Comparasion de Metodo Empiricos

SPE 162648

Comparison of Empirical Decline Curve Methods for Shale Wells Mohammed S Kanfar, and R.A. Wattenbarger, Texas A&M University

Copyright 2012, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Canadian Unconventional Resources Conference held in Calgary, Alberta, Canada, 30 October–1 November 2012. This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

Analyzing shale wells using traditional decline curve methods is problematic because of the nature of reservoir properties and flow behavior in typical shale wells. New empirical methods were developed to model the special production decline of shales. These methods were formulated using different mathematical and statistical bases and result in different forecasts. Hence, engineers have a variety of methods that may give different estimates for ultimate recovery when analyzing shale wells. In this work, four recently developed decline curve methods, along with the traditional Arps method, were compared.

The four recent methods compared here were empirically formulated for shale wells and tight gas wells. They are: a) the

Power Law Exponential Decline, b) the Stretched Exponential Decline, c) Duong’s Method, and d) the Logistic Growth Model. Each method has different tuning parameters and equation forms. In this work, the methods were programmed and automated by using nonlinear regression to match the production “history” of a well. In addition, they were compared in terms of “goodness of fit” to the history data and reliability of automation as well as production forecast and ultimate recovery estimation. These methods were compared with simulation models in addition to field data from Barnett Shale, Bakken Shale, and the Eagle Ford Shale.

Each of these methods may have application for different cases. It may be advisable to program each of these methods

for optional usage in applications. But this current paper should allow engineers to understand better the characteristics of each method and to choose the method that best models their wells under various circumstances. Introduction The ultimate recovery of conventional oil and gas wells can be reliably estimated using traditional Arps’ Decline Analysis equations. Arps’ equations are utilized to fit and extrapolate production rate-time plots during Boundary Dominated Flow (BDF) to abandonment. Shale wells, on the other hand, often exhibit long transient flow regimes, and don’t reach BDF for most of their production life. This long transient flow is due to the very low matrix permeability of shale. Traditional decline methods will usually overestimate reserves when applied on transient flow regimes of shale wells. This led to the development of new empirical equations more suitable for shale.

The new equations were formulated differently and will result in distinct estimations of ultimate recovery. The methods were reviewed and applied on simulation and field data for comparison. A program was developed on Microsoft Excel vba to automatically run and compare the methods. The software reads daily production data and finds the best match using non-linear regression. The match is found by minimizing the least square function using Levenberg-Marquardt Algorithm (LMA). LMA has the advantage of converging even if the starting point is far from the final minima. The Methods Arps.

Arps’ decline curve analysis (1945) is based on graphically extrapolating production semi-log plots (log q vs. t) to abandonment. Production rates show three types of declines during BDF: Exponential, Hyperbolic or Harmonic. Arps models these declines using the concept of loss-ratio and its derivative.

2 SPE 162648

The loss-ratio is defined as,

1!= −

!!" !"

∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ 1

Derivative of loss-ratio,

! =!!"

−!

!" !" ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ 2

The three declines have b values ranging from 0 to 1. Where b = 0 represents the exponential decline, 0 < b < 1 the

hyperbolic decline, and b = 1 the harmonic decline. The case of exponential decline can be derived on physical basis for slightly compressible fluids, constant bottom-hole pressure, and boundary dominated flow (BDF) (Fetkvovich 1971, Fetkovich 1980). The other cases were not purely derived on physical basis but can be observed in multilayered reservoirs during BDF (Fetkovich el al. 1990, El-banbi and Wattenbarger 1996). Fetkovich (1980) used Arps’ models combined with constant-pressure infinite-acting radial flow solutions to form a type curve. Arps’ hyperbolic model,

! = !! 1 + !"# !! ! ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ 3

The Arps/Fetkovitch approach usually limits the value of b to (0 ≤ ! ≤ 1). However, for shale wells it is often observed that values of b > 1 seem to match field data. For example, transient linear and bilinear flow regimes, often observed in shale wells, can be matched with Arps’ hyperbolic model with b values of 2 and 4 respectively. Lee and Sidle (2010) showed that b > 1 gives physically impossible results when Arps’ cumulative production equation is evaluated at infinite time. Nonetheless, Arps’ cumulative production equation gives acceptable results when finite values of producing time or economic rate limit are used.

