SPE 107981
Estimation of Reserves Using the Reciprocal Rate Method
P.D. Reese, SPE, D. Ilk, SPE, and T.A. Blasingame, SPE, Texas A&M U.
Copyright 2007, Society of Petroleum Engineers
This paper was prepared for presentation at the 2007 SPE Rocky Mountain Oil & Gas
Technology Symposium held in Denver, Colorado, U.S.A., 16–18 April 2007.
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Abstract
In this work we develop, validate, and apply the "reciprocal
rate method" to estimate oil reserves using only rate-time
production data. This approach requires the development of
boundary-dominated flow, and can be used to validate reserve
extrapolations from numerical/analytical reservoir models.
The methodology does presume that flowing well bottomhole
pressures are approximately constant — but we will
demonstrate that the method is tolerant of substantial changes
in the flowing bottomhole pressure.
This approach requires a plot of the reciprocal of flowrate
(1/q) and the so-called "material balance time" (cumulative
production/flowrate or Np/q). The "secret" to this approach is
the use of material balance time — this function accounts for
most variation in rate/pressure, and permits the extrapolation
of the 1/q function.
This methodology has been applied for oil and gas wells
(including oil wells with high water production) — and in all
cases, the reciprocal rate method has proven to be robust and
consistent.
The primary technical contributions of this work are:
● Direct method to estimate reserves using only rate-time
data (time, rate, and cumulative production).
● The reciprocal rate method is based on variable-rate
theory, and is more rigorous than Arps approach (exponential or hyperbolic rate relations).
Introduction
Simply put, the Reciprocal Rate Method is an unsophisticated,
yet theoretical approach for estimating reserves. The governing equation is derived in Appendix A — and for convenience
is given as the "Arps" form of the result: [Arps (1942)]
1 1 Di ⎡ Np
= +
⎢
q qi qi ⎢⎣ q
⎤
⎥ ............................................................. (1)
⎥⎦
Where the qi and Di parameters can be derived from theory for
the black oil case (see Appendix A).
For reference, the Arps exponential model is given as:
q = qi e− Dit ......................................................................... (2)
It is important (perhaps critical) to note that Eq. 1 (and 2) are
derived under the assumption that the well is producing at a
constant flowing bottomhole pressure, pwf. Our contention is
that the Reciprocal Rate Method is robust and will tolerate
changes in pwf, particularly smooth changes. We illustrate the
robustness of this method using appropriate field examples.
For convenience, we write Eq. 1 as a simple straight-line
relation with arbitrary coefficients. This form is given as:
⎡ Np
1
=c+m⎢
q
⎢⎣ q
⎤
⎥ ................................................................. (3)
⎥⎦
Multiplying through Eq. 3 by the flowrate term (q) yields:
1 = c q + m Np .................................................................... (4)
At depletion, the flowrate will decrease to zero (i.e., q → 0),
and Eq. 4 reduces to the following identity:
( Np ) q →0 ≡
1
(reserves)................................................... (5)
m
The procedure for this methodology is as follows:
Step 1: Plot 1/q versus Np/q.
Step 2: Estimate the slope of the straight-line portion of the
data trend, m. As advice, the "later" data should
yield the most consistent trend.
Step 3: Take the reciprocal of the slope (m) as the estimate
of the reserves which will be produced at depletion
(boundary-dominated flow regime) for this particular production scenario.
As noted above, the single most important constraint is the
assumption of the constant flowing bottomhole pressure —
however; we will demonstrate the utility of this approach,
even in the presence of erratic changes in the flowing
bottomhole pressures.
Demonstration
The purpose of this work is to provide the practicing engineer
with a theoretically robust, yet extraordinarily simple methodology to estimate reserves using production performance data
(in this case rate-time data only). Having said that, there are
limitations — in particular, for gas wells which do not exhibit
"liquid" character (exponential rate decline), our success has
been variable, and this case remains a work in progress. However; for the case of oil wells we have had considerable (almost universal) success — less-than-desired results for oil
cases tend to occur when some portion of the data is corrupted
by a substantial "non-ideal" condition (e.g., the continuous
2
P.D. Reese, D. Ilk, and T.A. Blasingame
SPE 107981
evolution of formation damage as will be indicated by one of
the example cases).
In this section we present 3 case histories (all oil wells) where
the Reciprocal Rate Method has been successfully applied.
These cases are presented in relevant detail below:
haps should) provide a simple reservoir signature. One "test"
would be to re-plot the data for this case as a log-log
reciprocal rate plot. This plot could serve as a diagnostic to
confirm that the straight-line observed on the Cartesian plot is
actually a relevant (reservoir) response.
