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Constant rate production

LaTeX Math Inline
bodyq_t = \rm const

Linear fluid flow

LaTeX Math Inline
bodyp(t, x)

Slightly compressible fluid flow

LaTeX Math Inline
bodyc_t(, p) = c_r +c = \rm const

Homogeneous reservoir

LaTeX Math Inline
bodyM(x, p)=M =\rm const

LaTeX Math Inline
body\phi(x, p)=\phi =\rm const

LaTeX Math Inline
bodyh(x)=h =\rm const

Infinite boundary

LaTeX Math Inline
bodyx \rightarrow \infty


Mathematical Model

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\frac{\partial p}{\partial t}  = \chi \, \frac{d^2 p}{dx^2}
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p(t = 0, x) = p_i
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p(t, x \rightarrow \infty ) = p_i
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\frac{\partial p(t, x )}{\partial x} \bigg|_{x \rightarrow 0} = \frac{q_t}{\sigma \, d}
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p(t,x) = p_i - \frac{q_t}{\sigma \, d} \bigg[ \sqrt{\frac{4 \chi t}{\pi}} \exp \bigg( -\frac{x^2}{4 \chi t} \bigg) - x \, \bigg[ 1- {\rm erf} \bigg(\frac{x}{\sqrt{4 \, \chi \, t}} \bigg) \bigg]  \bigg]



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p_{wf}(t) = p(t,x=0)= p_i - \frac{q_t}{\sigma \, d} \,  \sqrt{\frac{4 \chi t}{\pi}}



Expand
titleDerivation



LaTeX Math Block
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\frac{\partial p}{\partial t}  = \chi \, \frac{d^2 p}{dx^2}



LaTeX Math Block
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p(t = 0, x) = p_i



LaTeX Math Block
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p(t, x \rightarrow \infty ) = p_i



LaTeX Math Block
anchorEM415
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\frac{\partial p(t, x )}{\partial x} \bigg|_{x \rightarrow 0} = \frac{q_t}{\sigma \, d}





Scope of Applicability

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Pressure Testing

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