In linear formation approхimation the pressure response to the varying rates in the offset wells is subject to convolution equation:

$\begin{array}{l}\displaystyle p(t) = p_i + \int_0^t p_u(t - \tau) dq = p_i + \int_0^t p_u(t - \tau) \cdot q(\tau) d\tau\end{array}$

In case production history can be approximated by a finite sequence of constant rate production intervals (called Pressure Transients):

$\begin{array}{l}\displaystyle p(t) = p_i + \sum_{\alpha = 1}^{N} \left[ q^{(\alpha)} - q^{(\alpha-1)} \right] \cdot p_u(t - t^{\alpha})\end{array}$

where

$\begin{array}{l}p(t)\end{array}$	pressure at $\begin{array}{l}n\end{array}$ -th well at arbitrary moment of time $\begin{array}{l}t\end{array}$
$\begin{array}{l}p_i\end{array}$	initial pressure at $\begin{array}{l}n\end{array}$ -the well
$\begin{array}{l}\alpha = 1 .. N\end{array}$	index number of a pressure transient (period of time where rate was constant)
$\begin{array}{l}N\end{array}$	total number of transients
$\begin{array}{l}t^{\alpha}\end{array}$	starting point of the $\begin{array}{l}\alpha\end{array}$ -th transient
$\begin{array}{l}q^{(\alpha)}\end{array}$	rate value of $\begin{array}{l}\alpha\end{array}$ -th transient which starts at the time moment $\begin{array}{l}t^{(\alpha)}\end{array}$
$\begin{array}{l}p_u(t)\end{array}$	pressure transient response to the unit-rate production (DTR)

with assumption:

$\begin{array}{l}q^{(-1)} = 0\end{array}$ , which means that well was shut-in before it started the first transient $\begin{array}{l}\alpha =1\end{array}$
$\begin{array}{l}p_u(t) = 0\end{array}$ at $\begin{array}{l}t < 0\end{array}$ which means pressure drop is zero before the well starts unit-rate production

Hence, convolution is using initial formation pressure $\begin{array}{l}p_{i, n}\end{array}$ , unit-rate transient responses of wells and cross-well intervals $\begin{array}{l}p^u_{nk} (t)\end{array}$ and rate histories $\begin{array}{l}\{ q_k (t) \}_{k = 1 .. N}\end{array}$ to calculate pressure bottom-hole pressure response as function time $\begin{array}{l}p_n(t)\end{array}$ :

(1)	$\begin{array}{l}\displaystyle \big\{ p_{i, n}, \{ p^u_{nk} (t), q_k (t) \}_{k = 1 .. N} \big\} \rightarrow p_n(t)\end{array}$

The MDCV is a reverse problem to convolution and search for $\begin{array}{l}N^2\end{array}$ functions $\begin{array}{l}p^u_{nk} (t)\end{array}$ and $\begin{array}{l}N\end{array}$ numbers $\begin{array}{l}p_{i, n}\end{array}$ using the historical pressure and rate records $\begin{array}{l}\{ p_k(t), \ \{ q^{(\alpha)}_k \}_{\alpha = 1.. N_k} \}_{k = 1 .. N}\end{array}$ and provides the adjustment to the rate histories for the small mistakes $\begin{array}{l}\{ q_k \}_{\alpha = 1.. N_k} \rightarrow \{ \tilde q_k \}_{\alpha = 1.. N_k}\end{array}$ :

(2)	$\begin{array}{l}\displaystyle \big\{ p_k(t), q_k (t) \big\} _{k = 1 .. N} \rightarrow \big\{ p_{i, n}, \{ p^u_{nk} (t), \tilde q_k (t) \}_{k = 1 .. N} \big\}\end{array}$

The solution of deconvolution problem is based on the minimization of the objective function:

(3)	$\begin{array}{l}\displaystyle E(\{ p_{i,n}, p^u_{nk}(\tau), q^{(\alpha)}_n \}_{n=1..N}) \rightarrow {\rm min}\end{array}$

where

(4)

$\begin{array}{l}\displaystyle E(\{ p_{i,n}, p^u_{nk}(\tau), q^{(\alpha)}_n \}_{n=1..N}) = \sum_{n=1}^N \Big(p_{i,n} + \sum_{k = 1}^N \sum_{\alpha = 1}^{N_k} (q^{(\alpha)}_k - q^{(\alpha-1)}_k ) \ p^u_{nk}(t - t_{\alpha k})- p_n(t) \Big)^2 + w_c \, \sum_{n = 1}^N \sum_{k = 1}^{N_k} {\rm Curv} \big( p^u_{nk}(\tau) \big) + w_q \, \sum_{k = 1}^N \sum_{\alpha = 1}^{N_k} \big( q^{(\alpha)}_k - \tilde q^{(\alpha)}_k \big)^2\end{array}$

and objective function components have the following meaning:


$\begin{array}{l}\sum_{n=1}^N \Big(p_{i,n} + \sum_{k = 1}^N \sum_{\alpha = 1}^{N_k} (q^{(\alpha)}_k - q^{(\alpha-1)}_k ) \ p^u_{nk}(t - t_{\alpha k})- p_n(t) \Big)^2\end{array}$	is responsible for minimizing discrepancy between model and historical pressure data
$\begin{array}{l}w_c \, \sum_{n = 1}^N \sum_{k = 1}^{N_k} {\rm Curv} \big( p^u_{nk}(\tau) \big)\end{array}$	is responsible for minimizing the curvature of the transient response (which reflects the diffusion character of the pressure response to well flow)
$\begin{array}{l}w_q \, \sum_{k = 1}^N \sum_{\alpha = 1}^{N_k} \big( q^{(\alpha)}_k - \tilde q^{(\alpha)}_k \big)^2\end{array}$	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.

One of the efficeint regularizations has been suggested by Shroeter

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 MDCV also adjusts the rate histories for each well $\begin{array}{l}\{ q^{(\alpha)}_k \}_{\alpha = 1.. N_k} \rightarrow \{ \tilde q^{(\alpha)}_k \}_{\alpha = 1.. N_k}\end{array}$ to achieve the best macth of the bottom hole pressure readings.

The weight coefficients $\begin{array}{l}w_c\end{array}$ and $\begin{array}{l}w_q\end{array}$ control contributions from corresponding components and should be calibrated to the reference transients (manuualy or automatically).

The SDCV methodology constitute a big area of practical knowledge and not all the tricks and solutions are currenlty automated and require a practical skill.

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