Journal of Petroleum Science and Engineering

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1 Journal of Petroleum Science and Engineering 77 (2011) Contents lists available at ScienceDirect Journal of Petroleum Science and Engineering journal homepage: Rheological and yield stress measurements of non-newtonian fluids using a Marsh unnel Matthew T. Balhoff, Larry W. Lake, Paul M. Bommer, Rebecca E. Lewis, Mark J. Weber, Jennifer M. Calderin Petroleum and Geosystems Engineering, University of Texas at Austin, 1 University Station C0300 Austin, TX , USA article info abstract Article history: Received 30 September 2010 Accepted 10 April 2011 Available online 29 April 2011 Keywords: Marsh unnel Yield stress Rheology Drilling mud Polymer Non-Newtonian Accurate and simple techniques for measurement of fluid rheological properties are important for field operations in the oil industry, but existing methods are relatively expensive and the results can be subjective. This is particularly true for measurements of fluid yield stress, which are notoriously difficult to obtain. Marsh unnels are popular quality-control tools used in the field for drilling fluids and they offer a simple, practical alternative to viscosity measurement. In the normal measurements, a single point (drainage time) is used to determine an average viscosity; little additional information is extracted regarding the non-newtonian behavior of the fluid. Here, a new model is developed and used to determine the rheological properties of drilling muds and other non-newtonian fluids using data of fluid volume collected from a Marsh unnel as a function of time. The funnel results for viscosity and shear-thinning index compare favorably to the values obtained from a commonly-used ann 35 viscometer. More importantly, an objective, static method for determining yield stress is introduced, which has several advantages over dynamic, extrapolation techniques used for rheometer data. Published by Elsevier B.V. 1. Introduction 1.1. Non-Newtonian fluids in the petroleum industry Hydrocarbon production uses many fluids that are rheologically complex. Among these are cement, drilling muds, aqueous solutions of water-soluble polymer, and, of course, crude oil itself. Drilling fluids can be air or water, but most commonly they are muds or suspensions of solids in an aqueous or oleic fluid. The solids are suspended with one or more surfactants. The solids are to provide weight to the mud for pressure control, the main function of muds, but muds also lubricate the drill, carry drilling cuttings to the surface and cool the bit. Most muds are water-based as is the type used in this study. When fresh water is the liquid base, bentonite is the clay used for its superior properties necessary to achieve the goals stated for drilling mud. Salt water mud can be created using bentonite that has been pre-hydrated with fresh water so long as the salinity is not much more than that of sea water. Drilling mud exhibits several important rheological properties (Bourgoyne et al., 1991). The viscosity or consistency index of a mud is Corresponding author. Tel.: ; fax: address: Balhoff@mail.utexas.edu (M.T. Balhoff). a measure of flow resistance. Therefore viscosity should be as small as possible to limit friction pressure. However a certain amount of viscosity is required to improve the solids carrying capacity of the mud. If viscosity is too small, the mud may be unable to suspend drilled solids at the desired pump rate. This requires the pumps to be run faster to continue to circulate drilled solids out of the well. If viscosity is too high, an excessive pump pressure will be required to circulate the mud at the desired rate. Higher than necessary pump pressure is an added strain on the pumps and piping and an added pressure in the bore hole that can lead to well bore stability problems. Water-soluble polymers are also used in drilling fluids to improve the ability of muds to lift cuttings, but they are also used as fracturing fluids to improve the removal of solids after fracturing, and in enhanced oil recovery. Two common polymers used are xanthan gum (hereafter just xanthan) and partially hydrolyzed polyacrylamide (HPAM). These are also used in the current study. These non-newtonian fluids (drilling muds and polymers) may also exhibit a yield stress (or gel strength). or drilling operations, the higher the yield stress the more pump pressure will be required to initiate circulation. The yield stress can also be a desirable property because it will suspend the drilled solids and prevent or slow them from slipping back to the bottom of the hole during periods when there is no circulation. luid yield stress in fracturing fluids can help carry and suspend proppant, but can also make cleanup difficult (May et al., 1997; Balhoff and Miller, 2005) /$ see front matter. Published by Elsevier B.V. doi: /j.petrol

