Acoustic Sounding of Hydraulic Fractures in a Low-Permeability Reservoir

Abstract

Theoretical models indicate the possibility of diagnosing the presence and the conductivity of the hydraulic fractures in reservoirs of the permeability of order milli Darcy by means of an acoustic TV set, which is a cylindrical probe of the length of several meters with a generator of impulse signals and pressure sensors. It is suggested to generate an impulse signal in a fluid, filling the gap between the probe body and the outer wall of the borehole. It is supposed that the impulse signal is generated in a fluid located in the gap between the probe body and an outer wall of the borehole. The signal evolution recorded by means of the pressure sensors as a damping of its amplitude and the appearance of the reflected burst of pressure allow one to estimate the presence and the conductivity of the fractures in the bottomhole zone. We consider fractures, which are radial or longitudinal to an open part of the borehole. The length of the impulse signal is less than the length of the probe, but exceeds the width of the gap between the probe body and the outer wall of the borehole. We take into consideration the damping of the impulse signal due to the viscosity effects in a boundary layer near the borehole walls. The width of a fracture is much less than the wavelength, and this is why the radial fracture is admitted by the reflecting surface. We provide the results of the dispersion analysis and numerical experiments on the influence of the filtration characteristics of the fractures, the width of the gap and the type of fluid on the evolution of the impulse signals in the channel.

1. Introduction

It is known that the major part of an oil field is in a final stage of developing, which is characterized by a lower permeability of reservoirs of the order centi and milli Darcy and by a high oil viscosity. A popular operation for increasing the permeability of the reservoir is the hydraulic fracturing (HF) since the emerging fractures turn out to be involved into the process of oil filtration from the reservoir to the borehole. With respect to the open part of the borehole, which can have a vertical or a horizontal location, radial and longitudinal fractures are distinguished [1,2].

In work [3], a theoretical description of the fluid flow in a fracture and in a surrounding reservoir was provided, main filtration laws were given and there were derived the main motion equations of a homogeneous fluid in a medium with an inhomogeneous permeability. There was proposed a scheme of the fluid inflow from the reservoir into a vertical fracture.

In work [4], a simultaneous solution of hydrodynamics equations and elasticity theory was considered for problems with hydraulic fractures. The influence of the fractured porous reservoir, of the oil and water characteristics and reservoir pressure on the field data was studied under the HF for the case of the horizontal fractures.

In work [5], a semi-analytic solution was proposed for a problem on non-stationary oil filtration in an unbounded area in the length and width of the reservoir, from the reservoir to the borehole, with a vertical hydraulic fracture.

In work [6], there was presented a model of propagating a small perturbation in a channel having porous and permeable walls and filled by a fluid. The influence of the filtration processes, friction and inertia forces on the perturbations evolution in the channel was studied theoretically.

In work [7,8], there were considered quantitative and qualitative features of the evolution of acoustic waves of a small amplitude in channels of a cylindrical geometry located in a permeable porous medium. A wave equation and its analytic solution were presented; the transmission and reflection of the acoustic waves were studied on the boundary of a permeable part of the wall of the channel surrounded by a zone of a smaller permeability in the form of an annulus and also by a fissure–porous reservoir.

In work [9], the waves propagation was studied in a wide frequency range along a cylindrical cavity surrounded by a permeable porous medium saturated by a non-Newtonian power–law fluid. The propagation velocity, the wave attenuation coefficient in the channel, and the penetration depth of perturbations into the surrounding porous medium are compared for the Newtonian fluid approximation.

In works [10,11], the dynamics of the pressure waves was considered in a vertical hydraulic fracture, taking into consideration the inflow of the fluid through the walls of the borehole. A mathematical model was proposed as an integral–differential equation, and there were found analytic solutions for describing the pressure evolution in the fracture, taking into consideration the filtration of the fluid into the fracture and a surrounding porous and permeable medium under various regimes of the borehole work.

The work [12] presented analytical solutions to the problem of non-stationary pressure distribution around a well intersected by a vertical fracture; the model takes into account the compressibility of the fluid in the fracture, and fluid filtration in the fracture and reservoir.

In work [13], there were proposed approximate analytic solutions in elementary functions obtained by the method of successive change of the stationary states describing the laws of the pressure propagation in the fracture under various regimes of the borehole work, and the results obtained by approximate and exact formulae were compared.

In work [14], it was shown analytically that the penetration depth of the low-frequency components of the harmonic pressure waves in the hydraulic fracture located radially with respect to the borehole in the porous and permeable medium exceeds, by an order of magnitude, the penetration depth of the perturbations from an open borehole into the reservoir with no fractures.

