Suppose a well produces from all layers in an n-layer reservoir at a common wellbore pressure and a constant total rate, q, for $t>0$. Under the assumptions we made, the problem can be described as follows. The equations are

$\begin{array}{c}\hfill \frac{1}{{\eta }_i}\frac{\partial p_i}{\partial t}-\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial p_i}{\partial r}\right)+\frac{{\tilde{k}}_{i-1}}{k_i{h}_i}\left(p_i-p_{i-1}\right)+\frac{{\tilde{k}}_i}{k_i{h}_i}\left(p_i-p_{i+1}\right)=0,\hfill \\ \hfill i=1,2,\dots ,n;{\tilde{k}}_0={\tilde{k}}_n=0\hfill \end{array}$

where ${\eta }_i=\frac{k_i}{{\varphi }_i\mu c_t}$ and $\tilde{k}$ is the semipermeability between layers i and layers $i+1$.

The boundary conditions at the well are mixed, in other words, common to all layers. The flow conditions are

${\displaystyle \sum _{i=1}^n\frac{2\pi k_i{h}_i}{\mu }}{\left(r\frac{\partial p_i}{\partial r}\right)}_{r=r_w}=qB$

for $t>0$,

$p_1\left(r_w,t\right)=p_2\left(r_w,t\right)=\cdots =p_n\left(r_w,t\right)$

for $t>0$,

$\begin{array}{cc}\hfill p_i\to p_0,\hfill & \hfill i=1,2,\dots ,n\hfill \end{array}$

when $r\to \infty$, and

$\begin{array}{cc}\hfill p_i\to p_0,\hfill & \hfill i=1,2,\dots ,n\hfill \end{array}$

when $t\to 0$.

Using the dimensionless quantities given in Eqs. (3.1), (3.2), the problem becomes

$\begin{array}{c}\hfill \frac{1}{{\eta }_{Di}}\frac{\partial p_{Di}}{\partial t_D}-\frac{1}{r_D}\frac{\partial }{\partial r_D}\left(r_D\frac{\partial p_{Di}}{\partial r_D}\right)+\frac{1}{w_i}\left[{\tilde{k}}_{Di-1}\left(p_{Di}-p_{Di-1}\right)+{\tilde{k}}_{Di}\left(p_{Di}-p_{Di+1}\right)\right]=0,\hfill \\ \hfill {\tilde{k}}_{D0}={\tilde{k}}_{Dn}=0;i=1,2,\dots ,n\hfill \end{array}$

${\displaystyle \sum _{i=1}^n{w}_i}{\left(r_D\frac{\partial p_{Di}}{\partial r_D}\right)}_{r_D=1}=-{\displaystyle \sum _{i=1}^n{q}_{Di}}=-1$

$p_{D1}\left(1,t_D\right)=p_{D2}\left(1,t_D\right)=\cdots =p_{Dn}\left(1,t_D\right)$

$\begin{array}{cc}\hfill p_{Di}\to 0,\hfill & \hfill i=1,2,\dots ,n\hfill \end{array}$

when $r_D\to \infty$

$\begin{array}{cc}\hfill p_{Di}\to 0,\hfill & \hfill i=1,2,\dots ,n\hfill \end{array}$

The new dependent variables are

$p_D={\displaystyle \sum _{i=1}^n{w}_i}{p}_{Di}$

and

$\mathrm{\Delta }{p}_{Di}=p_{Di}-p_{Di+1},i=1,2,\dots ,n-1$

p_Di can be expressed in terms of p_D and Δp_Di as

$p_{Di}=p_D+{\displaystyle \sum _{j=1}^{n-1}\left(1-w_{sj}\right)}\mathrm{\Delta }{p}_{Dj}-{\displaystyle \sum _{j=1}^{i-1}}\mathrm{\Delta }{p}_{Dj},i=1,2,\dots ,n$

where

$w_{si}={\displaystyle \sum _{j=1}^i{w}_j,i=1,2,\dots ,n}$

Subtracting Eq. (B.4) for layer $i+1$ from Eq. (B.4) for layer i and using Eq. (3.4b), we get

