Treatment of time derivative and calculation of flow when solving groundwater flow problems by Galerkin finite element methods     

Snorri P. Kjaran  Sven T. Sigurdsson 《Advances in water resources》1981,4(1):23-33

Two techniques connected with the use of the finite element Galerkin method for solving the linear parabolic differential equation describing unsteady groundwater flow in an anisotropic non-homogeneous aquifer are introduced. The first is a mode superposition technique for dealing with the time derivative which involves computing the smallest eigenvalues and associated eigenvectors of the matrices arising from the Galerkin method. It is shown how such a technique allows us to interpret the response of the groundwater level to input in terms of parallel linear reservoirs. It is further argued that if properly implemented, the technique will have computational advantages over standard finite difference methods, e.g. in the case when the input function is constant over relatively large time subintervals. The second is a technique based on so-called generalized flow formulae for calculating flow values across external or internal boundaries, posterior to obtaining the groundwater level values. The implementation of the technique in the case of linear triangular elements on an irregular grid is discussed. It is finally argued from simplified cases that, apart from guaranteeing a match with prescribed input, the technique may often be expected to give more accurate flow values than those obtained directly from the groundwater gradients.  相似文献   

An experimental study of crystalline basalt dissolution from 2 ? pH ? 11 and temperatures from 5 to 75 °C     

Snorri Gudbrandsson  Domenik Wolff-Boenisch  Eric H. Oelkers 《Geochimica et cosmochimica acta》2011,75(19):5496-5525

Steady-state element release rates from crystalline basalt dissolution at far-from-equilibrium were measured at pH from 2 to 11 and temperatures from 5 to 75 °C in mixed-flow reactors. Steady-state Si and Ca release rates exhibit a U-shaped variation with pH where rates decrease with increasing pH at acid condition but increase with increasing pH at alkaline conditions. Silicon release rates from crystalline basalt are comparable to Si release rates from basaltic glass of the same chemical composition at low pH and temperatures ?25 °C but slower at alkaline pH and temperatures ?50 °C. In contrast, Mg and Fe release rates decrease continuously with increasing pH at all temperatures. This behaviour is interpreted to stem from the contrasting dissolution behaviours of the three major minerals comprising the basalt: plagioclase, pyroxene, and olivine. Calcium is primarily present in plagioclase, which exhibits a U-shaped dissolution rate dependence on pH. In contrast, Mg and Fe are contained in pyroxene and olivine, minerals whose dissolution rates decrease monotonically with pH. As a result, crystalline basalt preferentially releases Mg and Fe relative to Ca at acidic conditions. The injection of acidic CO₂-charged fluids into crystalline basaltic terrain may, therefore, favour the formation of Mg and Fe carbonates rather than calcite. Element release rates estimated from the sum of the volume fraction normalized dissolution rates of plagioclase, pyroxene, and olivine are within one order of magnitude of those measured in this study.  相似文献