Power Law Exponential Decline (PLE).

The concept of the PLE is to model the loss-ratio and its derivative in addition to modeling rate. The loss-ratio observed during transient linear and bilinear flow has a power law relation with time. Ilk et al. (2008) developed an equation to model this behavior.

PLE D model (loss-ratio),

! = !! + !!!!(!!!) ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ 4

The first term is a constant and it is the loss-ratio at infinite time while the second term is a power law function. In early times, the first term is negligible making the model a power law function to match linear or bilinear flow. While at late times, the first term is dominant and the model becomes constant to match the loss-ratio of exponential decline. The corresponding models for loss-ratio derivative and production rate are the following: PLE b model,

! = −!! ! − 1 !!

!!! + !!!! ! ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ (5)

PLE rate-time relation,

! = !! exp −!!! −!!!!! ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ (6)

Matching the loss-ratio and its derivative as well as production rate will give a more definite fit. However, loss-ratio

calculated from field data is often noisy and trends can be difficult to identify. A smoothing factor (SF) as that used in Bourdet’s derivative (1989) can help reduce noise. Stretched Exponential Decline (SEPD).

Valko (2009), and Valko and Lee (2010) independently proposed a similar model to the PLE. The model was originally developed to statistically evaluate large number of wells (Ilk et al. 2008).

SPE 162648 3

SEPD rate-time relation,

! = !! exp −!!

! ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ (7)

The SEPD model differs from the PLE model in not considering the behavior at late times (the !! term). In the SEPD

model, !! is always considered to be zero and ! is equivalent to ! !! ! !. One advantage of SEPD over PLE is the provided cumulative-time relation. This relation adds the option of fitting data to cumulative production, which is smoother and easier to regress on than the usually scattered production rate trends. SEPD cumulative-time relation,

! =q!!!

!1!− !

1! ,!!

! ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ (8)

The first term inside the brackets is the complete gamma function and the second term is the incomplete gamma

function. The natural logarithm of the gamma function can be calculated using a built-in function in Microsoft Excel. On the other hand, the incomplete gamma is not built-in in Excel. Duong’s Method.

Duong’s method (2010) was developed on the basis that production-rate and time would have a power law relation or form a straight line when plotted on a log-log scale. Integrating this relation with respect to time from (0 to t) gives the following material balance time and time relation:

!!!

= !!!! ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ (9)

Duong made the above equation more flexible to match field data by substituting m for the time exponent. The modified

equation is:

!!!

= !!!! ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ (10)

While this model is generally accurate for transient linear flow, production trends will curve from the log-log straight

line if BDF is reached. Duong adapts the concept of expanding Stimulated Reservoir Volume (Expanding SRV) which leads to infinite linear flow. This concept is based on reactivating existing faults and fractures caused by local stress changes during depletion (Warpinski and Branagan 1989). A constant SRV model can also show long linear flow to abandonment if matrix permeability is very low. Duong’s equations for production rate and cumulative are the following:

! = !!! !,! + !! ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ (11)

!! =!!! !,!!!!!

∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ (12) Where,

! !,! = !!!!"#!

1 −!!!!! − 1 ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ (13)

The original derived rate equation did not include the term !! (eqn. 11). Duong added the term to provide a better fit to

some field data that showed an intercept instead of a straight line to the origin when plotting (q vs. t(a,m)). The term can be positive or negative. Note that the cumulative production equation (eqn. 12) was not derived with !! present and should not be used if !! is not zero. Using the cumulative equation with !! other than zero will result in an erroneous cumulative when !! dominates the rate equation at long production times. Field data of shale wells usually show an increase in production at early times and Duong method models this behavior. This increase in production is most likely due frac fluid clean up and modeling this behavior is unnecessary. Logistic Growth Model (LGM).

The logistic models are based on the concept that growth is possible only to a certain size. The maximum growth size possible is referred to as the carrying capacity. LGMs are used to model population growth and its first adaptation in the petroleum industry was by Hubbert’s (1956). The Hubbert’s model was used to model production of a field or region. Clark

4 SPE 162648

el al. (2011) developed an LGM that models production of single well instead. The model was adapted from another LGM that models liver regrowth hyperbolically. The rate and cumulative equations are the following:

! =!"#!!!!

! + !! ! ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ (14)

! =!!!