$\begin{array}{l}\frac{\partial }{\partial t_D}\left(\frac{p_{Di}}{{\eta }_{Di}}-\frac{p_{Di+1}}{{\eta }_{Di+1}}\right)-\frac{1}{r_D}\frac{\partial }{\partial r_D}\left(r_D\frac{\partial \mathrm{\Delta }{p}_{Di}}{\partial r_D}\right)-\frac{{\tilde{k}}_{Di-1}}{w_i}\mathrm{\Delta }{p}_{Di-1}\hfill \\ +{\tilde{k}}_{Di}\left(\frac{1}{w_i}+\frac{1}{w_{i+1}}\right)\mathrm{\Delta }{p}_{Di}-\frac{{\tilde{k}}_{Di+1}}{w_{i+1}}\mathrm{\Delta }{p}_{Di+1}=0\text{,}\hfill \\ {\tilde{k}}_{D0}={\tilde{k}}_{Dn}=0;i=1,2,\dots ,n-1\hfill \end{array}$

Multiplying Eq. (B.4) for layer i by w_i and adding, we get

$\frac{\partial }{\partial t_D}\left({\displaystyle \sum _{j=1}^n\frac{w_i}{{\eta }_{Di}}}{p}_{Di}\right)-\frac{1}{r_D}\frac{\partial }{\partial r_D}\left(r_D\frac{\partial p_D}{\partial r_D}\right)=0$

The boundary conditions in Eq. (B.5) become

${\left(r_D\frac{\partial p_D}{\partial r_D}\right)}_{r_D=1}=-1$

$\mathrm{\Delta }{p}_{Di}\left(1,t_D\right)=0,i=1,2,\dots ,n-1$

and

$p_D\to 0,\mathrm{\Delta }{p}_{Di}\to 0,i=1,2,\dots ,n-1$

when $r_D\to \infty$. The initial conditions in Eq. (B.6) become

$p_D\to 0,\mathrm{\Delta }{p}_{Di}\to 0,i=1,2,\dots ,n-1$

when $t_D\to 0$. Because the wellbore pressure and initial pressure for all layers are the same and crossflows exist, the pressure differences $\mathrm{\Delta }{p}_{Di},i=1,2,\dots ,n-1$ should be very small compared with p_D, when time is not very short. This fact is confirmed by the numerical calculation. The approximate solution for p_D is obtained by neglecting all Δp_Di in Eq. (B.8b), compared with p_D. Using Eqs. (B.7), (3.3), we get

$\frac{\partial p_D}{\partial t_D}-\frac{1}{r_D}\frac{\partial }{\partial r_D}\left(r_D\frac{\partial p_D}{\partial r_D}\right)=0$

Using boundary conditions in Eq. (B.9) and initial conditions in Eq. (B.10), the approximate solution of Eq. (B.11) is $p_D=-\frac{1}{2}Ei\left(-\xi \right)$, where $\xi =\frac{r_D^2}{4t_D}$ is the effective Boltzmann variable and Ei is the exponential integral function, defined by

$-Ei\left(-\xi \right)={\displaystyle {\int }_{\xi }^{\infty }\frac{e^{-u}}{u}}du$

To obtain the asymptotic solution for Δp_Di, we substitute Eq. (3.10) into Eq. (B.7) and place the result into Eq. (B.8a) and neglect all derivatives of Δp_Di. This produces the following algebraic equations for determining $\mathrm{\Delta }{p}_{Di},i=1,2,\dots ,n-1$:

$\begin{array}{c}\hfill -\frac{{\tilde{k}}_{Di}}{w_i}\mathrm{\Delta }{p}_{Di-1}+{\tilde{k}}_{Di}\left(\frac{1}{w_i}+\frac{1}{w_{i+1}}\right)\mathrm{\Delta }{p}_{Di}-\frac{{\tilde{k}}_{Di+1}}{w_{i+1}}\mathrm{\Delta }{p}_{Di+1}=-\frac{D_i}{2t_D}{e}^{-\xi },\hfill \\ \hfill {\tilde{k}}_{D0}={\tilde{k}}_{Dn}=0;i=1,2,\dots ,n-1\hfill \end{array}$

where, $D_i=\frac{1}{{\eta }_{Di}}-\frac{1}{{\eta }_{Di+1}},i=1,2,\dots ,n-1$. Let

