Equations: Formation damage and stimulation

any consistent units: only the ratios $k/k_d$ and $r_d/r_w$ enter the formula; the logarithm is natural (base e)

Holds when: The damaged zone is a cylinder of uniform permeability, small compared with the drainage area (the part of the reservoir that feeds the well). The flow is radial over the whole thickness: open hole, or perforations that do not reach beyond the zone (then the actual skin is even larger, because the flow converges on the tunnels). Single-phase flow that follows Darcy's law (the rate is proportional to the pressure gradient).

any consistent units

Holds when: The same as for one zone. Inside each ring the permeability is uniform.

any consistent units

Holds when: Radial steady-state flow in the near-wellbore zone, and single-phase flow that follows Darcy's law.

SI

Holds when: A zone with a uniform $\beta$ and radial flow over the whole thickness. In the damaged zone, we use its own coefficient $\beta_d$ instead of $\beta$.

any consistent units: lengths in the same units

Holds when: The flow is one-dimensional and the sections work in series. The initial permeability is the same along the whole sample. Otherwise the sections are weighted not by the length $L_i$ but by $L_i/k_i^{\text{before}}$. The permeability before and after is measured with the same fluid at the same saturation.

SI; for a rate in m³/d, multiply by 86,400

Holds when: The inflow is spread evenly over the area. In reality it is stronger towards the ends of the tunnels and towards the few tunnels that are open, and the local velocity is higher. The criterion is the velocity of water in the test, where brine flows alone. When oil and water flow together, the water takes up only part of the pores and moves faster in them. So the allowable water rate is smaller, and the estimate is optimistic. The ends of the tunnels and tunnels that do not work are not counted.

concentrations in mol/L; the logarithm is decimal (base 10)

Holds when: SI > 0 means the water is supersaturated and salt can come out. SI < 0 means it is undersaturated. SI = 0 means equilibrium. In brines the activity coefficients are much smaller than one. They, and $K_{sp}$ at reservoir temperature and pressure, are calculated with brine models in special software. The index is thermodynamic: it tells whether the salt can come out, but not how fast.

mol/L; the mass of the precipitate is $x$ multiplied by the molar mass of barium sulfate (233.4 g/mol)

Holds when: The mixture has reached equilibrium. Other ions and precipitates are not counted. The activity coefficients do not change as the salt comes out.

SI. In metric oilfield units, $2\pi$ is replaced by 0.536, and the slope comes out in MPa·d/m³ with $k$ in mD, $h$ in m and $\mu$ in mPa·s

Holds when: Injection is close to steady state: the pressure at the distance $r_e$ equals the reservoir pressure $\bar p$ and hardly changes. The mobilities of the water and of the oil it displaces (how easily each fluid moves) are close. The wellhead pressure is often used instead of the bottomhole pressure. At a constant rate, hydrostatic pressure and friction give a constant correction. The slope also changes for other reasons: a rise of the reservoir pressure, or water saturation of the zone around the well. The most accurate measurement of the skin is a falloff test (a test that records how the pressure falls after injection stops).

dimensionless; the logarithm is natural

Holds when: The average reservoir pressure and the drawdown are the same before and after the treatment: the lift system holds the same bottomhole pressure at the higher rate. Pseudo-steady-state, single-phase flow in a circular drainage area with the well at the centre. The treatment changes only the skin. The permeability and thickness of the reservoir are the same.

SI (any consistent system of units)

Holds when: The same as for the formula for the ratio.

any consistent units: here days, m³/d and dollars

Holds when: The gain declines exponentially at a constant rate $a$, and the net income per cubic metre is constant. Discounting is not applied. For payback times of up to a year it changes the answer little. A solution exists only if $aC < \Delta q_0 v$. Otherwise all the future income from the gain, $\Delta q_0 v/a$, is smaller than the costs, and the treatment will never pay back.

any consistent units: the rate and the decline rate in the same units of time

Holds when: A closed drainage area, with no water influx and no injection. Single-phase inflow. A constant bottomhole pressure. The decline is exponential, and its rate is proportional to productivity. The economic limit in terms of rate is the same in both cases.