2 394 M.T. Balhoff et al. / Journal of Petroleum Science and Engineering 77 (2011) Table 1 Constitutive equations for rheological models and the resulting steady flow equations in a cylindrical tube of radius R and length L. Model Constitutive equation Cylindrical tube model Newtonian τ = μγ Q = π R4 8μ L ΔP Power-law Bingham (Skelland, 1967) Herschel Bulkley (Skelland, 1967) General luid (Carreau et al., 1997) τ = m γ n Q = + 1 = n nπr3 ð3n +1Þð2mLÞ 1 = n ΔP 1=n " # τ = τ 0 + m γ Q = π R4 8m L ΔP 1 4 2Lτ Lτ 4 0 ΔPR 3 ΔPR τ = τ 0 + m γ n Q = π R3 m 1 = n ðτ τ w τ 3 0 Þ 1 =n +1 ðτw τ 0 Þ τ 0 ðτw τ 0 Þ w = 1 + τ2 0 n +3 = 1 n +2 = n +1 τ = ηγ ð Þ γ Q = π R3 γ w 3 π 3 R τ w 3 γw η 3 γ 3 d γ Rheological models Purely viscous, non-newtonian fluids are often classified using constitutive models relating the shear stress to the shear rate (Table 1). Many of these models (e.g. power-law, Carreau) account for the shear-thinning (decrease in viscosity with shear rate) behavior observed in many non-newtonian fluids. or shear-thinning fluids, the power-law index, n, is less than one. The fluid is Newtonian and the model reduces to Newton's law if n equals one. Some fluids exhibit a yield stress, requiring a minimum stress to initiate flow. Below the yield stress the material is solid-like and has an infinite viscosity. The solid-like behavior is typically a result of a threedimensional microstructure at low stresses (Carreau et al., 1997). Above the yield stress the material deforms as a fluid and the viscosity is a function of shear rate. Many pastes, foodstuffs, gels, and drilling muds have a yield stress. The simplest yield-stress model is the Bingham model, in which the relationship between shear stress and shear rate is linear, with the yield stress defined as the extrapolated y- axis intercept. A more general model is the Herschel Bulkley constitutive equation, which has the shear-thinning (or shearthickening) behavior of power-law fluids and the yield-stress effect of the Bingham model (see Table 1). The Herschel Bulkley model reduces to the ideal Bingham model for the special case where n =1, and to the power-law model for no yield stress (τ 0 =0). Although more complex models may describe the rheology of specific fluids better, the Bingham and Herschel Bulkley models are widely used because of their mathematical simplicity. To mathematically describe the rheology of a fluid, a constitutive equation must be chosen and the empirical constants (e.g. m, n, τ 0 ) must be determined experimentally. Typically, experiments are performed using a rheometer with a couette, parallel plate, or cone and plate geometry. Shear stress (or viscosity) is measured dynamically as a function of shear rate and a best-fit match to the data determines the model constants. Accurate results are often obtained for the shear-thinning (n) and consistency (m) index provided enough data points are extracted over a wide range of shear rates. Although highly accurate rheometers are available, simple lab rheometers are commonly used in the oilfield, perhaps the most popular of which is the ann 35 viscometer (ann, 2010). The ann 35 is small, light, relatively inexpensive, and easy to use. It is often used to obtain quick rheological measurements of drilling muds and other oilfield fluids. The couette geometry allows for specification of revolutions per minute (RPM), which is proportional to shear rate, and measures the resulting torque, which is proportional to shear stress. Although a constitutive model can be fit to the data, only six shear rates (in a limited range) are obtainable, making parameter estimation (especially yield stress) difficult. Screen factor devices (Lake, 1989) are also popular for measuring the viscosity of polymers relative to brine, but only one data point is obtained. Curves of viscosity as a function of shear rate are not possible with the screen factor device Yield stress measurement The fluid yield stress is historically difficult to measure (Nguyen and Boger, 1983; Zhu et al., 2001; Balhoff, 2005) in any rheometer. Measurement of yield stress from a flow curve of shear stress versus shear rate can give faulty results for a number of reasons. irst, the yield stress must be determined by extrapolation of the curve to a shear rate of zero, since the lowest shear measurement is finite and limited by the rheometer. Second, measurements performed at low shear rates may be inaccurate because of slip at the walls (Nguyen and Boger, 1983). inally, the yield stress measured may be a dynamic yield stress and not the stress required to break the three-dimensional structure of the material and initiate flow (Carreau et al., 1997). Other tests such as creep, oscillatory shear, and stress ramp can be used to measure yield stress, but they can also be ambiguous and inconsistent (Nguyen and Boger, 1992; Carreau et al., 1997). Some authors even claim that a true yield stress does not exist (Barnes and Walters, 1985) while others claim otherwise (Peder et al., 2006; Moller et al., 2009). Certain direct tests, such as the vane (Nguyen and Boger, 1983) and plate (Carreau et al., 1997) method, tend to be more reliable and less subjective than the aforementioned dynamic tests. In each case, torque (proportional to stress) versus time (proportional to strain) is measured. Initially, the torque increases linearly with time, demonstrating the Hookian behavior of the material. After the linear region the torque may continue to increase non-linearly, indicating viscoelastic behavior. Once the torque reaches a maximum, the internal structure breaks down and the material yields. The yield stress can be determined from this maximum torque (Carreau et al., 1997). A slotted plate device has been developed by Zhu et al. (2001) that is particularly useful for suspensions that exhibit a low yield stress, when the vane or plate may be less reliable Marsh unnel The Marsh unnel was invented by Hallan N. Marsh in 1931 (Marsh, 1931). It is used to measure the time in seconds required to fill a set volume of fluid. (In the United States the volume is one quart.) The flow through the small tip at the end of the funnel is related to the rheological properties of the fluid being measured. The Marsh unnel viscosity is reported as seconds and used as an indicator of the relative consistency of fluids, the more viscous the fluid the longer the time to fill one quart. The calibration for Marsh unnel time is 25 seconds per quart for fresh water. The standard Marsh unnel is shown in ig. 1 (picture taken from Schlumberger). The Marsh unnel

