Equations: Sand production

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

Mohr–Coulomb strength criterion in principal effective stresses

\sigma'_1 = C_0 + q\,\sigma'_3,\qquad q = \tan^2\!\left(45^\circ + \frac{\varphi}{2}\right) = \frac{1 + \sin\varphi}{1 - \sin\varphi},\qquad C_0 = 2S_0\sqrt{q}

any identical units of stress. The stresses are effective, that is, less the pore pressure

Holds when: The intermediate principal stress $\sigma'_2$ does not affect the strength (in reality it affects it a little, usually strengthening the rock). The straight line is an approximation. At large confinements the strength curve flattens, and in loose sands the cohesion is close to zero. The strength is the peak one, the greatest stress of the test. After the peak a weak rock goes on carrying part of the load.

Sand production: types, prediction and cost of the problem →

Hoop stress at the wall of the hole of a thick-wall cylinder

\sigma_\theta = \frac{2\,P\,\rho^2}{\rho^2 - 1},\qquad \rho = \frac{D}{d}

any identical units of pressure; the diameters in the same units

Holds when: The rock is elastic up to failure. The pressure in the hole is atmospheric (the pore pressure is not maintained in the test). This is the Lamé solution for a thick-wall pipe under external pressure. The axial stress does not enter the formula.

Sand production: types, prediction and cost of the problem →

Effective hoop stress at the wall of an open wellbore during production

\sigma'_\theta = 2\sigma'_h + (2 - A)\,(\bar p - p_{wf}),\qquad A = \frac{\alpha(1 - 2\nu)}{1 - \nu}

any identical units of pressure

Holds when: A vertical open hole, with both horizontal stresses equal. The rock is elastic, and the Biot coefficient in the strength criterion is 1. Steady radial inflow, and the pore pressure at the wall equals the bottomhole pressure. The stress along the axis of the well at the moment of failure is intermediate and does not affect the failure. After noticeable depletion this may stop being true: the effective axial stress at the wall becomes the greatest (example below), and the formula is optimistic. With unequal horizontal stresses, in place of $2\sigma'_h$ we have $3\sigma'_H - \sigma'_h$ , and for a perforation tunnel $3\sigma'_1 - \sigma'_3$ , where $\sigma'_1$ and $\sigma'_3$ are the two effective stresses in the plane of its cross-section.

Sand production: types, prediction and cost of the problem →

Critical drawdown and critical depletion (open hole)

\Delta p_{\text{cr}} = \frac{C_w - 2\sigma'_{h,i} - 2(1 - A)(p_i - \bar p)}{2 - A},\qquad (p_i - \bar p)_{\text{cr}} = \frac{C_w - 2\sigma'_{h,i}}{2(1 - A)}

any identical units of pressure

Holds when: The same as for the formula for the wall. The depletion is uniform over the reservoir, and the reservoir deforms only vertically. $C_w$ does not change with time. In reality water cut and load cycles lower it. A negative $\Delta p_{\text{cr}}$ means that the wall has already failed without drawdown.

Sand production: types, prediction and cost of the problem →

Settling velocity of a grain in a fluid at rest (Stokes' law)

v_s = \frac{g\,d^2\,(\rho_s - \rho_l)}{18\,\mu}

Holds when: A single spherical grain. The Reynolds number of the grain, $\rho_l v_s d/\mu$ , is less than one. Otherwise the resistance is greater, and the formula overstates the velocity. The fluid is at rest and the wellbore is vertical. At a high concentration the grains hinder each other and settle more slowly, and in a deviated wellbore they quickly slide to the lower wall.

Sand production: types, prediction and cost of the problem →

The largest grain that passes through a pore of a packing of equal spheres

d_{\max} = \left(\frac{2}{\sqrt3} - 1\right)D \approx 0.155\,D\quad\text{(dense packing)},\qquad d_{\max} = \left(\sqrt2 - 1\right)D \approx 0.414\,D\quad\text{(cubic)}

any units of length

Holds when: The spheres are equal. In a dense packing three touching grains form a triangular constriction, and in a cubic packing a square one of four. Real gravel is a mixture of grains between two sieves, and it is not perfectly round. The estimate shows the order of magnitude and not an exact limit.

Sand control methods →

Uniformity coefficient of a sand

C = \frac{D_{40}}{D_{90}}

dimensionless

Holds when: The sizes are found from the cumulative curve by interpolation between the sieves in the logarithms of the size. The sample represents the productive interval. If there are several samples, we choose the protection by the finest of the productive layers.