There is a known downhole monitoring system [15] with using distributed acoustic sensors made of optical fiber and located along the wellbore, which tracks the integral characteristics of the hydraulic fractures by the intensity level, the frequency, and the frequency spread of acoustic perturbations near the point of the rock cracking. The noisiness of the process and the sufficient laboriousness in the processing of a large amount of data stimulated the idea of creating an acoustic TV set in reservoirs of the permeability of the order milli Darcy.

Theoretical models of the acoustic TV set were proposed in works [16,17] under the assumption that the characteristic depth of the filtration perturbations in the reservoir is much less than the radius of the borehole. The present work takes into consideration the radial geometry while describing the filtration into the reservoir and a hydraulic fracture perpendicular to the wellbore, and it is a continuation and a generalization of the cited works. In contrast to the known methods in the seismic exploration and acoustic emission [18,19] used in practice for studying the filtration characteristics of the reservoir, the diagnostic wave is generated and propagates in a hydrocarbon liquid with an informative frequency range determined by the characteristics of the channel and the liquid in it, and it is approximately 160–16,000 Hz. The proposed method of the acoustic TV set is new for diagnosing the quality of the made hydraulic fracture in the reservoir in low-permeability reservoirs. In the present work, we provide only analytic and numerical solutions according to the proposed model. The theoretical approach allows us to estimate a principal possibility before experimental studies and gives recommendations for the practical usage of the new method.

2. Materials and Methods

In order to study the possibility of diagnosis the hydraulic fractures in a reservoir, we propose a theoretical model of the dynamics of the impulse signal in the gap between the cylindrical probe and an open part of the borehole with an acoustically compressible fluid, taking into consideration the filtration of the fluid into fractures, which are longitudinal or radial to the borehole, as well as the filtration into the reservoir.

We analyze the influence of the filtration characteristics of the width of the gap and the type of the fluid on the dependence of the phase velocity and the damping coefficients as well as the reflection coefficient for the radial fractures on the circular frequency by means of analytic and numerical solutions.

The results of numerical experiments, based on the fast Fourier transform algorithms, on the evolution of the impulse signal in an annular channel surrounded by a reservoir of the permeability of milli Darcy and filled by the water allow us to diagnose the presence and the conductivity of the fractures.

3. Low-Permeable Reservoir with a Longitudinal Fracture

Let (Figure 1), in an open part of a borehole of radius R in a reservoir, there be a longitudinal cylindrical probe of radius a with a generator of acoustic waves $D_1$ and pressure sensors $D_2,D_3$. The axis $Oz$ coincides with the axis of the borehole. With respect to an open part of the borehole, a homogeneous reservoir contains a longitudinal hydraulic fracture of a width $d_f$. Here, x is the coordinate directed along the longitudinal fracture, and the reference point coincides with the borehole wall. r is the radial coordinate measured from the borehole axis. Figure 1a also shows a section in a plane perpendicular to the well; the axis is directed from the well wall along the longitudinal fracture.

We consider the propagation of pressure waves of a small amplitude in an annular channel of the radius $R-a$, which is the gap between the borehole and the probe. We adopt the following convention:

(1) The borehole and the skeleton of the porous medium as well as the fracture are filled by the same acoustically compressible fluid and at the initial state the fluid is at rest.

(2) The length of the scanning wave $\lambda$ is greater than the width of the fracture $(\lambda >d_f)$ and the width of the gap $(\lambda >R-a)$ but it is much less than the length of the probe $(\lambda \ll l)$.

(3) The width of the gap is much greater than the width of the boundary layer, in which the viscosity of the fluid manifests itself under the propagation of the wave in the gap ($R-a\gg 2\sqrt{\nu /\omega },$ where $\nu$ is the kinematic viscosity coefficient of the fluid and $\omega$ is a circular frequency).

The system of the main equations describing the propagation of a perturbation in the gap between the body of the probe and the borehole is the mass conservation and momentum conservation laws, and in the linearized approximation, it reads as follows [6,7]:

$$\pi \left(R^2-a^2\right)\left(\frac{1}{C^2}\frac{\partial P}{\partial t}+{\rho }_0\frac{\partial w}{\partial z}\right)=-2\pi R{\rho }_0{u}_p-2d_f{\rho }_0{u}_f,$$

(1)

$${\rho }_0\frac{\partial w}{\partial t}+\frac{\partial P}{\partial z}=-\frac{2\sigma }{R-a},\sigma ={\rho }_0\sqrt{\frac{\nu }{\pi }}\underset{-\infty }{\overset{t}{\int }}\frac{\left(\partial w/\partial \tau \right)}{\sqrt{t-\tau }}d\tau (\nu =\mu /{\rho }_0)$$

(2)

Here, P and w are the perturbations of the pressure and velocity; $u_p$ and $u_f$ are the velocities of the filtration from the permeable wells of the borehole into the reservoir and the fracture; ${\rho }_0$ is the density of the fluid in an unperturbed state; C is the sound velocity in the fluid; $\mu$ is the dynamical viscosity of the fluid; $\sigma$ is the tangential stress on the channel walls [20]; R is the radius of the well; and a is the radius of the probe.