! + !! ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ (15)

Where K is the carrying capacity or the technically recoverable resource.

Calculation and Matching Procedure This is a description of the calculation and matching procedure for each of the methods – all programed in Excel vba. The matching procedure for each case was done on a semi-log basis (log q vs. t). Each of the data cases had equally space time intervals (either daily or monthly) and each of the data points were weighted equally. The best fit for each case was determined by the least squares method, minimizing the sum of the squared differences between actual values (field or simulation values) and calculated values.

In addition to the log q vs. t match, the PLE method and the Duong method also require additional log-log plots. The PLE method fits log D and log b vs. log t and the Duong method fits log q/Gp vs. log t. A weighting factor was used for each point when matching these log-log plots to give each log t cycle equal weight. The weighting factor used for each point is the following:

!! =!"# !!!!

!!!"# !!"#$

!!

∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ (16)

Where !! is the first input time and !!"#$ the last. The matching parameters used for each method are constrained to avoid

unreasonable solutions. The constraints are shown in Table 1. Note that contrary to the original SPED paper, Tau (τ) has values in days instead of months because data are daily instead of monthly. Additional constraints are also enforced in the Duong and PLE methods. If production time is short, the best fit for Duong method is often accomplished using a large negative !! which often results in negative forecast rates near abandonment. Therefore, the !! parameter is set to (!! = 0) when production time is short. In the PLE method, the !! parameter is set to (!! = 0) if no BDF is observed to avoid a solution that underestimates reserves. The EUR calculated represents the actual cumulative plus remaining forecast production

Table 1 – Parameters Constraints for Each Method

Forecast Comparison against Simulation To understand the behavior of the aforementioned methods, they were applied on simulation and compared. Four cases were simulated to reflect the possible production trends that might be observed in the field. The four cases are the following:

1. Strictly linear flow for the whole production life of the well (Linear). 2. Linear flow followed by BDF (Linear-BDF). 3. Bilinear flow followed by linear flow for the remaining production life (Bilinear-Linear). 4. Bilinear flow followed by linear flow and ending with BDF (Bilinear-Linear-BDF).

Param Min Max Param Min Max Param Min MaxD 0 10000 τ (Days) 1E-‐09 10000 a 1E-‐09 10000b 0 5 n 0.006 1 n 1E-‐09 1qi (MSCF/D) 1E-‐09 1.E+10 qi (MSCF/D) 1E-‐09 1.E+10 k (MSCF) 1E-‐09 1.E+10

Param Min Max Param Min Maxn 0.02 1 a 1E-‐09 10D1 0.0001 1 m 1E-‐09 10D∞ 1E-‐09 1 qi (MSCF/D) 1E-‐09 1.E+10qi (MSCF/D) 10 1.E+15 q∞ (MSCF/D) -‐1000 10000

LGMSEPD

DuongPLE

Arps

SPE 162648 5

These cases were simulated using realistic properties typical of horizontal shale gas wells with multiple transverse hydraulic fractures. Linear flow is observed in all these wells and sometimes preceded by bilinear flow if natural fractures are present. Linear flow can also be followed by BDF decline if the boundary was reached. However, the BDF might not occur during the economic production life if matrix permeability is too low.

Two models were used for simulation. Model 1 was

used to simulate cases 1 & 2 and it assumes infinite-conductivity hydraulic fractures and a drainage volume equal to the SRV. Cases 3 & 4 were simulated using Model 2 which is similar to Model-1 but accounts for finite-conductivity natural fractures parallel to the horizontal well. The model schematics are shown in Figure 1 and the simulation parameters in Table 2. Both models assume a fixed SRV and flow is strictly sequential. In model 1, matrix flows strictly to hydraulic fractures and hydraulic fractures to the well. In model 2, matrix flows strictly to natural fractures, natural fractures to hydraulic fractures, and hydraulic fractures to the well.

The methods were compared in terms of modeling

accuracy. Also, the EUR was determined at different production times to assess when EUR calculation becomes accurate.

Table 2 – Simulation Parameters

a) Simulation Model 1 b) Simulation Model 2

Figure 1. a) Model 1: Horizontal well with infinite-conductivity hydraulic fractures b) Model 2: Horizontal well with infinite-conductivity hydraulic fractures in naturally fractured formation. Tivayanonda (2012)

Case 1 – Linear Flow.

