The XCRM model predicts the formation pressure p_{e,n} and bottom-hole pressure p_{wf,n} in the n-th oil producer in response to:
- its current oil/water production rate \left( q^{\uparrow}_{On}, \ q^{\uparrow}_{Wn} \right)
- its cumulative oil/water production \left( Q^{\uparrow}_{On}, \ Q^{\uparrow}_{Wn} \right)
- cumulative oil/water production from the offset wells \left( Q^{\uparrow}_{Om}, \ Q^{\uparrow}_{Wm} \right)_{m \neq n \, \in \uparrow}
- cumulative water injection in the offset wells \left( Q^{\downarrow}_{Om}, \ Q^{\downarrow}_{Wm} \right)_{m \in \downarrow}
using the following equations:
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where B_o, \, B_g, \, B_w, \, R_s, \, R_v are Dynamic fluid properties.
The value of cumulative Gas Cap influx Q^{\downarrow}_{GCAP} is modelled as in Gas Cap Drive @model.
The value of cumulative Aquifer influx Q^{\downarrow}_{GCAP} 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 : s_{o,n} = s_{or} \ , \quad s_{g,n} = 0 \ , \quad s_{w,n} = 1 - s_{or}.
In case of Gas Injector: s_{o,n} = 0 \ , \quad s_{g,n} = 1 - s_{wcg} \ , \quad s_{w,n} = s_{wcg}.
The history match objective function is:
(9) | 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 {\rm w}_e + {\rm w}_{\rm wf} = 1 are the weight coefficients for formation pressure and bottom-hole pressure correspondingly
and {\rm w}_k = {\rm w}(t_k) are the the weight coefficients for time (usually the weight of the later times is higher than that for early times).
The constraints are:
(10) | J_n \geq 0 , \quad \gamma_n \geq 0, \quad f_{nm} \ \geq 0 , \quad \sum_m f^{\uparrow}_{O, nm} \ \leq 1 , \quad \sum_m f^{\uparrow}_{G, nm} \ \leq 1, \quad \sum_m f^{\downarrow}_{W, nm} \ \leq 1, \quad \sum_m f^{\downarrow}_{G, nm} \ \leq 1 |
Normally, the initial formation pressure at datum is the same for all wells: p_{i,n}(0) = p_i = {\rm const}, \ \forall n.
The value of \gamma_n can be linked to the Dynamic drainage volume of a well V_{\phi, n} as:
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See Also
Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Capacitance Resistance Model (CRM)
[ Capacitance-Resistivity Model (CRM) @model ][ Slightly compressible Material Balance Pressure @model ]