In order to describe the outflow of the fluid from the permeable wall of the borehole under the propagation of the pressure waves in the gap, we write the equations of filtration into the porous medium around the borehole and the longitudinal fracture:

$$u_p=-\frac{k_p}{\mu }\frac{\partial P_p}{\partial r},\frac{\partial P_p}{\partial t}=\frac{{\varkappa }_p}{r}\frac{\partial }{\partial r}\left(r\frac{\partial P_p}{\partial r}\right),(R<r<\infty )$$

(3)

$$\begin{array}{c}\hfill u_f=-\frac{k_f}{\mu }\frac{\partial P_f}{\partial x},\\ \hfill \frac{\partial P_f}{\partial t}={\varkappa }_f\frac{{\partial }^2{P}_f}{\partial x^2}-2\frac{m_p}{m_f}\frac{\sqrt{{\varkappa }_p}}{d_f}\underset{-\infty }{\overset{t}{\int }}\frac{\partial P_f\left(\tau ,x\right)/\partial \tau }{\sqrt{\pi \left(t-\tau \right)}}d\tau ,(0<x<\infty )\end{array}$$

(4)

where $P_p$ and $P_f$ are the perturbation of the pressure in the reservoir and fracture, $m_s,k_s$ and ${\varkappa }_s=k_s{\rho }_0{C}^2/m_s\mu (s=f,p)$ are the porosity, the permeability and the piezoconductivity coefficients [21]. We note that the filtration equation in the fracture (4) was obtained in work [10]. Here and below, the subscript p corresponds to the parameters of the porous medium, and f corresponds to a fracture.

Initial and boundary conditions for Equations (1)–(4) are

$$P{\mid }_{t\to -\infty }=0,w{\mid }_{t\to -\infty }=0$$

(5)

$$P{\mid }_{z=0}=f\left(t\right)$$

(6)

$$P_p\left(r\to +\infty \right)=0,P_p\left(r=R\right)=P(t,z)$$

(7)

$$P_f\left(x\to +\infty \right)=0,P_f\left(x=0\right)=P(t,z)$$

(8)

Condition (5) means that the influence of the initial data is "forgotten" over time [22]. Condition (6) defines the pressure momentum at the input section of the gap $(z=0)$.

We seek solutions to Equations (1)–(4) as damped harmonic waves [23]:

$$\begin{array}{c}\hfill P,w=A^{\left(p\right)}\left(z\right),A^{\left(w\right)}\left(z\right)\left[exp\left[-i\left(\omega t-Kz\right)\right]\right]\\ \hfill P_p,u_p=A_p^{\left(p\right)}(z,r),A_p^{\left(u\right)}(z,r)\left[exp\left[-i\left(\omega t-Kz\right)\right]\right]\\ \hfill P_f,u_f=A_f^{\left(p\right)}(z,x),A_f^{\left(u\right)}(z,x)\left[exp\left[-i\left(\omega t-Kz\right)\right]\right]\end{array}$$

(9)

Here, $\omega$ is the circular frequency, $K=k+i\delta$ is the wave vector, $C_p$ and $\delta$ are the phase velocity and the damping coefficient. Here, $A^{\left(p\right)}$ is the amplitude of the parameter P, $A^{\left(w\right)}$ is the amplitude of the parameter w, and $A^{\left(u\right)}$ is the amplitude of the parameter u.

Substituting (9) into system (1) and (2), we obtain

$$-\frac{i\omega }{C^2}{A}^{\left(p\right)}+iK{\rho }_0{A}^{\left(w\right)}=-\frac{2R{\rho }_0}{{\tilde{R}}^2}{A}_p^{\left(u\right)}\left(R\right)-\frac{2d_f{\rho }_0}{\pi {\tilde{R}}^2}{A}_f^{\left(u\right)}\left(0\right)$$

(10)

$$-{\rho }_0\omega A^{\left(w\right)}+K{A}^{\left(p\right)}=\frac{2\sqrt{{\rho }_0\mu }}{R-a}\sqrt{i\omega }{A}^{\left(w\right)},{\tilde{R}}^2=R^2-a^2,\sqrt{i}=\frac{1}{\sqrt{2}}(1+i)$$

(11)

