Motivation
In many practical cases the reservoir flow created by well is getting aligned with a radial direction towards or away from well.
This type of flow is called radial fluid flow and corresponding pressure diffusion models provide a diagnostic basis for pressure-rate base reservoir flow analysis.
Although the actual flow may not have an axial symmetry around the well-reservoir contact or reservoir inhomogeneities (like boundary and faults and composite areas) but still the dominant part of wellbore and reservoir pressure variation is usually radial-flow or linear-flow and the two represent the basis for Pressure diffusion analysis.
Inputs & Outputs
Inputs | Outputs | ||
---|---|---|---|
q_t | total sandface rate | p(t,r) | reservoir pressure |
{p_i} | initial formation pressure | {p_{wf}(t)} | well bottomhole pressure |
\sigma | transmissibility | ||
\chi | pressure diffusivity |
Physical Model
Radial fluid flow | Homogenous reservoir | Infinite boundary | Zero wellbore radius | Slightly compressible fluid flow | Constant rate production |
---|---|---|---|---|---|
p(t, r) | M(r, p)=M =\rm const \phi(r, p)=\phi =\rm const h(r)=h =\rm const c_r(r)=c_r =\rm const | r \rightarrow \infty | r_w = 0 | c_t(r,p) = c_r(r) +c = \rm const | q_t = \rm const |
Mathematical Model
Applications
Pressure Testing – Infinite reservoir
Pressure Drop |
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Log derivative |
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Fig. 2. PTA Diagnostic plot for radial fluid flow |
Isobar Propagation
Isobar equation for a constant-rate production:
(9) | p(t,r) = p_i + \frac{q_t}{4 \pi \sigma} \, F \bigg( - \frac{r^2}{4 \chi t} \bigg) = {\rm const} \quad \rightarrow \quad \frac{r^2}{4 \chi t}= {\rm const} |
Since the pressure disturbance at
t=0 moment was at well walls
r=r_w then the formula for constant-pressure front propagation becomes:
(10) | r(t) = r_w + 2 \sqrt{\chi t} |
This leads to estimation of isobar velocity:
(11) | u_p(t) = \sqrt{\frac{\chi}{t}} |
See also
Physics / Fluid Dynamics / Radial fluid flow
[ Line Source Solution (LSS) @model ]
[ Linear Flow Pressure Diffusion @model ]