U.S. patent application number 13/902333 was filed with the patent office on 2014-11-27 for systems, methods, and computer-readable media for continuous capillary pressure estimation.
The applicant listed for this patent is Saudi Arabian Oil Company. Invention is credited to Johannes Jacobus Maria Buiting, Mustafa Al Ibrahim, Mokhles Mustapha Mezghani.
Application Number | 20140350860 13/902333 |
Document ID | / |
Family ID | 51022442 |
Filed Date | 2014-11-27 |
United States Patent
Application |
20140350860 |
Kind Code |
A1 |
Mezghani; Mokhles Mustapha ;
et al. |
November 27, 2014 |
SYSTEMS, METHODS, AND COMPUTER-READABLE MEDIA FOR CONTINUOUS
CAPILLARY PRESSURE ESTIMATION
Abstract
Provided are methods, systems, and computer-readable media for
determining capillary pressure in a basin/reservoir. Well log data
is obtained that includes permeability log data, porosity log data,
water saturation log data, and oil saturation log data. Thomeer
parameters for a multi-pore system of a Thomeer model are
determined by evaluating an objective function that measures the
mismatch between the well log data and modeled data having the
Thomeer parameters as input. The objective function is iteratively
evaluated using linear equality constraints, linear inequality
constraints, and nonlinear equality constraints until convergence
criteria are met.
Inventors: |
Mezghani; Mokhles Mustapha;
(Dhahran, SA) ; Ibrahim; Mustafa Al; (Dhahran,
SA) ; Buiting; Johannes Jacobus Maria; (Dhahran,
SA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Saudi Arabian Oil Company |
Dhahran |
|
SA |
|
|
Family ID: |
51022442 |
Appl. No.: |
13/902333 |
Filed: |
May 24, 2013 |
Current U.S.
Class: |
702/12 |
Current CPC
Class: |
E21B 49/00 20130101;
G01V 11/00 20130101; E21B 49/008 20130101 |
Class at
Publication: |
702/12 |
International
Class: |
E21B 49/00 20060101
E21B049/00 |
Claims
1. A computer-implemented method for determining capillary pressure
in a reservoir, the method comprising: accessing well log data from
a well log for a well, the well log data comprising permeability
log data, porosity log data, water saturation log data, and oil
saturation log data; determining Thomeer parameters from the
permeability log data, the porosity log data, the water saturation
log data, and the oil saturation log data, the Thomeer parameters
comprising a fractional bulk volume, a pore geometrical factor, and
a minimum entry pressure, the determining comprising: determining a
modeled permeability; determining a modeled porosity; determining a
modeled water saturation, and evaluating an objective function
based on one or more linear equality constraints, one or more
linear inequality constraints, and one or more nonlinear equality
constraints, the objective function comprising: F ( T ) = w 2 ( 1 -
Sw FAL ) - So ( T ) 2 + ( 1 - w ) 2 T - T ^ 2 ##EQU00017## wherein
T is the Thomeer parameters, Sw.sub.FAL is the value of the water
saturation data, So(T) is a modeled oil saturation; the one or more
linear equality constraints comprising: i = 1 n Bv i ( Pc ) =
.alpha. * .phi. FAL ##EQU00018## wherein Bv.sub.i is a fractional
bulk volume occupied by mercury, Pc is an applied capillary
pressure; .alpha. is the conversion factor from mercury-air to
oil-water, n is the number of pore systems in the reservoir,
.phi..sub.FAL is the porosity data; the one or more linear
inequality constraints comprising:
Bv.sub.i.sup.min.ltoreq.Bv.sub.i(Pc).ltoreq.Bv.sub.i.sup.max for
1.ltoreq.1.ltoreq.n
G.sub.i.sup.min.ltoreq.G.sub.i.ltoreq.G.sub.i.sup.max for
1.ltoreq.i.ltoreq.n wherein G.sub.i is the pore geometrical factor,
Pd.sub.i.sup.min.ltoreq.Pd.sub.i.ltoreq.Pd.sub.i.sup.max for
1.ltoreq.i.ltoreq.n wherein Pd.sub.i is a minimum entry pressure,
If Bv.sub.i(Pc).noteq.0 then Bv.sub.i+1(Pc)Bv.sub.i(Pc) for
1.ltoreq.i.ltoreq.n-1, Pd.sub.i.ltoreq.Pd.sub.i+1 for
1.ltoreq.i.ltoreq.n-1, and the one or more nonlinear equality
constraints comprising: K(T)=K.sub.FAL wherein K(T) is the modeled
permeability, K.sub.FAL is the permeability log data; and
determining the capillary pressure of the reservoir using a Thomeer
model having the determined Thomeer parameters.
2. The computer-implemented method of claim 1, wherein the modeled
permeability comprises: K ( T ) = 506 * i = 1 n Bv i ( Pc ) Pd i 2
exp ( - 4.43 G i ) . ##EQU00019##
3. The computer-implemented method of claim 1, wherein the modeled
porosity comprises: .phi. ( T ) = .alpha. i - 1 n Bv i .
##EQU00020##
4. The computer-implemented method of claim 1, wherein the modeled
oil saturation comprises: So i ( G i , Pd i ) = Bv .infin. * exp (
- G i log ( Pc ) - log ( Pd i ) ) ; ##EQU00021## and ##EQU00021.2##
So ( T ) = 1 .phi. i = 1 n Bv i * So i ( G i , Pd i ) .
##EQU00021.3##
5. The computer implemented method of claim 1, wherein the Thomeer
model comprises: B v ( P c ) .apprxeq. { .phi. exp ( - G log ( P c
) - log ( P d ) ) for P c > P d 0 elsewhere } ##EQU00022##
6. The computer-implemented method of claim 1, wherein evaluating
the objective function based on one or more linear equality
constraints, one or more linear inequality constraints, and one or
more nonlinear equality constraints comprises iteratively
evaluating the objective function until convergence criteria are
met.
7. The computer-implemented method of claim 1, wherein evaluating
the objective function based on one or more linear equality
constraints, one or more linear inequality constraints, and one or
more nonlinear equality constraints comprises iteratively
evaluating the objective function evaluating the objective function
using sequential quadratic programming (SQP).
8. The computer-implemented method of claim 1, wherein the well log
comprises a fluid analysis log.
9. The computer-implemented method of claim 1, wherein the
reservoir comprises an oil reservoir.
10. The computer-implemented method of claim 1, comprising
providing the capillary pressures to a reservoir modeling system, a
reservoir simulation system, or a combination thereof.
11. A non-transitory tangible computer-readable storage medium
having executable computer code stored thereon for determining
capillary pressure in a reservoir, the computer code comprising a
set of instructions that causes one or more processors to perform
the following operations: accessing well log data from a well log
for a well, the well log data including permeability log data,
porosity log data, water saturation log data, and oil saturation
log data; determining Thomeer parameters from the permeability log
data, the porosity log data, the water saturation log data, and the
oil saturation log data, the Thomeer parameters comprising a
fractional bulk volume, a pore geometrical factor, and a minimum
entry pressure, the determining comprising: determining a modeled
permeability; determining a modeled porosity; determining a modeled
water saturation, and evaluating an objective function based on one
or more linear equality constraints, one or more linear inequality
constraints, and one or more nonlinear equality constraints, the
objective function comprising: F ( T ) = w 2 ( 1 - Sw FAL ) - So (
T ) 2 + ( 1 - w ) 2 T - T ^ 2 ##EQU00023## wherein T is the Thomeer
parameters, Sw.sub.FAL is the value of the water saturation data,
So(T) is a modeled oil saturation; the one or more linear equality
constraints comprising: i = 1 n Bv i ( Pc ) = .alpha. * .phi. FAL
##EQU00024## wherein Bv.sub.i is a fractional bulk volume occupied
by mercury, Pc is an applied capillary pressure; .alpha. is the
conversion factor from mercury-air to oil-water, n is the number of
pore systems in the reservoir, .phi..sub.FAL is the porosity data;
the one or more linear inequality constraints comprising:
Bv.sub.i.sup.min.ltoreq.Bv.sub.i(Pc).ltoreq.Bv.sub.i.sup.max for
1.ltoreq.i.ltoreq.n
G.sub.i.sup.min.ltoreq.G.sub.i.ltoreq.G.sub.i.sup.max for
1.ltoreq.i.ltoreq.n wherein G.sub.i is the pore geometrical factor,
Pd.sub.i.sup.min.ltoreq.Pd.sub.i.ltoreq.Pd.sub.i.sup.max for
1.ltoreq.i.ltoreq.n wherein Pd.sub.i is a minimum entry pressure,
If Bv.sub.i(Pc).noteq.0 then Bv.sub.i+1(Pc).ltoreq.Bv.sub.i(Pc) for
1.ltoreq.i.ltoreq.n-1, Pd.sub.i.ltoreq.Pd.sub.i+1 for
1.ltoreq.i.ltoreq.n-1, and the one or more nonlinear equality
constraints comprising: K(T)=K.sub.FAL wherein K(T) is the modeled
permeability, K.sub.FAL is the permeability log data; and
determining the capillary pressures of the reservoir using a
Thomeer model having the determined Thomeer parameters.
