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DRAFT

Probabilistic Decline Curve Analysis of Barnett, Fayetteville, Haynesville, and

Woodford Gas Shales

J. R. Fanchi, M. J. Cooksey, K. M. Lehman and A. Smith

Department of Engineering and Energy Institute

Texas Christian University

and

A.C. Fanchi and C. J. Fanchi

Energy.Fanchi.com

June 25, 2013

Abstract

This paper presents a probabilistic decline curve workflow to model shale gas

production from the Barnett, Fayetteville, Haynesville, and Woodford shales. Ranges of

model input parameters for four gas shales are provided to guide the preparation of

uniform and triangle probability distributions. The input parameter ranges represent

realistic distributions of model parameters for specific gas shales.

Keywords: Shale Gas; Decline Curve Analysis; Monte Carlo Analysis; Probability

Distributions

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1. Introduction

Many people have developed techniques that were designed to forecast

production from unconventional resources, such as Brown, et al. (2009), Valkó and Lee

(2010), Duong (2011), Anderson, et al. (2012), and Esmaili, et al. (2012). Our ability to

forecast shale gas production is complicated by our inability to correctly account for all

of the mechanisms that affect production. We present a probabilistic workflow that is a

modification of reservoir simulation workflows and is designed for use with rate-time

decline curve models.

Different workflows exist for performing reservoir simulation projects. For

example, Fanchi has presented workflows for green fields (Fanchi, 2010 and 2011a) and

brown fields (Fanchi, 2010 and 2011b) that use reservoir flow models to generate a

distribution of recovery forecasts. The workflows are able to integrate uncertainty in the

development of recovery forecasts. The workflow presented here uses the probabilistic

decline curve analysis (DCA) workflow to develop model input parameter ranges for the

Barnett, Fayetteville, Haynesville, and Woodford shales. The methodology is automated

in the form of a software program that calculates the distribution of Estimated Ultimate

Recovery (EUR) for production of gas from unconventional gas shale. Using well data

from different shale gas plays, we determined realistic minimum and maximum values

for the uniform distribution of a parameter and triangular distribution of a parameter. The

realistic ranges of model input parameters can be used to guide the preparation of

uniform and triangle probability distributions.

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2. Decline Curve Models for Unconventional Resources

Decline curve models used here must have finite, bounded values of EUR. Not all

decline curve models satisfy this criterion. For example, Arps (1945) presented the

following empirical decline curve model for flow rate q as a function of time t and

parameters a, b:

1 baqdt

dq (1)

The Arps models are harmonic decline (b = 1), exponential decline (b = 0), and

hyperbolic decline with other positive values of b. The hyperbolic model typically has b

< 1 for conventional reservoir production. The Arps harmonic model (b = 1) and

hyperbolic model with b > 1 are not always applicable to unconventional reservoir

production forecasts because extrapolation of the decline curve can lead to unbounded

values of EUR and corresponding overestimates of EUR.

The Arps exponential model does not always adequately model the decline rate of

unconventional reservoir production. Valkó and Lee (2010) introduced the Stretched

Exponential Decline Model (SEDM) as a generalization of the Arps exponential model.

The SEDM is based on the idea that several decaying systems comprise a single decaying

system (Phillips, 1996; and Johnston, 2006). If we think of production from a reservoir as

a collection of decaying systems in a single decaying system, then SEDM can be viewed

as a model of the decline in flow rate. The SEDM has three parameters qi, n and τ (or a,

b, c):

] )exp[-(t/ a= ] )exp[-(t/ q=q cn

i b (2)

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Parameter qi is flow rate at initial time t. The Arps exponential decline model is the

special case of SEDM with n = 1.

A second decline curve model is based on the logarithmic relationship between

pressure and time in a radial flow system. We can use productivity index to link rate and

pressure to obtain the logarithmic decline curve model

b+lnt a=q (3)

with parameters a and b. This logarithm model is referred to as the LNDM model.

The third decline curve model used here is the Arps hyperbolic decline curve

model with the restriction that 0 < b < 1. The hyperbolic model

(-1/b)bct)+a(1=q (4)

is referred to as the HYDM model.

Model input parameters for each decline curve model are summarized in Table 1.

Cumulative gas production Q is given by the integral

T

T0

dtq=Q (5)

The lower limit T0 is the initial time and the upper limit T is the time when the economic

limit is reached.

3. Probabilistic DCA Workflow

Reserves estimates may be either deterministic or probabilistic (Lee, 2009). A

deterministic estimate of reserves is a single best estimate of reserves based on

geological, engineering, and economic data. A probabilistic estimate of reserves uses

geological, engineering, and economic data to generate a range of estimates and their

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associated probabilities. Linear regression can be used to obtain a deterministic estimate

of reserves. The probabilistic estimate of reserves is obtained here using the workflow for

probabilistic DCA of unconventional gas production outlined in Figure 1.

Each step of the probabilistic decline curve analysis method is briefly described

below. More discussion can be found in Fanchi (2012a, 2012b).

Step DCA1: Gather Rate-Time Data

Acquire production rate as a function of time. Remove significant shut-in periods

so rate-time data represents continuous production.

Step DCA2: Select a DCA Model and Specify Input Parameter Distributions

The number of input parameters depends on the DCA model chosen. The SEDM

model requires three parameters, and the LNDM model requires two parameters.

Parameter distributions may be either uniform or triangle distributions.

