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titleSolution

There is no universal analytical solution to the above problem

LaTeX Math Block Reference
anchorP1
–
LaTeX Math Block Reference
anchorPE
but it can be always presented as below:


LaTeX Math Block
anchorp_F
alignmentleft
p(t,r) = p_i - \frac{q_t}{4 \pi \sigma} \,  F \bigg( - \frac{r^2}{4 \chi t} \bigg)



LaTeX Math Block
anchorpwf
alignmentleft
p_{wf}(t) = p_i - \frac{q_t}{4 \pi \sigma} \, \bigg[2S +   F \bigg( - \frac{r_w^2}{4 \chi t} \bigg) \bigg]


where

LaTeX Math Inline
bodyF(\xi)
a single-argument function describing the peculiarities of the diffusion model (well geometry, penetration geometry, formation inhomogeneities, hydraulic fractures, boundary conditions, etc.).

The fact that solution of equations

LaTeX Math Block Reference
anchorP1
–
LaTeX Math Block Reference
anchorPE
can be represented presented as
LaTeX Math Block Reference
anchorp_F
–
LaTeX Math Block Reference
anchorpwf
finds a lot of practical applications in Well Testing.


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