3 M.T. Balhoff et al. / Journal of Petroleum Science and Engineering 77 (2011) (0.16 cm) (15.2 cm) 6 in. are often compared to the drainage time for water, but this can be very misleading since viscosity is a very nonlinear function for turbulent flow. urthermore, the drainage time for low viscosity fluids such as water are dominated by the resistance of the funnel constriction and independent of fluid viscosity (Pitt, 2000). Since the model developed here assumes laminar flow, a higher viscosity fluid than water is needed. Mineral oil (~0.20 Pa-s) was chosen instead as the Newtonian basis. provides a simple and effective tool to determine the relative viscosity of drilling mud. Here, we also use the funnel for additional oilfield fluids. Although rheological properties of these fluids can be measured by conventional rheometers, a simple method is often needed. The goal of this work is to develop such a method for determining rheological properties of non-newtonian fluids using a Marsh unnel. An experiment consists of filling the funnel to a pre-specified height and measuring the rate at which the test fluid drains. Pitt (2000) developed a model for non-newtonian flow in a Marsh unnel, but his approach was not used to estimate the rheological parameters uniquely. The model presented here can estimate multiple parameters for a fluid uniquely. Moreover, an objective measurement of yield stress is introduced using the final, non-zero fluid height in the funnel. We also show that the Marsh unnel could be used as a tool for field measurement of fluids used in enhanced oil recovery and fracturing as well. 2. Experimental (0.475 cm) 2.1. luid preparation 12 in. (30.5 cm) 2 in. (5 cm) ig. 1. Standard Marsh unnel. The height of cone-portion of the funnel is 12 in. (30.5 cm) and the diameter is 6 in. (15.2 cm). The copper tubing is 2 in. (5.08 cm) in length and has a diameter of 3/in. (0.48 cm). Many of the fluids used in these experiments must be mixed before testing. The fluids that are used in this project are mineral oil, polymer (xanthan, hydrolyzed polyacrylamide, carbopol), and bentonite. Before adding any solid particles or polymer to the water-based fluids used here, the water was adjusted to approximately a ph of 9 by adding droplets of 2 weight % NaOH (sodium hydroxide). or all tests, the fluid was allowed to cool to room temperature (~19 C), the density was measured using a density balance, and the rheology measured using a ann 35 viscometer. luid was then poured in the Marsh unnel for the tests. The sections below describe how each fluid is prepared Mineral oil A Newtonian fluid was used as a standard test for the Marsh unnel experiments. The viscosities of fluids using the Marsh unnel Xanthan polymer Xanthan polymer is widely used in several industries, including petroleum. Xanthan was chosen as a test non-newtonian fluid in this work because it is shear-thinning, may exhibit a yield stress (Song et al., 2006), and exhibits minimal elastic effects in comparison to polyacrylamides (Lake, 1989). Concentrations of 1 2 wt% xanthan polymer were used in this study. The fluid was mixed vigorously for several minutes and allowed to cool (the fluid heated as a result of mixing) for several hours before tests in the rheometer and funnel. The molecular weight of the xanthan used here is unknown Bentonite muds Bentonite muds are the most common used in oil drilling usually at concentrations of 5 lb/bbl (1.4 wt%) and larger. They are well known to be non-newtonian and may exhibit a yield stress at high concentrations. Here, low-gravity solids are added to simulate realworld effects of drilling mud as it comes out of a borehole carrying drilled cuttings. Low-gravity solids were mixed with the bentonite mixtures for approximately 30 minutes and then allowed to cool before using in the experimental tests. Bentonite exhibits some timedependent properties (including yield stress), but enough time elapsed (hours) so that the properties were not dynamic (Gray et al., 1980) HPAM Hydroxypolyacrylamide (HPAM) is widely used in the petroleum industry for chemical EOR processes (Lake, 1989). The fluid is shearthinning but also exhibits viscoelasticity, hysteresis, and other timedependent effects (Lake, 1989; Carreau et al., 1997). HPAM was created by adding polymer to deionized water with 1% NaCl. The solution was stirred for 72 hours and then filtered Carbopol Carbopol gel is used in household and personal care products. The polymer is well-known to be shear-thinning and is also reported to exhibit a yield stress (Chase and Dachavijit, 2003). Here, Carbopol 980 was obtained from Noveon Inc. The powder was added to 3% NaCl solution and stirred for approximately 24 hours. Ten percent NaOH was then added until the ph reached approximately 7.5 which crosslinked the gel and created a viscous, clear solution Rheology tests We used a ann 35 viscometer to measure the fluid's rheological properties which are then compared to results obtained using the Marsh unnel. The ann 35 cups are filled to the top of the dashed line and placed on the base of the rheometer. Once the cup is positioned, the rheometer is turned onto its highest setting of 600 rpm. The degree dial is allowed to stabilize before taking the first recording. Subsequent measurements at 300, 200, 100, 6, and 3 rpm are recorded as well along with the resulting dial reading. The data are then converted to shear stress and shear rate, respectively. Details can be found in the instruction manual (ann, 2010). Some fluids were also tested on an ARES LS-1, controlled-rate, temperature-controlled rheometer. Plots of shear stress versus shear rate are

