The XCRM model predicts the formation pressure and bottom-hole pressure in the -th oil producer in response to:

its current oil/water production rate
its cumulative oil/water production
cumulative oil/water production from the offset wells
cumulative water injection in the offset wells

using the following equations:

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]

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]

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}

Q(t) =  \int_0^t q(t) \, dt

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}

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}

B_{og} = \frac{B_o - R_s \, B_g}{1- R_s \, R_v}

B_{go} = \frac{ B_g - R_v \, B_o}{1- R_s \, R_v}

where are Dynamic fluid properties and is Normalized Pseudo-Pressure .

The value of cumulative Gas Cap influx is modelled as in Gas Cap Drive @model.

The value of cumulative Aquifer influx 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 : .

In case of Gas Injector: .

The history match objective function is:

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

where are the weight coefficients for formation pressure and bottom-hole pressure correspondingly

and are the the weight coefficients for time (usually the weight of the later times is higher than that for early times).

The constraints are:

	productivity is a positive number
	drainage volume is a positive number
	interference coefficients are all positive numbers
	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

Normally, the initial formation pressure at datum is the same for all wells: .

The value of can be linked to the Dynamic drainage volume of the n-th producer as:

\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

s_{w,n} + s_{o,n} + s_{g,n} = 1

See Also