The XCRM model predicts the formation pressure
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body | --uriencoded--p_%7Be,n%7D |
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and
bottom-hole pressure LaTeX Math Inline |
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body | --uriencoded--p_%7Bwf,n%7D |
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in
-th oil producer in response to:
- its current oil/water production rate
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body | --uriencoded--\left( q%5e%7B\uparrow%7D_%7BOn%7D, \ q%5e%7B\uparrow%7D_%7BWn%7D \right) |
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- its cumulative oil/water production rate
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body | --uriencoded--\left( Q%5e%7B\uparrow%7D_%7BOn%7D, \ Q%5e%7B\uparrow%7D_%7BWn%7D \right) |
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- cumulative oil/water production from the offset wells
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body | --uriencoded--\left( Q%5e%7B\uparrow%7D_%7BOm%7D, \ Q%5e%7B\uparrow%7D_%7BWm%7D \right)_%7Bm \neq n \, \in \uparrow%7D |
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- cumulative water injection in the offset wells
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body | --uriencoded--\left( Q%5e%7B\downarrow%7D_%7BOm%7D, \ Q%5e%7B\downarrow%7D_%7BWm%7D \right)_%7Bm \in \downarrow%7D |
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using the following equations:
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| p_{e,n} \ (t) = p_{i,n} \ (0) + \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] |
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| 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_{wn})}{k_{rw}(s_{wn})} \cdot q^{\uparrow}_{Wn}(t) \right] |
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| s_{wn} = \left[ 1 + \frac{B_o}{B_w} \cdot \frac{q^{\uparrow}_{On}}{f^{\uparrow}_{W,nn} \cdot q^{\uparrow}_{Wn}} \right]^{-1} |
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| Q(t) = \int_0^t q(t) \, dt |
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| 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}
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| 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}
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| B_{og} = \frac{B_o - R_s \, B_g}{1- R_s \, R_v} |
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| B_{go} = \frac{ B_g - R_v \, B_o}{1- R_s \, R_v} |
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where
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body | B_o, \, B_g, \, B_w, \, R_s, \, R_v |
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are
Dynamic fluid properties.
The value of cumulative Gas Cap influx
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body | --uriencoded--Q%5e%7B\downarrow%7D_%7BGCAP%7D |
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is modelled as in
Gas Cap Drive @model.
The value of cumulative Aquifer influx
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body | --uriencoded--Q%5e%7B\downarrow%7D_%7BGCAP%7D |
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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 :
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body | --uriencoded-- s_%7Bo,n%7D = s_%7Bor%7D \ , \quad s_%7Bg,n%7D = 0 \ , \quad s_%7Bw,n%7D = 1 - s_%7Bor%7D |
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.
In case of Gas Injector:
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body | --uriencoded-- s_%7Bo,n%7D = 0 \ , \quad s_%7Bg,n%7D = 1 - s_%7Bwcg%7D \ , \quad s_%7Bw,n%7D = s_%7Bwcg%7D |
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.
The objective function is:
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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
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body | --uriencoded--%7B\rm w%7D_e + %7B\rm w%7D_%7B\rm wf%7D = 1 |
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are the weight coefficients for
formation pressure and
bottom-hole pressure correspondingly
and
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body | --uriencoded--%7B\rm w%7D_k = %7B\rm w%7D(t_k) |
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are the the weight coefficients for time (usually the weight of the later times is higher than that for early times).
The constraints are:
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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:
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body | --uriencoded-- p_%7Bi,n%7D(0) = p_i = %7B\rm const%7D, \ \forall n |
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.
The value of
can be linked to the
Dynamic drainage volume of a well LaTeX Math Inline |
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body | --uriencoded--V_%7B\phi, n%7D |
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as:
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| \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 |
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| s_{w,n} + s_{o,n} + s_{g,n} = 1 |
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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 ]
[ Dynamic fluid properties ]