Mathematics
In linear formation approхimation the pressure response to the varying rates in the offset wells is subject to convolution equation:
| LaTeX Math Block |
|---|
|
p_n(t)
= p_{0,n} + \sum_{k = 1}^N \int_0^t p^u_{nk}(t - \tau) \, \dot q_k \, d\tau
\approx 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
|
|
|
|---|
| 1 | | pressure at -th well at arbitrary moment of time |
| 2 | |
{i,n} | 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
...
and objective function components have the following meaning:
|
|
|---|
| LaTeX Math Inline |
|---|
| 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 |
|---|
|
| is responsible for minimizing discrepancy between model and historical pressure data |
| LaTeX Math Inline |
|---|
| body | w_c \, \sum_{n = 1}^N \sum_{k = 1}^{N_k} {\rm Curv} \big( p^u_{nk}(\tau) \big) |
|---|
|
| 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 |
|---|
| body | w_q \, \sum_{k = 1}^N \sum_{\alpha = 1}^{N_k} \big( q^{(\alpha)}_k - \tilde q^{(\alpha)}_k \big)^2 |
|---|
|
| 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.
...
The weight coefficients
and
control contributions from corresponding components and should be calibrated to the reference transients (manuualy or automatically).
The
| Hint |
|---|
| 0 | MDCV |
|---|
| 1 | Multiwell Deconvolution |
|---|
|
XDCV 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
...
Petroleum Industry / Upstream / Subsurface E&P Disciplines / Production Analysis (PA) / Pressure Deconvolution / Multiwell deconvolution (MDCV) / XDCV