Example 1: Well NRU 3106 (Texas, USA)
In Fig. 3 we present the log-log reciprocal rate plot for this
case (Well NRU 3106 (Texas, USA)). We immediately note
that, while we might "adjust" the model constant, the translation of the reciprocal model to the log-log plot (straight line
trend → power-law trend) confirms the validity of the
(Cartesian) reciprocal rate plot.
This well is a production well in a low permeability reservoir
that is being waterflooded at pressures above the fracture
gradient (continuous fracture propagation is likely in the
injection wells). The oil and water production profiles for this
case are shown in Fig. 1.
Reciprocal Rate Method
Well NRU 3106 (Texas, USA) — Log-Log Reciprocal Rate Plot
Reciprocal Rate Method
Well NRU 3106 (Texas, USA) — Production History Plot
0
10
3
Legend: Well NRU 3106
qo Function
qw Function
qo Exponential Rate Model
Reciprocal of Oil Rate, 1/qo,1/STB/Day
2
10
1
10
1
-3
qo = 4.8771x10 exp(-5.61878x10 t), STB/D
(Np)max = Reserves = 86,000 STB
4000
3500
3000
2500
2000
1500
1000
500
-1
10
-2
10
-2
-3
10
-4
10
0
10
10
Production Time, Days
The Cartesian format plot of 1/q versus Np/q — i.e., the
Reciprocal Rate Method plot for this case is shown in Fig. 2.
We note an excellent data trend, and we can clearly see that
the method will yield relevant results for this case.
Reciprocal Rate Method
Well NRU 3106 (Texas, USA) — Reciprocal Rate Plot
-2
-5
0.10
(Np)max = Reserves = 86,000 STB
0.09
10
3
4
5
10
10
This case considers a production well in a high permeability
oil reservoir that is simultaneously experiencing waterflood
and water influx. Our rationale in presenting this case is to
validate that the proposed reciprocal rate model can be applied
to reservoir behavior that is experiencing addition of energy
from "outside" of the reservoir. The historical performance
data for this case is presented in Fig. 4.
Reciprocal Rate Method
Well WWL B41 (Louisiana, USA) — Production History Plot
4
0.06
1
10
2
(Np)max = Reserves = 360,000 STB
4000
3500
0
10
In Fig. 2 it is clear that the reciprocal rate method has some
merit — from empirical as well as theoretical standpoints.
The most common observation is that the 1/q versus Np/q
trend is excellent — but why? The most obvious answer is
theory — this case considers a black oil that is being repressured by waterflooding, which certainly could (and per-
-3
qo = 4.4009x10 exp(-1.22247x10 t), STB/D
3000
Figure 2 — Example 1: Cartesian reciprocal rate plot — Well
NRU 3106 (Texas, USA). Excellent match of the
data functions for this case.
2
10
2500
Oil Material Balance Time (Np/qo), Days
3
10
2000
9000
8000
7000
5000
4000
3000
2000
0.01
6000
Legend: Well NRU 3106
1/qo Function
1/qo Linear (Np/qo) Model
0.02
500
0.03
Legend: Well WWL B41
qo Function
qw Function
qo Exponential Rate Model
0
0.04
Oil and Water Rates, qoand qw, STB/D
10
0.05
0
2
Example 2: Well WWL B41 (Louisiana, USA)
0.07
0.00
10
Figure 3 — Example 1: Log-log format reciprocal rate plot —
Well NRU 3106 (Texas, USA). Excellent match of
data confirms reciprocal rate concept model.
0.08
1000
Reciprocal of Oil Rate, 1/qo,1/STB/Day
0.12
1/qo = 1.9545x10 + 1.15207x10 (Np/qo), 1/STB/D
1
Oil Material Balance Time (Np/qo), Days
Figure 1 — Example 1: Oil and water rate history plot — Well
NRU 3106 (Texas, USA).
0.11
-5
1/qo = 1.9545x10 + 1.15207x10 (Np/qo), 1/STB/D
(Np)max = Reserves = 86,000 STB
1500
0
0
10
Legend: Well NRU 3106
1/qo Function
1/qo Linear (Np/qo) Model
1000
Oil and Water Rates, qoand qw, STB/D
10
Production Time, Days
Figure 4 — Example 4: Oil and water rate history plot — Well
WWL B41 (Louisiana, USA).
The Cartesian format plot of 1/q versus Np/q — i.e., the
Estimation of Reserves Using the Reciprocal Rate Method
Reciprocal Rate Method
Well WWL B41 (Louisiana, USA) — Reciprocal Rate Plot
0.050
Legend: Well WWL B41
1/qo Function
1/qo Linear (Np/qo) Model
Reciprocal of Oil Rate, 1/qo,1/STB/Day
0.045
0.040
3
Reciprocal Rate Method
SE Asia Oil Well — Production and Pressure History Plot
4
5000
10
Legend: SE Asia Oil Well
Oil Flowrate (qo), STB/D
Wellbore Flowing Pressure (pwf), psia
4500
4000
3500
3000
3
2500
10
Oil Flowrate
2000
1500
0.035
1000
Wellbore Flowing
Pressure
500
0.030
0
225
200
175
150
125
100
75
50
25
0.025
0
2
10
Wellbore Flowing Pressure (pwf), psia
Reciprocal Rate Method plot for this case is shown in Fig. 5.
