U.S. patent application number 15/281799 was filed with the patent office on 2017-02-16 for permeability and inflow performance determination for horizontal wells.
This patent application is currently assigned to KING FAHD UNIVERSITY OF PETROLEUM AND MINERALS. The applicant listed for this patent is KING FAHD UNIVERSITY OF PETROLEUM AND MINERALS. Invention is credited to Sami Abdulaziz ALNUAIM, Muhammad Ali KHALID, Muzammil Hussain RAMMAY.
Application Number | 20170045642 15/281799 |
Document ID | / |
Family ID | 53368172 |
Filed Date | 2017-02-16 |
United States Patent
Application |
20170045642 |
Kind Code |
A1 |
KHALID; Muhammad Ali ; et
al. |
February 16, 2017 |
PERMEABILITY AND INFLOW PERFORMANCE DETERMINATION FOR HORIZONTAL
WELLS
Abstract
A method for assessing an inflow performance relationship for a
horizontal well in heterogeneous solution gas drives reservoirs. A
commercial simulator Eclipse is utilized to develop IPRs for
horizontal wells producing oil from solution gas drive reservoirs.
Firstly, a simulation model is developed where a base case is
considered with typical rock, fluid and reservoir properties using
a black oil model. Dimensionless IPR curves are generated by
obtaining a set of points relating to flowing bottom-hole pressures
to oil production rates. The effects of several reservoir and fluid
properties such as bubblepoint pressure, oil gravity, residual oil
saturation, critical gas saturation, initial water saturation,
porosity and absolute permeabilities on the calculated curves are
investigated. A new single empirical IPR model is obtained for
horizontal wells producing oil from heterogeneous solution gas
drive reservoirs suitable for systems with different reservoir
permeability.
Inventors: |
KHALID; Muhammad Ali;
(Dhahran, SA) ; ALNUAIM; Sami Abdulaziz; (Dhahran,
SA) ; RAMMAY; Muzammil Hussain; (Dhahran,
SA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
KING FAHD UNIVERSITY OF PETROLEUM AND MINERALS |
Dhahran |
|
SA |
|
|
Assignee: |
KING FAHD UNIVERSITY OF PETROLEUM
AND MINERALS
Dhahran
SA
|
Family ID: |
53368172 |
Appl. No.: |
15/281799 |
Filed: |
September 30, 2016 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
14132375 |
Dec 18, 2013 |
9470086 |
|
|
15281799 |
|
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Current U.S.
Class: |
1/1 |
Current CPC
Class: |
G01V 99/005 20130101;
E21B 49/00 20130101; E21B 47/06 20130101; E21B 41/0092
20130101 |
International
Class: |
G01V 99/00 20060101
G01V099/00; E21B 47/06 20060101 E21B047/06; E21B 41/00 20060101
E21B041/00 |
Claims
1: A method for assessing an inflow performance relationship (IPR)
for a horizontal well in heterogeneous solution gas drive
reservoirs, comprising: inputting permeability values of a
heterogeneous reservoir; determining spatial variability of the
heterogeneous reservoir based on the permeability values, wherein
determining the spatial variability of the heterogeneous reservoir
includes inputting a value for a number of pairs of permeability
values at a predetermined distance apart, determining a summation
including logarithms of the permeability values, and dividing the
summation by a value equal to twice the number of pairs of
permeability values; determining a spatial correlation of the
permeability values as a semi variogram defined as: .gamma. ( h ) =
1 2 n ( h ) i = 1 n ( h ) [ log ( k ) i + 1 - log ( k ) i ] 2
##EQU00008## where n (h) is a number of pairs of permeability
values at a lag distance h apart and k represents a permeability
value at i or i+1; measuring a bottom hole pressure of the
horizontal well; and determining a production rate of the
horizontal well based on a bottom hole pressure and the determined
spatial variability.
2. (canceled)
3: The method for assessing the IPR for a horizontal well as
claimed in claim 1, wherein the determining the summation including
the logarithms of the permeability includes: determining the
logarithms of permeability values; determining squared differences
between the logarithms of permeability values; and determining the
sum of the squared differences between the logarithms of
permeability values.