Figure 2.a shows the simulated production for case 1. A linear flow half slope is observed until abandonment at 30 years. The modeling accuracy is shown in Figure 2.b. All methods can be shaped into reasonably straight lines to model case 1 except for the SEPD method which tends to curve at long production times. Nonetheless, forcing the SEPD model into a straight line is possible since it is mathematically identical to the PLE model. For case 1, the equivalent τ that makes the SEPD model identical to the PLE is (τ = 8.2E-69). This value is very small and outside the iteration sensitivity range and therefore this solution was not found. The Arps model was shaped into a straight line using a relatively large D and b value of (b = 2).

The EUR comparison is shown in Figure 6.a. For this case, Arps and PLE methods give the best EUR since they were

shaped into a perfectly straight line. Arps gives very close EUR at very early production time (100 days) while the PLE method becomes accurate after fitting longer production times (500 days). Though not shown in Figure 6.a, the Duong

km (md) 0.00001 0.00002 0.00010 0.00010kF (md) infinite infinite infinite infinitekf (md) -‐ -‐ 20 40nF 15 27 18 18nf -‐ -‐ 1 2xF (ft) 400 400 600 440LF (ft) 180 100 150 150Lf (ft) -‐ -‐ 600 220xe (ft) 2700 2700 2700 2700w (ft) 0.0090 0.0100 0.0015 0.0011ω (Fraction) 5.00E-‐05 1.00E-‐04 2.50E-‐06 5.00E-‐06φ (Fraction) 0.07 0.07 0.07 0.07pi (psi) 4000 4000 4000 4000pwf (psi) 500 500 500 500

Linear Linear-‐BDFBilinear-‐Linear

Bilinear-‐Linear-‐BDF

6 SPE 162648

method is also accurate only if !! was constrained to zero. Figure 6.a shows a decrease in Duong’s EUR calculations after 1000 days of production. This is because at short production times (100 and 500 days) the !! parameter was constrained to zero to avoid negative forecast rates near abandonment. The LGM method provides reasonably close EUR but after longer production times. Note that the SEPD method consistently underestimates ultimate recovery because the model tends to curve in the forecast.

Case 1: Linear Flow

a) Simulated Production b) Comparison of Methods

Figure 2 Results for simulation Case 1 exhibiting Linear flow to abandonment. All method fit case 1 reasonably

except for the SEPD method which could’t be shaped into a straight line.

Case 2 – Linear-BDF. This case is similar to case 1 except it shows BDF in the last log cycle (Figure 3.a). The method comparison (Figure 3.b)

shows that most methods cannot model case 2. Arps and Duong methods are not flexible to match both linear and BDF. The LGM method accomplishes a reasonable fit for the late part of data but not for early data while the SEPD method fits early data but not BDF. The PLE method, on the other hand, is the best by reasonably fitting linear and BDF. The EUR comparison for case 2 is shown in figure 6.b. Similar to case 1, the SEPD method was observed to underestimates ultimate recover. In general, all methods become accurate after BDF is observed.

Case 2: Linear-BDF

a) Simulated Production b) Comparison of Methods

Figure 3 - Results for simulation Case 2 exhibiting Linear-BDF. Notice that the Arps and Duong methods do not fit

well either at early or late times. The PLE method appears to give the best fit.

SPE 162648 7

Case 3 – Bilinear-Linear. Case 3 shows bilinear flow prior to a long linear flow extending to abandonment (Figure 4.a). The method comparison

(Figure 4.b) shows that all methods can model the linear flow portion of data. However, only the Duong method fits both bilinear and linear flow accurately. The other methods fit the overall curve with an acceptable mismatch except for the Arps method which was shaped into a straight line and fitted linear flow only. The EUR comparison (figure 6.c) shows that the Duong method gives the most accurate EUR after only observing the transition to linear flow (100 days). However, as it was in case 1, Duong’s !! parameter has to be forced to zero. The PLE method also gives accurate EUR but only after linear flow is well established. The LGM and SEPD methods underestimate ultimate recover while the Arps method overestimates (Figure 6.c).

Case 3: Bilinear-Linear

a) Simulated Production b) Comparsion of Methods

Figure 4 - Results for simulation Case 3 exhibiting Bilinear-Linear flow. The Duong method fits this case the best.