This case also provides an extraordinary correlation trend for
the 1/q — Np/q data. This performance strongly confirms the
reciprocal rate method, and suggests that we should expect the
method to work for cases of external reservoir drive energy.
Oil Flowrate (qo), STB/D
SPE 107981
0.020
Production Time, days
0.015
Figure 7 — Example 3: Oil rate and pressure history plot —
Southeast Asia — Oil Well.
-3
-6
1/qo = 2.4024x10 + 2.77778x10 (Np/qo), 1/STB/D
0.005
14000
13000
12000
11000
10000
9000
8000
7000
6000
5000
4000
3000
0
1000
2000
(Np)max = Reserves = 360,000 STB
0.000
Oil Material Balance Time (Np/qo), Days
Figure 5 — Example 2: Cartesian reciprocal rate plot — Well
WWL B41 (Louisiana, USA). Very good correlation
of behavior.
As in the previous case, we would like to assess the general
character of the 1/q versus Np/q concept model. In Fig. 6 we
present the log-log reciprocal rate plot for this case (WWL
B41 (Louisiana, USA)). We immediately note that, while we
might "adjust" the model constant, the translation of the
reciprocal model to the log-log plot confirms the validity of
the (Cartesian) reciprocal rate plot.
Reciprocal Rate Method
Well WWL B41 (Louisiana, USA) — Log-Log Reciprocal Rate Plot
0
-3
-6
1/qo = 2.4024x10 + 2.77778x10 (Np/qo), 1/STB/D
(Np)max = Reserves = 360,000 STB
-1
10
In Fig. 8 we present "Arps" analysis (semilog rate-time plot
with the Arps exponential model imposed on the data).
Obviously the rate profile mimics the influences of the pressure behavior, except at the latest times, where damage
appears to be continuously evolving.
Reciprocal Rate Method
SE Asia Oil Well — Production History Plot
4
10
Legend: SE Asia Oil Well
qo Function
qo Exponential Rate Model
Change in Reservoir Model
(Increasing Damage)
3
10
-3
qo = 1546exp(-3.0x10 t), STB/D
(Np)max = Reserves = 515,000 STB
(Exponential Trend Not Applicable)
225
200
175
150
125
100
75
50
10
25
2
10
-2
0
Reciprocal of Oil Rate, 1/qo,1/STB/Day
10
From Fig. 7 we note that the late-time rate and pressure are
"off-trend," and while we do not know the cause, we do
observe a degradation in the rate performance with time
(evolving well damage may be the culprit).
Oil Flowrate, qo, STB/D
0.010
Production Time, Days
Figure 8 — Example 3: Oil rate history plot with exponential
rate model imposed — Southeast Asia — Oil Well.
-3
10
Legend: Well WWL B41
1/qo Function
1/qo Linear (Np/qo) Model
-4
10
0
10
1
10
2
10
3
10
4
10
5
10
Oil Material Balance Time (Np/qo), Days
Figure 3 — Example 2: Log-log reciprocal rate plot — Well
WWL B41 (Louisiana, USA). Excellent performance of the reciprocal rate concept model.
Example 3: Southeast Asia — Oil Well
This case is somewhat unique in that the data appear to be
relevant (comparison of multiple analyses), but the Arps ratetime relation clearly fails to represent the behavior for this
case. The original data for this case are shown in Fig. 7.
As noted, the trend shown in Fig. 8 appears to be valid — but
this trend is found to be inconsistent with all of the other
analyses that have been applied to these data (estimated
reserves are almost a factor of 2 too high).
In Fig. 9 we present the Cartesian format plot of 1/q versus
Np/q (i.e., the Reciprocal Rate Method plot). The obvious
features which are not aligned are the early-time (transient)
flow data, as well as the "late-time" data which appear to be
(strongly) affected by evolving reservoir damage (cause/
mechanism unknown).
