Nine water-based muds were prepared: three gel muds, three KCl-polymer muds, and three KCl-silicate muds. The full formulations are reported in Table 1 to allow reproduction of the rheological differences among samples. Concentrations are expressed in pounds per barrel, except silicate, which is reported in gallons per barrel. The general preparation and field-testing procedures followed standard drilling-fluid measurement practice [1].

Table 1.Mud sample formulations used to prepare gel, KCl-polymer, and KCl-silicate water-based muds.

Each mud was prepared at ambient laboratory temperature using fresh water as the continuous phase. Additives were weighed according to Table 1 and added sequentially under mechanical agitation in the following order: water conditioning additives, clay or polymer viscosifiers, shale inhibitors or silicate, fluid-loss-control additives, and weighting material where applicable. Each slurry was mixed until visually homogeneous, with a minimum mixing time of 1 h before testing. Density was measured using a calibrated mud balance, solid content was measured using a mud retort, and rheological measurements were collected using a calibrated 6-speed rotational viscometer.

Table 2.Geometric dimensions of the standard and modified Marsh funnels used in drainage analysis.

Six funnel geometries were used: one standard Marsh funnel and five modified funnels with interchangeable orifices. The geometric parameters used in the calculations are listed in Table 2. Here, Z1 is the cone height, Ro is the upper cone radius, Z2 is the orifice length, RL is the orifice radius, α is the funnel-wall slope, and V0 is the initial filled volume.

The experimental workflow used to convert Marsh-funnel drainage measurements into rheological indicators is shown in Figure 1.

Figure 1. Workflow for converting Marsh-funnel drainage data into rheological indicators for water-based muds.

The geometric parameters of each funnel, including cone height, top radius, orifice radius, orifice length, and wall slope, are summarized in Table 2. These dimensions were used to convert drained volume into instantaneous fluid height.

Hydrostatic pressure driving the flow was calculated using P = ρgh, where h is the instantaneous fluid height inside the funnel. The fluid head was updated with time from the measured volume-time data and funnel geometry, ensuring that the decreasing driving pressure during drainage was accounted for in the wall shear stress calculations.

Calibration requirement before rheological estimation. Before a funnel geometry is used as a rheological estimation tool, the orifice must be cleaned and visually inspected, the funnel must be leveled vertically, and freshwater drainage must be measured repeatedly at the test temperature. In this study, freshwater calibration was repeated 20 times for each modified funnel, and the mean drainage time and standard deviation were used as geometry-specific hydraulic reference values. For field use, the same funnel/orifice combination should be calibrated with fresh water and at least one reference mud previously characterized by a 6-speed rotational viscometer; the resulting calibration should be checked whenever the orifice is changed, cleaned, damaged, or used at a substantially different temperature.

Marsh-funnel discharge was treated as capillary-type pressure-driven flow because the fluid is discharged through a confined outlet under a hydrostatic pressure difference, analogous to gravity-driven capillary rheometry. This treatment assumes incompressible flow, no wall slip, and a usable laminar-flow region in which wall shear stress and corrected wall shear rate can be estimated from the drainage record. The Weissenberg-Rabinowitsch correction was applied to account for non-Newtonian behavior [11][12].

Data were screened before rheological fitting. Points were retained only when the instantaneous slope used in the Weissenberg-Rabinowitsch correction remained within the recommended range of approximately 0.2 < n' < 1.3, and when the log-log shear-stress/nominal-shear-rate trend did not show abrupt slope changes associated with non-laminar, entrance-loss, or orifice-dominated behavior [12]. Small-orifice responses, especially for very viscous samples, were rejected when they failed to represent bulk fluid flow.

Power-law behavior provided the more consistent description for the nine mud samples. The average power-law R² was 0.956, compared with 0.773 for the Bingham plastic model. Power-law R² exceeded Bingham R² for 8 of the 9 samples, and 6 of 9 power-law fits had R² values greater than 0.95.

Comparison with the 6-speed viscometer was used as the primary benchmark for evaluating funnel-derived rheograms. The funnel-derived curves reproduced the general shear-thinning tendency observed in the viscometer data, but the absolute values of shear stress and fitted parameters did not match perfectly because the Marsh funnel operates over a different shear-rate range and is affected by entrance and/or orifice losses. Therefore, agreement was interpreted in terms of rheological trend and model discrimination rather than direct replacement of standard viscometer measurements.

