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RATE-DECLINE RELATIONS FOR UNCONVENTIONAL RESERVOIRS AND DEVELOPMENT OF PARAMETRIC CORRELATIONS FOR
ESTIMATION OF RESERVOIR PROPERTIES
A Research Proposal
by
YOHANES AKLILU ASKABE
Submitted to the Office of Graduate Studies ofTexas A&M University
in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE
December 2012
Major Subject: Petroleum Engineering
Table of ContentsPage
1. Abstract......................................................................................................................................... 32. Objectives...................................................................................................................................... 43. Present Status of the Problem....................................................................................................... 44. Procedure...................................................................................................................................... 95. Analysis of Time-Rate Relations..................................................................................................
106. Development of Parametric Correlation.......................................................................................
167 Summary and Conclusion.............................................................................................................
238 Recommendation for Future Work...............................................................................................
239 Organization of the Research........................................................................................................
24Nomenclature.......................................................................................................................................
25References............................................................................................................................................
26
1. Abstract
Model-based production data analysis techniques are important tools for use in estimating the reservoir
properties of unconventional reservoirs. However, these techniques require knowledge of reservoir/well
parameters that are usually not readily available, which makes unique estimation of reservoir properties
difficult. Systematic implementation of empirical time-rate decline relations to study decline trends of
production data have been shown to provide an excellent match to the production data and provide a
reliable estimate of ultimate recovery. By using the parameters of the time-rate relation, it is possible to
formulate a parametric correlation that can provide a reliable estimate of reservoir/well properties.
Previously, Ilk et al. (2011) have shown that it is possible to correlate reservoir/well properties that are
estimated using model-based production data analysis with parameters of the power-law exponential (PLE)
time-rate relation by using limited number of wells from tight/shale gas reservoirs. From that study it was
apparent that synthetic data cases are required to obtain a more rigorous (perhaps semi-analytical)
correlation for the estimation of well/reservoir properties.
In this study, we identify relationships between reservoir properties and parameters of the time-rate
relations that are used to match time-rate data generated from a numerical simulation of a multi-fractured-
horizontal well model in a low/ultra-low permeability reservoir. Using these results, we show that the
time-rate model parameters are uniquely related to reservoir properties (specifically, we consider reservoir
permeability (k) and EUR over 30-years (EUR30-Yr)). We then formulate parametric correlations which
integrate the parameters of the time–rate relations with properties of the reservoir. We demonstrate that
when bottomhole pressure is constant (or gently changing), then the resulting correlations provide reliable
estimates of reservoir properties.
For reference, this study is performed using most "modern" time-rate relations:
● The Power Law Exponential model (PLE) (Ilk et al. 2008),
● The Duong model (Duong 2011), and
● The Logistic Growth Model (LGM) (Clark et al. 2011).
2. Introduction
In this work, we present:
● A detailed analysis of the performance of the PLE, Doung, and LGM time-rate relations in matching
time-rate data obtained from unconventional reservoirs.
● Diagnostic plots, where we show that the modern time-rate relations are effective at matching transient
and transition flow regimes observed from unconventional reservoirs. The Arps' hyperbolic model is
not used in this study as it is considered the current standard, and its strengths (excellent model for
linear and bi-linear flow) and limitations (need to be constrained (exponentially) to make realistic EUR
projections) are well-established.
(Page 3 of 28)
● Use of the "continuous EUR" approach for the PLE, Doung, and LGM time-rate relations.
● Modification of the Doung and LGM time-rate relations for boundary-dominated flow behavior.
This work showcases the performance of new time-rate relations in modeling production data from
unconventional reservoirs. The modifications we have developed for the time-rate relations provide
important constraining behavior to the Duong and Logistic Growth decline models. Time-rate data match
quality is improved, EUR projections and production forecasts are improved. This work also provides a
semi-analytical basis for correlating time-rate model parameters with fundamental reservoir properties via a
numerical simulation study.
2. Objectives
The objectives of this work are:
● To compare performance of modern time-rate relations in matching production data and estimating
reserves for wells in unconventional reservoirs. We use diagnostic plots for data matching and we
apply the "continuous EUR" approach to estimate ultimate recovery in time and to compare
performance of the time-rate relations.
● To identify diagnostic relations/plots for modern time-rate models that may allow us to identify data
characteristics for better data matching/representation.