Arps method fits only linear flow and cannot fit both bilinear and linear. Case 4 – Bilinear-Linear-BDF.

This case is similar to case 3 except it shows BDF in the last log cycle. The simulated production (Figure 5.a) shows a quarter slope bilinear flow followed by a half slope linear flow and then BDF. Figure 5.b shows the method comparison and, similar to case 2, the PLE is the best method to model case 4. The EUR comparison (Figure 6.d) shows that close estimation for ultimate recovery is not possible at short production times. However, all methods converge at longer production times and become most accurate after BDF is observed. The SEPD method gives the most conservative EUR at times prior to BDF.

Case 4: Bilinear-Linear-BDF

a) Simulated Production b) Comparsion of Methods

Figure 5. Results for simulation Case 4 exhibiting Bilinear-Linear-BDF flow. Notice that the Arps and Duong

methods do not fit well either at early or late times. The PLE method appears to be the best even though bilinear flow is not modeled accurately.

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EUR Comparison: EUR Determined At Different Production Times a) b)

c) d)

* Duong method parameter !! is fixed to zero. ** PLE method parameter !! is included in iteration and not fixed to (!!= 1E-9).

Figure 6 – EUR Determined at different production times a) EUR comparison for case1. b) EUR comparison for case2. c) EUR comparison for case3. d) EUR comparison for case4.

100 Days 500 Days 1000 Days 2000 Days 4000 DaysArps 2.0 2.0 2.0 1.9 1.9PLE 1.7 1.9 1.9 1.9 1.9SEPD 0.5 1.0 1.2 1.4 1.6Duong 1.4 1.8 1.2 1.6 1.8LGM 1.5 1.8 1.8 1.9 1.9

0

1

2

3

EUR, BSCF

Case 1: Linear Flow

Simulation EUR =1.9 BSCF

* *


0

1

2

3

4

5

6

EUR, BSCF

Case 2: Linear-‐BDF

Simulation EUR =3.7BSCF

* *

**


0

1

2

3

4

5

6

EUR, BSCF

Case 3: Bilinear-‐Linear

Simulation EUR = 3.7 BSCF

*


012345678910111213

EUR, BSCF

Case 4: Bilinear-‐Linear-‐BDF

Simulation EUR = 5.9 BSCF

*

**

SPE 162648 9

Field Data Applications The different decline methods were applied to 3 wells from different fields. These fields are the Barnett, Bakken, and Eagleford fields. Barnett Shale Gas - Well 314.

Well 314 is a shale gas well from the Barnett shale. The well shows linear flow half slope followed by a deviation that can be considered as BDF (Figure 7.a). At long production times some rates fall short from the linear flow half slope trend. These lower rates are cause by either liquid loading or partial shut-ins. Turner et. al (1969) developed a correlation to calculate critical rate and all rates below were filtered out (Figure 7.a).

The forecast and EUR is shown in Table 3 and Figure

7.b. The forecasted EUR should be accurate using any of the methods since boundary was reached. The PLE and LGM methods give comparable EURs while the SEPD and the Duong methods are different. The BDF is not pronounced which might explain the discrepancies. Al-Ahmadi and Almarzooq (2010) analytically estimated the OGIP for this well (OGIP = 2.74 BSCF).

Table 3 – Well 314: Forecast Results

Determined At 1600 Days

Barnett Shale Gas – Well 314 a) b)

Figure 7 Data for Barnett well 314. The well exhibits linear flow and BDF. Even though BDF was reached, the

methods don’t agree on EUR. The BDF is not pronounced which might explain the discrepancies. Bakken Shale Oil - Well 720.

Well 720 is an oil well in the Bakken shale. Its production shows a half slope indicating transient linear flow to 875 days (Figure 8.a). The forecast is shown in Figure 8.b. Well 720 is in transient linear flow and it might either continue showing linear flow until abandonment or go into BDF if boundary is reached. In the case of strictly linear flow the EUR by PLE, Duong, or Arps are the best estimates (Table 4). In the case of BDF, the forecast has to be rerun when the boundary is reached. However, since the SEPD method is the most conservative, a range for EUR can be established. The most optimistic EUR is the one determined assuming strictly linear flow and therefore the EUR range is (0.7 < EUR < 1.7 MMSTB).