4
P.D. Reese, D. Ilk, and T.A. Blasingame
reserves for this system based on:
Reciprocal Rate Method
SE Asia Oil Well — Reciprocal Rate Plot
( Np ) q →0 ≡
Legend: SE Asia Oil Well
1/qo Function
1/qo Linear (Np/qo) Model
0.00150
Change in Reservoir Model
(Increasing Damage)
0.00125
0.00100
(Np)max = Reserves = 313,000 STB
0.00075
-4
-6
1/qo = 2.676x10 + 3.1948x10 (Np/qo), 1/STB/D
0.00050
500
450
400
350
300
250
200
150
0
0.00000
100
0.00025
50
Reciprocal of Oil Rate, 1/qo,1/STB/Day
0.00200
0.00175
Oil Material Balance Time (Np/qo), Days
Figure 9 — Example 3: Cartesian reciprocal rate plot — Southeast Asia — Oil Well. Good correlation of mid-time
behavior (late-time data are affected by increasing
damage).
The final view of the data is provided by the log-log format
reciprocal rate plot (Fig. 10). In this case we find an acceptable match of the late-time data, but clearly the early time data
are not well represented by the reciprocal rate model (nor
should these data be). The issue for this analysis is that the
data for this case have been reviewed and analyzed repeatedly,
and were found to be consistent (with the noted exception of
the late time damage signature). The issue could be a slight
mis-match in the pressure and rate — but again, rigorous,
model-based analysis suggests that these data are valid.
Reciprocal Rate Method
SE Asia Oil Well — Log-Log Reciprocal Rate Plot
-1
10
Reciprocal of Oil Rate, 1/qo,1/STB/Day
-4
-6
1/qo = 2.676x10 + 3.1948x10 (Np/qo), 1/STB/D
(Np)max = Reserves = 313,000 STB
-2
10
-3
10
Legend: SE Asia Oil Well
1/qo Function
1/qo Linear (Np/qo) Model
0
1
10
2
10
3
10
4
10
We have provided field demonstrations of this methodology
for the following cases:
● Well NRU 3106 (Texas, USA) — The Reciprocal Rate
Method has significant value as an example — a black
oil system being re-pressured by waterflooding. This is
a very common scenario, and we expect this case to
become a typical application for this method.
● Well WWL B41 (Louisiana, USA) — An extraordinary
correlation trend for the 1/q — Np/q data is observed in
this case. The performance of the 1/q — Np/q data
confirms the validity of the Reciprocal Rate Method
where external reservoir drive energy is present.
● Southeast Asia (Oil Well) — The "early-time" and
"late-time" data are not aligned, which we believe
prevents the overall success of the Reciprocal Rate
Method for this case The issues in this case are not
clear but we suspect that the rate and pressure data are
affected by an evolving formation damage at late times.
Conclusions:
1. The Reciprocal Rate Method is a basic yet theoretical
approach for estimating reserves using only rate-time
data (time, rate, and cumulative production). The
reciprocal rate method is based on variable-rate theory,
and is more rigorous than the Arps approach.
2. The Reciprocal Rate Method is assumes a constant
flowing bottomhole pressure — but we have shown that
the method should tolerate arbitrary changes in pwf —
particularly smooth changes.
3. In this work we have validated the applicability of the
method using field examples — but we have limited
our application to black oil systems for simplicity and
clarity. Further development is required for cases of
compressible fluids (see Appendices B and C for efforts related to gas wells).
4. The method appears to be particularly robust for cases
where external reservoir drive energy is present.
Nomenclature
-5
10
1
(reserves)................................................... (5)
m
Recommendations/Comment: Reciprocal Rate Method should
be extended/generalized for cases where the flowing fluid is
compressible.
-4
10
10
SPE 107981
5
10
Oil Material Balance Time (Np/qo), Days
Figure 10 — Example 3: Log-log reciprocal rate plot — Southeast Asia — Oil Well. Fair performance of the reciprocal rate concept model (weak transient/
transition behavior).
Summary and Conclusions
Summary: We provide the systematic development of the
Reciprocal Rate Method, which has as a base the following
relation:
⎡ Np ⎤
1
=c+m⎢
⎥ ................................................................. (3)
q
⎣⎢ q ⎦⎥
The slope of the "reciprocal rate" plot yields an estimate of the
A
=
bpss
=
Bo
=
Boi
=
cA
=
ct
=
Di
=
EUR =
q
=
qi
=
h
=
k
=
mmb =
N
=
Np
=
(Np)q→0=
Reservoir drainage area, ft2
Intercept term in Eq. A-1, psi/STB/D
Oil formation volume factor, RB/STB
Initial oil formation volume factor, RB/STB
Reservoir shape factor, dimensionless
Total compressibility, psi-1
Coefficient in Eq. A-14, 1/D
Estimated ultimate recovery, STB
Production rate, STB/D
Initial production rate, STB/D
Net pay thickness, ft
Average reservoir permeability, md
Slope term in Eq. A-1, psi/STB
Original oil-in-place, STB
Cumulative oil production, STB
Maximum oil production, STB
SPE 107981
p
pi
pwf
rw
s
t
t
μo
γ
=
=
=
=
=
=
=
=
=
Estimation of Reserves Using the Reciprocal Rate Method
Average reservoir pressure, psia
Initial reservoir pressure, psia
Bottomhole flowing pressure, psia
Wellbore radius, ft
Skin factor, dimensionless
Time, days
Material balance time (Np/q), days
oil viscosity, cp
Euler's constant (0.577216 …)
References
Arps J.J.: "Analysis of Decline Curves," Trans. AIME (1945) 160,
228-247.
