Well Drilling Technology

Non-Newtonian fluid rheology

A non-Newtonian fluid is one whose shear stress is not directly proportional to shear rate, so its apparent viscosity \mu_a = \tau/\dot{\gamma} changes with how fast it is sheared. Drilling muds are almost always non-Newtonian, and the time-independent models used most often are the Bingham plastic, the power law and the yield-power law. The Bingham plastic model is \tau = \tau_0 + \mu_p\dot{\gamma}, where \tau_0 is the yield stress, the minimum stress needed to start flow, and \mu_p is the plastic viscosity, which is the slope of the line. The yield stress is the starting force divided by the plate area, and the plastic viscosity is found from the steady-state stress minus the yield stress, divided by the shear rate. For example, if the starting stress is 20 Pa and the steady stress is 50 Pa at a shear rate of 5 s^{-1}, then \mu_p = (50-20)/5 = 6 Pa·s.

The power law model is \tau = K\dot{\gamma}^n, where n<1 gives shear-thinning (pseudoplastic) behavior, n>1 gives dilatant behavior, and n=1 reduces to a Newtonian fluid. In a circular pipe the wall shear rate of a power law fluid is \dot{\gamma}_w = \left(\dfrac{3n+1}{4n}\right)\dfrac{8V}{D}, where V is the mean velocity and D is the pipe diameter. In a rotational viscometer the shear rate is proportional to rotor speed and the shear stress is proportional to the dial reading. Two readings at speeds N_1 and N_2 therefore give n = \log(\theta_2/\theta_1)/\log(N_2/N_1). For readings of 10 and 16 at 300 and 600 rpm, n = \log 1.6/\log 2 \approx 0.68. For a viscometer with a maximum dial reading, the highest measurable apparent viscosity at speed N is set by the full-scale stress divided by the shear rate at that speed.

Some fluids are also time-dependent. A thixotropic fluid loses viscosity under constant shear and regains it when at rest, whereas a rheopectic fluid does the opposite. Conventional water-based and clay-based muds are pseudoplastic and thixotropic. After shearing stops, a fluid with a relaxation equation of the form \frac{\mu_0}{\alpha}\frac{d\tau}{dt}+\tau = \mu_0\dot{\gamma} has shear stress that decays exponentially as \tau = \tau_0 e^{-\alpha t/\mu_0}, with relaxation time \mu_0/\alpha.

Duplex pump volumetric efficiency

A duplex pump has two cylinders, and each cylinder is double-acting, so fluid is displaced on both the forward and the backward stroke. On the forward stroke the piston displaces the full liner area A_l. On the return stroke the piston rod occupies part of the cylinder, so only the annular area A_l - A_r is swept, where A_r is the rod area. The displacement of each cylinder per stroke is therefore (2A_l - A_r)L, where L is the stroke length. The theoretical output of a duplex pump is Q_{th} = 2(2A_l - A_r)LN, with N in strokes per minute. The actual output is Q = \eta_v Q_{th}, where the volumetric efficiency \eta_v accounts for valve leakage and slippage. A single-acting pump would use only A_l per stroke, which is the key difference from the double-acting case.

When the question asks for the total volume per complete pump cycle, the same formula applies with both cylinders included, multiplied by \eta_v. For example, a two-cylinder pump with a liner area of 100 cm^2, a rod area of 20 cm^2, a stroke of 30 cm and \eta_v = 0.9 displaces 0.9 \times 2 \times (200 - 20) \times 30 = 9720 cm^3 per revolution.

Lost circulation and hole ballooning

Lost circulation occurs when drilling fluid escapes from the wellbore into a permeable or fractured formation. It is detected when the mud outflow at surface stays below the mud inflow pumped down the string for a sustained period, which means fluid is leaving the hole. Losses reduce the hydrostatic head, so they can lead to a kick if the wellbore is not refilled quickly.

Hole ballooning is a different phenomenon. A weak formation under high wellbore pressure accepts mud temporarily, because its pores or fractures open elastically, and then returns the fluid when the pressure is reduced. On the outflow record it appears as a short dip below the inflow, followed by an overshoot above the inflow, and then a return to the baseline. The key difference is that a ballooning formation gives back the mud, whereas a lost-circulation zone does not.

The pressure profile of a casing string shows whether the wellbore has full integrity. Full integrity requires the casing burst strength and the formation fracture strength to exceed the internal pressure at the casing setting depth (CSD). If the formation fracture strength falls below the internal pressure just beneath the shoe, the formation is the weak element. If the casing burst strength falls below the internal pressure at the CSD, the casing is the weak element.

