Asset Forecasting Formula — Arps, Duong, SEPD Decline Curve Analysis

Overview

A decline curve fits an empirical model to historical monthly (or daily) production data, then extrapolates to economic limit. Arps (1945) covers conventional boundary-dominated flow with a single hyperbolic exponent b. Duong (2011) was developed for unconventional shale wells where transient linear flow dominates early life and the Arps b apparent exceeds 1. SEPD (Valko, 2009) uses a stretched-exponential form that handles late-life behavior without the late-time blow-up Arps shows when b > 1.

The right model depends on flow regime, production history length, and reservoir architecture. In commercial workflows, all three models are fit, ranked by R²/RMSE on a held-out tail of the data, and the best-ranked model is used for forecasting — with a sanity check on the implied EUR vs analog wells.

Theory

All decline curve methods are empirical extrapolations of rate-time behavior. Arps derived hyperbolic decline from boundary-dominated flow in conventional reservoirs (single-phase, constant pressure drop). For unconventional reservoirs with hydraulic fractures and matrix-fracture transient flow, the Arps assumption breaks down, and the practitioner has to use one of the unconventional-specific models or cap the Arps b to prevent unphysical EUR.

Formulas

Arps (1945) Decline Models

Exponential (b = 0):
  q(t) = qi * exp(-Di * t)
  Np(t) = (qi - q(t)) / Di

Hyperbolic (0 < b < 1):
  q(t) = qi / (1 + b*Di*t)^(1/b)
  Np(t) = qi^b / ((1-b)*Di) * (qi^(1-b) - q(t)^(1-b))

Harmonic (b = 1):
  q(t) = qi / (1 + Di*t)
  Np(t) = qi / Di * ln(qi / q(t))

Modified hyperbolic (Robertson):
  use hyperbolic until D drops to D_min (typically 5%/yr),
  then switch to exponential at that effective rate.

qi = initial rate (STB/day or Mscf/day), Di = nominal initial decline rate (1/time), b = hyperbolic exponent (dimensionless), q(t) = rate at time t, Np(t) = cumulative production at time t.

Duong (2011) Decline Model

q(t) = q1 * t(t,a,m)
t(t,a,m) = t^(-m) * exp((a/(1-m)) * (t^(1-m) - 1))

q1 = rate at t = 1 (time unit consistent with a, m)
a, m = empirical Duong parameters

Cumulative:
Np(t) = (q(t) - q_inf) / a    (approximate; integrate t(t,a,m))

Designed for transient linear flow in hydraulically fractured shale; outperforms Arps for the first 3–5 years of unconventional well production.

SEPD / Stretched-Exponential (Valko, 2009)

q(t) = q0 * exp(-(t/tau)^n)

q0, tau, n = SEPD parameters
n = stretched exponent (typically 0.3-0.7 for unconventionals)

Cumulative:
Np(t) = q0 * tau / n * Gamma_lower(1/n, (t/tau)^n)
Gamma_lower = lower incomplete gamma function

Avoids the EUR blow-up Arps shows when b > 1; widely used as a "long-tail" model in unconventional reserve work.

Effective Decline Rate

D_effective (yr^-1) = 1 - q(t + 1 year) / q(t)
D_nominal (yr^-1)  = -dq/dt / q     (instantaneous rate of decline)

For hyperbolic:
  D_nominal(t) = Di / (1 + b*Di*t)
  D_effective(t) = 1 - (1 + b*Di*t)^(1/b) / (1 + b*Di*(t+1))^(1/b)

Economic Limit and EUR

q_econ = LOE_per_month / (Net_price * (1 - severance_rate))
       (rate at which net cash flow = 0)

t_econ: solve q(t) = q_econ for t
EUR = Np(t_econ)

Net_price = realized_price * NRI - transport - processing
LOE_per_month = lease operating expense per well per month

Model Ranking (held-out tail)

Fit on the first 70-80% of data; score on the held-out tail.

R^2 = 1 - sum((q_obs - q_pred)^2) / sum((q_obs - mean(q_obs))^2)
RMSE = sqrt(mean((q_obs - q_pred)^2))

Rank by R^2 first; tie-break by RMSE.
Sanity check: EUR within +/- 30% of analog wells.

Type Curves (P10 / P50 / P90)

For each well in the cohort:
  normalize to a common reference start date (first production)
  resample to monthly grid (or use raw monthly Aries data)

At each month t:
  P10 = 90th percentile of cohort rates at t
  P50 = median rate at t
  P90 = 10th percentile rate at t

Cohort stratification:
  - formation / reservoir
  - vintage (year drilled)
  - completion design (proppant intensity, fluid type)
  - basin / area

Key Symbols

Worked Example

Given: Horizontal oil well, qi = 1,200 BOPD initial rate, fit to 24 months of monthly Aries data. Best-fit Arps hyperbolic: qi = 1,200, Di = 0.85/yr (effective), b = 1.0 (cap applied), terminal D_min = 5%/yr. LOE = $9,000/well/month, net oil price = $58/bbl.

Step 1 — Economic limit rate:

q_econ = 9000 / (30.4 * 58) = 5.1 BOPD

Step 2 — Time to economic limit using harmonic (b=1):

q(t) = 1200 / (1 + Di * t) = 5.1
1 + Di*t = 1200/5.1 = 235.3
Di*t = 234.3
t = 234.3 / 0.85 = 275.6 years   (without modified hyperbolic switch)

With switch to exponential at D_min = 5%/yr:
  switch when D_nominal = 0.05
  Di / (1 + b*Di*t) = 0.05
  t_switch = (Di/0.05 - 1) / (b*Di) = (17 - 1) / 0.85 = 18.8 years
  q at switch: 1200 / (1 + 0.85 * 18.8) = 70.0 BOPD
  exponential continues until q = q_econ:
  70 * exp(-0.05*(t - 18.8)) = 5.1
  t = 18.8 + ln(70/5.1) / 0.05 = 18.8 + 52.4 = 71.2 years

Step 3 — EUR (cumulative oil to economic limit):

Hyperbolic Np to switch: ~ 600 MBO
Exponential tail Np: ~ 220 MBO
EUR ~ 820 MBO

Valid Ranges & Diagnostics

When the models do not apply

• Wells with significant well-event history (frac hits, workovers, shut-ins) — clean the production stream before fitting, or use event-by-event segmentation.
• Wells in transient flow with no boundary-dominated data — analytical RTA (e.g., Blasingame) is more appropriate than empirical decline curves.
• Heavy oil / SAGD — thermally enhanced production does not follow Arps geometry.
• Gas wells with significant liquid loading — modify rate stream for back pressure or use a gas-specific model.
• Reservoirs with active water influx or strong pressure support — material balance + simulation, not DCA.

References

1. Arps, J.J. (1945). "Analysis of Decline Curves." Trans. AIME, 160, 228–247.
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. Valko, P.P. & Lee, W.J. (2010). "A Better Way To Forecast Production From Unconventional Gas Wells." SPE 134231 (Stretched Exponential Production Decline).
4. Robertson, S. (1988). "Generalized Hyperbolic Equation." SPE 18731 (modified hyperbolic with terminal exponential).
5. SEC — Regulation S-X, Rule 4-10 (Reserve definitions, SEC PV10).
6. Society of Petroleum Engineers (SPE) — Petroleum Resources Management System (PRMS), 2018.
7. Cronquist, C. (2001). Estimation and Classification of Reserves of Crude Oil, Natural Gas, and Condensate. SPE Textbook.
8. Fetkovich, M.J. (1980). "Decline Curve Analysis Using Type Curves." JPT, 32(6), 1065–1077.