6iiMii
m
Soobty-of Petrolem Engheere
SPE 030793
Reservoir Performance History Matching Using Rate/Cumulative Type-Curves
J.G. Canard’, SPE, Louisiana State U., and P.A. Schenewerk’”,
Reservoir extent, continuity, and flow capacity are paramount
characteristics that are considered when developing models.
that predict reservoir performance while using alternative
depletion strategies, such as during fluid-injection projects or
enhanced recovery.
Reservoir producing conditions to which this technique can
be readily applied are those whose actual bottom-ho!e
flowing pressure (BHFP) closely approximates a constant
value. Most wells, however, produce with variable BHFP.
The work presented here focuses on an alternative ratecumulative type-curve format whereby variable BHFP is
. 1.,–– .- —L-:—:—
—L-.L .Lincorporateci into &lmensioniess varlames comammg oum uw
production rate and the cumulative production providing a
unified approach that can be applied to any reasonable
variability in the producing rate or flowing pressure history.
The proposed method, with application to single phase and
multiphase flow, provides the practicing engineer a better
method for decline curve analysis and therefore propagates
better reservoir characterization from production data.
‘Now at U. of Tulss
‘“Nw
at U. of Mis.souri-Rolls
LXpyright
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SPE, Louisiana State U.
U. S.A.. fas 01-214-952-94S5
Abstract
This paper presents an analysis technique for characterizing
reservoirs from production performance. Unique to this
technique is the incorporation of the instantaneous bottomhole flowing pressure (BHFP) to both the production rate
and to the cumulative production for a well depleting a
reservoir. This allows a single rate/cumulative analysis for
wells producing with constant BHFP, constant rate, and wells
with variable rate or variable BHFP (includlng wells with
shutins). This solution provides a powerful diagnostic typecurve which can be generated with almost any wellbore/reservoir situation encountered. Extension of the method to gas
reservoirs through use of pseudopressure and viscositycompressibility normalization allows these wells to be
analyzed using the slightly-compressible fluid solution. Well
performance during transient flow and depletion flow are
examined. Simulation results are compared with the analytic
solution. The use of spreadsheets to perform well test
analysis is also demonstrated.
Pressure Normalization
One advancement in decline-curve analysis presented here
inciudes pressure normalization of cumulative production.
Like pressure normalization of production rate, variations in
bottom-hole flowing pressure (BHFP) are accounted for by
dividing cumulative production by the pressure difference
between initial and bottom-hole flowing pressures. The
technique of combining pressure-normalized production rate
(PNR) and pressure-normalized
cumulative production
(PNC) is an improvement over rate normalization alone in
the analysis of reservoirs based on production data.
To apply this technique, determination of BHFP from
surface-measured flowing-tubing pressure (FTP) is required
along with determination of the original static reservoir
pressure. Data can then be presented by plotting PNR versus
PNC. This technique is then extended for use with gas
reservoirs by further incorporating changes in viscosity and
compressibility during reservoir depletion.
This technique relies heavily on either measured BHFP or
lTP. However, unlike with superposition techniques, it does
h~~~~~~ f~~ ~ Weiij
refit w~~II;r- the entire flndssu nreccnre
.~= . WY-. v . ..-- . . ...” -------~. -w--- -
Introduction
Recently, decline-curve analysis has expanded to permit
engineers to analyze a petroleum reservoir directly in regard
to its fluid-flow characteristics and its volumetric extent using
rate-time type-curves of the constant terminal pressure
soiution of the ilffusivity equation. Tiiis anaiysis is of
enormous value to reservoir managers whose goal is to
maximize oil and gas production from a petroleum reservoir.
947
RESERVOIR PERFORMANCE
2
HISTORY MATCHING USING RATE/CUMULATIVE
thus allowing for greater application to situations found in
the industry. The incorporation of PNR and PNC into decline-curve analysis provides a single-performance curve which
is applicable to wells producing at constant BHFP, to wells
producing at constant rate, and to wells with both varying
rate and varying flowing pressure.
The benefit of a single-performance type-curve is its
usefulness as a diagnostic tool. Identification of flow regimes,
geological heterogeneities or boundaries, and interference
from offset production or injection make it the ideal plot for
advanced decline-curve analysis. Although radial flow in
unbounded and bounded reservoirs are presented here, the
same diagnostic type-curve can be used ti[ii type-curves
generated for other common wellbore and reservoir conditions, such as hydraulically fractured wel~ naturally fractured reservoirs, dual-porosity systems, water-drive reservoirs, and other systems with pressure support at the outer
boundary.
