Power Law Exponential (PLE) Decline Model

Overview

The Power Law Exponential (PLE) decline model was introduced by Ilk et al. (2008) to address the limitations of Arps' hyperbolic model for unconventional reservoirs. Unlike traditional decline curves that assume boundary-dominated flow, PLE explicitly models the transition from transient linear flow to boundary-dominated flow, making it particularly suitable for tight gas and shale reservoirs.

Key Concepts

• Transient-to-BDF Transition: The PLE model captures early transient behavior with a power-law term and long-term boundary-dominated behavior with an exponential term
• Bounded EUR: Unlike hyperbolic decline with b > 1, PLE always produces finite cumulative production
• Loss-Ratio Foundation: Based on the inverse of the "loss-ratio" (D-parameter) behavior of time-rate data
• Four Parameters: Requires fitting Qi, Di, D∞, and n, which can be challenging with limited data

When to Use PLE

---

Theory

Physical Basis

The PLE model is constructed from the observation that the decline rate D(t) in unconventional wells follows a power-law relationship with time during transient flow, then transitions to a constant decline rate during boundary-dominated flow:

$$D(t) = D_1 t^{-n} + D_\infty$$

where:

• $D_1$ is the decline constant at t = 1 time unit
• $D_\infty$ is the terminal decline rate (BDF regime)
• $n$ is the time exponent (typically 0.1 to 0.4)

Behavior Characteristics

Early Time (Transient):

• The $t^{-n}$ term dominates
• Matches linear or bilinear flow regimes
• Rate decline is steeper than exponential

Late Time (BDF):

• The $D_\infty$ term dominates
• Transitions to exponential decline
• Ensures bounded cumulative production

---

Equations

Rate Equation

$$q(t) = \hat{q}_i \exp\left(-\hat{D}_i t^n - D_\infty t\right)$$

where:

• $\hat{q}_i$ = Rate intercept at t = 0 (L³/T)
• $\hat{D}_i$ = Decline constant defined as $D_1/n$ (1/T)
• $D_\infty$ = Decline constant at infinite time (1/T)
• $n$ = Time exponent (dimensionless, typically 0.1-0.4)
• $t$ = Time (T)

Alternative Formulation

Using $D_1$ directly:

$$q(t) = q_i \exp\left(-\frac{D_1}{n} t^n - D_\infty t\right)$$

Cumulative Production

The cumulative production requires numerical integration:

$$N_p(t) = \int_0^t q(\tau) \, d\tau$$

No closed-form analytical solution exists; Petroleum Office uses adaptive quadrature for accurate numerical evaluation.

Time to Economic Limit

Given an economic rate $q_{econ}$, solve numerically:

$$q_{econ} = \hat{q}_i \exp\left(-\hat{D}_i t^n - D_\infty t\right)$$

EUR to Economic Limit

$$EUR = N_p(t_{econ})$$

where $t_{econ}$ is the time when $q(t) = q_{econ}$.

---

Applicability & Limitations

Applicability Ranges

Physical Constraints

• $D_\infty \leq D_i$ (terminal decline cannot exceed initial)
• $n > 0$ (ensures declining rate)
• $q_i > 0$ (physical rate)

Limitations

1. Non-Unique Solutions: Four parameters can lead to multiple acceptable fits with limited data
2. Sensitivity to D∞: The terminal decline rate strongly influences EUR but is poorly constrained with short histories
3. No Flow Regime Identification: Assumes combined transient + BDF behavior without explicitly identifying transition
4. Numerical Integration: Cumulative production requires numerical methods, which may introduce minor errors

Comparison with Other Models

---

References

1. Ilk, D., Rushing, J.A., and Blasingame, T.A. (2008). "Exponential vs. Hyperbolic Decline in Tight Gas Sands – Understanding the Origin and Implications for Reserve Estimates Using Arps' Decline Curves." SPE-116731-MS.

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

3. Lee, W.J. and Sidle, R.E. (2010). "Gas Reserves Estimation in Resource Plays." SPE-130102-MS.

4. Fetkovich, M.J., Vienot, M.E., Bradley, M.D., and Kiesow, U.G. (1987). "Decline Curve Analysis Using Type Curves: Case Histories." SPE Formation Evaluation, 2(4): 637-656.