] An approximate solution of the problem can be obtained with
the use of the above-described analytical model, which utilizes
rates of increase of sensor temperatures.
[0065] The calorimetric law for the second perforated zone is
described by the equation
T 1 * Q 1 + T in , 2 Q 2 Q 1 + Q 2 = T 2 * ( 18 ) ##EQU00021##
where T.sub.1* are T.sub.2* are temperatures of the fluid below and
above the perforated zone. In accordance with the numeric
simulation, the difference between T.sub.1 and T.sub.1*, T.sub.2
and T.sub.2* remains practically constant and instead of Equation
(18) we can use the following equation for time derivatives of the
measured temperatures:
T . 1 Q 1 + T . in , 2 Q 2 Q 1 + Q 2 = T . 2 ( 19 )
##EQU00022##
[0066] Taking into consideration the above-presented relationships
(11) and (16), this formula can be written as an equation for the
dimensionless flow rate y.sub.2 of the second perforated zone
y.sub.2=Q.sub.2/Q.sub.1:
1 1 + y 2 { 1 + [ h 12 y 2 + y a 1 + y a ] y 2 } = T . 2 T . 1 = f
21 where h 12 = h 1 h 2 , y a = 4 .pi. a .chi. h 1 Q 1 . ( 20 )
##EQU00023##
[0067] If {dot over (T)}.sub.2>{dot over (T)}{dot over
(T.sub.1)} (f.sub.21>1), a unique solution exists. In the
opposite version (f.sub.21 <.sup.1), this equation has two
solutions. The physical sense of this peculiarity is quite evident
for f.sub.21=1, that corresponding to equal increase rates of
temperatures T.sub.2 and T.sub.1. Indeed, this may take place in
two cases: (1) Q.sub.2=0 (y.sub.2=0) and above the upper zone the
behavior of the temperature is the same as below it; (2)
Q.sub.2=Q.sub.1 (y.sub.2=1)--both zones are equal and they have the
same rate of temperature increase.
[0068] The possible solution of the problem of non-uniqueness of
solution uses the combination of two approaches. After evaluating
Q.sub.1 with the use of Equation (12) and determining y.sub.2 by
Equation (20), the true value of y.sub.2 can be chosen using the
known total flow rate Q (for two perforated zones):
Q=Q.sub.1+Q.sub.2=Q.sup.1(1+y.sub.2) (21)
[0069] Relative flow rates for perforated zones 3 and 4 can be
calculated using the dimensionless values y.sub.2, y.sub.3 and so
on, which were determined previously for the perforated zones
located down the wellbore.
1 1 + y 3 { 1 + [ y 3 ( 1 + y 2 ) h 13 + y a ( 1 + y a ) f 21 ] y 3
} = f 32 ( 22 ) 1 1 + y 4 { 1 + [ y 4 [ 1 + y 2 + y 3 ( 1 + y 2 ) ]
h 14 + y a ( 1 + y a ) f 21 f 32 ] y 4 } = f 43 where y 3 = Q 3 Q 1
+ Q 2 , y 4 = Q 4 Q 1 + Q 2 + Q 3 , f 32 = T . 3 T . 2 , f 43 = T .
4 T . 3 . ( 23 ) ##EQU00024##
[0070] The possibility of determining the inflow profile with the
use of the suggested method for a case where direct measurement of
temperatures of fluids inflowing into the wellbore from pay zones
is impossible was checked up on synthetic examples prepared with
the use of a numerical simulation software package for the
producing wellbore, which performs modeling of the unsteady-state
pressure field in the "wellbore-formation" system, flow of
non-isothermal fluids in a porous medium, mixing of the flows in
the wellbore, and heat transfer in the "wellbore-formation" system,
etc.
[0071] Modeling of the process operations carried out under the
following time schedule was performed: [0072] Circulation of the
well during 110 hours. The temperature of fluids at the formation
occurrence depth is assumed to be 40.degree. C. [0073]
Borehole-return-to-thermal-equilibrium time is 90 hrs. [0074]
Production for 6 hrs with flow rate Q=60 m.sup.3/day.
