Grid cell

Approaches used to model ozone formation include box, gradient transfer, and trajectoty methods. Another method, the particle-in-cell method, advects centers of mass (that have a specific mass assigned) with an effective velocity that includes both transport and dispersion over each time step. Chemistry is calculated using the total mass within each grid cell at the end of each time step. This method has the advantage of avoiding both the numerical diffusion of some gradient transfer methods and the distortion due to wind shear of some trajectory methods. 

The finite-element method (FEM) is based on shape functions which are defined in each grid cell. The imknown fimction O is locally expanded in a basis of shape fimctions, which are usually polynomials. The expansion coefficients are determined by a Ritz-Galerkin variational principle [80], which means that the solution corresponds to the minimization of a functional form depending on the degrees of freedom of the system. Hence the FEM has certain optimality properties, but is not necessarily a conservative method. The FEM is ideally suited for complex grid geometries, and the approximation order can easily be increased, for example by extending the set of shape fimctions. 

Figure 2.10 Multilamination flow oriented along the computational grid (left) and forming a tilted angle with the grid cells (right).

A face of a rectangular-grid cell overlaps with several faces of body-fitted grid cells. 

Typically, the interface obtained with the versions of the VOF method described above is smeared over a few grid cells, which, on sufficiently fine grids, allows one to identify uniquely the simply connected volumes belonging to the different phases. Instead of regarding the dynamic conditions of Eqs. (132)-(134) as boundary conditions, surface tension can be implemented as a volume force in those cells where c lies between 0 and 1. In the method developed by Brackbill et al. [176], a momentum source term of the form... 

Figure 2.68 Grid model of a porous medium (left) and renormalization group transformation replacing a cluster of grid cells by a unit cell of larger scale (right).

The mathematical model for a hydrocarbon reservoir consists of a number of partial differential equations (PDEs) as well as algebraic equations. The number of equations depends on the scope/capabilities of the model. The set of PDEs is often reduced to a set of ODES by grid discretization. The estimation of the reservoir parameters of each grid cell is the essence ofhistory matching. 

The water-oil ratio is a complex time-dependent function of the state variables since a well can produce oil from several grid cells at the same time. In this case the relationship of the output vector and the state variables is nonlinear of the form y(t,)=h(x(t,)). 

While prior information may be used to influence the parameter estimates towards realistic values, there is no guarantee that the final estimates will not reach extreme values particularly when the postulated grid cell model is incorrect and there is a large amount of data available. A simple way to impose inequality constraints on the parameters is through the incorporation of a penalty function as already discussed in Chapter 9 (Section 9.2.1.2). By this approach extra terms are added in the objective function that tend to explode when the parameters approach near the boundary and become negligible when the parameters are far. One can easily construct such penalty functions. For example a simple and yet very effective penalty function that keeps the parameters in the interval (kmjnkmaXil) is... 

Step 1. Construct a postulated grid cell model of the reservoir by using all the available information. 

Step 7. In order to modify and improve the postulated grid cell representation of the reservoir, analyze any zones with values close to these constraints. 

It should be noted that the nature of the problem is such that it is practically impossible to obtain a postulated model which is able to uniquely represent the reservoir. As a result, it is required to continuously update the match when additional information becomes available and possibly also change the grid cell description of the reservoir. 

By using automatic history matching, the reservoir engineer is not faced with the usual dilemma whether to reject a particular grid cell model because it is not a good approximation to the reservoir or to proceed with the parameter search because the best set of parameters has not been determined yet. 

Minimal and maximal shelf areas in GR15 are derived from minimal and maximal distances of the grid cell comers. 

The studies highlighted above have assessed model skill on a regional scale by averaging RCM results over an entire region and comparing them to areally averaged monthly observations (CRU). However, RCMs show considerable spatial variability in skill within individual experiments with adjacent grid-cells at times... 

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