Mathematics
In linear formation approхimation the pressure response to the varying rates in the offset wells is subject to convolution equation:
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p_n(t) = p_{i,n} + \sum_{k = 1}^N \sum_{\alpha = 1}^{N_k} \big( q^{(\alpha)}_k - q^{(\alpha-1)}_k \big) \ p^u_{nk}(t - t_{\alpha k}) |
where
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| 1 | | pressure at -th well at arbitrary moment of time |
| 2 | | initial pressure at -the well |
| 3 | | rate value of -th transient at -th well |
| 4 | | pressure transient response in -th wel to unit-rate production from -th well |
| 5 | | starting point of the -th transient in -th well |
| 6 | | number of wells in the test |
| 7 | | number of transients in -th well |
with assumption:
- – for any well
- at for any pair of wells
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and objective function components have the following meaning:
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| body | \sum_{n=1}^N \Big(p_{i,n} + \sum_{k = 1}^N \sum_{\alpha = 1}^{N_k} (q^{(\alpha)}_k - q^{(\alpha-1)}_k ) \ p^u_{nk}(t - t_{\alpha k})- p_n(t) \Big)^2 |
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| is responsible for minimizing discrepancy between model and historical pressure data |
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| body | w_c \, \sum_{n = 1}^N \sum_{k = 1}^{N_k} {\rm Curv} \big( p^u_{nk}(\tau) \big) |
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| is responsible for minimizing the curvature of the transient response (which reflects the diffusion character of the pressure response to well flow) |
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| body | w_q \, \sum_{k = 1}^N \sum_{\alpha = 1}^{N_k} \big( q^{(\alpha)}_k - \tilde q^{(\alpha)}_k \big)^2 |
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| is responsible for minimizing discrepancy between model and historical rate data (since historical rate records are not accurate at the time scale of pressure sampling) |
In practice the above approach is not stable.
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The
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| 0 | MDCV |
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| 1 | Multiwell Deconvolution |
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methodology constitute a big area of practical knowledge and not all the tricks and solutions are currenlty automated and require a practical skill.
See also
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Petroleum Industry / Upstream / Subsurface E&P Disciplines / Well Testing / Pressure Testing / Pressure convolution / Pressure Deconvolution / Multiwell deconvolution (MDCV)
