The XCRM model predicts the formation pressure $\begin{array}{l}p_{e,n}\end{array}$ and bottom-hole pressure $\begin{array}{l}p_{wf,n}\end{array}$ in the $\begin{array}{l}n\end{array}$ -th oil producer in response to:

its current oil/water production rate $\begin{array}{l}\left( q^{\uparrow}_{On}, \ q^{\uparrow}_{Wn} \right)\end{array}$
its cumulative oil/water production $\begin{array}{l}\left( Q^{\uparrow}_{On}, \ Q^{\uparrow}_{Wn} \right)\end{array}$
cumulative oil/water production from the offset wells $\begin{array}{l}\left( Q^{\uparrow}_{Om}, \ Q^{\uparrow}_{Wm} \right)_{m \neq n \, \in \uparrow}\end{array}$
cumulative water injection in the offset wells $\begin{array}{l}\left( Q^{\downarrow}_{Om}, \ Q^{\downarrow}_{Wm} \right)_{m \in \downarrow}\end{array}$

using the following equations:

(1)

$\begin{array}{l}\displaystyle P_{\Psi} \left[ p_{e,n} \ (t) \right] = P_{\Psi} \left[ p_{i,n} \ (0) \right] + \gamma_n^{-1} \cdot \left[ B_{og} \cdot Q^{\uparrow}_{O,nn} + f^{\uparrow}_{W,nn} \cdot B_w \cdot Q^{\uparrow}_{W,n} + \sum_{m \neq n} Q^{\uparrow}_{nm} + \sum_k Q^{\downarrow}_{nk} \right]\end{array}$

(2)	$\begin{array}{l}\displaystyle p_{{\rm wf}, n} \ \ \ (t) = p_e \ (t) + 0.5 \cdot J_{On}^{-1} \cdot \left[ q^{\uparrow}_{On}(t) + f^{\uparrow}_{W,nn} \cdot \frac{\mu_W}{\mu_O} \cdot \frac{k_{ro}(s_{w,n})}{k_{rw}(s_{w,n})} \cdot q^{\uparrow}_{Wn}(t) \right]\end{array}$

(3)	$\begin{array}{l}\displaystyle s_{w,n} = \left[ 1 + \frac{B_o}{B_w} \cdot \frac{q^{\uparrow}_{On}}{f^{\uparrow}_{W,nn} \cdot q^{\uparrow}_{Wn}} \right]^{-1}\end{array}$

(4)	$\begin{array}{l}\displaystyle Q(t) = \int_0^t q(t) \, dt\end{array}$

(5)	$\begin{array}{l}\displaystyle Q^{\uparrow}_{nm} \ = \ - \ f^{\uparrow}_{O,nm} \ \cdot B_{ob} \cdot \, Q^{\uparrow}_{O,m} \ - \ f^{\uparrow}_{G,nm} \ \cdot B_{go} \cdot Q^{\uparrow}_{G,m} \ - \ f^{\uparrow}_{W,nm} \ \cdot B_w \cdot Q^{\uparrow}_{W,m}\end{array}$

(6)	$\begin{array}{l}\displaystyle Q^{\downarrow}_{nk} \ = f^{\downarrow}_{G,nk} \ \cdot B_{go} \cdot Q^{\downarrow}_{G,k} \ + \ f^{\downarrow}_{W,nk} \ \cdot B_w \cdot Q^{\downarrow}_{W,k} \ + \ B_{go} \cdot Q^{\downarrow}_{GCAP,k} \ \ + \ B_w \cdot Q^{\downarrow}_{WAQ,k}\end{array}$

(7)	$\begin{array}{l}\displaystyle B_{og} = \frac{B_o - R_s \, B_g}{1- R_s \, R_v}\end{array}$

(8)	$\begin{array}{l}\displaystyle B_{go} = \frac{ B_g - R_v \, B_o}{1- R_s \, R_v}\end{array}$

where $\begin{array}{l}B_o, \, B_g, \, B_w, \, R_s, \, R_v\end{array}$ are Dynamic fluid properties and $\begin{array}{l}P_{\Psi} []\end{array}$ is Normalized Pseudo-Pressure .

