Duong Decline Model for Unconventional Wells

Overview

The Duong decline model was introduced by Duong (2010, 2011) specifically for multi-fractured horizontal wells (MFHW) in unconventional reservoirs. The model is based on the observation that these wells exhibit extended periods of linear or bilinear flow before transitioning to boundary-dominated flow, characterized by a straight-line relationship on a log-log plot of q/Gp versus time.

Key Concepts

• Linear Flow Foundation: Designed for fracture-dominated flow in tight/shale reservoirs
• q/Gp Analysis: Based on the ratio of rate to cumulative production declining as a power law
• Four Parameters: Uses a, m, q₁, and q∞ for maximum flexibility
• Asymptotic Rate: Includes an optional q∞ term for late-time stabilization

When to Use Duong

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Theory

Physical Basis

Multi-fractured horizontal wells in unconventional reservoirs typically exhibit long-duration linear flow from the fracture network before reaching boundary effects. Duong observed that for such wells, a log-log plot of q/Gp (rate divided by cumulative production) versus time yields a straight line:

$$\log\left(\frac{q}{G_p}\right) = \log(a) - m \cdot \log(t)$$

or equivalently:

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

where:

• $a$ is the intercept constant
• $m$ is the slope (typically > 1)

Flow Regime Characteristics

Infinite Conductivity Fractures (Linear Flow):

• Rate proportional to $t^{-0.5}$
• m typically around 1.0-1.2

Finite Conductivity Fractures (Bilinear Flow):

• Rate proportional to $t^{-0.25}$
• m may be lower

Transition to BDF:

• m increases during transition
• q∞ term captures late-time behavior

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Equations

Rate Equation

The standard Duong rate equation:

$$q = q_1 t^{-m} \exp\left[\frac{a}{1-m}(t^{1-m} - 1)\right]$$

With asymptotic rate term (modified form):

$$q = q_1 t^{-m} \exp\left[\frac{a}{1-m}(t^{1-m} - 1)\right] + q_\infty$$

where:

• $a$ = Intercept constant (1/T)
• $m$ = Slope parameter (dimensionless, typically 1.0-1.3)
• $q_1$ = Theoretical rate at t = 1 time unit (L³/T)
• $q_\infty$ = Asymptotic late-time rate (L³/T)
• $t$ = Time (T)

Cumulative Production

Without q∞:

$$G_p = \frac{q_1}{a} \exp\left[\frac{a}{1-m}(t^{1-m} - 1)\right]$$

With q∞:

$$G_p = \frac{q_1}{a} \exp\left[\frac{a}{1-m}(t^{1-m} - 1)\right] + q_\infty t$$

Time Function

The Duong model uses a special time function for curve fitting:

$$t(a,m) = t^{-m} \exp\left[\frac{a}{1-m}(t^{1-m} - 1)\right]$$

When plotting $q$ vs $t(a,m)$, the result should be a straight line with slope $q_1$ and intercept $q_\infty$.

Parameter Estimation

Step 1: Plot log(q/Gp) vs log(t) to find $a$ and $m$ Step 2: Calculate $t(a,m)$ function Step 3: Plot $q$ vs $t(a,m)$ to find $q_1$ (slope) and $q_\infty$ (intercept)

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Applicability & Limitations

Applicability Ranges

Physical Constraints

• $m > 1$ (otherwise cumulative becomes unbounded)
• $a > 0$ (positive intercept)
• $q_1 > 0$ (physical rate)
• $q_\infty \geq 0$ (non-negative asymptotic rate)

Limitations

1. Flow Regime Changes: May overestimate EUR if flow regime changes from linear to BDF during producing life
2. m < 1 Issues: If m < 1, cumulative production becomes unbounded
3. Linear Flow Assumption: Less accurate for wells not exhibiting linear flow
4. Vertical Wells: May not work well for vertical wells in classic shales

Cautions from Literature

Vanorsdale (2013) noted:

• Duong may overestimate recovery when flow regime changes during the well's life
• May provide conservative estimates for vertical, non-hydraulically fractured shale wells
• Best suited for MFHW with established linear flow

Comparison with Other Models

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References

1. Duong, A.N. (2010). "An Unconventional Rate Decline Approach for Tight and Fracture-Dominated Gas Wells." SPE-137748-MS, Canadian Unconventional Resources and International Petroleum Conference, Calgary, Alberta.

2. Duong, A.N. (2011). "Rate-Decline Analysis for Fracture-Dominated Shale Reservoirs." SPE Reservoir Evaluation & Engineering, 14(3): 377-387. SPE-137748-PA.

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

4. Vanorsdale, C. (2013). "Production Decline Analysis Lessons from Classic Shale Gas Wells." SPE-166205-MS.

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