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| r_{wf} < r \leq r_e |
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| \frac{\partial p}{\partial t}= \chi \left[ \frac{\partial^2 p}{\partial r^2} + \frac{1}{r} \frac{\partial p}{\partial r} \right] |
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| \left[ \frac{\partial p}{\partial r} \right]_{r=r_e} = 0 |
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| \left[ r\frac{\partial p(t,r)}{\partial r} \right]_{r \rightarrow= r_w} = \frac{q_t}{2 \pi \sigma} |
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| p_{wf}(t)= p(t, r_w) - S \cdot \left[ r \frac{\partial p(t,r)}{\partial r} \right]_{r=r_w \} = p(t, r_w) - \frac{q_t}{\partial 2 \pi \sigma} S |
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It is important to note that equations | LaTeX Math Block Reference |
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do not constitute a complete CVP as it does not specify the initial condition. |
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| p(t,r) = p_i - \frac{{\rm w \,} q_t }{ |
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\partial r} \Bigg|_{r=r_w} |
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title | SolutionV_e \, \phi \, c_t} \, t + \frac{{\rm w \,} q_t }{4\pi \sigma} \left[ 2 \ln \frac{r}{r_e} - \frac{r^2}{r_e^2} + 1 \right]
, \quad r_{wf} < r \leq r_e,
\quad {\rm w }= 1 - \frac{r_w^2}{r_e^2} |
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| p_e(t) = p_i - \frac{{\rm w \,} q_t}{V_e \phi c_t}t |
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| p_{wf}(t) = p_e(t) - \frac{q_t}{2 \pi \sigma} \, \left[ {\rm w\, } \ln \frac{r_e}{r_w} + 0.5 + S \right] |
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Approximations
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| {\rm w} = 1 - \frac{r_w^2}{r_e^2} \approx 1 |
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\approx p_i - \frac{q_t}{V_e \, \phi \, c_t} \, t + \frac{q_t}{4\pi \sigma} \bigg[ 2 \ln \frac{r}{r_e} - \frac{r^2}{r_e^2} + 1 \bigg]
, \quad r_{wf} < r \leq r_e |
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| p_{wf}(t) |
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\approx p_e(t) - \frac{q_t}{2 \pi \sigma} \, \left[ \ln \frac{r_e}{r_w} |
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\approx p_i - \frac{q_t}{V_e \phi c_t}t |
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Applications
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Equation
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shows how the
basic diffusion model parameters impact the relation between
drawdown | LaTeX Math Inline |
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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_e(t) - p_{wf}(t)} =\frac{2 \pi \sigma}{\ln \frac{r_e}{r_w} + 0.5 +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: | LaTeX Math Block |
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| J_t = \frac{q_t}{p_ir(t) - p_{wf}(t)} =\frac{2 \pi \sigma}{\ln \frac{r_e}{r_w} + 0.75 +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
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