Analysis of a post-fracture pressure buildup test with wellbore-storage distortion

Input(s)

$q_{g}$: Gas flow rate (MSCF/day)

$B_{g}$: Gas formation Volume Factor (RB/MSCF)

$p_{D}$: Dimensionless Pressure (dimensionless)

$\varnothing$: Porosity (dimensionless)

$c_{t}$: Compressibility $(1 / \mathrm{psi})$

$\mu$: Viscosity $(\mathrm{cP})$

$\mathrm{h}$: Formation Thickness (ft)

$t_{\left(L_{\rangle_{D}}\right.}$: Time of end of linear or pseudo radial flow from plot (dimensionless)

$C_{r D}$: Dimensionless fracture conductivity (dimensionless)

$C_{f D}$: Dimensionless Wellbore storage coefficient (dimensionless)

$L_{f}$: Length of fracture $(\mathrm{ft})$

$\mathrm{k}$: Permeability $(\mathrm{mD})$

$\Delta t_{A E}$: Equivalent Adjusted delta time from derivative curve (h)

Output(s)

C: Well bore storage coefficient (bbl/psi)

$w_{f k_{f}}$: Min Fracture conductivity for infinite conductive fracture $(\mathrm{mD} \mathrm{ft})$

$L_{f M P}$: Length of fracture from match point analysis (ft)

$\left(\Delta P_{a}\right)_{M P}$: Adjusted Pressure difference at match point from plot (psi)

Formula(s)

$$ \begin{gathered} C=\left(141.2 * q_{g} * B_{g} * \frac{\mu}{k * h}\right) *\left(p_{D}\right)_{M P} \\ w_{f k_{f}}=\left(\left(\frac{0.0002637 * k}{\varnothing * \mu * c_{t}}\right) *\left(\frac{\Delta t_{A E}}{t_{\left(L_{f}\right)_{D}}}\right)_{M P}\right)^{\frac{1}{2}} \\ L_{f M P}=\left(\emptyset * h * c_{t} * \frac{L_{f}^{2}}{0.8936}\right) * C_{f D} \\ \left(\Delta P_{a}\right)_{M P}=3.14 * k * C_{r D} * L_{f} \end{gathered} $$

Reference(s)

Lee, J., Rollins J.B., and Spivey J.P. 2003, Pressure Transient Testing, Vol. 9, SPE Textbook Series, Vol. 9 , Henry L. Doherty Memorial Fund of AIME, Richardson, Texas, SPE, Chapter: 6, Page: 127.