Substituting (9) into (3) and (4), we find an equation for the distribution of the amplitudes of the perturbations of the pressure and the velocity in the reservoir around the borehole and in the hydraulic fracture:

$$A_p^{\left(u\right)}\left(r\right)=-\frac{k_p}{\mu }\frac{d{A}_p^{\left(p\right)}\left(r\right)}{dr}\mathrm{and}\frac{1}{r}\frac{d}{dr}\left(r\frac{d{A}_p^{\left(p\right)}\left(r\right)}{dr}\right)={{\alpha }_p}^2{A}_p^{\left(p\right)}\left(r\right),$$

(12)

$$A_f^{\left(u\right)}\left(x\right)=-\frac{k_f}{\mu }\frac{d{A}_f^{\left(p\right)}\left(x\right)}{dx}\mathrm{and}\frac{d^2{A}_f\left(x\right)}{d{x}^2}={{\alpha }_f}^2{A}_f^{\left(p\right)}\left(x\right),$$

(13)

where ${{\alpha }_p}^2={\displaystyle \frac{-i\omega }{{\varkappa }_p}},{\alpha }_f^2=-i\left({\displaystyle \frac{\omega }{{\varkappa }_f}}+2{\displaystyle \frac{m_p}{m_f}}{\displaystyle \frac{\sqrt{{\varkappa }_p}}{{\ae }_f}}{\displaystyle \frac{\sqrt{i\omega }}{d_f}}\right)$.

It follows from boundary conditions (7), (8) that

$$A_p^{\left(p\right)}(z,R)=A^{\left(p\right)}\left(z\right)\mathrm{and}{A}_p^{\left(p\right)}(z,r\to \infty )\to 0$$

$$A_f^{\left(p\right)}(z,0)=A^{\left(p\right)}\left(z\right)\mathrm{and}{A}_f^{\left(p\right)}(z,x\to \infty )\to 0$$

Then for the solution of Equation (12), we can write

$$A_p^{\left(p\right)}(z,r)=A^{\left(p\right)}\left(z\right)\frac{K_0\left({\alpha }_{p}r\right)}{K_0\left({\alpha }_{p}R\right)}\mathrm{and}{A}_p^{\left(u\right)}(z,r)=\frac{k_p}{\mu }{\alpha }_p{A}^{\left(p\right)}\left(z\right)\frac{K_1\left({\alpha }_{p}r\right)}{K_0\left({\alpha }_{p}R\right)}$$

$$\left(K_1\left(S\right)=-\frac{d{K}_0\left(S\right)}{dS},K_0\left(S\right)={\int }_0^{\infty }exp(-Scosh\xi )d\xi \right)$$

where $K_0\left(S\right)$ and $K_1\left(S\right)$ are the McDonald functions of zero and first order [22].

In the same way for the solution of Equation (13), we can obtain

$$A_f^{\left(p\right)}(z,x)=A^{\left(p\right)}\left(z\right)exp\left[-{\alpha }_{f}x\right],A_f^{\left(u\right)}(z,x)=\frac{k_f}{\mu }{\alpha }_f{A}_{}^{\left(p\right)}\left(z\right)exp\left[-{\alpha }_{f}x\right]$$

Then Equation (10) becomes

$$-\frac{i\omega }{C^2}{A}^{\left(p\right)}+iK{\rho }_0{A}^{\left(w\right)}=-\frac{2R{\rho }_0}{{\tilde{R}}^2}\frac{k_p}{\mu }{\alpha }_p\frac{K_1\left({\alpha }_{p}R\right)}{K_0\left({\alpha }_{p}R\right)}{A}^{\left(p\right)}-\frac{2d_f{\rho }_0}{\pi {\tilde{R}}^2}\frac{k_f}{\mu }{\alpha }_f{A}^{\left(p\right)}$$

(14)

By the conditions on the existence of a nontrivial solutions and for Equations (11) and (14), we obtain a dispersion equation:

$$\begin{array}{c}\hfill K=\pm \frac{\omega }{C}{\left(1+\frac{2}{R-a}\sqrt{\frac{i\nu }{\omega }}\right)}^{1/2}\times \\ \hfill \times {\left(1+\frac{2R{m}_p}{{\tilde{R}}^2}\sqrt{\frac{i{\varkappa }_p}{\omega }}\frac{K_1\left({\alpha }_{p}R\right)}{K_0\left({\alpha }_{p}R\right)}+\frac{{2d}_f{m}_f}{\pi {\tilde{R}}^2}\sqrt{\frac{i{\varkappa }_f}{\omega }}\sqrt{1+\frac{{2m}_p}{d_f{m}_f}\sqrt{\frac{i{\varkappa }_p}{\omega }}}\right)}^{1/2}\end{array}$$

(15)

Here, the signs (+) and (−) in the right-hand side correspond to the waves propagating from the left to the right and from the right to the left.

4. Low-Permeable Reservoir with a Radial Fracture

Suppose that there exists a radial hydraulic fracture with respect to the open part of the borehole and it is located in section $z=z_f$, Figure 1b, and axis $Or$ is directed from the borehole wall along the fracture. Other assumptions are the same as in Section 3.

