Application of Darcy’s Law - Top Dog Engineer

Darcy’s law can be applied to model reservoir flow systems. The simplest is steady-state radial flow (steady-state just means that the flow rate and pressure does not vary with time). Consider the figure below:

In vertical wells, we often assume a cylindrial drainage area as depicted in the figure above. The flow is directed from the reservoir boundary to the wellbore. To model flow scenario, we apply Darcy’s Law:

STEP_1: Write Down Darcy’s Equation and define the variables

(Note: I excluded the negative. You’ll see why later)

$\frac{q}{A_c} =\frac{k}{u} \frac{dP}{dL}$

where:

${A_c} =2\pi rh$

For a cylindrical flow area.

STEP 2: Integrate by separation of variables

$\frac{q}{2\pi rh} = \frac{k}{u} \frac{dP}{dL}$

$\frac{q}{r}dr = \frac{2\pi hk}{u} {dP}$

$\int_{r_w}^{r_e}\frac{q}{r}dr = \frac{2\pi hk}{u} \int_{P_w}^{P_e}{dP}$

$q[\ln{r_e}-\ln{r_w}] = \frac{2\pi hk(P_e-P_w)}{u}$

STEP 3: Write the final expression

$\boxed{q = \frac{2\pi hk(P_e-P_w)}{u\ln{\frac{r_e}{r_w}}}}$

By inspection (if you plug in some numbers), the negative sign is not needed because the flow rate will be positive, indicating that the fluid is flowing to the wellbore.