Arps Decline Curve Analysis

Overview

Decline curve analysis (DCA) is the most widely used method for forecasting oil and gas production and estimating ultimate recovery (EUR). The technique fits mathematical models to historical production data and extrapolates future performance.

J.J. Arps (1945) introduced three empirical decline models — exponential, hyperbolic, and harmonic — which remain the industry standard. Modern extensions include the Stretched Exponential Production Decline (SEPD) model by Valko & Lee (2010) and the Duong model (2011) for unconventional wells.

Theory

All Arps models assume boundary-dominated flow. The general rate-time equation is:

q(t) = qi / (1 + b * Di * t)^(1/b)

where qi = initial rate, Di = initial decline rate, b = Arps decline exponent, and t = time.

The decline exponent b controls the shape:

• b = 0 → Exponential decline (constant % decline per period)
• 0 < b < 1 → Hyperbolic decline (decreasing decline rate)
• b = 1 → Harmonic decline (decline rate proportional to rate)

Formulas

Exponential Decline (b = 0)

q(t) = qi * exp(-Di * t)

Cumulative production:

Np(t) = (qi - q(t)) / Di

EUR (to economic limit q_el):

EUR = (qi - q_el) / Di

Hyperbolic Decline (0 < b < 1)

q(t) = qi / (1 + b * Di * t)^(1/b)

Cumulative production:

Np(t) = (qi^b / ((1-b) * Di)) * (qi^(1-b) - q(t)^(1-b))

Harmonic Decline (b = 1)

q(t) = qi / (1 + Di * t)

Cumulative production:

Np(t) = (qi / Di) * ln(qi / q(t))

Stretched Exponential (SEPD)

q(t) = qi * exp(-(t / tau)^n)

where tau = characteristic time constant and n = exponent (typically 0.3–0.8 for shale wells).

Duong Model (Unconventional)

q(t) = qi * t^(-m) * exp(a / (1-m) * (t^(1-m) - 1))

where a = intercept, m = slope (typically 1.0–1.5).

Worked Example

Given: A well produces 500 bbl/d initially with 15%/month nominal decline. Assume hyperbolic with b = 0.8.

Step 1: qi = 500 bbl/d, Di = 0.15/month, b = 0.8

Step 2: Rate at month 24:

q(24) = 500 / (1 + 0.8 * 0.15 * 24)^(1/0.8)
       = 500 / (1 + 2.88)^(1.25)
       = 500 / 3.88^1.25
       = 500 / 5.21
       = 96.0 bbl/d

Step 3: Cumulative production at month 24:

Np = (500^0.8 / (0.2 * 0.15)) * (500^0.2 - 96^0.2)
   = (158.5 / 0.03) * (3.466 - 2.479)
   = 5283.3 * 0.987
   = 5,215 bbl * 30.4 days/month ≈ 158,500 bbl (over 24 months)

Valid Ranges

Common Pitfalls

1. Using b > 1 without terminal decline → infinite EUR
2. Fitting transient flow data with boundary-dominated models
3. Ignoring operational shutdowns (zero-rate months) in the fit
4. Single-model reliance instead of comparing multiple models

References

1. Arps, J.J. (1945). "Analysis of Decline Curves." Trans. AIME, 160, 228–247.
2. Valko, P.P. & Lee, W.J. (2010). "A Better Way to Forecast Production from Unconventional Gas Wells." SPE-134231.
3. Duong, A.N. (2011). "Rate-Decline Analysis for Fracture-Dominated Shale Reservoirs." SPE-137748.
4. PetroWiki — Decline curve analysis: https://petrowiki.spe.org/Decline_curve_analysis
5. Fekete (IHS) — Decline Analysis Theory: https://www.ihsenergy.ca/support/documentation_ca/Harmony/content/html_files/reference_material/analysis_method_theory/decline_theory.htm