Fhid Dynamics, Vol 31, No . 6, 1

ON THE PROPAGATION OF PRESSURE WAVES IN SATURATED POROUS MEDIA M . P Lenu

UDC 532.546

The problem of the propagation of pressure waves through compressible porous media saturated with a slightly compressible fluid is considered. By using Darcy~ law the problem is reduced to a mixed initial-boundary value problem for an equation of the heat conduction type with a nonlinear term . The method of quasi-characteristics is used to solve this equation numerically . Solutions of the wave propagation problem for media with different permeability coefficients are presented . A solution of the inverse problem of determining the permeability coefficient using wave-pulse test data is constructed on the basis of a set of solutions of the direct problem .

Recently, various physical techniques have been applied to petroleum reservoirs in order to enhance oil recovery processes . Among these techniques is wave-pulse treatment which essentially consists in acting upon the reservoir with periodic l arge-amplitude pressure pulses generated in injection wells . However, the corresponding mechanism has not yet been studied, and this is hindering the wider dissemination of the accumulated experience . In the linear formulation, which applies to small-amplitude pulses, the problem reduces to the solution of an initial-boundary value problem for the classical heat conduction equation (the 'piezoconduction equation' [1, 2]) . However, for large-amplitude pulses, which are of practical interest, it is necessary to take into account the additional nonlinear terms in the piezoconduction equation [2] . The author is unaware of any studies of the problem in the nonlinear formulation . In this paper pressure wave propagation through compressible porous media saturated with a slightly compressible fluid is studied in the nonlinear formulation . 1 . Pulse-wave treatment of a saturated porous medium generates an unsteady-state fluid flow which causes changes in the densities of both the fluid and the porous medium . In the mathematical modeling of such phenomena a linear relation between the gradients of the fluid density Pf and pressure p is usually assumed [1, 2] . The porous medium movement is neglected . The dependence of the porosity m and rock density p, on pressure is assumed to be linear . At high pressures (p >> pfl~, V being the seepage velocity vector) it is possible to use Darcy's law [4] as the momentum conservation law for the porous medium, V=-(k/µ)0p . Here, k is the medium permeability, and µ is the dynamic viscosity of the fluid. Under these assumptions, the pressure must satisfy the nonlinear equation of slightly compressible fluid flow through a compressible porous medium [1] (the nonlinear piezoconduction equation) . This equation can be written in the following dimensionless form : - Y~f(Op) 2= YDp

t Y

=

1 + L 2m(if µ t'

p =p, ap

~m

mat X'

-

pl P r af '

t=-, x=-, y=-,

z'

z= L

Here x, y, z, p, t are the dimensionless spatial coordinates, pressure and tune ; x', y', z', p', t' are dimensional variables, z is a characteristic process time, L is a characteristic lengthscale, of is the fluid compressibility coefficient, and Rr is the rock compressibility coefficient (with respect to pressure change) . The nonlinear term in Eq .(U) cannot be neglected [2] if the dimensionless pressure pulse amplitude exceeds O.OOL This is precisely the case for the processes to be studied . We will consider the spatially one-dimensional problem of pressure wave propagation in the x-direction through a porous medium saturated with a slightly compressible fluid. We will restrict our analysis to the domain 0 s x s 1 . Then the pressure to be found should satisfy the following initial-boundary value problem

Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No . 6, pp . 81-84, November-December, 1996. Original article submitted January 30, 1995 . 0015-4628/96/3106-0865$15 .00 s 1997Plenum Publishing Corporation

865



p

21i 3

Fig. 2

Fig . 1

Fig. L Prescribed rectangular pressure profile at the left boundary of the computational domain (the Po plot) and computed pressure profiles at the right boundary (plots 1-7) for different values of the piezoconductivity coefficient . Fig. 2 . Prescribed nonrectangular pressure profile at the left boundary of the computational domain (the Po plot) and computed pressure profiles at the right boundary (plots 1-7) for different values of the piezoconductivity coefficient .

