Peaceman Well Model

The Peaceman well model provides the connection factor (well index) for coupling the wellbore to the reservoir grid. Based on Peaceman (1978, 1983).

Well Connection Factor (Well Index)

The well index $\mathrm{WI}$ quantifies the transmissibility between the wellbore and the reservoir grid block. It relates the volumetric flow rate to the pressure difference between the grid-block average pressure and the wellbore flowing pressure.

$$\mathrm{WI} = \frac{2\pi\,k\,h}{\ln \left(\dfrac{r_0}{r_\mathrm{w}}\right) + s}$$

where $kh$ is the effective permeability-thickness product, $r_0$ is the Peaceman equivalent radius, $r_\mathrm{w}$ is the wellbore radius, and $s$ is the skin factor.

Grid Block and Equivalent Radius

The diagram below shows a reservoir grid block with the wellbore at its centre. The equivalent radius $r_0$ is the distance at which the analytical steady-state pressure equals the numerically computed block-average pressure.

Figure 1 — Reservoir grid block with wellbore at centre, showing equivalent radius $r_0$ and well-block pressure definitions.

Pressure Profile

The radial pressure profile from the wellbore to the grid-block boundary shows the logarithmic relationship between pressure and radial distance:

Figure 2 — Radial pressure profile from wellbore to grid-block boundary, showing the logarithmic decay.

Peaceman Equivalent Radius

For a square grid block of side $\Delta x$ with isotropic permeability:

$$r_0 = 0.198\;\Delta x \qquad \text{(isotropic, square grid)}$$

Anisotropic Equivalent Radius

When the reservoir has different horizontal permeabilities $k_\mathrm{x}$ and $k_\mathrm{y}$:

$$r_0 = \frac{0.28\,\sqrt{\left(\frac{k_\mathrm{y}}{k_\mathrm{x}}\right)^{1/2} \Delta x^{\,2} \;+\; \left(\frac{k_\mathrm{x}}{k_\mathrm{y}}\right)^{1/2} \Delta y^{\,2}}}{\left(\frac{k_\mathrm{y}}{k_\mathrm{x}}\right)^{1/4} + \left(\frac{k_\mathrm{x}}{k_\mathrm{y}}\right)^{1/4}}$$

Flow Rate from Well Index

The mass injection rate into a grid block:

$$q = \mathrm{WI}\;\lambda\;\bigl(p_\mathrm{block} - p_\mathrm{wf}\bigr)$$

where $\lambda = k_\mathrm{r}/\mu$ is the fluid mobility.

Skin Factor

The skin factor $s$ accounts for near-wellbore damage ($s > 0$) or stimulation ($s < 0$). The effective wellbore radius:

$$r_\mathrm{w,eff} = r_\mathrm{w}\;e^{-s}$$

References

• Peaceman, D.W. (1978). Interpretation of well-block pressures in numerical reservoir simulation. SPE Journal, 18(03):183–194.
• Peaceman, D.W. (1983). Interpretation of well-block pressures in numerical reservoir simulation with nonsquare grid blocks and anisotropic permeability. SPE Journal, 23(03):531–543.