Well Productivity Index for Compressible Fluids and Gases

In this paper we discuss the notion of the diffusive capacity for the generalized Forchheimer flow of fluid through porous media. The diffusive capacity is an integral characteristic of the flow motivated by the engineering notion of the productivity index (PI), Dake 1983, Raghavan 1993, Christopher et al. 2014. The PI characterizes the well capacity with respect to drainage area of the well and in general is time dependent. We study its time dynamics for two types of fluids: slightly compressible and strongly compressible fluid (ideal gas). In case of the slightly compressible fluid the PI stabilizes in time to the specific value, determined by the so-called pseudo steady state solution, Aulisa et al. 2009, 2011, 2012. Here we generalize our results from Aulisa et al. 2012 on long term dynamics of the PI in case of arbitrary order of the nonlinearity of the flow. In this paper we study the mathematical model of the PI for compressible gas flow for the first time. In contrast to slightly compressible fluid this functional mathematically speaking is not time-invariant. At the same time it stays "almost" constant for a long period of time, but then it rapidly blows up as time approaches the certain critical value. This value depends on the initial data (initial reserves) of the reservoir. The "greater" are the initial reserves, the larger is this critical value. We present numerical and analytical results for the time asymptotic of the PI and its stability with respect to the initial data. Using comparison theorems for porous media equation from V\'azquez 2007 we obtain estimates between the PI's for the original gas flow and auxiliary flow with a distributed source. The latter one generates the time independent PI, and can be calculated using formula similar to one in case of slightly compressible fluid.