Modified Hyperbolic Decline Model

Overview

The Modified Hyperbolic Decline Model addresses a fundamental limitation of classical hyperbolic decline: when $b > 0$, the hyperbolic model predicts a continuously decreasing decline rate that can lead to physically unrealistic EUR estimates. The modified hyperbolic model solves this by transitioning from hyperbolic to exponential decline at a specified terminal (limiting) decline rate.

Key Characteristics

When to Use

• Conventional reservoirs with established decline but b-factor approaching or exceeding 1
• Long-term forecasts where hyperbolic alone would overestimate EUR
• Tight gas and unconventional wells during transition to boundary-dominated flow
• Economic limit studies requiring bounded EUR estimates

Historical Context

The modified hyperbolic model emerged from practical engineering experience showing that the decline rate for mature wells typically stabilizes at 5-10% per year, regardless of the early-time b-factor. This observation led to the "terminal decline" or "limiting decline" concept, which has become standard practice in many regulatory filings and reserve estimates.

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Theory

The Problem with Standard Hyperbolic Decline

From the classical hyperbolic decline, the instantaneous decline rate is:

$$D(t) = \frac{D_i}{1 + b D_i t}$$

For $b > 0$, as $t \to \infty$, the decline rate $D(t) \to 0$. This means:

• Production rate never quite reaches zero
• Cumulative production continues to infinity
• EUR depends entirely on the economic limit rate

When $b \geq 1$, the cumulative production integral does not converge, leading to theoretically infinite EUR. This is physically unrealistic.

The Modified Hyperbolic Solution

The modified hyperbolic model is a piecewise function that switches from hyperbolic to exponential decline when the instantaneous decline rate reaches a specified terminal value $D_{lim}$:

$$q(t) = \begin \frac{(1 + b D_i t)^{{1/b}} & t < t}* \[1em] q_i^ \cdot \exp[-D_(t - t^)] & t \geq t^ \end$$

where $t^*$ is the transition time and $q_i^{exp}$ is the rate at the transition point.

Transition Time

The transition from hyperbolic to exponential occurs when the instantaneous decline rate equals the terminal decline rate:

$$D(t^*) = \frac{D_i}{1 + b D_i t^*} = D_{lim}$$

Solving for $t^*$:

$$t^* = \frac{1}{b D_i}\left(\frac{D_i}{D_{lim}} - 1\right)$$

This can be simplified to:

$$t^* = \frac{D_i - D_{lim}}{b D_i D_{lim}}$$

Rate at Transition

The rate at the transition point is:

$$q(t^*) = \frac{q_i}{(1 + b D_i t^*)^{1/b}}$$

For continuity, this becomes the initial rate for the exponential portion:

$$q_i^{exp} = q(t^*) = \frac{q_i}{\left(\frac{D_i}{D_{lim}}\right)^{1/b}}$$

Terminal Decline Rate from Nominal Annual Decline

In practice, the terminal decline rate is often specified as a percentage per year. For a nominal annual decline of $p$ (as a decimal):

$$D_{lim} = -\frac{\ln(1 - p)}{365}$$

For example:

• 5% per year: $D_{lim} = 0.000141$ per day
• 10% per year: $D_{lim} = 0.000289$ per day
• 15% per year: $D_{lim} = 0.000445$ per day

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Equations

Rate-Time Relation

Hyperbolic period ($t < t^*$):

$$q(t) = \frac{q_i}{(1 + b D_i t)^{1/b}}$$

Exponential period ($t \geq t^*$):

$$q(t) = q_i^{exp} \cdot \exp[-D_{lim}(t - t^*)]$$

where:

$$q_i^{exp} = \frac{q_i}{(1 + b D_i t^*)^{1/b}} \cdot \frac{1}{\exp[-D_{lim} t^*]}$$

Note: The second factor accounts for the exponential function being evaluated from time zero.

Cumulative Production

Hyperbolic period ($t < t^*$):

$$N_p(t) = \frac{q_i}{(1-b)D_i}\left[1 - (1 + b D_i t)^{1-(1/b)}\right]$$

Exponential period ($t \geq t^*$):

$$N_p(t) = N_p(t^*) + \frac{q_i^{exp}}{D_{lim}}\left[\exp(-D_{lim} t^*) - \exp(-D_{lim} t)\right]$$

where $N_p(t^*)$ is the cumulative production at the transition time.

EUR Calculation

To a specified economic limit rate $q_{econ}$:

If $q_{econ} \geq q(t^*)$ (limit reached during hyperbolic phase):

$$EUR = \frac{q_i}{(1-b)D_i}\left[1 - \left(\frac{q_{econ}}{q_i}\right)^{1-b}\right]$$

If $q_{econ} < q(t^*)$ (limit reached during exponential phase):

$$EUR = N_p(t^*) + \frac{q(t^*) - q_{econ}}{D_{lim}}$$

Time to Economic Limit

If limit reached during hyperbolic phase:

$$t_{econ} = \frac{1}{b D_i}\left[\left(\frac{q_i}{q_{econ}}\right)^b - 1\right]$$

If limit reached during exponential phase:

$$t_{econ} = t^* + \frac{1}{D_{lim}}\ln\left(\frac{q(t^*)}{q_{econ}}\right)$$

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Applicability & Limitations

Typical Parameter Ranges

Terminal Decline Rate Guidelines

When to Apply Terminal Decline

1. b > 0.3: Always consider using modified hyperbolic
2. b > 1.0: Must use modified hyperbolic or alternative model
3. Long forecast periods: > 10 years
4. Regulatory filings: Many agencies require bounded EUR

Selection of Terminal Decline Rate

The terminal decline rate should be based on:

1. Analogous mature fields in the same basin
2. Company/operator experience with similar reservoirs
3. Regulatory requirements (some jurisdictions specify limits)
4. Physical reasoning about ultimate drainage

Limitations

1. Arbitrary transition: The switch point is user-specified, not derived from physics
2. Discontinuous derivative: The rate-time derivative has a discontinuity at $t^*$
3. Not suitable for transient flow: Assumes boundary-dominated behavior
4. Single-phase assumption: Does not account for changing fluid properties
5. Stationary parameters: $D_i$ and $b$ assumed constant before transition

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Related Documentation

Related Decline Models

• Arps Decline Curves - Classical exponential, hyperbolic, harmonic
• Power Law Exponential - Physics-based alternative for tight reservoirs
• Stretched Exponential - Statistical physics approach

Selection Guidance

• Decline Overview - Model selection guide (planned)

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References

1. Arps, J.J. (1945). "Analysis of Decline Curves." Transactions of the AIME, 160: 228-247.

2. Johnson, R.H. and Bollens, A.L. (1927). "The Loss Ratio Method of Extrapolating Oil Well Decline Curves." Transactions of the AIME, 77: 771. SPE-927771-G.

3. Currie, S.M. (2010). User Guide for the Rate Time Relations Spreadsheet. Texas A&M University.

4. Ilk, D., Perego, A.D., Rushing, J.A., and Blasingame, T.A. (2008). "Exponential vs. Hyperbolic Decline in Tight Gas Sands — Understanding the Origin and Implications for Reserve Estimates Using Arps' Decline Curves." SPE 116731, SPE Annual Technical Conference and Exhibition, Denver, Colorado.

5. Robertson, S. (1988). "Generalized Hyperbolic Equation." SPE 18731, unpublished.

6. Society of Petroleum Evaluation Engineers (SPEE). (2011). Monograph 3: Guidelines for the Practical Evaluation of Undeveloped Reserves in Resource Plays.

7. Harmony Enterprise. Online Help: Decline Analysis - Modified Hyperbolic.