# 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)

## Output(s)

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

## Formula(s)

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

## Reference(s)

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

## Related

### Mudcake permeability - Formation invasion

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