[0075] Geothermal gradient equals 0.02 K/m. The temperature of the
undisturbed reservoir at the depth of sensor 1 (274 m) is
65.5.degree. C. and at the depth of sensor 3 (230 m) is
64.6.degree. C. Thermal diffusivity of the reservoir is
.alpha.=10.sup.-6 m.sup.2/s and .chi.=0.86.
[0076] FIG. 1 shows the scheme of a well with three perforated
intervals (#1: 280-290 m, #2: 260-270 m, #3: 240-250 m) and three
temperature sensors: T.sub.1 at the depth of 274 m, T.sub.2 at the
depth of 254 m and T.sub.3 at the depth of 230 m.
[0077] Two options were considered with different combinations of
formation permeabilities and the following flow rate parameters:
[0078] Option 1 (FIG. 2a): Q.sub.1=10 m.sup.3/day, Q.sub.2=23.4
m.sup.3/day, Q.sub.3=26.6 m.sup.3/day; and [0079] Option 2 (FIG.
2b): Q.sub.1=46 m.sup.3/day, Q.sub.232 13 m.sup.3/day, Q.sub.3=1
m.sup.3/day.
[0080] During circulation and the return-to-thermal-equilibrium
time, the reservoir/wellbore temperature is the same in both cases
under consideration. At the end of the
return-to-thermal-equilibrium time, the rate of temperature growth
was {dot over (T)}.sub.s (200 h)=0.034 K/hr.
[0081] FIGS. 3 and 4 show temperatures of the produced fluids (thin
curves) and temperatures of the corresponding sensors (bold
curves). The difference between T.sub.in,1 and T.sub.1 remains
practically constant after .about.1 hr of production. Time
derivatives of fluid temperature and temperature of sensor #1 are
presented in FIGS. 5 and 6. One can see that approximately 3 hours
after start of production, the difference between dT.sub.in,1/dt
and {dot over (T)}.sub.1 amounts to about 6-8%, confirming our
assumption made in the analysis presented above.
[0082] The correlation between time derivative T.sub.in and
specific flow rate q (data for each of the perforated intervals are
utilized) is presented in FIG. 9. For flow rate q tending to zero,
the linear regression equation gives: {dot over
(T)}.sub.in(q.fwdarw.0)=0.0374 K/hr. This value is close to the
rate of temperature recovery {dot over (T)}.sub.s(200 h)=0.034 K/hr
due to the conductive heat transfer. This result confirms Formula
(14) suggested above for correlation between flow rate and rate of
temperature growth of the produced fluid.
[0083] The values of the flow rate can be estimated from the
lowermost perforated zone. With duration of production equaling 4
hours, FIGS. 5 and 8 give: Option #1--T.sub.1=0.067 K/hr, Option
#2--T.sub.1=0.17 K/hr. Substituting these values in Formula (1), we
find:
[0084] Option #1: Q.sub.1=11 m.sup.3/day (the true value is
Q.sub.1=10 m.sup.3/day);
[0085] Option #2: Q.sub.1=46.5 m.sup.3/day (the true value is
Q.sub.1=46 m.sup.3/day).
[0086] Flow rate values for other perforated zones are determined
by Formulas (20), (23).
[0087] Option #1: For the estimated value Q.sub.1=11 m.sup.3/day
presented above, we find) y.sub.a=1.1. For production duration of 4
hours, FIG. 7 gives f.sub.21.apprxeq.1.45, while Equation (2) gives
one positive solution y.sub.2=2.346 and flow rate
Q.sub.2=Q.sub.1y.sub.2=25.8 m.sup.3/day.
[0088] For the third perforated zone, FIG. 7 gives
f.sub.32.apprxeq.1.08 and from Equation (22) we find one positive
solution y.sub.3=0.75 and Q.sub.3=(Q.sub.1+Q.sub.2)y.sub.3=27.6
m.sup.3/day.
