Mathematical model of Capacitance Resistance Model (CRM)
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CRM – Single-Injector Capacitance Resistance Model
The model equation is:
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q^{\uparrow}(t) + \tau \cdot \frac{ d q^{\uparrow}}{ dt } = f \cdot q^{\downarrow}(t) - \gamma \cdot \frac{d p_{wf}}{dt} |
where
The
and
constants are related to some primary well and reservoir properties:
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\gamma = c_t \, V_\phi |
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\tau = \frac{\gamma}{J} = \frac{c_t V_\phi}{J} |
where
| total formation-fluid compressibility |
| drainable reservoir volume |
| total rock volume within the drainage area |
| average effective reservoir porosity |
| total fluid productivity index |
Total formation compressibility is a linear sum of reservoir/fluid components:
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c_t = c_r + s_w c_w + s_o c_w + s_g c_g |
where
| rock compressibility |
| water, oil, gas compressibilities |
| water, oil, gas formation saturations |
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| The first assumption of CRM is that productivity index of producers stays constant in time: | LaTeX Math Block |
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| J = \frac{q_{\uparrow}(t)}{p_r(t) - p_{wf}(t)} = \rm const |
which can be re-written as explicit formula for formation pressure: | LaTeX Math Block |
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| p_r(t) = p_{wf}(t) + J^{-1} q_{\uparrow}(t) |
The second assumption is that drainage volume of producers-injectors system is finite and constant in time: | LaTeX Math Block |
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| V_\phi = V_r \phi = \rm const |
The third assumption is that total formation-fluid compressibility stays constant in time: | LaTeX Math Block |
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| c_t \equiv \frac{1}{V_{\phi}} \cdot \frac{dV_{\phi}}{dp} = \rm const |
which can be easily integrated: | LaTeX Math Block |
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| V_{\phi}(t) =V^\circ_{\phi} \cdot \exp \big[ - c_t \cdot [p_i - p_r(t)] \big] |
where is field-average initial formation pressure, is initial drainage volume,
– field-average formation pressure at time moment , is drainage volume at time moment .
Equation | LaTeX Math Block Reference |
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can be rewritten as:| LaTeX Math Block |
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| dV_{\phi} = c_t \, V_{\phi} \, dp |
The dynamic variations in drainage volume
are due to production/injection:| LaTeX Math Block |
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| dV_{\phi}= \int_0^t q_{\uparrow}(\tau) d\tau - f \int_0^t q_{\downarrow}(\tau) d\tau |
and leading to corresponding formation pressure variation: | LaTeX Math Block |
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| dp = p_i - p_r(t) |
thus making | LaTeX Math Block Reference |
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become:| LaTeX Math Block |
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| \int_0^t q_{\uparrow}(\tau) d\tau - f \int_0^t q_{\downarrow}(\tau) d\tau = c_t \, V_\phi \, [p_i - p_r(t)] |
and differentiated | LaTeX Math Block |
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| q_{\uparrow}(\tau) = f q_{\downarrow}(\tau) - c_t \, V_\phi \, \frac{d p_r(t)}{d t} |
and substituting from productivity equation | LaTeX Math Block Reference |
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:| LaTeX Math Block |
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| q_{\uparrow}(\tau) = f q_{\downarrow}(\tau) - c_t \, V_\phi \, \left[ \frac{d p_{wf}(t)}{d t} + J^{-1} \frac{d q_{\uparrow}}{d t} \right] |
which leads to | LaTeX Math Block Reference |
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The equation
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can be integrated explicitly:
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q^{\uparrow} (t) =\exp(-t/\tau) \cdot \left[ \ q^{\uparrow} (0) + \tau^{-1} \cdot \int_0^t \exp(s/\tau) \left[ f \cdot q^{\downarrow}(s) - \gamma \frac{dp}{ds} \right] ds \ \right] |
and written in equivalent form:
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q^{\uparrow} (t) =\exp(-t/\tau) \cdot \left[ \ q^{\uparrow} (0) +
\tau^{-1} \gamma \cdot \big( p(0) - p(t) \cdot \exp(t/\tau) \big)
+\tau^{-1} \cdot \int_0^t \exp(s/\tau) \left[ f \cdot q^{\downarrow}(s) + \gamma \cdot p(s) \right] ds \ \right] |
The objective function is:
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E[\tau, \gamma, f] = \sum_k \big[ q^{\uparrow}(t_k) - \tilde q^{\uparrow}(t_k) \big]^2 \rightarrow \min |
The basic constraints are:
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\tau \geq 0 , \quad \gamma \geq 0, \quad f \geq 0 |
The additional constraints may be imposed as:
which means that a part of injection (
) is going away from the reservoir drained by producer.
