Classical verigin problem as a limit case of verigin problem with surface tension at free boundary

In this paper we consider Verigin problem with surface tension at free boundary: $$\begin{array}{*{20}c} {\partial p_i - \nabla \cdot \left( {\frac{k}{{u_i }}\nabla p_i } \right) = 0{\text{in }}Q_i \equiv \cup _{t > 0} \Omega _i (t),(i = 1,2),} \\ {p_1 - p_2 = \varepsilon K,(\varepsilon > 0)on{\text{ }}\Gamma \equiv \cup _{t > 0} \Gamma _t ,} \\ { - \frac{k}{{\mu _1 }}\frac{{\partial p_1 }}{{\partial n_t }} = - \frac{k}{{\mu _2 }}\frac{{\partial p_2 }}{{\partial n_t }} = \phi V_n on{\text{ }}\Gamma ,} \\ \end{array} $$ where Ω₁ (t),Ω ₂ (t) are regions of water and oil respectively, Γ_t is a free boundary between Ω₁(t) and Ω₂ (t). Let Ω = Ω₁ (t) U Γ _t U Ω₂ t) be a bounded annular domain in R², Ω₂ (t) is inside.n _t is a normal of Γ _t , pointing inwards Ω₂(t),P ₁ andP ₂ are pressures of water and oil, μ₁ and μ₂ are viscosities of water and oil respectively,k is the permeability,? is the porosity,K andV _n are the curvature and the normal velocity of Γ _t in the direction ofn _t . We prove that the classical Verigin problem is the limit case (ε→ 0) of Verigin problem with surface tension at free boundary.