## Other Mathematical Models of Petrophysical Properties of Rocks

Two important problems of petroleum geology (modeling of petro-physical properties of oil- and gas-bearing formations) have been solved: (a) identification of reservoir beds and (b) identification of productive formations. Any of the parameters listed below may be used as a cut-off (identifying) parameters:

 Table 12-3 Corrections to the (1) Resistivity and (2) Formation Resistivity Factor for Effective Pressure and Temperature In Situ Depth, Correction in Rt Correction in Rt Correction in F m for peff for peff and T for peff and T Clay Cement 2,000-3,000 1.7S 0.66 1.61 3,000-4,000 1.SS 0.62 1.74 4,000-5,000 2.06 0.5S 1.7S 5,000-6,000 2.05 0.52 1.75 Clay-Carbonate Cement 2,000-3,000 1.45 0.61 1.4S 3,000-4,000 1.52 0.57 1.69 4,000-5,000 1.51 0.52 1.61 5,000-6,000 1.45 0.47 1.5S Carbonate Cement 2,000-3,000 1.23 0.57 1.32 3,000-4,000 1.25 0.52 1.43 4,000-5,000 1.25 0.4S 1.3S 5,000-6,000 1.22 0.43 1.33

1. true resistivity, Rt

2. interval transit time, At

3. macroscopic cross-section of thermal neutron absorption, E

4. formation bulk density, y

Information about the types of fluid saturating the rocks is provided by resistivity. Although various models of petrophysical properties of rocks are used, the most commonly used are the following relationships:

where Rt, Rt,a, At, I, and Y are the true resistivity of isotropic (Rt) and heterogeneous (Rta) rocks, interval transit time in rocks, macroscopic cross-section of thermal neutron capture (absorption), and formation bulk density, respectively. Subscripts f, w, o, and ma denote the fluid, water, and oil filling the pore space, and the rock matrix, respectively; Rw and Rsh are the resistivities of formation water and shale; ksh is volume content of clay in an laminated shaly reservoir rock; t is the electrical tortuosity of pore channels; and m and n are the empirical coefficients (m = 1.6 and n = 2.0 for the rocks of region under study; average values).

Heuristic formulas were used for determining the cut-off (separating) values of parameters. To distinguished the water- and oil-saturated beds using their resistivity, the following criteria for determining the cutoff values of Rtcr were used:

where Rw, Ro, Rwmd, and Ro,md are average and median values, whereas Dw and Do are standard deviations of resistivity in water- and oil-saturated layers.

The calculations show that the cut-off values of petrophysical parameters depend considerably upon the geometry of the pore space, the degree of cementation, the clay content in reservoir rocks, and the salinity of formation water. For example, with the increase in clay content in the reservoir rocks and transition from isotropic to aniso-tropic layers, the cut-off value of resistivity of the oil- and gas-bearing formations decreases from 10-15 to 4-8 ohm • m.

Thus, the use of static (for identification) as well as dynamic (for forecast) stochastic models in geologic studies allows solution of a number of important practical problems when data are scarce by using artificial distributions of various geologic and petrophysical parameters.

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