5.1 Assumption and Approximate Theoretical Solution of the Problem

In 1949, Van Everdingen and Hurst (1949) gave the unsteady flow solutions for both the constant terminal rate case and the constant terminal pressure case when a well penetrates a homogeneous layer. Since then, the transient well test methods, based on the constant terminal rate solution, were studied deeply and used widely in practice. Paralleling to the constant-rate flow test method, Jacob and Lohman (1952) developed a method for analyzing the constant-pressure flow test in a single layer reservoir, which could be used to determine the (kh) value and skin factor. The constant-pressure flow test was described by Earlougher (1977), also, in his monograph. Unfortunately, this test method was seldom used in practice. The reason might be that it is much easier to measure wellbore pressure accurately than it is to measure flow rate accurately. However, constant-rate tests may inadvertently become constant-pressure tests, so we still need a method for analyzing such tests.

In Chapters 1 and 3, we have studied the constant total flow rate case when each layer is perforated in a two-layer reservoir or an arbitrary n-layer reservoir. We shall use the results obtained in Chapters 1 and 3.

In this chapter, we will examine the unsteady flow with crossflow in an n-layer reservoir when each layer produces under a constant wellbore pressure. Simple theoretical expressions for the asymptotic crossflow and the total flow rate are developed and compared with the numerical simulation results. It will be shown that these theoretical expressions agree very well with the numerical results. The simple expression for the total flow rate can be used to determine the total kh value of the reservoir.

The following assumptions are used in this work: The n-layer reservoir is infinite and each layer is homogeneous; the reservoir is filled with a slightly compressible fluid; the semipermeable model can be used to approximate the actual reservoir; Gravity can be neglected. Suppose a well penetrates all the n layers, and each layer begins to produce under a constant wellbore pressure from time $t=0$ (see Fig. 5.1). The problem can be expressed as follows:

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

The boundary conditions are

$p_i=p_{wi}\mathrm{at}r=r_w,i=1,2,\dots ,n$

$p_i=p_0\mathrm{at}r=\infty ,i=1,2,\dots ,n$

The initial conditions are

$p_i=p_0\mathrm{at}t=0,i=1,2,\dots ,n$

where $b_i={\varphi }_i\mu c{h}_i$, ${\alpha }_i=\frac{k_i}{{\varphi }_i\mu c}$, and c is compressibility. Using the following dimensionless expressions,

$k_{Di}=\frac{{\tilde{k}}_i{r}_w^2}{4{\left(kh\right)}_t},w_i={\frac{k_i{h}_i}{\left(kh\right)}}_t\mathrm{where}{\left(kh\right)}_t={\displaystyle \sum _{i=1}^n{k}_i{h}_i},r_{Dd}=\frac{r_d}{r_w},d_i=\frac{{\alpha }_i}{{\displaystyle \sum _{i=1}^n{\alpha }_i}}\overline{\alpha }=\frac{1}{n}{\displaystyle \sum _{i=1}^n{\alpha }_i},p_{Di}=\frac{p_0-p_i}{\frac{q_0\mu }{4\pi {\left(kh\right)}_t}},t_D=\frac{4\overline{\alpha }t}{r_w^2}$

and introducing the new variables:

$f_i=p_{Di}-p_{Di+1},i=1,2,\dots ,n-1$

$F={\displaystyle \sum _{i=1}^n{w}_i}{p}_{Di},r_D=\frac{r}{r_w}$

Eq. (5.1) becomes

$\begin{array}{r}\hfill \frac{1}{n{d}_i}\frac{\partial p_{Di}}{\partial t_D}-\frac{1}{4r_D}\frac{\partial }{\partial r_D}\left(r_D\frac{\partial p_{Di}}{\partial r_D}\right)+\frac{1}{w_i}\left[{\tilde{k}}_{Di-1}\left(p_{Di}-p_{Di-1}\right)+{\tilde{k}}_{Di}\left(p_{Di}-{p_{Di}}_{+1}\right)\right]=0,\\ \hfill i=1,2,\dots ,n;{\tilde{k}}_{D0}={\tilde{k}}_{Dn}=0\end{array}$

$p_{D1}=p_{Dwi}\mathrm{at}{r}_D=1$

$p_{Di}=0\mathrm{at}{r}_D=\infty$

$p_{Di}=0\mathrm{at}{t}_D=0$

An approximate analytical solution can be obtained as follows: the total rate q_D diminishes with time when the well produces with a constant wellbore pressure for each layer. If the form of q_D(t_D) is known, the above problem can be replaced by a variable-rate flow problem.