The system of the main equations 2describing the propagation of the impulse signal in the gap coincides with (2) and (3), while in Equation (1), the second term in the right-hand side is absent. Similar to Section 3, for damped harmonic waves of form (9), we obtain the dispersion equation:

$$K=\pm \frac{\omega }{C}{\left(\left(1+\frac{2}{R-a}\sqrt{\frac{i\nu }{\omega }}\right)\left(1+\frac{2R{m}_p}{{\tilde{R}}^2}\sqrt{\frac{i{\varkappa }_p}{\omega }}\frac{K_1\left({\alpha }_{p}R\right)}{K_0\left({\alpha }_{p}R\right)}\right)\right)}^{1/2}$$

(16)

We note that for the case of a cased borehole, the dispersion equation reads as [7,8,9]

$$K=\pm \frac{\omega }{C}{\left(1+\frac{2}{R-a}\sqrt{\frac{i\nu }{\omega }}\right)}^{1/2}$$

(17)

We write the conditions on the section of the fracture $(z=z_f)$ treated as a reflecting surface as

$$P_{(-)}=P_{(+)}=P_f,\pi {\tilde{R}}^2{\rho }_0{w}_{(-)}=2\pi R{\rho }_0{d}_f{u}_f+\pi {\tilde{R}}^2{\rho }_0{w}_{(+)},$$

(18)

where the second equation follows from the mass conservation law and the subscripts (+) and (−) correspond to the perturbations of the parameters before and after the reflecting surface. The first term in the right-hand side of the second Equation (18) takes into consideration the filtration of the fluid into the fracture. In order to find $P_f$ and $u_f$, we need to consider the filtration problem for the fracture. Under the radial geometry of the fracture, we write the fluid filtration equation with the boundary conditions as

$$u_f=-\frac{k_f}{\mu }\frac{\partial P_f}{\partial r},\frac{\partial P_f}{\partial t}=\frac{{\varkappa }_f}{r}\frac{\partial }{\partial r}\left(r\frac{\partial P_f}{\partial r}\right)-2\frac{m_p}{m_f}\frac{\sqrt{{\varkappa }_p}}{d_f}\underset{-\infty }{\overset{t}{\int }}\frac{\partial P_f\left(\tau ,r\right)/\partial \tau }{\sqrt{\pi \left(t-\tau \right)}}d\tau$$

(19)

$$P_f\left(r\to +\infty \right)=0,P_f\left(r=R\right)=P(t,z_f),R<r<\infty$$

For damped harmonic waves in the borehole, we seek a solution of problem (19) as

$$P_f,u_f=A_f^{\left(p\right)}(z_f,r),A_f^{\left(u\right)}(z_f,r)\left[exp\left[-i\left(\omega t-K{z}_f\right)\right]\right]$$

and we obtain

$$A_f^{\left(u\right)}(z_f,r)=-\frac{k_f}{\mu }\frac{d{A}_f^{\left(p\right)}(z_f,r)}{dr}\mathrm{and}\frac{1}{r}\frac{d}{dr}\left(r\frac{d{A}_f^{\left(p\right)}(z_f,r)}{dr}\right)={{\alpha }_f}^2{A}_f^{\left(p\right)}(z_f,r),$$

$$A_f^{\left(p\right)}(z_f,R)=A^{\left(p\right)}\left(z_f\right)\mathrm{and}{A}_f^{\left(p\right)}(z_f,r\to \infty )\to 0$$

This yields

$$A_f^{\left(p\right)}(z_f,r)=A^{\left(p\right)}\left(z_f\right)\frac{K_0\left({\alpha }_{f}r\right)}{K_0\left({\alpha }_{f}R\right)},A_f^{\left(u\right)}(z_f,r)=\frac{k_f}{\mu }{\alpha }_f{A}^{\left(p\right)}\left(z_f\right)\frac{K_1\left({\alpha }_{f}r\right)}{K_0\left({\alpha }_{f}R\right)},$$

(20)

$${\alpha }_f^2=-i\left(\frac{\omega }{{\varkappa }_f}+2\frac{m_p}{m_f}\frac{\sqrt{{\varkappa }_p}}{{\varkappa }_f}\frac{\sqrt{i\omega }}{d_f}\right)$$