Table 4 – Well 720: Forecast Results Determined At 875 Days

Abd Rate EUR Abd Time(MSCF/D) BSCF Days

Arps 209 4.8 10950

PLE 44 3.4 10950

SEPD 15 2.7 10950

Duong 7 2.9 10950LGM 54 3.3 10950

10

100

1,000

10,000

100,000

1 10 100 1,000 10,000

Prod

uctio

n Ra

te, M

SCF/Da

y

Time, Days

Log-‐Log Rate Plot

Filtered Out Data

Data

1/2 Slope Line

10

100

1,000

10,000

100,000

1 10 100 1,000 10,000

qg, M

SCF/Da

y

Time, Days

Forecast Comparison

LGM

SEPDDuong

PLEArps

Abd Rate EUR Abd Timebbls/D MMbbls Days

Arps 84 1.5 10950

PLE 97 1.7 10950

SEPD 11 0.7 10950

Duong 67 1.4 10950LGM 74 1.4 10950

10 SPE 162648

Bakken Shale Oil – Well 720 a) b)

Figure 8 – Linear flow for Bakken shale oil well 720. Arps, Duong, LGM and PLE methods all comparable forecasts

assuming linear flow regime until abandonment. Eagleford Shale Gas - Well 204.

Monthly data for this shale gas well was gathered from HPDI database. The monthly rates were converted to daily rates and plotted at mid-month. The well shows linear flow behavior for 1000 days (Figure 9.a). Forecast and EUR are shown in Figure 9.b and Table 5. This well is similar to the previous one except less data points are available which resulted in a wider range of forecasts. An EUR range can be established with linear flow extrapolation as being the most optimistic while the SEPD method the most conservative (3.0 < EUR < 8.8 BSCF). The forecast has to be rerun if BDF is reached for more accurate estimations.

Table 5 – Well 720: Forecast Results Determined At 1000 Days

Eaglefrod Shale Gas – Well 204

a) b)

Figure 9 – Monthly data gathered from HPDI database for Eagleford well 204. Linear flow is observed. The methods

show a wide range of forecast due to the low density of monthly data.

Abd Rate EUR Abd Time(MSCF/D) BSCF Days

Arps 477 8.8 10950

PLE 27 4.1 10950

SEPD 0 3.0 10950

Duong 185 5.7 10950LGM 40 4.1 10950

SPE 162648 11

Discussion and Conclusions The decline methods showed here have different models and equation forms, thus often provide different forecasts. Arp’s hyperbolic equation can be shaped into a straight line to fit either bilinear or linear flow with b values of 2 and 4 respectively. However, it cannot model multiple flow regimes. Most methods give reasonably accurate forecasts if flow is strictly transient except for the SEPD method which seems to always underestimate ultimate recovery. If BDF is expected, the EUR cannot be accurately established until BDF is observed. Four cases that represent the typical flow regimes of shale wells were simulated and below is a summary:

• Linear: if a well is expected to show linear flow to abandonment, the most accurate EUR is determined using Arps, PLE, or Duong. Though, both the !! and !! parameters of the PLE and Duong methods should be constrained to zero.

• Bilinear-Linear: if a well is expected to show linear flow to abandonment but was preceded with bilinear flow, the most accurate EUR is determined by the PLE or Duong methods. Also, in this case, both !! and !! constants should be constrained to zero.

• Linear-BDF or Bilinear-Linear-BDF: if the well is expected to show BDF prior to abandonment, a reliable forecast cannot be established. The BDF must be observed to reliable estimate ultimate recovery. The method that best models both transient and BDF of shale wells is the PLE method.

Acknowledgement The authors would like to thank the sponsors of the Computer Modeling Consortium at Texas A&M University for their support. The authors are also grateful to HPDI, LLC for providing access to their production database. Nomenclature ! = Duong or LGM constant. ! = derivative of loss-ratio (Arps’ decline exponent), dimensionless ! = loss-ratio (Arps’ decline constant), Days-1 !! = loss-ratio at (! = 1), Days-1