Figure 2 compares Bingham plastic and power-law model fits obtained from the rotational viscometer data.

Figure 2. Comparison of Bingham plastic and power-law model fit to 6-speed viscometer data.

Freshwater drain time varied strongly with funnel geometry, from 12.5 seconds for funnel #2 to 78.2 seconds for funnel #6. This confirms that orifice geometry materially changes hydraulic response and must be included when interpreting funnel drainage data.

Funnel #2 provided the most reliable rheological response among the tested geometries. Its larger orifice radius (0.360 cm) and short freshwater drainage time (12.463 s) reduced orifice resistance and allowed the drainage curve to reflect bulk fluid behavior more clearly than smaller-orifice funnels. By contrast, Funnel #6, with the smallest orifice radius (0.150 cm), produced long drainage times and invalid or poor responses for viscous samples, indicating that entrance/orifice losses can dominate the measured response when the outlet is too restrictive.

Accordingly, field application should be restricted to calibrated funnels that produce complete, monotonic drainage curves within the tested fluid-property range. The present correlations should not be extrapolated to oil-based muds, high-temperature fluids, highly weighted systems, or muds that do not drain smoothly through the selected orifice without additional calibration.

Figure 3 summarizes the freshwater drainage-time response for each funnel geometry.

Figure 3.Freshwater drainage time for each funnel geometry.

The final multivariable regression model for the consistency index K retained solid content, drainage-time ratio, freshwater drainage time, the square of freshwater drainage time, initial funnel volume, and the square of solid content as predictors:

K = 7.65S − 72.65(DivT) − 1.66T_w + 0.01266T_w² − 0.0492V₀ − 3.145S² + 169.2

where S is solid content (%), DivT is the mud/water drainage-time ratio, T_w is the time required to drain 1000 mL of fresh water, and V₀ is the initial funnel volume. The K model produced R² = 74.77% and p = 0.024.

The final regression model for the flow-behavior index n retained K, drainage-time ratio, drainage-time difference, and K² as predictors:

n = 0.698 − 0.04502K + 0.0947 (DivT) − 0.000931(ΔT) + 0.000611K²

The n model produced R² = 87.26% and p = 0.029. Because only nine mud systems were available, overfitting was addressed by retaining only statistically useful predictors, rejecting funnel-geometry dimensions when they did not improve the model, and interpreting the regressions as screening correlations rather than universal predictive equations. Additional validation with independent mud systems is required before field deployment.

Because the dataset contains only nine mud systems, the regression models and gel-response indicator should be interpreted as preliminary calibrated relationships for the tested water-based muds. The Marsh funnel should therefore be considered a supplementary field rheometry tool, not a standalone replacement for a rotational viscometer, until the method is validated using a larger independent dataset covering broader mud chemistries, temperatures, densities, and solids contents.

Figure 4 shows the consistency-index relationship with solid content and mud density.

Figure 4. Power-law consistency index as a function of solid content and mud density.

Freshwater calibration runs were repeated 20 times for each modified funnel geometry; mean drainage time and standard deviation were reported. Mud drainage tests were performed in duplicate, and the average drainage-volume/time response was used to reduce operator timing error. For gel-response testing, each valid sample-funnel combination could generate three waiting-period comparisons: immediate versus 1 min, immediate versus 5 min, and immediate versus 10 min.

A “valid attempt” was defined as a drainage comparison in which the mud passed through the funnel without blockage, produced a complete and monotonic volume/time record, allowed calculation of flow rate, shear stress, corrected shear rate, and apparent viscosity, and remained within the accepted capillary-flow screening range. Runs were rejected when the sample was too viscous to pass through the nozzle properly, when no complete drainage curve was obtained, when the response was dominated by orifice resistance rather than fluid-flow behavior, or when slope changes indicated departure from the usable laminar-flow range. The denominator of 145, therefore, represents only valid waiting-period comparisons, not all physically attempted combinations.

The gel-response method detected a positive response in 94 of 145 valid attempts, giving a success rate of 64.8%. Using a binomial proportion treatment, the approximate 95% Wilson confidence interval is 56.8% - 72.1%. This uncertainty should be reported with the success value to avoid overstatement of the gel-indicator reliability.

Figure 5 reports the gel-strength indication success by mud sample and funnel geometry.

Figure 5.Gel-strength indication success by mud sample and funnel geometry.