Semi-Analytical Solutions for a Partially Penetrated Well with Wellbore Storage and Skin Effects in a Double-Porosity System with a Gas Cap

Abstract

We have studied the effect of a constant top pressure on the pressure transient analysis of a partially penetrated well in an infinite-acting fractured reservoir with wellbore storage and skin factor effects. Semi-analytical solutions of a two-dimensional diffusivity equation have been obtained by using successive applications of the Laplace and modified finite Fourier sine transforms. Both pseudo-steady-state and transient exchanges between the matrix and the fractures have been considered. Solutions are presented that can be used to generate type curves for pressure transient analysis or can be used as a forward model in parameter estimation. The presented analysis has applications in well testing of fractured aquifers and naturally fractured oil reservoirs with a gas cap.

Appendix: Derivation of Solutions

The details of the derivations for obtaining the dimensionless average pressure response of a partially penetrated well in the oil zone of an infinite-acting double-porosity reservoir including wellbore storage and skin effects are presented in this appendix.

First, we define the Laplace transforms with respect to $t_\mathrm{D}$ for dimensionless fracture and matrix pressures, respectively, as follows:

$$\begin{aligned} \bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})&= \ell _{t_\mathrm{D} } \{p_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},t_\mathrm{D})\}=\int \limits _0^{+\infty } {e^{-st_\mathrm{D} }p_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},t_\mathrm{D} ) dt_\mathrm{D} } \end{aligned}$$

(41)

$$\begin{aligned} \bar{{p}}_\mathrm{mD} (z_\mathrm{mD},\mathrm{s})&= \ell _{t_\mathrm{D} } \{p_\mathrm{mD} (z_\mathrm{mD} ,t_\mathrm{D} )\}=\int \limits _0^{+\infty } {e^{-st_\mathrm{D} }p_\mathrm{mD} (z_\mathrm{mD},t_\mathrm{D}) dt_\mathrm{D} } \end{aligned}$$

(42)

where $\bar{{p}}_\mathrm{fD}$ and $\bar{{p}}_\mathrm{mD}$ are the dimensionless fracture and matrix pressures in the Laplace domain, respectively; “s” is the variable of the Laplace transform; $\ell _{t_\mathrm{D} }$ is the sign for the Laplace transform with respect to $t_\mathrm{D}$.

After applying the Laplace transform with respect to $t_\mathrm{D}$ for the boundary conditions in Eqs. (25–29), we have:

$$\begin{aligned}&\quad \left( {\frac{\partial \bar{{p}}_{\mathrm{{fD}}} (r_{\mathrm{{D}}},z_{\mathrm{{D}}}, \mathrm{s})}{\partial r_{\mathrm{{D}}} }} \right) _{r_{\mathrm{{D}}} ={r_{\mathrm{{wD}}}}} =0\quad \quad 0<z_D <h_{\mathrm{{oD}}} -h_{\mathrm{{pD}}}\end{aligned}$$

(43)

$$\begin{aligned}&\quad \left\{ {\begin{array}{l} \bar{{p}}_{\mathrm{{fD,S}}} (r_{\mathrm{{wD}}},z_{\mathrm{{D}}},\mathrm{s})\!=\!\left[ {\bar{{p}}_{\mathrm{{fD}}} (r_{\mathrm{{D}}},z_{\mathrm{{D}}}, \mathrm{s})-h_{\mathrm{{D}}} \left( {\frac{\partial \bar{{p}}_{\mathrm{{fD}}} (r_{\mathrm{{D}}},z_{\mathrm{{D}}},\,\mathrm{s})}{\partial r_{\mathrm{{D}}}}} \right) {S}} \right] _{r_{\mathrm{{D}}} ={r_{\mathrm{{wD}}}}} \\ s C_{\mathrm{{sD}}} \bar{{p}}_{\mathrm{{fD},\mathrm{S}}} (r_{\mathrm{{wD}}},z_{\mathrm{{D}}}, \mathrm{s})-h_{\mathrm{{D}}} \left( {\frac{\partial \bar{{p}}_{\mathrm{{fD}}} (r_{\mathrm{{D}}} ,z_{D},\, \mathrm{s})}{\partial r_D }} \right) _{r_{\mathrm{{D}}} ={r_{\mathrm{{wD}}}}} =\frac{1}{\mathrm{{s}}} \\ \end{array}} \right. \, h_{\mathrm{{oD}}}\! -\!h_{\mathrm{{pD}}} <z_D \!<\!h_{\mathrm{{oD}}}\nonumber \\ \end{aligned}$$

(44)

$$\begin{aligned}&\bar{{p}}_\mathrm{fD} (+\infty ,z_\mathrm{D},\mathrm{s})=0\end{aligned}$$

(45)

$$\begin{aligned}&\bar{{p}}_\mathrm{fD} (r_\mathrm{D},\mathrm{0,s})=0\end{aligned}$$

(46)

$$\begin{aligned}&\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D}, h_\mathrm{oD},\mathrm{s})}{\partial z_\mathrm{D} }=0 \end{aligned}$$

(47)

Applying the Laplace transform with respect to $t_\mathrm{D}$ to Eq. (23) results in:

$$\begin{aligned}&\frac{\partial ^{2}\bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D}^2 }+\frac{1}{r_\mathrm{D} }\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D}, z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D} }+\frac{\partial ^{2}\bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial z_\mathrm{D}^2 } \nonumber \\&\quad =k_\mathrm{fD} \left[ {\omega \ell _{t_\mathrm{D} } \left\{ {\frac{\partial p_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},t_\mathrm{D} )}{\partial t_\mathrm{D} }} \right\} +(1-\omega ) \ell _{t_\mathrm{D} } \left\{ {\frac{\partial p_\mathrm{mD} (z_\mathrm{mD},t_\mathrm{D} )}{\partial t_\mathrm{D} }} \right\} } \right] \end{aligned}$$

