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.