Well Test Analysis for Multilayered Reservoirs with Formation Crossflow

Appendix C

Solution of the Crossflow Caused by Different Boundary Pressures (Chapter 4)

Denote

$p_D=\left[\begin{array}{c}\hfill p_{D1}\hfill \\ \hfill p_{D2}\hfill \\ \hfill ⋮\hfill \\ \hfill p_{Dn}\hfill \end{array}\right]B=\left[\begin{array}{c}\hfill w_1/4\hfill \\ \hfill w_2/4\hfill \\ \hfill \ddots \hfill \\ \hfill w_n/4\hfill \end{array}\right]A=\left[\begin{array}{cc}{A}_1\hfill & -A_1\hfill \\ -A_1\hfill & A_1+A_2\hfill & -A_2\hfill \\ \ddots \hfill & \ddots \hfill & \ddots \hfill \\ -A_{n-1}\hfill \\ -A_{n-1}\hfill & A_{n-1}\hfill \end{array}\right]$

Neglecting the differential part respect to time, Eq. (4.7) can be written as

$B\frac{1}{r_D}\frac{d}{d{r}_D}\left(r_D\frac{d{p}_D}{d{r}_D}\right)=A{p}_D$

  (C.1)

Its general solution has the form

$p_D=\left({\alpha }_1+{\beta }_{1}ln{r}_D\right)D_1+{\displaystyle \sum _{i=2}^n\left[{\alpha }_i{I}_0\left(\sqrt{{\lambda }_i}{r}_D\right)+{\beta }_i{K}_0\left(\sqrt{{\lambda }_i}{r}_D\right)\right]}{D}_i$

  (C.2)

where I₀, K₀ are modified Bessel functions, and λ_i, $i=2,3,\dots ,n$, are the nonzero eigenvalues of the ...