Well Test Analysis for Multilayered Reservoirs with Formation Crossflow

Appendix I

Limiting Behavior for f=α_D(p_wD1-p_wD2) (Chapter 9)

I.1 Early Time Period

In the early time period after the well is being produced, the flow is controlled almost completely by wellbore storage. The rates of the formations are negligible. From Eqs. (H.9)–(H.11) we have

${\alpha }_D\left(p_{wD1}-p_{wD2}\right)+C_{D1}\frac{d{p}_{wD1}}{d{t}_D}=1$

  (I.1)

${\alpha }_D\left(p_{wD1}-p_{wD2}\right)-C_{D2}\frac{d{p}_{wD2}}{d{t}_D}=0$

  (I.2)

and boundary conditions

${\left.p_{wDj}\right|}_{t_D=0}=0,j=1,2$

  (I.3)

Subtracting Eqs. (I.1), (I.2) and integrating, we ...