Instability of Buckley-Leverett flow in a heterogeneous medium

We study the simultaneous one-dimensional flow of water and oil in a heterogeneous medium modelled by the Buckley-Leverett equation. It is shown both by analytical solutions and by numerical experiments that this hyperbolic model is unstable in the following sense: Perturbations in physical parameters in a tiny region of the reservoir may lead to a totally different picture of the flow. This means that simulation results obtained by solving the hyperbolic Buckley-Leverett equation may be unreliable.Symbols and Notation f fractional flow function varying withs andx - value off outsideI - value off insideI - local approximation off around¯x - f ^–,f ⁺ values of - f _j ⁿ value off atS _j ⁿ andx _j - g acceleration due to gravity [ms^–2] - I interval containing a low permeable rock - k dimensionless absolute permeability - k ^* absolute permeability [m²] - k _c ^* characteristic absolute permeability [m²] - k _ro relative oil permeability - k _rw relative water permeability - L ^* characteristic length [m] - L ₁ the space of absolutely integrable functions - L the space of bounded functions - P _c dimensionless capillary pressure function - P _c ^* capillary pressure function [Pa] - P _c ^* characteristic pressure [Pa] - S similarity solution - S _j ⁿ numerical approximation tos(x_j, t_n) - S ₁, S₂,S ₃ constant values ofs - s water saturation - value ofs at - s ^L left state ofs (wrt. ) - s ^R right state ofs (wrt. ) - s s for a fixed value of in Section 3 - T value oft - t dimensionless time coordinate - t ^* time coordinate [s] - t _c ^* characteristic time [s] - t _n temporal grid point,t _n=n t - v ^* total filtration (Darcy) velocity [ms^–1] - W, , v dimensionless numbers defined by Equations (4), (5) and (6) - x dimensionless spatial coordinate [m] - x ^* spatial coordinate [m] - x _j spatial grid piont,x _j=j x - discontinuity curve in (x, t) space - right limiting value of¯x - left limiting value of¯x - angle between flow direction and horizontal direction - t temporal grid spacing - x spatial grid spacing - length ofI - parameter measuring the capillary effects - argument ofS - _o dimensionless dynamic oil viscosity - _w dimensionless dynamic water viscosity - _c ^* characteristic viscosity [kg m^–1s^–1] - _o ^* dynamic oil viscosity [kg m^–1s^–1] - _w ^* dynamic water viscosity [k gm^–1s^–1] - _o dimensionless density of oil - _w dimensionless density of water - _c ^* characteristic density [kgm^–3] - _o ^* density of oil [kgm^–3] - _w ^* density of water [kgm^–3] - porosity - dimensionless diffusion function varying withs andx - ^* dimensionless function varying with s andx ^* [kg^–1m³s] - _j ⁿ value of atS _j ⁿ andx _j This research has been supported by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap a.s. (Statoil).     

Biological N Removal from Wastes Generated from Amine-Based CO2 Capture: Case Monoethanolamine

Large-scale amine-based CO₂ capture will generate waste containing large amounts of ammonia, in addition to contaminants such as the actual amine as well as degradation products thereof. Monoethanolamine (MEA) has been a dominant amine applied so far in this context. This study reveals how biological N removal can be achieved even in systems heavily contaminated by MEA in post- as well as pre-denitrification treatment systems, elucidating the rate-limiting factors of nitrification as well as aerobic and denitrifying biodegradation of MEA. The hydrolysis of MEA to ammonia readily occurred both in post- and pre-denitrification treatment systems with a hydraulic retention time of 7 h. MEA removal was ≥99?±?1 % and total nitrogen removal 77?±?10 % in both treatment systems. This study clearly demonstrates the advantage of pre-denitrification over post-denitrification for achieving biological nitrogen removal from MEA-contaminated effluents. Besides the removal of MEA, the removal efficiency of total nitrogen as well as organic matter was high without additional carbon source supplied.     

A note on the efficiency of the conjugate gradient method for a class of time‐dependent problems

We discuss the efficiency of the conjugate gradient (CG) method for solving a sequence of linear systems; Auⁿ⁺¹ = uⁿ, where A is assumed to be sparse, symmetric, and positive definite. We show that under certain conditions the Krylov subspace, which is generated when solving the first linear system Au¹ = u⁰, contains the solutions {uⁿ} for subsequent time steps. The solutions of these equations can therefore be computed by a straightforward projection of the right‐hand side onto the already computed Krylov subspace. Our theoretical considerations are illustrated by numerical experiments that compare this method with the order‐optimal scheme obtained by applying the multigrid method as a preconditioner for the CG‐method at each time step. Copyright © 2007 John Wiley & Sons, Ltd.     

