Block-centered grid

The simulations were performed using a finite-difference model for solution of the diffusion equation in porous media (. For modeling purposes, the cross section (Figure 1) was divided into a 3-layered, 36 by 53 block-centered grid. The layers represented the Toulon Member, the Radnor Till Member, and eolian silts (Roxana Silt and Peoria Loess). Physical properties of the unsaturated zone that were used for modeling were identical to those listed in Table I. 

The point (y, Zj) is considered to be at the center of block (i, j). Such block-centered grid systems are common in reservoir simulators since each block can be associated with an average pressure and average phase saturations. Also, injection and production wells can be placed in appropriate blocks. We will assume that one-quarter of an injection well lies in node (1,1) and one quarter of a producer lies in node (N,N). 

Different positions of the probe in the xy plane were examined, while the probe was fuUy inserted into the channel. Furthermore, for the case in which the probe is in the center of a channel, nine axial positions of the probe tip were explored. For each configuration, a block-structured grid with a typical size of 700,000 prism cells was generated. 

Next assume that we are dissatisfied with the flow rates. We refer back to the LAYER. DRL pictures, and we decide to drill a horizontal drainhole starting from Layer 4 in Well 1, which cuts a four grid block path through the pay zone. Nine blocks define the revised path for Well 1. The simulator provides the existing coordinates of well block centers. In what follows, we also reconstrain the well at a new 55 psi, shaking it up to test numerical stability ... 

The reader is invited to examine this phenomenon by running the models described above, by varying these two sets of parameters. The solute is modeled as a 10 X 10 block of 100 cells in the center of a 55 x 55 cell grid. The water content of the grid is 69% of the spaces around the solute block, randomly placed at the beginning of each run. The water temperature (WW), solute-solute afiinity (SS), and hydropathic character of the solute (WS) are presented in the parameter setup for Example 4.4. The extent of dissolution as a function of the rules and time (5000 iterations) is recorded as the fo and the average cluster size of the solute (S). 

Fig. 20.2. Indexing of nodal blocks in a finite difference grid, showing point (7,./) and it nearest neighbors. Properties of the nodal blocks are projected onto nodal points located at the center of each block.

Cellular automata simulations of the dissolution process have been described.42 The solute molecules, S, started in a solid block of cells at the center of the grid. They are endowed with rules PB(S), /(S), PB(WS), and /(WS). The attributes recorded from the dynamics were the /o(S), fraction of solutes unbound to other solute molecules, plus the average number of solute-solute joined faces, T(S), and the average distance that solute molecules have traveled from the center of the block at some specific iteration, D(S). The /o(S) values were interpreted to represent the extent of dissolution of the solute. The decrease in the T(S) values characterize the extent of disruption of the solute block, whereas the D(S) values quantify the extent of diffusion of solutes into the surrounding water. 

Figure 1. A single realization of a synthetic reservoir cross-section containing sand (light) and shales (dark). The lengths, thicknesses and center coordinates of the shales are independent, random events taken from known distribution functions. Shales are accumulated until a pre-specified global degree of shale area/total area (f 0.24 in this case) is attained. The light boxes correspond to grid blocks. (Reproduced from Ref. 2.)...

Until the early 1990s, nearly all products were laid out in what is now referred to as the conventional layout. In this layout, which is still used for most products, individual test plots (ground boards and concrete slabs) are separated in a grid on 1.5-m centers. Each of ten replications contains a mixture of rates, products, test methods and control plots arranged in a randomized block design (Figure 4). Replications are laid out in two tiers of five replications per tier. The conventional layout utilizes about 0.08 ha. 

In contrast to Chapter 1, we have explicitly introduced q(x,y,x,t), representing the local source volume flow rate per unit volume produced by any infinitesimal element of a general well. It is a three-dimensional, point singularity that applies to both injector and producer applications. For example, when q is a semi-infinite line, cylindrical radial flow is obtained over most of the source distribution, while spherical flow effects apply at the tip. In other words, partial penetration and spherical flow are modeled exactly. In this section, subscripts are used in three different contexts. First, they represent partial derivatives for example, Px is the partial derivative of p(x,y,z,t) with respect to the spatial coordinate x. Second, they are used as directional markers for example, ky (x,y,z) is the anisotropic permeability in the y direction. Finally, subscript indexes (i,j,k) in pijrepresent the centers of grid block volumes used in our finite difference discretizations. As usual. Ax, Ay, Az, and At are used to denote grid sizes for the independent variables x, y, z, and t. 

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