(48)

Where

$$\begin{aligned} \ell _{t_\mathrm{D} } \left\{ {\frac{\partial p_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},t_\mathrm{D})}{\partial t_\mathrm{D} }} \right\} =s\bar{{p}}_\mathrm{fD} (r_\mathrm{D} ,z_\mathrm{D},\mathrm{s})-p_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},0) \end{aligned}$$

(49)

And

$$\begin{aligned} \ell _{t_\mathrm{D} } \left\{ {\frac{\partial p_\mathrm{mD} (z_\mathrm{mD} ,t_\mathrm{D} )}{\partial t_\mathrm{D} }} \right\} =s\bar{{p}}_\mathrm{mD} (z_\mathrm{mD}, \mathrm{s})-p_\mathrm{mD} (z_\mathrm{mD} ,0) \end{aligned}$$

(50)

Using Eqs. (24) and (49), Eq. (50) reduces to:

$$\begin{aligned} \ell _{t_\mathrm{D} } \left\{ {\frac{\partial p_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},t_\mathrm{D})}{\partial t_\mathrm{D} }} \right\} = \mathrm{s} \bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s}) \end{aligned}$$

(51)

And

$$\begin{aligned} \ell _{t_\mathrm{D}} \left\{ {\frac{\partial p_\mathrm{mD} (z_\mathrm{mD},t_\mathrm{D} )}{\partial t_\mathrm{D} }} \right\} =\mathrm{s} \bar{{p}}_\mathrm{mD} (z_\mathrm{mD},\mathrm{s}) \end{aligned}$$

(52)

Combining Eqs. (48), (51) and (52), we obtain:

$$\begin{aligned} \frac{\partial ^{2}\bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D}^2 }+\frac{1}{r_\mathrm{D} }\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D} }+\frac{\partial ^{2}\bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial z_\mathrm{D}^2 }\nonumber \\ =k_\mathrm{fD} [\omega _\mathrm{s} \bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})+(1-\omega )\mathrm{s} \bar{{p}}_\mathrm{mD} (z_\mathrm{mD},\mathrm{s})] \end{aligned}$$

(53)

The following relation between $\bar{{p}}_\mathrm{fD}$ and $\bar{{p}}_\mathrm{mD}$ can be obtained for the pseudo-steady-state matrix–fracture exchange (Warren and Root 1963):

$$\begin{aligned} \bar{{p}}_\mathrm{mD} (z_\mathrm{mD}, \mathrm{s})=\frac{\lambda }{\lambda +(1-\omega )\mathrm{s} }\bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s}) \end{aligned}$$

(54)

Also, the following relation between $\bar{{p}}_\mathrm{fD}$ and $\bar{{p}}_\mathrm{mD}$ can be obtained for the transient matrix–fracture exchange (Serra et al. 1983; Ozkan et al. 1987; Stewart and Asharsobbi 1988; Da Prat 1990; Sabet 1991; Al-Bemani and Ershaghi 1997; Hassanzadeh et al. 2009):

$$\begin{aligned} \bar{{p}}_\mathrm{mD} (z_\mathrm{mD}, \mathrm{s})=\frac{\tanh \left( {\sqrt{\frac{3(1-\omega )\mathrm{s} }{\lambda }}} \right) }{\sqrt{\frac{3(1-\omega )\mathrm{s} }{\lambda }}}\bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s}) \end{aligned}$$

(55)

By inserting Eqs. (54) and (55) into Eq. (53), it is possible to arrive at:

$$\begin{aligned} \frac{\partial ^{2}\bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D}^2 }+\frac{1}{r_\mathrm{D} }\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D} ,z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D} }+\frac{\partial ^{2}\bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial z_\mathrm{D}^2 }=k_\mathrm{fD} \mathrm{s} f(\mathrm{s} )\bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})\nonumber \\ \end{aligned}$$

(56)

where $f (\mathrm{s} )$ for the pseudo-steady-state matrix–fracture exchange is presented by (Warren and Root 1963):

$$\begin{aligned} f(s)=\omega +\frac{(1-\omega )\lambda }{\lambda +(1-\omega )s} \end{aligned}$$

(57)

and $f (\mathrm{s} )$ for the transient matrix–fracture exchange, assuming a slab-shaped matrix block, is given by (Serra et al. 1983; Ozkan et al. 1987; Stewart and Asharsobbi 1988; Da Prat 1990; Sabet 1991; Al-Bemani and Ershaghi 1997; Hassanzadeh et al. 2009):

$$\begin{aligned} f(\mathrm{s} )=\omega +\sqrt{\frac{\lambda (1-\omega )}{3\mathrm{s} } } \tanh \left( {\sqrt{\frac{3(1-\omega )\mathrm{s} }{\lambda }}} \right) \end{aligned}$$

(58)

where $\lambda$ is the matrix–fracture interporosity flow coefficient, defined as (Warren and Root 1963):

$$\begin{aligned} \lambda =\sigma \frac{r_\mathrm{w}^2 k_\mathrm{m} }{k_\mathrm{fh} } \end{aligned}$$

(59)

where $\sigma$ is the matrix–fracture transfer coefficient or the so-called shape factor, and $k_\mathrm{m}$ is the matrix permeability.