Supposing that the harmonic wave approaching section $z=z_f$ of the fracture splits into the reflected and transmitted wave, that is, on the segment $0<z<z_f$, there exist two waves (incident wave and reflected wave) corresponding to the signs (+) and (−) in the right-hand side of the dispersion Equation (16), while on the segment $z>z_f$, there is just one wave (transmitted wave) corresponding to the sign (+). For these waves of form (9), under the incidence near the section of the fracture $z=z_f$, we can write

$$\begin{array}{cc}\hfill & P_{\left(O\right)}=A_{\left(O\right)}^{\left(p\right)}{e}^{-i\left(\omega t-K(z-z_f)\right)},P_{\left(R\right)}=A_{\left(R\right)}^{\left(p\right)}{e}^{-i\left(\omega t-K(z-z_f)\right)},\hfill \\ \hfill & P_{\left(G\right)}=A_{\left(G\right)}^{\left(p\right)}{e}^{-i\left(\omega t-K(z-z_f)\right)},w_{\left(O\right)}=A_{\left(O\right)}^{\left(w\right)}{e}^{-i\left(\omega t-K(z-z_f)\right)},\hfill \\ \hfill & w_{\left(R\right)}=A_{\left(R\right)}^{\left(w\right)}{e}^{-i\left(\omega t-K(z-z_f)\right)},w_{\left(G\right)}=A_{\left(G\right)}^{\left(w\right)}{e}^{-i\left(\omega t-K(z-z_f)\right)}\hfill \end{array}$$

(21)

$$P_{(-)}=P_{\left(O\right)}+P_{\left(R\right)},P_{(+)}=P_{{}^{\left(G\right)}},w_{(-)}=w_{\left(O\right)}+w_{\left(R\right)},w_{(+)}=w_{{}^{\left(G\right)}}$$

(22)

Here, the subscripts (+) and (−) indicate the values of the perturbations of the pressure and velocity before and after the reflecting surface $(z=z_f)$, while the subscripts $\left(O\right),\left(R\right)$ and $\left(G\right)$ correspond to the incident, reflected and transmitted waves.

From the first condition (18) in view of (22), it follows for solutions of form (21) that

$$A_{\left(O\right)}^{\left(p\right)}+A_{\left(R\right)}^{\left(p\right)}=A_{\left(G\right)}^{\left(p\right)}$$

(23)

Then, on the base of momentum Equation (2) and in view of (16) for the amplitudes of the velocities $A_{\left(O\right)}^{\left(w\right)},A_{\left(R\right)}^{\left(w\right)}$ and $A_{\left(G\right)}^{\left(w\right)}$ corresponding to the incident, transmitted and reflected waves, we can write

$$A_{\left(J\right)}^{\left(w\right)}=\pm \frac{A_{\left(J\right)}^{\left(P\right)}}{{\rho }_{0}C}{\left(\frac{1+{\displaystyle \frac{2R{m}_p}{{\tilde{R}}^2}}\sqrt{{\displaystyle \frac{i{\varkappa }_p}{\omega }}}{\displaystyle \frac{K_1\left({\alpha }_{p}R\right)}{K_0\left({\alpha }_{p}R\right)}}}{1+{\displaystyle \frac{2}{R-a}}\sqrt{{\displaystyle \frac{i\nu }{\omega }}}}\right)}^{1/2}$$

(24)

Here the plus sign corresponds to $J=O,G,$ the minus sign does to $J=R$.

By the second boundary condition (18) and in view of expression (24), we can obtain the expression for the reflection coefficient $N\left(N=A_{\left(R\right)}^{\left(P\right)}/A_{\left(O\right)}^{\left(P\right)}\right)$ and the transmission coefficient $M\left(M=A_{\left(G\right)}^{\left(P\right)}/A_{\left(O\right)}^{\left(P\right)}\right)$:

$$\begin{array}{c}\hfill N=M-1,\\ \hfill M={\left(1+\frac{R{m}_f}{{\tilde{R}}^2}\frac{{\varkappa }_f}{C}{d}_f{\alpha }_f\frac{K_1\left({\alpha }_{f}R\right)}{K_0\left({\alpha }_{f}R\right)}{\left(\frac{1+{\displaystyle \frac{2}{R-a}}\sqrt{{\displaystyle \frac{i\nu }{\omega }}}}{1+{\displaystyle \frac{2R{m}_p}{{\tilde{R}}^2}}\sqrt{{\displaystyle \frac{i{\varkappa }_p}{\omega }}}{\displaystyle \frac{K_1\left({\alpha }_{p}R\right)}{K_0\left({\alpha }_{p}R\right)}}}\right)}^{1/2}\right)}^{-1}\end{array}$$

(25)

5. Numerical Analysis

The dynamics of the impulse signal in the annular cylindrical channel is studied by means of the Fourier transform [22], and the fast transform program is employed for the numerical realization [24,25]. As a diagnostic signal, at a time $t=t_0$, a bell-shaped signal of a duration $\Delta t$ with an amplitude $\Delta P_0$ is sent from $D_1$, having the coordinate $(z=0)$

$${\tilde{P}}^{\left(0\right)}=\Delta P_{0}exp\left(-{\left(\frac{t-t_0}{\Delta t/6}\right)}^2\right)$$

