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p_n(t) = p_{i0,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}) |
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| 1 | | pressure at -th well at arbitrary moment of time |
| 2 | | LaTeX Math Inline |
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| body | --uriencoded--p_{i%7B0,n}n%7D |
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| 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 |
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\big\{ p_{i0, n}, \{ p^u_{nk} (t), q_k (t) \}_{k = 1 .. N} \big\} \rightarrow p_n(t) |
The
| Hint |
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| 0 | MDCV |
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| 1 | Multiwell Deconvolution |
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The RDCV is a reverse problem to convolution and search for
functions
and
numbers
using the historical pressure and rate records
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| body | \{ p_k(t), \ \{ q^{(\alpha)}_k \}_{\alpha = 1.. N_k} \}_{k = 1 .. N} |
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and provides the adjustment to the rate histories for the small mistakes
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| body | \{ q_k \}_{\alpha = 1.. N_k} \rightarrow \{ \tilde q_k \}_{\alpha = 1.. N_k} |
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:
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\big\{ p_k(t), q_k (t) \big\} _{k = 1 .. N} \rightarrow \big\{ p_{i0, n}, \{ p^u_{nk} (t), \tilde q_k (t) \}_{k = 1 .. N} \big\} |
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E(\{ p_{i0,n}, p^u_{nk}(\tau), q^{(\alpha)}_n \}_{n=1..N}) \rightarrow {\rm min} |
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E(\{ p_{i0,n}, p^u_{nk}(\tau), q^{(\alpha)}_n \}_{n=1..N}) = \sum_{n=1}^N \Big(p_{i0,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
+ w_c \, \sum_{n = 1}^N \sum_{k = 1}^{N_k} {\rm Curv} \big( p^u_{nk}(\tau) \big) +
w_q \, \sum_{k = 1}^N \sum_{\alpha = 1}^{N_k} \big( q^{(\alpha)}_k - \tilde q^{(\alpha)}_k \big)^2 |
and objective function components have the following meaning:
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| body | --uriencoded--\sum_{n=1}^N %7Bn=1%7D%5eN \Big(p_{i%7B0,n} n%7D + \sum_{k = 1}^N %7Bk = 1%7D%5eN \sum_{%7B\alpha = 1}^{N_k} (q^{1%7D%5e%7BN_k%7D (q%5e%7B(\alpha)}%7D_k - q^{q%5e%7B(\alpha-1)}%7D_k ) \ p^up%5eu_{nk}%7Bnk%7D(t - t_{%7B\alpha k}k%7D)- p_n(t) \Big)^2 %5e2 |
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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) |
| LaTeX Math Inline |
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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) |
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In practice the above approach is not stable.
One of the efficeint efficient regularizations has been suggested by Shroeter
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One of the most efficient method in minimizing the above objective function is the hybrid of genetic and quasinewton algorithms in parallel on multicore workstation.
The
| Hint |
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| 0 | MDCV |
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| 1 | Multiwell Deconvolution |
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The RDCV also adjusts the rate histories for each well
| LaTeX Math Inline |
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| body | \{ q^{(\alpha)}_k \}_{\alpha = 1.. N_k} \rightarrow \{ \tilde q^{(\alpha)}_k \}_{\alpha = 1.. N_k} |
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to achieve the best macth of the bottom hole pressure readings.
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