The modified finite Fourier sine transform with respect to $z_\mathrm{D}$ is defined for the dimensionless fracture pressure as follows:

$$\begin{aligned} \tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s})=\wp _{z_\mathrm{D} } \{\bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})\}=\int \limits _0^{h_\mathrm{oD} } {\bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})\sin (a_n z_\mathrm{D} ) dz_\mathrm{D} } \end{aligned}$$

(60)

where $\tilde{\bar{{p}}}_\mathrm{fD}$ is the modified finite Fourier sine transform of $\bar{{p}}_\mathrm{fD} ,\,n$ is the variable of the modified finite Fourier sine transform, $\wp _{z_\mathrm{D} }$ is the sign for the modified finite Fourier sine transform with respect to $z_\mathrm{D}$, and $a_{n}$ is a function of the variable of the modified finite Fourier sine transform, defined as:

$$\begin{aligned} a_n =\left( {n-\frac{1}{2}} \right) \frac{\pi }{h_\mathrm{oD} },\quad \quad \quad n=1, 2, 3, \ldots \end{aligned}$$

(61)

By applying the modified finite Fourier sine transform with respect to $z_\mathrm{D}$ for the inner and outer boundary conditions (44) and (45), which are defined in the Laplace domain, we have:

$$\begin{aligned}&\int \limits _0^{h_\mathrm{oD} } {\bar{{p}}_\mathrm{fD,S} (r_\mathrm{wD} ,z_\mathrm{D},\mathrm{s})\sin (a_n z_\mathrm{D}) dz_\mathrm{D} } =\left[ {\int \limits _0^{h_\mathrm{oD} } {[\bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})]_{r_\mathrm{D} =r_\mathrm{wD} } \sin (a_n z_\mathrm{D}) dz_\mathrm{D} } } \right] \nonumber \\&\quad -h_\mathrm{D} \left[ {\int \limits _0^{h_\mathrm{oD} } {\left( {\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D} }} \right) _{r_\mathrm{D} =r_\mathrm{wD} } \sin (a_n z_\mathrm{D}) dz_\mathrm{D} } } \right] { S} \end{aligned}$$

(62)

$$\begin{aligned}&\quad \mathrm{s} C_\mathrm{sD} \left[ {\int \limits _0^{h_\mathrm{oD} } {\bar{{p}}_\mathrm{fD,S} (r_\mathrm{wD},z_\mathrm{D},\mathrm{s})\sin (a_n z_\mathrm{D}) dz_\mathrm{D} } } \right] \nonumber \\&\qquad \qquad -h_\mathrm{D} \left[ {\int \limits _0^{h_\mathrm{oD} } {\left( {\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D}}} \right) _{r_\mathrm{D} =r_\mathrm{wD} } \sin (a_n z_\mathrm{D}) dz_\mathrm{D} } } \right] = \quad \left[ {\int \limits _0^{h_\mathrm{oD} } {\frac{1}{\mathrm{s} }\sin (a_n z_\mathrm{D}) dz_\mathrm{D}} } \right] \nonumber \\ \end{aligned}$$

(63)

$$\begin{aligned}&\qquad \qquad \int \limits _0^{h_\mathrm{oD} } {\bar{{p}}_\mathrm{fD} (+\infty ,z_\mathrm{D},\mathrm{s})\sin (a_n z_\mathrm{D}) dz_\mathrm{D} } =\tilde{\bar{{p}}}_\mathrm{fD} (+\infty ,n,\mathrm{s})=0 \end{aligned}$$

(64)

The integrals within interval 0 $< {z}_\mathrm{D}< h_\mathrm{oD}$, which includes $[{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}/{\partial r_\mathrm{D} }]_{r_\mathrm{D} =r_\mathrm{wD} }$, in Eqs. (62) and (63) are partitioned into two integrals for non-perforated interval, 0 $< z_\mathrm{D}< h_\mathrm{oD}-h_\mathrm{pD}$, and perforated interval, $h_\mathrm{oD}-h_\mathrm{pD}< z_\mathrm{D } < h_\mathrm{oD}$, as follows:

$$\begin{aligned} \tilde{\bar{{p}}}_\mathrm{fD,S} (r_\mathrm{wD},n,\mathrm{s})&= [\tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s})]_{r_\mathrm{D} =r_\mathrm{wD} } \nonumber \\&-h_\mathrm{D} \left[ \int \limits _0^{h_\mathrm{oD} -h_\mathrm{pD} } {\left( {\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D}}} \right) _{r_\mathrm{D} =r_\mathrm{wD} } \sin (a_n z_\mathrm{D}) dz_\mathrm{D} }\right. \nonumber \\&\left. +\int \limits _{h_\mathrm{oD} -h_\mathrm{pD} }^{h_\mathrm{oD} } {\left( {\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D} }} \right) _{r_\mathrm{D} =r_\mathrm{wD} } \sin (a_n z_\mathrm{D}) dz_\mathrm{D} } \right] {S} \end{aligned}$$

(65)

$$\begin{aligned}&sC_\mathrm{sD} \tilde{\bar{{p}}}_\mathrm{fD,S} (r_\mathrm{wD},n,\mathrm{s}) \nonumber \\&\quad - h_\mathrm{D} \left[ \int \limits _0^{h_\mathrm{oD} -h_\mathrm{pD} } {\left( {\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D}}} \right) _{r_\mathrm{D} =r_\mathrm{wD} } \sin (a_n z_\mathrm{D}) dz_\mathrm{D} }\right. \nonumber \\&\quad \left. +\int \limits _{h_\mathrm{oD} -h_\mathrm{pD} }^{h_\mathrm{oD} } {\left( {\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D}}} \right) _{r_\mathrm{D }=r_\mathrm{wD} } \sin (a_n z_\mathrm{D}) dz_\mathrm{D} } \right] =\frac{1}{s a_n }\qquad \end{aligned}$$

