I.2 Las actuaciones arqueológicas
I.2.6 Bóveda bien se merece una Misión científica
All fluids encountered in drilling and production operations can be character-ized as either Newtonian fluids or non–Newtonian fluids. Newtonian fluids, like water, gases, and thin oils (high API gravity) show a direct proportional relation-ship between the shear stress τ and the shear rate ˙γ, assuming pressure and temperature are kept constant. They are mathematically defined by:
τ = µ ˙γ , (8.1)
where τ is shear stress, ˙γ is shear rate. The proportionality coefficient µ is the (dynamic) viscosity of the fluid. A dimensional analysis shows that dynamic viscosity has the dimension [F L−2 T ] or [M L−1 T−1]. Typical units are N sm2 = P a s, P (poise in regard to Poiseuille), cP (centipoise), and lbf sf t2 . Conversion factors are:
1 P = 1 g cm s , 1 N s
m2 = 10 P = 1000 cP , 1 cP = 1
47880.259 lbf s
ft2 .
A plot of τ versus ˙γ produces a straight line that passes through the origin and has a slop µ. Viscosity usually changes with pressure and temperature.
(See Figure 8.1).
Most fluids encountered at drilling operations like drilling mud, cement slur-ries, heavy oil, and gelled fracturing fluids do not show this direct relation-ship between shear stress and shear rate. They are characterized as non–
Newtonian fluids. To describe the behavior of non-Newtonian fluids, various
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Figure 8.1: Typical graph of Newtonian fluids.
models like the Bingham plastic and the Power Law fluid models (two parame-ter models), the Herschel–Bulkley and the Robertson–Stiff fluid models, which are time–independent models. There exist time–dependent fluid models, which present change of viscosity and other parameters based in time and shear history. Time–dependent fluids model are sub-classified as thixotropic(time–
thinning) and rheopetic (time-thickening).
It shall be understood that all the models mentioned above are based on different assumptions that are hardly valid for all drilling operations, thus they are valid to a certain extend only. Most commonly drilling fluids are treated behaving either aa a Bingham plastic or a power–law fluid. These two models can describe relatively well most of the common drilling fluid for all ordinary drilling operations.
8.1.0.1 Bingham–Plastic Fluid Model:
The Bingham plastic fluid model is a linear model (although not proportional), and is expressed mathematically as follows.
τ = τy+ µp ˙γ ˙γ > 0 τ = −τy+ µp ˙γ ˙γ < 0
˙γ = 0 −τ y ≤ τ ≤ τy
(8.2)
The constant τy is called the yield point usually denoted “YP” and µy is called plastic viscosity, usually denoted “PV”. The typical graph of a Bingham plastic fluid is shown in Figure 8.2.
Figure 8.2: Typical graph of Bingham-plastic fluids.
8.1.0.2 Power–Law Fluid Model:
The power law fluid model is non–linear and do not present a yield stress, and can be expressed as follows.
τ = K ( ˙γ)n ˙γ ≥ 0
τ = −K (− ˙γ)n ˙γ < 0 (8.3)
The constant K is called the consistency index and n is called behavior in-dex.The typical graph of a power–law fluid is shown in Figure 8.3.
Figure 8.3: Typical graphs of power–law fluids.
When the plot is done on a log–log scale it results in a straight line. Here the slope determines the flow behavior index n and the intercept with the verti-cal the value of the consistency index (log K). The flow behavior index, which ranges from 0 to 1.0, declares the degree of non–Newtonian behavior, where n = 1.0 indicates a Newtonian fluid. Mathematically n can be greater than 1.0, but drilling fluids do not present this characteristic. Another characteristic (er-roneously shown in the graph) is that the power–law presents an infinite slope at ˙γ = 0. This, and other experimental results lead to the proposal of other fluid models. The consistency index K on the other hand gives the thickness fluid,
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where larger K represent thicker (more viscous) fluids. It should be under-stood, however, that K has not the dimension of viscosity. Consistency index K is usually expressed in equivalent viscosity units.
8.2 Rheometry
To determine the rheological properties of a particular fluid a rotational vis-cometer or rheometer is commonly used. Rheometers may have two, six, or continuous (variable) speeds. The typical rotational rheometer (see Figure 7.3 has an arrangement of two concentric cylinders in which the outer cylinder ro-tates and the inner cylinder actuate a torsion spring with a dial, as depicted in Figure 8.4.
Figure 8.4: Arrangement of a rotational viscometer.
The rotation of the rotor (external cylinder) shears the fluid between the rotor and the bob (internal cylinder), which transmits torque from the rotor to the bob.
The torque causes a deflection of the dial (proportional to the torque), against the resistance of the torsion spring. The wide use of rheometers in the field leaded to the design of special dimensions that made easy the measurement of fluid parameters.
8.2.1 Viscosity of Newtonian Fluids
The dimensions of the rotor and bob, and the coefficient of the torsion spring are determined such that the viscosity of a Newtonian fluid in centipoise (cP) is obtained directly by reading the deflection of the dial in degrees, when the rotor rotates at 300 rpm.
8.2.2 Parameters of Bingham–Plastic Model Fluids
Two measurements at different rotation speeds are required. The parameters are obtained with the following formulas (please note the units):
µp = 300
N2 − N1(θN 2− θN 1) [cP]
τy = θN 1− N1
300µp [lbf/100 ft2]
Note that any two rotation speeds can be selected, but 300 rpm and 600 rpm lead to the simplification of the calculation, in addition to cover a range of shear rates that typically occurs in drilling operations. The simplified expressions for N1 = 300 rpmand N2 = 600 rpm are:
µp = θ600− θ300 [cP]
τy = θ300− µp [lbf/100 ft2]
(8.4)
The general formulas are useful to determine the parameters in other ranges of interest (too small or too large).
8.2.3 Parameters of Power–Law Model Fluids
The parameters are obtained with the following general formulas:
n = log (θN 2/θN 1)
log (N2/N1) [1]
K = 510 θN
(1.703 N )n [eq.cP]
Again, any two rotation speeds can be selected. For 300 rpm and 600 rpm the expressions reduce to:
n = log (θ600/θ300)
log 2 [1]
K = 510 θ300
511n [eq.cP]
(8.5)
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Example 31: A fluid rheometry test results in dial readings of θ300 = 13 and θ600 = 22. Calculate the parameters using power–law and Bingham–plastic models Solution:
(a) Power–law
n = log (22/13)
log 2 = 0.759 K = 510 × 13
5110.759 = 58.3 eq.cP (b) Bingham
µp = 22 − 13 = 9 cP τy = 13 − 9 = 4 lbf/100 ft2
8.2.4 Gel Strength
Gel strength is the shear stress measured at low shear rate after a mud has set quiescently for a period of time (10 seconds and 10 minutes in the standard API procedure, although measurements after 30 minutes or 16 hours may also be made).
To measure the gel strength a rheometer with low rotary speed (3 rpm) is needed. The sample is sheared at 600 rpm for a period (1 to 5 minutes) and set to rest for the determined time (10 seconds or 10 minutes). Then the shear is applied at 3 rpm and the maximum deflection of the dial is read. The value indicates the gel strength at 10s or 10m in lbf/100ft2.