Hoisting system mechanical efficiency

The drawworks supply power to the fast line of the hoisting system. If n lines run between the crown block and the traveling block, the hook moves at v_{hook} = v_f/n, where v_f is the fast-line speed, and the hook load is supported by n line segments. The hoisting efficiency \eta_m is the ratio of hook power to the input power, and in practice the power balance for a constant output is T_f\,v_f = \text{constant}, where T_f is the fast-line tension.

Because the power is fixed, the pullout speed varies inversely with the fast-line tension. If the tension rises from 20 000 lb to 25 000 lb at the same output power, the speed falls from 60 ft/min to 60 \times 20\,000/25\,000 = 48 ft/min.

The static derrick load is the load carried by the derrick structure and equals the hook load plus the dead-line tension. With ideal sheaves the tension is the same in every line, so each line carries W/n and the derrick load is F_d = W\left(1 + \dfrac{1}{n}\right). For a hook load of 2\times10^5 kg with n = 4 supporting lines, F_d = 2.5\times10^5 kg.

Stokes' law particle settling

Stokes' law gives the terminal settling velocity of a small sphere in a laminar, Newtonian fluid as v_s = \dfrac{g\,d_s^2(\rho_s - \rho_l)}{18\,\mu}, where d_s is the particle diameter, \rho_s and \rho_l are the particle and fluid densities, and \mu is the fluid viscosity. The velocity is proportional to the square of the diameter and to the density difference, and it is inversely proportional to viscosity. Doubling the diameter therefore increases the settling velocity fourfold. The law is valid only for low particle Reynolds numbers.

When two spheres of different density and size settle at the same velocity in the same fluid, g, \mu and \rho_l cancel, so d_s^2(\rho_s - \rho_l) must be equal for both. A denser or heavier particle can therefore match the settling velocity of a larger but lighter one.

Since the velocity scales with d^2, the ratio of settling rates for two sizes in the same fluid is the square of the ratio of their diameters. A particle three times as large as another settles nine times as fast, which is the basis for comparing the settling behavior of different batches of barite.

Drilling rig and BHA equipment functions

Solids-control equipment removes cuttings from the circulating mud, and the separating mechanism distinguishes each unit. A shale shaker uses screening, a vibrating screen that separates solids by size. Hydrocyclones and centrifuges use forced settling, because centrifugal force many times greater than gravity drives the denser solids outward and downward faster than they would settle naturally. Chemical flocculation and dilution are other methods that are used in combination with mechanical equipment, but they are not the basic mechanism of a hydrocyclone or centrifuge.

The bottom-hole assembly (BHA) contains the components that are closest to the bit. Drill collars add weight to the bit, and stabilizer subs are short sections with blades or ribs sized close to the hole diameter. A stabilizer keeps the drill collars centralized in the wellbore, which helps control hole inclination, reduces lateral vibration and lowers the risk of differential sticking. It does not hold the drill string up or isolate formations, which are functions of the hoisting system and of casing and cement respectively.

d-exponent drilling efficiency

The d-exponent is a normalized drilling parameter introduced by Jorden and Shirley to detect changes in formation pressure while drilling. It is defined as d = \dfrac{\log(R/60N)}{\log(12W/1000D)}, where R is the rate of penetration in ft/hr, N is the rotary speed in rpm, W is the weight on bit in klb and D is the bit diameter in inches. The equation shows that the d-exponent increases with penetration rate and decreases with rotary speed, and it is normalized for the effects of weight and bit size.

As an example, a bit of 8 in diameter drilling at 100 rpm with 20 klb on bit and a penetration rate of 60 ft/hr gives \log(60/6000) = -2 in the numerator and \log(240/8000) = -1.523 in the denominator, so d \approx 1.31. The d-exponent is usually corrected for mud weight with d_c = d\,\rho_n/\rho_e, where \rho_n is the normal pore-pressure mud weight and \rho_e is the actual mud weight. A departure of the corrected value from its normal trend with depth indicates abnormal pressure.

Drilled volume and hole enlargement

The volume of rock drilled equals the cross-sectional area of the hole multiplied by the drilled length. The rate of penetration is the drilled length divided by the drilling time, so ROP = V/(A\,t), where V is the drilled rock volume, A is the hole area and t is the time. The area must be computed from the actual hole diameter, not from the bit diameter, because the hole may have been enlarged by washout, mechanical erosion or a swelling formation.

For example, if 200 ft^3 of rock is drilled in 4 hours through a hole with a diameter of 9.0 in, the area is \frac{\pi}{4}(0.75)^2 = 0.4418 ft^2, the drilled length is 200/0.4418 = 452.7 ft and the penetration rate is 113.2 ft/hr. Using the 8.5-in bit diameter instead would give a larger and incorrect value, which is why the enlarged diameter must be used.