HYDRAULICS

A. G. TUKAEV

ON THE PROBLEM OF DETERMINING THE PRESSURE FUNCTION IN STRATA OF VARIABLE THICKNESS UNDER ELASTIC CONDITIONS

(Presented by Academician P. Ya. Kochina on 1 VI 1960)

The unsteady motion of a homogeneous fluid according to the linear law of filtration is considered in a circular stratum of unit radius \((R = 1)\), which is exploited by means of \(n\) arbitrarily located wells.

It is required to determine the pressure \(P(r,\theta,t)\) at any moment of time (after the well is put into operation) at any point of the stratum, if on the supply contour \(r = 1\) the initial pressure \(P_0 = \mathrm{const}\) is maintained and the thickness of the stratum \(H(r,\theta) > 0\) satisfies the equation

\[ \Delta \sqrt{H} - \alpha \sqrt{H} = 0, \tag{1} \]

where \(\alpha\) is any real number and \(\Delta\) is the Laplace operator. In addition, the function \(P(r,\theta,t)\) at the well points with coordinates \((\delta_i,\theta_i)\), which are regarded as vertical linear sinks or sources, must have a singularity of the type of a heat source or sink of constant strength \(Q_i\).

If one introduces a new function \(u\), related to \(P\) by the relation

\[ P = P_0 + \frac{u}{\sqrt{H}}, \tag{2} \]

and takes (1) into account, then the problem is mathematically formulated as follows \((^{1,2})\): it is required to solve the differential equation

\[ \Delta u - \alpha u = \frac{1}{\omega}\frac{\partial v}{\partial t} \tag{3} \]

subject to the following initial and boundary conditions:

\[ u(r,\theta,0) = 0; \tag{4} \]

\[ u(1,\theta,t) = 0; \tag{5} \]

\[ \lim_{\rho \to 0} \rho \frac{\partial}{\partial \rho}\left(\frac{u}{\sqrt{H}}\right) = \mathrm{const}, \tag{6} \]

where \(\omega\) is the piezoconductivity coefficient, \(\rho = \sqrt{r^2 - 2r\delta_i \cos(\theta - \theta_i) + \delta_i^2}\) is the distance from the center of the well.

First the case is considered in which the stratum is exploited by only one fixed well with coordinates \((\delta,0)\). To solve the formulated problem, the Laplace transform with respect to the variable \(t\) is used \((^4)\)

\[ L[u(r,\theta,t)] = \bar{u}(r,\theta,q) = \int_0^\infty e^{-qt} u(r,\theta,t)\,dt. \]

The solution of equation (3), after applying the Laplace transform to it with allowance for the initial condition (4), has the form

\[ \bar u(r,\theta,q)=AK_0\!\left(\sqrt{s(r^2-2r\delta\cos\theta+\delta^2)}\right) +\sum_{m=0}^{\infty} B_m I_m(r\sqrt{s})K_m(\sqrt{s})\cos m\theta, \tag{7} \]

where \(s=\alpha+q/\omega\); \(I_m\) and \(K_m\) are the modified Bessel functions of the first and second kind of order \(m\).

The arbitrary constants \(A, B_m\) are determined according to condition (5) and the condition of constancy of the rate \(Q\), which is given by the formula

\[ \bar Q=\frac{Q}{q}=?-\frac{k}{\mu}\lim_{r_c\to0}\int_0^{2\pi} \left[H\rho\,\frac{\partial}{\partial \rho}\left(\frac{\bar u}{\sqrt{H}}\right)\right]_{\rho=r_c}\,d\varphi, \]

where \(r_c\) is the well radius, \(k\) is the permeability coefficient, and \(\mu\) is the dynamic viscosity of the liquid.

Then the transform of the function \(u(r,\theta,t)\) satisfying the boundary conditions (5), (6) will be

\[ \bar u(r,\theta,q)=\frac{\mu Q}{2\pi k\sqrt{H_c}}\,\frac{1}{q} \left[ K_0\!\left(\sqrt{s(r^2-2r\delta\cos\theta+\delta^2)}\right) -\right. \]

\[ \left. -\sum_{m=0}^{\infty}\varepsilon_m \frac{I_m(\delta\sqrt{s})K_m(\sqrt{s})}{I_m(\sqrt{s})} I_m(r\sqrt{s})\cos m\theta \right], \tag{8} \]

where \(\varepsilon_m=1\) for \(m=0\), \(\varepsilon_m=2\) for \(m\geqslant1\).

By the inversion theorem we have

\[ L^{-1}[\bar u(r,\theta,q)] =\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty} e^{qt}\bar u(r,\theta,q)\,dq. \tag{9} \]

It is easy to show that the function \(\bar u(r,\theta,q)\) has no singularities other than the countable set of poles: \(q=0\), \(q=-\omega(\beta_1^2+\alpha),-\omega(\beta_2^2+\alpha),\ldots\), where \(\beta_1,\beta_2,\ldots\) are the roots of the equation \(J_m(\beta)=0\) \((s=\alpha+q/\omega=-\beta^2)\). Therefore the line \((\gamma-i\infty,\gamma+i\infty)\) in (9) may be replaced by a circle \(\Gamma\) containing inside it all the poles of the integrand.