the coefficients are numbers of moles

Holds when: The acid is fully spent. The reaction with calcite is very fast, and with dolomite it is slower, especially at low temperature. Siderite, $\mathrm{FeCO_3}$, reacts in the same way as calcite, but it puts iron into solution.

the coefficients are numbers of moles

Holds when: The acid is in excess: silicon goes into the fluorosilicate ion and aluminium into complexes with two fluorides. If fluorine is in short supply the products are different, and less HF is used per mole of mineral. The reactions with clays and feldspars are fast, and those with quartz are slow. The speed grows with temperature. With calcium minerals HF gives insoluble calcium fluoride. This is why they are removed with hydrochloric acid before the hydrofluoric acid.

ratios of masses and volumes; the units of densities and molar masses are the same in the numerator and the denominator

Holds when: The acid is fully spent, and only on this mineral. For HF the coefficient $\nu_{\text{a}}$ depends on which fluorides form. Here silicon goes into the fluorosilicate ion. In tables where $\mathrm{SiF_4}$ is taken as the product, the values are 1.3–1.5 times larger. The density of the solution is taken at the temperature at which it is prepared.

SI (m, m³/m)

Holds when: The acid enters the reservoir evenly over the thickness and around the circumference. The mineral dissolves completely behind a sharp front (a fast reaction). The slow reaction of HF with quartz, mixing and the growth of porosity are not counted. The volume is underestimated, so a margin is added to the calculation. For mud acid, $x$ is the fraction of fast-reacting minerals that the acid can reach: clays on the pore walls, fines, and particles of drilling fluid.

any consistent units; the logarithm is natural

Holds when: The acid is displaced like a piston, without mixing. The precipitate is spread evenly over the ring and lowers the permeability by the same amount wherever the ring ends up. The flow is radial, and the damage near the wellbore has already been removed.

SI; $\ln$ is the natural logarithm

Holds when: Steady radial flow, with the pressure at the distance $r_e$ equal to the reservoir pressure. Injection lasts hours, and the pressure disturbance has no time to reach $r_e$. So the pressure difference is overstated, and the rate comes out with a safety margin. The viscosities of the acid and of the reservoir fluid are close. If the reservoir fluid is much more viscous, the rate calculated from the viscosity of the acid is too high. The reservoir is uniform. In a layered interval the rate is divided unevenly between the layers.

SI; in metric units $\rho g H = 0.00981\,\rho H$ (MPa, with $\rho$ in g/cm³ and $H$ in m)

Holds when: The pipes are filled with one fluid. When fluids of different densities replace each other in the pipes, the hydrostatic pressure changes during pumping. Friction grows almost as the square of the rate. Losses in the wellhead equipment and in the perforations are small, or are counted separately.

SI; in metric oilfield units $2\pi$ is replaced by 0.536 (the rate in m³/d)

Holds when: The flow is steady. Transient effects at the start of pumping and after a change of rate distort the value. The apparent skin includes the pseudo-skins, the viscous slug of acid (a batch of thickened acid), and the action of the diverter.

dimensionless

Holds when: The same as for the calculation of the volume with a sharp front.

any consistent units; the logarithm is natural

Holds when: The viscosity is constant. In reality gels and foams thin under shear: near the wellbore, where the velocity is high, their viscosity is lower, and deeper in it is higher. The diversion is based on this. The permeability of the zone does not change.

SI (dimensionless)

Holds when: The acid is spent on the walls at a rate proportional to its concentration. This is an estimate. The diameter and the length of a wormhole are measured only approximately in a test.

any consistent units of velocity

Holds when: The constants $v_{opt}$ and $PV_{bt,opt}$ are taken from core tests of the same rock with the same acid at reservoir temperature. The formula smooths the minimum: at $v_i = v_{opt}$ it gives $PV_{bt} \approx 1.04\,PV_{bt,opt}$. At high velocities $PV_{bt}$ grows as $v_i^{1/3}$, and at low velocities very steeply. On larger samples the optimum velocity and $PV_{bt}$ are smaller than on standard ones. So laboratory values are transferred to the reservoir with a correction, or are calibrated against the results of treatments.

