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 μg(p)·cg(p) product can be approximated by a first-order polynomial function of pressure, enabling semi-analytical solutions.

The model couples:

Gas material balance equation — relates cumulative production to reservoir pressure
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	Description	Units
$q_{gi}$	Initial gas rate	L³/T (e.g., Mscf/day)
$\alpha$	Dimensionless pressure ratio	dimensionless
$\beta$	Decline parameter	1/T (e.g., 1/day)
$t$	Time	T (e.g., days)

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:

Symbol	Description	Units
$p_{wf}$	Flowing bottomhole pressure	psia
$z_{wf}$	Gas deviation factor at $p_{wf}$	dimensionless
$p_i$	Initial reservoir pressure	psia
$z_i$	Gas deviation factor at $p_i$	dimensionless
$J_g$	Gas productivity index	Mscf/day/psi²
$c_{ti}$	Total compressibility at initial conditions	1/psi
$G$	Original gas in place	Mscf

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:

Symbol	Description	Units
$k$	Permeability	md
$h$	Net pay thickness	ft
$\mu_i$	Gas viscosity at initial conditions	cp
$B_{gi}$	Gas formation volume factor at $p_i$	rcf/scf
$A$	Drainage area	ft²
$C_A$	Dietz shape factor	dimensionless
$r_{wa}$	Apparent wellbore radius (including skin)	ft

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:

Parameter	Description	Units
`Qi`	Initial production rate	L³/T
`alpha`	Dimensionless pressure ratio	dimensionless
`beta`	Decline parameter	1/T
`time`	Evaluation time	T

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

Cumulative Production

AnsahKnowlesBubaDeclineCumulative — Cumulative production to time t

Parameters:

Parameter	Description	Units
`Qi`	Initial production rate	L³/T
`alpha`	Dimensionless pressure ratio	dimensionless
`beta`	Decline parameter	1/T
`time`	Evaluation time	T

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

Economic Life

AnsahKnowlesBubaDeclineTime — Time to reach economic limit

Parameters:

Parameter	Description	Units
`Qi`	Initial production rate	L³/T
`alpha`	Dimensionless pressure ratio	dimensionless
`beta`	Decline parameter	1/T
`econRate`	Economic rate limit	L³/T

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

EUR Calculation

AnsahKnowlesBubaDeclineEUR — Estimated Ultimate Recovery

Parameters:

Parameter	Description	Units
`Qi`	Initial production rate	L³/T
`alpha`	Dimensionless pressure ratio	dimensionless
`beta`	Decline parameter	1/T
`econRate`	Economic rate limit	L³/T

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

Curve Fitting

AnsahKnowlesBubaDeclineFitParameters — Fit model to production data

Parameters:

Parameter	Description	Type
`timeValues`	Array of time samples	Range or array
`rateValues`	Array of observed rates	Range or array

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

Weighted Curve Fitting

AnsahKnowlesBubaDeclineWeightedFitParameters — Fit with weighted samples

Parameters:

Parameter	Description	Type
`timeValues`	Array of time samples	Range or array
`rateValues`	Array of observed rates	Range or array
`weightsValues`	Per-sample weights	Range or array

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

Scenario	Reason
Gas wells in depletion	Derived from gas material balance
Boundary-dominated flow	Assumes pseudo-steady state
Known reservoir properties	Can estimate $\alpha$ and $\beta$ from reservoir data
Physics-based forecasting	More rigorous than empirical Arps

Comparison with Arps Models

Aspect	AKB	Arps Hyperbolic
Derivation	Semi-analytical (material balance)	Purely empirical
b-parameter	Time-varying	Constant
Gas compressibility	Explicitly handled	Ignored
Parameter meaning	Tied to reservoir properties	Curve-fit only
Best for	Gas wells, physics-based	Oil wells, quick estimates

Limitations

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

References

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

Ansah-Knowles-Buba (AKB) Decline Model

Theory

Physical Basis

Rate Equation

Parameter Definitions

Gas Productivity Index

Cumulative Production

Special Case: α → 0

Diagnostic Functions

Loss-Ratio (D-function)

b-Parameter

Functions

Rate Calculation

Cumulative Production

Economic Life

EUR Calculation

Curve Fitting

Weighted Curve Fitting

When to Use AKB

Best Applications

Comparison with Arps Models

Limitations

References

See Also

Related Excel Functions