Forward and inverse modeling of near-well flow using discrete edge-based vector potentials

Abstract

Homogeneous effective permeabilities in the near-well region are generally obtained using analytical solutions for transient flow. In contrast, this paper focuses on heterogeneous permeability obtained from steady flow solutions, although extensions to unsteady flow are introduced too. Exterior calculus and its discretized form have been used as a guide to derive the system of algebraic equations. Edge-based vector potentials describing 3-D steady and unsteady flow without mass balance error stabilize the solution. After a physics-oriented introduction, relatively simple analytic examples of forward and inverse discrete modeling demonstrate the applicability. Nomenclature(Vectors are denoted as lower case bold face symbols; their corresponding k-forms as lower case italic symbols. Matrices and arrays (rows, columns) are denoted as upper case symbols.) a vector potential [m 2 s −1 ] a 1-form corresponding to vector potential a [m 3 s −1 ] a i physical component of vector a and 1-form a [m 2 s −1 ] a i = m i a i , apparent component of 1-form a a l =(a 1 , a 2 , a 3 ), apparent vector potential A column array of vector potentials integrated along edges [m 3 s −1 ] A e =∫ e a, vector potential integrated along edge e [m 3 s −1 ] b = ω + q, divergence-free vector (∇ · b = 0) [m s −1 ] 116 Transp Porous Med (2007) 67:115-133matrix relating volumes to faces, discrete d-operator corresponding to div [−] G matrix relating edges to nodes, discrete d-operator corresponding to gradmatrix in which all components equal zero p pore pressure [Pa] P column array of pressures p v in volume centers v [Pa] q flux density vector, specific discharge [m s −1 ] q 2-form corresponding to flux density vector q [m 3 s −1 ] Q column array of normal components of flux densities q integrated over faces [m 3 s −1 ] Q f = ∫ f q, normal component of flux density q integrated over face f [m 3 s −1 ] r radial coordinate in circular cylinder coordinate system [m] r v radial coordinate in center of tetrahedron v [m] R matrix relating faces to edges, discrete d-operator corresponding to curl (rot) [−] S storage matrix relating column Ω to column ∂P/∂τ x = (x, y, z), position vector [m] Greek symbols γ isotropic physical resistivity [Pa s m −2 ] γ i = (m i+1 m i−1 /m i )γ ii , apparent resistivity in ζ i -direction for vector anisotropy γ ii diagonal component of physical resistivity matrix [Pa s m −2 ] -γ (physical) resistivity tensor (viscosity divided by absolute permeability) [Pa s m −2 ] -γ a = diag(γ 1 , γ 2 , γ 3 ), apparent resistivity tensor Galerkin Hodge matrix containing volume resistivities and grid metrics [Pa s m −3 ] Γ i j component (row = i, column = j) of Galerkin Hodge matrix [Pa s m −3 ] δ(x) Dirac delta function of coordinate x [m −1 ] δ i j =1 if i = j, otherwise δ i j = 0, Kronecker delta [−] ζ axial coordinate in circular cylinder coordinate system [m] θ polar angle, azimuthal coordinate in circular cylinder coordinate system [rad] κ isotropic physical permeability [m 2 Pa −1 s −1 ] κ i = (m i+1 m i−1 /m i )κ ii , apparent per...

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