● To propose modifications of the time-rate relations to improve the quality of match and for a better
estimation of EUR, particularly for transition and boundary-dominated flow behavior.
● To develop a methodology for integration of reservoir/well properties — specifically, to demonstrate
the rigorous correlation of k and EUR30-Yr with time-rate model parameters using production data
generated from numerical simulation solutions.
3. Present Status of the Problem
Modern production data analysis techniques have become standard tools in analyzing production data from
shale gas and "liquid-rich" shale reservoirs. Analytical techniques can provide valuable information about
the formation including permeability, extent of hydraulic fracture and reserve estimates — however, in
most cases, the analysis becomes a challenge because of complex nature of the reservoir and fracture
properties. In particular, permeability is the most challenging variable to try to estimate using analytical
techniques.
In addition, the complexity of fracture networks, reservoir heterogeneity, stress dependence of permeability
and porosity and changes in conductivity can make unique estimation of reservoir parameters a difficult, if
not non-unique task. In-addition, operational complications such as liquid loading can result in erratic
production rate and surface pressure data, which are not altogether representative of the conditions at the
formation. As noted, these circumstances can (and do) lead to non-unique, if not erroneous solutions.
(Page 4 of 28)
On the other hand, empirical time-rate decline relations are used regularly because these require only time-
rate data as inputs. However, since most time-rate decline models have an empirical basis (or highly
idealized conditions of application), these relations do not convey fundamental characteristics of the
reservoir. Statistical analyses of the power-law exponential (or PLE) time-rate model parameters have been
shown to provide a somewhat unique insight in to the performance of the reservoir, and it is logical to
assume that other models may do so as well. Specifically, factors that may affect well performance such as
the effectiveness of well completions and characteristics of the reservoir for example reservoir boundary
conditions can be identified/isolated by statistically studying the time-rate model parameters (Arps 1945;
Clark et al. 2011; Duong 2011; Fetkovich et al. 1996).
Arps (1945) rate decline models have been used widely to forecast production and estimate reserves from
conventional oil and gas wells. However, it has been shown that Arps hyperbolic model overestimates
reserves by more than 100 percent when used to match wells producing from unconventional low/ultra-low
permeability reservoirs (Rushing et al. 2007). In the past decade several new time-rate relations have been
presented with improvements to analyze time-rate data from unconventional reservoirs. Ilk et al. (2008)
have presented a power law exponential model (PLE) based on the "loss-ratio" and the "loss-ratio
derivative" definitions (Johnson and Bollens 1927). The PLE model provides a superior match quality to
the transient, transitional and boundary dominated flow-regimes observed from wells producing from
unconventional resources.
A stretched exponential model (SEPD) was independently presented by Valko (2009) after studying flow
characteristics of over 7000 wells in the Barnet Shale. The SEPD model is the same as the base form of the
PLE model (without the terminal decline term); the SEPD models the long transient and transitional flow
regimes observed in production data from unconventional gas reservoirs — however; it lacks the boundary-
dominated flow term present in the PLE model.
Duong (2011) presented a time-rate decline model for fracture linear flows producing at a constant flowing
bottomhole pressure. Duong derived his relation based on the observation of a straight line behavior in
q/Gp vs. time on a log-log plot. The Duong model can match transient characteristics that are dominant in
low/ultra-low permeability fractured reservoirs, but this model is incapable of representing boundary-
dominated flow performance. The slope and intercept of the straight-line portion of the log(q/Gp) vs.
log(time) data show narrow ranges for reservoirs with similar reservoir rock types and fracture
characteristics.
Clark et al. (2011) adapted the hyperbolic form of the "logistic growth model" for reserve estimation of oil
and gas reservoirs. Blumberg (1968) suggested this hyperbolic form to characterize regenerative growth in
biological systems. The hyperbolic nature of the LGM provides a good match for transient and transition
flow regimes of wells from unconventional reservoirs, but this relation does not model boundary-
dominated flow well.
(Page 5 of 28)
There have been attempts in the past to link fundamental reservoir properties with time-rate model
parameters. Fetkovich (1980) has shown that time-rate models which are analogous to Arps empirical
models can be derived analytically from solutions of oil and gas wells producing from a reservoir at a
constant flowing bottomhole pressure condition. This means that Arps time-rate model parameters are
more than simple empirical parameters. Instead these parameters are characteristics (or proxies of
characteristics) of the reservoir and can convey fundamental reservoir properties. Another approach to link
empirical time-rate model parameters with reservoir properties was proposed by Ilk et al. (2011). In their
study, the authors demonstrated that it is possible to correlate time-rate model parameters (PLE) with
reservoir parameters (k and kxf) by using small number of wells producing from tight/shale gas reservoirs.