$\mathrm{\Delta }{p}_{Di}={\alpha }_{si}\frac{e^{-\xi }}{t_D},i=1,2,\dots ,n-1$

Substituting into Eq. (B.12) and canceling $\frac{e^{-\xi }}{t_D}$, we get

$-\frac{{\tilde{k}}_{Di-1}}{w_i}{\alpha }_{si-1}+{\tilde{k}}_{Di}\left(\frac{1}{w_i}+\frac{1}{w_{i+1}}\right){\alpha }_{si}-\frac{{\tilde{k}}_{Di+1}}{w_{i+1}}{\alpha }_{si+1}=-\frac{D_i}{2},i=1,2,\dots ,n-1$

After some simple algebraic manipulations, we get

${\alpha }_{si}=\frac{1}{2{\tilde{k}}_{Di}}\left(w_{si}-{\displaystyle \sum _{j=1}^i\frac{w_j}{{\eta }_{Dj}}}\right)=\frac{1}{2{\tilde{k}}_{Di}}{\displaystyle \sum _{j=1}^i\left(w_j-{\omega }_j\right)}$

We define the net crossflow velocity, v_cDi, and the area crossflow rate, q_cDi, of layer i as

$v_{cDi}={\tilde{k}}_{Di-1}\mathrm{\Delta }{p}_{Di-1}-{\tilde{k}}_{Di}\mathrm{\Delta }{p}_{Di}$

and

$q_{cDi}=r_D^2{v}_{cDi}$

respectively. The asymptotic steady value of q_cDi is then the following from Eqs. (B.13), (3.14):

$q_{\mathit{csDi}}=2w_i\left(\frac{1}{{\eta }_{Dj}}-1\right)\xi e^{-\xi }=2\left({\omega }_i-w_i\right)\xi e^{-\xi },i=1,2,\dots ,n$

which is independent of ${\tilde{k}}_{Di}$.

Setting $\frac{\partial q_{\mathit{csDi}}}{\partial r_D}=0$, we see that the position of the peak of q_csDi is at

$\xi =\frac{r_D^2}{4t_D}=1$

Substituting Eq. (B.15) into Eq. (3.21), the steady peak value can be obtained:

$q_{\mathit{cspDi}}=2w_i\left(\frac{1}{{\eta }_{Di}}-1\right)/e,i=1,2,\dots ,n$

The numerical solution shows that the peak value, q_cpDi of q_cDi, will change with time when t_D is short and approach constant q_cspDi when t_D is long, but the shapes of q_cDi for different t_D are quite similar to each other. To get an approximate Δp_Di at short t_D, we assume Δp_Di has the form

$\mathrm{\Delta }{p}_{Di}={\alpha }_i\left(t_D\right)\frac{e^{-\xi }}{t_D},i=1,2,\dots ,n-1$

where α_i(t_D) are arbitrary functions that need to be determined. Differentiating, we get

$\frac{\partial \mathrm{\Delta }{p}_{Di}}{\partial t_D}=\left[{\alpha }_i^{\prime }\left(t_D\right)+{\alpha }_i\left(t_D\right)\frac{\left(\xi -1\right)}{t_D}\right]\frac{e^{-\xi }}{t_D}$

and

$\frac{1}{r_D}\frac{\partial }{\partial r_D}\left(r_D\frac{\partial \mathrm{\Delta }{p}_{Di}}{\partial t_D}\right)=\frac{{\alpha }_i\left(t_D\right)}{t_D^2}{e}^{-\xi }\left(\xi -1\right)$

Because we want to find the behavior of the peak point that locates near $\xi =1$, we can take

$\frac{\partial \mathrm{\Delta }{p}_{Di}}{\partial t_D}\approx {\alpha }_i^{\prime }\left(t_D\right)\frac{e^{-\xi }}{t_D},\frac{1}{r_D}\frac{\partial }{\partial r_D}\left(r_D\frac{\partial \mathrm{\Delta }{p}_{Di}}{\partial t_D}\right)\approx 0,i=1,2,\dots ,n-1$