(26)

Using the Fourier transform, for the signal approaching section z of the channel, we have

$$P(z,t)=\frac{1}{\pi }\underset{0}{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}{\tilde{P}}^{\left(0\right)}\left(\tau \right)exp\left(iK\left(\omega \right)z\right)exp\left[i\omega (t-\tau )\right]d\omega d\tau$$

(27)

Under the presence of the radial fracture the incident, reflected and transmitted signals on the boundary $z=z_f$ are written as

$$\begin{array}{c}\hfill P_{\left(O\right)}(z_f,t)=\frac{1}{\pi }\underset{0}{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}{\tilde{P}}^{\left(0\right)}\left(\tau \right)exp\left(iK\left(\omega \right)z_f\right)exp\left[i\omega (t-\tau )\right]d\omega d\tau \end{array}$$

(28)

$$\begin{array}{c}\hfill P_{\left(R\right)}(z_f,t)=\frac{1}{\pi }\underset{0}{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}{P}_{\left(O\right)}(z_f,\tau )N\left(\omega \right)exp\left[i\omega (t-\tau )\right]d\omega d\tau \end{array}$$

(29)

$$\begin{array}{c}\hfill P_{\left(G\right)}(z_f,t)=\frac{1}{\pi }\underset{0}{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}{P}_{\left(O\right)}(z_f,\tau )M\left(\omega \right)exp\left[i\omega (t-\tau )\right]d\omega d\tau \end{array}$$

(30)

It follows from dependence (27) in view of (15) and from dependencies (28)–(30) in view (16), (25) that the propagation of the impulse signal in the gap goes with a dispersion under the presence of hydraulic fractures radial and longitudinal to the borehole.

Figure 2 demonstrates the influence of the conductivity $(C_f=d_f{k}_f)$ of the longitudinal hydraulic fracture in a low-permeable reservoir on the dispersion dependence according to Equation (15). The lines 1, 2 and 3 correspond to the fracture of the porosity $m_f=0.2$ and conductivities $C_f={10}^{-13},5·{10}^{-13}\mathrm{and}{10}^{-12}{\mathrm{m}}^3.$ The admitted parameters of the borehole and reservoir are as follows: $R=7.5·{10}^{-2}\mathrm{m},a=5.5·{10}^{-2}\mathrm{m};m_p=0.1;k_p={10}^{-15}{\mathrm{m}}^2.$ The borehole and the reservoir are filled by (a) water and (b) oil with the physical parameters ${\rho }_0={10}^3,890\mathrm{kg}/{\mathrm{m}}^3;C=1.5·{10}^3,1.23·{10}^3\mathrm{m}/\mathrm{s};\mu ={10}^{-3},2·{10}^{-2}\mathrm{Pa}·\mathrm{s}.$ We see that the curve of the phase velocity in the reservoir with a fracture of a low permeability (curve 1) almost coincides with the curve in the reservoir without fracture (dot-dash line), and the differences in the damping coefficients under the presence (curve 1) and absence of the fracture (dot line) are inessential both in the water and the oil. However, for the conductivity of the fracture $C_f\ge 2·{10}^{-13}{\mathrm{m}}^3$, we observe the decreasing of the phase velocity in the field of low frequencies and the increasing of the damping coefficient, which is more than 20 times for the water and more than 5 times for the oil. For instance, for the circular frequency $\omega ={10}^4{\mathrm{s}}^{-1}$, in the channel filled by the water (oil) under the presence of the fracture, the damping coefficient is approximately equal to $\delta =3·{10}^{-2}{\mathrm{m}}^{-1}(\delta =2·{10}^{-2}{\mathrm{m}}^{-1})$, while in the reservoir with a fracture of the conductivity $C_f=5·{10}^{-13}\mathrm{and}{10}^{-12}{\mathrm{m}}^3-8·{10}^{-1}\mathrm{and}2{\mathrm{m}}^{-1}(\delta ={10}^{-1}\mathrm{and}2·{10}^{-1}{\mathrm{m}}^{-1})$. The value of the phase velocity for the circular frequency $\omega ={10}^4{\mathrm{s}}^{-1}$ under the presence of the fracture of the conductivity $C_f={10}^{-13}\mathrm{and}{10}^{-12}{\mathrm{m}}^3$ is approximately $C=1460\mathrm{and}1360\mathrm{m}/\mathrm{s}$ for the water and $C=1220\mathrm{and}1210\mathrm{m}/\mathrm{s}$ for the oil.