SI

Holds when: The acid enters the interval evenly over the thickness and around the circumference. The wormhole front is taken as a cylinder. The wormholes themselves take a negligible share of the pore volume.

SI (m, m³/m)

Holds when: The wormhole front advances at the velocity $v_{wh}(v_i)$, as in a linear test at the same pore velocity. The rate is constant. The second form is a “volumetric model” with one value of $PV_{bt}$. It is good if the velocity along the whole path is close to the optimum. Otherwise the integral is calculated by rings. The rock is uniform. Natural fractures and vugs take acid faster and change the picture.

fractions; the units of $k$ and $h$ are the same for all layers

Holds when: Steady injection, as in the formula for the limit on the rate. The reservoir pressure and the viscosity of the fluid are the same in all layers, and there is no crossflow between the layers at the wellbore. A slug of diverting agent (a batch of it pushed into the layer) adds $(\mu_{\text{v}}/\mu - 1)\ln(r_{\text{v}}/r_w)$ to the skin of the layer, where $\mu_{\text{v}}$ and $r_{\text{v}}$ are its apparent viscosity and radius (skin of a viscous diverter).

SI

Holds when: All the acid has reacted on the walls within the fracture, with no leak-off (loss of acid into the reservoir rock) and no use near the wellbore. In reality the width is smaller. The etching is even. But the conductivity is created by unevenness, so $w_i$ is a measure of the volume of work that the acid has done, not the width of the channel after closure.

consistent units: the rate and the decline rate in the same units of time

Holds when: A closed drainage area with no pressure support and a constant bottomhole pressure (the drawdown falls together with the reservoir pressure). The decline is exponential, and its rate is proportional to productivity. With pressure support by injection the gain lasts longer, and the formula understates it. With water breakthrough after the treatment, it overstates it.

dimensionless; the logarithm is natural

Holds when: The wormhole zone conducts infinitely well, and the pressure in it equals the bottomhole pressure. So the skin comes out as a limit, and in reality it is somewhat higher. The damage is uniform in the ring from $r_w$ to $r_d$. At $r_{wh} = r_w$ the formula turns into the Hawkins formula (Skin factor).

any consistent units: $k_f$ and $k$ in the same units, and $w$ and $x_f$ in the same units too

Holds when: A vertical fracture over the whole thickness of the reservoir, with two equal wings. The conductivity of the fracture and the permeability of the reservoir are the same along the length. Single-phase flow that follows Darcy's law. Inertial losses and multiphase flow in the fracture reduce its effective conductivity.

any units of length, the same for all the radii

Holds when: The fracture has high conductivity: $F_{CD}$ is 10 and above. At a lower conductivity the effective wellbore radius is smaller (the next section). The fracture is small compared with the drainage area (the part of the reservoir that feeds the well): the half-length is much smaller than $r_e$. Pseudo-steady-state flow (the pressure falls at the same pace everywhere in the drainage area). In the first months, while the flow to the fracture is linear, productivity is higher. The fracture bypasses the damage, so the damage skin is not included in the sum.

dimensionless; the logarithm is natural

Holds when: The average reservoir pressure and the drawdown are the same. The lift system and the surface equipment pass the increased rate. Pseudo-steady-state flow in a circular drainage area.

dimensionless; the logarithms are natural

Holds when: A vertical fracture over the whole thickness of the reservoir, the well at the centre of the drainage area, and a fracture that is small compared with the area. The approximation fits the chart with an error of a few per cent in the practically important range, from about 0.3 to hundreds. At a lower conductivity it overstates $r_w'$ (at $F_{CD} = 0.1$ by about 12%, in more detail in the lesson Inflow to a hydraulically fractured well). At large $F_{CD}$ the right-hand side approaches $\ln 2$, that is, $r_w'$ approaches $x_f/2$.

SI (any consistent system of units)

Holds when: A single-phase, slightly compressible fluid. The hydraulic diffusivity $k/(\phi\mu c_t)$ is the same as in the diffusivity equation (Diffusivity equation).

dimensionless; natural logarithms

Holds when: The flow is pseudo-radial: the pressure disturbance has moved away from the fracture by several of its half-lengths. The fracture is small compared with the drainage area (the penetration ratio is below about 0.3, see the next section). The conductivity is uniform along the length. Choking at the wellbore and damage to the faces are accounted for by separate skins (last section).