4 396 M.T. Balhoff et al. / Journal of Petroleum Science and Engineering 77 (2011) generated and rheological properties (i.e. viscosity, power index, and yield stress) are obtained through curve fitting to various rheological models (Table 1) Marsh unnel test After the fluid rheology is measured, the fluid is placed in the Marsh unnel (ig. 1). The Marsh unnel is designed so that 1500 ml of fluid can be poured into the funnel. A small stopper is placed in the orifice at the bottom to prevent flow out while the fluid is poured into the funnel. Once it reaches the bottom of the screen, this indicates that 1500 ml now rests in the funnel. The purpose of the screen is to remove any unmixed solid particles from the rest of the fluid. The fluids are allowed to rest in the funnel for a few minutes before being tested to potentially build up gel strength. Once the funnel is filled, a beaker is placed on a scale positioned below the funnel. The scale is connected to a computer that records weight versus time at intervals of 1 second. The weight is converted to a volume using the fluids density. The plug is removed from the orifice and the fluid is allowed to flow until it comes to a complete stop or completely drains. or fluids that exhibit a yield stress, a steady-state height is recorded which can be related to the yield stress. The final height is measured using a submersible measuring stick. Most tests were repeated 2 3 times and excellent repeatability was obtained. Some of the more viscous fluids adhere to the sides of the funnel wall making the volume collected plus volume remaining in the static head less than 1500 ml. or these cases, the volume versus time data is corrected by assuming accumulation occurs uniformly along the walls with funnel height. The corrected volume versus time curve is then converted to a height versus time curve using the mathematical equations developed in Section Model development 3.1. Model assumptions A model for fluid height (h) as a function of time (t) in the funnel is developed and rheological properties are then determined from a best fit of the data to the model. The analysis is based on a solution to an ordinary differential equation that is based on the following assumptions: 1. The density of air is negligible, compared to the fluid densities, as is capillary pressure between air and the existing fluid. 2. The fluids being tested are incompressible. 3. There is no viscous resistance to flow in the funnel itself. This assumption implies that pressure at the top of the nozzle region is given by the hydrostatic head of the funnel and is verified in the Appendix. The hydrostatic head is independent of the funnel geometry for fluids without a yield stress. The assumption also eliminates entrance effects at the bottom of the funnel, which could lead to inaccuracies since L/D of the capillary tube is approximately Elastic effects are negligible. 5. low in the nozzle region is fully developed, steady-state Poiseuille flow. The basic measurements are necessarily transient and this assumptions means that flow is quasi-static. This assumption is verified with calculations in the Appendix. The flow is laminar in all tests conducted here. 6. There is no loss of fluids to evaporation or to the sides of the funnel wall. While fluid loss is not included directly into the ODE, it is corrected during data reduction as described in Section 2.3 to achieve material balance Model derivation or these assumptions, a mass balance for funnel fluid can be written as: dv dt = QðhÞ The volume of fluid in the funnel can be written as a function of height using the formula for a cone (V=πr 2 h/3). The radius can be related to the height using similar triangles: r = V = π 3 h 2 h 3 = αh 3 ð3þ Where is the maximum funnel radius and is the maximum height. The above equation has a coefficient α=0.065 using the dimensions of the funnel. A calibration curve for volume versus height has shown α=0.078 is more accurate and is used in this work. Substituting Eq. (3) into the mass balance and using the chain rule yields: π 2 h 2 dh dt = QðhÞ Eq. (4) is an ODE that describes the height of fluid in the Marsh unnel as a function of time. An appropriate equation for Q(h) in a cylindrical tube is needed. The equation for flow, Q, in a capillary (bottom of funnel) for various non-newtonian fluid models is given in Table 1. The pressure drop is the hydrostatic head; the total fluid height being the sum of the tube length, L, and funnel height, h. Where the shear stress at the wall of the tube is given by (see the Appendix): τ w = ΔPR 2L ρg h+ L = ½ ð ÞŠR 2L ð1þ ð2þ ð4þ 2 τ 0 ð5þ The goal is to determine rheological properties using the time elapsed and height of fluid displaced. or Newtonian flow, the Hagen Poiseuille equation can be substituted into the ODE. Separation of variables gives: π R 2 8μL πr 4 h h 0 2 h dh = t ð6þ ½ρgðh+ LÞŠ Integrating Eq. (6) relates the funnel height as a function of time. L 2 ln h + L 1 h i " h 2 2hL h 2 h 0 + L 2 0 2h 0 L = R 4 # H 2 ρg t 8μ L h=0 at t=t f (drainage time). Also, using the fact that: L 2 L ln bblh h 0 + L h2 0 The viscosity of a Newtonian fluid can then be estimated by the following equation and the total drainage time, t f, for laminar flow. " μ R 4 #" # H 2 ρg 8L Lh h2 0 t f or non-newtonian fluids, Q(h) is more complicated (Table 1) and in general Eq. (4) must be integrated numerically (e.g. 4th order ð7þ ð8þ ð9þ