.An advan[age of using Ci!hcr rate.!irn-e Qr ra[c-ctunu!afive
decline-curve analysis is that reservoir size, formation
capacity, and wellbore effectiveness can be determined
without either closing in the well or running costly instruments down the wellbore. This capability is greatly extended
by the use of rate-cumulative analysis because pressure
normalization of cumulative production allows for variable
BHFP in the producing well.
re
~cD
qD =
141.2qBp
kh(Pi-Pwf)
““““
. . . . . . . . . . . . . . . . . . . . ...(1)
Where q is the production rate (STB/d), B is the formation
volume factor (rb/STB), p is the fluid viscosity (cp), k is the
permeability (red), his the formation height (ft), Pi and Pti
are the initial reservoir pressure and the wellbore flowing
pressure (psia) respectively. Dimensionless time, t~, is
defined ax
tD
=
.006328M
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)
Ovc?:a
The additional terms used in this expression are t for time
l..J-....\
(uays), @ fGi pilidy
(fF~&d),
~ k tk
kjd
SySklii
Compressibility (psi-l), and rm is the apparent wellbore
radius (ft). The dimensionless external radlu~ re~ is defined
ax
_
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(3)
.
~
wa
Where the external radius is re (ft) and the apparent
wellbore radius is rw (ft). Apparent wellbore radius is a
measure of effectiveness and is related to the actual wellbore
radius, rW(ft) by
r Mu = rw, exp (-s)
. . . . . . . . . . . . . . . . . . . . . . . . . . (4)
Use of the apparent wellbore radius and the van Everd:__Sl
:“ Gullrxam
..fi”cto”* p-#.ace,,*a
—m,,
?-l,n.,ar;u-u, w t,ms
Lyp,W–UU.
. w . U. Sm~GII ..1,:..,rO,.,,.shm kautul, S, m
ables was investigated by Uraite and Raghavan- to allow for
near wellbore damage ( +s) or improvement (-s).
Dimensionless flow rate, qD, and dimensionless cumulative
production, QD, are related using
~
QD
I
.
qDdtD
““””” . (5)
”----””””””””””””””””””
Where dimensionless cumulative production, QD, is defined
by
QD
Definitions
Dimensionless variables are used as they provide a general
solution to any number of specific problems. Actual rate and
time can be calculated from dimensionless rate and time for
n“., .nn,.:c..
fir .a.,a*,m:.
-0.-...
-+,?...
,.,.-.m:maA In
:.. *ha
vuu pm
CUUGLGI
a wlmanl=u
LIIG
aJJy
OPVUJWC.=*
-w UI
1*OVS
dimensionless variables. The single-phase dimensionless rate,
q~, is defined (in field units) as
SPE 030793
TYPE-CURVES
0.8936QB
.
. . . . . . . . . . . . . . . . . . . . . . (6)
4JK?;.(R%+
And Q is the cumulative production (STB).
Tsarevich and Kuranovq (1966) are credited with being the
fk.t
,s.
0.
tn
.“
nhcanw=
that
““o”.
. “
. ..-.
~~~
hnlmAaru.dnmin~td
““. . ..-.
,
..” . . . ...-.”-
~~!~
~~~
exponential in the rate decline, giving credence to the semilog decline-curve plot used by industry for decades. This
discovery allowed a much simpler analytic expression for
flow rate during the boundary-dominated flow period. The
exponential decline equation using dimensionless variables
normaliid by area and geometry is:
qdD ‘exp(-tdD)
. . ..-.
o -----------
‘--..0----(7)
These variables have an additional lower case “d”for declinecurve and are more convenient for type-curve presentation
during boundary-dominated flow. Decline-curve dimensionless time, rate, and cumulative become:
tD
=—.
. . . . . . . .OOOoo. . ...”””
‘dD
(a13)
““”””””
“ . . (8)
q~~ =J3qD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...(9)
QdD
_
QD
~
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(lo)
3
J.G. CALLARD AND P.A. SCHENEWERK
SPE 030793
Table
1-
Area and Geometry Normalizing Factors for Type-Curves
Normalizing
Factors
Circular
Circular
&&!Q
Lw!M!!
a
(reD2 - 1)/2
reD2/2
A/(21rrw2)
ln(reD)-+
+ln 2.24?
B
C*r,#@
Where the area and geometry normalizing
circular reservoirs are defined by
a=—
factors for
‘eD2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
late the transition from infinite-actimz to boundary-dominated flow periods as a function of dimensionle& external
radius and also state that for all dimensionless external
radius the transition can be approximated by a dimensionless
time based on drainage area of 0.1. Were this dimensionless
time is defined ax
(11)
2
J3=ln(reD)-*
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
(12)
~
‘DA =tD2jL”
when reD >30
””””””””””””””’”””
. .
