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In many practical cases the Radial Flow Pressure Diffusion is getting
reservoir fluid flow created by well is getting aligned with a radial direction towards or away from well.
This type of reservoir fluid flow is called radial fluid flow and corresponding pressure diffusion models provide a diagnostic basis for pressure-rate base reservoir flow analysis.
The radial flow can be infinite acting or boundary dominated or transiting from one to another.
Although the actual reservoir fluid flow may not have an axial symmetry around the well-reservoir contact or around reservoir inhomogeneities (like boundary and faults and composite areas) but still in many practical cases the long-term correlation between the flowrate and bottom-hole pressure response can be approximated by a radial flow pressure modelevolving towards pressure stabilization and can be efficiently analyzed using the steady state flow model.
Inputs & Outputs
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Physical Model
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| r_{wf} < r \leq r_e |
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| p(t, r ) = p(r) \Leftrightarrow \frac{\partial p}{\partial t} = 0 |
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| \frac{\partial^2 p}{\partial r^2} + \frac{1}{r} \frac{\partial p}{\partial r} =0 |
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| p(r_e ) = p_i |
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| \left[ r\frac{\partial p(r )}{\partial r} \right]_{r \rightarrow r_w} = \frac{q_t}{2 \pi \sigma} |
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| p_{wf}= p(r_w ) - S \cdot r_w \, \frac{\partial p}{\partial r} \Bigg|_{r=r_w} |
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Equation
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shows how the
basic diffusion model parameters impact the relation between
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| body | \Delta p = p_i - p_{wf} |
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and
total sandface flowrate and plays important methodological role as they are used in many algorithms and express-methods of
Pressure Testing.
It also called Dupuis
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| title | Productivity Index Analysis |
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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: | LaTeX Math Block |
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| J_t = \frac{q_t}{p_i - p_{wf}(t)} =\frac{2 \pi \sigma}{\ln \frac{r_e}{r_w} + S} = {\rm const} |
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See Also
Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Radial fluid flow / Pressure diffusion / Pressure Diffusion @model / Radial Flow Pressure Diffusion @model
Petroleum Industry / Upstream / Subsurface E&P Disciplines / Well Testing / Pressure Testing
[ Well & Reservoir Surveillance ]
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but this only works for the middle-times and long-times as early times are influenced by wellbore storage and non-linear effects of skin.
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p(t,r) = p_i + \frac{q_t}{4 \pi \sigma} \, {\rm Ei} \bigg( - \frac{r^2}{4 \chi t} \bigg) |
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Диффузия давления описывается решением уравнения однофазного радиального течения в бесконечном однородном пласте:
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\frac{\partial p}{\partial t} = \chi \, \Delta p = \chi \, \frac{1}{r} \frac{\partial}{\partial r} \bigg( r \frac{\partial p}{\partial r} \bigg) |
с начальным условием:
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p(t = 0, r) = p_i |
и граничными условиями:
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p(t, r \rightarrow \infty ) = p_i |
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r \frac{\partial p(t, x )}{\partial r} \bigg|_{r \rightarrow 0} = \frac{q_t}{2 \pi \sigma} |
где
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| body | \sigma = \frac{k \, h}{\mu} |
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– гидропроводность пласта, | LaTeX Math Inline |
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| body | \chi = \frac{k}{\mu} \, \frac{1}{\phi \, c_t} |
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– пьезопроводность пласта, – проницаемость пласта, – пористость пласта, – сжимаемость пласта, – сжимаемость порового коллектора, – сжимаемость насыщающего пласт флюида, – вязкость насыщающего пласт флюида.При анализе отклика давления на самой скважине (
) после включения на достаточно больших временах, удовлетворяющих условию:| LaTeX Math Block |
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t \gg \frac{r_w^2}{4 \chi}
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которые на практике наступают очень быстро, можно воспользоваться приближением
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| body | {\rm Ei}(-x) \sim \ln (x) + \gamma \sim \ln (1.781 x) |
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, где – постоянная Эйлера. ...
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p(t,r_w) = p_i + \frac{q_t}{4 \pi \sigma} \, \ln \bigg( 1.781 \, \frac{r_w^2}{4 \chi t} \bigg) |
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\delta p = p_i - p_{wf}(t) \sim { \rm const } + \frac{q_t}{4 \pi \sigma} \, \ln t |
а логарифмическая производная становится постоянной во времени:
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t \frac{d (\delta p)}{dt} \sim \frac{q_t}{4 \pi \sigma} |
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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: | LaTeX Math Block |
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| 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} |
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See Also
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Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Radial fluid flow / Pressure diffusion / Pressure Diffusion @model / Radial Flow Pressure Diffusion @model
Petroleum Industry / Upstream / Subsurface E&P Disciplines / Well Testing / Pressure Testing
[ Well & Reservoir Surveillance ]