From that study it was concluded that more data is required for conclusively unique results.
The main objective of this study is to investigate performance of modern time-rate models in matching
time-rate data from unconventional reservoirs. Our strategy in this investigation is:
● Prepare diagnostic relations/analyses using each rate decline models.
● Compare the performance of each models for transient, transition and boundary dominated flow.
● Use the "continuous EUR" approach to estimate reserves for both synthetic and a field data.
● Compare the relative accuracy of reserve estimates made with the modern time-rate models.
● Propose modified time-rate relations that better capture flow regimes for unconventional reservoirs.
Also, we present more rigorous basis for the work presented by Ilk et al. (2011) using time-rate data
generated from a numerical simulator to demonstrate a methodology for integrating fundamental reservoir
properties (k and EUR) with time-rate model parameters. We show that the derived parametric correlations
provide a reliable estimate of the reservoir properties for wells producing with similar production
constraints.
As background regarding our diagnostic approach, we begin with the work of Ilk et al. (2008) where they
demonstrated the application of the "loss-ratio" and the "loss-ratio" derivative relations as diagnostic tools
for matching time-rate data. The definitions of the D(t) function (i.e., the reciprocal of the "loss ratio") and
b(t) function (i.e., the "loss-ratio" derivative) are given by Eqs. 1 and 2.
D( t )=− 1qg
dqg
dt ......................................................................................................................................(1)
b ( t )= ddt [ 1
D( t ) ] .......................................................................................................................................(2)
Multiplying the D(t) function by the production time (t) provides another diagnostic relation that we can
also use to match the rate time data. This relation is known as the "beta function" (β(t)) and is given by:
β ( t )=tD ( t )..............................................................................................................................................(3)
(Page 6 of 28)
The time–rate empirical models considered in this research include:
● The Power Law Exponential model (PLE) (Ilk et al. 2008),
● The Duong model (Duong 2011), and
● The Logistic Growth Model (LGM) (Clark et al. 2011).
Following we will provide a short description of the time–rate models.
Power Law Exponential Model
The power–law exponential model was derived by observing the "loss-ratio" behavior of wells producing
from low/ultra–low permeability reservoirs producing through fracture stimulation. Ilk et al. (2008)
demonstrated that by describing the "loss–ratio" of the data using a power–law function, it is possible to
match the dominant transient and transition flow regimes observed from unconventional reservoirs. In
addition, boundary dominated flow regimes are represented by adding a decline term (D∞) to the power law
relation. The PLE model "loss-ratio" relation is given by Eq. 4.
D( t )=D∞+ Di ntn−1................................................................................................................................(4)
During late-time flow periods, the power-law term becomes less significant and D(t) approaches a constant
term (D∞) similar to the case in Arps exponential decline model. Furthermore, the authors observed that the
derivative of the "loss-ratio" is not constant as was the case in Arps' rate decline relations ( i.e., exponential,
hyperbolic and harmonic equations), but instead it is a function of time. The "loss-ratio" derivative ( b-
parameter) is given by:
b ( t )=−Di (n−1 )ntn
( D∞ t+Di ntn )2............................................................................................................................(5)
And the PLE "beta" relation is given by:
β ( t )=D∞+ Di nt n....................................................................................................................................(6)
And the power-law exponential rate relation is given by Eq. 7.
q=qgi exp[−D∞ t−Di tn ] ..........................................................................................................................(7)
As mentioned earlier a diagnostic plot of D(t) ,b(t) and β(t) vs. time will help diagnose the time-rate data
matching process
Duong’s Model
Duong (2011) presented a new empirical rate decline model based on the long-term linear or bilinear flow
regimes observed in hydraulically fractured ultra-low permeability reservoirs. On a log-log plot of rate vs.
(Page 8 of 28)
time, the early time data shows half slope for linear flow and quarter slope for bilinear flow regimes. A
log-log plot of rate over cumulative production vs. time will result in a straight line for wells producing
from unconventional reservoirs. This straight line behavior is described by a power-law relation given by
Eq. 8.
qGp
=at−m
.................................................................................................................................................(8)
Where:
a = straight line intercept on log-log plot of q/Gp vs. time
m = negative slope of straight line on log-log plot of q/Gp vs. time.