““”””””””
(13)
Bounded Reservoirs: Rate-Cumulative Type-CurveS
The alternative constant pressure type-curve for flow rate
data is the rate-cumulative type-curve shown in Fig. 2. Ratecumuiative type-curves tili be shown to offer a enormous
advantage over rate-time type-curves because they are
equally applicable for constant pressure performance as well
as variable pressure performance.
For non-circular reservoirs the Dietz Shape factor4, Ca, is
included. Definitions in the general case and for circular
reservoirs with r=n < 30 are given by Chen and Poston5 in
Table 1.
‘Eqs. 11 and 12 can be obtained from the General column
by substitution of appropriate definitions of area and value
for Dietz Shape factor for circular reservoirs.
I
3’,
i
ma i
&ml
.-J—<,
I
0$01
al
Onl
QTIIl
Fig. 1- Rate-Time Decline Type-Curve
Ehlig-Economides and Ramey7)
Fig. 2-
(RTDTC) (after Fetkovich* and
Rate-Cumulative
Oaotine Type-Curve
(RCDTC)
For wells that are produced at constant back-pressure, rate
versus cumulative data can be plotted and matched just as
they would be using the rate versus time data. Wells that
have variabie flowing pressure histories, including shut-in
periods, can plotted using PNR and PNC. This data plotting
technique greatly extends the use of type-curves for most of
the conditions encountered in the field.
Rate-time type-curves based on decline-curve dimensionless variables are shown in Fig. 1. Fetkovich6 and Ehlig_,__ ______ ._> _,_,,__ cf______
.____ :A_- __A n-__..7 l_____
EwmJImum iinu Kdmcy
niivc iusu prfsxxrwu slmnar ngurtss.
In Fig. 1 the unbounded curves converge and at that inflection, boundary-dominated
data becomes concave to the
origin. Uraite and Raghavan2 provide expressions to calcu-
949
RESERVOIR PERFORMANCE
4
HISTORY MATCHING
USING RATE/CUMULATIVE
dimensionless rate or reciprocal dimensionless pressure over
the dimensionless time period displayed.
Secondly, while dimensionless rate and dimensionless
reciprocal pressure diverge at the end of the infinite-acting
period (inflection from convex to concave) on the RTDTC,
they continue to track during the boundary-dominated
portion on the RCDTC.
To examine the abtlity to predict flow rates as function of
dimensionless cumulative production, the exponential decline
equation:
sexp(-fdD)
!?dD(fdD)
iscomblned
. . . . . . . . . . . . . . . .
. . . . . .. (14)
with the cumulative-time relationship:
f&)(t~D)
=1
which yields
relationship:
qdD(QdD)= 1
_ew(-fdD)
the
. . . . . . . . . . . . . . . . . ..(15)
boundary-dominated
10
1
k
rate-cumulative
a
t
.
_QdD
. . . . . . . . . . . . . . . . . . . . . . .
SPE 030793
TYPE-CURVES
‘:
Mi
‘1
‘1
;;:.%%0
reD = 1128
,/ ‘
~’.,
‘.
,
reD = 1000
\
W)
10,000
ii 0.,
Eq. 16 infers that thedlmensionless rate during the boundary-dominated flow period is a function of dimensionless
cumulative and is not dependent on the pressure and rate
hktory. To illustrate this point with a variable BHFP case,
the constant rate solution is presented on both the constant
pressure rate-time decline type-curve (RTDTC) and the
constant pressure rate-cumulative decline type-curve (RCDTC). In order to make this comparison, decline-curve
dimensionless pressure is defined as:
*dD=
pfi
T
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
‘37,
0.01,
Mm
0,s
0.08
am1
,-. ,!!7:
10
W
Fig. 4- RCDTC: Constant
Rate/Constant
Pressure Comparison
Type-Curve Matching Techniques
Reservoir parameters such as permeability, apparent
wellbore radks, and drainage area are determined conventionally, using rate-time type-curves and the graphkal
technique of plotting rate-time field data on tracing paper
with a log-log scale equivalent to the scale used for the typecurve. The field data are aligned keeping the grids parallel
to the type-curve and a match point is seiected. ‘The match
hnth mnnhc and rnntaks
-_:-. --- L.-...-., . ..-...*.-*-,-9tn
L“,,,,,.”,.
.“
““...
&..=...
. . ..---------pulm call UC Cllly p“,,lt
(17)
(1!)
Dimensionless tabular data from Earlougher et al.8 for a well
in the center of a closed square with an equivalent dlmen~ion!e~~e~-~rnal radhls of IIM is shown in F@. 3 and 4.
ordinate and abscissa for both curves. This method is
outlined by Earlougher 4. For RCDTC matching field data
are plotted as PNR vs PNC. The match point from the
pressure normalized field data and the RCDTC are selected
as above.