(66)

Combination of Eqs. (43), (65) and (66) results in:

$$\begin{aligned}&\tilde{\bar{{p}}}_\mathrm{fD,S} (r_\mathrm{wD},n,\mathrm{s})=[\tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s})]_{r_\mathrm{D} =r_\mathrm{wD} } \nonumber \\&\quad -h_\mathrm{D} \left[ {\int \limits _0^{h_\mathrm{oD} -h_\mathrm{pD} } {(0)\sin (a_n z_\mathrm{D}) dz_\mathrm{D} } +\int \limits _{h_\mathrm{oD} -h_\mathrm{pD} }^{h_\mathrm{oD} } {\left( {\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D} }} \right) _{r_\mathrm{D} =r_\mathrm{wD} } \sin (a_n z_\mathrm{D}) dz_\mathrm{D} } } \right] {S}\nonumber \\ \end{aligned}$$

(67)

$$\begin{aligned}&\quad sC_\mathrm{sD} \tilde{\bar{{p}}}_\mathrm{fD,S} (r_\mathrm{wD},n,\mathrm{s})-h_\mathrm{D} \left[ \int \limits _0^{h_\mathrm{oD} -h_\mathrm{pD} } {(0)\sin (a_n z_\mathrm{D}) dz_\mathrm{D} }\right. \nonumber \\&\qquad \left. +\int \limits _{h_\mathrm{oD} -h_\mathrm{pD} }^{h_\mathrm{oD} } {\left( {\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D} }} \right) _{r_\mathrm{D} =r_\mathrm{wD} } \sin (a_n z_\mathrm{D}) dz_\mathrm{D}} \right] =\frac{1}{\mathrm{s} a_n } \end{aligned}$$

(68)

By simplifying Eqs. (67) and (68), it is possible to arrive at:

$$\begin{aligned}&\tilde{\bar{{p}}}_\mathrm{fD,S} (r_\mathrm{wD},n,\mathrm{s})=[\tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s})]_{r_\mathrm{D} =r_\mathrm{wD} } -h_\mathrm{D} \nonumber \\&\quad \left[ {\int \limits _{h_\mathrm{oD} -h_\mathrm{pD} }^{h_\mathrm{oD} } {\left( {\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D} }} \right) _{r_\mathrm{D} =r_\mathrm{wD} } \sin (a_n z_\mathrm{D}) dz_\mathrm{D} } } \right] S\end{aligned}$$

(69)

$$\begin{aligned}&s C_\mathrm{sD} \tilde{\bar{{p}}}_\mathrm{fD,S} (r_\mathrm{wD} ,n,\mathrm{s})-h_\mathrm{D} \left[ {\int \limits _{h_\mathrm{oD} -h_\mathrm{pD} }^{h_\mathrm{oD} } {\left( {\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D} }} \right) _{r_\mathrm{D} =r_\mathrm{wD} } \sin (a_n z_\mathrm{D}) dz_\mathrm{D} } } \right] =\frac{1}{\mathrm{s}, a_n }\nonumber \\ \end{aligned}$$

(70)

In order to compute the integrals in Eqs. (69) and (70), it is necessary to differentiate with respect to $r_{D}$ from both sides of Eq. (60) as follows:

$$\begin{aligned} \frac{\partial \tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s})}{\partial r_\mathrm{D} }=\int \limits _0^{h_\mathrm{oD} } {\left( {\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D} ,z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D} }} \right) \sin (a_n z_\mathrm{D}) dz_\mathrm{D} } \end{aligned}$$

(71)

It is possible to write Eq. (31) at $r_\mathrm{D}=r_\mathrm{wD}$:

$$\begin{aligned} \left( {\frac{\partial \tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s})}{\partial r_\mathrm{D} }} \right) _{r_\mathrm{D} =r_\mathrm{wD} } =\int \limits _0^{h_\mathrm{oD} } {\left( {\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D} }} \right) _{r_\mathrm{D} =r_\mathrm{wD} } \sin (a_n z_\mathrm{D}) dz_\mathrm{D} } \end{aligned}$$

(72)

The integral within interval 0 $< {z}_\mathrm{D} < h_\mathrm{oD}$ in right hand side of Eq. (72) is partitioned into two different integrals for non-perforated interval, 0 $< z_\mathrm{D}< h_\mathrm{oD}-h_\mathrm{pD}$, and perforated interval, $h_\mathrm{oD}-h_\mathrm{pD} < z_\mathrm{D }< h_\mathrm{oD}$, as following:

$$\begin{aligned} \left( {\frac{\partial \tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s})}{\partial r_\mathrm{D} }} \right) _{r_\mathrm{D} =r_\mathrm{wD} }&= \int \limits _0^{h_\mathrm{oD} -h_\mathrm{pD} } {\left( {\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D} ,z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D} }} \right) _{r_\mathrm{D} =r_\mathrm{wD} } \sin (a_n z_\mathrm{D}) dz_\mathrm{D} } \nonumber \\&+\int \limits _{h_\mathrm{oD} -h_\mathrm{pD} }^{h_\mathrm{oD} } {\left( {\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D} ,z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D} }} \right) _{r_\mathrm{D} =r_\mathrm{wD} } \sin (a_n z_\mathrm{D}) dz_\mathrm{D} }\qquad \end{aligned}$$

(73)