The slope and intercept parameters of Duong model show narrow ranges for wells producing from similar
rock-types and similar fracture stimulation practices and operational conditions (Duong 2011). The
parameters can be directly estimated from q/Gp vs. time diagnostic plot.
The Duong rate and cumulative relations are given by Eq. 9 and Eq. 10 respectively.
qq1
=t−m exp [ a1−m
( t1−m−1)]..................................................................................................................(9)
Gp=q1
aexp [ a
1−m( t1−m−1)] ................................................................................................................(10)
In addition to q/Gp vs. time log-log diagnostic plot, we will use the "loss-ratio" and the "loss-ratio"
derivative definitions to estimate the model parameters and guide the data matching process. The b, D and
β-parameters of Duong model are given by Eqs. 11, 12 and 13 respectively.
D( t )=mt−1−at−m..................................................................................................................................(11)
b ( t )=mtm( tm−at )(at−mtm )2
...............................................................................................................................(12)
β ( t )=m−at−m......................................................................................................................................(13)
We can also estimate the m parameter independently using the following diagnostic relation:
tGp
qddt [ q
Gp ]=m..................................................................................................................................(14)
Logistic Growth Model
(Page 9 of 28)
Yet another model, the Logistic Growth Model (LGM) is adapted to model time-rate data from oil/gas
reservoirs. The LGM is originally developed to model growth trends of various populations in nature. A
form of the logistic growth model has been used to model growth of yeast and to study market penetration
of new products and technologies (Martinez et al. 2008). Tsoularis and Wallace (2002) have derived a
general form of the logistic growth model. They have also presented a summary of various logistic growth
models for various applications.
Clark (2011) adapted the hyperbolic form of the logistic growth model to match time-rate data of oil/gas
reservoirs. The hyperbolic form was suggested by (Blumberg 1968) to study regenerative growth in nature.
Eq. 15 shows the hyperbolic form of the logistic growth model.
N ( t )= K ( t+b )n
a+( t+b)n.................................................................................................................................(15)
Clark et al. (2011) applied this model to several oil and gas wells and noted that the b parameter
consistently becomes zero. After eliminating the parameter b and rearranging, the cumulative form of the
hyperbolic logistic growth model is given by:
Q( t )= Kt n
a+ tn.........................................................................................................................................(16)
The time-rate form is given by Eq. 17.
q ( t )= a K n tn−1
( a+t n )2.....................................................................................................................................(17)
Similarly the "loss-ratio" (b) and the "loss-ratio" derivatives (D) of the logistic growth model will be used
to perform diagnostic match of time-rate data. Logistic growth model b, D and β parameters are given by
Eqs. 18 , 19 and 20 respectively.
D( t )= a−a n+(1+ n) t n
t( a+t n ) ......................................................................................................................(18)
b ( t )=−a2 ( n−1 )−2 a( n2−1 )t n+( n+1) t2n
( a−a n+( n+1) tn )2.......................................................................................(19)
β ( t )= a−a n+(1+ n) t n
( a+t n ) ......................................................................................................................(20)
And the q/Gp formulation of LGM is given by:
qGp
=t ( a+tn
a n )........................................................................................................................................(21)
(Page 10 of 28)
If the initial gas in place is known, we can rewrite the rate relation to find a diagnostic relation that will
allow a diagnostic plot estimation of the remaining model parameters (Clark et al. 2011).
KQg ( t )
−1=a t−n
.......................................................................................................................................(22)
4. Procedure
As an introduction to the time-rate models discussed here, prior to developing the parametric correlations,
we will investigate the performance of the rate decline models in matching time-rate data from
unconventional reservoirs. We will perform data driven model matching using the "loss-ratio", the "loss-
ratio" derivative relations as well as other diagnostic relations. We will also study performance of the time-
rate relations using the "continuous EUR" approach. Then we will modify the time-rate models to better
estimate EUR and accurately forecast production.
We will demonstrate development of a parametric correlation using a set of production data generated from
a simulated multi-fractured horizontal well model. First, we will study a cross-plot of the rate-decline
model parameters and reservoir properties, particularly, permeability (k) and EUR30-Yr. This will allow us
identify a parametric function that best correlates the rate-decline model parameters with the reservoir
properties. Once we know the relationship between the model parameters and the reservoirs properties, we
will develop the correlating function for estimation of reservoir properties.
