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We consider a fully-implicit formulation for two-phase flow in a porous medium with capillarity, gravity, and compressibility in three dimensions. The method is implicit in time and uses the multiscale mortar mixed finite element method for spatial discretization in a non-overlapping domain decomposition context. The interface conditions between subdomains are enforced in terms of Lagrange multiplier variables defined on a mortar space. There are two novelties in our approach: first, we linearize the coupled system of subdomain and mortar variables simultaneously to form a global Jacobian and eliminate variables by taking Schur complements; and second, we adapt a two-stage preconditioning strategy to solve the resulting formulation. The new formulation exploits the local invertibility of mortar block matrices to obtain a system with subdomain variables as unknowns. The two-stage preconditioner uses the …