Derivation of an exact formula for oil flow rate for a hydrodynamically imperfect flat-bottomed well based on Maxwell’s filtration law depending on the degree of formation opening

The article derives a formula for the exact total oil flow rate of a flat-bottomed production oil well, which is hydrodynamically imperfect in terms of the degree of formation penetration, but hydrodynamically perfect in terms of the nature of penetration. Approximate flow rates for a hydrodynamically imperfect flat-bottomed well depending on the degree of formation penetration were obtained by Academician L.S. Leibenson, Professor V.N. Shelkachev and Morris Musket. Leibenson simply assumed a simple hemispherical radial flow in the unexposed lower part of the layer. Other scientists used the gamma function. The gamma function is given in the table in the article. Formulas for the values x<1 and x>1 argument of this function are proposed. According to a new idea put forward by the authors of the article, there is a certain analogy (similarity) between the laws of friction of liquids and the laws of filtration of these liquids. Using this analogy, they determined the laws of filtration of liquids for which the laws of friction are known, namely the laws of friction of Caisson, Shulman and Maxwell. The bottom of borehole 2 was drilled with a new double-bladed axe with rounded edges, giving it the shape of a concave hemisphere. The remaining well parameters were kept the same as in well 1. Two hydrodynamic – flat-radial simple and hemispherical radial simple flows stationary theoretical problems were solved, and flow rate formulas were derived for both flows. These flow rates were summed to obtain the exact total flow rate of the flat-bottomed well.