8.1 Introduction

In developing a suitable exploitation method for oil and gas reservoirs, knowledge of the vertical permeability of the reservoir is needed. This is particularly true for a thick reservoir or a layered reservoir undergoing secondary or tertiary recovery.

The vertical interference testing is a method for determining horizontal and vertical properties of a reservoir near the well by analyzing pressure transients opposite an isolated set of perforations while the well is flowed or pulsed at another depth. Vertical interference tests have been used to determine the vertical permeability in a single zone or in the low-permeability layer separating two productive zones, to determine crossflow between the two zones and to detect leakage behind the casing.

Methods for determination of vertical permeability in a single zone were provided by several authors (Burns, 1969; Prats, 1970; Falade and Brigham, 1974; Hirasaki, 1974; Earlougher, 1980; Kamal, 1986). Earlougher compared the various methods and considered wellbore storage at both the active and observation zones, leakage behind the casing, and vertical flow across a permeability barrier in the formation. Kamal provided a new type of curve set for the early time testing data. However, these methods are not suitable for stratified reservoirs with several layers having quite different properties.

In Chapters 2 and 7, we provided an interpretation method of the reservoir properties based on semilog straight line portions of both the active and observation pressures. We also presented a detailed discussion of the crossflow behavior between layers. Our method neglects the wellbore storage and uses the early time data to determine layer parameters. Several authors (Bremer et al., 1985; Ehlig-Economides and Ayoub, 1986; Lee et al., 1984; Wijesinghe and Kececioglu (1988) have studied the vertical interference test in a three-layered reservoir with the middle layer being a thin tight one. Bremer et al., (1985) presented a curve method for the observation pressure to estimate reservoir parameters. This method assumed identical formation properties except thickness for both the active and observation layers, and it neglected the skin factor and wellbore storage. Ehlig-Economides and Ayoub (1986) eliminated the above restrictions. They presented a detailed analysis of the asymptotic behavior of the pressure and used both the active and observation pressures in type curve matching. In the above studies, steady-state crossflow in the tight middle layer is assumed. Lee et al. (1984) presented an improved analysis for pulse tests, which assumed unsteady crossflow in the tight middle layer. Instead of using type curve matching, they used nonlinear parameter estimation. Wijesinghe and Kececioglu (1988) compared the behavior of the steady-state crossflow and the unsteady crossflow in the tight layer and showed their influence on active and observation pressures. They found that the pressures of the active layer for unsteady crossflow and steady-state crossflow are close to each other at an early time and are almost the same at a long time. But the observation pressure for unsteady crossflow is far different from that for steady-state crossflow in the early time, which is called the middle time period in Chapter 7. Therefore, the interpretation methods, which are based on the steady-state crossflow assumption and use the observation pressure in type curve matching or analysis, may provide the wrong results. The interpretation methods, which are based on the steady-state crossflow assumption and use only the pressure of the active layer in analysis, should still work.

Because of bad cementing or hydraulic fracturing, leakage behind the casing is found in many wells. Earlougher (1980) showed that a micro-annulus of only 25 μm (0.001 in.) wide will give enough vertical flow between observation and active perforations to cause the appearance of a high vertical permeability. He simulated the micro-annulus by adding a very large vertical permeability at the nodes close to the well in a program, but he did not give a detailed analysis. To the author's knowledge, this is the only treatment of the leakage behind the casing. Overall, the study of the transient testing on wells with leakage behind the casing is very poor and a theory is earnestly needed in practice. The objective of this chapter is to give a well testing theory to analyze and interpret the vertical interference test conducted at a three-layered reservoir, including skin factors, wellbore storage of the active layer, the compressibility of the low-permeability middle layer, and the fluid leakage behind the casing.

8.2 Model Description

Suppose a reservoir consists of two permeable layers, separated by a third layer of comparatively low permeability. A well drilled the reservoir completely. A packer is located at the low-permeability middle layer and separates the two productive layers, and a micro-annulus behind the casing allows communication of fluid between the two productive layers under the pressure difference between them.