- YPf

f\2 =YI

0 s x s 1:

6 Y

p(x, 0) =p, p(0, t) =P0(t)

x =0 :

at

PI

fl

P(l, t) =Pmin ,

l2-

s0

Y~ p l, () `

> 0

tax)

Here, pis a prescribed pressure in the unperturbed porous medium . Physically, the boundary condition (L5) corresponds to propagation of the pressure waves across the domain boundary without reflection . 2 . Simulation experience indicates that higher-order finite-difference schemes should be used for solving the initial-boundary value problem formulated because first-order schemes generally have high numerical viscosity, which distorts the solution wave pattern . In this study a hybrid modification of the explicit quasi-characteristic scheme [4, 5] which takes into account the characteristic properties of the nonlinear operator on the r .hs . of Eq .(L2) was used to solve the problem (L2)-(L5) . A uniform mesh was introduced in the calculation domain . The time step was chosen to satisfy the finite-difference scheme stability condition . Let us consider the results of computations based on the approach described . Figure 1 shows the pressure at the right boundary of the domain versus time p(l, t) for the case of pressure at the left boundary P0 (t) in the shape of a rectangular pulse . Figure 2 shows the similar results for the function P0(t) modeling an experimentally observed pressure pulse . The plots 1-7 correspond to the values of the coefficient y shown below 866



H

N 35.

70

2S'70

75.70 1

42

0,1

y

0,3

Fig . 4

Fig. 3

Fig. 3 . Comparison of the predictions with experiment . Fig. 4 . Maximum value of the partial time derivative of the pressure at the right boundary versus the piezoconductivity coefficient .

n yn

1 0.11636

2 0 .14546

3 0.17455

4 0.20364

5 0.23273

6 0 .26182

7 0.29091

Figure 3 presents a comparison of the predicted and experimental results for y = y t . The continuous lines correspond to the calculated data and broken lines to the experimental ones . This comparison shows that the model used gives a quite satisfactory qualitative description of the process of pressure wave propagation through saturated porous media. Figure 4 shows the maximum values pf the partial derivative (ap(l, t)/at) versus the piezoconductivity coefficient y for two types of pressure profiles at the left boundary. The plot A corresponds to a rectangular profile, and the plot B to a non-rectangular one. The derivative ap(l, t)/ar was computed using a second-order finite-difference scheme . The quantity M=max {ap(1, t)/dr} is a monotonic function of the piezoconductivity coefficient y on the parameter range considered ; the piezoconductivity itself is directly proportional to the permeability coefficient k . Hence the problem of determining the permeability coefficient using wave test data has a unique solution on the parameter range . This solution can be found using the graphs in Fig . 4 and the experimentally measured value of M . Conclusion . The results provide a theoretical basis for the possibility of experimentally determining the permeability coefficient of porous media under real conditions using wave-pulse test data. This is of importance for the estimation of the real reserves of oil and gas reservoirs as they depend primarily on the permeability coefficient . The author wishes to thank V S . Zamakhaev who drew his attention to the problem and provided the experimental data. REFERENCES 1. 2. 3. 4. 5.

Yu . P. Zheltov, Mechanics of Oil and Gas Reservoirs, [in Russian], Nedra, Moscow (1976) . G . I . Barenbalatt, V M. Entov and V M . Ryzhik, Flow of Fluids and Gases through Natural Reservoirs, [in Russian], Nedra, Moscow (1984) . A . N. Konoalov Problems of the Flow of Multiphase Incompressible Fluids through Porous Media, [in Russian], Nauka, Novosibirsk (1988) . M . P. Levin, 'A quasi-characteristic difference scheme and its application to computing supersonic gas flows', Zhurn . Vychisl. Mathematiki i Mathem . Fizild, 33, 131 (1993). M. P. Levin and M. Yu . Zheltov The quasi-characteristics scheme for two-phase flows through porous media', Comput. Fluid Dynamics 1., 2, 363 (1993).

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