Sand control methods →

Choosing the gravel by the median sizes

D_{50,g} = (5\text{–}6)\,D_{50,\text{s}}

any units of length, the same for both sizes

Holds when: The sand is well or moderately sorted. The gravel is of a narrow size: almost all the mass is between two neighbouring sieves. For poorly sorted sands the criterion by the median is not reliable. We then use the Schwartz criterion: the size of the gravel $D_p$ is taken as six times the same size $D_p$ of the sand, and the point of the curve is chosen by the sorting: $D_{10}$ for $C$ up to 5, $D_{40}$ for $C$ from 5 to 10, and $D_{70}$ for $C$ above 10 (the worse the sorting, the finer the point of choice). And we run sand retention tests.

Sand control methods →

Calculated mass of gravel for a pack in casing

M = \rho_g(1 - \phi_g)\left[\frac{\pi}{4}\left(D_c^2 - d_s^2\right)(L_s + L_r) + n\,\frac{\pi}{4}d_p^2\,l_p\right]

Holds when: The reserve above the screen makes up for settling and compaction of the pack as the well works. Tunnels and voids behind the casing usually take more gravel than their dimensions give, especially if the reservoir has already produced sand. So a placed mass larger than calculated is a good sign, and a smaller one is a reason to check.

Sand control methods →

Symbols

Symbol	Meaning and context
$\bar p$	average (current) reservoir pressure, MPaSand production: types, prediction and cost of the problem
$(p_i - \bar p)_{\text{cr}}$	critical depletion: the fall of the reservoir pressure at which the wall fails even without drawdown, MPaSand production: types, prediction and cost of the problem
$A$	poroelastic stress coefficient; $\alpha$ is the Biot coefficient and $\nu$ is Poisson's ratio, —Sand production: types, prediction and cost of the problem
$C$	uniformity coefficient, —Sand control methods
$C_0$	unconfined compressive strength (UCS), MPaSand production: types, prediction and cost of the problem
$C_w$	effective strength of the cavity wall; a first estimate is $2.25\,\sigma_{\text{TWC}}$ , MPaSand production: types, prediction and cost of the problem
$\Delta p_{\text{cr}}$	critical drawdown $\bar p - p_{wf}$ , at which the wall starts to fail, MPaSand production: types, prediction and cost of the problem
$d$	diameter of the grain, mSand production: types, prediction and cost of the problem
$D$	diameter of the gravel grains, mmSand control methods
$D_{40},\ D_{90}$	sizes above which lie 40% and 90% of the mass of the sand, mmSand control methods
$D_{50,\text{s}}$	median size of the reservoir sand, mmSand control methods
$D_{50,g}$	median size of the gravel, mmSand control methods
$D_c,\ d_s$	inside diameter of the casing and outside diameter of the screen, mSand control methods
$d_{\max}$	diameter of the largest sphere that passes through a pore between neighbouring grains, mmSand control methods
$D,\ d$	outer diameter of the sample and diameter of the hole, mmSand production: types, prediction and cost of the problem
$g$	acceleration of gravity, 9.81, m/s²Sand production: types, prediction and cost of the problem
$L_s,\ L_r$	length of the screen and the reserve of gravel above it, mSand control methods
$M$	mass of gravel, kgSand control methods
$n,\ d_p,\ l_p$	number of perforation tunnels, their diameter and their length behind the casing, —, m, mSand control methods
$P$	external pressure on the sample, MPaSand production: types, prediction and cost of the problem
$p_i,\ \bar p$	initial and current reservoir pressure, MPaSand production: types, prediction and cost of the problem
$p_{wf}$	bottomhole pressure during production, MPaSand production: types, prediction and cost of the problem
$q$	slope of the line: how much confinement strengthens the rock, —Sand production: types, prediction and cost of the problem
$S_0$	cohesion: the shear strength with no normal load, MPaSand production: types, prediction and cost of the problem
$v_s$	settling velocity of the grain, m/sSand production: types, prediction and cost of the problem
$\mu$	viscosity of the fluid, Pa·sSand production: types, prediction and cost of the problem
$\rho_g,\ \phi_g$	density of the gravel grains and porosity of the pack, kg/m³, —Sand control methods
$\rho_s,\ \rho_l$	density of the grain and of the fluid, kg/m³Sand production: types, prediction and cost of the problem
$\sigma'_1,\ \sigma'_3$	largest and smallest effective principal stresses at failure, MPaSand production: types, prediction and cost of the problem
$\sigma'_h$	effective horizontal stress in the reservoir at the current reservoir pressure, $\sigma_h - \bar p$ , MPaSand production: types, prediction and cost of the problem
$\sigma'_{h,i}$	effective horizontal stress at the initial reservoir pressure, MPaSand production: types, prediction and cost of the problem
$\sigma'_\theta$	effective hoop stress at the wall (less the pore pressure at the wall, which equals $p_{wf}$ ), MPaSand production: types, prediction and cost of the problem
$\sigma_\theta$	hoop stress at the wall of the hole, MPaSand production: types, prediction and cost of the problem
$\varphi$	angle of internal friction, °Sand production: types, prediction and cost of the problem