Because the observation pressure may be used in the analysis, the compressibility of fluid in the low-permeability middle layer cannot be neglected (Gao, 1984; Bremer et al., 1985; Ehlig-Economides and Ayoub, 1986), but the horizontal flow in it can be neglected. Therefore, the following assumptions are used:

(1) A three-layered reservoir is infinite, homogeneous, isotropic, and filled with a single-phase fluid of slight compressibility and constant viscosity. The middle layer has low permeability.

(2) A well penetrates the reservoir completely. A packer is located at the middle layer and separates the upper and lower ones. Vertical communication may occur through the micro-annulus between the casing and the wall of the well or through the packer.

(3) The horizontal flow in the low-permeability middle layer is negligible.

(4) Flow obeys Darcy's law. The gravity force is neglected.

(5) The semipermeable wall mode is used in replace of the actual reservoir.

According to the semipermeable wall model, the semipermeabilities of the semipermeable walls at the top and bottom of the middle layer are, respectively,

${\tilde{k}}_1=2/\left(\frac{h_1}{k_{v1}}+\frac{h_3}{k_{v3}}\right),{\tilde{k}}_2=2/\left(\frac{h_2}{k_{v2}}+\frac{h_3}{k_{v3}}\right)$

Define semipermeability

$\tilde{k}={\left(\frac{1}{{\tilde{k}}_1}+\frac{1}{{\tilde{k}}_2}\right)}^{-1}=2/\left(\frac{h_1}{k_{v1}}+\frac{h_2}{k_{v2}}+\frac{2h_3}{k_{v3}}\right)$

which is the semipermeability when the low-permeability layer is not treated as a layer in the model.

Since $k_{v3}\ll k_{v1}$, $k_{v3}\ll k_{v2}$, from Eq. (8.1) we have ${\tilde{k}}_1\approx {\tilde{k}}_2$. For simplicity, we take:

${\tilde{k}}_1={\tilde{k}}_2=2\tilde{k}$

where $\tilde{k}$ is calculated by Eq. (8.2).

Suppose the initial pressure, p_i, is constant everywhere. The well begins to produce from $t=0$. Under these assumptions, the problem can be described mathematically as follows.

$\begin{array}{r}\hfill {\left(\varphi c_{t}h\right)}_1\frac{\partial p_1}{\partial t}-\frac{k_1{h}_1}{\mu }\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial p_1}{\partial r}\right)+\frac{2\tilde{k}}{\mu }\left(p_1-p_3\right)=0\\ \hfill {\left(\varphi c_{t}h\right)}_2\frac{\partial p_2}{\partial t}-\frac{k_2{h}_2}{\mu }\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial p_2}{\partial r}\right)+\frac{2\tilde{k}}{\mu }\left(p_2-p_3\right)=0\\ \hfill {\left(\varphi c_{t}h\right)}_3\frac{\partial p_3}{\partial t}+\frac{2\tilde{k}}{\mu }\left(2p_3-p_1-p_2\right)=0\end{array}$

The initial conditions are

$p_j=p_i,j=1,2,3\mathrm{at}t=0$

The boundary conditions at the well are

$p_j=p_i,j=1,2\mathrm{f}\mathrm{o}\mathrm{r}r\to \infty$

The boundary conditions at the well are

$\frac{2\pi k_j{h}_j}{\mu }{\left(r\frac{\partial p_j}{\partial r}\right)}_{r=r_w}=q_j\left(t\right),j=1,2$

Infinitely thin skin for each layer is assumed:

$p_{wj}={\left(p_j-s_{j}r\frac{\partial p_j}{\partial r}\right)}_{r=r_w},j=1,2$

The flow rate, q₂, of Layer 2, which flows to Layer 1 through the micro-annulus behind the casing, is described by a dimensionless function of p_w1 and p_w2:

$q_2\left(t\right)=q_{r}Bf\left(p_{w1},p_{w2}\right)$

Eq. (8.8) shows that we have assumed that the communication between the two productive layers can only happen when fluid flows through the skins of the two layers.

