Duong Decline Model

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

Reservoir Type	Suitability	Notes
Shale gas MFHW	✅ Excellent	Original target application
Shale oil MFHW	✅ Excellent	Works well for unconventional liquids
Tight gas	✅ Good	If linear flow regime present
Conventional	❌ Poor	Not designed for radial flow
Vertical wells	⚠️ Limited	May underestimate for non-MFHW

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

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)

Functions Covered

Function	Description	Returns
DuongDeclineRate	Rate at time t	q(t), L³/T
DuongDeclineCumulative	Cumulative production at time t	Gp(t), L³
DuongDeclineTime	Time to reach economic rate limit	t, T
DuongDeclineEUR	EUR to economic rate limit	Gp(tₑcon), L³
DuongDeclineFitParameters	Fit [a, m, q1, qInf] to data	Array[4]
DuongDeclineWeightedFitParameters	Weighted fit [a, m, q1, qInf]	Array[4]

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

Applicability & Limitations

Applicability Ranges

Parameter	Recommended Range	Notes
Slope m	1.0 - 1.3	m ≈ 1.0 for strong linear flow
Intercept a	0.5 - 2.0	Field-specific
q∞/q₁ ratio	0 - 0.2	Often set to 0
Production history	> 6 months	Shorter may work for MFHW

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

Flow Regime Changes: May overestimate EUR if flow regime changes from linear to BDF during producing life
m < 1 Issues: If m < 1, cumulative production becomes unbounded
Linear Flow Assumption: Less accurate for wells not exhibiting linear flow
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

Aspect	Duong	Arps Hyperbolic	PLE	SEPD
Target wells	MFHW	Conventional	Unconventional	Population
Flow regime	Linear	BDF	Transient→BDF	Statistical
Parameters	4	3	4	3
EUR bounded	⚠️ If m > 1	❌ If b > 1	✅ Always	✅ Always
q∞ term	✅ Optional	❌ No	✅ D∞	❌ No

References

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.
Duong, A.N. (2011). "Rate-Decline Analysis for Fracture-Dominated Shale Reservoirs." SPE Reservoir Evaluation & Engineering, 14(3): 377-387. SPE-137748-PA.
Ali, T.A. and Sheng, J.J. (2015). "Production Decline Models: A Comparison Study." SPE-177300-MS.
Vanorsdale, C. (2013). "Production Decline Analysis Lessons from Classic Shale Gas Wells." SPE-166205-MS.
Lee, W.J. and Sidle, R.E. (2010). "Gas Reserves Estimation in Resource Plays." SPE-130102-MS.