Stretched Exponential Production Decline (SEPD) Model

Overview

The Stretched Exponential Production Decline (SEPD) model was introduced by Valkó and Lee (2010) as a statistically-grounded approach to decline curve analysis for unconventional reservoirs. Unlike other empirical models, SEPD has a foundation in statistical physics, specifically the theory of "fat-tailed" distributions, making it particularly suitable for analyzing large populations of wells.

Key Concepts

• Statistical Physics Basis: Derived from stretched exponential (Kohlrausch) functions used in material science
• Bounded EUR: Always produces finite cumulative production, avoiding the b > 1 problem
• Population-Based: Originally designed for analyzing entire field populations, not individual wells
• Three Parameters: Uses Qi, τ (characteristic time), and n (stretching exponent)

When to Use SEPD

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Theory

Physical Basis

The SEPD model is based on the stretched exponential function (also called Kohlrausch function), which arises naturally in systems with a distribution of relaxation times. In reservoir context, this represents heterogeneity in drainage patterns and flow geometries.

The key insight is that production from unconventional wells can be viewed as a superposition of multiple exponential decays with varying time constants:

$$q(t) = \int_0^\infty q_0 e^{-t/\tau'} f(\tau') d\tau'$$

When $f(\tau')$ follows a specific "fat-tail" distribution, the result is the stretched exponential form.

Stretched Exponential Function

The stretched exponential differs from a standard exponential by the exponent n on the time term:

• Standard exponential: $e^{-t/\tau}$
• Stretched exponential: $e^{-(t/\tau)^n}$

For $n < 1$, the function decays more slowly at early times and faster at late times compared to a standard exponential, capturing the transition from high initial decline to slower long-term decline.

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Equations

Rate Equation

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

where:

• $q_i$ = Initial (peak) production rate (L³/T)
• $\tau$ = Characteristic time parameter (T)
• $n$ = Stretching exponent (dimensionless, typically 0.2-0.5)
• $t$ = Time (T)

Cumulative Production

The cumulative production involves the incomplete gamma function:

$$N_p(t) = \frac{q_i \tau}{n} \left\{\Gamma\left[\frac{1}{n}\right] - \Gamma\left[\frac{1}{n}, \left(\frac{t}{\tau}\right)^n\right]\right\}$$

where:

• $\Gamma[a]$ = Complete gamma function
• $\Gamma[a, x]$ = Upper incomplete gamma function

Ultimate Recovery (EUR)

As $t \to \infty$, the incomplete gamma function approaches zero:

$$EUR = \frac{q_i \tau}{n} \Gamma\left[\frac{1}{n}\right]$$

This closed-form EUR is a key advantage of the SEPD model.

Time to Economic Limit

Solving for time when $q = q_{econ}$:

$$t_{econ} = \tau \left[\ln\left(\frac{q_i}{q_{econ}}\right)\right]^{1/n}$$

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Applicability & Limitations

Applicability Ranges

Physical Constraints

• $n > 0$ (required for declining rate)
• $n \leq 1$ (physically meaningful stretching)
• $\tau > 0$ (positive time constant)
• $q_i > 0$ (physical rate)

Limitations

1. No Physical D∞ Term: Unlike PLE, SEPD does not explicitly model terminal decline rate
2. Late-Time Behavior: May not accurately capture boundary-dominated flow behavior
3. Statistical Nature: Best suited for population analysis rather than individual well prediction
4. Gamma Function: Cumulative calculation requires special mathematical functions

Comparison with Other Models

Relationship to PLE

When $D_\infty = 0$ in the PLE model:

$$q = q_i \exp(-D_i t^n)$$

This is equivalent to SEPD with $\tau = D_i^{-1/n}$. The key difference is that PLE explicitly includes a terminal decline term $D_\infty$ for boundary-dominated flow.

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References

1. Valkó, P.P. and Lee, W.J. (2010). "A Better Way to Forecast Production from Unconventional Gas Wells." SPE Annual Technical Conference and Exhibition, Florence, Italy. SPE-134231-MS.

2. Valkó, P.P. (2009). "Assigning Value to Stimulation in the Barnett Shale: A Simultaneous Analysis of 7000 Plus Production Histories and Well Completion Records." SPE-119369-MS.

3. Ali, T.A. and Sheng, J.J. (2015). "Production Decline Models: A Comparison Study." SPE-177300-MS.

4. Abramowitz, M. and Stegun, I.A. (1972). Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. New York: Dover Publications.