$k_f w/k$ has the dimension of length: with $k_f w$ in mD·m and $k$ in mD, the radius comes out in metres

Holds when: $F_{CD} \lesssim 0.1$. At $F_{CD}$ = 0.1 the exact curve lies 3.3% below the asymptote, and at 0.05 it lies 1.6% below. The flow is pseudo-radial. The fracture is longer than the part that really works (lengths of about $k_f w/k$).

dimensionless

Holds when: The fracture passes through the centre of the area, parallel to its side. At $I_x$ = 1 the ends of the fracture reach the boundaries.

SI (any consistent system of units)

Holds when: The regime is pseudo-steady-state in a closed area (Pseudo-steady-state flow and the shape factor). In the transient regime $J_D$ depends on time (section on the first rates). The inflow is single-phase. For a vertical well in a square, $1/J_D$ equals $\ln(x_e/r_w) - 1.311 + S$. Here $-1.311$ is $\tfrac12\ln\big(4/(\gamma C_A)\big)$ at $C_A$ = 30.88, where $C_A$ is the Dietz shape factor (a number that depends on the shape of the area and the position of the well).

dimensionless

Holds when: The area is closed and rectangular, the fracture runs through the centre parallel to the side $x_e$, and the regime is pseudo-steady-state. There is one vertical fracture over the whole thickness of the reservoir. The derivation is in the advanced section.

dimensionless

Holds when: These are early times. For $F_{CD} \ge 3$ the regime lasts until about $t_{Dxf} \approx 0.1/F_{CD}^2$. For $F_{CD}$ from 1.6 to 3 it lasts longer than this formula gives (at $F_{CD}$ = 1.6, up to $t_{Dxf} \approx 0.7$). For $F_{CD} < 1$ it lasts less: only a short stretch of the fracture near the wellbore works, and already at $t_{Dxf}$ of about $F_{CD}^2/400$ the inflow starts to turn into pseudo-radial flow, skipping the linear regime. The fracture is uniform and is not choked at the wellbore. The effect of the wellbore has ended.

SI (any consistent system of units)

Holds when: $F_{CD}$ is above about 300. The regime lasts until $t_{Dxf} \approx 0.016$. At lower conductivity bilinear flow takes its place. The fracture runs over the whole thickness of the reservoir. If the fracture is less tall than the reservoir, the first part of the record is distorted.

dimensionless; natural logarithms

Holds when: The derivative reaches the plateau 0.5 to within 7–8% at $t_{Dxf} = 3$ and to within 5% at $t_{Dxf} = 5$, for any $F_{CD}$ from 1 to infinity. The reservoir is infinite: the disturbance has not yet reached the boundaries of the area.

dimensionless

Holds when: The area is closed and homogeneous, and the inflow is single-phase. Before the treatment the well worked in the pseudo-steady-state regime. At constant bottomhole pressure the rate changes with time in a similar way, but not exactly the same. This formula is a measure, not a forecast of the rate day by day.

dimensionless; permeabilities and conductivities in consistent units (mD and mD·m)

Holds when: The whole flow of a wing passes along the fracture through the choked stretch, so each wing carries half of the rate. This is an upper estimate. If $k_{f,ch}w$ is not much larger than $k\,x_s$, part of the fluid bypasses the choked stretch through the reservoir, and the actual skin is smaller (in the example below, by about 40%).

dimensionless

Holds when: The layer is thin, $b_s \ll x_f$, and the same along the whole fracture. The flow through it is perpendicular to the face. In the linear regime $s_{fs}$ is added to $p_{wD}$ exactly. In the pseudo-radial regime the total skin grows by about $s_{fs}$, and never by less. The model gives 5–20% more at $s_{fs}$ from 0.5 to 1, and relatively even more at small $s_{fs}$, because the inflow is redistributed along the fracture.

any identical units of stress and modulus (MPa or psi); the strains are dimensionless

