# American Institute of Mathematical Sciences

March  2016, 5(1): 1-36. doi: 10.3934/eect.2016.5.1

## Well productivity index for compressible fluids and gases

 1 Department of Mathematics and Statistics, Texas Tech University, Lubbock TX, 79409-1042 2 SUNY New Paltz, Department of Mathematics, 1 Hawk Dr, New Paltz, NY 12561, United States, United States

Received  September 2015 Revised  December 2015 Published  March 2016

We discuss the notion of the well productivity index (PI) for the generalized Forchheimer flow of fluid through porous media. The PI characterizes the well capacity with respect to drainage area of the well and in general is time dependent. 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, [5,3,4]. Here we generalize our results from [4] in case of arbitrary order of the nonlinearity of the flow. In case of the compressible gas flow the mathematical model of the PI is studied for the first time. In contrast to slightly compressible fluid the PI stays almost'' constant for a long period of time, but then it 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 theoretical results for the time asymptotic of the PI and its stability with respect to the initial data.
Citation: Eugenio Aulisa, Lidia Bloshanskaya, Akif Ibragimov. Well productivity index for compressible fluids and gases. Evolution Equations & Control Theory, 2016, 5 (1) : 1-36. doi: 10.3934/eect.2016.5.1
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