The Marsh Funnel and Drilling Fluid Viscosity: A New Equation for Field Use M.J. Pitt,* U. of Leeds Summary The flow behavior of a Marsh funnel was numerically simulated for power law fluids and was found to give good agreement with experimental measurements. As a result, the simulation provides a general picture of the meaning of the Marsh funnel time and a correlation enabling this to be converted into a value for effec- tive viscosity of non-Newtonian fluids. For field use, the follow- ing equation relates the effective viscosity m e ~cp! to the Marsh funnel ~quart! time t ~seconds! and the density r (g/cm 23 ) m e 5r ( t 225). Introduction The funnel devised by Marsh 1 in the late 1920’s is the classical method of defining the viscosity of drilling mud in day to day use. It consists of an inverted cone, to the vertex of which a standard orifice tube is fixed ~Fig. 1!. The operator holds it vertically and covers the orifice with a finger. It is filled with the fluid to be tested. The operator releases the orifice and the time taken for a set volume of liquid ~usually a quart or a liter! to be discharged is recorded as the Marsh funnel time. Its great advantage is its sim- plicity and reliability under field conditions. There is, however, no recognized method of converting Marsh funnel times to viscosity in conventional units. Moreover, since drilling muds are generally highly non-Newtonian in character, their flow characteristics can- not be defined by a single viscosity. Marsh himself said, ‘‘Its readings, however, are only comparative ....this is really a com- bination measure of yield value and plasticity that gives only a practical indication of fluidity.’’ 1 More complex instruments ~rhe- ometers! provide multiple measurements of shear stress at a set of shear rates, i.e., rheograms. It has therefore been usual to regard the Marsh funnel as a purely empirical measurement of no fundamental significance. However, generations of mud engineers have adjusted and con- trolled drilling muds with this device, and the author’s own expe- rience of using the funnel led him to feel that there was a genuine pattern, probably the recognized intuitively by experts in the field. Apparatus The Marsh funnel used was a plastic one, with a brass orifice, provided by NL Baroid, and made according to API specification 13B. 2 For water at 21°C it was found to have a quart discharge time of 25.9 seconds ~standard deviation 0.18 seconds! compared with the API specification of 26.060.5 seconds. Its dimensions are shown in Fig. 1. Marsh himself measured the time for collection of 500 cm 3 , but the practice today is to collect either a quart (946 cm 3 ) or a liter. ~The relevant times for water are 18.5, 26.0, and 27.7 seconds.! For the present experimental work it was found convenient to collect a liter sample by using a 2 L conical flask as a receptacle. Rheograms of non-Newtonian fluids were measured on a Weis- senberg rheogoniometer, model R16, manufactured by Sangamo Controls Ltd., with a cone and plate geometry, cone angle 0° 58 ft 02 min. The viscosity of Newtonian fluids was determined in Ostwald U-tube viscometers ~sizes C and D!. Density was deter- mined by comparison with water in a 50 mL specific gravity bottle ~a pycnometer!. Materials Mixtures with distilled water were made of glycerol for use as Newtonian fluids. Because glycerol is very prone to absorbing moisture, these were not used as standards, but instead nominal mixtures were made and their viscosities found experimentally. Non-Newtonian fluids were made with two polymers com- monly used in drilling fluids, namely, xanthan cellulose ~XC! and hydroxyethylcellulose ~HEC!, supplied by Milchem Drilling Flu- ids. These were made in tap water, with the pH adjusted to 9 to 10, and aged for 1 week prior to the experiments. In each case, when a rheogram or viscosity determination was carried out, it was immediately followed by a Marsh funnel mea- surement. The laboratory temperature was 20 to 23°C. Theory To model the flow of liquid through the Marsh funnel it was assumed that an inverted cone of liquid provided a hydrostatic head that caused a drop in pressure through the working orifice. This drop in pressure is partially converted to kinetic energy ~fluid discharging from the orifice! and partially dissipated in fluid fric- tion going through the orifice. Using the formula of Skelland, 3 for a power law fluid in lami- nar flow through a tube ~neglecting entrance effects! the energy balance gives h r g 5 1 2 r v 2 12 k S 3 n 11 n D n S v n L r n11 D , ~1! where h is the height of the liquid above the orifice, r is the density, and g is the acceleration due to gravity; L and r are the length and radius of the tube, respectively, and k and n are the power law constants for the fluid. Method of Solution The fluid flow out of the funnel was calculated numerically by means of a Fortran program for intervals of 0.1 second until the requisite volume ~a quart or a liter! had been discharged, then the accumulated time was found. There was very little difference ~,0.1 second total! if intervals of 0.01 second were used. @A quart discharged through the orifice gives a stream about 53 m long, so a funnel time of 53 seconds means a velocity of about 1 m/s. A funnel time of 35 seconds is equivalent to about 1.5 m/s. The program used a start value for v of 1.5 m/s then iterated to find the value that solved Eq. 1 for the value of h for a full funnel. This flow was operated for the set time interval and a new height calculated, then a new velocity was found by iteration from the previous value. For short times the kinetic energy term was larger, for long times the viscosity term was larger. It was difficult to get convergence for those cases ~quart time about 42 seconds! in which they became equal during discharge, so this small section of the curve was interpolated.# Reynolds Number The initial and final generalized Reynolds numbers N Re for flow through the tube could be calculated 3 from N Re 5 d n v 22n r k 8 n21 , ~2! where d is the diameter. It was found that for discharge times below 30 seconds, the Reynolds number then exceeded 2,000 for some of the time, but for times in excess of this N Re was always below this value, so laminar flow is a fair assumption for the normal range of drilling fluids. *Now with the U. of Sheffield. Copyright © 2000 Society of Petroleum Engineers This paper (SPE 62020) was revised for publication from paper SPE 31018. Original manu- script received for review 12 April 1995. Revised manuscript received 17 November 1999. Revised manuscript approved 6 December 1999. SPE Drill. & Completion 15 ~1!, March 2000 1064-6671/2000/15~1!/3/4/$5.0010.50 3