For the purpose of the present paper the analytical formulations of Herschel & Bulkley rheological model including the most relevant expressions of pressure drops, valid both for circular and annular sections, are applied in order to determine the three characteristic parameters of a known drilling fluid, having yield pseudoplastic behaviour, and flowing in the drilling hydraulic circuit, starting from field circulation tests.

As it is well known, a typical standard drilling hydraulic circuit consists of the surface circuit (stand pipe, rotary hose, swivel and kelly), the circular section (inside the drill string with variable diameters), the bit and the annular section (the gap between the borehole wall or casing and the drill string). In this circuit the drilling mud enters the drill pipe, comes out from the bit, flows up through the annulus up to the surface, where (after a short time for cleaning) it is put back in the circuit.

The parameters to be inspected are the yield point, the consistency index k, the flow behaviour index n.

By means of at least 3 flow tests at a certain drilling depth, with the bit off bottom, the pump rates and the corresponding stand pipe pressures are recorded.

The obtained N couple of values of stand pipe pressure and pump rate, the geometry of the hydraulic circuit and the fluid density are the input data for a numerical procedure to determine the three parameters of the considered drilling fluid.

In this way, using this numerical process, a nonlinear system of N equations (with N 3) having 3 unknowns (the three parameters of the fluid: n, k and) is solved determining the Herschel & Bulkley rheological parameters.

This procedure inspects the most probable solution for each tentative value of the flow behaviour index np, considering the infinite couples of and k satisfying the input value of the stand pipe pressure. Through the computation of the mean square deviation for each of the tentative values of np, the solution tern of the nonlinear system of N equations can be obtained. The computed results are compared to the ones obtained using the readings on the same drilling mud performed on Fann VG 35 viscometer Not always the rheological tern outcoming from the viscometer data exactly fits the equivalent rheological tern found considering the drilling well as viscometer Computed SPP data using the equivalent rheological tern and the rheological parameters from viscometer readings, using different rheological models such as Bingham, Ostwald & de Waele and Herschel & Bulkley, are compared to field stand pipe pressure data. It can be seen that the overall average error between measured and computed SPP (using the Herschel & Bulkley equivalent tern) has drastically reduced to very small values, while the computed SPP using viscometer readings with most of the rheological models today used could lead to large errors, thus misleading an accurate evaluation of the SPP on the rig floor site. This method shows to be useful also to carefully evaluate the annular pressure drop and the corresponding ECD, in order to have the maximum allowable pump rates without fracturing the crossed formations. Besides it could be used to monitor the SPP behaviour for potentially occurring problems in the hydraulic circuit, such as wash out, plugged nozzles, and also in the case of gas kicks in the well. If applied to different drilling depths, the method can usefully inspect the influence of pressure and temperature existing in the well upon the rheology of the mud. This particular characterization is going to be inspected in the forthcoming research stages, which have already been designed to date.


While drilling it is always very important to exactly know the pressure drop within the hydraulic circuit for many reasons.

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