4: The method for assessing the IPR for a horizontal well as
claimed in claim 1, wherein the determining the summation including
the logarithms of the permeability includes: determining a value
for .SIGMA..sub.i=1.sup.n(h)[log(k).sub.i+1-log(k).sub.i].sup.2,
wherein, n(h) corresponds to the number of pairs of permeability
values at h distance apart and k corresponds to one of the
permeability values.
5: The method for assessing the IPR for a horizontal well as
claimed in claim 1, wherein determining, the spatial variability of
the heterogeneous reservoir includes: determining a value for 1 2 n
( h ) i = 1 n ( h ) [ log ( k ) i + 1 - log ( k ) i ] ##EQU00009##
wherein, n(h) corresponds to a number of pairs of permeability
values at h distance apart and k corresponds to one of the
permeability values.
6: The method for assessing the IPR for a horizontal well as
claimed in claim 1, wherein the determining the production rate of
the horizontal well is further based on an average pressure of the
heterogeneous reservoir.
7: The method of assessing the IPR for a horizontal well as claimed
in claim 1, wherein the determining the production rate of the
horizontal well includes: determining a value for q o q o ( max ) =
1 - ( 0.63788 - 0.0278 .gamma. ) ( Pwf Pr ) - ( 0.0278 .gamma. +
0.36212 ) ( Pwf Pr ) 2 , ##EQU00010## wherein, .gamma. corresponds
to a spatial variability value, and Pwf Pr ##EQU00011## corresponds
to a ratio of the bottom hole pressure and an average pressure of
the heterogeneous reservoir.
8: The method for assessing the TPR for a horizontal well as
claimed in claim 7, wherein the spatial variability value is
determined by: determining a value for 1 2 n ( h ) i = 1 n ( h ) [
log ( k ) i + 1 - log ( k ) i ] 2 , ##EQU00012## wherein, n(h)
corresponds to a number of pairs of permeability values at h
distance apart and k corresponds to one of the permeability
values.
9. The method for assessing the IPR for a horizontal well as
claimed in claim 7, wherein the production rate for a homogenous
well is determined by substituting zero for .gamma. in the
equation: 1 - ( 0.63788 - 0.0278 .gamma. ) ( Pwf Pr ) - ( 0.0278
.gamma. + 0.36212 ) ( Pwf Pr ) 2 . ##EQU00013##
10: The method for assessing the IPR for a horizontal well as
claimed in claim 1, wherein the horizontal well is a two-phase
horizontal well.
11-19. (canceled)
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This application is a continuation of Ser. No. 14/132,375,
allowed.
FIELD OF THE DISCLOSURE
[0002] The invention pertains to the field of oil well productivity
modeling, and more particularly, to modeling the inflow performance
relationship for horizontal wells in heterogeneous solution gas
drive reservoirs. More specifically, the invention pertains to
assessing the inflow performance relationship by modeling the
relationship between the flowing pressure of the horizontal well
and the flowing rate of the horizontal well.
DESCRIPTION OF THE RELATED ART
[0003] The "background" description provided herein is for the
purpose of generally presenting the context of the disclosure. Work
of the presently named inventors, to the extent it is described in
this background section, as well as aspects of the description
which may not otherwise qualify as prior art at the time of filing,
are neither expressly or impliedly admitted as prior art against
the present invention,
[0004] Oil well performance is a very important matter to oil
companies due to its direct impact on their fields' total oil
production and future development investment. Hence, it is
essential to drill and maintain oil wells as healthy as possible.
Oil well performance is measured by the assessment of its Inflow
Performance & Outflow Performance Relationships (IPR & OPR,
respectively). Since 1968, the Vogel equation has been used
extensively for analyzing the IPR of flowing oil wells under a
solution gas drive mechanism. However, the Vogel curve was
originally developed for vertical wells and may not be applicable
to horizontal wells due to the fact that the flow into a horizontal
well, with an overlying gas cap, is different than flow into a
vertical well. In addition, currently used inflow performance
relationship models for horizontal wells are impractical in nature,
mainly developed for homogeneous reservoirs, and not suitable for
multi-layered systems with different permeability. Thus, there is a
need for a new practical IPR model that considers the effects of
reservoir heterogeneity on IPR curves for horizontal wells
producing oil from two-phase reservoirs overlaid by a gas cap.