Solving for the drainage area or external radius, freed by
the shift in horizontal axes (using eqs. 6, 10, & 11):
an
A=—
\i
00:.
O.Cal
O.vol
0,3
0.01
1
5.615B
(Q/A~)M
@~lc, (QdD)M
10
>fl
z
................”..(18)
WI
This can be rearranged to solve for the pore volume> Vp:
--
Cmmnarisan
=b.
r my.n
.J -- nTnTP.
“ .“. ”. PAW.*-A
. . .. . . .. .❑..-4a
.. .lPAne**&
. . .. . . .. Dra-nma
. --------. --r-------
‘P
F@. 3 and 4 reveal two very important properties. First,
infinite-acting data lying on the dimensionless external radius
of 1000 branch fits either type-curve equally well. This is due
in part to the logarithmic approximation being valid for
= ~
(Q/A~)M
-,
.,
r
Jmf......
(10)
. . . . . . . . . . . . . . . ~-.,
Cr (QdD) M
Eq. 18 can also be used to determine the external drainage
radlux
950
‘e
=
a suitable match of the data and the type-curve are made.
One specific advantage of this technique is the match
between the field data and the analytic solution can be
displayed on one graph. Dimensionless rate and cumulative
production data during the infinite-acting period used in Fig.
2 obtained from Ehlig-Economides9 can alternative
obtained by combining van Everdingen and Hurst J,:
Sengulll. With infinite-acting dimensionless rate and cumulative tabular dat~ branches for specific dimensionless external
radks can be generated using eqs. 8 through 12. The
exponential solution, Eq. 7, can be used to generate boundary-dominated data after a tDA >0.1.
J$fi ”””””””””’”””””””””””””””””””@’)
To calculate permeability and skh, enough early time data
must be available to determine a dimensionless external
radius. Selecting a dimensionless external radius combined
with the effective external radius calculated from the area
(eq. 18) provides the apparent wellbore radius. Rearrangement of eq. 3
‘e
r wa =—.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(21)
‘eD
Application to Gas Reservoirs
Two major assumptions, constant fluid compressibility and
constant fluid viscosity, inherent to the development of the
liquid solution require additional handling for the prediction
of flow rates and pressures for gas reservoirs. In 1%7 AlHussainy et al. 12 defined gas pseudopressure as
Allows skb to be calculated using rearrangement of eq. 4.
An assumption of reservoir geometry is not required to
solve for reservoir size or skin effect because the reservoir
shape factor is not involved. To determine permeabMy, an
assumed geometry (usually radial) is used to calculate B (eq.
12 or Table 1- General). No signiihnt difference occurs
selecting among other symmetrical drainage patterns such as
a well in the center of a square.
The vertical axes alignment along with a calculated or
approximated value of fi is used to determine permeability
~ _
5
J.G. CALIARD AND P.A. SCHENEWERK
SPE 030793
14123UJ3
11
@?/Ap)M
.
p
P* = Z &p
( pz
Where the compressibility factor, z, and the viscosity, p (cp),
are pressure dependent functions.
Gas pseudopressure represents the potential difference or
driving force of fluid flow in the reservoir. Substitution of
pseudopressure in dimensionless rate results in the following
definition for gas reservoirs
,md . . . . . . . . . . . . . . . . . . (22)
.
@dD)M
Rate-Cunmlative
data
Dimensionless
D
h
Ad
u
— Olatiw
c
1422qOT
n
%)=
F%mneter
mock
of Spread-sheet
ueed for Type-Curve
‘a
k#(pPi-pPwf)
feA\
““”” ”””” ”””” ”””” ”””” ”””” (L+)
Where q is the gas production rate (MCF/d), T is temperature ~R ! and k is the permeabtity to gas (red). Declinecurve dlmenslo
“1$ ess rate can be obtained by eq. 9.
By replacing pressure with pseudopressure, drawdowns of
gas reservoirs during the infinite-acting time period can be
analyzed using semilog and type-curve matchktg techniques.
During boundary-dominated flow, gas wells producing at
constant pressure do not follow the exponential decline
predicted by the liquid solution. This was demonstrated in
1985 by C~er13, who presented a family of type curves
correlated by a parameter describing the severity of the
t+mwthun
~h~ detiadra.+JdoW~;the geater tht=
-“ .-.