Blasingame, T.A. and Lee, W.J.: "Variable-Rate Reservoir Limits
Testing," paper SPE 15028 presented at the SPE Permian Basin Oil
and Gas Recovery Conference, Midland, TX, 13-14 March 1986.
Blasingame, T.A. and Rushing, J.A.: "A Production-Based Method
for Direct Estimation of Gas-in-Place and Reserves," paper SPE
98042 presented at the 2005 SPE Eastern Regional Meeting held in
Morgantown, W.V., 14–16 September 2005.
Doublet, L.E., Pande, P.K., McCollum, T.J., and Blasingame, T.A.:
"Decline Curve Analysis Using Type Curves — Analysis of Oil
Well Production Data Using Material Balance Time: Application to
Field Cases," paper SPE 28688 presented at the 1994 Petroleum
Conference and Exhibition of Mexico held in Veracruz, MEXICO,
10-13 October 1994.
Fetkovich, M.J.: "Decline Curve Analysis Using Type Curves," JPT
(March 1980) 1065-1077.
Pratikno, H., Rushing, J.A., and Blasingame, T.A.: "Decline Curve
Analysis Using Type Curves — Fractured Wells," paper SPE 84287
presented at the SPE annual Technical Conference and Exhibition,
Denver, Colorado, 5-8 October 2003.
Appendix A: Derivation of the Reciprocal-Rate
Method for Estimating Reserves — Oil Case
The purpose of this derivation is to provide the rigorous basis
for the Reciprocal-Rate Method using a fundamental material
balance and the pseudosteady-state flow relation. Starting
with the oil material balance relation, we have:
Black Oil Material Balance Relation: (p > pb)
p = pi − mmb Np ............................................................ (A-1)
Where mmb is defined as:
mmb =
1 Bo
.............................................................. (A-2)
Nct Boi
We next utilize the pseudosteady-state flow relation (also for a
black oil). This relation is given as:
Black Oil Pseudosteady-State Flow Relation: (p > pb)
p = pwf + bpss q ............................................................. (A-3)
Where bpss is defined as:
μ B ⎡ 1 ⎡ 4 1 A ⎤ ⎤⎥
⎥ + s ......................... (A-4)
bpss = 141.2 o o ⎢ ln ⎢
kh ⎢ 2 ⎢ eγ CA rw2 ⎥ ⎥
⎣
⎦ ⎦
⎣
Combining Eqs. A-1 and A-3, and solving for (pi-pwf), we obtain the following identity:
( pi − pwf ) = m mb Np + b pss q ......................................... (A-5)
5
Rearranging Eq. A-5 yields the "material balance time" formulation, which is given as:
⎡ Np
Δp
= b pss + m mb ⎢
q
⎢⎣ q
⎤
⎥ .................................................. (A-6)
⎥⎦
Where Δp=(pi-pwf). Eq. A-6 is the basis of modern production
data analysis — and is known as the "material balance time"
formulation. Material balance time is defined as:
⎡ Np ⎤
t =⎢
⎥ ....................................................................... (A-7)
⎣⎢ q ⎦⎥
A plot of Δp/q versus Np/q yields a straight-line trend where
the slope of the line (mmb) is inversely proportional to the inplace fluid volume of the reservoir system. This is an older
approach for the analysis of production data [Blasingame and
Lee (1986)], but we note that more recent implementations of
this approach use the Δp/q and Np/q plotting functions (and
auxiliary functions) as diagnostic functions on log-log plots
[Doublet et al (1994)].
Returning to Eq. A-7, we would like to assume a constant
bottomhole flowing pressure (pwf), which yields a constant
pressure drop Δpcon. Dividing through Eq. A-7 by Δpcon yields
the following result:
⎡ Np
1 ˆ
= b pss + mˆ mb ⎢
q
⎢⎣ q
⎤
⎥ .................................................... (A-8)
⎥⎦
Where:
b pss
bˆ pss =
................................................................. (A-9)
Δpcon
mˆ mb =
m mb
............................................................... (A-10)
Δpcon
As noted, we have assumed a constant pressure drop (Δpcon) to
develop Eq. A-8 — however; this result provides the basis for
the reciprocal rate method, that can be used to estimate the
reserves (NOT in-place fluid volume for a given reservoir
system). Multiplying through Eq. A-8 by the flowrate term
(q), we have:
1 = bˆ pss q + mˆ mb Np ...................................................... (A-11)
As the flowrate decreases to zero (i.e., q → 0), Eq. A-11
reduces to the following identity:
( Np ) q →0 ≡
1
........................................................ (A-12)
mˆ mb
Where the maximum cumulative production (Np)q→0 is typically referred to as the Estimated Ultimate Recovery (EUR) or
the reserves of the system (NOT the in-place fluids — but
rather, the quantity of fluids that will be produced at infinite
producing time, at the current producing condition.