12. The non-transitory tangible computer-readable storage medium of
claim 12, wherein the modeled permeability comprises: So i ( G i ,
Pd i ) = Bv .infin. * exp ( - G i log ( Pc ) - log ( Pd i ) ) ;
##EQU00025## and ##EQU00025.2## So ( T ) = 1 .phi. i = 1 n Bv i *
So i ( G i , Pd i ) . ##EQU00025.3##
13. The non-transitory tangible computer-readable storage medium of
claim 12, wherein the modeled porosity comprises: .phi. ( T ) =
.alpha. i - 1 n Bv i . ##EQU00026##
14. The non-transitory tangible computer-readable storage medium of
claim 12, wherein the modeled oil saturation comprises: K ( T ) =
506 * i = 1 n Bv i ( Pc ) Pd i 2 exp ( - 4.43 G i ) .
##EQU00027##
15. The computer implemented method of claim 1, wherein the Thomeer
model comprises: B v ( P c ) .apprxeq. { .phi. exp ( - G log ( P c
) - log ( P d ) ) for P c > P d 0 elsewhere } ##EQU00028##
16. The non-transitory tangible computer-readable storage medium of
claim 12, wherein evaluating the objective function based on one or
more linear equality constraints, one or more linear inequality
constraints, and one or more nonlinear equality constraints
comprises iteratively evaluating the objective function until
convergence criteria are met.
17. The non-transitory tangible computer-readable storage medium of
claim 12, wherein evaluating the objective function based on one or
more linear equality constraints, one or more linear inequality
constraints, and one or more nonlinear equality constraints
comprises iteratively evaluating the objective function evaluating
the objective function using sequential quadratic programming
(SQP).
18. The non-transitory tangible computer-readable storage medium of
claim 12, wherein the well log comprises a fluid analysis log.
19. The non-transitory tangible computer-readable storage medium of
claim 12, wherein the reservoir comprises an oil reservoir.
20. A system for determining capillary pressure in a basin and
reservoir, the system comprising: well log data, the well log data
comprising permeability log data, porosity log data, water
saturation log data, and oil saturation log data; one or more
processors; a tangible non-transitory computer-readable memory
having executable computer code stored thereon for determining
capillary pressure in a reservoir, the computer code comprising a
set of instructions that causes the one or more processors to
perform the following operations: determining Thomeer parameters
from the permeability log data, the porosity log data, the water
saturation log data, and the oil saturation log data, the Thomeer
parameters comprising a fractional bulk volume, a pore geometrical
factor, and a minimum entry pressure, the determining comprising:
evaluating an objective function based on one or more linear
equality constraints, one or more linear inequality constraints,
and one or more nonlinear equality constraints, the objective
function comprising: F ( T ) = w 2 ( 1 - Sw FAL ) - So ( T ) 2 + (
1 - w ) 2 T - T ^ 2 ##EQU00029## wherein T is the Thomeer
parameters, Sw.sub.FAL is the value of the water saturation data,
So(T) is a modeled oil saturation; the one or more linear equality
constraints comprising: i = 1 n Bv i ( Pc ) = .alpha. * .phi. FAL
##EQU00030## wherein Bv.sub.i is a fractional bulk volume occupied
by mercury, Pc is an applied capillary pressure; .alpha. is the
conversion factor from mercury-air to oil-water, n is the number of
pore systems in the reservoir, .phi..sub.FAL is the porosity data;
the one or more linear inequality constraints comprising:
Bv.sub.i.sup.min.ltoreq.Bv.sub.i(Pc).ltoreq.Bv.sub.i.sup.max for
1.ltoreq.i.ltoreq.n
G.sub.i.sup.min.ltoreq.G.sub.i.ltoreq.G.sub.i.sup.max for
1.ltoreq.i.ltoreq.n wherein G.sub.i is the pore geometrical factor,
Pd.sub.i.sup.min.ltoreq.Pd.sub.i.ltoreq.Pd.sub.i.sup.max for
1.ltoreq.i.ltoreq.n wherein Pd.sub.i is a minimum entry pressure,
If Bv.sub.i(Pc).noteq.0 then Bv.sub.i+1(Pc).ltoreq.Bv.sub.i(Pc) for
1.ltoreq.i.ltoreq.n-1, Pd.sub.i.ltoreq.Pd.sub.i+1 for
1.ltoreq.i.ltoreq.n-1, and the one or more nonlinear equality
constraints comprising: K(T)=K.sub.FAL wherein K(T) is the modeled
permeability, K.sub.FAL is the permeability log data; and
determining the capillary pressures of the reservoir using a
Thomeer model having the determined Thomeer parameters.
21. The system of claim 20, tangible non-transitory
computer-readable memory storing a modeled permeability, a modeled
porosity, and a modeled water saturation.
22. The system of claim 21, wherein the modeled permeability
comprises: K ( T ) = 506 * i = 1 n Bv i ( Pc ) Pd i 2 exp ( - 4.43
G i ) . ##EQU00031##
23. The system of claim 21, wherein the modeled porosity comprises:
.phi. ( T ) = .alpha. i - 1 n Bv i . ##EQU00032##
24. The system of claim 21, wherein the modeled oil saturation
comprises: So i ( G i , Pd i ) = Bv .infin. * exp ( - G i log ( Pc
) - log ( Pd i ) ) ; ##EQU00033## and ##EQU00033.2## So ( T ) = 1
.phi. i = 1 n Bv i * So i ( G i , Pd i ) . ##EQU00033.3##
25. The system of claim 20, the tangible non-transitory
computer-readable memory storing a the Thomeer model, the Thomeer
model comprising: B v ( P c ) .apprxeq. { .phi. exp ( - G log ( P c
) - log ( P d ) ) for P c > P d 0 elsewhere } ##EQU00034##
26. The system of claim 20, wherein evaluating the objective
function based on one or more linear equality constraints, one or
more linear inequality constraints, and one or more nonlinear
equality constraints comprises iteratively evaluating the objective
function until convergence criteria are met.
27. The system of claim 20, wherein evaluating the objective
function based on one or more linear equality constraints, one or
more linear inequality constraints, and one or more nonlinear
equality constraints comprises iteratively evaluating the objective
function evaluating the objective function using sequential
quadratic programming (SQP).
28. The system of claim 20, wherein the well log comprises a fluid
analysis log.
29. The system of claim 20, wherein the reservoir comprises an oil
reservoir.
30. The system of claim 20, comprising a network coupled to the one
or more processor.
31. The system of claim 20, comprising providing, over the network,
the capillary pressures to a reservoir modeling system, a reservoir
simulation system, or a combination thereof.
