Motivation

In many practical cases the Radial Flow Pressure Diffusion is evolving towards pressure stabilization and can be efficiently analyzed using the steady state flow model.

Inputs & Outputs

Radial fluid flow	Homogenous reservoir	Finite reservoir flow boundary	Slightly compressible fluid flow	Constant rate	Constant skin

r_{wf} < r \leq r_e

p(t, r ) = p(r) \Leftrightarrow  \frac{\partial p}{\partial t}  =  0

 \frac{\partial^2 p}{\partial r^2} + \frac{1}{r} \frac{\partial p}{\partial r} =0

p(r_e ) = p_i

\left[ r\frac{\partial p(r)}{\partial r} \right]_{r \rightarrow r_w} = \frac{q_t}{2 \pi \sigma}

p_{wf}= p(r_w ) - S \cdot r_w \, \frac{\partial p}{\partial r} \Bigg|_{r=r_w}

p(r) = p_i + \frac{q_t}{2 \pi \sigma} \, \ln \frac{r}{r_e} = p(r_w) + \frac{q_t}{2 \pi \sigma} \, \ln \frac{r}{r_w} , \quad r_{wf} < r \leq r_e

p_{wf} = p_i - \frac{q_t}{2 \pi \sigma} \, \bigg[ S + \ln \frac{r_e}{r_w} \bigg]

Equation shows how the basic diffusion model parameters impact the relation between drawdown and total sandface flowrate and plays important methodological role as they are used in many algorithms and express-methods of Pressure Testing.

The Total Sandface Productivity Index for low-compressibility fluid and low-compressibility rocks does not depend on formation pressure, bottomhole pressure and the flowrate and can be expressed as:

J_t = \frac{q_t}{p_i - p_{wf}(t)} =\frac{2 \pi \sigma}{\ln \frac{r_e}{r_w} + S} = {\rm const}

The Field-average Productivity Index for low-compressibility fluid and low-compressibility rocks does not depend on formation pressure, bottomhole pressure and the flowrate and can be expressed as:

J_t = \frac{q_t}{p_r(t) - p_{wf}(t)} =\frac{2 \pi \sigma}{\ln \frac{r_e}{r_w} + 0.5 +S} = {\rm const}