5 M.T. Balhoff et al. / Journal of Petroleum Science and Engineering 77 (2011) Runge-Kutta method). The rheological parameters can be found by fitting the model solution of h(t) to the dynamic funnel data; nonlinear regression can be used to find the best fit. Microsoft Excel's SOLVER function was used in this work to find the model parameters. The funnel data is never converted to shear rate and shear stress in the present work (although such a conversion could be performed). Parameter estimation is performed by finding a least-squares fit of the height versus time data to the solution of the model (Eq. (4)). The fluid yield stress could be determined in the same fashion as the other rheological properties, i.e. as a fitted parameter in the leastsquares optimization. However, the yield stress is a fundamental fluid property and should be measured under static conditions (Zhu et al., 2001) if possible. Moreover, a goal of this work is to identify a simple, objective measurement technique for yield stress. The yield stress is the minimum shear stress required to induce flow, or equivalently the maximum stress that can be imposed before flow occurs. Here, the shear stress can be related to the height of fluid in the funnel; the static height where the forces are in balance (i.e. dh/dt=0). The derivation is given in the Appendix. τ 0 = τ w = ρgh ð ss + LÞ 2L R + 2 ð10þ If there is no static height, the fluid exhibits no yield stress or the yield stress is smaller than can be measured with the capillary radius of the funnel. The smallest yield stress that can be measured in the Marsh unnel is 12.5 Pa. Once the yield stress is determined under the static conditions, it can be substituted directly into the ODE (Eq. (4)); the consistency index (m) and shear-thinning index (n) are then determined from nonlinear regression by minimizing the sum of squared errors between the model and data. 4. Results and discussion 4.1. Mineral oil (Newtonian standard) The shear stress, shear rate data obtained from the rheometers can be plotted to determine the rheological model that best describes the fluid and the best-fit rheological properties for that model. The results can be compared to those obtained using the Marsh unnel. ig. 2 shows shear stress versus shear rate data for mineral oil. The data are clearly linear validating Newtonian behavior with a viscosity of 0.19 Pa-s (from the ann 35 viscometer). A slightly higher viscosity ig. 3. Height versus time data for mineral oil. The Newtonian model gives an excellent fit to the data with a viscosity of 0.18 Pa-s. The predicted curve based on the ann 35 estimated viscosity (0.19 Pa-s) shows a slightly longer drainage time. (0.20 Pa-s) was obtained using ARES rheometer. ig. 3 is a plot of fluid height versus time in the funnel for mineral oil. An excellent match between the data and the best-fit solution to model ODE (assuming a Newtonian fluid in laminar flow) is obtained. Moreover, the viscosity, 0.18 Pa-s, is close to the values found using traditional rheometers. ig. 3 also shows the predicted curve based on the ann 35 estimated rheology, which is shifted to the right. Note that the curve in ig. 3 is concave down for the entire time scale of the experiment. This indicates a high shear-thinning index (near one); other experiments showed concave-up behavior (at least at early times) suggesting a low shear-thinning index Xanthan polymer ig. 4a shows the rheological data for 1.1 wt% xanthan polymer. A good fit can be obtained using a power-law model and rheological parameters n=0.10 and m=15 Pa-s n using the ann 35. Some authors (Song et al., 2006) report that xanthan exhibits a yield stress and a Herschel Bulkley model would fit the data equally as well by extrapolating the shear stress curve to zero shear rate and using parameters n=0.51, m=0.52 Pa-s n, and τ 0 =17 Pa. The ambiguity in the stress at zero shear rate demonstrates the difficulty in yield stress measurement using a traditional rheometer. Any number of values for τ 0 between 0 and 17 Pa would result in an acceptable fit, each with different Herschel Bulkley parameters (n and m). ig. 4b plots the same rheology data as viscosity versus shear rate on a log log plot. The funnel data for 1.1 wt% xanthan is shown in ig. 5. All of the fluid drains from the funnel suggesting no yield stress (or at least below the measurable value of τ 0 =12.5 Pa). However, a power-law model does not adequately fit the funnel data either (significant systematic deviation is observed). Inspection of the shear rates in the funnel show many values well above /s, where a Newtonian plateau may exist (ig. 4b), suggesting a Carreau model (Eq. (11)) may be better suited for the data. No analytical solution is available for flowrate in a tube for Carreau fluids, but the integral shown in Table 1 can be evaluated numerically if the wall shear rate (γ w ) is known. Given the shear stress at the wall (Eq. (5)), the wall shear rate can be found from the nonlinear Eq. (12). ig. 2. Rheological data for mineral oil. A Newtonian fit to the data gives a viscosity of 0.19 Pa-s using the ann 35 and 0.20 Pa-s using the ARES rheometer. ηðγ Þ = η + ðη 0 η h Þ 1+ðλγ Þ2 i n 1 2 ð11þ