32. A computer-implemented method for determining capillary
pressure in a reservoir, the method comprising: accessing well log
data from a well log for a well, the well log data comprising
permeability log data, porosity log data, water saturation log
data, and oil saturation log data; evaluating an objective function
measuring the different between the permeability log data and a
modeled permeability, the porosity log data and a modeled porosity,
and the oil saturation log data and a modeled oil saturation, the
modeled permeability, the modeled porosity, and the modeled oil
saturation each a function of Thomeer parameters; and determining
the capillary pressures of the reservoir using a Thomeer model
having the Thomeer parameters, the Thomeer parameters comprising a
fractional bulk volume, a pore geometrical factor, and a minimum
entry pressure for each pore system.
33. The computer-implemented method of claim 32, wherein evaluating
the objective function measuring the different between the
permeability log data and a modeled permeability, the porosity log
data and a modeled porosity, and the oil saturation log data and a
modeled oil saturation comprises evaluating the objective function
based on a linear equality constraint dependent on the porosity log
data, a linear inequality constraint, and a nonlinear equality
constraint dependent on the permeability log data.
Description
BACKGROUND OF THE INVENTION
[0001] 1. Field of the Invention
[0002] This invention relates generally to sedimentary basin
exploration and reservoir hydrocarbon production and, more
particularly, to estimating capillary pressure for basin
exploration and reservoir development and production.
[0003] 2. Description of the Related Art
[0004] Various exploration and production systems are employed to
find and extract oil, natural gas, and other resources from natural
basins and reservoirs in the earth. Capillary pressure is a
property used for basin exploration and reservoir development and
production. For example, capillary pressure may be used to compute
the original hydrocarbon in place (OHIP) and to estimate a recovery
factor. Reservoir modeling and simulation enables characterization
of static reservoir properties and prediction of reservoir dynamic
behavior. Exploration and production activities and strategies are
strongly impacted by reservoir modeling workflows and techniques.
Accordingly, accurate capillary pressure estimation will contribute
in reducing uncertainties on reservoir model predictions and in
optimizing the exploration activities and the reservoir development
plan. Capillary pressure may be used to determine the saturation
distribution and the total in situ volumes of fluids (e.g., oil,
water, and gas).
[0005] Capillary pressure data is obtained using special core
analysis (SCAL). However, due to the expense and time associated
with SCAL procedures, only a relatively few measurements of
capillary pressure are typically performed. Additionally, the
capillary pressure measurement support size used for SCAL is
significantly smaller than the modeling support size used in
reservoir simulation that uses the capillary pressure data.
Capillary pressure is typically measured by mercury injection
capillary pressure (MICP). Other techniques for measuring capillary
pressure may include core analysis using computer tomography (CT)
and nuclear magnetic resonance (NMR).
SUMMARY OF THE INVENTION
[0006] Various embodiments of methods, computer-readable media, and
systems for determining capillary pressure in a basin and a
reservoir are provided herein. In some embodiments, a method is
provided that includes accessing well log data from a well log for
a well, the well log data comprising permeability log data,
porosity log data, water saturation log data, and oil saturation
log data and determining Thomeer parameters from the permeability
log data, the porosity log data, the water saturation log data, and
the oil saturation log data. The Thomeer parameters include, for
each pore system, a fractional bulk volume, a pore geometrical
factor, and a minimum entry pressure. Determining the Thomeer
parameters includes determining a modeled permeability, determining
a modeled porosity, and determining a modeled water saturation.
Additionally, determining the Thomeer parameters includes
evaluating an objective based on one or more linear equality
constraints, one or more linear inequality constraints, and one or
more nonlinear equality constraints. The objective function
includes
F ( T ) = w 2 ( 1 - Sw FAL ) - So ( T ) 2 + ( 1 - w ) 2 T - T ^ 2
##EQU00001##
wherein T is the Thomeer parameter, Sw.sub.FAL is the value of the
water saturation data, and So(T) is a modeled oil saturation for
the current Thomeer parameter T. The one or more linear equality
constraints include:
i = 1 n Bv i ( Pc ) = .alpha. * .phi. FAL ##EQU00002##
wherein Bv.sub.i is a fractional bulk volume occupied by mercury,
Pc is an applied capillary pressure, .alpha. is the conversion
factor from mercury-air to oil-water, n is the number of pore
systems in the reservoir, and .phi..sub.FAL is the porosity data.
The one or more linear inequality constraints include:
Bv.sub.i.sup.min.ltoreq.Bv.sub.i(Pc).ltoreq.Bv.sub.i.sup.max for
1.ltoreq.i.ltoreq.n
G.sub.i.sup.min.ltoreq.G.sub.i.ltoreq.G.sub.i.sup.max for
1.ltoreq.i.ltoreq.n
wherein G.sub.i is the pore geometrical factor,
Pd.sub.i.sup.min.ltoreq.Pd.sub.i.ltoreq.Pd.sub.i.sup.max for
1.ltoreq.i.ltoreq.n
wherein Pd.sub.i is a minimum entry pressure,
If Bv.sub.i(Pc).noteq.0 then Bv.sub.i+1(Pc).ltoreq.Bv.sub.i(Pc) for
1.ltoreq.i.ltoreq.n-1
Pd.sub.i.ltoreq.Pd.sub.i+1 for 1.ltoreq.i.ltoreq.n-1,
and the one or more nonlinear equality constraints include:
K(T)=K.sub.FAL (20)
wherein K(T) is the modeled permeability and K.sub.FAL is the
permeability from log data. The method further includes determining
the capillary pressure of the basin/reservoir using a Thomeer model
having the Thomeer parameters.
[0007] In other embodiments, a non-transitory tangible
computer-readable storage medium having executable computer code
stored thereon for determining capillary pressure in a basin
reservoir is provided. The computer code includes a set of
instructions that causes one or more processors to perform the
following operations: accessing well log data from a well log for a
well, the well log data comprising permeability log data, porosity
log data, water saturation log data, and oil saturation log data
and determining Thomeer parameters from the permeability log data,
the porosity log data, the water saturation log data, and the oil
saturation log data. The Thomeer parameters include, for each pore
system, a fractional bulk volume, a pore geometrical factor, and a
minimum entry pressure. Determining the Thomeer parameters includes
determining a modeled permeability, determining a modeled porosity,
and determining a modeled water saturation. Additionally,
determining the Thomeer parameters includes evaluating an objective
based on one or more linear equality constraints, one or more
linear inequality constraints, and one or more nonlinear equality
constraints. The objective function includes
F ( T ) = w 2 ( 1 - Sw FAL ) - So ( T ) 2 + ( 1 - w ) 2 T - T ^ 2
##EQU00003##
wherein T is the Thomeer parameter, Sw.sub.FAL is the value of the
water saturation data, and So(T) is a modeled oil saturation for
the current Thomeer parameter T. The one or more linear equality
constraints include:
i = 1 n Bv i ( Pc ) = .alpha. * .phi. FAL ##EQU00004##
wherein Bv.sub.i is a fractional bulk volume occupied by mercury,
Pc is an applied capillary pressure, .alpha. is the conversion
factor from mercury-air to oil-water, n is the number of pore
systems in the reservoir, and .phi..sub.FAL is the porosity data.
The one or more linear inequality constraints include:
Bv.sub.i.ltoreq.Bc.sub.i(Pc).ltoreq.Bv.sub.i.sup.max for
1.ltoreq.i.ltoreq.n
G.sub.i.sup.min.ltoreq.G.sub.i.ltoreq.G.sub.i.sup.max for
1.ltoreq.i.ltoreq.n
wherein G.sub.i is the pore geometrical factor,
Pd.sub.i.sup.min.ltoreq.Pd.sub.i.ltoreq.Pd.sub.i.sup.max for
1.ltoreq.i.ltoreq.n
wherein Pd.sub.i is a minimum entry pressure,
If Bv.sub.i(Pc).noteq.0 then Bv.sub.i+1(Pc)Bv.sub.i(Pc) for
1.ltoreq.i.ltoreq.n-1,
Pd.sub.i.ltoreq.Pd.sub.i+1 for 1.ltoreq.i.ltoreq.n-1
and the one or more nonlinear equality constraints include:
K(T)=K.sub.FAL (20)
wherein K(T) is the modeled permeability and K.sub.FAL is the
permeability log data. The computer code includes a set of
instructions that causes one or more processors to perform the
following operations: determining the capillary pressure of the
reservoir using a Thomeer model having the Thomeer parameters.