Therefore a plot of 1/q versus Np/q yields a straight-line trend
where the slope of the line (mˆ mb ) is inversely proportional to
the EUR or reserves of the reservoir system. Again we note
that a constant bottomhole pressure was assumed, but we will
demonstrate that the "reciprocal rate method" is robust and
useful method for estimating reserves.
As an alternative derivation of this method, one can use the
"exponential decline result" derived using the assumption of a
constant flowing bottomhole pressure (see Blasingame and
6
P.D. Reese, D. Ilk, and T.A. Blasingame
Rushing [Blasingame and Rushing (2005)] for generic detail
regarding the derivation of the exponential decline result).
Without derivation, the "exponential decline result" is given
as:
q = qi e
− Dit
................................................................... (A-13)
Where the following coefficients are given without derivation:
m
Di = mb .................................................................... (A-14)
bpss
qi =
1
( pi − pwf ) ..................................................... (A-15)
bpss
Integration of Eq. A-13 with respect to time yields:
Np =
1
(qi − q) ........................................................... (A-16)
Di
Solving Eq. A-16 for the flowrate, q, yields:
q = qi − Di Np ............................................................... (A-17)
Dividing through Eq. A-17 by the flowrate, q, gives us:
q
1 = i − Di
q
⎡ Np
⎢
⎢⎣ q
⎤
⎥ .......................................................... (A-18)
⎥⎦
Dividing through Eq. A-18 by the initial flowrate, qi, we have:
1 1 Di ⎡ Np ⎤
= −
⎢
⎥ ....................................................... (A-19)
qi q qi ⎣⎢ q ⎦⎥
Solving Eq. A-19 for the reciprocal flowrate (1/q) yields:
1 1 Di ⎡ Np ⎤
= +
⎢
⎥ ........................................................ (A-20)
q qi qi ⎣ q ⎦
Recalling our "rigorous result" (Eq. A-8) for comparison:
⎡ Np ⎤
1 ˆ
= b pss + mˆ mb ⎢
⎥ ..................................................... (A-8)
q
⎢⎣ q ⎦⎥
Substituting Eqs. A-9 and A-10 into Eq. A-8, we obtain:
b pss
1
m
=
+ mb
q Δpcon Δpcon
⎡ Np
⎢
⎢⎣ q
⎤
⎥ ............................................. (A-21)
⎥⎦
Rearranging Eq. A-15 yields:
bpss
bpss
1
=
=
............................................. (A-22)
qi ( pi − pwf ) Δpcon
Combining and solving Eqs. A-14 and A-15 for the quantity
(mmb/Δpcon) yields:
bpss
Di mmb
=
qi bpss ( pi − pwf )
mmb
=
( pi − pwf )
=
mmb
Δpcon
..................................................................................... (A-23)
Substitution of Eqs. A-22 and A-23 into Eq. A-21 gives us:
1 1 Di ⎡ Np
= +
⎢
q qi qi ⎢⎣ q
⎤
⎥ ....................................................... (A-24)
⎥⎦
Where Eqs. A-20 and A-24 are identical — which confirms
the "reciprocal rate formulation" from separate (albeit theoreti-
SPE 107981
cally tied) results.
As a final comment, the Di/qi term is the reciprocal of reserves
— that is:
Di
1
=
........................................................... (A-25)
qi ( Np ) q → 0
Or, in proper form, we have:
q
( Np ) q →0 = i ........................................................... (A-26)
Di
Substituting Eq. A-26 into Eq. A-24 gives us:
1 1
1
= +
q qi ( Np ) q → 0
⎡ Np
⎢
⎢⎣ q
⎤
⎥ ........................................... (A-27)
⎥⎦
As was noted earlier in this derivation (i.e., Eq. A-12), as q →
0 we obtain the maximum cumulative production at a
particular flow conditions (i.e., (Np)q→0) — where (Np)q→0, is
by defini-tion, the reserves.
