A Compressible Two-Phase Model with Pressure-Dependent Well-Reservoir Interaction

Abstract

Abstract. This paper deals with a two-phase compressible gas-liquid model relevant for modeling of gas-kick flow scenarios in oil wells. To make the model more realistic we include a natural pressure-dependent well-formation interaction term allowing for modeling of dynamic gas influx/efflux. More precisely, the interaction between well and surrounding formation is controlled by a term of the form A = qw(Pw − P ) which appears in the gas continuity equation where qw is a rate constant, and Pw is a critical pressure, whereas P is pressure in the well. Consequently, an additional coupling mechanism is added to the mass and momentum equations. We obtain a global existence result for the new model. One consequence of the existence result is that as long as the well initially is filled with a mixture of gas and liquid, the system will regulate itself (in finite time) in such a way that there does not exist any point along the well where all the gas vanishes, e.g., by escaping into the formation. Similarly, the result guarantees that neither will any pure gas region appear in finite time, despite that gas is free to enter the well from the formation as long as the well pressure P is lower than the critical pressure Pw.

Asymptotic study of the initial value problem to a standard one pressure model of multifluid flows in nondivergence form

“…The convolution in (26) permits that the derivative ∂ ∂x Φ i in (25) makes sense: thus the fact the equations (19,20) are not in divergence form does not cause any trouble for the approximating sequences. We recall α is defined in (15), ρ 1 = r 1 α , and one will prove α i (x, t, ) > 0 ∀ > 0; ρ 2 is given in (12,21), u i = r i u i r i and one will prove r i (x, t, ) > 0 ∀ > 0. This will follow from (33) below, which, from (13), implies α = 0 and α = 1.…”

“…Then one notices that b 1 − b 2 > 0, K i > 0, r i > 0 and the result follows from (29). For ρ 2 one uses (12).…”

“…In this way the weak asymptotic methods presented here can be a tool for mathematical and numerical investigations of discontinuous solutions despite this system is in nondivergence form. Various systems in divergence form have been obtained by replacing (10,11) below by their sum, and then by introducing a new equation [10,11,12,15,17,18,19].…”