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).     

Robust convergence of multi point flux approximation on rough grids

This paper establishes the convergence of a multi point flux approximation control volume method on rough quadrilateral grids. By rough grids we refer to a family of refined quadrilateral grids where the cells are not required to approach parallelograms in the asymptotic limit. In contrast to previous convergence results for these methods we consider here a version where the flux approximation is derived directly in the physical space, and not on a reference cell. As a consequence, less regular grids are allowed. However, the extra cost is that the symmetry of the method is lost.     

A cheap versatile ultraviolet photoelectron spectrometer

The design, construction and operation of an ultraviolet photoelectron spectrometer are reported. It has a readily accessible target chamber and a versatile inlet system, which can cope with a rapid throughput of relatively involatile samples. The electrons emerging from the target chamber pass through two slits which are subject to a variable electric field, into a magnetic field. Thus the spectrum is linear. The resolution based on argon ²P_{$\text{3}\text{2}$} peak is 18 meV. The instrument is easy to make and operate. It is proving adequate for many chemical and analytical purposes and may be coupled directly to a gas-liquid chromatograph. Problems encountered during construction and optimization are discussed.     

An ESCA investigation of some thiocyanato complexes

A series of simple thiocyanato complex ions have been investigated as their tetraphenylphosphonium and potassium salts. The binding energies of N, C and S were determined as well as those of the metals. From the first-mentioned data effective charge-values were estimated for the atoms of the ligands. For this purpose linear relations E_b = kq + E_b0 were used that had been previously established within a scheme having C_1s (phenyl) as the internal standard.From the data thus obtained the effective charge on the metal atoms was estimated. For elements where we have sufficient data the same type of linear relation seems to he followed. Tentatively valid examples are E_b(Ni) = 6.74q_Ni + 848.3 eV and E_b(Pd) = 4.45q_Pd + 333.9 eV.In this interpretation the atoms of the metals are considered to be positively charged and surrounded by the negative charge of the electrons occupying the s band.It is further suggested for complexes with pronounced π backbonding (Pt(SCN)₄^2?, Pd(SCN)₄^2? and Hg(SCN)₄^2?) that the C 1s binding energy measured for the carbon atom of the SCN^? ligand is composed of the ionisation energy from the 1s level and an additional term corresponding to the intraligand π → π* transition.     

Determination of selenium in pyrite by the atomic absorption-hydride generation technique

Summary Pyrite is digested in a (3 + 1) nitric-hydrochloric acid mixture. Nitrate is removed by warming the solution after addition of sulphuric acid. Strong hydrochloric acid is added to hydride prior generation to eliminate interference from iron. The result is compared with that obtained by neutron activation analysis. Standard deviation was in the range of 1–2 ppm.

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Selenbestimmung in Pyrit über das Hydrid mit Hilfe der AAS

Zusammenfassung Die Pyritprobe wird mit Salpetersäure/ Salzsäure aufgeschlossen und anschließend werden die Nitrationen durch Erhitzen mit Schwefelsäure entfernt. Zur Hydriderzeugung wird starke Salzsäure zugesetzt, um Störungen durch Eisen zu verhindern. Die Ergebnisse wurden mit denen der Neutronenaktivierung verglichen, wobei sich gute Übereinstimmung ergab. Die Standardabweichung liegt im Bereich von 1–2 ppm.

     

Convergence properties of generalized inner projections to dynamic polarizabilities

Generalized inner projections to dynamic polarizabilities, α_A (iω), are shown to give bounds to dispersion coefficients, C_AB, which are improved as the dimensionality of the projections is increased. Error bounds in the regular half-planes of α_A±(ω) are found from the maximum principle. Comparison with related approximations to polarizabilities is made.