A binary type permeability distribution with spatial autocorrelation is introduced to model the transition between shales and sand in a reservoir. The contrast between the two modal permeability values can be made realistically high, and the autocorrelation ranges can be made realistically large and anisotropic. A steady state, single phase flow simulation is run over a network whose block grids are informed from the previous permeability distribution. The resulting network effective permeabilities are plotted vs- the shale proportion and show permeabilities are plotted vs- the shale proportion and show that a power averaging process would yield a good estimate much more accurate than either the arithmetic average (power 1) or geometric average (power 0) traditionally used. Connections with percolation theory results are indicated.
One of the most pervasive problems in the description of an heterogeneous medium is the problem of averaging from one scale to another. A medium property is observed at one scale on a particular property is observed at one scale on a particular support (volume) of measurement, but the value of that property "averaged" over a different volume size at a different location is needed.
By "average" it is meant the unique value of the latter volume that could replace the set of all smaller measurements that could be obtained and processed within it, were there no limitations of processed within it, were there no limitations of resolution, money, and/or CPU time.
For example, actual measurements can be performed on a small support such as permeabilities on performed on a small support such as permeabilities on core plugs and the effective permeability of a simulation block is required.
If the averaging process of the particular variable under study were known the problem would be much alleviated. For example, the average porosity of a volume is simply the arithmetic average of the porosities of all the samples that constitute it. porosities of all the samples that constitute it. The same arithmetic averaging process holds true for additive variables such as saturations and more generally grades or volume/weight percentages of various phases.
Unfortunately, many other medium characteristics are not additive, i.e. the corresponding averaging process is not a mere arithmetic average and, worse, process is not a mere arithmetic average and, worse, most often it is unknown. For example, the average or effective permeability of a block remains unknown even if all the thousands of core plugs that constitute it were accessible for analysis: it is neither the arithmetic average nor the geometric average of these core permeabilities, although in many situations practice has adopted the second averaging process. process. Compounding the problem is the fact that not all plugs (or supports) constituting the block are plugs (or supports) constituting the block are available for measurements. This second problem calls for interpolation or the unknown plug values from neighboring known values. Unfortunately, interpolating values before knowing how they average is like putting the cart before the horse and it does not matter how fancy the cart is, whether called kriging or splines. The key problem is the averaging process not the choice of the interpolation process.
Another word of caution should be addressed to the people in charge of data gathering, geologists and geophysicists. The goals of planning and monitoring production are in many ways fundamentally different from those of exploration.
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