Holds when: The layer is linearly elastic and isotropic, and wide compared with its thickness. The vertical stress is a principal stress. The strains are measured from the state in which the horizontal stresses were created only by the weight of the rocks. $\varepsilon_h$ and $\varepsilon_H$ are the same for all layers of the section, but the stresses they cause are different in each layer. The moduli are static: measured by slowly loading a core sample (a cylinder of rock taken from the well), or converted from dynamic moduli (found from sound speeds) with a relationship that has been checked on core (Sonic log). $\sigma_H$ is found with the same formula by swapping $\varepsilon_h$ and $\varepsilon_H$. At $\varepsilon_h = \varepsilon_H = 0$ the two horizontal stresses are equal, and at $\alpha = 1$ we get the Eaton formula.

any identical units of pressure

Holds when: The pressure changes over the whole wide reservoir, and the reservoir deforms only vertically (uniaxial strain). The rock is elastic, and the overburden pressure does not change during development. The ratios $\Delta\sigma_h/\Delta p_p$ measured in fields are scattered, and they can differ noticeably from the calculated one. We check the ratio by repeated measurements of the closure pressure.

any identical units of stress; compression is positive

Holds when: The rock is linearly elastic, homogeneous and isotropic. The well is vertical, and the vertical stress is a principal stress. The pore pressure at the wall is the same as the formation pressure. Either a filter cake (a thin layer of solids deposited on the wall) separates the wall from the well, or the pressure changes so fast that the fluid has no time to enter the rock (the effect of filtration is below). The temperature of the wall equals the formation temperature. Plane strain along the axis of the well (the rock does not stretch along it). With distance from the wall the disturbance dies out quickly, and already a few radii from the axis the stresses hardly differ from $\sigma_H$ and $\sigma_h$ (advanced section).

any identical units of pressure

Holds when: The same as for the Kirsch solution, but the pore pressure at the wall equals the pressure in the well, and the change of the pore pressure around the well is the same in all directions (axisymmetric). The wall splits when the effective hoop stress $\sigma_\theta - p_w$ reaches $-T$. The loading is slow: at every moment the rock has time to come to equilibrium with the pressure in the pores. The real breakdown pressure lies between this value and the value without penetration. Which one it is closer to depends on the injection rate, the viscosity of the fluid, the filter cake and the permeability.

any identical units of pressure; all pressures are at the depth of the reservoir

Holds when: The wellhead pressure is converted to the bottom of the well by adding the hydrostatic head of the column of fluid and subtracting the friction losses in the pipes. At the moment of shut-in, the friction losses in the pipes, in the perforations and near the wellbore vanish, and the net pressure has not yet had time to change.

dimensionless; $\Delta t$ and $t_p$ in the same units

Holds when: The injection is at a constant rate. After shut-in the area of the fracture does not change. Leak-off from the fracture into the reservoir follows Carter's law: the leak-off rate is inversely proportional to the square root of the time since the given part of the wall opened. The stiffness of the fracture is constant: the net pressure is proportional to the volume of fluid in it. There is also a second limiting form of the function, for a fracture whose area grew as the square root of time. The two forms differ by no more than about 20% (the later the closure, the smaller the difference), and the choice does not affect the closure pressure that is found.

consistent: $p_{net}$ and $E'$ in the same units, $w$ in the units of $h_f$

Holds when: The fracture is much longer than its height, and its vertical section is an ellipse (plane strain). The net pressure is the same over the height. The rock is homogeneous and elastic. This is an order-of-magnitude estimate. In reality the pressure falls towards the tip of the fracture, and layers of different stiffness open differently.

