Theoretical analysis of a model of fluid flow in a reservoir with the Caputo–Fabrizio operator

Abstract

We obtain the exact analytic solutions of a fluid flow model that includes the Caputo–Fabrizio operator and new constitutive equations in its definition. Formulas are obtained for a slightly compressible fluid in an infinite single-porosity reservoir with the inner boundary having a constant pressure. The flow equation is given by0CFCDtαp+τ0CFCDtβp=1r∂∂r(r∂p∂r),0<α,β≤1or1≤α,β≤2,where p is the pressure, which depends on the position r and the time t, τ is the relaxation time, and 0CFCDtα is the Caputo–Fabrizio operator of order α. We show that this equation is local and does not involve the flow properties associated with a fractal medium. Consequently, the mean square displacement at both short and long time scales exhibits normal behavior when 0 < α, β ≤ 1, while it shows a combination of ballistic and normal behaviors when 1 ≤ α, β ≤ 2. Indeed, in general, the discretized models with the CF operator exhibit the absence of a memory process, with the potential of speeding up the computations. On the other hand, we show that the solutions have a finite propagation velocity only when both α and β (or either of them) are equal to 2. In addition, the known solutions of the Helmholtz, diffusion, wave, and Cattaneo equations are recovered by our generalized solutions given the appropriate integer orders of α and β.