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titleDerivation



Applications

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titleLine Source Solution


In simplest case of infinite homogeneous reservoir, produced by a vertical well the 

LaTeX Math Inline
bodyF
 function has an exact analytical formula, given by exponential integral 
LaTeX Math Inline
bodyF(z) = {\rm Ei}_1 (z)
 (see Line Source Solution (LSS) @model).



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titlePTAPressure Testing


Pressure Testing – Infinite reservoir



Pressure Drop


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anchor1EWTY
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\delta p = p_i - p_{wf}(t) \sim  \ln t + {\rm const}



Log derivative


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t \frac{d (\delta p)}{dt}  \sim \rm const







Fig. 2. PTA Diagnostic plot for radial fluid flow




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titleLine Source SolutionProductivity Index Analysis


The Productivity Index for single-phase low-compressibility fluid and low-compressibility rocks  does not depend on formation pressure, bottom-hole pressure and the flow rate and can be expressed as:

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anchorJ
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J(t) = \frac{q_t}{p_i - p_{wf}(t)} =\frac{ 4 \pi \sigma }{ 2S - F \bigg( - \frac{r_w^2}{4 \chi t} \bigg)  }
In simplest case of infinite homogeneous reservoir, produced by a vertical well the 
LaTeX Math Inline
bodyF
 function has an exact analytical formula, given by exponential integral 
LaTeX Math Inline
bodyF(z) = {\rm Ei}_1 (z)
 (see Line Source Solution (LSS) @model).


 Expand
titleIsobar Propagation

Isobar equation for a constant-rate production:
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anchorQ7VZX
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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 
 LaTeX Math Inline
bodyt=0
 moment was at well walls 
 LaTeX Math Inline
bodyr=r_w
 then the formula for constant-pressure front propagation becomes:
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anchorH09BI
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r(t) = r_w + 2 \sqrt{\chi t}
This leads to estimation of isobar velocity:
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anchorNX4O7
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u_p(t) = \sqrt{\frac{\chi}{t}}



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