!! = loss-ratio at (! = ∞), Days-1

EUR = estimated ultimate recovery, Bscf or MMSTB !! = cumulative production. Mscf ! = carrying capacity of LGM method ! = time exponent or hyperbolic exponent. ! = flow rate, STB/Day or Mscf/Day !! = flow rate at (! = 0), STB/Day or Mscf/Day !! = flow rate at (! = 1), STB/Day or Mscf/Day !! = flow rate at (! = ∞), STB/Day or Mscf/Day ! = cumulative production, Mscf ! = time, Days !! = first input time, Days !!"#$ = last input time, Days ! !,! = Duong’s time function based on Eqn. 13 ! = characteristic time parameter for SEPD model, Days !! = weighting factor References Ahmadi, H.A.A., Almarzooq, A.M., and Wattenbarger, R.A. 2010. Application of Linear Flow Analysis to Shale Gas Wells -

Field Cases. Paper SPE 130370 presented at the SPE Unconventional Gas Conference, Pittsburgh, Pennsylvania, USA, 23-25 February. doi: 10.2118/130370-MS.

Arps, J.J. 1945. Analysis of Decline Curves. SPE-945228-G. Trans., AIMS 160: 228-247. Clark, A.J., Lake, L.W., and Patzek, T.W. 2011. Production Forecasting with Logistic Growth Models. Paper SPE 144790

presented at the SPE Annual Technical Conference and Exhibition, Denver, Colorado, USA, 30 October-2 November. doi: 10.2118/144790-MS.

Duong, A.N. 2011. Rate-Decline Analysis for Fracture-Dominated Shale Reservoirs. SPE Reservoir Evaluation & Engineering (3): pp. 377-387. doi: 10.2118/137748-PA.

El-Banbi, A.H. and Wattenbarger, R.A. 1996. Analysis of Commingled Tight Gas Reservoirs. Paper SPE 36736 presented at the SPE Annual Technical Conference and Exhibition, Denver, Colorado, USA, 6-9 October. doi: 10.2118/36736-MS.

Fetkovich, M.J. 1971. A Simplified Approach to Water Influx Calculations-Finite Aquifer Systems. SPE Journal of Petroleum Technology 23 (7): 814-828. doi: 10.2118/2603-PA.

Fetkovich, M.J. 1980. Decline Curve Analysis Using Type Curves. SPE Journal of Petroleum Technology 32 (6): 1065-1077. doi: 10.2118/4629-PA.

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Fetkovich, M.J., Bradley, M.D., Works, A.M. et al. 1990. Depletion Performance of Layered Reservoirs without Crossflow. SPE Formation Evaluation 5 (3): 310-318. doi: 10.2118/18266-PA.

Hubbert, M.K. 1956. Nuclear Energy and the Fossil Fuel. American Petroleum Institute API-56-007. Ilk, D., Rushing, J.A., Perego, A.D. et al. 2008. Exponential Vs. Hyperbolic Decline in Tight Gas Sands - Understanding the

Origin and Implications for Reserve Estimates Using Arps' Decline Curves. Paper SPE 116731 presented at the SPE Annual Technical Conference and Exhibition, Denver, Colorado, USA, 21-24 September. doi: 10.2118/116731-MS.

Lee, W.J. and Sidle, R.E. 2010. Gas Reserves Estimation in Resource Plays. Paper SPE 130102 presented at the SPE Unconventional Gas Conference, Pittsburgh, Pennsylvania, USA, 23-25 February. doi: 10.2118/130102-MS.

Tivayanonda, V. 2012. Comparison of Single, Double, and Triple Linear Flow Models For Shale Gas/Oil Reservoirs. M.Sc. Thesis, Texas A&M U., College Station, Texas, USA.

Valko, P.P. 2009. Assigning Value to Stimulation in the Barnett Shale: A Simultaneous Analysis of 7000 Plus Production Histories and Well Completion Records. Paper SPE 119369 presented at the SPE Hydraulic Fracturing Technology Conference, The Woodlands, Texas, USA 19-21 January. doi: 10.2118/119369-MS.

Valko, P.P. and Lee, W.J. 2010. A Better Way to Forecast Production from Unconventional Gas Wells. Paper SPE 134231 presented at the SPE Annual Technical Conference and Exhibition, Florence, Italy, 19-22 September. doi: 10.2118/134231-MS.

Warpinski, N.R. and Branagan, P.T. 1989. Altered-Stress Fracturing. SPE Journal of Petroleum Technology 41 (9): 990-997. doi: 10.2118/17533-PA.

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