Now we try to find the expression for q_D(t_D). From Chapter 3 we know that

$F\left(r_D,t_D\right)={\displaystyle {\int }_{\frac{r_D^2}{{\mathit{d}}_e{t}_D}}^{\infty }\frac{e^{-x}}{x}dx}=-Ei\left(-\frac{r_D^2}{{\mathit{d}}_e{t}_D}\right)$

is a good approximation of the kh-weighted pressure F when all the layers are perforated and produce with a constant total rate, where

$d_e=\frac{1}{\frac{1}{n}{\displaystyle \sum _{i=1}^n\frac{w_i}{d_i}}}$

According to the superposition principle, function F can be approximated by

$F={\displaystyle {\int }_0^{\tau }\frac{q_D\left(\sigma \right)}{t_D-\sigma }{e}^{-\frac{r_D^2}{de\left(t_D-\sigma \right)}}d\sigma }$

when the total rate q_D changes with time. If q_D(σ) is replaced by q_D(t_D) in Eq. (5.6), the integral value will decrease, since q_D(t_D) is a monotonically decreasing function. As a remedy, we can increase the integral interval. There exists a value T for each t_D such that

$F={\displaystyle {\int }_0^{\tau }\frac{q_D\left(\sigma \right)}{t_D-\sigma }{e}^{-\frac{r_D^2}{de\left(t_D-\sigma \right)}}d\sigma }=q_D\left(t_D\right){\displaystyle {\int }_0^T\frac{1}{T-\sigma }{e}^{-\frac{r_D^2}{de\left(T-\sigma \right)}}d\sigma }$

Using the transformation $x=\frac{r_D^2}{de\left(T-\sigma \right)}$ the above formula becomes

$F=q_D\left(\tau \right){\displaystyle {\int }_{\frac{r_D^2}{deT}}^{\infty }\frac{e^{-x}}{x}dx}=-Ei\left(-\frac{r_D^2}{deT}\right)q_D\left(\tau \right)$

Here, T is a function of t_D and needs to be determined. We can simply let $T=t_D$. In this case, Eq. (5.7) is a good approximation of F. This might be useful in some cases, but a more reasonable choice of T is to allow the following:

$T=\frac{{\displaystyle {\int }_0^{\tau }{q}_D\left(\sigma \right)d\sigma }}{q_D\left(t_D\right)}$

The relationship of T and t_D is shown in Fig. 5.2, which is obtained from numerical simulation for some cases. In practice it is better to calculate T from Eq. (5.8) directly, since constant wellbore pressures cannot be maintained strictly. From Eq. (5.7) we have

$\frac{F}{q_D\left(\tau \right)}=-Ei\left(-\frac{R^2}{deT}\right)$

This formula agrees quite well with numerical results, as shown in Fig. 5.7. Using the boundary conditions in Eqs. (5.4)–(5.9), we get

$\frac{p_{wD}}{q_D\left(\tau \right)}=-Ei\left(-\frac{1}{deT}\right)$

where

$p_{wD}={\displaystyle \sum _{i=1}^n{w}_i{p}_{wDi}}$

Eq. (5.10) at times of interest reduces to

$\frac{p_{wD}}{q_D\left(t_D\right)}=-ln\left(\frac{\gamma }{deT}\right)$

where $\gamma =1.781$ is Euler's constant. When we express Eq. (5.12) in practical oilfield units of psi, B/D, cp, md, and ft, so it becomes

$\frac{p_0-p_w}{q\left(t\right)}=\frac{162.6\mu B}{{\left(kh\right)}_t}log\left(\frac{4{\alpha }_e{t}^{\prime }}{\gamma r_w^2}\right)$

where

$p_w={\displaystyle \sum _{i=1}^n{w}_i{p}_{wi}}$

is the kh-weighted wellbore pressure and

${\alpha }_e=\frac{{\displaystyle \sum _{i=1}^n{k}_i{h}_i}}{c\mu {\displaystyle \sum _{i=1}^n{\varphi }_i{h}_i}}$

is the effective diffusivity of the reservoir. B is the formation volume factor, and t′ is the dimension time corresponding to T.