.....-., . . ..$ the
---- lar~e~
.- ~-
L
Fig. 5- Sohemetic
. . . . . . . . . . . . . . . . . . . . . . . . . . . .. (23)
Matohing
Another technique, promoted here, is to obtain performance history matches in a computer spread-sheet. Incorporating the elements of Fig. 2 with the field data and a
parameter block, containing all reservoir parameters used in
the dlmensiordess variables, can be utiliied to non-dimensionalize the field data and compare it to the dimensionless
liquid solution. Fig. 5 shows the spread-sheet schematically.
External radius, permeability and skin can be adjusted until
tion from the liquid solution for gas reservoirs producing
under the condition of constant BHFP.
To account for the changes in viscosity and compressibility
in dimensionless time, Fraim and Wattenbarger14 in 1987 introduced a normaliid time function that drew together the
family of curves presented by Carter 13 into a single curve,
the liquid solution.
951
RESERVOIR PERFORMANCE
6
HISTORY MATCHING
USING RATE/CUMUb4TlVE
SPE 030793
TYPE-CURVES
Viscosity-Compressibility normtilzed time is defined as:
fn(p -c)
‘OQdf
. .. .. ...
[ WI
=
—
Q.(U -c) =
. . . . . . . . . . . . . . . . (25)
=
“~s~gfn(lf
@(Pc~)i
-c)
9.WQn(U.c)T
. . . . . . . . . . . . . . . . . . . . . .
~toaan
.(27)
. . . . . . . . . . . . . . . . . . . . .
—
I Pcl
A derivation for normalized cumulative paralleling that of
normalized time by Fraim and Wattenbarger14 can be found
in reference 15 and results in the definition of viscosity-compressibility normalized decline-curve cumulative:
In eq. 25 viscosity and compressibility are evaluated at
average reservoir pressure. Dimensionless normalized
decline-curve dimensionless time becomes
t~~
‘@cf)idQ
(26)
..
. . . . . . . . . . . . . (28)
Q@ =
2 (Ppl.-PPWf)a
~]~(~cl) i ‘Wa
Fu. 6 presents simulator generated production versus both
dimensionless time and versus dimensionless normalized
time for “Case 1 - Circular reservoir” from Fraim and
Wattenbarger 14. This technique involves successive approximations of gas in place (GIP) using the gas material balance,
to interrelate average pressure through cumulative production to time. The method of computation for normalized
time requires a summation of time steps that is sensitive to
step size.
The additional subscript “(p-c)” in the variables defined in
eqs. 25 and 27 indicate Viscosity-compressibility normalization.
Handling viscosity and compressibility in the cumulative
term also provides a simpler computation method for
normalization since fractional recovery, Q/GIp and p/z are
linearly related by the material balance equation:
. . . . . (29)
0
The integration in Eq. 27 can then be evaluated at intervals
of P/z as shown in Fig. 7.
A..\\i\\\%.[z
.
—
0.01 j
‘.
1,.
1.
4.
lquid sdutii
:
i
OdOJ& , ,
,i h ,
Ml
0.1
1
h
.; ~
.
‘.
I
gr’atity
= 0.601
Gas’
‘ \.
~
n.k
0.5
104
.
p
0.3
10
tdo
‘“’
02
\\
.. ..
0.1
Fig. 6- RTDTC:
Wattenbsrger’4)
Gas Well with Constant
j,,
BHFP (after Fraim and
,,,
o
0.1
o
0.2
\
,,,
41
0.4
4.6
;01
0,8
0?
08
.\,
,
+0
0.$
1
Q/GIP or (1-(P/z)/(P/zh)
,
Normalized Cumulative.
The constant rate/constant pressure identity revealed in
Fig. 4 suggest that it would be desirable to handle pressure
dependent viscosity and compressibility in the dimensionless
cumulative term. Using this technique, gas wells with
variable rate and variable flowing pressure could be plotted
as pseudopressure normalized production rate (PPNR) and
. . . ..Aa...,a.....-a
..a.-.. nA-nAcumumuvc
“..-..1 -*:.... p---4..
-.:-- [r
/DmNTm\
maul
G IIUI UICIWXU
UUUCUUII
r INU~
y.=uuup,
on the RCDTC. This was investigated and found to be
effective. Viscosit y-compressibility normalization of cumula-
952
Fig. 7 - Viscosity-Compreasibiiity
Raoovery
Produet
Rstio and Fn Versus
Also shown in Fig 7 is the ratio of normalized cumulative
production to actual cumulative production, or the viscosityctxnpressib]iity normalizing factor F-,..
..:
n~~-cj
Qn(u -c)
‘n(fl
-c)
=—
.
Q
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(30)
7
J.G. CALLARD AND P.A. SCHENEWERK
SPE 030793
limited amount of flowing pressure data available and
because the drawdown is variable in pressure and variable in
rate. Table 2 presents reservoir and production data.