6 398 M.T. Balhoff et al. / Journal of Petroleum Science and Engineering 77 (2011) ig. 5. Height versus time funnel data for 1.1 wt% xanthan. A best-fit solution for the model was found using η 0 =220 Pa-s, η =0.014, m=18 Pa-s n, and n=0.12 were found. The predicted curve using the ann 35 estimated parameters is also included. xanthan (Table 2) was tested which also deviated from power-law behavior. At longer times, the 1.4 wt% solution was dominated by low shear rates (b0.1 1/s) where a low-shear Newtonian plateau exists Bentonite ig. 4. Rheological data for 1.1 wt% xanthan presented as (a) shear stress versus shear rate and (b) viscosity versus shear rate on a log log plot. Best fit parameters n=0.10 and m=15 Pa-s n for the ann 35 and n=0.09 and m=16 Pa-s n for the ARES were found. τ w = ηð γ wþ γw ð12þ The parameter λ can be converted to the consistency index, m, used in the power-law model by: m = ðη 0 η Þλ n 1 ð13þ The algorithm for solving the ODE for a Carreau fluid requires several additional numerical approximations; (1) provide initial guess the Carreau rheological properties, (2) at a given stepsize in the numerical ODE solver, calculate the wall stress using Eq. (5); (3) solve for the wall shear rate in Eq. (12) using Newton's method; (4) compute the integral in Table 1 numerically and calculate the flowrate, Q. Update the height for the next timestep using Q. ig. 5 shows a good Carreau model fit to the data for 1.1 wt% xanthan using parameters η 0 =220 Pa-s, η =0.014 Pa-s, m=18 Pas n, and n=0.12. These values now compare favorably to the values obtained from the ann 35 for the power-law regime (m=15 Pa-s n and n=0.10). One advantage of the funnel is the ability to estimate Carreau model parameters that are relevant outside of the shear rate range of the ann 35 viscometer. A higher concentration (1.4 wt%) Solutions with bentonite are often reported as exhibiting a yield stress. ig. 6 shows the ann 35 rheology data for 8.6 wt% (30 lb/bbl) bentonite; both a power-law and Herschel Bulkley model fits the data well. However, less subjectivity is observed in the Marsh unnel compared to the rheometers. The height versus time curve approaches an asymptote (ig. 7) and the flow is observed to come to a complete stop at h=16 cm (not even a drop was observed for several days), which corresponds to a yield stress of τ 0 =47 Pa. The remaining Herschel Bulkley parameters (m and n) were then found from a best fit to the funnel model. The ann 35 data show a minimum stress of 11 Pa suggesting that τ 0 11 Pa, contradicting the funnel data. A yield stress of 11 Pa is not large enough to suspend fluid in the funnel and the model predicts (ig. 8) that fluid drains relatively quickly for the estimated ann 35 parameters compared to the observed behavior. The ann 35 Table 2 Rheological parameters obtained via best fit to various models (power-law, Carreau, or Herschel Bulkley). luid Rheometer Shear rate (1/s) m (Pa-s n ) n η 0 (Pa-s) η (Pa-s) τ 0 (Pa) Mineral oil unnel ann ARES Xanthan (1.1 wt%) Xanthan (1.4 wt%) Bentonite (7.2 wt%) Bentonite (8.6 wt%) Carbopol (2 wt%) HPAM (1 wt%) unnel ann ARES unnel ann ARES unnel ann unnel ann unnel ann unnel ann ARES

7 M.T. Balhoff et al. / Journal of Petroleum Science and Engineering 77 (2011) ig. 6. Rheological data for 8.6 wt% bentonite using the ann 35 rheometer. Best-fit parameters, n=0.74, m=0.69 Pa-s n, and τ 0 =11 Pa, were found. ig. 8. Rheological data for 1 wt% HPAM. Best fit ann 35 parameters n=0.27 and m= 7.0 Pa-s n for the ann 35 and n=0.25 and m=4.9 Pa-s n for the ARES were found. rheometer may underestimate the yield stress because the shear rate (as opposed to shear stress) is imposed. As a result, the threedimensional microstructure of the fluid may break even at low rates, thus yielding the material. On the other hand, the stress is controlled in the funnel by the fluid height. The stress is slowly reduced and fluid comes to a complete stop. As a result, the three-dimensional microstructure remains intact. After reaching a static height, the bentonite fluid was manually disturbed in the funnel; the microstructure broke and began to flow again until it reached a new static height (not shown in the figure) Carbopol Experiments using 2 wt% and 3 wt% carbopol were conducted since it was expected to exhibit a yield stress. 3 wt% carbopol did not flow at all from the original height of 27.5 cm (not even a drop was observed) indicating τ 0 N75 Pa. One limitation of the funnel is the limited range of imposed stress in the upper cone (12 75 Pa). This problem could easily be addressed by using funnels of different size and shape. 2 wt% carbopol was found to have a yield stress of 31 Pa and the funnel flow behavior was similar to that of bentonite. Estimated parameters are shown in Table HPAM igs. 8 and 9 show the rheometer and funnel data, respectively, for 1 wt% HPAM. The fluid did not exhibit a yield stress and appeared to be mostly in a shear-thinning regime; therefore a power-law model seemed to be sufficient. Power-law parameters n =0.27 and m=7.0 Pa-s n were found for the ann 35 which matched relatively well to the ARES (n=0.24 and m=4.9 Pa-s n ). Although the funnel data matched a power-law model very well, the best-fit parameters were very different from the rheometers' (n =0.56 and m=2.5 Pas n ); HPAM was the only fluid which resulted in a significantly different shear-thinning index (n) for the funnel when compared to the ann 35. ig. 7. Height versus time data for 8.6 wt% bentonite. Best-fit parameters to the funnel model were found as n=0.64, m=0.76 Pa-s n, and τ 0 =47 Pa. The best-fit model is also compared to the prediction based on the ann 35 estimates. n ig. 9. Height versus time data for 1 wt% HPAM. A best-fit solution for the model was found using m=2.4 Pa-s, and n=0.57.