Using Eq. (3.10) as approximate p_D, we have

$\frac{\partial p_D}{\partial t_D}\approx \frac{e^{-\xi }}{2t_D}$

Substituting Eq. (B.7) into Eq. (B.8a) and using Eqs. (B.17), (B.18), we get (after canceling $\frac{e^{-\xi }}{t_D}$)

$\begin{array}{l}{D}_i\left[{\displaystyle \sum _{j=1}^{n-1}\left(1-w_{sj}\right){\alpha }_{ij}^{\prime }-{\displaystyle \sum _{j=1}^{i-1}{\alpha }_j^{\prime }}}\right]+\frac{{\alpha }_i^{\prime }}{{\eta }_{Di+1}}-\frac{{\tilde{k}}_{Di-1}}{w_i}{\alpha }_{i-1}\hfill \\ +{\tilde{k}}_{Di}\left(\frac{1}{w_i}+\frac{1}{w_{i+1}}\right){\alpha }_i-\frac{{\tilde{k}}_{Di+1}}{w_{i+1}}{\alpha }_{i+1}=-\frac{D_i}{2},i=1,2,\dots ,n-1\hfill \end{array}$

The initial conditions can be set as

${\alpha }_i\left(0\right)=0,i=1,2,\dots ,n-1$

because the crossflow is zero at $t_D=0$.

Eq. (B.19) includes first-order differential equations with constant coefficients. The general solution is the sum of a particular solution and the general solution for the corresponding homogeneous equations. If we let ${\alpha }_j^{\prime }=0$, then Eq. (B.14) is obtained again to determine the particular solution, which is α_sj. The corresponding homogeneous equations are

$\begin{array}{l}{D}_i\left[{\displaystyle \sum _{j=1}^{n-1}\left(1-w_{sj}\right){\alpha }_j^{\prime }-{\displaystyle \sum _{j=1}^{i-1}{\alpha }_j^{\prime }}}\right]+\frac{{\alpha }_i^{\prime }}{{\eta }_{Di+1}}-\frac{{\tilde{k}}_{Di-1}}{w_i}{\alpha }_{i-1}\hfill \\ +{\tilde{k}}_{Di}\left(\frac{1}{w_i}+\frac{1}{w_{i+1}}\right){\alpha }_i-\frac{{\tilde{k}}_{Di+1}}{w_{i+1}}{\alpha }_{i+1}=0,i=1,2,\dots ,n-1\hfill \end{array}$

The solution of Eq. (B.21) has the form

${\alpha }_i\left(t_D\right)=x_i{e}^{-\lambda t_D},i=1,2,\dots ,n-1$

Substituting Eq. (B.22) into Eq. (B.21), we get

$\begin{array}{l}-D_i\left[{\displaystyle \sum _{j=1}^{n-1}\left(1-w_{sj}\right)\lambda x_j-{\displaystyle \sum _{j=1}^{i-1}\lambda x_j}}\right]-\frac{\lambda x_i}{{\eta }_{Di+1}}-\frac{{\tilde{k}}_{Di-1}}{w_i}{x}_{i-1}\hfill \\ +{\tilde{k}}_{Di}\left(\frac{1}{w_i}+\frac{1}{w_{i+1}}\right)x_i-\frac{{\tilde{k}}_{Di+1}}{w_{i+1}}{x}_{i+1}=0,i=1,2,\dots ,n-1\hfill \end{array}$

Let λ_j and $X_j={\left(x_{1,j},x_{2,j},\dots ,x_{n,j}\right)}^T$, and $j=1,2,\dots ,n-1$ be the eigenvalues and the corresponding eigenvectors of Eq. (B.23). The general solution of Eq. (B.19) is then

${\alpha }_i\left(t_D\right)={\alpha }_{si}+{\displaystyle \sum _{j=1}^{n-1}{a}_j}{x}_{i,j}{e}^{-{\lambda }_j{t}_D},i=1,2,\dots ,n-1$

where a_j are arbitrary constants determined by the boundary conditions (Eq. B.20)

${\alpha }_{si}+{\displaystyle \sum _{j=1}^{n-1}{a}_j}{x}_{i,j}=0,i=1,2,\dots ,n-1$

The explicit theoretical results for two- and three-layer cases can be obtained readily with the above theory. For more than three layers, the solution is generally obtained numerically because of the eigenvalue problem (Eq. B.23).