The numerically simulated data was generated for a well in
the center of a square. The data plot for this is presented in
Fig. 9 showing PPNR versus PPNC. The immediate observation is that all data is concave to the origin indicating
boundary-dominated data and therefore the RCDTC can be
used.
The normalizing factor (upper curve) and the viscositycompressibdity product ratio (lower curve) are shown versus
fractional recovery for the fluid properties associated with
“Case 1 - Circular reservoir”. Also shown as solid triangles
along the lower curve are viscosity-compressibility product
ratio data from Fraim and Wattenbarger 14. Techniques for
calculating viscosity and compressibility are developed in
Reference 15. Normalized cumulative production of field
data can then obtained by rearrangement of eq. 30:
Qn(fl-c) =~,,(u-c)
Q”””””””””-”””-”-”
.“--”””
(31)
Table 2- Reservoir and Production
“Case 1“
Therefore, cumulative production combined with a choice of
GIP yields fraction recovery. And fractional recovery yields
the viscosity-compressibility normalization factor by numerical integration of gas fluid properties.
Rate data from Fig. 6 was used with cumulative production
obtained by re-simulating Fraim and Wattenbarger14 “Case
1 - Circular reservoir” using a personal computer (PC)
version of Boast II lb and is presented on the RCDTC show
in Fig. 8.
0.3
Permeability to gas
-.. —
UP
Height
Temperature
Porosity
4.0s
80
636
10
Rate
md
BCF
ft
“R
%
0.7
Gas gravity
Gas Saturation
Initial Pressure
Year
Data for Garb
75
2500
Cumulative
Ww
%
psia
BHFP
*
PP
JwU!2E!
10~—–——-—–———-
o
0
0
1
2
Im
10C4)
365
730
3
4
800
800
600
m
400
4002044
5
6
7
8
2500
1604
1361
1022
1352
1153
1216
1071
1197
1107
1314
1533
1752
1898
.4767 + E9
.2108+E9
.1538+E9
.1519+E9
.1116+E9
.1238+E9
.9762 + E8
.1200+E9
.1032+E9
.4
UJ{!W-,
–---r-,
~~lTmJ--Y
,,lr,
-
W
Fig. 8- RCDTC: Gas Well With Constant
BHFP
Two distinct advantages of using the RCDTC have now
been demonstrated. Most importantly, constant pressure and
constant rate solutions are identical, providing the basis for
variable pressure variable rate analysis using PNR and PNC
for single phase liquid flow and PPNR and PPNC for single
phase gas flow. Secondly, for gas reservoirs, accounting for
viscosity-compressibility normalization in the dimensionless
cumulative term gives unique results without regard to stepsize of the field data and normalizes singie phase gas fiow to
the liquid solution. Both of these advantages will be demonstrated in the following application.
Id
MOOl
—T——r-
y
0.001
Q/(m-m’n MrFhi--%
Fig. 9- oats Plot for Garb ~.’7
Case 1.
The cumulative normalization factor was determined as a
function of gas fluid properties similar to Fig. 7 and a
polynomial curve fit of the factor as a function of fractional
Example Application: Gas Well
Data for this example comes from Garb et al. 17, and also
Rodgers et al. 18. This e~mple was selected because of the
953
Reservoir
8
PERFORMANCE
HISTORY MATCHING
USINGRATE/CUMUUTIVE
/
/
SPE
‘- 030793
ty normalized cumulative using the rate-cumulative typecurve or semilog techniques. Boundary-dominated data,
concave to the origin, can be analyzed with the RCDTC
(Fig. 2) using viscosity-compressibility normalized cumulative. Fermeabdity and skin can be determined from a match
of the infinite-acting data and Area (or GIP) can be determined from boundary-dominated data. A flow chart for this
procedure is presented in Fig. 11.
recovery was generated:
Fn(ll-c) = a
+’[%J+’[&+’[&
WPE-CURVES
F””’32’
With a = 0.990
b = -0.579
C =
0.358
d = -0.238
“
Known permeability, GIP, and apparent wellbore radius
-----:--..4 :...
- .LZ,-n . allluLU
. . . .. . kl-,-~
crm=arl.chw=t
.,,;*
;“ tk- o=.
WG1 G Illpul
111LU L1lG pal
u,uem
w,. !’. . . . .Hw
wau
o..””.
Compik hit.ml Completion Data
G+mlate BtIfT’s
Generate PM Pro@.v Tab!+
Poiymxniai
resulting in the match shown in Fig. 10,
~
Cimwt
\
Pklt Pm
\
iltt
Bate
and
+
id
rhiiti-t)
Ma
to PF?W and PPN(
If
Bxmdarv
DOnun.9kl
->
(-
J
Fp
Fhsrum
w PPNC
Mennine
:
RR
,
Yes
No
MA from Garb’sMe 1
‘:-----’’-~””
+
c“
%“
i
%
.