Finding the residues with respect to these poles and summing them, for all \(r<\delta\), we obtain

\[ u(r,\theta,t)=\frac{\mu Q}{2\pi k\sqrt{H_c}} \left[ K_0\!\left(\sqrt{\alpha(r^2-2r\delta\cos\theta+\delta^2)}\right)-\right. \]

\[ -\sum_{m=0}^{\infty}\varepsilon_m \frac{I_m(\delta\sqrt{\alpha})K_m(\sqrt{\alpha})}{I_m(\sqrt{\alpha})} I_m(r\sqrt{\alpha})\cos m\theta - \]

\[ \left. -2\sum_{m=0}^{\infty}\varepsilon_m\cos m\theta \sum_{j=1}^{\infty} \frac{J_m(\delta\beta_j)J_m(r\beta_j)} {(\beta_j^2+\alpha)J_m'^2(\beta_j)} e^{-\omega(\beta_j^2+\alpha)t} \right]; \tag{10} \]

by virtue of the symmetry of expression (10) with respect to \(r\) and \(\delta\), it is also valid for \(r>\delta\). Here the following identities were used \((^3)\):

\[ I_m(iz)=i^mJ_m(z), \qquad I_m'(iz)=i^{m-1}J_m'(z), \]

\[ K_m(iz)=\frac{1}{2}\pi(-i)^{m+1}\,[J_m(z)-iY_m(z)], \]

\[ J_m(\alpha z)Y_m'(\alpha z)-Y_m(\alpha z)J_m'(\alpha z)=\frac{2}{\pi\alpha z}, \]

where \(J_m, Y_m\) are Bessel functions of order \(m\) of the first and second kind.

By virtue of the linearity of equation (3), as well as of conditions (4)—(6), the superposition principle can be applied for \(n\) wells with coordinates \((\delta_i,\theta_i)\). Then, passing at the same time to \(P(r,\theta,t)\) with allowance for (2), we obtain the final formula for determining the pressure function in a stratum that is operated by means of \(n\) arbitrarily placed wells:

\[ P=P_0+\frac{\mu}{2\pi k\sqrt{H}}\sum_{i=1}^{n}\frac{Q_i}{\sqrt{H_i}} \left[ K_0\!\left(\sqrt{\alpha\left(r^2-2r\delta_i\cos(\theta-\theta_i)+\delta_i^2\right)}\right) -\right. \]

\[ \left. -\sum_{m=0}^{\infty}\varepsilon_m \frac{I_m(\delta_i\sqrt{\alpha})K_m(\sqrt{\alpha})}{I_m(\sqrt{\alpha})} I_m(r\sqrt{\alpha})\cos m(\theta-\theta_i) -\right. \]

\[ \left. -2\sum_{m=0}^{\infty}\varepsilon_m\cos m(\theta-\theta_i) \sum_{j=1}^{\infty} \frac{J_m(\delta_i\beta_j)J_m(r\beta_j)} {(\beta_j^2+\alpha)J_m^{\prime\,2}(\beta_j)} e^{-\omega(\beta_j^2+\alpha)t} \right]. \tag{11} \]

The formula for determining the pressure function in a circular stratum with one central well, derived by V. Yu. Kim \((^5)\) for the case when \(\sqrt{H(r)}\) is a harmonic function of one variable \(r\), is obtained from (11) under quite special assumptions. Indeed, for this it is sufficient to put \(n=1\) in (11) and pass to the limit as \(\delta_i\to0,\ \alpha\to0\). We obtain

\[ P=P_0-\frac{\mu Q}{2\pi k\sqrt{H\cdot H_0}} \left[ \ln r+2\sum_{j=1}^{\infty} \frac{J_0(r\beta_j)e^{-\omega\beta_j^2t}} {\beta_j^2J_1^2(\beta_j)} \right]. \tag{12} \]

We further note in passing that, according to the well-known thermohydrodynamic analogy, it may be considered that the solution found for the problem under study in filtration theory is simultaneously a solution of a related problem of heat-conduction theory \((^6)\).

Kazan State
Pedagogical Institute

Received
25 V 1960

CITED LITERATURE

\(^1\) M. Musket, Flow of Homogeneous Fluids in a Porous Medium, 1949.
\(^2\) G. S. Salekhov, DAN, 105, No. 6, 1174 (1955).
\(^3\) G. N. Watson, A Treatise on the Theory of Bessel Functions, Part I, IL, 1949.
\(^4\) A. V. Lykov, Theory of Heat Conduction, Moscow, 1952.
\(^5\) V. Yu. Kim, Izv. Kazan. branch of the Academy of Sciences of the USSR, Ser. phys.-math. and techn. sciences, issue 13, 23 (1959).
\(^6\) V. I. Shchelkanov, DAN, 79, No. 5, 751 (1951).