SUMMARY OF THE INVENTION
[0005] The invention investigates the effects of reservoir
heterogeneity on IPR curves for horizontal wells drilled in
heterogeneous solution gas drive reservoirs. To achieve the desired
objective, a commercial simulator Eclipse is utilized to develop
IPRs for horizontal wells producing oil from solution gas drive
reservoirs. Firstly, a simulation model is developed where a base
case is considered with typical rock, fluid and reservoir
properties using, a black oil model. Dimensionless IPR carves are
generated by obtaining a set of points relating to flowing
bottom-hole pressures to oil production rates. The effects of
several reservoir and fluid properties such as bubblepoint
pressure, oil gravity, residual oil saturation; critical gas
saturation, initial water saturation, porosity and absolute
permeabilities on the calculated carves are investigated.
[0006] Reservoir heterogeneity is included in the simulation model
by incorporating a semi-variogram function. Finally, a new single
empirical IPR model is obtained for horizontal wells producing oil
from heterogeneous solution gas drive reservoirs suitable for
systems with different reservoir permeability. The new empirical
IPR model is then compared to published correlations and is found
to have a small and acceptable average absolute error of less than
2%. Furthermore, the invention also shows that bubble point
pressure has significant effect on dimensionless IPR curves.
However, plots for other properties indicate that although the
curves are not identical, they are similar in shape and demonstrate
much less variance than the bubble point pressure plot. Therefore,
these properties have only a minor effect on calculated,
dimensionless IPR curves.
[0007] An embodiment of the present disclosure includes a method of
assessing an inflow performance relationship (IPR) for a horizontal
well producing from heterogeneous solution gas drive
reservoirs.
[0008] A further embodiment of the present disclosure includes a
method of assessing an inflow performance relationship (IPR) for a
horizontal well that includes inputting permeability values of a
heterogeneous reservoir; determining spatial variability of the
heterogeneous reservoir based on the permeability values; measuring
a bottom hole pressure of the horizontal well; and determining a
production rate of the horizontal well based on the measured bottom
hole pressure and the determined spatial variability.
[0009] A further embodiment of the present disclosure includes
determining the production rate based on at least one of reservoir
flowing composition, well characteristics, existence of well zones,
behavior of fluid phases under reservoir flowing conditions and an
average pressure of the heterogeneous reservoir.
[0010] A further embodiment of the present disclosure includes
determining the spatial variability of the heterogeneous reservoir
includes inputting a value for a number of pairs of permeability
values at a predetermined distance apart; determining logarithms of
the permeability values; determining a summation including the
logarithms of the permeability values; and dividing the summation
by a value equal to twice the number of pairs of permeability
values.
[0011] A further embodiment of the present disclosure includes
determining the summation including the logarithms of the
permeability may include determining the logarithms of permeability
values; determining squared differences between the logarithms of
permeability values; and determining the sum of the squared
differences between the logarithms of permeability values.
[0012] According to an embodiment of the present disclosure the
spatial variability of the heterogeneous reservoir may be
calculated using the equation
.gamma. ( h ) = 1 2 n ( h ) i = 1 n ( h ) [ log ( k ) i + 1 - log (
k ) i ] 2 ##EQU00001##
where n(h) corresponds to a number of pairs of permeability values
at distance h (lag distance) apart and k corresponds to one of the
permeability values.
[0013] According to an embodiment of the present disclosure the
production rate of the horizontal well in heterogeneous solution
gas reservoir may be calculated using the equation
qo / qomax = 1 - ( 0.63788 - 0.0278 .gamma. ) ( Pwf Pr ) - ( 0.0278
.gamma. + 0.36212 ) ( Pwf Pr ) 2 ##EQU00002##
where, .gamma. corresponds to a spatial variability value, and
Pwf Pr ##EQU00003##
corresponds to a ratio of the bottom hole pressure and an average
reservoir pressure of the heterogeneous reservoir.
[0014] Another embodiment of the present invention includes a
computer implemented method for assessing an inflow performance
relationship (IPR) for a horizontal well that includes determining,
on a computer processor, spatial variability of a heterogeneous
reservoir based on permeability values and saving spatial
variability in a computer memory; and determining, on a computer
processor, a production rate of the horizontal well based on a
bottom hole pressure and the determined spatial variability and
saving production rate in the computer memory.
[0015] According to an embodiment of the present disclosure the
permeability values of the heterogeneous reservoir are read into
the computer memory and wherein the bottom hole pressure is read
into the computer memory.