[0008] Additionally, in some embodiments, a system is provided that
includes well log data, the well log data comprising permeability
log data, porosity log data, water saturation log data, and oil
saturation log data, one or more processors, and a tangible
non-transitory computer-readable memory having executable computer
code stored thereon for determining capillary pressure in a basin
and a reservoir. The computer code includes a set of instructions
that causes one or more processors to perform the following
operations: accessing well log data from a well log for a well, the
well log data comprising permeability log data, porosity log data,
water saturation log data, and oil saturation log data and
determining Thomeer parameters from the permeability log data, the
porosity log data, the water saturation log data, and the oil
saturation log data. The Thomeer parameters include, for each pore
system, a fractional bulk volume, a pore geometrical factor, and a
minimum entry pressure. Determining the Thomeer parameters includes
determining a modeled permeability, determining a modeled porosity,
and determining a modeled water saturation for the current Thomeer
parameter T. Additionally, determining the Thomeer parameters
includes evaluating an objective based on one or more linear
equality constraints, one or more linear inequality constraints,
and one or more nonlinear equality constraints. The objective
function includes
F ( T ) = w 2 ( 1 - Sw FAL ) - So ( T ) 2 + ( 1 - w ) 2 T - T ^ 2
##EQU00005##
wherein T is the Thomeer parameter, Sw.sub.FAL is the value of the
water saturation data, and So(T) is a modeled oil saturation. The
one or more linear equality constraints include:
i = 1 n Bv i ( Pc ) = .alpha. * .phi. FAL ##EQU00006##
wherein Bv.sub.i is a fractional bulk volume occupied by mercury,
Pc is an applied capillary pressure, .alpha. is the conversion
factor from mercury-air to oil-water, n is the number of pore
systems in the reservoir, and .phi..sub.FAL is the porosity data.
The one or more linear inequality constraints include:
Bv.sub.i.ltoreq.Bc.sub.i(Pc).ltoreq.Bv.sub.i.sup.max for
1.ltoreq.i.ltoreq.n
G.sub.i.sup.min.ltoreq.G.sub.i.ltoreq.G.sub.i.sup.max for
1.ltoreq.i.ltoreq.n
wherein G.sub.i is the pore geometrical factor,
Pd.sub.i.sup.min.ltoreq.Pd.sub.i.ltoreq.Pd.sub.i.sup.max for
1.ltoreq..ltoreq.n
wherein Pd.sub.i is a minimum entry pressure,
If Bv.sub.i(Pc).noteq.0 then Bv.sub.i+1(Pc)Bv.sub.i(Pc) for
1.ltoreq.i.ltoreq.n-1,
Pd.sub.i.ltoreq.Pd.sub.i+1 for 1.ltoreq.i.ltoreq.n-1
and the one or more nonlinear equality constraints include:
K(T)=K.sub.FAL
wherein K(T) is the modeled permeability and K.sub.FAL is the
permeability from log data. The computer code further includes a
set of instructions that causes one or more processors to perform
the following operations: determining the capillary pressure of the
basin/reservoir using a Thomeer model having the Thomeer
parameters.
[0009] Further, in some embodiments, a computer-implemented method
for determining capillary pressure is provided. The method includes
accessing well log data from a well log for a well, the well log
data including permeability log data, porosity log data, water
saturation log data, and oil saturation log data. The method
further includes evaluating an objective function measuring the
different between the permeability log data and a modeled
permeability, the porosity log data and a modeled porosity, and the
oil saturation log data and a modeled oil saturation, the modeled
permeability, the modeled porosity, and the modeled oil saturation
each a function of Thomeer parameters. Additionally, the method
includes determining the capillary pressures of the basin and a
reservoir using a Thomeer model having the Thomeer parameters, the
Thomeer parameters comprising a fractional bulk volume, for each
pore system, a pore geometrical factor, and a minimum entry
pressure.
BRIEF DESCRIPTION OF THE DRAWINGS
[0010] FIG. 1 is a graph of a Thomeer function representing an MICP
B.sub.v-curve for a mono-modal pore system according to the prior
art;
[0011] FIG. 2 is a block diagram of a process for estimating
Thomeer parameters from standard well log data and determining
capillary pressure in accordance with an embodiment of the present
invention;
[0012] FIG. 3 is a block diagram of an optimization process for the
objective function and constraints for estimating Thomeer
parameters in accordance with an embodiment of the present
invention;
[0013] FIG. 4 is a block diagram of a system for estimating Thomeer
parameters from standard well log data and determining capillary
pressure in accordance with an embodiment of the present
invention;
[0014] FIG. 5 is a block diagram of a computer in accordance with
an embodiment of the present invention;
[0015] FIGS. 6-10 are graphs depicting the results of validation
testing of the estimation of Thomeer parameters in accordance with
an embodiment of the present invention;
[0016] FIGS. 11-15 are plots of modeled data and actual data for
the five synthetic wells used for the validation testing of the
estimation of Thomeer parameters in accordance with an embodiment
of the present invention.
[0017] While the invention is susceptible to various modifications
and alternative forms, specific embodiments thereof are shown by
way of example in the drawings and will herein be described in
detail. The drawings may not be to scale. It should be understood,
however, that the drawings and detailed description thereto are not
intended to limit the invention to the particular form disclosed,
but to the contrary, the intention is to cover all modifications,
equivalents, and alternatives falling within the spirit and scope
of the present invention as defined by the appended claims.
DETAILED DESCRIPTION
[0018] As discussed in more detail below, provided in some
embodiments are systems, methods, and computer-readable media for
determining capillary pressure in a basin and a reservoir. Well log
data is obtained from a well log for a well and used to determine
Thomeer parameters for each pore system, i.e., minimum entry
pressure, pore geometrical factor, and fractional bulk volume
occupied, used in a Thomeer model for determining capillary
pressure. The well log data may include permeability log data,
porosity log data, water saturation log data, and oil saturation
log data. The Thomeer parameters are determined by evaluating an
objective function that measures the mismatch between the well log
data and modeled data having the Thomeer parameters as input. The
objective function is iteratively evaluated using linear equality
constraints, linear inequality constraints, and nonlinear equality
constraints until convergence criteria are met. In some
embodiments, the evaluation may be performed using sequential
quadratic programming. The determined capillary pressures for a
basin and a reservoir may be provided for basin exploration,
prospect evaluation, and reservoir modeling and reservoir
simulation.
[0019] As will be appreciated, mercury injection capillary pressure
(MICP) may be used with core samples to determine capillary
pressures and pore size distributions. The mercury-air systems used
with MICP may be converted to oil-water systems that typically
exist in basin and a reservoir. The Thomeer model is based on an
observed hyperbolic relationship between the amount of mercury
entering a pore system in an MICP experiment and the applied
mercury pressures. The Thomeer model provides an empirical formula
for the occupied fractional bulk volume B.sub.v and the mercury
pressures P.sub.c, as shown in Equation 1 as follows:
B v ( P c ) .apprxeq. { B v , .infin. exp ( - G log ( P c ) - log (
P d ) ) for P c > P d 0 elsewhere } ( 1 ) ##EQU00007##
[0020] Where Bv.sub.i(Pc) is the fractional bulk volume occupied by
mercury at capillary pressure Pc, Bv,.sub..infin. is the fractional
bulk volume occupied by mercury at infinitely high capillary
pressure, G is the pore geometrical factor, Pc is the applied
capillary pressure, and Pd is the minimum entry pressure.