[0089] The total flow rate calculated by temperature data amounts
to Q.sub.e=Q.sub.1+Q.sub.2+Q.sub.3=64.4 m.sup.3/day (the true value
is 60 m.sup.3/day).
[0090] Using this value for determining relative flow rates, we
find:
Y 1 = Q 1 Q e = 0.17 ; Y 2 = 0.4 ; Y 3 = 0.43 ##EQU00025##
[0091] The corresponding flow rate values for different zones
are:
[0092] Q.sub.1=QY.sub.1=10.2 m.sup.3/day (the true value is 10
m.sup.3/day)
[0093] Q.sub.2=QY.sub.2=24 m.sup.3/day (the true value is 23.4
m.sup.3/day)
[0094] Q.sub.1=QY.sub.1=25.8 m.sup.3/day (the true value is 26.6
m.sup.3/day)
[0095] Relative errors (related to the total flow rate) are 0.3%,
1%, and 1.3%.
[0096] Option #2: For the above-estimated flow rate value
Q.sub.1=46.5 m.sup.3/day, y.sub.a=0.25. FIG. 8 gives a production
duration of 4 hours f.sub.21.apprxeq.0.85. In this case, Equation
(20) has no solution and as the approximate solution we have to
take the value of y.sub.2 that corresponds to the minimum value of
f.sub.21 (f.sub.21 min.apprxeq.0.863), which provides for the real
solution: y.sub.2=0.413. The corresponding flow rate is
Q.sub.2=19.85 m.sup.3/day.
[0097] For the third perforated zone, Equation 8 gives
f.sub.32.apprxeq.0.96 , while from Equation (22) we find two roots:
[0098] y.sub.3=0.5, Q.sub.3=(Q.sub.1+Q.sub.2)y.sub.3=34 m.sup.3/day
and total flow rate Q.sub.e=102 m.sup.3/day, and [0099]
y.sub.3=0.062, Q.sub.3=(Q.sub.1+Q.sub.2)y.sub.3=4.18 m.sup.3/day
and total flow rate Q.sub.e=72 m.sup.3/day.
[0100] As the approximate solution of the problem, we will take the
value of y.sub.3=0.062, which gives the total flow rate value
Q.sub.e=72 m.sup.3/day that is closer to the true value.
[0101] In the second case the estimate of Q.sub.1 is more reliable
than the estimate of Q.sub.2 and Q.sub.3, hence, we fix the value
of Q.sub.1 and use the previously determined values of Q.sub.2 and
Q.sub.3 for distributing the remaining flow rate Q-Q.sub.1 between
these zones:
Q 2 ' = Q 2 Q 2 + Q 3 ( Q - Q 1 ) = 11.2 m 3 / day ##EQU00026## and
##EQU00026.2## Q 3 ' = Q 3 Q 2 + Q 3 ( Q - Q 1 ) = 2.3 m 3 / day
##EQU00026.3##
[0102] The determined flow rate values are as follows:
[0103] Q.sub.1=46.5 m.sup.3/day (the true value is 46
m.sup.3/day)
[0104] Q.sub.2=11.2 m.sup.3/day (the true value is 13
m.sup.3/day)
[0105] Q.sub.3=2.3 m.sup.3/day (the true value is 1
m.sup.3/day)
Relative errors (related to the total flow rate) are 0.8%, 3% and
2.2%.
[0106] For solving the inverse problem, this inflow profile (a low
inflow rate of the uppermost zone) is complex. Nonetheless, results
of solving the inverse problem are consistent with the data
utilized in direct simulation.
[0107] In general, a reliable inversion of temperature measured
among perforated intervals immediately after perforating can be
made with the use of a specialized numerical model and fitting the
transient temperature data with consideration of absolute values of
temperature as well as time derivatives of temperature.
* * * * *