$p_{net}$ and $\Delta\sigma$ in the same units; $h_f$ in the units of $h$; the angle is in radians

Holds when: The fracture is symmetric about the middle of the reservoir. The barriers are identical and thick enough, with no layers of low stress behind them. The pressure is the same over the height of the fracture (the weight of the fluid and the friction losses are not taken into account). The fracture toughness of the rock (its resistance to the growth of a crack) is not taken into account (with it, the fracture goes into the barriers a little less). The elastic moduli of the layers are the same.

any identical units of stress; compression is positive

Holds when: The same as for the formula for the wall. At $r = r_w$ we get $\sigma_r = p_w$, the formula for $\sigma_\theta$ at the wall, and $\tau_{r\theta} = 0$.

any identical units of pressure

Holds when: There are faults of every orientation in the rock mass, with no cohesion. The effective stress is the simple one of Terzaghi (stress minus pore pressure; the Biot coefficient for faults is taken as 1).

any identical units of pressure

Holds when: Near the wellbore the pressure in the fracture equals the bottomhole pressure. Farther along the fracture it is higher, and the load is smaller. As the reservoir is depleted (its pressure falls), the closure pressure falls with it, by $A\,\Delta\bar p$, where $A$ is the poroelastic coefficient (Rock mechanics for hydraulic fracturing). This is usually less than the fall of the bottomhole pressure, so we calculate the design load from the initial closure pressure and the lowest planned bottomhole pressure. This is on the safe side.

conductivity in mD·m with $k$ in mD and the width in m

Holds when: The proppant lay evenly, without settling to the bottom of the fracture. The factors $R$ and $e$ are found by tests with the fracturing fluid and with samples of the reservoir rock. Without tests, we choose cautious values.

SI

Holds when: The inflow into the fracture is uniform along the length, which is how a fracture of high conductivity behaves. At low conductivity the inflow is concentrated near the wellbore, and the losses are closer to the value at the wellbore, $k_f/(1 + \text{Fo}_w)$. The $\beta_f$ of a pack grows when grains crush, with gel residue, and when a second phase (for example water) is present.

dimensionless; $k_f$ and $k$ in the same units

Holds when: A vertical fracture in the centre of a square drainage area, parallel to its side. Only the proppant within the reservoir goes into $V_{prop}$. If the fracture is taller than the reservoir and the proppant is spread evenly over the height, this is the share $h/h_f$ of all the proppant. $k_f$ is the permeability of the pack in the reservoir, taking into account the load, polymer residue and embedment (previous section).

dimensionless; natural logarithm

Holds when: $N_{prop}$ is up to 0.1. At 0.1 the formula overstates $J_{D,\max}$ by 0.7%, and at 0.3 already by 3%. The area is a square and the regime is pseudo-steady-state. The derivation is in the advanced section.

consistent; $k_f$ and $k$ in the same units

Holds when: It follows from $V_{prop} = 2x_f w h$ and $F_{CD} = k_f w/(k x_f)$. An equivalent form is $x_f = I_x x_e/2$ with $I_x = \sqrt{N_{prop}/F_{CD}}$.

SI (any consistent system of units)

Holds when: The fracture is much longer than its height, the height is constant, and every vertical cross-section is an ellipse. The fluid is Newtonian and the flow is laminar. The factor 2.75 and the ratio 0.58 are the midpoint between the solution without leak-off (2.82 and 0.59) and the limiting case of strong leak-off (2.68 and 0.57). The derivation is in the advanced section. For fluids whose viscosity falls as the shear rate grows, we take the apparent viscosity at the shear rate in the fracture. The factor refers to the total rate into both wings. If we use the rate into one wing instead, the factor is $2^{1/4} \approx 1.19$ times larger, about 3.27. The formula is often given in this form, so when comparing formulas, check which rate the factor refers to.

SI (any consistent system of units)

Holds when: Carter's law: the leak-off rate is inversely proportional to the square root of the time since the part of the wall opened. It goes through both walls, hence the factor 2. Leak-off goes only into the permeable reservoir. The proppant is part of the pumped volume, but it does not leak off. $\kappa$ comes from averaging $2\sqrt{t-\tau}$ over the area, when the area grows by a power law. In the examples of the lesson $\kappa$ = 1.5, and spurt loss is not counted.