CRMP – Multi-Injector Capacitance Resistance Model
The model equation is:
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q^{\uparrow}_n (t) + \tau_n \cdot \frac{ d q^{\uparrow}_n}{ dt }= \sum_m f_{nm} \cdot q^{\downarrow}_m(t) - \gamma_n \cdot \frac{d p_n}{dt} |
This equation can be integrated explicitly:
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q^{\uparrow}_n (t) =\exp(-t/\tau_n) \cdot \left[ \ q^{\uparrow}_n (0) + \tau_n^{-1} \cdot \int_0^t \exp(s/\tau_n) \left[ \sum_m f_{nm} \cdot q^{\downarrow}_m(s) - \gamma_n \frac{dp_n}{ds} \right] ds \right] |
The objective function is:
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E[\tau_n, \gamma_n, f_{nm}] = \sum_k \sum_n \big[ q^{\uparrow}_n(t_k) - \tilde q^{\uparrow}_n(t_k) \big]^2 \rightarrow \min |
The constraints are:
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\tau_n \geq 0 , \quad \gamma_n \geq 0, \quad f_{nm} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{nm} \leq 1 |
ICRM – Integrated Multi-Injector Capacitance Resistance Model
The model equation is:
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Q^{\uparrow}_n (t) = \sum_n f_{nm} Q^{\downarrow}_n(t) - \tau_n \cdot \big[ q^{\uparrow}_n(t) - q^{\uparrow}_n(0) \big] - \gamma_n \cdot \big[ p_n(t) - p_n(0) \big] |
The objective function is:
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E[\tau_n, \gamma_n, f_{nm}] = \sum_k \sum_n \big[ Q^{\uparrow}_n(t_k) - \tilde Q^{\uparrow}_n(t_k) \big]^2 \rightarrow \min |
The constraints are:
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\tau_j \geq 0 , \quad \gamma_n \geq 0, \quad f_{ij} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{ij} \leq 1 |
QCRM – Liquid-Control Multi-Injector Capacitance Resistance Model
The model equation is:
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p_n(t) = p_n(0) - \tau_n / \gamma_n \cdot \big[ q^{\uparrow}_n(t) - q^{\uparrow}_n(0) \big] - \gamma_n^{-1} \cdot Q^{\uparrow}_n (t) + \gamma_n^{-1} \cdot \sum_m f_{nm} \ Q^{\downarrow}_m(t) |
The objective function is:
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E[\tau_n, \gamma_n, f_{nm}] = \sum_k \sum_n \big[ p_n(t_k) - \tilde p_n(t_k) \big]^2 \rightarrow \min |
The constraints are:
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\tau_n \geq 0 , \quad \gamma_n \geq 0, \quad f_{nm} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{ij} \leq 1, \quad p_{nr}(0) > 0 |
where
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p_{nr}(0) = p_n(0) + (\tau_n / \gamma_n) \cdot q^{\uparrow}_n(0) |
is the initial formation pressure.
The equation
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can be re-written with explicit form of initial formation pressure:
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p_n(t) = p_{nr}(0) + (\tau_n / \gamma_n) \cdot q^{\uparrow}_n(t) + \gamma_n^{-1} \cdot \sum_m f_{nm} \ Q_m^{\downarrow}(t) |
where could be both producer | LaTeX Math Inline |
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| body | --uriencoded--Q_m%5e%7B\uparrow%7D |
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or injector | LaTeX Math Inline |
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| body | --uriencoded--Q_m%5e%7B\downarrow%7D |
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If
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| body | --uriencoded--p_%7Bnr%7D(0) |
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is known then it can be fixed during
the search loop which normally improves
the quality of future production forecasts.
XCRM – Liquid-Control Cross-well Capacitance Resistance Model
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| p_e \ (t) = p_{nr} \ (0) + \gamma_n^{-1} \cdot \sum_m \left( Q^{\uparrow}_{nm} + Q^{\downarrow}_{nm} \ \right) |
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| p_{{\rm wf}, n} \ \ \ (t) = p_e \ (t) + J_n^{-1} \cdot q_n(t) |
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| Q^{\uparrow}_{nm} \ =
\ - \ f^{\uparrow}_{O,nm} \ \cdot B_{ob} \cdot \, Q^{\uparrow}_O
\ - \ f^{\uparrow}_{G,nm} \ \cdot B_{go} \cdot Q^{\uparrow}_G
\ - \ f^{\uparrow}_{W,nm} \ \cdot B_w \cdot Q^{\uparrow}_W
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| Q^{\downarrow}_{nm} \ =
f^{\downarrow}_{G,nm} \ \cdot B_{go} \cdot Q^{\downarrow}_G
\ + \ f^{\downarrow}_{W,nm} \ \cdot B_w \cdot Q^{\downarrow}_W
\ + \ B_{go} \cdot Q^{\downarrow}_{GCAP} \
\ + \ B_w \cdot Q^{\downarrow}_{WAQ}
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| Q_m(t) = \int_0^t q_m(t) \, dt |
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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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The objective function is:
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E[ \ \tau_n, \gamma_n, f_{nm} \ ] = \sum_k \sum_n \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 |
The constraints are:
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J_n \geq 0 , \quad \gamma_n \geq 0, \quad f_{nm} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{nm} \leq 1 |
In regular case , the initial formation pressure at datum is the same for all wells:
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| body | --uriencoded-- p_%7Bnr%7D(0) = p_i = %7B\rm const%7D, \ \forall n |
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See Also
Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Capacitance Resistance Model (CRM)
Production – Injection Pairing @ model
[ Slightly compressible Material Balance Pressure @model ]
References
| Sayarpour, M., Zuluaga, E., Kabir, C.S., and Lake, L.W. (2007). The Use of Capacitance-Resistive Models for Rapid Estimation of Waterflood Performance and Optimization. Paper SPE 110081 presented at the SPE Annual Technical Conference and Exhibition, Anaheim, California, 11-14 November, doi.org/10.2118/110081-MS |
| Nguyen, A. P., Kim, J. S., Lake, L. W., Edgar, T. F., & Haynes, B. (2011, January 1). Integrated Capacitance Resistive Model for Reservoir Characterization in Primary and Secondary Recovery. Society of Petroleum Engineers, doi.org/10.2118/147344-MS |
| Holanda, R. W. de, Gildin, E., & Jensen, J. L. (2015, November 18). Improved Waterflood Analysis Using the Capacitance-Resistance Model Within a Control Systems Framework. Society of Petroleum Engineers, doi.org/10.2118/177106-MS |
