In this work, the finite element solution of the flow of miscible and immiscible incompressible fluids is addressed. A block-iterative scheme is used to solve the resulting finite element equations in time, combined with element-by-element iterative techniques, a dynamic mesh partition algorithm where the pressure equation is always implicit and the transport equation may be treated as implicit or explicit, and an adaptive time step control strategy. Numerical experiments on the CRAY Y-MP for five-spot patterns and the analysis of the problem of water injection in a rectangular oil reservoir with small scale permeability variations are performed to demonstrate that the proposed techniques are accurate and result in a very fast code.

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