Combination of Eqs. (43) and (73) leads to:

$$\begin{aligned} \left( {\frac{\partial \tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s})}{\partial r_\mathrm{D} }} \right) _{r_\mathrm{D} =r_\mathrm{wD} }&= \int \limits _0^{h_\mathrm{oD} -h_\mathrm{pD} } {(0)\sin (a_n z_\mathrm{D}) dz_\mathrm{D} }\nonumber \\&+\int \limits _{h_\mathrm{oD} -h_\mathrm{pD} }^{h_\mathrm{oD} } {\left( {\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D} }} \right) _{r_\mathrm{D} =r_\mathrm{wD} } \sin (a_n z_\mathrm{D}) dz_\mathrm{D} }\qquad \end{aligned}$$

(74)

After simplification, Eq. (74) turns to:

$$\begin{aligned} \left( {\frac{\partial \tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D} ,n,\mathrm{s})}{\partial r_\mathrm{D} }} \right) _{r_\mathrm{D} =r_\mathrm{wD} } =\int \limits _{h_\mathrm{oD} -h_\mathrm{pD} }^{h_\mathrm{oD} } {\left( {\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D} ,z_\mathrm{D},\mathrm{s})}{\partial r_\mathrm{D} }} \right) _{r_\mathrm{D} =r_\mathrm{wD} } \sin (a_n z_\mathrm{D}) dz_\mathrm{D}}\nonumber \\ \end{aligned}$$

(75)

Combination of Eqs. (69), (70) and (75) results in:

$$\begin{aligned} \tilde{\bar{{p}}}_\mathrm{fD,S} (r_\mathrm{wD} ,n,\mathrm{s})=\left[ {\tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s})-h_\mathrm{D} \left( {\frac{\partial \tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D} ,n,\mathrm{s})}{\partial r_\mathrm{D} }} \right) S} \right] _{r_\mathrm{D} =r_\mathrm{wD} }\end{aligned}$$

(76)

$$\begin{aligned} s C_\mathrm{sD} \tilde{\bar{{p}}}_\mathrm{fD,S} (r_\mathrm{wD} ,n,s)-h_\mathrm{D} \left( {\frac{\partial \tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D} ,n,\mathrm{s})}{\partial r_\mathrm{D} }} \right) _{r_\mathrm{D} =r_\mathrm{wD} } =\frac{1}{\mathrm{s}a_n } \end{aligned}$$

(77)

Applying the modified finite Fourier sine transform with respect to $z_\mathrm{D}$ to Eq. (56), which is defined in the Laplace domain, results in:

$$\begin{aligned} \frac{\partial ^{2}\tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D} ,n,\mathrm{s})}{\partial r_\mathrm{D}^2 }+\frac{1}{r_\mathrm{D} }\frac{\partial \tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D} ,n,s)}{\partial r_\mathrm{D} }+\wp _{z_\mathrm{D} } \left\{ {\frac{\partial ^{2}\bar{{p}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s})}{\partial z_\mathrm{D}^2 }} \right\} =k_\mathrm{fD} \mathrm{s} f(\mathrm{s}) \tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D} ,n,\mathrm{s})\nonumber \\ \end{aligned}$$

(78)

where

$$\begin{aligned} \wp _{z_\mathrm{D} } \left\{ {\frac{\partial ^{2}\bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial z_\mathrm{D}^2 }} \right\} =-a_n^2 \tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s})+a_n \bar{{p}}_\mathrm{fD} (r_\mathrm{D},0,\mathrm{s})-(-1)^{n}\frac{\partial \bar{{p}}_\mathrm{fD} (r_\mathrm{D},h_\mathrm{oD},\mathrm{s})}{\partial z_\mathrm{D}}\nonumber \\ \end{aligned}$$

(79)

By applying the boundary conditions at the top $(z_\mathrm{D} = 0)$ and bottom $(z_\mathrm{D} = h_\mathrm{oD})$ of the oil zone of an infinite-acting fractured reservoir in the Laplace domain, i.e., Eqs. (46) and (47), it is possible to simplify Eq. (79) to:

$$\begin{aligned} \wp _{z_\mathrm{D} } \left\{ {\frac{\partial ^{2}\bar{{p}}_\mathrm{fD} (r_\mathrm{D},z_\mathrm{D},\mathrm{s})}{\partial z_\mathrm{D}^2 }} \right\} =-a_n^2 \tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s}) \end{aligned}$$

(80)

After inserting Eq. (80) into Eq. (79) and replacing $\partial$ by $d$, we arrive at:

$$\begin{aligned} \frac{d^{2}\tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D} ,n,\mathrm{s})}{dr_\mathrm{D}^2 }+\frac{1}{r_\mathrm{D} }\frac{d\tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s})}{dr_\mathrm{D}}-a_\mathrm{n}^2 \tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s})=k_\mathrm{fD} \mathrm{s}f(\mathrm{s}) \tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s}) \end{aligned}$$

(81)

With some rearrangements, Eq. (81) becomes:

$$\begin{aligned} \frac{d^{2}\tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D} ,n,\mathrm{s})}{dr_\mathrm{D}^2 }+\frac{1}{r_\mathrm{D} }\frac{d\tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s})}{dr_\mathrm{D} }-\left[ {a_n^2 +k_\mathrm{fD} s f(s)} \right] \tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D} ,n,\mathrm{s})=0 \end{aligned}$$

(82)

The square root of the expression inside the bracket in Eq. (82) is defined by $\eta$, as given by:

$$\begin{aligned} \eta =\sqrt{a_n^2 +k_\mathrm{fD} \mathrm{s} f(\mathrm{s})} \end{aligned}$$

(83)

Combining Eqs. (82) and (83) results in:

$$\begin{aligned} \frac{d^{2}\tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s})}{dr_\mathrm{D}^2 }+\frac{1}{r_\mathrm{D} }\frac{d\tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s})}{dr_\mathrm{D} }-\eta ^{2} \tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s})=0 \end{aligned}$$

(84)

Equation (84) is a modified homogeneous Bessel differential equation that has the following general solution:

$$\begin{aligned} \tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D} ,n,\mathrm{s})=A_1 I_0 (\eta r_\mathrm{D} )+A_2 K_0 (\eta r_\mathrm{D}) \end{aligned}$$

(85)

where $A_{1}$ and $A_{2}$ are the first and second constants of the general solution (85), and $I_{0}$ and $K_{0}$ are the modified Bessel functions of the first and second kinds of order 0.

