Dynamically coupled linear flow - Formation invasion

Input(s)

$x_{f}$ : Transient Invasion Front (in.)

$x_{f, o}$ : Initial Displacement, i.e., Spurt (in.)

L: Lineal Core Length (in.)

$p_{m}$ : Constant Mud Pressure (psi)

$p_{r}$ : Constant Reservoir Pressure (psi)

$\emptyset_{\text {eff: }} \quad$ Effective Rock Porosity (fraction)

$\varnothing_{c}$ : Mudcake Porosity (fraction)

$k_{1}$ : Mudcake Permeability to Filtrate (mD)

$k_{2}$ : Rock Permeability to Filtrate $(\mathrm{mD})$

$k_{3}$ : Rock Permeability to "Oil" (mD)

$\mu_{f}$ : Mud Filtrate Viscosity $(\mathrm{cP})$

$\mu_{o}$ : Viscosity of "Oil" or Formation Fluid (cP)

$f_{s}$ : Mud Solid Fraction (fraction)

$\mathrm{x}_{\mathrm{f}}(\mathrm{t})$ : Minimum Number of Jobs to Survive in a Minimum Chance Scenario (dimensionless)

\begin{gathered} \mathrm{x}_{\mathrm{f}}(\mathrm{t})=-H+\sqrt{\left\{H^{2}+2\left(H x_{f, o}+1 / 2 x_{f, o}^{2}+G t\right)\right\}} \\ G=-\left\{k_{1}\left(p_{m}-p_{r}\right) / \mu_{f} \emptyset_{e f f}\right\} /\left\{\frac{\mu_{o} k_{1}}{\mu_{f} k_{3}}-\frac{k_{1}}{k_{2}}-\frac{\emptyset_{e f f} f_{s}}{\left\{\left(1-\emptyset_{c}\right)\left(1-f_{s}\right)\right\}}\right\} \\ H=\left[\frac{x_{f, o} \emptyset_{e f f} f_{s}}{\left\{\left(1-\emptyset_{c}\right)\left(1-f_{s}\right)\right\}}-\frac{\mu_{o} k_{1} L}{\mu_{f} k_{3}}\right] /\left\{\frac{\mu_{o} k_{1}}{\mu_{f} k_{3}}-\frac{k_{1}}{k_{2}}-\frac{\emptyset_{e f f} f_{s}}{\left\{\left(1-\emptyset_{c}\right)\left(1-f_{s}\right)\right\}}\right\} \end{gathered}

Chin, W. C. (1995). Formation Invasion, Page: 16.