The apparent rate, q, of Layer 1, which is measured at the bottom of the well as a known function of time, should be the total rate of the two layers:

$q_1\left(t\right)+q_2\left(t\right)=q\left(t\right)$

If the well produces at a constant surface rate q_r, we have

$-q_{r}B=C\frac{d{p}_{w1}}{dt}-2\pi r_w{\displaystyle \sum _{j=1}^2\frac{k_j{h}_j}{\mu }{\left(\frac{\partial p_j}{\partial r}\right)}_{r=r_w}}$

Instead of boundary condition Eq. (8.10a), Eqs. (8.4)–(8.10) give the complete mathematical description of the problem. To simplify the analysis, the following dimensionless parameters are used, which are similar to those used in reference Ehlig-Economides and Ayoub (1986):

$p_{Dj}=\frac{{\left(kh\right)}_t}{{\alpha }_p{q}_{r}B\mu }\left(p_i-p_j\right),j=1,2,3$

$t_D=\frac{{\alpha }_t{\left(kh\right)}_{t}t}{{\left(\varphi c_{t}h\right)}_t\mu r_w^2},r_D=r/r_w,{\sigma }_D=\frac{{\left(\varphi c_{t}h\right)}_t\mu r_w^2}{{\alpha }_t{\left(kh\right)}_t}\sigma$

$q_D=\frac{q}{B{q}_r},q_{Dj}=\frac{q_j}{B{q}_r},j=1,2$

$w_j=\frac{k_j{h}_j}{{\left(kh\right)}_t},j=1,2;{\omega }_j=\frac{{\left(\varphi c_{t}h\right)}_j}{{\left(\varphi c_{t}h\right)}_t},j=1,2,3$

${\tilde{k}}_D=\frac{\tilde{k}{r}_w^2}{{\left(kh\right)}_t},C_D=\frac{{\alpha }_{c}C}{{\left(\varphi c_{t}h\right)}_t{r}_w^2},{\alpha }_D=\frac{{\alpha }_p{q}_r\mu B}{{\left(kh\right)}_t}\alpha$

where

${\left(kh\right)}_t=k_1{h}_1+k_2{h}_2,{\left(\varphi c_{t}h\right)}_t={\left(\varphi c_{t}h\right)}_1+{\left(\varphi c_{t}h\right)}_2$

The values for α_p, α_t, and α_c depend on the units used in the analysis. Values for several systems of units are given in Table 8.1.

Table 8.1

Unit Conversions

Symbol	Customary Units	SI Metric Units	Darcy Units
q	STB/D	dm3/s	cm3/s
μ	cp	Pa. s	cp
k	md	μm2	darcy
$\tilde{k}$	md/ft	μm2/m	darcy/cm
r, h	ft	m	cm
p	psia	kPa	atm
t	hours	hours	s
ct	psia−1	kPa−1	atm−1
C	bbl/psi	dm3/kPa	cm3/atm
αp	141.2	106/2π	1/2π
αt	.000264	3.6×10−6	1
αc	.8936	159.2	1/2π

Eqs. (8.1)–(8.10) are written in darcies. From Eqs. (8.14), (8.16), we have

$w_1+w_2=1,{\omega }_1+{\omega }_2=1$

8.3 The Interpretation Theory When q(t) is Known

Suppose F(t) is a function of t. Its dimensionless form is given by

$F_D\left(t_D\right)={\alpha }_{F}F\left(t\right)$

where α_F is a constant, depending on what physical quality F(t) represents. Define the following dimension and dimensionless Laplace transformation:

$\overline{F}\left(\sigma \right)={\displaystyle {\int }_0^{\infty }F\left(t\right)}{e}^{-\sigma t}dt$

and

${\overline{F}}_D\left({\sigma }_D\right)={\displaystyle {\int }_0^{\infty }{F}_D\left(t_D\right)}{e}^{-{\sigma }_D{t}_D}d{t}_D$

