Analysis of a flow test with smoothly varying rates

Input(s)

$t_{1}$: Time at $P_{w f 1}$ from Given Values or Trendline (h)

$t_{2}$: Time at $P_{w f 2}$ from Given Values or Trendline (h)

$\mathrm{q}_{1}$: Flow Rate at $\mathrm{P}_{\mathrm{wf} 1}(\mathrm{STB} /$ day $)$

$\mathrm{q}_{2}$: Flow Rate at $\mathrm{P}_{\mathrm{wf} 2}(\mathrm{STB} /$ day $)$

$\mathrm{p}_{\mathrm{i}}$: Initial Pressure (psi)

$\mathrm{p}_{\mathrm{wf} 2}$: Wellflow Pressure at Point 2 from Given Values or Trendline (psi)

$\mathrm{p}_{\mathrm{wf} 1}$: Wellflow Pressure at Point 1 from Given Values or Trendline (psi)

$\mathrm{p}_{\mathrm{wf}}$: Pressure Value at $t=1 \mathrm{~h}(\mathrm{psi})$

q: Flow Rate (STB/day)

B: Volume Factor (RB/STB)

$\mathrm{h}$: Thickness of Reservoir (ft)

$\mu$: Viscosity of Oil (cP)

$\varnothing$: Porosity (fraction)

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

$\mathrm{r}_{\mathrm{w}}$: Wellbore Radius (ft)

Output(s)

$\mathrm{m}$: Slope of Line (dimensionless)

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

s: Skin Factor (dimensionless)

Formula(s)

$$ \begin{gathered} \mathrm{m}=\frac{\left(\frac{\mathrm{p}_{\mathrm{i}}-\mathrm{p}_{\mathrm{wf} 2}}{\mathrm{q}_{2}}\right)-\left(\frac{\mathrm{p}_{\mathrm{i}}-\mathrm{p}_{\mathrm{wf} 1}}{\mathrm{q}_{1}}\right)}{\log \left(\mathrm{t}_{2}\right)-\log \left(\mathrm{t}_{1}\right)} \\ \mathrm{k}=162.6 * \mathrm{~B} * \frac{\mu}{\mathrm{m} * \mathrm{~h}} \\ \mathrm{~s}=1.151 *\left(\frac{1}{\mathrm{~m}\left(\frac{\mathrm{p}_{\mathrm{i}}-\mathrm{p}_{\mathrm{wf}}}{\mathrm{q}}\right)}-\log \left(\frac{\mathrm{k}}{\emptyset * \mu * \mathrm{c}_{\mathrm{t}} * \mathrm{r}_{\mathrm{w}}^{2}}\right)+3.23\right) \end{gathered} $$

Reference(s)

Lee, J., Rollins, J. B., \& Spivey, J. P. (2003). Pressure Transient Testing (Vol. 9). Richardson, Texas: Society of Petroleum Engineers, Page: 31.