5. Analysis of Modern Time-Rate Relations
Time-rate data generated from a set of numerical models of a multi-fractured horizontal well is matched
using the empirical rate decline relations discussed above. A set of numerical models, with permeability
values ranging from 0.25µD-5µD, are used to generate the production data. Other reservoir, fluid and well
parameters are kept constant.
Fig. 1 shows a schematic of the reservoir model we are considering in this study. It depicts a horizontal
well that is producing from low/ultra-low permeability reservoir producing through multistage fracture
stimulation.
Transverse fractures Horizontal well
(Page 11 of 28)
Figure 1 — A schematic of a numerical simulation model. (Horizontal well with multiple transverse
fractures)
Table 1 shows a summary of the model input parameters.
Table 1 — Reservoir and fluid properties for simulated horizontal shale gas well model with multiple transverse fractures.
Reservoir Properties:Net pay thickness, h = 130 ftFormation permeability, k = 0.25µD-5µDWellbore Radius, rw = 0.35 ftFormation compressibility, cf = 3 x 10-6 psi-1
Porosity, = 0.09 (fraction)Initial reservoir pressure, pi = 5000 psiGas saturation, sg = 1.0 fractionSkin factor, s = 0.01 (dimensionless)Reservoir temperature, Tr = 212 °F
Fluid properties:Gas specific gravity, γg = 0.6 (air = 1)
Hydraulically fractured well model parameters:Fracture half-length, xf = 164.0 ftNumber of fractures = 15Horizontal well length = 4921.3 ft
Production parameters:Flowing pressure, pwf = 500 psiaProduction time, t = 10,958 days (30 years)
Fig. 2 and Fig. 3 shows a "qDb" type diagnostic plot of a time-rated data match using the PLE, LGM and
Duong time-rate relations. The diagnostic relations will provide a systematic and data driven estimation of
the model parameters. When there is no restriction on the degree of freedom, it is possible to obtain a wide
range of parameters that result in a model fit.
(Page 12 of 28)
Figure 2 — (Log-log Plot): qDb type plot ─ flow rate (qg), D- and b- parameter versus production time and time-rate model (PLE, LGM, Duong) matches ─ simulated case (k=800 nD)
(Page 13 of 28)
Figure 3 — (Log-log Plot): qDb type plot ─ flow rate (qg), D- and b- parameter versus production time and time-rate model (PLE, LGM, Duong) matches ─ simulated case (k=50 nD)
.
Fig. 3 shows that the PLE model is capable of modeling all flow regimes including boundary dominated
flow regimes whereas Duong and LGM time-rate relations match transient and transition flow regimes. Fig.
2 and Fig. 3 show that the new time-rate relations are capable of modeling the long transient flow regimes
observed from unconventional reservoirs.
Modified time-rate relations
We have seen that LGM and Duong models fail to match long-time boundary behaviors which leads to
overly optimistic reserve estimate. We have developed new relations based on LGM and Duong time-rate
models of these relations derived by considering long term boundary behaviors. These relations will
provide better time-rate data matches and more reliable EUR estimate. Following is a summary of the
modified new time-rate relations.