Appendix B: Derivation of the Reciprocal-Rate
Method for Estimating Reserves — Gas Case
In this Appendix we attempt to rationalize the use of the
reciprocal rate concept for the case of a single well in a dry
gas reservoir. We begin by recalling the result used by
Blasingame and Rushing [Blasingame and Rushing (2005)]
for rate-cumulative production (where the well is produced at
a constant bottomhole flowing pressure). This result is given
as:
q g = q gi − Di Gp +
1 Di
Gp 2 ........................................... (B-1)
2 G
Dividing through Eq. B-1 by the gas flowrate, qg, we have:
⎤
⎡
⎡ Gp ⎤ 1 D
i q ⎢ Gp ⎥
1=
− Di ⎢
⎥+
g
qg
⎢⎣ q g ⎥⎦
⎢⎣ q g ⎥⎦ 2 G
q gi
2
............................. (B-2)
Dividing through Eq. B-2 by the initial gas flowrate, qgi, yields
the following result:
D
1
1
=
− i
q gi q g q gi
2
⎤
⎡
⎡ Gp ⎤ 1 1 D
i q ⎢ Gp ⎥ .................. (B-3)
⎥+
⎢
g
⎢⎣ q g ⎥⎦
⎢⎣ q g ⎥⎦ 2 G q gi
Solving Eq. B-3 for the reciprocal gas flowrate, 1/qg, gives:
D
1
1
=
+ i
q g q gi q gi
2
⎤
⎡
⎡ Gp ⎤ 1 1 D
i q ⎢ Gp ⎥ .................. (B-4)
⎥−
⎢
g
⎢⎣ q g ⎥⎦
⎢⎣ q g ⎥⎦ 2 G q gi
Recalling the "oil" result, Eq. A-24, we have:
1 1 Di ⎡ Np ⎤
= +
⎢
⎥ ....................................................... (A-24)
q qi qi ⎣⎢ q ⎦⎥
Comparing Eqs. B-4 and A-24, it does not appear that we can
obtain a simple extrapolation form for the gas case. We can
modify the "reciprocal rate" formulation by moving the
reciprocal initial gas flowrate, 1/qgi, to the left-hand-side of
Eq. B-4, we obtain:
2
⎡ Gp ⎤
1
1
D Gp 1 1 Di
−
= i
−
qg ⎢
⎥ ....................... (B-5)
q g q gi q gi q g 2 G q gi
⎢⎣ q g ⎥⎦
Estimation of Reserves Using the Reciprocal Rate Method
Appendix C: Estimation of Reservoirs for the Case
of a Gas Well
Production Data Analysis Plot for East Texas Gas Well
"Summary" History Plot — Rate and Pressure Functions
Gas Flowrate, qg, MSCF/D
10
5
12000
Legend: East Texas Gas Well
qg Production Data
pw Production Data (measured surface/converted bottomhole)
10
10000
4
8000
6000
10
3
4000
2000
0
2200
2100
2000
1900
1800
1700
1600
1500
1400
1300
1200
1100
900
1000
800
700
600
500
400
300
200
0
2
100
10
(Computed) Flowing Bottomhole Pressure, pwf, psia
In this Appendix we provide the analysis of the gas well case
first evaluated by Pratikno et al. [Pratikno et al. (2002)]. In
this analysis we consider the behavior of the "East Texas Gas
Well" from initial production (July 2001) to present (April
2007) (see Fig. C.1, below). This work is in contrast to
previous analyses which only considered the first year of
production, but we believe that addressing the entire
production history will be more relevant than just the first year
— particularly because our present goal is to estimate
reserves, not reservoir properties.
Production Time, Day
Fig. C.1 — Production and pressure history for the "East
Texas Gas Well" (Texas, USA).
7000
6500
6000
5500
4500
4000
3500
3000
2500
2000
1500
1000
500
5000
0.0040
0.0035
0.0030
0.0030
0.0025
0.0025
0.0020
0.0020
0.0015
(1/G) = 1/(2.2x10 )
0.0010
0.0010
6
0.0005
0.0000
7000
6500
6000
5500
5000
4000
3500
3000
2500
4500
G = 2.2x10 MSCF
= 2.2 BSCF
0.0005
0.0000
0.0015
6
Approximate Material Balance Time (Gp/qg), Days
Fig. C.2 — Cartesian format "Reciprocal Rate" plot for the
"East Texas Gas Well" (Texas, USA).
To confirm the Cartesian format match, we also present the
reciprocal rate plot in log-log format (i.e., log(1/qg) versus
log(Gp/qg)) (see Fig. C.3). Although our model match favors
the late-time data (this view is exaggerated in the log-log
format), we are satisfied that our trend provides a representative estimate of the gas reserves for this case.
Production Data Analysis Plot for East Texas Gas Well
Reciprocal Rate Plot — Log-Log Format
Reciprocal Gas Flowrate (1/qg), 1/MSCFD
After somewhat exhaustive efforts with Eq. B-8, it was concluded that this result cannot be formulated as an extrapolation
technique to estimate reserves directly. For application of Eq.