[0021] Thus, the shape of the function described above in Equation
1 is determined by the three Thomeer parameters described above:
P.sub.d, G, and B.sub.v,.infin.. The Thomeer parameterization for
MICP experiments may be used to describe the internal architecture
of a basin and a reservoir pore system. FIG. 1 depicts a Thomeer
function representing an MICP Bv-curve for a mono-modal pore
system.
[0022] As shown in the figure, the asymptotes for the Thomeer
parameters B.sub.v and P.sub.d are indicated. The pore geometrical
factor G determines the curvature of the illustrated MICP Bv-curve,
such that a large value of G results in a gradual onset and small
value of G results in a sudden and sharp onset. For multi-modal
systems (e.g., carbonates), a summation of up to three different
Thomeer hyperbolas may be used to constitute a complete MICP
B.sub.v-curve. Because each such hyperbola would upscale
independently, multi-modality is left out of the upscale
derivation. Thus, unless otherwise noted, pore systems employed in
the techniques described below may be assumed to be mono-modal.
However, the techniques described below are not restricted to
mono-modal pore systems and may be used with multi-modal pore
systems.
[0023] Additionally, in some embodiments B.sub.v,.infin. may be
assumed to equal .phi.. Thus, applying this assumption to Equation
1, in such embodiments the Thomeer formula is shown in Equation 2
as follows:
B v ( P c ) .apprxeq. { .phi. exp ( - G log ( P c ) - log ( P d ) )
for P c > P d 0 elsewhere } ( 2 ) ##EQU00008##
[0024] Where Bv(Pc) is the fractional bulk volume occupied by
mercury at capillary pressure Pc, .phi. is the porosity, G is the
pore geometrical factor, Pc is the applied capillary pressure, and
Pd is the minimum entry pressure.
[0025] Accordingly, Equation 2 may be expressed in terms of
porosity and mercury saturations, as shown below in Equations 3 and
4 as follows:
B.sub.v(P.sub.c).apprxeq..phi.S.sub.Hg(P.sub.c) (3)
[0026] Where Bv(Pc) is the fractional bulk volume occupied by
mercury at capillary pressure Pc, .phi. is the porosity,
S.sub.Hg(Pc) is the mercury saturation at capillary pressure Pc,
and Pc is the applied capillary pressure.
S Hg ( P c ) .apprxeq. exp ( - G log ( P c ) - log ( P d ) ) ( 4 )
##EQU00009##
[0027] Wherein S.sub.Hg(Pc) is the mercury saturation at capillary
pressure Pc, Pc is the applied capillary pressure, G is the pore
geometrical factor, and Pd is the minimum entry pressure.
[0028] As will be appreciated, the conversion from mercury
saturation to actual reservoir fluid saturation (i.e., S.sub.HG to
S.sub.oil) may be performed using the interfacial tension values
for oil-brine-rock and for Hg-air-rock (or Hg-vapor-rock).
[0029] Typically estimations using Thomeer analysis involve the
fitting of the Thomeer hyperbola to core plug MICP data by
determining the Thomeer parameters from such data. This results in
the full characterization of the entire pore space of each core
plug. However, such core plug MICP data is very sparse and Thomeer
parameters estimation from MICP data will generate capillary
pressure data with a high degree of uncertainty. For example, a
core plug may have a typical volume of about 10 cm.sup.3 and
reservoir element, as probed by wire-line logs such as resistivity,
may represent a reservoir volume equivalent billions of core plugs,
while a typical reservoir model grid cell is about 250 m.times.250
m.times.1 m. Consequently, reservoir modeling and simulation (e.g.,
for reserve estimation and production forecast) based on such
capillary pressure data may also be highly uncertain.
[0030] FIG. 2 depicts a process 200 for estimating the Thomeer
parameters from standard well logs, such as those obtained from
fluid analysis logs. Some or all steps of the process 200 may be
implemented as executable computer code stored on a non-transitory
tangible computer-readable storage medium and executed by one or
more processors of a special-purpose machine, e.g., a computer
programmed to execute the code. Initially, well log data is
obtained (block 202). The obtained well log data 204 may include
porosity data, water saturation data, oil saturation data, and
permeability data. Next, as described in detail below, the Thomeer
parameters for the Thomeer model are estimated from the well log
data (block 206) by minimizing an objective function measuring a
mismatch between observed data and predicted data from a
theoretical model having the Thomeer parameters as an input. Next,
the estimated Thomeer parameters are inputted into the Thomeer
formula (block 208). Based on these parameters, capillary pressure
is estimated for a reservoir using the Thomeer equation (block
210). The estimated capillary pressures may then be used in further
basin exploration, prospect evaluation, and reservoir modeling and
simulations (block 212).
[0031] As mentioned above, embodiments of the present invention may
estimate Thomeer parameters from "standard" well logs having, for
example, porosity data, water saturation data, oil saturation data,
and permeability data. As described in detail below, the estimation
of Thomeer parameters from well log data is based on an inverse
problem theory. Accordingly, the Thomeer parameters are estimated
by minimizing an objective function measuring a mismatch between
observed data and predicted data from a theoretical model having
the Thomeer parameters as an input. In the techniques described
below, the Thomeer parameters may be abbreviated using Equation 5
as follows:
T=(Bv.sub.i(Pc),G.sub.i,Pd.sub.i).sub.1.ltoreq.i.ltoreq.n (5)
[0032] wherein T is the abbreviation for the Thomeer parameters, n
is the number of pore systems in the reservoir,
Bv.sub.i(Pc).sub.1.ltoreq.i.ltoreq.n is the fractional bulk volume
occupied by mercury at infinitely high capillary pressure, Pc is
the applied capillary pressure, (G.sub.i).sub.1.ltoreq.i.ltoreq.n
is the pre geometrical factor and
(Pd.sub.i).sub.1.ltoreq.i.ltoreq.n is the minimum entry
pressure.
[0033] As explained above, the well log data may include a porosity
log, water saturation log, oil saturation log, and permeability
log. A porosity model using the multi-pore system porosity
definition may be represented by Equation 6 as follows:
.phi. ( T ) = .alpha. i - 1 n Bv i ( 6 ) ##EQU00010##
[0034] Where .phi. is the porosity, .alpha. is the conversion
factor from mercury-air to oil-water, n is the number of pore
systems in the reservoir, and Bv.sub.i is the fractional bulk
volume occupied by mercury.
[0035] An oil saturation model using the Thomeer equations may be
represented by Equations 7 and 8 as follows:
So i ( G i , Pd i ) = Bv .infin. * exp ( - G i log ( Pc ) - log (
Pd i ) ) ( 7 ) ##EQU00011##
[0036] Where So.sub.i (G.sub.i, Pd.sub.i) is the modeled oil
saturation using Thomeer equations for the pore system i described
by the Thomeer parameters (Bv.sub.i(Pc), G.sub.i, Pd.sub.i),
Bv.sup..infin. is the fractional bulk volume occupied by mercury at
infinitely high capillary pressure, Pc is the applied capillary
pressure, G.sub.i is the pore geometrical factor and Pd.sub.i is
the minimum entry pressure.
So ( T ) = 1 .phi. i = 1 n Bv i * So i ( G i , Pd i ) ( 8 )
##EQU00012##
[0037] Where So(T) is the modeled oil saturation using Thomeer
equations for the multi-modal pore system described by the Thomeer
parameters T at capillary pressure Pc, .phi. is the porosity,
So.sub.i (G.sub.i, Pd.sub.i) is the modeled oil saturation using
Thomeer equations for the pore system i described by the Thomeer
parameters (Bv.sub.i(Pc), G.sub.i, Pd.sub.i), Bv.sub.i is the
fractional bulk volume occupied by mercury, G.sub.i the pore
geometrical factor and Pd.sub.i is the minimum entry pressure.
