Equations: Offshore and subsea drilling

The equations of the “Offshore and subsea drilling” lessons in “Drilling engineering”, with their conditions and units, and what each symbol means in them.

Least tension at the top of the riser

T_{\min} = w_s\,h_w + A_{\text{bore}}\,(p_m - p_{sw}), \qquad p_m = \rho_m\,g\,(h_w + h_a), \quad p_{sw} = \rho_{sw}\,g\,h_w

SI; in the examples, kN, MPa and g/cm³ ( $p = 0.00981\,\rho\,h$ MPa with $\rho$ in g/cm³)

Holds when: The riser is vertical and does not move. Current, waves and heave are not included. We neglect the height of the preventer stack above the seabed. The mud stands as a column from the flowline at the height of the rotary table. While mud circulates, friction losses add to the pressure inside. This is a lower limit. The tensioners are set higher, with a margin for the inaccuracy of the weight and of the buoyancy, for the friction and the tilt of the wire ropes, and for the sudden failure of a tensioner.

Offshore and subsea drilling →

Tensioner setting with a margin for a failure

T_{\text{set}} = \frac{N}{(N - n)\,\eta}\;T_{\min}

any one unit of force

Holds when: The load is shared equally between the tensioners. A failed tensioner stops pulling at once. $\eta$ and the allowed number of failures are taken from the technical data of the rig and from the work plan. The values in the example are assumed.

Offshore and subsea drilling →

Mud density with a riser margin

\rho_{\text{rm}} = \frac{\rho_{\text{p}}\,H - \rho_{sw}\,h_w}{H - h_a - h_w}, \qquad \Delta\rho_{\text{rm}} = \rho_{\text{rm}} - \rho_{\text{p}} = \frac{\rho_{\text{p}}\,(h_a + h_w) - \rho_{sw}\,h_w}{H - h_a - h_w}

densities in the same unit (g/cm³), depths vertical, in metres

Holds when: After the disconnect there is sea water from sea level down to the wellhead, and the same mud stays below the seabed. The height of the stack above the seabed is not included. The pressures are hydrostatic: there is no circulation, and there is no gas in the mud. We calculate for the formation with the highest pressure in the open hole. If $\rho_{\text{rm}}$ is larger than the equivalent mud weight of the fracture pressure under the shoe, the margin cannot be reached.

Offshore and subsea drilling →

Angles at the bottom and at the top of a tensioned riser

\tan\theta_{\text{bot}} = \frac{T_{\text{top}} - T_{\text{bot}}}{T_{\text{bot}}\,\ln(T_{\text{top}}/T_{\text{bot}})}\cdot\frac{x}{L}, \qquad \tan\theta_{\text{top}} = \frac{T_{\text{bot}}}{T_{\text{top}}}\,\tan\theta_{\text{bot}}

any consistent units: the tensions in one unit, $x$ and $L$ in one unit

Holds when: The riser is a taut string: its bending stiffness matters only at the ends. There is no current and no waves, and the offset is static. The effective weight of one metre is the same along the length, and the tension grows linearly from the bottom up: $T_{\text{top}} - T_{\text{bot}} = w\,L$ . The top of the riser is taken at the water surface. The part above the water up to the upper joint is not included. With it, the angles are a little smaller for the same offset, so the calculation is on the safe side. The angles are small: the tangent is close to the angle in radians.

Offshore and subsea drilling →

Capacity of a conductor along its side surface

Q = \pi D\,\alpha\int_0^{L} s_u(z)\,\dd z = \pi D\,\alpha\left(s_{u0}\,L + \frac{k\,L^2}{2}\right) \quad \text{for}\ s_u = s_{u0} + k\,z

kN, m, kPa

Holds when: Soft clay. We ignore the support under the end of the open pipe, which is on the safe side. We take $s_u$ and $\alpha$ from site surveys and from the experience of installation on neighbouring wells. The values in the example are assumed. Cyclic bending (bending that repeats again and again) from the riser and the stack weakens the soil at the top of the conductor with time. We take it into account separately.

Offshore and subsea drilling →

Symbols

Symbol	Meaning and context
$A_{\text{bore}}$	area of the inner bore of the riser, m²Offshore and subsea drilling
$D$	outer diameter of the conductor, mOffshore and subsea drilling
$H$	true vertical depth of the formation from the rotary table, mOffshore and subsea drilling
$h_a$	height of the rotary table (and of the flowline) above sea level, mOffshore and subsea drilling
$h_a,\ h_w$	height of the rotary table above sea level and depth of the sea, mOffshore and subsea drilling
$h_w$	depth of the sea; the length of the riser is taken equal to it, mOffshore and subsea drilling
$k$	increase of strength with depth, kPa/mOffshore and subsea drilling
$L$	length of the riser (here, the sea depth), mOffshore and subsea drilling
$L,\ z$	length of the conductor in the soil and the depth below the seabed, mOffshore and subsea drilling
$N,\ n$	number of tensioners and the number of those that may fail at the same time, —Offshore and subsea drilling
$p_m,\ p_{sw}$	pressure of the column of mud inside and of the sea water outside, at the bottom of the riser, PaOffshore and subsea drilling
$Q$	force that the soil holds by friction and adhesion along the wall, kNOffshore and subsea drilling
$s_u,\ s_{u0}$	undrained shear strength of the clay at depth $z$ and at the seabed, kPaOffshore and subsea drilling
$T_{\min}$	tension at the top of the riser at which the effective axial force at its bottom is zero, NOffshore and subsea drilling
$T_{\text{set}}$	total tension to which the tensioners are set, NOffshore and subsea drilling
$T_{\text{top}},\ T_{\text{bot}}$	effective tension at the top (at the water surface) and at the bottom of the riser, kNOffshore and subsea drilling
$w_s$	weight of one metre of the riser structure in sea water: pipes, lines, flanges and buoyancy modules together, without the fluid in the bore, N/mOffshore and subsea drilling
$x$	offset of the rig from the point above the wellhead, mOffshore and subsea drilling
$\alpha$	the fraction of the clay strength that is used at the wall: small right after jetting, growing with time, —Offshore and subsea drilling
$\Delta\rho_{\text{rm}}$	riser margin, g/cm³Offshore and subsea drilling
$\eta$	share of the force of the tensioners that reaches the riser along the vertical (friction, tilt of the wire ropes), —Offshore and subsea drilling
$\theta_{\text{bot}},\ \theta_{\text{top}}$	angles of the riser with the vertical at the lower and at the upper joint, °Offshore and subsea drilling
$\rho_m,\ \rho_{sw}$	density of the mud and of the sea water (1030 kg/m³), kg/m³Offshore and subsea drilling
$\rho_{\text{p}}$	pore pressure of the formation as an equivalent mud weight, measured from the rotary table, g/cm³Offshore and subsea drilling
$\rho_{\text{rm}}$	mud density that holds the formation even after the riser is disconnected, g/cm³Offshore and subsea drilling
$\rho_{sw}$	density of sea water, about 1.03, g/cm³Offshore and subsea drilling