Eq. (5.13) tells us that the total productivity of the multilayer reservoir can be determined by the slope of $\frac{p_0-p_w}{q\left(t\right)}$ versus log(t′). This is quite similar to the drawdown test analysis. The only difference is that log(t′) is used instead of log(t).

It is indicated in Chapter 1 that there are two kinds of crossflow in a homogeneous layered reservoir. One is caused by different boundary pressures and another is caused by different diffusivities for different layers. In the cases we study here, these two crossflows may exist if the constant wellbore pressures for different layers are different.

The crossflow caused only by different diffusivities when a well produces with a constant flow rate is studied in detail in Chapter 3. The theory and formulas can be used here if variables F and f_i in Chapter 3 are replaced by $\frac{F}{q_D\left(t_D\right)}$ and $\frac{f_i}{q_D\left(t_D\right)}$, the corresponding quantities per unit output rate. From Chapter 3 we have

$\frac{f_i}{q_D\left(t_D\right)}=a_i\left(t_D\right)\frac{1}{t_D}{e}^{-\frac{r_D^2}{d_e{t}_D}}$

$a_i\left(t_D\right)=a_{si}\left(1+{\displaystyle \sum _{k=1}^{n-1}{a}_{0k}{x}_{i,k}{e}^{-{\lambda }_k{t}_D}}\right),i=1,2,\dots ,n-1$

$a_{si}=\frac{1}{{\tilde{k}}_{Di}}\left(\frac{w_{si}}{d_e}-\frac{1}{n}{\displaystyle \sum _{k=1}^i\frac{w_k}{d_k}}\right),i=1,2,\dots ,n-1$

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

and λ_k, X_i,k, $k=1,2,\dots ,n-1$ is the kth eigenvalue and corresponding eigensolution of the following equations:

$-{\displaystyle \sum _{j=1}^{n-1}\left(1-w_{sj}\right)}\lambda x_j-{\gamma }_1\lambda x_j+\frac{{\tilde{k}}_{D1}}{D_1}\left(\frac{1}{w_1}+\frac{1}{w_2}\right)x_1-\frac{{\tilde{k}}_{D2}}{D_1{w}_2}{x}_2=0$

$\begin{array}{l}-{\displaystyle \sum _{j=1}^{n-1}\left(1-w_{sj}\right)}\lambda x_j+{\displaystyle \sum _{j=1}^{i-1}\lambda x_j}-\lambda {\gamma }_i{x}_i-\frac{{\tilde{k}}_{Di-1}}{D_i{w}_i}{x}_{i-1}+\frac{{\tilde{k}}_{Di}}{D_i}\left(\frac{1}{w_i}+\frac{1}{w_{i+1}}\right)x_i\hfill \\ -\frac{{\tilde{k}}_{Di+1}}{D_i{w}_{i+1}}{x}_{i+1}=0,i=2,3,\dots ,n-2\hfill \end{array}$

$-{\displaystyle \sum _{j=1}^{n-1}\left(1-w_{sj}\right)}\lambda x_j+{\displaystyle \sum _{j=1}^{n-2}\lambda x_j}-\lambda {\gamma }_{n-1}{x}_{n-1}-\frac{{\tilde{k}}_{Dn-2}}{D_{n-1}{w}_{n-1}}{x}_{n-2}+\frac{{\tilde{k}}_{Dn-1}}{D_{n-1}}\left(\frac{1}{w_{n-1}}+\frac{1}{w_n}\right)x_{n-1}=0$

where

${\gamma }_i=\frac{d_i}{d_{i+1}-d_i},D_i=\frac{1}{n}\left(\frac{1}{d_i}-\frac{1}{d_{i+1}}\right),i=1,2,\dots ,n-1$

a_0k, $k=1,2,\dots ,n-1$ are determined by the following equations:

${\displaystyle \sum _{k=1}^{n-1}{x}_{i,k}}{a}_{0k}=-1,i=1,2,\dots ,n-1$

after x_i,k is determined by Eq. (5.20). Based on numerical examples, all the eigenvalues are positive. The terms including $e^{-{\lambda }_k\tau }$ in Eq. (5.17) will approach zero when t_D is long, and a_i(t_D) will converge to the steady state solution a_si.