[0016] According to an embodiment of the present disclosure may
include an input performance relationship device comprising a
computer usable medium having a processing circuitry stored therein
for causing a computer to perform a method of assessing an inflow
performance relationship (IPR) for a horizontal well, the
processing circuitry configured to receive permeability values of a
heterogeneous reservoir; determine spatial variability of the
heterogeneous reservoir based on the permeability values; and
determine a production rate of the horizontal well based on a
bottom hole pressure and the determined spatial variability.
[0017] The foregoing paragraphs have been provided by way of
general introduction, and are not intended to limit the scope of
the following claims. The described embodiments, together with
further advantages, will be best understood by reference to the
following detailed description taken in conjunction with the
accompanying drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
[0018] A more complete appreciation of the disclosure and many of
the attendant advantages thereof will be readily obtained as the
same becomes better understood by reference to the following
detailed description when considered in connection with the
accompanying drawings, wherein:
[0019] FIG. 1 shows an effect of critical gas saturation on a
dimensionless IPR curve.
[0020] FIG. 2 shows an effect of residual oil saturation on a
dimensionless IPR curve.
[0021] FIG. 3 shows an effect of initial water saturation on a
dimensionless IPR curve.
[0022] FIG. 4 shows an effect of porosity on a dimensionless IPR
curve.
[0023] FIG. 5 shows an effect of API gravity on a dimensionless IPR
curve.
[0024] FIG. 6 shows an effect of permeability on a dimensionless
IPR curve.
[0025] FIG. 7 shows an effect of bubble point pressure on a
dimensionless IPR curve.
[0026] FIG. 8 shows an inclusion of reservoir heterogeneity in a
simulation model generated by the Eclipse simulator.
[0027] FIG. 9 shows an effect of reservoir heterogeneity
(permeability variations) on a dimensionless IPR curve.
[0028] FIG. 10 illustrates a flowchart for assessing IPR for a
horizontal well.
[0029] FIG. 11 illustrates a flowchart for determining spatial
variability for heterogeneous reservoirs.
[0030] FIG. 12 shows a hardware description of the inflow
performance relationship device.
DETAILED DESCRIPTION OF THE EMBODIMENTS
[0031] Referring now to the drawings, wherein like reference
numerals designate identical or corresponding parts throughout the
several views.
[0032] Deliverability of a well is analyzed by estimating the
production rate for any given bottom hole pressure. Bottom hole
pressure is the pressure at a sand face of the reservoir in the
well. Inflow into a well is directly proportional to a pressure
drop between the reservoir and the wellbore and can be represented
by the straight line IPR relationship for single-phase
under-saturated reservoirs. However, this relationship is no longer
linear in two-phase flow of oil and gas in saturated oil
systems.
[0033] To decide whether to drill a conventional vertical or a
horizontal well, the type of well that will result in the highest
productivity and economic return must be selected. Common practice
in the oil industry for making this selection is to calculate well
productivity by using the well inflow performance relationship
(TPR). IPR of a well is an essential tool to assess the well
performance as it indicates the production behavior of a well and
assists in determining the feasibility of a producing well. The IPR
curve visualizes the relationship between the well's producing
bottom hole pressures and its corresponding oil production rates
under a given reservoir condition. The shape of the curve is
influenced by many factors such as the reservoir fluid composition,
well characteristics, existence of well zones and behavior of fluid
phases under reservoir flowing conditions. The simplest and most
widely used IPR is the straight-line TPR, implying that the rate is
directly proportional to the pressure drawdown for under-saturated
reservoirs. The constant of proportionality is defined as the
"Productivity Index" (PI), another way to define inflow
performance. One of the main objectives of production engineering
is to maximize PI which can be obtained by maximizing the flow rate
for a given pressure drawdown.