[0038] An absolute permeability model may be represented using the
Buiting-Clerke equation, as shown below in Equation 9:
K ( T ) = 506 * i = 1 n Bv i ( Pc ) Pd i 2 exp ( - 4.43 G i ) ( 9 )
##EQU00013##
[0039] Wherein K(T) is the modeled permeability using the using
Thomeer equations for the multi-modal pore system described by the
Thomeer parameters T at capillary pressure Pc, Bv.sub.i(Pc) is the
fractional bulk volume occupied by mercury at infinitely high
capillary pressure, Pc is the applied capillary pressure, G.sub.i
is the pore geometrical factor and Pd.sub.i is the minimum entry
pressure.
[0040] As mentioned above, the Thomeer parameters are estimated
from well log data based on an inverse problem theory. The inverse
problem formulation includes an objection function minimized using
linear and nonlinear constraints. The objection function is shown
in Equation 10 as follows:
F ( T ) = w 2 ( 1 - Sw FAL ) - So ( T ) 2 + ( 1 - w ) 2 T - T ^ 2 (
10 ) ##EQU00014##
[0041] Wherein F(T) is the objective function using Thomeer
equations for the multi-modal pore system described by the Thomeer
parameters T, w is the, Sw.sub.FAL is the water saturation log
value (e.g., from a fluid analysis log), So(T) is the modeled oil
saturation using Thomeer equations for the multi-modal pore system
described by the Thomeer parameters T at capillary pressure Pc, and
T is the abbreviation for the Thomeer parameters.
[0042] The linear equality constraints are shown by Equation 11 as
follows:
i = 1 n Bv i ( Pc ) = .alpha. * .phi. FAL ( 11 ) ##EQU00015##
[0043] Wherein Bv.sub.i(Pc) is the fractional bulk volume occupied
by mercury at capillary pressure Pc, Pc is the applied capillary
pressure, .alpha. is the conversion factor from mercury-air to
oil-water, n is the number of pore systems in the reservoir, and
.phi..sub.FAL is the porosity log data (e.g., from a fluid analysis
log).
[0044] The linear inequality constraints are shown by Equations
12-16 as follows:
Bv.sub.i.sup.min.ltoreq.Bv.sub.i(Pc).ltoreq.Bv.sub.i.sup.max for
1.ltoreq.i.ltoreq.n (12)
[0045] Wherein Bv.sub.i is the fractional bulk volume occupied by
mercury, Bv.sub.i(Pc) is the fractional bulk volume occupied by
mercury at capillary pressure Pc, Pc is the applied capillary
pressure, and n is the number of pore systems in the reservoir.
G.sub.i.sup.min.ltoreq.G.sub.i.ltoreq.G.sub.i.sup.max for
1.ltoreq.i.ltoreq.n (13)
[0046] Wherein G.sub.i is the pore geometrical factor and n is the
number of pore systems in the basin/reservoir.
Pd.sub.i.sup.min.ltoreq.Pd.sub.i.ltoreq.Pd.sub.i.sup.max for
1.ltoreq.i.ltoreq.n (14)
[0047] Wherein Pd.sub.i is the minimum entry pressure and n is the
number of pore systems in the reservoir.
if Bv.sub.i(Pc).noteq.0 then Bv.sub.i+1(Pc).ltoreq.Bv.sub.i(Pc) for
1.ltoreq.i.ltoreq.n-1 (15)
[0048] Wherein Bv.sub.i is the fractional bulk volume occupied by
mercury, Bv.sub.i(Pc) is the fractional bulk volume occupied by
mercury at capillary pressure Pc, Pc is the applied capillary
pressure, and n is the number of pore systems in the reservoir.
Pd.sub.i.ltoreq.Pd.sub.i+1 for 1.ltoreq.i.ltoreq.n-1 (16)
[0049] Wherein Pd.sub.i is the minimum entry pressure and n is the
number of pore systems in the reservoir.
[0050] The nonlinear equality constraints are shown by Equation 17
as follows:
K(T)=K.sub.FAL (17)
[0051] Wherein K(T) is the modeled permeability using the using
Thomeer equations for the multi-modal pore system described by the
Thomeer parameters T and K.sub.FAL is the permeability log data
(e.g., from a fluid analysis log).
[0052] In some embodiments, the objective function is evaluated
using sequential quadratic programming (SQP). By way of background,
SQP may be used to solve differentiable nonlinear programming
problems having forms shown by Equations 18-22 as follows:
min f(x) (18)
x.epsilon..sup.n (19)
x.sub.i.ltoreq.x.ltoreq.x.sub.u (20)
g.sub.j(x)=0,j=1, . . . ,m.sub.e (21)
g.sub.j(x).ltoreq.,j=m.sub.e+1, . . . ,m (22)
[0053] Wherein x is an n-dimensional parameter vector and all
problems functions f(x) and g.sub.j(x), j=1, . . . , m are assumed
to be continuously differentiable. As will be appreciated, SQP is
the standard general purpose technique to solve smooth nonlinear
optimization problems under the following assumptions: the problem
is not large, functions and gradients can be evaluated with
sufficiently high precision, and the problem is smooth and
well-scaled. Accordingly, in such embodiments a quadratic
programming sub-problem may at solved at any iteration by
linearizing the constraints and quadratically approximating the
Lagrangian function shown in Equation 23 as follows:
L ( x , u ) = f ( x ) - j = 1 m u j g j ( x ) ( 23 )
##EQU00016##
[0054] Wherein x is the primal variable and u=(u.sub.1, . . . ,
u.sub.m).sup.T is the Lagrange multiplier vector.
[0055] FIG. 3 depicts the optimization process 300 for the
objective function and constraints described above. Some or all
steps of the process 300 may be implemented as executable computer
code stored on a non-transitory tangible computer-readable storage
medium and executed by one or more processors of a special-purpose
machine, e.g., a computer programmed to execute the code. The
process 300 begins at initialization block 302 and is an iterative
process that is stopped after convergence criteria are met. The
initialization may include obtaining the well log data and
receiving any other data or parameters used in the subsequent
determinations. As shown in FIG. 3, Equation 5, the abbreviation
for the Thomeer parameters, is depicted in block 304 and represents
the Thomeer parameters to be estimated via the process 300. Next,
the various equations for the well log data as a function of the
Thomeer parameters are determined. As shown in FIG. 3, Equation 7
for estimating the oil saturation is depicted in block 306.
Subsequently, the modeled oil saturation as a function of the
Thomeer parameters is estimated, as shown by Equation 8 depicted in
block 308. Additionally, as mentioned above, the modeled porosity
as a function of the Thomeer parameters is estimated, as shown by
Equation 6 depicted in block 310. Similarly, the modeled
permeability as a function of the Thomeer parameters is estimated,
as shown by Equation 9 depicted in block 312.
[0056] The evaluation of the equations described above is performed
by minimizing the objective function described above in Equation 10
using the linear equality constraints described above in Equation
11, the linear inequality constraints described above in Equations
12-16, and the nonlinear inequality constraints described above in
Equation 17 (block 314). As mentioned above, in some embodiments
the objection function may be optimized using SQP techniques. Next
convergence criteria are evaluated to determine if the criteria are
met (decision block 316). If the convergence criteria are not met
(line 318), the next iteration of the process 300 is executed
(block 320). In this manner, the equations described above are
evaluated until the convergence criteria of the minimization of the
objective function are met. If the convergence criteria are met
(line 322), the process may stop (block 324) and the values for the
Thomeer parameters may be used in subsequent calculations of
capillary pressure.
[0057] FIG. 4 depicts a system 400 for determining capillary
pressure in accordance with an embodiment of the present invention.
The system 400 may include well log data 402 and server 404 having
a Thomeer parameter estimation process 406 and a capillary pressure
estimation process 408. Additionally, FIG. 4 also illustrates a
network 410 and a reservoir modeling and simulation system 412. As
will appreciated, well log data 402 may be obtained from "standard"
well logging techniques, such as a fluid analysis log, Facimage,
and other techniques. Advantageously, no additional procedures,
such as core plugs, are required to obtain data for estimating the
Thomeer parameters and the capillary pressure. As noted above, in
some embodiments the well log data includes porosity data 414,
saturation data 416 (e.g., water saturation data and oil saturation
data), and permeability data 418.
