Semi-analytical solution for unsteady fluid flow to a partially penetrating well
P.E. Morozov
Institute of Mechanics and Engineering, Kazan Science Center, Russian Academy of Sciences, Kazan, 420111 Russia
E-mail: morozov@imm.knc.ru
Received July 3, 2017
Abstract
An analytical solution of the problem of unsteady fluid flow to a partially penetrating well flowing at constant rate in an anisotropic reservoir with the impermeable top and bottom boundaries has been obtained. The problem reduces to a system of integral equations in the Laplace transform domain that connects the pressure drop and flux distribution along the open interval. The arbitrary number and position of the opening intervals relative to the top and bottom boundaries have been taken into account, as well as the wellbore storage effect and non-uniform skin effect. By using the superposition method, the solution for unsteady fluid flow to a partially penetrating well after its shut down has been obtained. Simulations have showed that the fluid overflow takes place through the opening intervals after a well is shut down at the bottomhole.
Keywords: semi-analytical solution, unsteady fluid flow, partially penetrating well, anisotropic reservoir, non-uniform skin effect, wellbore storage effect, “overflow” effect
Figure Captions
Fig. 1. The scheme of a layer that is partially penetrated by a vertical well.
Fig. 2. The comparison of the semi-analytical and approximate analytical solutions with the numerical solution from [19]: 1 - hd = 100 , S = 0 , Cd = 50 , z2d - z1d = 0:5 ; 2 { hd = 100 , S = 2:5 , Cd = 50 , z2d - z1d = 0:5 ; 3 - hd = 500 , S = 0 , Cd = 250 , z2d - z1d = 0.25 ; 4 - hd = 500 , S = 5 , Cd = 250 , z2d - z1d = 0.25 .
Fig. 3. The pressure field near the partially penetrating well at the uniform (a) and linear (b) distributions of the skin effect along the length of the penetration interval.
Fig. 4. The distribution of the fluid inflow along the length of the penetration interval after the start-up (a) and shut-down (b) of the well (1 – at the uniform distribution of the skin effect, 2 – at the linear distribution of the skin effect).
Fig. 5. The pressure field near the partially penetrating well with three penetration intervals.
Fig. 6. The distribution of the fluid inflow along the penetration intervals after the start-up (a) and shut-down (b) of the the partially penetrating well.
Fig. 7. The pressure filed near the vertical well in the isotropic (a) and anisotropic (b) layer at the linear distribution of the skin effect.
Fig. 8. The distribution of the fluid inflow along the wellbore after the start-up (a) and shut-down (b) of the well.
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For citation: Morozov P.E. Semi-analytical solution for unsteady fluid flow to a partially penetrating well. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2017, vol. 159, no. 3, pp. 340–353. (In Russian)
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