8 400 M.T. Balhoff et al. / Journal of Petroleum Science and Engineering 77 (2011) HPAM is well-known to be viscoelastic and exhibit hysteresis and other time-dependent effects (Lake, 1989; Carreau et al., 1997; Delshad et al., 2008). The funnel model does not currently account for any of these phenomena and flow through the constriction from the upper cone to the capillary tube may be largely influenced by elasticity. The elastic fluid elongates as it passes through the constriction and then relaxes in the tube. Moreover, the stress response to a change in shear rate is not instantaneous and rheological measurements should be taken at steady state. In the funnel, a steady state shear rate is never reached (it decreases as fluid drains) which could lead to inaccurate predictions of steady-shear behavior. or fluids with minimal elastic effects, the slow change of shear rates may not significantly affect results, but these effects may partially explain the discrepancy between the rheometers and funnel results for HPAM. The model could be potentially improved by including an elastic component in the model. Several constitutive equations for stress (Maxwell, Oldroyd-B, etc.) exist, but derivation of resulting flow equations through a constriction can be difficult. uture work might focus on computational fluid dynamics modeling of viscoelastic flow in the funnel to investigate its effect on drainage. 5. Discussion Table 2 summarizes the estimated rheological properties for all fluids tested assuming a shear-thinning (Power-Law or Carreau) or yield stress (Herschel Bulkley) model. In most cases, very good agreement occurs between the ann 35 and funnel rheometers for shear-thinning fluids (elastic HPAM being the exception), despite several assumptions. The shear-thinning index (n) was very consistent between the ann 35 and funnel for most fluids. The consistency index (m) was comparable, but higher in the funnel (~30%) on average. The additional resistance could be due to the many model assumptions (resistance in the tube alone, no entrance or exit effects, and elasticity ignored). The funnel method has some advantages over other portable lab rheometers (e.g. ann 35) of more data points, ability to calculate parameters describing Newtonian plateaus at low and high shear rates, and a more objective measurement of fluid yield stress. Table 2 also shows reasonable agreement for yield stress fluids (8.6 wt% bentonite and 2 wt% carbopol). Although yield stress measurements are notoriously difficult to obtain, the funnel provides a simple and objective measurement of yield stress. Unlike extrapolation techniques (on curves of stress versus shear rate) or even static tests (e.g. Creep), there is little ambiguity in measurement. The table shows that the funnel estimated a higher yield stress than the ann 35 viscometer for bentonite, possibly because the threedimensional microstructure broke in the controlled-rate viscometer. The height versus time data necessary for solution to the model is easy to obtain and, in theory, could be obtained with a beaker and stopwatch. Solution to the ODE is robust and convergence to a minimum sum of squared residuals for parameter estimation is good provided a decent initial guess is given (e.g. the ann 35 values). However, alternatives to the solution of the ODE for data reduction are possible. or example, plots of flowrate (dv/dt) versus hydrostatic head data could be generated and the flow equations in Table 1 fit to the data. Equivalently, the data could be converted to curves of apparent viscosity versus shear rate. Data reduction might be even easier using these approaches, but requires differentiating (instead of integrating) the data and more scatter is observed (especially near the end of the experiment when Q approaches zero). Another alternative to the approach presented in this work is to use only a few data points of height to estimate parameters. In most cases the model fit of height versus time was excellent suggesting that only two data points would be necessary to obtain parameters for a power-law fluid. Two experiments could be conducted with different initial volumes (e.g ml and 1500 ml) and the power-law parameters extracted explicitly from only the total drainage time of the experiments. The ODE (Eq. (4)) would still need to be solved, but it eliminates the need to obtain entire curves of height versus time. 6. Conclusions We present a new method for obtaining rheological properties of drilling muds and other non-newtonian fluids that uses height versus time data in a draining Marsh unnel. Currently the Marsh unnel is used in the oilfield industry to estimate a relative viscosity of drilling fluids, but current practice uses only one data point the total drainage time. Consequently, the procedure can be used to unambiguously estimate only one rheological parameter. The new method can be used to estimate several parameters, including the fluid yield stress. The method is simple and very inexpensive; in theory it can be performed with a funnel, beaker, and a stopwatch. The new funnel approach appears to be as accurate as the popular ann 35 viscometer. The Marsh unnel could be used as a tool in the field for areas in the oil industry other than drilling (e.g. enhanced oil recovery and fracturing). A few major conclusions of this work are as follows. 1. There are excellent fits between the height versus time data and the model (typically the curves lie on top of each other). This suggests that a full data set used for least-squares fit may not be necessary. Instead for a two-parameter model, drainage time could be measured for two different initial heights and the parameters calculated explicitly. 