[0058] As described above, the server 404 may execute a Thomeer
parameter estimation process 406 to estimate Thomeer parameters 420
from the well log data 402, according to the equations and
processes described above. As described above, the Thomeer
parameters 420 are estimated from the well log data 402 by
minimizing an objective function measuring the mismatch between the
observed well log data 402 and the predicted data from a
theoretical model having the Thomeer parameters as the input. The
server 404 may also execute the capillary pressure estimation
process 408 to estimate capillary pressures 422 from the estimated
Thomeer parameters 420 using the Thomeer equations described above.
The estimated capillary pressures 422 may be provided to the
reservoir modeling and simulation system 412, such as via the
network 410. The reservoir modeling and simulation system may
produce a reservoir model 424 and a reservoir simulation 426 using
the capillary pressure as an input, in addition to other inputs
typically provided for such models and simulations.
[0059] Advantageously, the determination of capillary pressure
according to the techniques described above is relatively quick as
they are based on numerical modeling and not laboratory
experiences. Additionally, the cost of estimating capillary
pressure may be reduced by using standard well log data for the
estimation of capillary pressure as described above instead of
laboratory experiments. Moreover, the capillary pressure for the
entire well is estimated and is not limited to only the relatively
small number of score samples having SCAL. Accordingly, the
estimated capillary pressure reflects the capillary pressures in
the well and not just for the small plug volume used for SCAL.
Consequently, the capillary pressure is more appropriate for
reservoir simulation and modeling wherein the modeling cell size
may be closer to the well neighborhood scale than to the core
scale.
[0060] FIG. 5 depicts a computer 500, such as a server, in
accordance with an embodiment of the present invention. The
computer depicted in FIG. 5, and other computers providing
comparable capabilities, may be used in conjunction with the
present techniques. The computer 500 may communicate over a network
502, described further below. The computer 500 may include various
internal and external components that contribute to the function of
the device and which may allow the computer 500 to function in
accordance with the techniques discussed herein. As will be
appreciated, various components of computer 500 may be provided as
internal or integral components of the computer 500 or may be
provided as external or connectable components. It should further
be noted that FIG. 5 depicts merely one example of a particular
implementation and is intended to illustrate the types of
components and functionalities that may be present in computer
500.
[0061] In various embodiments, the computer 500 may be a server, a
desktop computer, a laptop computer, a tablet computer, a
smartphone, or other types of computers. As shown in FIG. 5, the
computer 500 may include one or more processors 504 and memory 506.
Additionally, the computer 500 may include, for example, an
interface 508, a display 510, an input device 512, input/output
ports 514 and a network interface 516.
[0062] The display 510 may include a cathode ray tube (CRT)
display, a liquid crystal display (LCD), an organic light emitting
diode (OLED) display, or other types of displays. The display 510
may display a user interface (e.g., a graphical user interface) and
may display various function and system indicators to provide
feedback to a user, such as power status, call status, memory
status, etc. In some embodiments, the display 510 may include a
touch-sensitive display (referred to as a "touch screen). In such
embodiments, the touch screen may enable interaction with the
computer via a user interface displayed on the display 510. In some
embodiments, the display 510 may display a user interface for
implementing the techniques described above, such as, for example,
selecting well log data, initiating determination of Thomeer
parameters, viewing the status of the processes described above,
viewing the determined Thomeer parameters, viewing the determined
capillary pressures, and so on.
[0063] The one or more processors 504 may provide the processing
capability required to execute the operating system, programs, user
interface, and functions of the computer 500. The one or more
processors 500 may include microprocessors, such as
"general-purpose" microprocessors, a combination of general and
special purpose microprocessors, and Application-Specific
Integrated Circuits (ASICs). The computer 500 may thus be a single
processor system or a multiple processor system. The one or more
processors 500 may include single-core processors and multicore
processors and may include graphics processors, video processors,
and/or related chip sets.
[0064] The memory 506 may be accessible by the processor 502 and
other components of the computer 500. The memory 506 (which may
include tangible non-transitory computer readable storage mediums)
may include volatile memory and non-volatile memory accessible by
the processor 502 and other components of the computer 500. The
memory 506 may store a variety of information and may be used for a
variety of purposes. For example, the memory 506 may store the
firmware for the computer 500, an operating system for the computer
500, and any other programs or executable code necessary for the
computer 500 to function. The memory 506 may include volatile
memory, such as random access memory (RAM) and may also include
non-volatile memory, such as ROM, a solid state drive (SSD), a hard
drive, any other suitable optical, magnetic, or solid-state storage
medium, or a combination thereof.
[0065] The memory may store executable computer code that includes
program instructions 518 executable by the one or more processors
502 to implement one or more embodiments of the present invention.
For example, the processes 100, 200, and 300 described above may be
implemented in program instructions 518. Thus, in some embodiments
program instructions 518 may include instructions 520 for Thomeer
parameter estimation (e.g., the Thomeer parameter estimation
process 406) and instructions 522 for capillary pressure estimation
(e.g., the capillary pressure estimation process 408). The program
instructions 518 may include a computer program (which in certain
forms is known as a program, software, software application,
script, or code). A computer program may be written in a
programming language, including compiled or interpreted languages,
or declarative or procedural languages. A computer program may
include a unit suitable for use in a computing environment,
including as a stand-alone program, a module, a component, a
subroutine, etc., that may or may not correspond to a file in a
file system. The program instructions 518 may be deployed to be
executed on computers located locally at one site or distributed
across multiple remote sites and interconnected by a communication
network (e.g., network 502).
[0066] The interface 508 may include multiple interfaces and may
couple various components of the computer 500 to the processor 502
and memory 504. In some embodiments, the interface 508, the
processor 502, memory 504, and one or more other components of the
computer 500 may be implemented on a single chip. In other
embodiments, these components and/or their functionalities may be
implemented on separate chips.
[0067] The computer 500 also includes a user input device 512 that
may be used to interact with and control the computer 500. In
general, embodiments of the computer 500 may include any number of
user input devices 512, such as a keyboard, a mouse, a trackball, a
digital stylus or pen, buttons, switches, or any other suitable
input device. The input device 512 may be operable with a user
interface displayed on the computer 500 to control functions of the
computer 500 or of other devices connected to or used by the
computer 500. For example, the input device 500 may allow a user to
navigate a user interface, input data to the computer 500, select
data provided by the computer 500, and direct the output of data
from the computer 500.
[0068] The computer 500 may also include an input and output port
514 to enable connection of devices to the computer 500. The input
and output 514 may include an audio port, universal serial bus
(USB) ports, AC and DC power connectors, serial data ports, and so
on. Further, the computer 500 may use the input and output ports to
connect to and send or receive data with other devices, such as
other computers, printers, and so on.
[0069] The computer 500 depicted in FIG. 5 also includes a network
interface 516, such as a network interface card (NIC), wireless
(e.g., radio frequency) receivers, etc. For example, the network
interface 516 may include known circuitry for communicating with
communication networks via electromagnetic signals transmitted over
a wired or wireless connection. Such circuitry may include, for
example, antennas, amplifiers, transceivers, receivers, processors,
and so on. The network interface 516 may communicate with various
communication networks (e.g., network 502), such as the Internet,
an intranet, a cellular telephone network, a wireless local area
network (LAN) a metropolitan area network (MAN), or other suitable
communication networks. The network interface 516 may implement any
suitable communications standard, protocol and/or technology,
including wired Ethernet, wireless Ethernet (Wi-Fi) ((e.g., IEEE
802.11a, IEEE 802.11b, IEEE 802.11g and/or IEEE 802.11n), a 3G
network (e.g., based upon the IMT-2000 standard), high-speed
downlink packet access (HSDPA), wideband code division multiple
access (W-CDMA), code division multiple access (CDMA), time
division multiple access (TDMA), a 4G network (e.g., IMT Advanced,
Long-Term Evolution Advanced (LTE Advanced), etc.), and any other
suitable communications standard, protocol, or technology.