We define the area crossflow as

$q_{cDi}=r_D^2\left({\tilde{k}}_{Di-1}{f}_{i-1}-{\tilde{k}}_{Di}{f}_i\right)$

and the area crossflow rate as

$q_{ci}=q_{cDi}/q_D\left(t_D\right)$

From Eq. (5.16) we have

$q_{ci}=d_e\left[{\tilde{k}}_{Di-1}{a}_{i-1}\left(t_D\right)-{\tilde{k}}_{Di}{a}_i\left(t_D\right)\right]\xi e^{-\xi }$

where

$\xi =\frac{r_D^2}{d_e{t}_D}$

q_ci has a maximum at $\xi =1$. The peak value is

$q_{cpi}=d_e\left[{\tilde{k}}_{Di-1}{a}_{i-1}\left(\tau \right)-{\tilde{k}}_{Di}{a}_i\left(\tau \right)\right]/e$

where $e=2.71828$. When t_D is long, a_i converges to a_si. Using Eq. (5.18), we get the steady peak value of q_ci

$q_{cpi}^S=\frac{w_i}{e}\left(\frac{d_e}{n{d}_i}-1\right)$

These formulas are exactly the same as those in Chapter 3.

The crossflow caused by different boundary pressures develops with time and gradually converges to a steady state. The analysis solution for the steady pressure differences between layers is

$f_i={\displaystyle \sum _{i=2}^n{\beta }_i{K}_0\left(\sqrt{{\omega }_i}{r}_D\right)}\left(d_{ij}-d_{ij+1}\right),j=1,2,\dots ,n-1$

and is given in the Appendix C. Here, ω_i and d_ij, $j=1,2,\dots ,n-1$ are the eigenvalues and eigensolutions of Eqs. (C.3), (C.4) respectively, and β_i is determined by Eq. (5.30). K₀ is the modified Bessel function.

${\displaystyle \sum _{i=2}^n\left(d_{ij}-d_{ij+1}\right)K_0\left(\sqrt{{\lambda }_i}\right){\beta }_i}=p_{wj}-p_{wj+1},j=1,2,\dots ,n-1$

The total crossflow should be the sum of these two kinds of crossflow.

5.2 Numerical Results and Comparison with the Approximation Theory

In order to study the crossflow phenomenon and to test the above approximation theory, some two- and three-layer cases were calculated using a standard finite difference method. For convenience, a new pair of variables,

$u=ln{r}_D^2,v=ln\left(1+t_D\right)$

is used in the numerical calculation. Part of the calculation results will be shown below. All parameters are dimensionless.

5.2.1 Flow Rate of Each Layer Changes with Time

Fig. 5.3 shows how the flow rate of each layer changes with time in a two-layer reservoir. The flow rate diminishes very quickly at first and diminishes slower and slower when time increases. When the wellbore pressures for different layers are different, q_Di/q_D, the ratio of the rate for layer i to the total rate changes constantly with time. When the wellbore pressures are the same, q_Di/q_D changes with time when time is short and converges to the productivity ratio w_i when time is long enough.

5.2.2 kh-Weighted Pressure Changes with Time

Figs. 5.4 and 5.5 show the distributions of kh-weighted pressure F and $\frac{F}{q_D\left(t_D\right)}$ at a different time t_D. It can be seen how the constant wellbore pressure problem is changed to a constant terminal rate problem when F/q_D is used. Figs. 5.6 and 5.7 show that the numerical calculation of $\frac{F}{q_D\left(t_D\right)}$ agrees very well with the Ei function.

5.2.3 Distributions of Pressure Difference Changes with Time

Figs. 5.8 and 5.9 show the distributions of pressure difference at different times for a two-layer reservoir when the constant wellbore pressures are different and the same, respectively. When the wellbore pressures are different, the pressure difference will gradually converge to a steady state. This steady state coincides with the theoretical solution. When wellbore pressures are the same, the pressure difference will go to zero with increasing time.

