Why the Modified Hyperbolic Works
Unique Characteristics of the Modified Hyperbolic Equation Make It Ideal for Unconventional Resources
The modified hyperbolic equation (MHE), while singular in name, is two linked Arps equations, where the first segment is used to model linear flow and the second a combination of boundary influenced & boundary dominated flow. The transition from one segment to the next is smooth with neither change in rate nor slope.
The modified hyperbolic equation must have the following characteristics:
- preserve the log-log straight line of linear flow, that is to behave as a negative ½ slope power law, commencing as early in time as necessary.
- approximate the rate-time profile during the transition from linear to boundary dominated flow.
- Preserve the log-log straight line of late time boundary dominated flow, that behaves as a power law with slope equal to the negative inverse of the b‑value.
The transformation of linear flow from a log-log straight line is not as simple as using a hyperbolic b-exponent of two. In Equation 6, we equate the linear and hyperbolic equations then, eliminate one in the denominator when bdit >> 1. The condition is true for practical ranges of accuracy: bdit >> 49 for difference <= 1%. Inspecting Equation 7, the power law equation has the same form as the hyperbolic, confirming that the hyperbolic b‑exponent is the negative inverse of the power law n‑exponent.
Rearranging to solve for the initial hyperbolic decline factor, di, we reduce the hyperbolic degrees of freedom by making di a function of other parameters (Equation 8). Rearranging, the hyperbolic coefficients are functions of only the linear flow equation (Equation 9).
In Equation 10, we see a relationship derived from Poston & Poe in their book Analysis of Production Decline Curves (SPE Books, 2008). This relationship is that qb / d = constant and we discover the value of the constant is a function of only parameters from the linear flow equation.
Equation 10 tells us that we are free to choose any initial rate to model linear flow, providing we choose a corresponding decline factor that will honor the equation. We cannot assume that the initial rate (from the linear flow equation) occurs at time zero because that rate is infinite.
If our software is capable, we could time-shift our data and allow negative time in the calculation. For instance, if qi is calculated at mid-month (15 days) using the power law equation, then physical day 15 will be the reference time, tref, at which qi and di are calculated and the hyperbolic equation will have the form q = qi (1+b di [t-tref])(-1/b). This equation will perfectly match the linear flow equation for any time.
In Figure 1, the blue line shows behavior consistent with the way some evaluators use declines to model conventional reservoirs. The average rate for the first month (calculated from the linear flow equation) is assumed to occur at mid‑month and is represented as qi in the hyperbolic equation. This initial hyperbolic rate is a maximum that acts as an asymptote and the calculated rates deviate from the desired ½ slope toward the asymptote. This method is okay if the initial rate is calculated at a very small time.
For example, the red line of Figure 1 is calculated using qi at 1 minute from the linear flow equation and di calculated from Equation 8. Because we are approximating the linear flow equation at small times, the values for qi and di are very large.
We need guidance toward determining the value one should choose for qi. We start with two premises from which we derive the relationship shown in Figure 2.
- there is a lower limit of time for which an accurate hyperbolic representation is desired.
- the value of this accuracy will depend on the situation.
The time ratio tells how much time it will take for the effects of the asymptote to become negligible. It is defined as the reference time divided by the time at which accurate results are desired. The desired accuracy is the difference in rate calculated using the two equations (linear flow minus hyperbolic) divided by the linear flow rate.
Suppose you would like the hyperbolic equation to be within 1% of the power law equation at 1 day. From Figure 2, we obtain the time ratio of 0.02055. Then the value for qi is calculated from the linear flow equation at approximately 30 minutes (0.02055 x 1 day).
Earlier we saw that the initial hyperbolic rate acted as an asymptote, causing the rate-time profile to bend away from the straight line on a log-log scale. This is the characteristic we are looking for to model the combined boundary influenced (transition) and boundary dominated flow. Given that the transition from linear to boundary influenced flow should be smooth, the only unknowns to complete the forecast are the time at which the switch between segments occurs and the b-exponent for the second segment. The transition time will shift the second segment laterally and change the slope by a modest amount. The b-exponent will adjust the second segment up and down and adjust the curve of the forecast.
We observe in Figure 3 that the asymptotic behavior of the hyperbolic equation does a very good job of modelling the boundary flow. Here the transition occurs at about 5 months with a second segment b-exponent of 0.77 and the rate-time profile approaches the final slope inferred by the b-exponent.
In other work not referenced in this white paper, it was observed that fitting the second segment alone results in a similar rate-time profile as that obtained by locking the values for qi and di based on the end of linear flow. However, this alternative requires much more post linear flow data.
The final b-value is on the high end based on the work of the Fetkovitch (SPE 28628). Also, Fetkovitch shows that it may take a long time for the post linear flow stem to straighten on a log-log scale. Thus, we cannot assume that the time to reach a second straight line is comprised of only transition time from 1st fracture pair interference to last. Some of this time is the delay for the change in flow character to be observed as a change in decline.
We started this white paper as an examination of the modified hyperbolic equation. We have seen that it is perfect for capturing the nuances of flow in thigh multi-fractured horizontal wells. The equation may be adapted to follow the power law relationship of linear flow. Its asymptotic behavior toward an early time makes it perfect to approximate the transition from linear to fully bounded flow and permits very early detection of the character of the second segment. Don’t fight perfection – stop looking for other unconventional equations.
About the Author
Randy Freeborn is an SPE Distinguished Lecturer on type wells and a subject matter expert in empirical forecasting. Currently, he is a Fellow at Aucerna where he is a thought leader, responsible for identifying and inventing engineering technology and assisting clients. He has been a professional engineer for 45 years and is a member of SPEE and SPE. Freeborn has prepared numerous technical papers for presentation at conferences, workshops and industry meetings and has been a guest lecturer at the University of Houston and Texas A&M.