The value of cumulative Gas Cap influx $\begin{array}{l}Q^{\downarrow}_{GCAP}\end{array}$ is modelled as in Gas Cap Drive @model.

The value of cumulative Aquifer influx $\begin{array}{l}Q^{\downarrow}_{GCAP}\end{array}$ is modelled as in Aquifer Drive Models (most popular being Carter-Tracy model for infinite-volume aquifer and Fetkovich for finite-volume aquifer).

In case of Water Injector : $\begin{array}{l}s_{o,n} = s_{or} \ , \quad s_{g,n} = 0 \ , \quad s_{w,n} = 1 - s_{or}\end{array}$ .

In case of Gas Injector: $\begin{array}{l}s_{o,n} = 0 \ , \quad s_{g,n} = 1 - s_{wcg} \ , \quad s_{w,n} = s_{wcg}\end{array}$ .

The history match objective function is:

(9)

$\begin{array}{l}\displaystyle E[ \ \tau_n, \gamma_n, f_{nm} \ ] = \sum_n {\rm w}_k \sum_k \left[ {\rm w}_e \cdot \left( p_{e,n} \ \ (t_k) - \tilde p_{e,n} \ \ (t_k) \right)^2 + {\rm w}_{\rm wf} \ \ \cdot \left( p_{{\rm wf},n} \ \ (t_k) - \tilde p_{{\rm wf},n} \ \ (t_k) \right)^2 \right] \rightarrow \min\end{array}$

where $\begin{array}{l}{\rm w}_e + {\rm w}_{\rm wf} = 1\end{array}$ are the weight coefficients for formation pressure and bottom-hole pressure correspondingly

and $\begin{array}{l}{\rm w}_k = {\rm w}(t_k)\end{array}$ are the the weight coefficients for time (usually the weight of the later times is higher than that for early times).

The constraints are:

$\begin{array}{l}J_n \geq 0\end{array}$	productivity is a positive number
$\begin{array}{l}\gamma_n \geq 0\end{array}$	drainage volume is a positive number
$\begin{array}{l}f_{nm} \ \geq 0\end{array}$	interference coefficients are all positive numbers
$\begin{array}{l}0 \leq f_{W,nn} \leq 1\end{array}$	total water production from a given well is a sum of good water and bad water and as such the good water share is always less or equal to one
$\begin{array}{l}\sum_m f^{\uparrow}_{O, nm} \ \leq 1\end{array}$
$\begin{array}{l}\sum_m f^{\uparrow}_{G, nm} \ \leq 1\end{array}$
$\begin{array}{l}\sum_m f^{\downarrow}_{W, nm} \ \leq 1\end{array}$
$\begin{array}{l}\sum_m f^{\downarrow}_{G, nm} \ \leq 1\end{array}$

Normally, the initial formation pressure at datum is the same for all wells: $\begin{array}{l}p_{i,n}(0) = p_i = {\rm const}, \ \forall n\end{array}$ .

The value of $\begin{array}{l}\gamma_n\end{array}$ can be linked to the Dynamic drainage volume of the n-th producer $\begin{array}{l}V_{\phi, n}\end{array}$ as:

(10)	$\begin{array}{l}\displaystyle \gamma_n = c_{t,n} \cdot V_{\phi, n} = (c_r + s_{w,n} \cdot c_w + s_{o,n} \cdot c_o + s_{g,n} \cdot c_g) \cdot \phi_n \cdot V_n\end{array}$

(11)	$\begin{array}{l}\displaystyle s_{w,n} + s_{o,n} + s_{g,n} = 1\end{array}$

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See Also

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XCRM – Liquid-Control Cross-well Capacitance Resistance Model @model

See Also