By applying Eq. (64) to Eq. (85), we can conclude that $A_{1} = 0$; therefore, Eq. (85) reduces to:

$$\begin{aligned} \tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s})=A_2 K_0 (\eta r_\mathrm{D}) \end{aligned}$$

(86)

By differentiation with respect to $r_\mathrm{D}$ from Eq. (86), we obtain:

$$\begin{aligned} \frac{\partial \tilde{\bar{{p}}}_\mathrm{fD} (r_\mathrm{D},n,\mathrm{s})}{\partial r_\mathrm{D} }=-A_2 \eta K_1 (\eta r_\mathrm{D}) \end{aligned}$$

(87)

where $K_{1}$ is the modified Bessel function of the second kind of order 1.

By combining Eqs. (76), (86) and (87), it is possible to arrive at:

$$\begin{aligned} \tilde{\bar{{p}}}_\mathrm{fD,S} (r_\mathrm{wD} ,n,\mathrm{s})=A_2 K_0 (\eta r_\mathrm{wD} )+{S}h_\mathrm{D} A_2 \eta K_1 (\eta r_\mathrm{wD} ) \end{aligned}$$

(88)

By combining Eqs. (77), (86) and (87), we have:

$$\begin{aligned} s C_\mathrm{sD} \tilde{\bar{{p}}}_\mathrm{fD,S} (r_\mathrm{wD} ,n,\mathrm{s})+h_\mathrm{D} A_2 \eta K_1 (\eta r_\mathrm{wD} )=\frac{1}{\mathrm{s} a_n } \end{aligned}$$

(89)

From Eqs. (88) and (89), it is possible to cancel $\tilde{\bar{{p}}}_\mathrm{fD,S} (r_\mathrm{wD},n,\mathrm{s})$, resulting in:

$$\begin{aligned} \mathrm{s} C_\mathrm{sD} A_2 \left[ {K_0 (\eta r_\mathrm{wD} )+S h_\mathrm{D} \eta K_1 (\eta r_\mathrm{wD} )} \right] +h_\mathrm{D} A_2 \eta K_1 (\eta r_\mathrm{wD} )=\frac{1}{\mathrm{s}a_n } \end{aligned}$$

(90)

From Eq. (90), we can obtain $A_{2}$ given by:

$$\begin{aligned} A_2 =\frac{1}{\mathrm{s} a_n \left[ {\mathrm{s} C_\mathrm{sD} \left[ {K_0 (\eta r_\mathrm{wD} )+S h_\mathrm{D} \eta K_1 (\eta r_\mathrm{wD} )} \right] +h_\mathrm{D} \eta K_1 (\eta r_\mathrm{wD} )} \right] } \end{aligned}$$

(91)

By combining Eqs. (88) and (91), we arrive at:

$$\begin{aligned} \tilde{\bar{{p}}}_\mathrm{fD,S} (r_\mathrm{wD} ,n,\mathrm{s})=\frac{K_0 (\eta r_\mathrm{wD} )+S h_\mathrm{D} \eta K_1 (\eta r_\mathrm{wD} )}{\mathrm{s}a_n \left[ {\mathrm{s} C_\mathrm{sD} \left[ {K_0 (\eta r_\mathrm{wD} )+Sh_\mathrm{D} \eta K_1 (\eta r_\mathrm{wD} )} \right] +h_\mathrm{D} \eta K_1 (\eta r_\mathrm{wD} )} \right] } \end{aligned}$$

(92)

The inverse modified finite Fourier sine transform with respect to $n$ is defined for $\tilde{\bar{{p}}}_\mathrm{fD,S}$ as follows:

$$\begin{aligned} \bar{{p}}_\mathrm{fD,S} (r_\mathrm{wD},z_\mathrm{D},\mathrm{s})&= \wp _n^{-1} \{\tilde{\bar{{p}}}_\mathrm{fD,S} (r_\mathrm{wD},n,\mathrm{s})\}=\frac{2}{h_\mathrm{oD} }\sum _{n=1}^{+\infty } {\tilde{\bar{{p}}}_\mathrm{fD,S} (r_\mathrm{wD},n,\mathrm{s})\sin (a_n z_\mathrm{D} ) } \nonumber \\&0<z_\mathrm{D} <h_\mathrm{oD} \end{aligned}$$

(93)

where $\wp _n^{-1}$ is the sign for the inverse modified finite Fourier sine transform with respect to $n$.