Limits of Pure-Injective Cotilting Modules

The set of pure-injective cotilting modules over an artin algebra is shown to have a monoid structure. This monoid structure does not restrict down to a monoid structure on the finitely generated cotilting modules in general, but it does whenever the algebra is of finite representation type. Pure-injective cotilting modules are also constructed from any set of finitely generated cotilting modules with bounded injective dimension. Presented by Y. Drozd Mathematics Subject Classifications (2000) 16G10, 16P20, 16E30.     

A Maximum Principle for an Explicit Finite Difference Scheme Approximating the Hodgkin-Huxley Model

We analyze an explicit finite difference scheme for the general form of the Hodgkin-Huxley model, which is a nonlinear partial differential equation coupled to a set of ODEs. The system of equations describes propagation of an electrical signal in excitable cells. We prove that the numerical solution is bounded in the L^∞-norm and L² converges to a unique solution. The L^∞-bound, which is the key point of our analysis, is proved by showing that the discrete solutions are invariant in a physically relevant bounded region. For the convergence proof we use the compactness method. AMS subject classification (2000) 65F20     

Denominators in cluster algebras of affine type

The Fomin–Zelevinsky Laurent phenomenon states that every cluster variable in a cluster algebra can be expressed as a Laurent polynomial in the variables lying in an arbitrary initial cluster. We give representation-theoretic formulas for the denominators of cluster variables in cluster algebras of affine type. The formulas are in terms of the dimensions of spaces of homomorphisms in the corresponding cluster category, and hold for any choice of initial cluster.     

A numerical study of optimized sparse preconditioners

Preconditioning strategies based on incomplete factorizations and polynomial approximations are studied through extensive numerical experiments. We are concerned with the question of the optimal rate of convergence that can be achieved for these classes of preconditioners.Our conclusion is that the well-known Modified Incomplete Cholesky factorization (MIC), cf. e.g., Gustafsson [20], and the polynomial preconditioning based on the Chebyshev polynomials, cf. Johnson, Micchelli and Paul [22], have optimal order of convergence as applied to matrix systems derived by discretization of the Poisson equation. Thus for the discrete two-dimensional Poisson equation withn unknowns,O(n ^1/4) andO(n ^1/2) seem to be the optimal rates of convergence for the Conjugate Gradient (CG) method using incomplete factorizations and polynomial preconditioners, respectively. The results obtained for polynomial preconditioners are in agreement with the basic theory of CG, which implies that such preconditioners can not lead to improvement of the asymptotic convergence rate.By optimizing the preconditioners with respect to certain criteria, we observe a reduction of the number of CG iterations, but the rates of convergence remain unchanged.Supported by The Norwegian Research Council for Science and the Humanities (NAVF) under grants no. 413.90/002 and 412.93/005.Supported by The Royal Norwegian Council for Scientific and Industrial Research (NTNF) through program no. STP.28402: Toolkits in industrial mathematics.     

Tame concealed algebras and cluster quivers of minimal infinite type

The well-known list of Happel-Vossieck of tame concealed algebras in terms of quivers with relations, and the list of A. Seven of minimal infinite cluster quivers are compared. There is a 1-1 correspondence between the items in these lists, and we explain how an item in one list naturally corresponds to an item in the other list. A central tool for understanding this correspondence is the theory of cluster-tilted algebras.     

A system of conservation laws including a stiff relaxation term; the 2D case

We analyze a system of conservation laws in two space dimensions with a stiff relaxation term. A semi-implicit finite difference method approximating the system is studied and an error bound of order measured inL ¹ is derived. This error bound is independent of the relaxation time > 0. Furthermore, it is proved that the solutions of the system converge towards the solution of an equilibrium model as the relaxation time tends to zero, and that the rate of convergence measured inL ¹ is of order . Finally, we present some numerical illustrations.This research has been supported by the Norwegian Research Council (NFR), program no. STP 110673/420, at the Department of Applied Mathematics, SINTEF, Oslo, Norway