## 2 Statement of Problem  
A porous medium considered consists of two parts: (1) macroporous cylindrical medium (macropore) with a radius \(a\) (i.e., region \(\Omega_{1}\)\(\{0\leq x<\infty,\ 0\leq r\leq a\}\)), with large pores, characterized by a relatively high porosity and average fluid velocity in it, (2) surrounding cylindrical microporous medium (micropore) occupying the region \(\Omega_{2}\)\(\{0\leq x<\infty,\ a\leq r\leq b\}\), with low porosity (here and futher 1 stands for the macropore and 2 for the micropore) (Figure 1). Macropore is considered to be a two-dimensional medium, i.e., pressure distributions and filtration velocity in this zone will be heterogeneous. To determine them, it is necessary to derive corresponding equations and solve together with the transport equations. Assume that the outer cylindrical region \(\Omega_{2}\) has permeability \(k_{2}\), and internal \(\Omega_{1}-k_{1}\), where \(k_{2}<<k_{1}\). External surface of the cylindrical porous media \(\Omega_{2}\) is impermeable. To determine solute concentration fields, it is necessary to determine the distribution of pressure and filtration velocity in cylindrical regions. Let the point \((0,\ 0)\) is the inlet point for fluid. Inhomogeneous fluid inflows with constant pressure \(p_{c}=\text{const}\). Initially there was a constant pressure in the medium \(p_{0}\), \(p_{c}>p_{0}\). Components of the filtration velocity in \(\Omega_{1}\) and \(\Omega_{2}\) defined as  
\[v_{1x}=-\frac{k_{1}}{\mu}\frac{\partial p_{1}}{\partial x},v_{1r}=-\frac{k_{1 }}{\mu}\frac{\partial p_{1}}{\partial r},(r,x)\in\Omega_{1}, \tag{1}\]  
\[v_{2x}=-\frac{k_{2}}{\mu}\frac{\partial p_{2}}{\partial x},v_{2r}=-\frac{k_{2 }}{\mu}\frac{\partial p_{2}}{\partial r},(r,x)\in\Omega_{2}, \tag{2}\]  
where \(v_{1x}\), \(v_{2x}\)--components of the filtration velocity in the \(x\) direction in \(\Omega_{1}\) and \(\Omega_{2}\), \(v_{1r},\ v_{2r}\)--radial components of the filtration velocity in \(\Omega_{1}\) and \(\Omega_{2}\), respectively, \(p_{1},\ p_{2}\)--pressure in zones \(\Omega_{1}\), \(\Omega_{2}\), \(\mu\)--viscosity coefficient of the fluid. To determine pressures in zones \(\Omega_{1}\), \(\Omega_{2}\) we use piezoconductivity equations  
\[\frac{\partial p_{1}}{\partial t}=\chi_{1}\left\{\frac{1}{r}\frac{\partial}{ \partial r}\left(r\frac{\partial p_{1}}{\partial r}\right)+\frac{\partial^{2 }p_{1}}{\partial x^{2}}\right\},\ \ (r,x)\in\Omega_{1}, \tag{3}\]  
\[\frac{\partial p_{2}}{\partial t}=\chi_{2}\left\{\frac{1}{r}\frac{\partial}{ \partial r}\left(r\frac{\partial p_{2}}{\partial r}\right)+\frac{\partial^{2 }p_{2}}{\partial x^{2}}\right\},\ \ (r,x)\in\Omega_{2}, \tag{4}\]  
\[\chi_{1}=\frac{k_{1}}{\mu\beta^{*}},\ \ \ \chi_{2}=\frac{k_{2}}{\mu\beta^{*}},\]  
where \(\chi_{1}\), \(\chi_{2}\)--coefficients of piezoconductivity, \(\beta^{*}\)--coefficient of elastic capacity of the media (\(\beta^{*}=m\beta_{f}+\beta_{r}\), \(\beta_{f}\)--elasticity coefficient of the fluid and \(\beta_{r}\)--of the rock of porous media). For the  
Figure 1: Cylindrical medium with cylindrical macropore. sake of simplicity here we assume that rocks of both media \(\Omega_{1}\) and \(\Omega_{2}\) have same coefficient of elasticity--\(\beta_{r}\).