Equation (93) is a function of $z _\mathrm{D}$. To obtain a dimensionless average fracture pressure in the Laplace domain along the dimensionless perforation interval of the well, Eq. (93) must be integrated with respect to $z _\mathrm{D}$ over the limits of the dimensionless perforation interval, $h _\mathrm{oD} \hbox {-}h _\mathrm{pD} < z _\mathrm{D } < h _\mathrm{oD}$, divided by the dimensionless perforated interval of the oil zone, $h _\mathrm{pD}$. Gerami and Pooladi-Darvish (2009) used the same approach to present a 2D mathematical model for a constant-rate drawdown test performed in a partially penetrated well completed in the free-gas zone of a hydrate-capped gas reservoir during the early time production. Therefore, by using this approach it is possible to write:

$$\begin{aligned} \bar{{p}}_{\mathrm{fD},\mathrm{S}} (r_\mathrm{wD},\mathrm{s})=\frac{1}{h_\mathrm{pD} }\int \limits _{h_\mathrm{oD} -h_\mathrm{pD} }^{h_\mathrm{oD} } {\bar{{p}}_{\mathrm{fD},\mathrm{S}} (r_\mathrm{wD} ,z_\mathrm{D},\mathrm{s}) dz_\mathrm{D}} \end{aligned}$$

(94)

By substituting Eq. (93) into Eq. (94), we get:

$$\begin{aligned} \bar{{p}}_{\mathrm{fD},\mathrm{S}} (r_\mathrm{wD},\mathrm{s})=\frac{1}{h_\mathrm{pD} }\int \limits _{h_\mathrm{oD} -h_\mathrm{pD} }^{h_\mathrm{oD} } {\frac{2}{h_\mathrm{oD} }\sum _{n=1}^{+\infty } {\tilde{\bar{{p}}}_{\mathrm{fD},\mathrm{S}} (r_\mathrm{wD},n,\mathrm{s})\sin (a_n z_\mathrm{D}) } dz_\mathrm{D} } \end{aligned}$$

(95)

Eq. (95) can be simplified as:

$$\begin{aligned} \bar{{p}}_{\mathrm{fD},\mathrm{S}} (r_\mathrm{wD},\mathrm{s})=\frac{2}{h_\mathrm{oD} h_\mathrm{pD} }\sum _{n=1}^{+\infty } {\tilde{\bar{{p}}}_\mathrm{fD},S (r_\mathrm{wD},n,\mathrm{s})\int \limits _{h_\mathrm{oD} -h_\mathrm{pD} }^{h_\mathrm{oD} } {\sin (a_n z_\mathrm{D})dz_\mathrm{D}}} \end{aligned}$$

(96)

The integral in Eq. (96) is calculated as:

$$\begin{aligned} \int \limits _{h_\mathrm{oD} -h_\mathrm{pD} }^{h_\mathrm{oD} } {\sin (a_n z_\mathrm{D}) dz_\mathrm{D} } =\frac{2h_\mathrm{oD} }{(2n-1)\pi }\cos \left[ {\left( {\frac{2n-1}{2}} \right) \left( {1-h_\mathrm{D}} \right) \pi } \right] \end{aligned}$$

(97)

By replacing Eq. (97) in Eq. (96), we have:

$$\begin{aligned} \bar{{p}}_{\mathrm{fD},\mathrm{S}} (r_\mathrm{wD},\mathrm{s})=\frac{1}{h_\mathrm{pD} }\sum _{n=1}^{+\infty } {\left[ {\frac{4}{\pi (2n-1)}\cos \left[ {\left( {\frac{2n-1}{2}} \right) \left( {1-h_\mathrm{D} } \right) \pi } \right] } \right] \tilde{\bar{{p}}}_{\mathrm{fD},\mathrm{S}} (r_\mathrm{wD},n,\mathrm{s}) } \nonumber \\ \end{aligned}$$

(98)

By substituting Eq. (92) into Eq. (98), it is possible to arrive at:

$$\begin{aligned}&\bar{{p}}_{\mathrm{fD},\mathrm{S}} (r_\mathrm{wD},\mathrm{s})=\nonumber \\&\quad \frac{1}{ h_\mathrm{pD} }\sum _{n=1}^{+\infty } {\left[ {\frac{\frac{4}{\pi (2n-1)}\left[ {K_0 (\eta r_\mathrm{wD} )+S h_\mathrm{D} \eta K_1 (\eta r_\mathrm{wD} )} \right] \cos \left[ {\left( {\frac{2n-1}{2}} \right) \left( {1-h_\mathrm{D} } \right) \pi } \right] }{\mathrm{s} a_n \left[ {s C_\mathrm{sD} \left[ {K_0 (\eta r_\mathrm{wD} )+S h_\mathrm{D} \eta K_1 (\eta r_\mathrm{wD} )} \right] +h_\mathrm{D} \eta K_1 (\eta r_\mathrm{wD} )}\right] }} \right] }\nonumber \\ \end{aligned}$$

(99)

Eq. (99) is only a function of “s”. Therefore, Eq. (99) can be written as below:

$$\begin{aligned}&\bar{{p}}_{\mathrm{fD},\mathrm{S}} (\mathrm{s})= \nonumber \\&\quad \frac{1}{ h_\mathrm{pD} }\sum _{n=1}^{+\infty } {\left[ {\frac{\frac{4}{\pi (2n-1)}\left[ {K_0 (\eta r_\mathrm{wD} )+S h_\mathrm{D} \eta K_1 (\eta r_\mathrm{wD} )} \right] \cos \left[ {\left( {\frac{2n-1}{2}} \right) \left( {1-h_\mathrm{D} } \right) \pi } \right] }{\mathrm{s} a_n \left[ {\mathrm{s} C_\mathrm{sD} \left[ {K_0 (\eta r_\mathrm{wD} )+{S} h_\mathrm{D} \eta K_1 (\eta r_\mathrm{wD} )} \right] +h_\mathrm{D} \eta K_1 (\eta r_\mathrm{wD} )} \right] }} \right] }\nonumber \\ \end{aligned}$$

(100)