\,
0.1;
J
Detennme
pe&eatility’ and dim
L
_——’
,
Ftg. 11-
.
I
0.0 :r
01
Conclusions
Use of the liquid solution constant pressure rate-cumulative decline type-curve (RCDTC) can be extended to singlephase flow of compressible gases via the use of the viscositycompressibility normalization factor and gas pseudopressure.
Like gas pseudopressure,
the viscosity-compressibility
normalization factor can be determined from fluid properties
alone.
Because of the independence in step size of time intervals
in the determination of the viscosity-compressibility normalization factor, use of the RCDTC is superior to use of the
rate-time decline type-curve (RTDTC) even for wells
producing at constant BHFP.
10
Gas Well with Variable
Flow Chart for Gas Well Analysis
I
Qdn
Fig. 10- RCDTC:
i
5,
b
BHFP
The data show excellent agreement with the liquid solution
constant pressure RCDTC demonstrating the ability to
handle the variable BHFP case for gas reservoirs.
Type-Curve Matching Techniques: Gas Wells
Two preparation steps are required to analyze field declinecurves for gas wells. First, calculation of BHFP from lTP
must be performed for all data. This can be done most
efficiently in a programming language and the results
imported to a spread-sheet that contain the rate and cumulative data as described in Fig. 5.
The second step is to, again, use a program to calculate
compressibility factors, compressibility, and viscosity for the
gas gravity and temperature of the reservoir. Integrations can
be performed in the program to obtain gas pseudopressure
and viscosity compressibility normalizing factor. polynomial
tits, such as the one presented in the example application for
the normalizing factor, can also be made for gas pseudopressure as a function of BHFP. The coefficients for these two
fits can then be incorporated into the spread-sheet.
A data plot of PPNR versus PPNC is then made and flow
periods present are determined. Infinite-acting data, convex
to the origin, can be analyzed without viscosity-compressiblli-
Nomenclature
A= area (sq ft)
BHFP= bottom-hole flowing pressure (psi) same as P~
B= formation volume factor (rb/STB)
Bbl= barrel (5.615 ft3)
CA= Dietz shape factor
et= system total compressibility (psi-l)
FTP= flowing tubing pressure (psia)
F n(p-c) = viscosity-compressibility normalizing factor
‘n(m-c = mobility-compressibility normalizing factor
Gd=
gas in place (Mcf)
h= formation thickness (ft)
k= permeability (red)
kg= permeability to gas (red)
954
SPE 030793
PNR = pressure normalized production rate (STB/d/psi)
PNC= pressure
normalized
cumulative
production
(STB/psi)
PPNR = pseudopressure normalized rate
PPNC= pseudopressure normalized cumulative
pD . dimensionless pressure
pdt) = decline-curve dimensionless pressure
gas pseudopressure (psiz/cp)
}:
initial pressure (psia)
Pp;= initial pseudopressure (psi2/cp)
P*= flowing bottom-hole pressure (psia)
Ppd= flowing bottom-hole pseudopressure (psi2/cp)
q= flow rate (STB/d)
9g= gas flow rate (MCF/d)
qD . dimensionless flow rate
qdD = decline-curve dimensionless flow rate
Q= cumulative production (STB for oil, MCF for gas)
Qn(u.c)= viscosity-compressibility normalized cumulative
production (MCF)
dimensionless cumulative production
~QD :
=Ctil
RCDTC=
RTDTC=
rw=
rm =
re=
reD=
s=
STB =
T=
t=
*n(u-c) =
tD =
fDA =
tdD=
Vp=
z=
~~~!~~~-rllweAim
---
. - ~A...
en.innlec.c,,m,,l
”..”.
”...
w,”uu
...
2
3
4
5
6
7
8
-.=..nti.,~p~~~~~f~~fi
. “
decline-curve normalizing factor
factor
13= decline-curve normaltilng
@= porosity (fraction)
p.
fluid viscosity (cp)
Subscripts
M=
the Production Capacity of a Well,” Trans. AIME (1953)
198, 171-176.
Uraite, A.A. and Raghavan, R.: “Unsteady Flow to a
Well Producing at a Constant Pressure,’’.lPT (Oct. 1980)
1803-12.
Tsarevich, K.A. and Kuranov, I.F.: “Calculation of the
Flow Rates for the Center Well in a Circular Reservoir
under Elastic Condhions,” Problents of Reservoir Hydrodynamics, Part Z, Leningrad (1956) 9-34.