Step DCA3: Specify Constraints

Available rate-time production history is used to decide which DCA trials are

acceptable. Every DCA model run that uses a complete set of model input parameters

constitutes a trial. The results of each trial are then compared to user-specified criteria.

Criteria options include an objective function, rate at the end of history, and cumulative

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production at the end of history. The objective function quantifies the quality of the

match by comparing the difference between model rates and observed rates. Objective

functions with smaller values are considered better matches than objective functions with

larger values.

Step DCA4: Generate Decline Curve Trials

Decline curve trials are obtained by running the DCA model. The number of trials is

specified by the analyst.

Step DCA5: Determine Subset of Acceptable Trials

The trials generated in Step DCA4 are compared to the criteria specified in Step DCA3.

Each trial that satisfies the user-specified criteria is included in a subset of acceptable

trials.

Step DCA6: Generate Distribution of Performance Results

The distribution of EUR values for the subset of acceptable trials is analyzed and the 10th

(PC10), 50th (PC50), and 90th (PC90) percentiles are determined.

4. Model Results

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The workflow in Figure 1 is applied using rate-time data for gas production for a

sampling of wells from the Barnett (30 wells), Fayetteville (10 wells), Haynesville (10

wells), and Woodford shales (60 wells). Three DCA models (SEDM, LNDM, and

HYDM) are used to match rate-time data for each well. The Monte Carlo analysis uses

1000 trials initially. The value of parameter b in the HYDM model is restricted to the

range 0.01 < b < 0.99. A match of cumulative gas production at the end of the historical

production period is used as the constraint for selecting the subset of trials. The constraint

requires that the decline curve model and its set of model parameters must generate a

cumulative gas production that matches actual cumulative gas production to within 1%

by the end of the historical production period.

Figures 2 through 5 illustrate the quality of the matches for each shale gas play.

Each figure shows the match of the 50th

percentile (PC50) trial to well data. The

parameters for the PC50 trial are collected and a minimum and maximum value for each

model parameter is determined for the set of wells in each play. Model trials that do not

satisfy the cumulative gas constraint are not used to determine minimum and maximum

model input parameter values.

Tables 2 through 5 present minimum and maximum values for parameters of each

decline curve model.

5. Concluding Remarks

We have analyzed shale gas production from the Barnett, Fayetteville,

Haynesville, and Woodford shales using a probabilistic decline curve workflow.

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Minimum and maximum values of model input parameters were determined from the

model input parameters for the 50th

percentile trials. The model input parameter ranges

are realistic distributions of model parameters that can be used to guide the preparation of

uniform and triangle probability distributions.

Acknowledgement

We thank drillinginfo.com for access to their database of well production data.

References

Anderson, D.M., P. Liang, and V. Okouma, 2012. Probabilistic Forecasting of

Unconventional Resources Using Rate Transient Analysis: Case Studies. Paper SPE

155737. Society of Petroleum Engineers, Richardson, Texas.

Arps, J.J., 1945. Analysis of Decline Curves. Paper SPE 945228-G. Trans. AIME,

Volume 160, 228-247.

Brown, M., E. Ozkan, R. Raghavan, and H. Kazemi, 2009. Practical Solutions for

Pressure Transient Responses of Fractured Horizontal Wells in Unconventional

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Duong, A.N, 2011. Rate-Decline Analysis for Fracture-Dominated Shale Reservoirs, SPE

Reservoir Evaluation and Engineering (June) 377-387.

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Table 1

Decline Model Parameters

Parameter SEDM LNDM HYDM

DCMA a or qi a a or qi

DCMB b or τ b b

DCMC c or n NA c or Di

Table 2

Barnett Shale Gas Wells

SEDM LNDM HYDM

Case Parameter MIN MAX MIN MAX MIN MAX

PC50 A 973170 1014700 -43398 -5976 31002 239570

PC50 B 5.3E-07 0.1821 34214 223090 0.869 0.9736

PC50 C 0.08147 0.22797 0 0 0.0212 0.244

Table 3

Fayetteville Shale Gas Wells

SEDM LNDM HYDM

Case Parameter MIN MAX MIN MAX MIN MAX

PC50 A 951660 1037500 -53075 -22994 122020 364690

PC50 B 0.00372 0.39793 95847 178610 0.7372 0.9294

PC50 C 0.12784 0.28141 0 0 0.1011 0.3803

Table 4

Haynesville Shale Gas Wells

SEDM LNDM HYDM

Case Parameter MIN MAX MIN MAX MIN MAX

PC50 A 975540 1034800 -91907 -4235.4 33319 523580

PC50 B 4.6E-12 1.1276 34839 339460 0.65501 0.96419

PC50 C 0.04457 0.3553 0 0 0.01153 0.52415

Table 5

Woodford Shale Gas Wells

SEDM LNDM HYDM

Case Parameter MIN MAX MIN MAX MIN MAX

PC50 A 961150 1046400 -215090 -1979 10518 966990

PC50 B 6.63E-20 32.7740 10859 1120800 0.60790 1.42630

PC50 C 0.0282 0.7969 0 0 0.02601 0.72473

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Figure 1. Probabilistic DCA Workflow

Figure 2. Barnett Shale A – PC50

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Figure 3. Fayetteville Shale A – PC50

Figure 4. Haynesville Shale A – PC50

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Figure 5. Woodford Shale A – PC50