(Page 14 of 28)
Modified Duong Model (Model 1)
Rate relation:
qg ( t )=q1 t−mexp [ a1−m
( t1−m−1 )−DDNG t ].......................................................................................(23)
D-parameter
D( t )=DDNG+ mt−at−m
.........................................................................................................................(24)
b-parameter
b ( t )= mtm(−at+tm)(at−tm(m+DDNG t ))2
..............................................................................................................(25)
Modified Duong Model (Model 2)
Rate relation:
qg ( t )=q1
t m exp [DDNG(1−t )+aDDNGm−1 (Γ [1−m, DDNG ]−Γ [ 1−m ,DDNG t ])] .......................................(26)
D-parameter:
D( t )=DDNG+ mt−at−m exp[−DDNG t ]
...............................................................................................(27)
b-parameter:
b ( t )=exp[ DLGM t ] tm (exp[ DLGM t ]mtm−at (m+DLGM t ))
(at−exp[ DLGM t ] tm(m+DLGM t ))2 .........................................................................(28)
Modified LGM (Model 3)
Rate relation:
qg ( t )=aKnt (n−1)
(a+ tn )2 exp[−DLGM t ]..........................................................................................................(29)
D-parameter:
D( t )=a (1−n+DLGM t )+tn(1+n+DLGM t )
t (a+tn ) ....................................................................................(30)
b-parameter
b ( t )= −a2(n−1)−2a (n2−1) tn+(n+1) t2 n
(a(1−n+DLGM t )+tn(1+n+DLGM t ))2................................................................................(31)
Modified LGM (Model 4)
Rate relation:
qg ( t )=Ktn−1 a exp[ DLGM t ](n+DLGM t )
(a+(1+R )exp [DLGM t ] tn)2 ...............................................................................................(32)
Cumulative production:
(Page 15 of 28)
Qg( t )=Kt n exp[−DLGM t ]
a+(1+R)exp [ DLGM t ] tn.......................................................................................................(33)
The modified relations are capable of matching long-term time production data obtained from
unconventional reservoirs. These new models constrain the EUR estimates and provide more reliable
production forecasts. Figs. 4 and 5 show the improved match quality of the modified LGM and Duong
time-rate models.
Figure 4 — (Log-log Plot): qDb type plot ─ flow rate (qg), D- and b- parameter versus production time and time-rate model (Duong Model, Modified Duong Model (Model 3), Modified Duong Model (Model 4)) matches ─ simulated case (k=8 µD)
(Page 16 of 28)
Figure 5 — (Log-log Plot): qDb type plot ─ flow rate (qg), D- and b- parameter versus production time and time-rate model (LGM, Modified LGM (Model 3), Modified LGM (Model 4)) matches ─ simulated case (k=8 µD)
We have done model matches of 14 synthetic well models generated using varying permeability values. In
addition, we will investigate the performance of the empirical models in matching production data from
unconventional reservoirs. Also, we will estimate reserves using the “continuous EUR” approach and
compare the performance of the time-rate relations.
6. Development of parametric correlations
Here, we will demonstrate the methodology used to study the relationship between the time-rate model
parameters and the reservoir parameters, specifically permeability (k) and EUR30-Yr. As mentioned earlier,
using the time-rate model diagnostic relations, we can match the time-rate data and estimate the
corresponding model parameters.
(Page 17 of 28)
Figs. 2 and 3 show the diagnostic approach used to model the time-rate data trends. As mentioned earlier,
matching production data is subjective in nature due to the degree of freedom provided by the model
parameters. To obtain consistent and repeatable data matches, we rely on plots of diagnostic relations to
"magnify" trends in the early, middle, and late time periods of the production data. We aim to obtain
quality matches to all flow regimes thereby testing the performance of the rate decline models in matching
the production data. Table 2 summarizes the observed reservoir properties (k and EUR30-Yr) and the
corresponding time-rate model matching parameters when using the PLE model.
Table 2 — Power Law Exponential Model Parameters (14 numerical simulation cases)
Permeability (k),md
qgiMSCF/D
DiD-1
ndimensionless
D∞D-1
EUR30-Yr
BSCF
0.00025 88,607 2.468
0.091 1.8E-05 5.219
0.00050 355,248 3.265
0.078 5.0E-05 7.033
0.00075 896,261 3.834
0.072 7.5E-05 8.113
0.00100 1,792,552 4.273
0.068 9.7E-05 8.829
0.00125 2,503,938 4.419
0.068 1.2E-04 9.343
0.00150 3,249,731 4.529
0.068 1.4E-04 9.739
0.00175 4,608,106 4.742
0.066 1.6E-04 10.018
0.00200 5,937,051 4.875
0.066 1.7E-04 10.240
0.00250 9,396,501 5.139
0.065 2.0E-04 10.585
0.00300 13,723,998
5.362
0.064 2.4E-04 10.799
0.00350 18,706,208
5.543
0.064 2.7E-04 10.967
0.00400 23,653,940
5.666
0.064 3.0E-04 11.082
0.00450 30,171,081
5.816
0.063 3.3E-04 11.170
0.00500 38,246,539
5.973
0.062 3.6E-04 11.245
Next, to provide a basis for developing the parametric correlation, we need to demonstrate how the rate
decline model parameters are related to the reservoir parameters. We will construct a series of cross-plots
to determine how the rate decline model parameters are varying with the reservoir/well properties. Figs. 4 -
6 show permeability plotted against the PLE model parameters and Figs. 7-9 Show EUR30-Yr plotted against
the PLE model parameters. The plots also show results of a regression analysis along with the best fit trend
line function. From the cross-plots it is evident that the PLE model parameters show behaviors that could
(Page 18 of 28)
be described by parametric functions. We will apply similar analysis using Duong and LGM time-rate
relations.