[0034] Horizontal drilling has rapidly come into its own as a
viable alternative to conventional exploration and production
techniques. In the late 70's and early 80's, with oil prices around
$35 a barrel, interest in horizontal wells was reignited. With
technological advances in horizontal well drilling, horizontal well
technology has emerged as a promising method to boost well
productivity and reserves. The purpose of the horizontal wells was
to enhance well productivity, reduce water and gas coning,
intersect natural fractures and to improve well economics. For
Horizontal Wells, Inflow performance serves as an important
component with outflow performance to quantify hydrocarbon
production from a reservoir. Both TPR & OPR are essential
factors to generate a well deliverability curve which enables to
predict an optimal well production rate under certain operating
conditions. The application of horizontal drilling technology made
the prior set of well equation rules obsolete, as hydrocarbon
reservoirs are typically laterally extensive but thin. The
proximity of reservoir boundaries in horizontal wells required new
relationships since the set of well equations for vertical wells
may not be applicable to horizontal wells.
[0035] Inflow performance relationships of horizontal wells are
different than the ones for vertical wells. The two most pronounced
factors for horizontal wells are flow streamline and permeability.
Flow streamline is a combination of radial flow and linear flow
with linear flow dominating and, permeability includes not only
horizontal permeability, but also vertical permeability. Therefore
the anisotropic ratio of the reservoir becomes important when
modeling a horizontal well performance. These yield additional
difficulty to obtain analytical models of horizontal well inflow
performance. In addition, using a single-phase inflow relationship
(straight line IPR) to predict a two-phase flow well performance
can result in significant deviation in flow rate and pressure
distribution in the wellbore, and deliver misleading information
for well performance and decision making.
[0036] Several investigators have utilized reservoir simulators to
study the behavior of a horizontal well producing from oil solution
gas-drive reservoirs. These investigations have led to proposed
empirical inflow performance relationships (IPRs) to predict the
rate-pressure behavior of horizontal oil wells. However, all the
previously developed EPR models for horizontal wells assume
homogeneous reservoirs and are not suitable for a multilayered
reservoir with varying permeability. Therefore, the available IPR
relationships do not provide accurate performance of such
reservoirs. It is the object of this invention to consider the
effects of reservoir heterogeneity on IPR curves for horizontal
wells producing oil from solution gas-drive reservoirs.
Accordingly, a new empirical model is generated for such
reservoirs, which is discussed in more detail below.
[0037] The following describes a reservoir model and grid
characteristics. Eclipse, a three phase, three dimensional, general
purpose black oil simulator is used for modeling solution gas-drive
reservoirs. A horizontal well is placed in the center of the
reservoir. There are 15 cells in x-direction, 15 cells in
y-direction and 5 cells in z-direction representing a reservoir.
The size of a grid in the x-direction is 500 feet (per grid), 500
feet (per grid) in the y-direction and 50 feet (per grid) in the
z-direction. The location of the well is shown in Table 1
below.
TABLE-US-00001 TABLE 1 Location of Well X.sub.1 X.sub.2 Y.sub.1
Y.sub.2 Z.sub.1 Z.sub.2 6 10 8 8 3 3
[0038] The following describes reservoir characteristics. A base
case is modeled as a box-shaped reservoir with a constant thickness
of 250 ft. The porous medium has a homogenous porosity of 0.25
allowing areal permeability isotropy and vertical anisotropy with
values in x, y and z directions of 200, 200 and 50 md,
respectively. A horizontal well of radius 0.33 ft is located in the
middle of the reservoir. A constant and immobile water saturation
of 22% is assigned to all cases. A bubble point pressure of 4000
psi (pounds per square inch) is used in all cases. The black oil
properties used for the base case are shown in Table 2 below. The
base case data used for TR developments is shown in Table 3 below.
Non-darcy flow effects and capillary pressure effects are
neglected. Neither damage nor stimulation is present in the
vicinity of the wellbore (R.sub.s is solution gas oil ratio,
B.sub.o is oil formation volume factor, B.sub.g is gas formation
volume factor, .mu..sub.o is oil viscosity, and .mu..sub.g is gas
viscosity).