2. luids that have height versus time data that do not match the theoretical model well after a least-squares fit, probably should be described by a more complicated rheological model. or example, the funnel shear rate may be in a Newtonian plateau, which means that a better constitutive equation (e.g. 4-parameter Carreau model) is needed. Additionally, some fluids exhibit strong elastic effects (e.g. HPAM) and additional work should be performed to account for the behavior in the model. 3. The ultimate static height in the funnel is a method to objectively estimate yield stress. High-concentration solutions of bentonite and carbopol drained to a static height from which calculation of fluid yield stress is simple and unambiguous. There is no extrapolation. This approach has several advantages over other existing methods that are subjective and/or difficult to perform. 7. List of variables g gravity (cm/s 2 ) h height of fluid in the funnel (cm) h 0 initial height in the funnel (cm) h ss steady state height in the funnel (cm) total height of the cone portion of the funnel (cm) L length of the capillary tube (cm) m consistency index (Pa-s n ) n shear-thinning index P pressure (Pa) Q flow rate (cm 3 /s) r radius of fluid in the funnel (cm) R radius of capillary tube (cm) maximum radius of the funnel (cm) t time (s) t f total drainage time (s) V volume of fluid in the funnel (cm 3 ) α coefficient relating volume and height γ shear rate (1/s) γ w shear rate at the wall of capillary tube (1/s) η non-newtonian viscosity (Pa-s) η 0 low shear rate viscosity (Pa-s) high shear rate viscosity (Pa-s) η

9 M.T. Balhoff et al. / Journal of Petroleum Science and Engineering 77 (2011) λ Carreau model parameter (s) μ viscosity (Pa-s) ρ fluid density (g/cm 3 ) τ w shear stress at the wall of capillary tube (Pa) yield stress (Pa) τ 0 Acknowledgements We would like to thank the Petroleum and Geosystems Department and UT-Austin as well as the Yates oundation for funding this project. We would also like to thank Chun Huh and Do Hoon Kim for the many thoughtful discussions on the subject. Larry W. Lake holds the W.A. (Monty) Moncrief Centennial Chair. Appendix. Verification of assumptions in funnel model The mathematical model for fluid height in the funnel was derived in Eqs. (1) (10) assuming that the all of the resistance for a flowing fluid is in the lower tube and none in the upper cone. Additionally, it is assumed that flow is quasi-static so the steady-state momentum equations can be utilized. Newtonian fluids without a yield stress The total pressure drop in the funnel is the sum of pressure drops in series of the tube and upper cone: ΔP total = ΔP tube + ΔP cone ða 1Þ or a Newtonian fluid, the pressure drop in the lower tube is Poiseuille flow, fluids. This is mainly because the radius quickly becomes large and the resistance is inversely proportional to r 4. Resistance in the upper cone is even smaller for shear thinning fluids without a yield stress because resistance is inversely proportional to r 3+1/n. Since n is less than one, the pressure drop because of viscous flow is even less significant. Yield-stress fluids or yield stress fluids there is additional pressure loss (outside of the viscous forces derived in Eq. (A-7) in the upper cone). At steady state (h=h ss ) there is no flow in the funnel (Q=0) and Eq. (A-1) can then be written as: ΔP total = 2Lτ 0 R Eq. (A-4) gives: ΔP total = + 2hτ 0 rh ð Þ funnel 2L R + 2H τ 0 This leads to the following expression for yield stress ða 8Þ ða 9Þ τ 0 = ρgh ð ss + L Þ ða 10Þ 2L R + 2 This also implies that the total pressure drop should not be used for the Bingham and Herschel Bulkley flow equations (Table 1). or relatively large flow rates the pressure drop in the cone is small but becomes significant as the drainage rate decreases and yield stress effects become important. The pressure loss in the tube should be corrected as follows: ΔP tube = 8μL πr 4 tube Q ða 2Þ ΔP tube = ρgðh+ LÞ 2 τ R 0 ða 11Þ In the upper cone, flow is not unidirectional but a lubrication approximation can be used to give a good approximation: P funnel h 8μ = Q πrðhþ 4 funnel ða 3Þ Where the cone radius increases linearly from the connection of the upper cone and capillary tube. rh ð Þ = R + R h h Substituting and separating variables gives P = 8μ π ða 4Þ H H 4 Q R 0 h 4 dh ða 5Þ Quasi-steady flow assumption Another assumption included in the model is quasi-steady flow; that is time-dependent effects associated with the fluid head are neglected. A transient momentum balance describing flow in the capillary tube is given as: ðρv z Þ = ρg t z + 1 r ð r rτ rzþ ða 12Þ The assumption is that the time derivative is small compared to the other two terms on the right hand side (gravity and viscous resistance). or constant density fluids, this requires: ðv z Þ bbg t z ða 13Þ And solving the resulting ODE, gives: ΔP cone = 8μ H 4 Q 1 π 3H 3 ða 6Þ The assumption is valid for all fluids tested. or all experiments conducted in this work, the unsteady term is factors of ten smaller than gravity. or example mineral oil has one the largest time derivatives and it is on average only g z. The ratio of pressure drops is now: ΔP cone = R 4 tube ΔP tube 3L = bb1 ða 7Þ This calculation is based on the height of the top of funnel which represents the most possible resistance of the cone. The pressure drop because of viscous flow in the upper cone is negligible for Newtonian References Balhoff, M., Modeling the low of non-newtonian fluids at the Pore Scale. PhD Dissertation. Louisiana State University. Balhoff, M.T., Miller, M.J., An analytical model for cleanup of yield-stress fluids in hydraulic fractures. Soc. Pet. Eng. J. 10 (1), Barnes, H., Walters, K., The yield stress myth? Rheol Acta 24, Bourgoyne Jr., A.T., Chenevert, M.E., Millheim, K.K., Young Jr.,.S., Applied Drilling Engineering, SPE Textbook Series, Vol. 2. Society of Petroleum Engineers, Richardson, TX.

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