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.

Dynamically coupled linear flow - Formation invasion

Input(s)

Output(s)

Formula(s)

Reference(s)

Related

Mudcake permeability - Formation invasion

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