TABLE-US-00002 TABLE 2 Reservoir fluid properties data P R.sub.s
B.sub.o B.sub.g .mu..sub.o .mu..sub.g psi SCF/STB bbl/STB SCF/bbl
cp cp 14.7 1.34 1.0488 4.735205 2.7463 0.011799 280.38 44.26 1.0691
93.7407 2.1087 0.012107 546.07 97.64 1.0943 189.5318 1.6657
0.012543 811.76 156.32 1.122 292.2744 1.3725 0.013062 1077.44
218.77 1.1516 401.558 1.1684 0.013656 1343.13 284.19 1.1825 516.099
1.0193 0.014321 1608.82 352.09 1.2146 633.557 0.9059 0.015059
1874.5 422.13 1.2477 750.812 0.8168 0.015872 2140.19 494.06 1.2817
864.66 0.7451 0.016764 2405.88 567.68 1.3166 972.524 0.6861
0.017739 2671.57 642.84 1.3521 1072.83 0.6366 0.018804 2937.25
719.41 1.3883 1164.945 0.5945 0.019966 3202.94 797.29 1.4251
1248.922 0.5583 0.021231 3468.63 876.39 1.4625 1325.212 0.5268
0.022608 3734.31 956.63 1.5005 1394.469 0.4991 0.024106
TABLE-US-00003 TABLE 3 Reservoir data Variable Base Case Values
Units Bubble Point Pressure 4000 Psi Oil Gravity 50 API Reservoir
"x" coordinate 7500 ft. Reservoir "y" coordinate 7500 ft. Reservoir
thickness 250 ft. Permeability - x direction 200 md Permeability -
y direction 200 md Permeability - z direction 20 md Porosity 25 %
Critical gas saturation 5 % Residual Oil Saturation 30 % Initial
water saturation 22 % Well diameter 0.6667 ft. Well length 2500 ft.
Gas density 0.06 lb/ft.sup.3
[0039] Bottom-hole pressure and production rates are required for
IPR calculations. The simulation results were generated starting
from an initial pressure that is less than the bubble point
pressure. Dimensionless IPR curves were generated by dividing the
pressure coordinate of each point on an IPR curve by the average
reservoir pressure and the oil rate coordinate by the maximum oil
rate, corresponding to 100% pressure drawdown. Dimensionless IPR
curves are made in order to compare their curvature or the rate of
change of oil production rate with flowing bottom hole
pressure.
[0040] Two types of simulation runs are examined. In the first
simulation nm, the well is constrained by a constant flowing
bottom-hole pressure. In the second simulation run, a constant oil
production rate is specified. For the same number of simulation
runs, constant pressure rims result in better IPR curve resolution
than constant oil rate runs. For this reason, all runs were done at
a constant wellbore pressure constraint. The performance of each
case was simulated using 9 different bottom-hole pressures as
illustrated in Table 4 below.
TABLE-US-00004 TABLE 4 Bottomhole pressures No. P.sub.wf 1 14.7 2
400 3 800 4 1200 5 1600 6 2000 7 2400 8 2800 9 3200
[0041] FIGS. 1 through 7 show the effects of several variables on
generated TPR curves. The effects of bubble point pressure, oil
gravity, residual oil saturation, critical gas saturation, initial
water saturation, porosity and absolute permeability are
investigated. It is clear that bubble point pressure has a
significant effect on dimensionless IPR curves. However, plots for
other properties indicate that although the curves are not
identical, they are similar in shape and demonstrate much less
variance than the bubble point pressure plot. Therefore, these
variables have only minor effect on calculated, dimensionless IPR
curves.
[0042] An embodiment of the present invention includes permeability
variations in the simulation model. For example, a horizontal well
is placed in 5 z-direction (layers) grids, and heterogeneity is
added in each of the grids by assuming different permeability
values ranging from 0.1 md to 5000 md, Dykstra Parson (1950)
coefficient, V.sub.Dp, was considered as a non-spatial measure of
heterogeneity. The method, using the Dykstra Parson coefficient,
assumes that permeability data is log-normal distributed. However,
spatial correlation of permeability data (permeability values and
permeability data are used interchangeably in the disclosure)s
important for heterogeneous reservoirs. The semi-variogram,
.gamma.(h), is one way to measure or quantify spatial
variability/continuity. For logarithms of permeability data, log
(k), a semi variogram is defined as:
.gamma. ( h ) = 1 2 n ( h ) i = 1 n ( h ) [ log ( k ) i + 1 - log (
k ) i ] 2 ##EQU00004##
Where n (h) is the number of pairs of permeability values at
distance h (lag distance) apart and k represents a permeability
value at i or i+1 Alternatively, if permeability values are not
known, .gamma. (semi-variogram) can be estimated by multirate well
test.
