Radial Inflow: Steady & Pseudo-Steady State
Every drop of oil heading for the wellbore must converge radially inward - the closer it gets, the faster the rush - and Darcy's law in radial form gives the steady-state rate, with a small correction for a closed, bounded reservoir.
Every drop of oil heading for the wellbore has to converge radially inward - the closer it gets, the faster the rush, and Darcy's law in radial form tells you exactly how fast.
Radial Flow Toward a Well
A production well is a tiny hole at the centre of a large reservoir, so every drop of oil or gas heading for it has to flow radially inward - converging from a wide drainage radius $r_e$ down to the small wellbore radius $r_w$. Because the same fluid is squeezed through ever-smaller concentric cylinders as it approaches the well, the velocity rises and most of the pressure drop is concentrated near the wellbore, not spread evenly across the reservoir. The radial geometry - not the rock physics - is what makes well inflow behave so differently from linear flow in a core plug. Darcy's law, rewritten for this converging geometry, gives the engineer the single most-used inflow equation in the business.
The Steady-State Radial Inflow Equation
For steady-state radial inflow, where pressure is held constant at the outer boundary, the radial Darcy equation in field units gives the oil rate directly: $q = \dfrac{k\,h\,(p_e - p_{wf})}{141.2\,\mu\,B\,(\ln(r_e/r_w) + s)}$ STB/d. Here $k$ is permeability in millidarcies (md), $h$ is net pay in feet (ft), $p_e$ is the pressure at the drainage boundary and $p_{wf}$ the flowing wellbore pressure (both in psi), $\mu$ is viscosity in centipoise (cp), $B$ is the formation volume factor in reservoir barrels per stock-tank barrel (rb/STB), $r_e$ and $r_w$ are the drainage and wellbore radii in feet, and $s$ is the skin factor (taken as zero for now; the next lesson develops it). The constant 141.2 bundles the field-unit conversions. The rate is driven by the pressure difference $(p_e - p_{wf})$ - the drawdown - and resisted by the log-radius term.
The equation above assumes steady state: pressure at the outer boundary $p_e$ stays fixed, as if an aquifer or injector holds it up. Most real bounded reservoirs are not steady-state - they are pseudo-steady state, meaning the reservoir is closed and pressure declines uniformly everywhere once the disturbance reaches the boundary. For pseudo-steady state you swap the constant boundary pressure for the average reservoir pressure $\bar p$ and replace $\ln(r_e/r_w)$ with $\ln(r_e/r_w) - 0.75$, reflecting that the relevant driving pressure is now the reservoir average rather than a fixed outer value. The $-0.75$ correction is the price of a closed boundary. Both regimes use the same equation skeleton; the boundary condition is what changes.
- Drawdown: $p_e - p_{wf} = 3000 - 2000 = 1000$ psi.
- Numerator: $k \times h \times (p_e - p_{wf}) = 50 \times 30 \times 1000 = 1{,}500{,}000$.
- Log term: $\ln(r_e/r_w) = \ln(660/0.33) = \ln(2000) = 7.60$ (skin $s = 0$ adds nothing).
- Denominator: $141.2 \times \mu \times B \times (\ln(r_e/r_w) + s) = 141.2 \times 1.2 \times 1.25 \times 7.60 = 1{,}610$.
- Rate: $q = 1{,}500{,}000 / 1{,}610 \approx 932$ STB/d.
Check your understanding
- Fluid converges radially inward toward the wellbore, so most of the pressure drop sits near the well - the geometry that makes well inflow distinct from linear flow.
- The steady-state radial Darcy equation in field units is q = k h (p_e - p_wf) / (141.2 mu B (ln(r_e/r_w) + s)) STB/d, driven by drawdown and resisted by the log-radius term.
- Pseudo-steady state (closed reservoir) swaps the fixed boundary pressure p_e for the average pressure p-bar and replaces ln(r_e/r_w) with ln(r_e/r_w) - 0.75; for the worked well, ln(r_e/r_w) = 7.60 and q is about 932 STB/d.