Arps Decline Curve Analysis

Overview

Decline curve analysis (DCA) is the most widely used technique for estimating ultimate recovery and forecasting future production from oil and gas wells. The methodology assumes that historical production trends can be characterized mathematically and extrapolated into the future.

J.J. Arps published his seminal paper "Analysis of Decline Curves" in 1945 [1], establishing three fundamental decline models based on the behavior of the decline rate $D$ :

Model	Decline Rate Behavior	b Value
Exponential	Constant	$b = 0$
Hyperbolic	Decreasing proportionally	$0 < b < 1$
Harmonic	Decreasing linearly with rate	$b = 1$

The Arps equations remain the industry standard for conventional reservoirs in boundary-dominated flow (BDF).

Theory

The Loss Ratio Concept

Arps defined the decline rate $D$ as the fractional change in production rate per unit time:

D = -\frac{1}{q}\frac{dq}{dt}

He also defined the loss ratio $D^{-1}$ and observed that for many wells, the derivative of the loss ratio with respect to time was approximately constant:

b = \frac{d}{dt}\left(\frac{1}{D}\right)

This parameter $b$ (sometimes called the Arps exponent or hyperbolic exponent) determines which decline model applies:

$b = 0$ : Decline rate is constant → Exponential decline
$0 < b < 1$ : Decline rate decreases with time → Hyperbolic decline
$b = 1$ : Special case → Harmonic decline

Physical Interpretation

The value of $b$ relates to the drive mechanism and reservoir characteristics:

b Value	Typical Reservoir Type
$b \approx 0$	Strong water drive, high-pressure gas
$b = 0.3 - 0.5$	Solution gas drive (typical)
$b = 0.4 - 0.5$	Gas expansion drive
$b = 0.5 - 1.0$	Gravity drainage
$b > 1$	Transient flow (not boundary-dominated)

Important: Arps equations are only valid during boundary-dominated flow. Values of $b > 1$ indicate transient flow conditions where alternative models (PLE, SEPD, Duong) should be used.

Equations

Exponential Decline ( $b = 0$ )

When the decline rate $D$ is constant, integration of the decline rate equation yields:

q(t) = q_i e^{-D_i t}

Cumulative Production:

N_p(t) = \frac{q_i - q(t)}{D_i} = \frac{q_i}{D_i}\left(1 - e^{-D_i t}\right)

Time to Reach Economic Limit:

t_{econ} = \frac{1}{D_i}\ln\left(\frac{q_i}{q_{econ}}\right)

Estimated Ultimate Recovery (EUR):

EUR = \frac{q_i - q_{econ}}{D_i}

Hyperbolic Decline ( $0 < b < 1$ )

For a time-varying decline rate where $b$ is constant:

q(t) = \frac{q_i}{(1 + b \cdot D_i \cdot t)^{1/b}}

Cumulative Production:

N_p(t) = \frac{q_i}{(1-b) D_i}\left[1 - (1 + b \cdot D_i \cdot t)^{(1-1/b)}\right]

Or equivalently:

N_p(t) = \frac{q_i^b}{(1-b) D_i}\left(q_i^{1-b} - q(t)^{1-b}\right)

Time to Reach Economic Limit:

t_{econ} = \frac{1}{b \cdot D_i}\left[\left(\frac{q_i}{q_{econ}}\right)^b - 1\right]

EUR:

EUR = \frac{q_i^b}{(1-b) D_i}\left(q_i^{1-b} - q_{econ}^{1-b}\right)

Harmonic Decline ( $b = 1$ )

The harmonic decline is the limiting case of hyperbolic decline as $b \to 1$ :

q(t) = \frac{q_i}{1 + D_i \cdot t}

Cumulative Production:

N_p(t) = \frac{q_i}{D_i}\ln(1 + D_i \cdot t) = \frac{q_i}{D_i}\ln\left(\frac{q_i}{q(t)}\right)

Time to Reach Economic Limit:

t_{econ} = \frac{1}{D_i}\left(\frac{q_i}{q_{econ}} - 1\right)

EUR:

EUR = \frac{q_i}{D_i}\ln\left(\frac{q_i}{q_{econ}}\right)

Note: For harmonic decline, cumulative production approaches infinity as $t \to \infty$ . This is physically unrealistic for finite reservoirs, which is why harmonic decline should only be used for forecasting over finite time periods.

Parameter Definitions

Symbol	Description	Units
$q$	Production rate at time $t$	STB/d, Mscf/d, or any consistent [L³/T]
$q_i$	Initial production rate	STB/d, Mscf/d, or any consistent [L³/T]
$D_i$	Initial (nominal) decline rate	1/d, 1/month, 1/yr (must match time units)
$b$	Arps hyperbolic exponent	dimensionless
$t$	Time since start of decline	d, month, yr (must match decline rate units)
$N_p$	Cumulative production	STB, Mscf, or consistent [L³]
$q_{econ}$	Economic limit rate	Same as $q$
$EUR$	Estimated Ultimate Recovery	Same as $N_p$

Unit Consistency

The Arps equations are unit-flexible. The key requirement is that:

[D_i] \times [t] = \text{dimensionless}

For example:

If $D_i$ is in 1/day, then $t$ must be in days
If $D_i$ is in 1/year, then $t$ must be in years

Functions Covered

The following Petroleum Office functions implement Arps decline curve analysis:

Rate Calculation

Function	Description
ExponentialDeclineRate	Rate using exponential decline: $q = q_i e^{-D_i t}$
HyperbolicDeclineRate	Rate using hyperbolic decline: $q = q_i (1 + b D_i t)^{-1/b}$
HarmonicDeclineRate	Rate using harmonic decline: $q = q_i / (1 + D_i t)$