[0043] A total of ten cases of different permeability values are
used. FIG. 8 shows a simulator generated image when spatial
permeability variations are included in the simulation model. The
simulation model with added spatial permeability variations
(heterogeneity) is run and dimensionless IPR curves are plotted for
each case. Moreover, a semi-variogram value is calculated for each
case. The calculated semi-variogram values represent the spatial
variability of the permeability data points for that particular
case.
[0044] FIG. 9 depicts the effect of different spatial permeability
variations (reservoir heterogeneity) on dimensionless IPR curves
plotted for ten assumed cases. As heterogeneity values are
changing, there is a deviation in the curvature of the
dimensionless IPR curve. Moreover, the shape of IPR curve is not
similar to those of homogenous reservoirs.
[0045] A new IPR model is developed that considers the effect of
permeability variations in two-phase horizontal wells. Linear
regression techniques are applied to develop an empirical equation
that fits dimensionless flowrate as a function of dimensionless
pressure. The following empirical equation is found to best fit the
IPR data obtained from simulation for horizontal wells producing
oil from heterogeneous solution gas-drive reservoirs given as:
q o q o ( max ) = 1 - ( 0.63788 - 0.0278 .gamma. ) ( Pwf Pr ) - (
0.0278 .gamma. + 0.36212 ) ( Pwf Pr ) 2 ##EQU00005##
[0046] In the above proposed IPR model, `.gamma.` represents the
semi-variogram function, q.sub.o represents the oil flow rate,
q.sub.o(max) represents the maximum oil flow rate, Pwf represents
the bottom hole pressure and Pr represents the average reservoir
pressure. Moreover, the above equation can be used for homogeneous
reservoirs by substituting zero in the semi-variogram function.
[0047] The above-illustrated IPR model is then compared to the
published correlations of Cheng, Retnanto & Economides,
Harrison and Wiggins (hereby incorporated by reference), using
simulation results of three data sets for different
heterogeneities. Table 5 presents the summary of the statistical
accuracy of the above-illustrated IPR model with other published
correlations. It is evident from the table that the empirical IPR
model is in good agreement with the actual obtained data when
compared to the above-noted published correlations, as evidenced by
an acceptable absolute average error of less than 2%.
TABLE-US-00005 TABLE 5 Comparison of statistical accuracy for IPR
correlation Author Data set Abs Average Error Relative Error St.
Dev. Cheng 01 23.391 -23.05 15.48 02 20.917 -20.58 13.94 03 22.267
-21.93 14.75 Retnanto & 01 17.26 -17.26 12.55 Economides 02
14.92 -14.92 11.15 03 16.23 -16.23 11.92 Harrison 01 13.72 -13.72
11.18 02 11.47 -11.47 9.97 03 12.72 -12.72 10.63 Wiggins 01 86.44
-86.44 91.23 02 82.87 -82.87 89.05 03 85.15 -85.15 90.89 Proposed
IPR 01 1.68 1.68 1.43 Model 02 1.91 16.44 9.44 03 0.25 -0.08
0.44
[0048] Next, flowcharts with regard to the present invention will
be described with respect to the above-described equations. In FIG.
10, a flowchart is illustrated with regard to the assessment of an
IPR for a horizontal well. In step S 101, spatial variability of
the heterogeneous reservoir is determined based on permeability
values. Permeability values are input for the heterogeneous
reservoir. As noted above, the semi-variogram, .gamma.(h), is one
way to measure or quantify spatial variability/continuity. In step
S102, a production rate is determined based a bottom hole pressure
and the determined spatial variability. The equation that is found
to best fit the production rate data for horizontal wells is shown
below:
q o q o ( max ) = 1 - ( 0.63788 - 0.0278 .gamma. ) ( Pwf Pr ) - (
0.0278 .gamma. + 0.36212 ) ( Pwf Pr ) 2 , ##EQU00006##
where q.sub.o is the oil flow rate, q.sub.o(max) is the maximum oil
flow rate, .gamma. is the semi-variogram value, Pwf is the bottom
hole pressure and Pr is the average reservoir pressure,
[0049] In FIG. 11, a flowchart is illustrated with regard to
determining spatial variability for heterogeneous reservoirs. A
value for a number of pairs of permeability values at a
predetermined distance apart is input in step S111. Additionally,
logarithms of the permeability values and a summation including the
logarithms of the permeability values are determined in steps S112
and S113. Finally, in step S114, the summation is divided by a
value equal to twice the number of pairs of permeability values.
