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If $b$ is in the range, $0<b<1$, then the integration of **Equation 4.68** and **Equation 4.69** results in the rate-time relationship:

$${q}_{o}=\frac{{q}_{oi}}{{\left(1+b{D}_{i}t\right)}^{\left(\frac{1}{b}\right)}}$$

While a second integration with respect to time results in the rate-cumulative production relationship:

$${{\displaystyle q}}_{o}^{\left(1-b\right)}={{\displaystyle q}}_{oi}^{\left(1-b\right)}-\frac{{N}_{p}{D}_{i}\left(1-b\right)}{{{\displaystyle q}}_{oi}^{b}}$$

When the constant $b$ is in the range $0<b<1$, we refer to the resulting production decline as * Hyperbolic Decline*. In hyperbolic decline, we have all three parameters, ${q}_{oi}$, ${D}_{i}$ , and $b$ , with which to match the field data.