Asymptotic production behavior in waterflooded oil reservoirs: Decline curves on a simplified model

In this paper we consider a very simplified model of a mature waterflooded oil reservoir and study the asymptotic behavior in time of well’s oil production. More precisely, under assumptions on stationary points, we mathematically justify and precise classical decline laws: the oil production rate decreases like C1t−γ for some γ>1 if the nonlinear front velocity vanishes when the oil concentration S is close to vacuum (ϕ(S)=Sα with α>0). A more general law is obtained for general vanishing function ϕ at vacuum. It decreases exponentially fast like C2exp(−t) if the nonlinear front velocity does not vanish. Our calculations allow us to express constants C1, C2 in terms of physical and geometrical features of the reservoir through PDEs resolution. To the authors’ knowledge, this is the first result taking into account space variables. This could be of particular interest for the optimization process of oil production.