B-8, the reader is referred to the work by Blasingame and
Rushing [Blasingame and Rushing (2005)].
0.0045
0.0035
1
q gi ⎤
⎡ Gp ⎤ (1− b) ⎡
q g = q gi ⎢1 −
⎥
⎢G ≡
⎥ ........................ (B-8)
G ⎦⎥
(1 − b) Di ⎦⎥
⎣⎢
⎣⎢
0.0050
1/qg ≈ intercept + (1/G) Gp/qg
0.0040
2000
The extrapolation of Eq. B-7 for qg→0 will yield a factor of 2
times the reserves (note the 2G term in Eq. B-7).
Another formulation which was considered is that of the socalled "hyperbolic" rate decline relation [Arps (1942),
Fetkovich (1980)], given in rate-cumulative production form
by Blasingame and Rushing [Blasingame and Rushing
(2005)]. This result is given as:
0.0045
1500
qg ⎤ 1
D
1 1 Di
= i −
Gp .................................. (B-7)
⎥
q gi ⎥⎦ Gp q gi 2 G q gi
0.0050
1000
⎡
⎢1 −
⎢⎣
Production Data Analysis Plot for East Texas Gas Well
Reciprocal Rate Plot — Log-Log Format
0
Eq. B-6 is simply a modified form of the Blasingame and
Rushing extrapolation formula. We will provide a demonstration of this relation for a gas reservoir case in as part of the
example application given in Appendix C. We will use a
slight simplification of Eq. B-6 for our demonstration example
in Appendix C — this result is:
0
⎡ 1
1 ⎤ qg
D
1 1 Di
= i −
−
Gp ............................... (B-6)
⎢
⎥
q
q
G
q
gi ⎦⎥ p
gi 2 G q gi
⎣⎢ g
7
As with the other cases evaluated in this work, we will use the
Cartesian format "reciprocal rate" plot (1/qg versus Gp/qg —
and this graph is shown in Fig. C.2. While the model match
shown in Fig. C.2 favors the latest production data, this is
appropriate as we wish to estimate the reserves at this
particular point in time (i.e., the present is the latest time in the
well history).
500
Then multiplying through Eq. B-5 by the (qg/Gp) function
gives us:
Reciprocal Gas Flowrate (1/qg), 1/MSCFD
SPE 107981
10
10
10
10
10
-1
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
1/qg ≈ intercept + (1/G) Gp/qg
-2
10
-3
10
6
(1/G) = 1/(2.2x10 )
-4
10
-1
-2
-3
-4
6
G = 2.2x10 MSCF
= 2.2 BSCF
10
-5
10
-1
0
1
2
3
4
10
10
10
10
10
Approximate Material Balance Time (Gp/qg), Days
10
10
-5
5
Fig. C.3 — "Reciprocal Rate" plot (log-log format ) for the
"East Texas Gas Well" (Texas, USA).
In Fig. C.4 we present the "Gas Reciprocal Rate" plot, which
is derived from Eq. B-7. This plot is simply a variation of the
result given by Blasingame and Rushing [Blasingame and
Rushing (2005)] for the case of a gas well producing at a
constant flowing bottomhole pressure. We also noted that the
x-axis intercept is (from Eq. B-7) a factor of 2 times the gas
reserves (G). In this case (i.e., Fig. C.4), our results confirm
the estimate of 2.2 BSCF for the reserves estimate.
While we had hoped to develop a "reciprocal rate" analog for
the gas case from Eq. B-4, it simply is not possible to develop
8
P.D. Reese, D. Ilk, and T.A. Blasingame
a (direct) reserves extrapolation formula from Eq. B-4. This
remains a topic for further investigation.
Production Data Analysis Plot for East Texas Gas Well
Reciprocal Rate Function Versus Cumulative Gas Production Plot
0.9
qgi = 1650 MSCFD
⎡
qg ⎤ 1
D
1 1 Di
= i −
Gp
⎥
⎢1 −
⎣⎢ q gi ⎥⎦ Gp q gi 2 G q gi
0.8
0.7
0.6
0.5
0.4
Intercept:
2G = 4.4 BSCF
G = 2.2 BSCF
0.3
0.2
5.00
4.75
4.50
4.25
4.00
3.75
3.50
3.25
3.00
2.75
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0.75
0.00
0.0
0.50
0.1
0.25
Rate-Cumulative Production Function:
[1-(qg/qgi)](1/Gp), 1/BSCF
1.0
Cumulative Gas Production, Gp, BSCF
Fig. C.4 — "Gas Reciprocal Rate" plot for the "East Texas Gas
Well" (Texas, USA).
SPE 107981