After integration by parts, Eq. (8.19) becomes

$\overline{F}\left(\sigma \right)=\frac{F\left(0\right)}{\sigma }-\frac{1}{{\sigma }^2}{\displaystyle {\int }_0^{\infty }{F}^{\prime }\left(t\right)}d\left(e^{-\sigma t}\right)$

For simplicity, we also use $\overline{F}$ for $\overline{F}\left(\sigma \right)$ and ${\overline{F}}_D$ for ${\overline{F}}_D\left({\sigma }_D\right)$. From Eqs. (8.18) to (8.20) and Eq. (8.12), we have

$\overline{F}\left(\sigma \right)=\frac{{\left(\varphi c_{t}h\right)}_t\mu r_w^2}{{\alpha }_F{\alpha }_t{\left(kh\right)}_t}{\overline{F}}_D\left({\sigma }_D\right)$

The dimensionless form of Eqs. (8.4)–(8.10) is given in Appendix F and solved in the Laplace space. We have the following governing equations of the problem from Appendix F:

$A\left({\sigma }_D\right){\alpha }_2{w}_2+D\left({\sigma }_D\right)w_2=\tilde{f}\left({\sigma }_D\right)$

$A\left({\sigma }_D\right)\left(w_1+{\alpha }_2{w}_2\right)+D\left({\sigma }_D\right)\left({\alpha }_1{w}_1+w_2\right)={\overline{q}}_D\left({\sigma }_D\right)$

$A\left({\sigma }_D\right)\left[K\left({\lambda }_1\right)+s_1\right]+D\left({\sigma }_D\right){\alpha }_1\left[K\left({\lambda }_2\right)+s_1\right]={\overline{p}}_{wD1}\left({\sigma }_D\right)$

$A\left({\sigma }_D\right){\alpha }_2\left[K\left({\lambda }_1\right)+s_2\right]+D\left({\sigma }_D\right)\left[K\left({\lambda }_2\right)+s_2\right]={\overline{p}}_{wD2}\left({\sigma }_D\right)$

where A(σ_D) and D(σ_D) are arbitrary functions, being determined by any two equations in Eqs. (8.22)–(8.25). Eigenvalue λ_j and coefficient α_j are given by (F.23), (F.25). Function K(x) is

$K\left(x\right)=K_0\left(x\right)/x{K}_1\left(x\right)\approx ln\frac{2}{\gamma x}\mathrm{f}\mathrm{o}\mathrm{r}x\ll 1$

where $\gamma =1.78107$. For most reservoirs and reasonable observation time, we have ${\lambda }_1\ll 1$ and ${\lambda }_2\ll 1$. So we can use the approximation in Eq. (8.26).

The Laplace transform of a known function F(t) can be calculated approximately if the upper limit of the Laplace transform is taken as $\frac{6}{\sigma }~\frac{7}{\sigma }$. Thus we can take

$\overline{F}\left(\sigma \right)={\displaystyle {\int }_0^{\frac{7}{\sigma }}F\left(t\right)}{e}^{-\sigma t}dt\approx \frac{F\left(0\right)}{\sigma }-\frac{1}{{\sigma }^2}{\displaystyle {\int }_0^{\frac{7}{\sigma }}{F}^{\prime }\left(t\right)}d\left(e^{-\sigma t}\right)$

This integral can be calculated very quickly by the following formula:

$\overline{F}\left(\sigma \right)=\frac{F\left(0\right)}{\sigma }+\frac{1}{{\sigma }^2}{\displaystyle \sum \left[F\left(t_{j+1}\right)-F\left(t_j\right)\right]}\left[e^{-\sigma t_j}-e^{-\sigma t_{j+1}}\right]/\left(t_{j+1}-t_j\right)$

where t_j are net points in region $\left(0,\frac{7}{\sigma }\right)$.