5.2.4 Distributions of Area Crossflow

Fig. 5.10 shows the distributions of area crossflow for a two-layer reservoir when wellbore pressures are different. The area crossflow develops from a single peak curve into a two-peak curve. The peak near well represents the crossflow caused by different wellbore pressures and becomes stable when time is long. When the wellbore pressures are the same, this peak will not appear. The peak away from the wellbore represents the crossflow caused by different diffusivities. The height of the second peak diminishes with time. From Fig. 5.10 we can see that the theory agrees quite well with numerical results.

Figs. 5.11 and 5.12 show the distributions of the area crossflow rate q_c1 for a two-layer reservoir when the wellbore pressures are different and the same, respectively. The distribution of q_c1 will develop from a one-peak curve into a two-peak curve if the wellbore pressures are different. The first peak, which represents the crossflow caused by different boundary pressures, will increase with time. The second peak, which represents the crossflow caused by different diffusivities, moves forward like an unchanged wave when time is long. This phenomenon has been seen when the constant terminal rate flow was studied. Comparing Figs. 5.10–5.12 we can conclude that when time is long, the crossflow caused by different wellbore pressures is independent of the change of flow rate. The crossflow caused by different diffusivities is proportional to the total flow rate.

5.2.5 The Peak Value of q_ci Changes with Time

Figs. 5.13 and 5.14 show the change of the peak value of q_ci with time for a two-layer reservoir and a three-layer reservoir respectively. When time is short, $log\left(q_{cpi}/q_{cpi}^S\right)~ln\left(1+t_D\right)$ is almost a straight line. When time is long, the peak value converges to a constant value. From these figures it can be seen that the theory agrees very well with the numerical results. When the wellbore pressure differences are not very large, the crossflow, caused by the wellbore pressures, has only a small influence on the peak value of the crossflow caused by different diffusivities. This is the reason why the theoretical curves agree very well with the cases when wellbore pressures are not that much different from each other.

5.4 Summary

From what we have shown above, the following conclusions about the crossflow behavior can be drawn:

(1) The constant wellbore pressure problem can be replaced by a corresponding constant total flow rate problem if F/q_D, the kh-weighted pressure per unit rate, and f_i/q_D, the pressure differences per unit rate, are used.

(2) The part of area crossflow rate, caused by different diffusivities, at first develops with time and then gradually reaches a quasisteady state. In the quasisteady state, it moves forward like an unchanged wave. Its behavior is described by the theory given in Chapter 3 for the case when all the layers are perforated and produce with a constant total rate.

(3) In the quasisteady state period, the pressure differences between layers and the crossflow caused by different wellbore pressures will reach a steady state and will not be influenced by the flow rate change.

(4) The ratio of the flow rate, q_Di/q_D, changes constantly when wellbore pressures are different for different layers, and it will converge to the productivity ratio w_i when the wellbore pressures are the same.

(5) The total flow rate diminishes with time and can be expressed by Eqs. (5.10) or (5.12) when each layer produces under a constant wellbore pressure in a multilayer reservoir. The total productivity of the multilayer reservoir can be determined by the slope of $\frac{p_0-p_w}{q\left(t\right)}$ versus log(t) curve at the straight line part.

(6) Three different regimes occur in a two-layer reservoir. At an early time, before crossflow is established, the response is the same as that for the two-layer reservoir without crossflow. At a later time, q_jD/q_D converges to a constant. At an intermediate time, a transition behavior appears. For smaller vertical permeability, the transition occurs later. If $s_1=s_2$, q_1D/q_D will converge to γ. If $s_1\ne s_2$, q_1D/q_D doesn't converge to γ. We can find that if $s_1>s_2$, the constant of Layer 1 will be more than γ. If $s_1<s_2$, the constant of Layer 1 will be less than γ.

(7) Curves of 1/q_D with time have two lines; between two lines is an intermediate regime. The smaller the γ, the smaller 1/q_D is. The smaller the λ, the later the transition occurs. At a later time, the 1/q_D of different parameters will converge to a line.