Cumulative Production

Function	Description
ExponentialDeclineCumulative	Cumulative production, exponential decline
HyperbolicDeclineCumulative	Cumulative production, hyperbolic decline
HarmonicDeclineCumulative	Cumulative production, harmonic decline

Time to Economic Limit

Function	Description
ExponentialDeclineTime	Time to reach economic rate, exponential
HyperbolicDeclineTime	Time to reach economic rate, hyperbolic
HarmonicDeclineTime	Time to reach economic rate, harmonic

EUR Calculation

Function	Description
ExponentialDeclineEUR	EUR to economic limit, exponential
HyperbolicDeclineEUR	EUR to economic limit, hyperbolic
HarmonicDeclineEUR	EUR to economic limit, harmonic

Parameter Fitting

Function	Description
ExponentialDeclineFitParameters	Fit $[q_i, D_i]$ to rate-time data
HyperbolicDeclineFitParameters	Fit $[q_i, D_i, b]$ to rate-time data
HarmonicDeclineFitParameters	Fit $[q_i, D_i]$ to rate-time data
ExponentialDeclineWeightedFitParameters	Weighted fit $[q_i, D_i]$
HyperbolicDeclineWeightedFitParameters	Weighted fit $[q_i, D_i, b]$
HarmonicDeclineWeightedFitParameters	Weighted fit $[q_i, D_i]$

See each function page for detailed parameter definitions, Excel syntax, and usage examples.

Applicability and Limitations

Valid Conditions

Arps decline curves are valid when:

Boundary-dominated flow has been established
Operating conditions remain constant (bottomhole pressure, completion, artificial lift)
Drainage area is fixed (no infill drilling or recompletions)
The well is producing at or near capacity

Typical Parameter Ranges

Parameter	Typical Range	Notes
$b$	0 - 1.0	Values > 1 indicate transient flow
$D_i$	5% - 95% per year	Higher for tight formations
$q_i$	Well-dependent	Must be positive

Limitations

Transient Flow: When $b > 1$ , the well is still in transient flow and Arps equations will overestimate reserves [2, 3].
Infinite EUR for Harmonic: Harmonic decline ( $b = 1$ ) predicts infinite cumulative production as $t \to \infty$ . Use modified hyperbolic or economic limits.
Changing Conditions: Arps assumes constant operating conditions. Significant changes in:
- Bottomhole pressure
- Artificial lift
- Completion/stimulation
- Regulatory constraints
will invalidate the forecast.
Unconventional Reservoirs: Tight gas and shale wells exhibit extended transient flow with $b > 1$ . Use modern decline models (PLE, SEPD, Duong) for these reservoirs.

Historical Background

J.J. Arps developed his decline curve methodology while working for Shell Oil Company in the 1940s. His 1945 paper [1] in Transactions of AIME introduced the mathematical framework that remains the foundation of production forecasting today.

The approach was empirical—Arps observed that production decline could be characterized by a few parameters and developed equations to describe the three main types of decline behavior. The simplicity and robustness of the Arps equations led to their widespread adoption.

Subsequent work by Fetkovich (1980) [4] provided a theoretical basis for Arps equations by linking them to material balance and flow equations, showing that:

Exponential decline corresponds to constant-pressure, single-phase liquid flow
Hyperbolic decline arises from changing pressure, two-phase flow, or layered reservoirs

Modified Hyperbolic Decline - Transition to exponential at limiting $D$
Power Law Exponential Decline - For tight/unconventional reservoirs
Stretched Exponential Decline - Statistical physics approach
Duong Decline Model - Fracture-dominated unconventional flow
Decline Overview - Model selection guide

References

Arps, J.J. (1945). "Analysis of Decline Curves." Transactions of AIME, 160, 228-247.
Ali, T.A. and Sheng, J.J. (2015). "Production Decline Models: A Comparison Study." SPE-177300-MS, SPE Eastern Regional Meeting, Morgantown, West Virginia, 13-15 October 2015.
Lee, W.J. and Sidle, R.E. (2010). "Gas Reserves Estimation in Resource Plays." SPE-130102, SPE Unconventional Gas Conference, Pittsburgh, Pennsylvania, 23-25 February 2010.
Fetkovich, M.J. (1980). "Decline Curve Analysis Using Type Curves." Journal of Petroleum Technology, 32(6), 1065-1077. SPE-4629-PA.
Blasingame, T.A. and Rushing, J.A. (2005). "A Production-Based Method for Direct Estimation of Gas-in-Place and Reserves." SPE-98042, SPE Eastern Regional Meeting, Morgantown, West Virginia, 14-16 September 2005.

Arps Decline Curve Analysis

Overview

Theory

The Loss Ratio Concept

Physical Interpretation

Equations

Exponential Decline (b=0b = 0b=0)

Hyperbolic Decline (0<b<10 < b < 10<b<1)

Harmonic Decline (b=1b = 1b=1)

Parameter Definitions

Unit Consistency

Functions Covered

Rate Calculation

Cumulative Production

Time to Economic Limit

EUR Calculation

Parameter Fitting

Applicability and Limitations

Valid Conditions

Typical Parameter Ranges

Limitations

Historical Background

Related Topics

References

Related Excel Functions

Exponential Decline ( $b = 0$ )

Hyperbolic Decline ( $0 < b < 1$ )

Harmonic Decline ( $b = 1$ )