When the leakage rate behind the casing is not negligible, both p_w1 and p_w2 change remarkably with time from the very beginning of production and are strongly influenced by all the parameters of the reservoir. This is quite different from case $f\left(t\right)=0$. From the previous works (Bremer et al., 1985; Ehlig-Economides and Ayoub, 1986; Lee et al., 1984; Wijesinghe and Kececioglu, 1988), which studied cases $f\left(t\right)=0$, we know that in the short time period, the pressure p_w1 of the active layer only depends on the parameters of the active layer and the observation pressure, p_w2, will not change with time, as though the active layer did not produce. p_w1 and p_w2 gradually influence each other, and p_w2 begins to change with time only after the short time period.

We can make use of the communication rate between layers. For example, if there is no leakage behind the casing, we can use a specially designed leakage packer to create it in drawdown tests. The leakage rate, q₂(t), through the leakage packer at a different time can be calculated by its function f(p_w1,p_w2), which is determined by the design and measured wellbore pressures, p_w1⁰(t) and p_w2⁰(t), in the test. If the middle layer of the reservoir is an impermeable one, that is ${\tilde{k}}_D=0$, and a leakage packer is used, the two layers can be tested and interpreted by one drawdown test. This case will be studied later.

Suppose we have obtained the expressions for ${\overline{p}}_{wD1}$ and ${\overline{p}}_{wD2}$, which depend on σ_D and reservoir parameters and are different in different cases, p_wD1 and p_wD2 can be obtained by Stehfest's numerical inversion method. We construct the following objective function for nonlinear estimation of the reservoir parameters:

$E\left(\overrightarrow{R}\right)={\displaystyle \sum _{j=1}^2{\displaystyle \sum _{k=1}^n{\left[\frac{{\alpha }_p{q}_r\mu B}{{\left(kh\right)}_t}{p}_{wDj}\left(t_{Dk};\overrightarrow{R}\right)-\mathrm{\Delta }{p}_{wj}^0\left(t_k\right)\right]}^2}}$

in real space or

$L\left(\overrightarrow{R}\right)={\displaystyle \sum _{j=1}^2{\displaystyle \sum _{k=1}^n{\left[c_p{\overline{p}}_{wDj}\left({\sigma }_{Dk};\overrightarrow{R}\right)-\mathrm{\Delta }{\overline{p}}_{wj}^0\left({\sigma }_k\right)\right]}^2}}$

in Laplace space, where

$c_p=\frac{{\alpha }_p{q}_{r}B{\mu }^2{r}_w^2{\left(\varphi c_{t}h\right)}_t}{{\alpha }_t{\left(kh\right)}_t^2}$

If only one pressure, p_w1 or p_w2, is used in analysis, Eqs. (8.28), (8.29) should include one corresponding square term. $E\left(\overrightarrow{R}\right)$ or $L\left(\overrightarrow{R}\right)$ can be minimized to give the best parameter estimation, where $\overrightarrow{R}$ is the dimensionless parameter vector being estimated. The parameters represented by vector $\overrightarrow{R}$ may be different for different cases and will be explained later in each case. In this chapter, we assume p_i, (ϕc_th)₁, and (ϕc_th)₂ are known in all cases. Therefore, ω₁ and ω₂ are known parameters, they do not include in unknown parameters vector $\overrightarrow{R}$, but, w₁, s₁, and s₂ always do. Vector $\overrightarrow{R}$ and (kh)_t should be determined. Later we shall discuss how to choose the initial values of vector $\overrightarrow{R}$ in the nonlinear estimation. Barua and Horne (1988) discussed the different nonlinear parameter estimation methods used in well test analysis and compared them.

8.4 The Interpretation Theory When the Well Produces with a Constant Surface Rate

In practice, the most important case in well testing is the surface rate of the well being constant. Although the surface rate of the well is constant, the bottom hole rate changes with time because of the wellbore storage. From Appendix G we know that the Laplace transform of the changing bottom hole rate can be written as

${\overline{q}}_D=\frac{1}{{\sigma }_D}\left(1-C_D{\sigma }_D^2{\overline{p}}_{wD1}\right)$

which was obtained by other authors before. Substituting Eq. (8.45) into the previous results for the changing rate, q(t), we can obtain parallel results for the constant surface rate, as it was done in Appendix G.