Abstract
A mixed element-characteristic finite element method is put forward to approximate three-dimensional incompressible miscible positive semi-definite displacement problems in porous media. The mathematical model is formulated by a nonlinear partial differential system. The flow equation is approximated by a mixed element scheme, and the pressure and Darcy velocity are computed at the same time. The concentration equation is treated by the method of characteristic finite element, where the convection term is discretized along the characteristics and the diffusion term is computed by the scheme of finite element. The method of characteristics can confirm strong computation stability at the sharp fronts and avoid numerical dispersion and nonphysical oscillation. Furthermore, a large step is adopted while small time truncation error and high order accuracy are obtained. It is an important feature in numerical simulation of seepage mechanics that the mixed volume element can compute the pressure and Darcy velocity simultaneously and the accuracy of Darcy velocity is improved one order. Using the form of variation, energy method, $L^2$ projection and the technique of priori estimates of differential equations, we show convergence analysis for positive semi-definite problems. Then a powerfu tool is given to solve international famous problems.
Highlights
The incompressible miscible positive semi-definite displacement problem in porous media consists of two partial different equations: an elliptic equation for the pressure, a convection-diffusion equation for the concentration, where the concentration equation has strong hyperbolic feature (Douglas, 1983; Dougals, Ewing & Wheeler, 1983; Ewing, Russell & Wheeler, 1984; Russell, 1985), − ∇ · ( κ(X) μ(X) (∇ p γ(c)∇d(X))) ≡ ∇· u =
The flow equation is approximated by a mixed element scheme, and the pressure and Darcy velocity are computed at the same time
The concentration equation is treated by the method of characteristic finite element, where the convection term is discretized along the characteristics and the diffusion term is computed by the scheme of finite element
Summary
The incompressible miscible positive semi-definite displacement problem in porous media consists of two partial different equations: an elliptic equation for the pressure, a convection-diffusion equation for the concentration, where the concentration equation has strong hyperbolic feature (Douglas, 1983; Dougals, Ewing & Wheeler, 1983; Ewing, Russell & Wheeler, 1984; Russell, 1985),. While in actual numerical simulation applications such as oil-gas resources basin assessment (Yuan & Han, 2008; Yuan, Wang & Han, 2010) and numerical computation of enhanced (chemical) oil recovery (Yuan, Cheng, Yang & Li, 2014,2015), the diffusion matrix is only positive semi-definite (Dawson, Russell & Wheeler, 1989; Ewing, 1983; Yuan, 2013), D(X, u) ≥ 0. For two-dimensional positive definite problems, Douglas and Russell presented well-known numerical methods such as characteristic finite difference and characteristic finite element (Russell, 1985; Douglas, 1983). Based on the above discussions, we present a method of mixed element-characteristic finite element to simulate three-dimensional incompressible miscible positive semi-definite displacement problem of (1)(4). The symbols K and ε denote a generic positive constant and a generic small positive number, respectively They have different definitions at different places
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