Eadougher, R.C. Jr.: Advances in Wel[ Test Analysis,
Henry L. Doherty Series, SPE, Richardson, TX (1977)
5.
Chen, H.Y. and Poston, S.W.: “Application of a Pseudotime Function To Permit Better Decline-Curve Analysis,” SPEFE (Sep 1989) 421-428.
Fetkovich, M.J.,: “Decline Curve Analysis Using Type
Curves,” JPT (June 1980) 1065-77.
Ehlig-Economides, C.A. and Ramey, H.J. Jr.: ‘Transient
Rate Decline Analysis for Wells Produced at Constant
Pressure,” SPEJ (Feb 1981) 98-104.
&li~Uuh~r.
0----,
R
-----C
.Tr
-..,
. . .. Rame.v
-._--. -J, H
---- .1 Jr
Miller
C,
. .. ... . . . .F .-.,
and
----
Mueller, T.D.: “Pressure Distributions in Rectangular
Reservoirs,” JPT (Feb. 1%8) 199-208.
9 Ehlig-Economides, C.A.: “Well Analysis for Wells
Produced at a Constant Pressure,” PhD dissertation,
Stanford U., Stanford, CA (June 1979).
10 van Everdingen, A.F. and Hurst, W.: ‘The Application
of the Laplace Transformation to Flow Problems in
Reservoirs,” Trans., AIME (1949) 186,305-324.
11 Sengul, M.M.: “Analysis of Step-Pressure Tests,” paper
SPE 12175 presented at the 1983 Annual Technical
Conference and Exhibition, San Francisco, Oct. 5-8.
12 A1-Hussainy, R., Ramey, H.J., Jr., and Crawford, P.B.:
ml-.-..–]m–
--- n. .>..
“ rr,w
ml-n---- –L-n –-1 n---HOW 01 Keal Uases
I nrougn rorus Meala, Jr f
I m
(May 1966) 637-64% Trans., AIME, 237.
13 Carter, R. D.: “Type Curves for Finite Radial and Linear
Gas-Flow Systems Constant-Terminal-Pressure
Case,”
SPEI (Ott 1985) 719-28.
14 Fraim, M.L. and Wattenbarger, R.A.: “Gas Reservoir
Decline-Curve Analysis Using Type Curves with Real
Gas Pseudopressure and Normalized Time,” SPEFE
(Dee 1987) 671-682.
15 Canard, J.G.: “Reservoir Performance History Matching
Using Type-Curves,” PhD Dissertation, Louisiana State
U., Baton Rouge, LA (1994).
16 Stapp, L.G. and Allison, E.C.: “Handbook for Personal
Computer Version of Boast II: A Three-Dimensional,
Three-Phase Black Oil Applied Simulation Tool,” U.S.
Department of Energy Bartlesville Project Office,
Bartlesville, Ok. (Jan. 1989).
17 Garb, F.A., Rodgers, J.S., and Prasad, R.K.: “13nd Gas
In-Place from Shut-In or Flowing Pressures,” Oil& Gas
J. (July 1973) 58-64.
rate-cumulative decline type-curve
rate-time decline type-curve
wellbore radius (ft)
apparent wellbore radius (ft)
external radius (ft)
dimensionless external radius
dimensionless skin
stock tank barrel (5.615 ft3)
reservoir temperature ~R)
time, days
viscosity-compressibility normalized time (days)
dimensionless time
ciimensioniess time based on drainage area
decline-curve dimensionless time
pore volume (Bbl)
gas compressibility factor (dimensionless)
Greek
a=
9
J.G. CALIARD AND P.A. SCHENEWERK
match point in type-curve matching
Acknowledgments
The author recognizes the Department of Energy grant
SBIR/DOE
DE-FG05-90ER80976
and the Society of
Petroleum Engineers for financial contributions.
References
1 van Everdingen, A.F.:” The Skin Effect and Its Influence
955
.
10
RESERVOIR PERFORMANCE
HISTORY MATCHING
USING RATE/CUMULATIVE
18 Rodgers, J.S., Iloykin, R.S., and Cobie, L.E.: T?onstatic
Pressure History Analyses for Gas Reservoirs,” SPEY
(April 1983) 209-18.
S1 Metric Conversion Factors
E-01 = m3
bbl x 1.599873
Cpx 1.0
E-03 = Pas
CU ft x 2.831685
E-02 = m3
E-01 =m
ft x 3.048”
E-04 = pmz
md x 9.869233
EOO =kPa
psi x 6.894757
‘R X 5/9
= ‘K
‘G3nvaraion
factor is exact,
956
TYPE-CURVES
SPE 030793