Figure 3 demonstrates the dynamics of the impulse signal of the duration ($\Delta t={10}^{-4}\mathrm{s}$) in the channel. The parameters coincide with those on Figure 2. The length of the probe is $l=6\mathrm{m}$, while the sensors $D_1,D_2\mathrm{and}{D}_3$ have coordinates $z=0;2\mathrm{and}4\mathrm{m}$. The amplitude decreasing phenomenon in the channel under the presence of the fractures in the reservoir longitudinal to the borehole is more noticeable for the water than for the oil in the channel.

Figure 4 shows the influence of the conductivity of the radial fracture in the reservoir on the dependence of the absolute value and the argument of the reflection coefficient in accordance with Equation (23). We see that the growth of the amplitude of the reflected signal occurs under the increasing of the conductivity of the fracture for the water, while for the oil, this is not sufficient. For instance, for the circular frequency $\omega ={10}^4{\mathrm{s}}^{-1}$, the amplitude of the reflected signal under the presence of the fracture of the conductivity $C_f=5·{10}^{-13}{\mathrm{m}}^3$ for the water is approximately 12% of the amplitude of the scanning signal, while for the oil, this is just about 3%.

As we see in Figure 5, the sensibility of the method for the water in the channel can be increased by an appropriate choice of the duration of the signal and the width of the gap. For instance, for the duration of the signal $\Delta t=6·{10}^{-4}\mathrm{s}$ and the fracture of the conductivity $C_f=5·{10}^{-13}{\mathrm{m}}^3$ the amplitude of the reflected signal does not exceed 15% of the amplitude of the scanning signal even under the decreasing of the width of the gap. However, for shorter signals ($\Delta t=6·{10}^{-5}\mathrm{s}$), the decreasing of the width of the gap in two times from $R-a=4\mathrm{cm}$ to $R-a=2\mathrm{cm}$ produces the increasing of the amplitude of the reflected signal approximately into five times from 5% to 25%. It follows from the calculations that under the duration of the signal $\Delta t=6·{10}^{-5}\mathrm{s}$ and the width of the gap $R-a=2\mathrm{cm}$ for the fractures of the conductivity $C_f=5·{10}^{-13}{\mathrm{m}}^3$, the echo of the reflected signal is more than 25% of the scanning signal.

In Figure 6, we show the influence of the conductivity of the fracture on the dynamics of the signal in the channel filled by the water. The parameters for the borehole and the reservoir coincide with those for Figure 2. Oscillograms 1, 2 and 3, 4 are the readings of the algorithmic sensors $D_1,D_2\mathrm{and}{D}_3,D_4$ (Figure 1b) with the coordinates $z=0;2;2;4\mathrm{m}$. On oscillogram $D_1$, the first burst is the initial impulse, which approaches the section of the fracture a bit weakened; this is the first burst on the oscillogram $D_2$. The impulse reflected from the section of the fracture is fixed on oscillogram $D_2$, as well as one returned back to the sensor $D_1$ (the second burst). On the calculated oscillograms, the amplitude and duration of the initial pulse are taken as scales along the pressure and time axes. We see that the fractures of a sufficient conductivity (curves 2 and 3) form the unloading impulse with the decreasing of the amplitude of the signal returned back to the producing site. However, the fracture of a low conductivity (curve 1) produces a weak echo, which, due to the damping, almost do not reach the beginning of the scanned part of the borehole.

We note that for the permeability of the reservoir of the order centi Darcy, the given calculations are identical.

6. Conclusions

We propose a method of acoustic TV set for diagnosing filtration characteristics of hydraulic fractures in reservoirs of the permeability of order milli Darcy, which, in our opinion, is simpler and less costly than the known methods used in the seismic exploration. The analytic and numerical solutions obtained in the work show that by the evolution of the impulse signal in the liquid between the body of the cylindrical probe and the outer well of the borehole, one can estimate the permeability of the fractures and hence the quality of the made hydraulic fracturing of the reservoir. In the case of the fractures longitudinal to the borehole, this parameter is determined by the damping of the impulse signal, while in the case of the radial fractures, it is determined by the amplitude of the signal reflected from the section of the borehole with the radial fracture. We establish that if the fracture is absent, then there is almost no damping of the impulse signal of the duration of order 0.1 ms on the distances of order 2–4 m, while under the presence of the longitudinal fracture of the conductivity $C_f\ge 2·{10}^{-13}{\mathrm{m}}^3$ the decreasing of the amplitude is more than 3 times. For the fractures radial to the borehole, under the same parameters of the reservoir and the fracture, the unloading impulse is formed, and the amplitude of the reflected signal is 4–6 times less than the amplitude of the initial signal. We find that once the gap between the body of the probe and the outer wall of the borehole becomes smaller, the sensibility of the signal on the presence of the hydraulic fracturing fractures increases. The phenomena found in the work are more noticeable for water than for oil.