# Investigation on Interference Test for Wells Connected by a Large Fracture
## 3 Model and Solution for the Interference Test
### Extremum Analysis of the Interference Test  
When Well A shuts after a period of production time \(t_{p}\) with a production rate \(q\), the pressure transient of Well B can be derived by the superposition principle because the governing Equation (1) is a linear equation, as follows  
\[p(t)=p_{i}-\frac{q\mu B}{2 Khb}\left[\frac{2\sqrt{\eta t}}{\sqrt{\pi}}e^{- \frac{x^{2}}{4\eta t}}+xerf\left(\frac{x}{2\sqrt{\eta t}}\right)-\frac{2\sqrt {\eta(t-t_{p})}}{\sqrt{\pi}}e^{-\frac{x^{2}}{4\eta(t-t_{p})}}-xerf\left(\frac{ x}{2\sqrt{\eta(t-t_{p})}}\right)\right] \tag{28}\]To the pressure transient of Well B in the period \((0,t_{p})\), the ratio of pressure change can be derived using Equation (26) as follows  
\[\frac{dp}{dt}=-\frac{q\mu B\sqrt{\eta}}{2 Khb\sqrt{\pi t}}e^{-\frac{x^{2}}{4\eta t }}<0 \tag{29}\]  
Therefore, in the period \((0,t_{p})\), the pressure of Well B is monotonically decreasing. When \(t>t_{p}\), according to Equation (27),  
\[ \begin{array}{l}\frac{dp}{dt}=-\frac{q\mu B}{2 Khb}\left[\begin{array}{c} \frac{\sqrt{\eta}}{\sqrt{\pi t}}e^{-\frac{x^{2}}{4\eta t}}+\frac{x^{2}}{2\sqrt{ \eta}\pi t}e^{-\frac{x^{2}}{4\eta t}}-\frac{\sqrt{\eta}}{\sqrt{\pi(t-t_{p})}} e^{-\frac{x^{2}}{4\eta(t-t_{p})}}-\frac{x^{2}}{2(t-t_{p})\sqrt{\eta\pi(t-t_{p})}}e^{- \frac{x^{2}}{4\eta(t-t_{p})}}\\ -\frac{x^{2}}{2\sqrt{\eta}\pi t}e^{-\frac{x^{2}}{4\eta t}}+\frac{x^{2}}{2(t-t_ {p})\sqrt{\eta\pi(t-t_{p})}}e^{-\frac{x^{2}}{4\eta(t-t_{p})}}\end{array} \right]\\ =-\frac{q\mu B}{2 Khb}\left[\frac{\sqrt{\eta}}{\sqrt{\pi t}}e^{-\frac{x^{2}}{4 \eta t}}-\frac{\sqrt{\eta}}{\sqrt{\pi(t-t_{p})}}e^{-\frac{x^{2}}{4\eta(t-t_{p })}}\right]\end{array}  \tag{30}\]  
Setting  
\[f(t)=\frac{\sqrt{\eta}}{\sqrt{\pi t}}e^{-\frac{x^{2}}{4\eta t}} \tag{31}\]  
then  
\[\frac{dp}{dt}=-\frac{q\mu B}{2 Khb}\big{[}f(t)-f(t-t_{p})\big{]} \tag{32}\]  
Using Equation (31), the following equation is derived  
\[\frac{df}{dt}=\bigg{(}\frac{x^{2}}{2\eta t}-1\bigg{)}\frac{\sqrt{\eta}}{2t \sqrt{\pi t}}e^{-\frac{x^{2}}{4\eta t}} \tag{33}\]  
It can easily be found that when \(t<x^{2}/2\eta,f(t)\) monotonically increases, while when \(t>x^{2}/2\eta\), \(f(t)\) monotonically decreases. Combining the above equation with Equation (32), it can be concluded that when \(t>t_{p}+x^{2}/2\eta t\), \(\frac{dp}{dt}>0\), the pressure \(p\) of Well B monotonically increases, and when \(t<x^{2}/2\eta\), \(\frac{dp}{dt}<0\), the pressure \(p\) of well B monotonically decreases. The continuously differentiable function \(p\) monotonically decreases in the region \([0,\max\big{(}t_{p},x^{2}/2\eta\big{)}]\), and monotonically increases in the region \(t>t_{p}+x^{2}/2\eta t\). Therefore, there must be a point \(t_{m}\in[\max(t_{p},x^{2}/2\eta),t_{p}+x^{2}/2\eta]\) when the pressure \(p\) of Well B reaches a minimal value. According to the extremum condition of a function,  
\[\frac{dp}{dt}=-\frac{q\mu B}{2 Khb}\left[\frac{\sqrt{\eta}}{\sqrt{\pi t_{m}}}e ^{-\frac{x^{2}}{4\eta t_{m}}}-\frac{\sqrt{\eta}}{\sqrt{\pi(t_{m}-t_{p})}}e^{- \frac{x^{2}}{4\eta(t_{m}-t_{p})}}\right]=0 \tag{34}\]  
where \(t_{m}\) is the time when the pressure reaches the minimal value.