Equation (100) shows the dimensionless average pressure at the wellbore $(r _\mathrm{D} = r _\mathrm{wD})$ in the Laplace domain. Therefore, it is possible to replace $\bar{{p}}_{\mathrm{fD},\mathrm{S}}$ by $\bar{{p}}_\mathrm{wD}$ in Eq. (100), resulting in:

$$\begin{aligned}&\bar{{p}}_\mathrm{wD} (\mathrm{s})= \nonumber \\&\quad \frac{1}{ h_\mathrm{pD} }\sum _{n=1}^{+\infty } {\left[ {\frac{\frac{4}{\pi (2n-1)}\left[ {K_0 (\eta r_\mathrm{wD} )+S h_\mathrm{D} \eta K_1 (\eta r_\mathrm{wD} )} \right] \cos \left[ {\left( {\frac{2n-1}{2}} \right) \left( {1-h_\mathrm{D} } \right) \pi } \right] }{\mathrm{s} a_n \left[ {\mathrm{s} C_\mathrm{sD} \left[ {K_0 (\eta r_\mathrm{wD} )+S h_\mathrm{D} \eta K_1 (\eta r_\mathrm{wD} )} \right] +h_\mathrm{D} \eta K_1 (\eta r_\mathrm{wD} )} \right] }} \right] }\nonumber \\ \end{aligned}$$

(101)

By applying a numerical Laplace inversion method (Hassanzadeh and Pooladi-Darvish 2007; Mashayekhizadeh et al. 2011) for Eq. (101), the dimensionless average pressure response of the well ($p_\mathrm{wD})$ in the oil zone of an infinite-acting double-porosity reservoir, including wellbore storage and skin effects, can be obtained as follows:

$$\begin{aligned}&p_\mathrm{wD} (t_\mathrm{D} )= \nonumber \\&\quad \ell _\mathrm{s}^{-1} \left\{ {\frac{1}{ h_\mathrm{pD} }\sum _{n=1}^{+\infty } {\left[ {\frac{\frac{4}{\pi (2n-1)}\left[ {K_0 (\eta r_\mathrm{wD} )+S h_\mathrm{D} \eta K_1 (\eta r_\mathrm{wD} )} \right] \cos \left[ {\left( {\frac{2n-1}{2}} \right) \left( {1-h_\mathrm{D} } \right) \pi } \right] }{\mathrm{s} a_n \left[ {s C_\mathrm{sD} \left[ {K_0 (\eta r_\mathrm{wD} )+S h_\mathrm{D} \eta K_1 (\eta r_\mathrm{wD} )} \right] +h_\mathrm{D} \eta K_1 (\eta r_\mathrm{wD} )} \right] }} \right] } } \right\} \nonumber \\ \end{aligned}$$

(102)

where

$$\begin{aligned} p_\mathrm{wD} (t_\mathrm{D} )=\ell _\mathrm{s}^{-1} \{\bar{{p}}_\mathrm{wD} (\mathrm{s})\} \end{aligned}$$

(103)

The time derivative of the dimensionless average pressure can be applied to analyze and interpret the well test data more precisely. After replacing $p_\mathrm{fD}$ by $p_\mathrm{wD}$ and $\bar{{p}}_\mathrm{fD}$ by $\bar{{p}}_\mathrm{wD}$ in Eq. (51), we have:

$$\begin{aligned} \ell _{t_\mathrm{D} } \left\{ {\frac{\mathrm{d}p_\mathrm{wD} (t_\mathrm{D} )}{\mathrm{d}t_\mathrm{D} }} \right\} =\mathrm{s} \bar{{p}}_\mathrm{wD} (\mathrm{s}) \end{aligned}$$

(104)

Eq. (104) can be written as follows:

$$\begin{aligned} \frac{\mathrm{d}p_\mathrm{wD} (t_\mathrm{D} )}{\mathrm{d}t_\mathrm{D} }=\ell _\mathrm{s}^{-1} \{s \bar{{p}}_\mathrm{wD} (\mathrm{s})\} \end{aligned}$$

(105)

To obtain $dp_\mathrm{wD}/\mathrm{d}t_\mathrm{D}$, we need to multiply $\bar{{p}}_\mathrm{wD} (\mathrm{s})$ from Eq. (101) by “s” and then take the Laplace inverse. However, in well testing, $\mathrm{d}p_\mathrm{wD}/\mathrm{d}\hbox {ln}t_\mathrm{D}$ is used as the derivative of the dimensionless average pressure response of the well. $\mathrm{d}p_\mathrm{wD}/\mathrm{d}\hbox {ln}t_\mathrm{D}$ can be simplified as below:

$$\begin{aligned} \frac{\mathrm{d}p_\mathrm{wD} (t_\mathrm{D} )}{\mathrm{d}\ln t_\mathrm{D} }=t_\mathrm{D} \frac{\mathrm{d}p_\mathrm{wD} (t_\mathrm{D} )}{\mathrm{d}t_\mathrm{D} } \end{aligned}$$

(106)

By substituting Eq. (105) into Eq. (106), it is possible to find $\mathrm{d}p_\mathrm{wD}/\mathrm{d}\hbox {ln}t_\mathrm{D}$ as follows:

$$\begin{aligned} \frac{\mathrm{d}p_\mathrm{wD} (t_\mathrm{D} )}{\mathrm{d}\ln t_\mathrm{D} }=t_\mathrm{D} \ell _\mathrm{s}^{-1} \{\mathrm{s} \bar{{p}}_\mathrm{wD} (\mathrm{s})\} \end{aligned}$$

(107)

In this work, the Stehfest (1970) method is applied for numerical Laplace inversion.