Figure 6 — (Cartesian Plot): Cross plot of permeability and PLE n-Parameter—simulated case.
(Page 19 of 28)
Figure 7 — (Cartesian Plot): Cross plot of permeability and PLE Di -parameter—simulated case.
(Page 20 of 28)
Figure 8 — (Cartesian Plot): Cross plot of permeability and PLE qgi -parameter —simulated case.
Figure 9 — (Cartesian Plot): Cross plot of EUR30-Yr and PLE n-parameter —simulated case.
Figure 10 — (Cartesian Plot): Cross plot of EUR30-Yr and PLE Di -parameter—simulated case.
(Page 21 of 28)
Figure 11 — (Cartesian Plot): Cross plot of EUR30-Yr and PLE-qgi -parameter—simulated case.
The cross-plots describe the relationship between the model parameters and the reservoir properties. The
parametric functions we choose to correlate the individual model parameters with the reservoir properties
should result in a best-fit to the data. The coefficient of determination ("R-squared") value indicates the
accuracy of the selected parametric function in describing the correlation.
The final task is to formulate a function with the rate decline model parameters as variables to estimate the
reservoir properties. Since the rate decline model parameters represent characteristic of the production data
and of the diagnostic relations, it is fair to assume a form of the function that contains the parametric
functions we defined for each model parameter earlier (Figs. 6-11). For example, from Figs. 6-8 we can
see that permeability (k) is related to PLE model parameters, n,Di , and qgi via a power function. From
this we can suggest the following integrating parametric function.
k=a01na02 D
ia03
^qgi
a04 ......................................................................................................................(34)
Where:
a01 = model parameter, md-D2/MSCFa02 = model parameter, n-parameter exponent, dimensionless
a03 = model parameter Di -parameter exponent, dimensionless
a04 = model parameter qgi -parameter exponent, dimensionless
(Page 22 of 28)
Similarly, EUR can be estimated by the parametric correlation:
EUR=a05 ln ( a06 D i)exp(a07n ) ............................................................................................................(35)
Where:
a05 = model parameter, BSCFa06 = model parameter, Daysa07 = model parameter, dimensionless
Once we establish a form of the parametric equation, we will perform regression analysis to determine the
coefficients and exponents of the parametric correlations. It should be noted that it may not be necessary to
have all time-rate model parameters present in the correlation. Coefficients and exponents that gave best
estimate of the reservoir/well properties are selected. In some cases the exponents or coefficients can
become zero. In such cases, the parameter is eliminated from the final equation. It can be seen that the
qgimodel parameter does not appear in the EUR parametric equation (Eq. 35).
A correlation plot of permeability (k) and EUR30-Yr is shown on Figs. 12 and 1,3 respectively. The figures
show that it is possible to formulate a parametric correlation that estimates reservoir properties from time-
rate relation parameters when reservoir properties such as flowing bottomhole pressure (pwf) and
completion parameters (e.g., number of fracture stages) remain constant. The figures show that the
developed correlations were able to provide a reliable estimate of the reservoir properties.
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Figure 12 — (Log-log Plot): Comparison of permeability (k) calculated using permeability correlation versus simulated case model permeability.
Figure 13 — (Log-log Plot): Comparison of EUR30-Yr calculated using EUR correlation versus simulated case model EUR30-Yr.
We require that the time-rate relations provide reliable reserve estimates before we attempt to develop the
parametric correlation. Reliable reserve estimate and quality time-rate data match indicates that the model
parameters are related to fundamental reservoir characteristics.