[0070] FIGS. 6-10 depict the results of various validation tests to
estimate reservoir capillary pressure in accordance with the
techniques described herein. The validation testing is based on
reference sets of Thomeer parameters that are used to populate five
synthetic wells. For each synthetic well, porosity, permeability,
and saturation logs are generated using the models described above
in Equations 6-10. For validation, the actual Thomeer parameters
are assumed to be unknown and the synthetic porosity, permeability,
and saturation log data is used to estimate Thomeer parameters and,
thus, the capillary pressure.
[0071] FIG. 6 depicts graphs of the model well data vs. the
estimated well data and illustrates the efficiency of the
optimization of the objective function using the linear equality
constraints, linear inequality constraints, and nonlinear equality
constraints. FIG. 6 depicts a first graph 600 illustrating
estimated saturation data vs. modeled saturation data, a second
graph 602 illustrating estimated permeability data vs. modeled
permeability data, and a third graph 604 illustrating estimated
porosity data vs. modeled porosity data. The process took less than
one minute to reach the convergence criteria and determine a
solution that matches the saturation data and honors the linear
constraints on the porosity and the non-linear constraints on the
permeability
[0072] FIG. 7 depicts graphs of the estimated Thomeer parameters
(G, Bv, and Pd) vs. the actual Thomeer parameters for a macro
system. FIG. 7 depicts a first graph 700 illustrating estimated
pore geometry factor G vs. actual pore geometry factor G, a second
graph 702 illustrating estimated fractional bulk volume By vs.
actual fractional bulk volume By, and a third graph 704
illustrating estimated minimum entry pressure Pd vs. actual minimum
entry pressure Pd. FIG. 7 also includes bar graphs that depict the
distribution of the estimated Thomeer parameters as compared to the
distribution of the actual Thomeer parameters obtained from a model
database. As shown in FIG. 7, the figure includes a first bar graph
706 that depicts the distribution of estimated and actual pore
geometry factor G, a second bar graph 708 that depicts the
distribution of estimated and actual fractional bulk volume Bv, and
a third bar graph 710 that depicts the distribution of estimated
and actual minimum entry pressure Pd. As shown from these figures,
the distribution of the estimated Thomeer parameters is consistent
with the distribution of the actual Thomeer parameters.
Additionally, as shown from these figures, the range for the
Thomeer parameters is also consistent.
[0073] FIG. 8 depicts graphs of the estimated Thomeer parameters
(G, By, and Pd) vs. the actual Thomeer parameters for a second pore
system having n=2. FIG. 8 depicts a first graph 800 illustrating
estimated pore geometry factor G vs. actual pore geometry factor G,
a second graph 802 illustrating estimated fractional bulk volume Bv
vs. actual fractional bulk volume Bv, and a third graph 804
illustrating estimated minimum entry pressure Pd vs. actual minimum
entry pressure Pd. As shown in FIG. 8, the figure also includes a
first bar graph 806 that depicts the distribution of estimated and
actual pore geometry factor G, a second bar graph 808 that depicts
the distribution of estimated and actual fractional bulk volume Bv,
and a third bar graph 810 that depicts the distribution of
estimated and actual minimum entry pressure Pd. The figures
illustrate a higher difficulty of estimation due to the
insensitivity of the permeability to the pore system 2
characteristics. However, these figures show that a consistent
trend for fractional bulk volume Bv is obtained as the porosity is
sensitive to the Bv values for the three pore systems. For the
minimum entry pressure Pd and the pore geometry factor G, the low
sensitivity of the saturation to these two parameters increases the
difficulty of their estimation and the optimization program may
generate a solution consistent with the prior term included in the
objective function. As shown in the bar graphs 806, 808, and 810,
the prior term will control the consistency of these parameters'
distributions.
[0074] FIG. 9 depicts graphs of the estimated Thomeer parameters
(G, Bv, and Pd) vs. the actual Thomeer parameters for a second pore
system having n=3. FIG. 9 depicts a first graph 900 illustrating
estimated pore geometry factor G vs. actual pore geometry factor G,
a second graph 902 illustrating estimated fractional bulk volume Bv
vs. actual fractional bulk volume Bv, and a third graph 904
illustrating estimated minimum entry pressure Pd vs. actual minimum
entry pressure Pd. The figures illustrate a higher difficulty of
estimation due to the insensitivity of the permeability to the pore
system 23 characteristics. Here again, however, the graphs show a
consistent trend for fractional bulk volume Bv as the porosity is
sensitive to the Bv values for the three pore systems. Similarly,
as noted above, the low sensitivity of the saturation to the
minimum entry pressure Pd and the pore geometry factor G increases
the difficulty of their estimation, and the optimization program
may generate a solution consistent with the prior term included in
the objective function. FIG. 9 also depicts a first bar graph 806
that depicts the distribution of estimated and actual pore geometry
factor G, a second bar graph 808 that depicts the distribution of
estimated and actual fractional bulk volume Bv, and a third bar
graph 810 that depicts the distribution of estimated and actual
minimum entry pressure Pd that depict the control the prior term of
the objective function over the parameters' distributions.
[0075] FIG. 10 depicts the Thomeer parameters' distributions from
for the model parameters and database parameters for the three pore
systems discussed above and in accordance with an embodiment of the
present invention. FIG. 10 illustrates the consistency between the
modeled and actual Thomeer parameters and thus validates the
utility of the capillary pressures derived from the Thomeer
equation using Thomeer estimated according the techniques described
herein.
[0076] FIGS. 11-15 are plots of modeled data and actual data for
the five synthetic wells used for the validation testing described
above and in accordance with an embodiment of the present
invention. Each plot includes: the match of the modeled porosity
vs. the actual porosity data, the match of the modeled saturation
vs. the actual saturation data; the match of the modeled
permeability vs. the actual permeability data; the geometry factor
G for the three pore systems; the bulk volume Bv for the three pore
systems, and the minimum entry pressure Pd for the three pore
systems. The plots illustrate that the techniques described
estimate the Thomeer parameters for a multi-pore system that is
consistent with the actual Thomeer parameters and well log data
used for validation testing.
[0077] Further modifications and alternative embodiments of various
aspects of the invention will be apparent to those skilled in the
art in view of this description. Accordingly, this description is
to be construed as illustrative only and is for the purpose of
teaching those skilled in the art the general manner of carrying
out the invention. It is to be understood that the forms of the
invention shown and described herein are to be taken as examples of
embodiments. Elements and materials may be substituted for those
illustrated and described herein, parts and processes may be
reversed or omitted, and certain features of the invention may be
utilized independently, all as would be apparent to one skilled in
the art after having the benefit of this description of the
invention. Changes may be made in the elements described herein
without departing from the spirit and scope of the invention as
described in the following claims. Headings used herein are for
organizational purposes only and are not meant to be used to limit
the scope of the description.
[0078] As used throughout this application, the word "may" is used
in a permissive sense (i.e., meaning having the potential to),
rather than the mandatory sense (i.e., meaning must). The words
"include", "including", and "includes" mean including, but not
limited to. As used throughout this application, the singular forms
"a", "an" and "the" include plural referents unless the content
clearly indicates otherwise. Thus, for example, reference to "an
element" includes a combination of two or more elements. Unless
specifically stated otherwise, as apparent from the discussion, it
is appreciated that throughout this specification discussions
utilizing terms such as "processing", "computing", "calculating",
"determining" or the like refer to actions or processes of a
specific apparatus, such as a special purpose computer or a similar
special purpose electronic processing/computing device. In the
context of this specification, a special purpose computer or a
similar special purpose electronic processing/computing device is
capable of manipulating or transforming signals, typically
represented as physical electronic or magnetic quantities within
memories, registers, or other information storage devices,
transmission devices, or display devices of the special purpose
computer or similar special purpose electronic processing/computing
device.
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