Equations: Perforating

The equations of the “Perforating” lessons in “Oil and gas production”, with their conditions and units, and what each symbol means in them.

Speed of the tunnel deepening and pressure at the front of the jet

\tfrac12\,\rho_j\,(v - u)^2 = \tfrac12\,\rho_t\,u^2 \;\Rightarrow\; u = \frac{v}{1 + \sqrt{\rho_t/\rho_j}},\qquad p = \tfrac12\,\rho_t\,u^2

Holds when: The jet and the target are incompressible fluids with no strength (the pressure is much larger than the strength). The penetration is steady: the speed of the element of the jet is constant. The jet of a powder liner consists of particles, and its effective density is lower than the density of the metal.

Shaped-charge perforating →

Density law: depth of the tunnel from the length of the jet

P = L\,\sqrt{\frac{\rho_j}{\rho_t}}

any consistent units of length and density

Holds when: The jet is continuous and has no strength, and all its elements penetrate in the steady regime. It is an upper estimate: the stretching and breakup of the jet, the resistance of the target and water on the way reduce the depth.

Shaped-charge perforating →

Pressure opposite the reservoir before the shot, and the underbalance

p_{\text{well}} = p_{wh} + \rho\,g\,h, \qquad \Delta p_{\text{ub}} = \bar p_R - p_{\text{well}}

Holds when: The liquid is at rest, and its density is constant over the height. If there are several liquids in the well, their columns are added. Gas above the liquid level is hundreds of times lighter than the liquid, and its weight is ignored (at a high gas pressure it is included). The reservoir pressure is the current one. In a depleted reservoir it is lower than the initial one, and the same column of liquid that earlier gave an underbalance may now give an overbalance.

Perforating operations: gun conveyance and procedures →

Minimum underbalance for an oil-bearing sandstone (field criterion)

\Delta p_{\min} \approx \frac{3100}{k^{0.37}}

empirical: $\Delta p_{\min}$ in psi, $k$ in mD

Holds when: Oil-bearing sandstones, perforating on tubing with a static underbalance. For gas a different relation was found, and for carbonates the link with permeability does not work. It is an approximate fit to scattered well data, with permeability mostly from a few to several hundred millidarcies. Different treatments of the same data give a coefficient of about 2900 to 3500 psi with an exponent of 0.36–0.37, and outside this range the formula is an extrapolation. It is a lower limit based on an indirect sign (the response to acid), not a guarantee of clean tunnels. Later laboratory criteria call for an underbalance 2–4 times larger.

Perforating operations: gun conveyance and procedures →

Pressure loss in the perforation holes

\Delta p_{\text{perf}} = \frac{\rho\,q^2}{2\,C_d^2\,n^2 A^2}, \qquad A = \frac{\pi d^2}{4}

Holds when: The flow is like flow through a short nozzle: pressure turns into jet speed, and friction along the length of the tunnel is ignored. The rate is divided equally between the holes. Erosion of the holes by proppant increases both $C_d$ and $d$ , and the losses fall several times.

Perforating operations: gun conveyance and procedures →

Symbols

Symbol	Meaning and context
$C_d$	discharge coefficient: about 0.6 for new holes with sharp edges, and it grows to about 0.9 when proppant rounds the edges, —Perforating operations: gun conveyance and procedures
$k$	reservoir permeability, mDPerforating operations: gun conveyance and procedures
$L$	length of the continuous jet, mmShaped-charge perforating
$\Delta p_{\min}$	smallest static underbalance at which the tunnels were cleaned, psiPerforating operations: gun conveyance and procedures
$n,\ d$	number of open holes and their diameter, —, mPerforating operations: gun conveyance and procedures
$p$	pressure at the front of the jet, PaShaped-charge perforating
$P$	depth of the tunnel in the target, mmShaped-charge perforating
$p_{\text{well}}$	well pressure opposite the perforated interval at the moment of the shot (gauge), PaPerforating operations: gun conveyance and procedures
$p_{wh}$	pressure at the wellhead in the pipes through which the pressure is passed to the interval (above the liquid), PaPerforating operations: gun conveyance and procedures
$\Delta p_{\text{perf}}$	pressure loss as the liquid passes from the casing into the reservoir through the holes, PaPerforating operations: gun conveyance and procedures
$q$	flow rate of liquid through these holes, m³/sPerforating operations: gun conveyance and procedures
$\bar p_R$	reservoir pressure at the depth of the interval (gauge), PaPerforating operations: gun conveyance and procedures
$u$	speed at which the tunnel deepens, m/sShaped-charge perforating
$\Delta p_{\text{ub}}$	underbalance; a negative value is an overbalance, PaPerforating operations: gun conveyance and procedures
$v$	speed of an element of the jet, m/sShaped-charge perforating
$\rho$	density of the liquid (of the mixture, if it carries proppant), kg/m³Perforating operations: gun conveyance and procedures
$\rho,\ h$	density of the liquid and the true vertical height of its column above the middle of the interval, kg/m³, mPerforating operations: gun conveyance and procedures
$\rho_j$	density of the metal of the jet, kg/m³Shaped-charge perforating
$\rho_j,\ \rho_t$	densities of the jet and of the target, kg/m³Shaped-charge perforating
$\rho_t$	density of the target: steel, cement, rock, kg/m³Shaped-charge perforating