7. Summary and Conclusion
Summary:
In this work we have executed a performance analysis of modern rate decline models (PLE, LGM, and
Duong) using diagnostic relations and diagnostic plots, in matching the production data and estimating
reserves of unconventional reservoirs. We have shown that new time-rate relations are capable of
modeling the dominant transient and transition flow regimes observed from production data analysis of
hydraulically fractured wells in low/ultra-low permeability reservoirs. We have shown that PLE model is
capable of modeling boundary dominated flow regimes whereas LGM and Duong models lack boundary
characteristics. The "continuous EUR" approach is used to study performance of the time-rate models in
estimating ultimate recovery. Finally, we have proposed new time rate-relations to improve the reserve
estimates and production forecasts of these (Duong, LGM) models. The proposed time-rate relations
include boundary conditions to model observed long-time boundary-dominated flow regimes and to
constrain reserve estimates.
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In addition, we have developed a methodology to formulate a parametric correlation to integrate reservoir
properties with rate decline model parameters by analyzing time-rate data generated from a reservoir
simulation of a multi-fractured horizontal well in low/ultra-low permeability reservoir. The developed
correlations will allow estimation of reservoir properties using parameters of the time-rate models. When
unique estimates of reservoir parameters are missing to allow a complete model based production data
analysis, these correlations will permit fast estimation of fundamental reservoir properties (permeability (k)
and EUR) by using time-rate data which is readily available. This theoretical consideration shows that
when bottomhole flowing pressure is constant, it is possible to correlate empirical time-rate model
parameters with reservoir properties (k and EUR).
Conclusions:
● We have compared performance of PLE, LGM and Duong time-rate relations when used to model
production data from unconventional reservoirs. We have demonstrated various diagnostic relations
and a diagnostic plot method of estimating the model parameters. We have also used the "continuous
EUR" approach to compare degrees of convergence of reserve estimates from these new time-rate
relations.
● We have modified the LGM and Duong time-rate models to obtain relations that can better model
transient, transition and boundary-dominated flow regimes.
● Theoretical consideration for integrating time-rate model parameters with reservoir model parameters
is presented using time-rate data generated from a numerical simulator. The parametric correlations
can provide fast estimates of fundamental reservoir properties when unique estimation is not possible
from model-based production data analysis.
8. Recommendations for Future Work
We have presented a theoretical validation for estimation of reservoir properties using parametric
correlations. Future study will focus on developing the method using a high quality field data.
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9. Organization of the Research
Following is outline of the proposed research:
● Chapter I — Introduction
■ Statement of problem■ Objectives■ Organization
● Chapter II — Literature Review
■ Empirical, Semi-Analytical, and Empirical Production Data Analysis ■ Integrated Production Data Analysis■ Data Diagnosis■ Reservoir properties from time-rate data analysis
● Chapter III — Analysis of Modern Time-Rate Relations
■ Modern Time-Rate Relations■ Analysis of Time-Rate Data■ Reserve Estimation-"Continuous EUR"■ Modified time-rate relation
● Chapter IV — Development of Parametric Correlation
■ Methodology■ Analysis of Model Parameters■ Parametric Correlation
● Chapter V — Summary and conclusion
■ Summary■ Conclusion■ Recommendation for future work
● Nomenclature
● References
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Nomenclaturea = Duong model intercept defined by Eq. 7, 1/Db = "Loss-ratio" derivative, dimensionlessa = LGM time-rate equation model parameter (Eq. 13), Daysa01 = Model parameter for empirical correlation, md-D2/MSCFa02 = Model parameter for empirical correlation, dimensionlessa03 = Model parameter for empirical correlation, dimensionlessa04 = Model parameter for empirical correlation, dimensionlessa05 = Model parameter for empirical correlation, BSCFa06 = Model parameter for empirical correlation, Daysa07 = Model parameter for empirical correlation, dimensionless
b = LGM time-rate equation model parameter, DD = "Loss-ratio", 1/DD∞ = Power-law exponential decline relation at infinite time, 1/DDDNG = Modified Duong model decline parameter, 1/D
Di = Power-law exponential decline relation, 1/D DLGM = Modified Duong model decline parameter, 1/DEUR = Estimated ultimate recovery, MSCFEUR30-Yr = Estimated ultimate recovery after 30 years, MSCFk = Model permeability, mdK = LGM model parameter (Carrying capacity), MSCFm = Duong model Slope defined by Eq. 7, dimensionlessn = Power-law exponential relation time exponent, dimensionlessn = LGM time-rate relation model parameter (Eq. 13), dimensionlessq1 = Rate at day 1, MSCF/Dqgi = Power-law exponential relation rate intercept, MSCF/Dt = Production time, daysxf = Fracture half-length, ft
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