SI; $C_L$ in m/s$^{1/2}$

Holds when: The $G$ function has the form used in the lesson on rock mechanics: the area of the fracture grew in proportion to time during injection. With the second limiting form the efficiency comes out a few per cent less. After shut-in the area does not change, there is no spurt loss, and by the time of closure there is no fluid left in the fracture. The compliance is that of a PKN fracture. For short fractures we need the compliance of the KGD model, and the leak-off coefficient will be different.

consistent

Holds when: Leak-off follows Carter's law, and the area of the fracture grows during injection. The proppant does not settle and moves together with the fluid. The concentration is the mass of proppant per unit volume of slurry. In field practice it is given per volume of clean fluid: $c_{\text{fl}} = c/(1 - c/\rho_p)$. The efficiency is taken from the calculation of the main injection, calibrated against the calibration injection. When in doubt, we make the pad a little larger: a screen-out costs more than extra fluid.

dimensionless; natural logarithms

Holds when: The fracture is small compared with the area ($I_x$ is below about 0.3), the regime is pseudo-steady-state, and the area is a square.

SI; $K$ in Pa·sⁿ (in oilfield units lbf·sⁿ/ft², with 1 lbf·sⁿ/ft² = 47.88 Pa·sⁿ)

Holds when: The power-law model holds in the range of the measurements. At very low shear rates the viscosity of a real fluid stops growing, and extrapolation overstates it. The points are taken at reservoir temperature, after a hold that matches the injection time.

SI

Holds when: The flow is laminar. The slot is two parallel walls (the cross-section of a fracture is really an ellipse, and the formula gives a characteristic shear rate). At $n = 1$ we get the Newtonian $6v/w$ and $8v/d$ (shear rate in a pipe). The factors take into account the flatter velocity profile of a thinning fluid.

any consistent units: $C_w$ comes out in units of length per square root of the unit of time

Holds when: The leak-off rate through the cake is $C_w/\sqrt t$. This gives the factor 2 on integration: $\int_0^t C_w\,dt'/\sqrt{t'} = 2C_w\sqrt t$. The pressure drop in the test is close to that in the reservoir (otherwise the coefficient is recalculated; for polymer cakes it grows with the pressure drop more slowly than the square root of it). The sample is of the same rock and permeability.

SI ($k$ in m²: 1 mD = $0.987\cdot10^{-15}$ m²). In oilfield units the coefficients are more often given in ft/min$^{1/2}$: 1 m/s$^{1/2}$ = 25.4 ft/min$^{1/2}$

Holds when: The filtration into the reservoir is linear (one-dimensional) at a constant pressure drop. The filtrate displaces the reservoir fluid like a piston. The overall coefficient is an exact combination of three resistances in series at one common pressure drop, if the coefficients of the cake and of the filtrate grow as the square root of their share of the pressure drop, and the compression coefficient grows in proportion to it. The approximation $1/C_L = 1/C_w + 1/C_v + 1/C_c$ is also often used. It always gives a smaller value (in the example below, by 15%). The derivation of the formulas is in the advanced section.

SI

Holds when: A single spherical grain at a low Reynolds number, $\rho_f v_s d/\mu \ll 1$. The walls of the fracture are far away, and the fluid is at rest. The second formula is Stokes' law with the apparent viscosity at the shear rate $3v_s/d$. At $n = 1$ it coincides with the first. It is an estimate: at very low shear rates the real viscosity stops growing, and the formula understates the settling velocity. In a concentrated slurry the grains hinder each other and settle more slowly. By the empirical law of Richardson and Zaki, at low Reynolds numbers the velocity is multiplied by about $(1 - c_v)^{4.65}$, where $c_v$ is the volume fraction of proppant.

SI

Holds when: Before the screen-out the area of the fracture grew as the square root of time (the limit of strong leak-off), and every part of the wall filters by Carter's law from the moment it opened. After the screen-out the area does not change, and the spurt loss all happened before the screen-out. The same function describes the leak-off after the pumps stop in a calibration injection: $g(\delta) - \pi/2 = \tfrac{\pi}{4}G(\delta)$, where $G$ is the second limiting form of Nolte's function, for a fracture whose area grew as the square root of time (the G function is written there for growth in proportion to time). The derivation is in the advanced section.

SI. The concentration per cubic metre of fluid is given in field practice in pounds per gallon: 1 lb/gal = 119.8 kg/m³

Holds when: The density of the fluid is constant. Gas in the slurry (foams, nitrogen) is taken into account separately.