Quasi-implicit treatment of velocity-dependent mobilities in underground porous media flow simulation
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ORIGINAL PAPER
Quasi-implicit treatment of velocity-dependent mobilities in underground porous media flow simulation Leonardo Patacchini1,2
· Romain de Loubens1
Received: 9 October 2018 / Accepted: 24 July 2020 © Springer Nature Switzerland AG 2020
Abstract Quasi-implicit schemes for treating velocity-dependent mobilities in underground porous media flow simulation, occurring when modeling non-Newtonian and non-Darcy effects as well as capillary desaturation, are presented. With low-order finitevolume discretizations, the principle is to evaluate mobilities at cell edges using normal velocity components calculated implicitly, and transverse velocity components calculated explicitly (i.e., based on the previously converged time-step); the pressure gradient driving the flow is, as usual, treated implicitly. On 3D hexahedral meshes, the proposed schemes require the same 7-point stencil as that of common semi-implicit schemes where mobilities are evaluated with an entirely explicit velocity argument. When formulated appropriately, their higher level of implicitness however places them, in terms of numerical stability, closer to “real” fully implicit schemes requiring at least a 19-point stencil. A von Neumann stability analysis of these proposed schemes is performed on a simplified pressure equation, representative of both single-phase and multiphase flows, following an approach previously used by the authors to study semi-implicit schemes. Whereas the latter are subject to stability constraints which limit their usage in certain cases where the logarithmic derivative of mobility with respect to velocity is large in magnitude, the former are unconditionally stable for 1D and 2D flows, and only subject to weak restrictionsfor 3D flows. Keywords Stability analysis · Non-Newtonian · Non-Darcy · Capillary desaturation · Semi-implicit · Quasi-implicit
1 Introduction 1.1 Inadequacy of the two-point flux approximation The modeling of underground porous media flows at the macroscopic scale conventionally uses a constitutive law relating the superficial velocity of each phase ϕ, uϕ , to the volumetric forces it is subjected to, as [1]: uϕ = ϕ · −∇pϕ + ρϕ g , ∀ϕ ∈ P . (1) In the above, P is the phases ensemble (typically, P = {gas, oil, water}), pϕ and ρϕ are phase ϕ’s pressure and mass density, respectively, and g is the gravity acceleration. ϕ is referred to as phase ϕ’s mobility tensor, characterizing its “friction” on the pore walls as well as the hydraulic obstruction caused by the presence of other phases. Leonardo Patacchini
[email protected] 1
Total S.A., Pau, France
2
Present address: Stone Ridge Technology, Milan, Italy
Equation 1 is used as a first-order closure relationship to the system of mass conservation equations, which for immiscible flows writes ∂(φSϕ ρϕ ) + ∇ · ρϕ uϕ = 0, ∀ϕ ∈ P, (2) ∂t where φ is the medium porosity, and Sϕ is phase ϕ’s saturation. For Newtonian flows at low Reynolds and low capillary numbers, Eq. 1 can be considered an extension of Darcy’s law to multi
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