Ansah-Knowles-Buba (AKB) Decline Model

The Ansah-Knowles-Buba (AKB) model is a semi-analytical decline model specifically developed for gas wells under boundary-dominated flow conditions. Unlike empirical Arps models, AKB is derived from material balance principles coupled with stabilized gas deliverability equations.

---

Theory

Physical Basis

The AKB model addresses the challenge of non-linear gas flow by linearizing the pressure-dependent gas properties. The key insight is that the μ_g(p)·c_g(p) product can be approximated by a first-order polynomial function of pressure, enabling semi-analytical solutions.

The model couples:

1. Gas material balance equation — relates cumulative production to reservoir pressure
2. Stabilized gas deliverability — valid for boundary-dominated flow (pseudo-steady state)

This approach provides a more rigorous treatment of gas compressibility changes than purely empirical methods.

Rate Equation

The semi-analytical rate-time relationship is:

$$q_g(t) = q_{gi} \cdot \frac{4\alpha^2 \exp(-\beta t)}{\left[(1+\alpha) - (1-\alpha)\exp(-\beta t)\right]^2}$$

Where:

Parameter Definitions

The dimensionless parameter $\alpha$ represents the ratio of wellbore to initial reservoir conditions:

$$\alpha = \frac{p_{wf}/z_{wf}}{p_i/z_i}$$

The decline parameter $\beta$ incorporates reservoir and well properties:

$$\beta = \frac{p_{wf}}{z_{wf}} \cdot \frac{J_g}{p_i/z_i} \cdot \frac{1}{c_{ti} \cdot G}$$

Where:

Gas Productivity Index

The productivity index $J_g$ for stabilized flow is:

$$J_g = \frac{2kh}{141.2 \mu_i B_{gi} \ln\left(\frac{2.2458A}{C_A r_{wa}^2}\right)}$$

Where:

Cumulative Production

Integrating the rate equation yields cumulative production:

$$G_p(t) = q_{gi} \cdot \frac{2\alpha(\exp(\beta t) - 1)}{\beta\left[\alpha - 1 + (1+\alpha)\exp(\beta t)\right]}$$

Special Case: α → 0

When $\alpha$ approaches zero (very low flowing pressure relative to initial pressure), the rate equation simplifies to:

$$q_g(t) = \frac{q_{gi}}{\left(1 + \frac{\beta t}{2}\right)^2}$$

This limit avoids numerical instability from the 0/0 indeterminate form.

---

Diagnostic Functions

Loss-Ratio (D-function)

The instantaneous decline rate for AKB:

$$D(t) = \frac{\beta\left[1 + \exp(\beta t) + \alpha(\exp(\beta t) - 1)\right]}{\alpha + \exp(\beta t) + \alpha\exp(\beta t) - 1}$$

b-Parameter

The time-varying b-parameter:

$$b(t) = \frac{2\exp(\beta t)(1 - \alpha^2)}{\left[1 - \alpha + (1+\alpha)\exp(\beta t)\right]^2}$$

Unlike Arps hyperbolic where $b$ is constant, AKB exhibits time-varying $b$ that captures the physics of gas reservoir depletion.

---

Functions

Rate Calculation

AnsahKnowlesBubaDeclineRate — Production rate at time t

Parameters:

Returns: Production rate at the specified time, [L³/T]

---

Cumulative Production

AnsahKnowlesBubaDeclineCumulative — Cumulative production to time t

Parameters:

Returns: Cumulative production from time 0 to t, [L³]

---

Economic Life

AnsahKnowlesBubaDeclineTime — Time to reach economic limit

Parameters:

Returns: Time when production rate falls to the economic limit, [T]

---

EUR Calculation

AnsahKnowlesBubaDeclineEUR — Estimated Ultimate Recovery

Parameters:

Returns: Cumulative production to the economic limit, [L³]

---

Curve Fitting

AnsahKnowlesBubaDeclineFitParameters — Fit model to production data

Parameters:

Returns: Array [Qi, Alpha, Beta] of fitted parameters

---

Weighted Curve Fitting

AnsahKnowlesBubaDeclineWeightedFitParameters — Fit with weighted samples

Parameters:

Returns: Array [Qi, Alpha, Beta] of fitted parameters

Use weights to emphasize recent production data or de-emphasize periods with operational issues.

---

When to Use AKB

Best Applications

Comparison with Arps Models

Limitations

• Requires boundary-dominated flow (not valid for transient or fracture-dominated flow)
• Assumes stabilized deliverability equation applies
• Linear approximation of μ_gc_g may lose accuracy at very low pressures
• For unconventional wells with extended transient flow, consider Duong or SEPD models

---

References

1. Ansah, J., Knowles, R.S., and Blasingame, T.A. (1996). A Semi-Analytic (p/z) Rate-Time Relation for the Analysis and Prediction of Gas Well Performance. SPE 35268, SPE Mid-Continent Gas Symposium, Amarillo, Texas.

2. Currie, S. (2010). User Manual for Rate-Time Analysis Spreadsheet. Texas A&M University.

---

See Also

• Decline Overview — Model selection guide
• Arps Decline Models — Traditional empirical models
• Duong Decline — Fracture-dominated unconventional wells
• Stretched Exponential (SEPD) — Alternative for unconventional