The following equation more clearly quantifies spatial
variability:
.gamma. ( h ) = 1 2 n ( h ) i = 1 n ( h ) [ log ( k ) i + 1 - log (
k ) i ] 2 , ##EQU00007##
where n(h) is the number of pairs of permeability values at
distance h (lag distance) apart, and log(k) is the logarithm of
permeability data.
[0050] Next, a hardware description of the Inflow performance
relationship device according to exemplary embodiments is described
with reference to FIG. 12. In FIG. 12, the Inflow performance
relationship device includes a CPU 100 which performs the processes
described above. The process data and instructions may be stored in
memory 102. These processes and instructions may also be stored on
a storage medium disk 104 such as a hard drive (HDD) or portable
storage medium or may be stored remotely. Further, the claimed
advancements are not limited by the form of the computer-readable
media on which the instructions of the inventive process are
stored. For example, the instructions may be stored on CDs, DVDs,
in FLASH memory, RAM, ROM, PROM, EPROM, EEPROM, hard disk or any
other information processing device with which the inflow
performance relationship device communicates, such as a server or
computer.
[0051] Further, the claimed advancements may be provided as a
utility application, background daemon, or component of an
operating system, or combination thereof, executing in conjunction
with CPU 100 and an operating system such as Microsoft Windows 7,
UNIX, Solaris, LINUX, Apple MAC-OS and other systems known to those
skilled in the art.
[0052] CPU 100 may be a Xenon or Core processor from Intel of
America or an Opteron processor from AMD of America, or may be
other processor types that would be recognized by one of ordinary
skill in the art. Alternatively, the CPU 100 may be implemented on
an FPGA, ASIC, PLD or using discrete logic circuits, as one of
ordinary skill in the art would recognize. Further, CPU 100 may be
implemented as multiple processors cooperatively working in
parallel to perform the instructions of the inventive processes
described above.
[0053] The inflow performance relationship device in FIG. 12 also
includes a network controller 106, such as an Intel Ethernet PRO
network interface card from Intel Corporation of America, for
interfacing with network 10. As can be appreciated, the network 10
can be a public network, such as the Internet, or a private network
such as an LAN or WAN network, or any combination thereof and can
also include PSTN or ISDN sub-networks. The network 10 can also be
wired, such as an Ethernet network, or can be wireless such as a
cellular network including EDGE, 3G and 4G wireless cellular
systems. The wireless network can also be WiFi, Bluetooth, or any
other wireless form of communication that is known.
[0054] The Inflow performance relationship device further includes
a display controller 108, such as a NVIDIA GeForce GTX or Quadro
graphics adaptor from NVIDIA Corporation of America for interfacing
with display 110, such as a Hewlett Packard HPL2445w LCD monitor. A
general purpose I/O interface 112 interfaces with a keyboard and/or
mouse 114 as well as a touch screen panel 116 on or separate from
display 110. General purpose I/O interface also connects to a
variety of peripherals 118 including printers and scanners, such as
an OfficeJet or DeskJet from Hewlett Packard.
[0055] A sound controller 120 is also provided in the inflow
performance relationship device, such as Sound Blaster X-Fi
Titanium from Creative, to interface with speakers/microphone 122
thereby providing sounds and/or music.
[0056] The general purpose storage controller 124 connects the
storage medium disk 104 with communication bus 126, which may be an
ISA, EISA, VESA, PCI, or similar, for interconnecting all of the
components of the Inflow performance relationship device. A
description of the general features and functionality of the
display 110, keyboard and/or mouse 114, as well as the display
controller 108, storage controller 124, network controller 106,
sound controller 120, and general purpose I/O interface 112 is
omitted herein for brevity as these features are known.
[0057] Thus, the foregoing discussion discloses and describes
merely exemplary embodiments of the present invention. As will be
understood by those skilled in the art, the present invention may
be embodied in other specific forms without departing from the
spirit or essential characteristics thereof. Accordingly, the
disclosure of the present invention is intended to be illustrative,
but not limiting of the scope of the invention, as well as other
claims. The disclosure, including any readily discernible variants
of the teachings herein, define, in part, the scope of the
foregoing claim terminology such that no inventive subject matter
is dedicated to the public.
* * * * *