Implicit finite difference schemes for a linear model of well-reservoir coupling

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Standard reservoir models usually consider wells as Dirac measures over an interval length. Moreover, the well-reservoir coupling is taken into account under quite simplified assumptions. Most recently, however, attention has been drawn to the fact that in some situations, such as those related to non-vertical wells, these simplifications do not allow us to model some relevant mechanisms of the coupled flow. Therefore, more complex alternative models have been proposed recently in the oil reservoir simulation literature. A linearized version for the well-reservoir coupling can be written, in an appropriate functional setting, in the form U′(t) + AU(t) = F(t). In this work we discuss implicit in time discretizations of this equation, of the form { Un+1h - Unh÷τ+AhUn+1h = Fn=1h, U0h=U0.h We propose two different approximations, corresponding to first- and second-order spatial truncation errors, and we establish the convergence of both approximations.