# Decline Curve Analysis

Oil and gas production rates decline as a function of time. Loss of reservoir pressure or the changing relative volumes of the produced fluids are usually the cause. Fitting a line through the existent production data and assuming this same line extends similarly into the future forms the basis for the decline curve analysis (DCA) concept. Lets review popular decline curve models.

### Arps Decline Curves

Arps decline curve analysis is an empirical model based on the plot of flow rate versus time to the abandonment time. There are three types of production rate decline characterized based on the way in which rate declines with time: hyperbolic, harmonic, and exponential. Arps applied the equation of hyperbola to define three general equations to model production declines. In order to locate a hyperbola in space one must know the following three variables. The starting point on the “y” axis which in this case is the initial rate (qi), the initial decline rate (Di), and the degree of curvature of the line, controls how the change of the decline rate with time, known as decline exponent constant (b).

These three models are related through the following relative decline rate equation:

$D=-\ \frac{1}{q}\frac{dq}{dt}$
$D=\ k\ q^b$

The behaviour of the production data can be characterized based on the way in which the nominal decline rate (D) varies with production rate (q), based on the value of the decline exponent constant (b).

• Exponential: b = 0
• Hyperbolic: b is a value other than 0 or 1
• Harmonic: b = 1

Using the values of b, and with mathematical manipulation for Eq-01 & 02 we can obtain the following governing equation for each model:

• Exponential Decline
$q=q_{i\ }{exp\left(-D_it\right)}^\$
• Hyperbolic Decline
$q\ =\ \frac{q_i}{\ \left(1+bD_it\right)^\frac{1}{b}}$
• Harmonic Decline
$q\ =\ \frac{q_i}{\ \left(1+D_it\right)^\ }$

Basic assumptions that should be considered while using Arp’s:

• Constant bottom hole pressure production
• The future behavior of a well will be governed by whatever trend or mathematical relationship is apparent in its past performance.
• Drainage area remains constant.

Limitations:

• Should be used in case of boundary dominated flow, as it might give unrealistic b value greater than 1 in case of transient flow.
• Limited application in tight gas and unconventional reservoirs which also yield unrealistic b value greater than 1.

## Power Law Exponential (PLE) Model

This decline model is based on Arps exponential decline model developed by Ilk et al. in 2008 to fit and predict tight gas and shale production. The only difference is that the decay in the Arps exponential decline model is constant, whereas the power law model considers the decay to be a power law function. This model is flexible enough to model transient, transition, and boundary-dominated flow, but at long times, the relation deduces to the traditional exponential decline relation (i.e., At late times the D inf will govern the decline behaviour, whereas the contribution of the power-law term is relatively smaller).

$q\left(t\right)={\hat{q}}_{i\ }{exp\left(-D_\infty t-{\hat{D}}_it^n\right)}^\$
$D\left(t\right)=\left(D_\infty+D_1t^{-\left(1-n\right)}\right)^\$

where qi is the rate intercept or q at (t = 0), Di is the decline constant, Dinf is the decline constant at infinite time, and n is a time exponent, D1 is the Decline constant intercept at t = 1 day.

Limitations:

• Practical application of the power law model is not as simple as an empirical fit with the Arps equation.
• The value of is arbitrary, and with insufficient data the value has a lot of uncertainty.
• The power law model can result in non-unique solution because of four degrees of freedom resulting from the four unknown parameters.

## Stretched Exponential Production Decline (SEPD) Model

Valko (2009) proposed the model to avoid the arbitrariness associated with long-term reserve estimates from the hyperbolic model. The model, unlike other models, does have a physical basis and is governed by a defining differential equation. The production rate obeys a decaying exponential relation as follows:

$q=q_0 \exp \left[-\left(\frac{t}{\tau}\right)^n\right]$

where q is the time varying production rate, q0 is the initial production rate, tie parameter is the characteristic time parameter and n is the exponent parameter.

Application of the stretched exponential model for decline curve analysis is more complicated to apply compared to other models. The stretched exponential model is used for large scale evaluation of entire fields and does not emphasize the individual well analysis.

## Logistic Growth Model (LGM)

Clark et al. propose a new empirical model for production forecasting in extremely low permeability oil and gas reservoirs based on logistic growth models. This model incorporates known physical volumetric quantities of oil and gas into the forecast to constrain the reserve estimation to a reasonable quantity. The formulation presented relies on the assumption that cumulative production grows logistically to a previously known or estimated initial hydrocarbon volume in-place.

$q(t)=\ \ \frac{K\ n\ a\ t^{n-1}}{\left(a+t^n\right)^2}$

Where: q(t) is the production rate, 𝐾 is the carrying capacity or estimated ultimate recovery, a is a model constant, n is hyperbolic exponent, and t is time.

## Duong Model

Duong (2011) developed a model that fits production performance of fractured wells producing from super tight and shale reservoirs. This approach accounted for the presence of fracture-dominated flow regimes. It can predict production performance and estimate reserve accurately for those reservoirs. Duong (2011) has shown that a log-log plot of cumulative production vs. time will yield a straight line with a unity slope for finite and infinite fractures.

$\frac{q}{G_p}\ =\ a\ t^{-m}$

Where a is the intercept constant and m is the negative slope parameter. Production rate equation based on Duong model is as the following:

$q\left(t\right)=q_{1\ }t^{-m\ }{exp\left({\frac{a}{1-m}\ \left(t^{1-m}\ -\ 1\right)}^\ \ \right)}^\$

Where isq1 production rate at t = 1 day.

## References

• Arps, J. J. (1945). Analysis of decline curves. Transactions of the AIME, 160(01), 228-247.
• Ali, T. A., & Sheng, J. J. (2015, October). Production decline models: A comparison study. In SPE Eastern Regional Meeting. OnePetro.
• Clark, A. J., Lake, L. W., & Patzek, T. W. (2011, October). Production forecasting with logistic growth models. In SPE annual technical conference and exhibition. OnePetro.
• Collins, P. W. (2016). Decline Curve Analysis for Unconventional Reservoir Systems-Variable Pressure Drop Case (Doctoral dissertation).
• Ilk, D., Rushing, J. A., Perego, A. D., & Blasingame, T. A. (2008, September). Exponential vs. Hyperbolic Decline in Tight Gas Sands—Understanding the Origin and Implications for Reserve Estimates Using Arps’ Decline Curves. In SPE annual technical conference and exhibition. OnePetro.
• Valkó, P. P. (2009, January). Assigning value to stimulation in the Barnett Shale: a simultaneous analysis of 7000 plus production hystories and well completion records. In SPE hydraulic fracturing technology conference. OnePetro.
• Wahba, A. M., Khattab, H. M., Tantawy, M. A., & Gawish, A. A. (2022). Modern Decline Curve Analysis of Unconventional Reservoirs: A Comparative Study Using Actual Data. Journal of Petroleum and Mining Engineering, 24(1), 51-65.