A subgrid model for the flow in fractal porous media (Q2760698)

The authors study the flow of incompressible fluid through fractal porous rock. Physically, the problem is formulated as follows: the incompressible fluid steadily flows through a medium with varying permeability coefficient \(\varepsilon(\vec x)\), the velocity is given by Darcy law \(\vec v = \varepsilon(\vec x)\nabla p\), \(p\) stands for the pressure, and pressure is governed by the equation \(\frac{\partial}{\partial x_j} [\varepsilon(\vec x)\frac{\partial} {\partial x_j}p(\vec x)] = 0\). The authors assume that there are fluctuations of \(\varepsilon\) in an extremely wide range of scales. In this case the following question can be posed: is it possible to derive an equation which describes only the fluctuating field in the largest scale from the above-mentioned equation for pressure? Similar problems arise in various fields of physics, in particular, in the theory of turbulence where the main information provide the pulsations in the largest scale.NEWLINENEWLINENEWLINETo solve the problem, the authors decompose the fluctuating permeability coefficient \(\varepsilon(\vec x)\) into two components. The large-scale component of \(\varepsilon(\vec x)\) is obtained via the so-called spatial smoothing, where a Fourier filter makes it possible to neglect all Fourier harmonics whose wavelength is shorter than some threshold value \(l\). The short wave (subgrid) component of \(\varepsilon(\vec x)\) is given by the formula \(\varepsilon' = \varepsilon - \varepsilon_l\). The authors seek the large-scale pressure field \(p_l(\vec x)\) as a solution to the equation for \(p\), where the large scale component \(\varepsilon_l\) is fixed or, in other words, \(p_l(\vec x) = \langle p_l(\vec x)\rangle_l\). The corresponding filtered equation has the form NEWLINE\[NEWLINE \frac{\partial}{\partial x_j}\left[\varepsilon_l(\vec x)\frac{\partial} {\partial x_j}p_l(\vec x) + \left\langle\varepsilon'(\vec x),\frac{\partial} {\partial x_j}p'(\vec x)\right\rangle_l\right] = 0, NEWLINE\]NEWLINE where the choice of the second term determines the subgrid model.NEWLINENEWLINENEWLINEThe final part of the paper presents a derivation of a subgrid model using the scale arguments, and validates the invariance hypothesis for fractal porous media.