Abstract. A methodology for incorporating subgrid effects in coarse-scale numerical models of flow in heterogeneous porous media is presented. The method proceeds by upscaling the deterministic fine-grid permeability description and then solving the pressure equation over the coarse grid to obtain coarse-scale velocities. The coarse-grid saturation equation is formed through a volume average of the fine-scale equations and includes terms involving both the average and fluctuating components of the velocity field. The terms involving the fluctuating components are subgrid effects that appear as length -and time-dependent dispersivities. A simplified model for the coarse-scale dispersivity, in terms of these subgrid velocity fluctuations, is proposed, and a numerical scheme based on this model is implemented. Results using the new method are presented for a variety of twodimensional heterogeneous systems characterized by moderate permeability correlation length in the dominant (horizontal) flow direction and small correlation length in the vertical direction. The new method is shown to provide much more accurate results than comparable coarse-grid models that do not contain the subgrid treatment. Extensions of the overall methodology to handle more general systems are discussed. Using the second, Monte Carlo approach, multiple realizations of the heterogeneous formation are generated, and each is then simulated deterministically [Hewett and Behrens, 1990]. In this approach, there is no uncertainty associated with the formation properties or the flow results for any single realization (other than errors due to numerical discretization). The probability distribution associated with the flow problem is determined through the simulation of multiple realizations of the formation. The complication in this case is that each realization must be highly detailed in order to capture all of the relevant scales of heterogeneity. This renders the simulation of even a single realization demanding. The handling of multiple realizations represents an even greater challenge.Upscaling procedures are commonly used to coarsen these highly detailed geostatistical realizations to scales more suitable for flow simulation. A variety of such approaches exists, and there are advantages and disadvantages associated with the different methodologies (for recent reviews, see Christie [1996] and Wen and Gomez-Hernandez [1996]). In general, there is somewhat of a trade-off between robustness and computational efficiency. Robustness in this context refers to process independence, meaning the same coarse-scale description is applicable for different flow processes; for example, different boundary and initial conditions. Less robust approaches typically achieve higher degrees of coarsening, but their use for problems which significantly deviate from the base case (used 2031

We have fabricated a vertically aligned 4-domain nematic liquid crystal display cell with thin film transistor. Unlike the conventional method constructing 4-domain, i.e., protrusion and surrounding electrode which needs additional processes, in this study the pixel design forming 4-domain with interdigital electrodes is suggested. In the device, one pixel is divided into two parts. One part has a horizontal electric field in the vertical direction and the other part has a horizontal one in the horizontal direction. Such fields in the horizontal and vertical direction drive the liquid crystal director to tilt down in four directions. In this article, the electro-optic characteristics of cells with 2 and 4 domain have been studied. The device with 4 domain shows faster response time than normal twisted-nematic and in-plane switching cells, wide viewing angle with optical compensation film, and more stable color characteristics than 2-domain vertical alignment cell with similar structure.