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Definition
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Specific type of Production Analysis (PA) workflow based on correlation between injection rates history and production rate history with account of bottomhole pressure history.
In case the
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bottomhole pressure
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data is not available it is considered constant over time.
The CRM is trained over historical records of production rates, injection rates and bottom-hole pressure variation.
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Application
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Advantages
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Limitations
CRM does not pretend to predict reserves distribution as dynamic model does.
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Motivation
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Production rate in producing well depends on its productivity index , current formation pressure and current BHP LaTeX Math Inline |
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body | --uriencoded--p_%7Bwf%7D |
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CRM can only be tuned for injectors with a rich history of rates variations.
CRM only works at long times and only in areas with limited drainage volume.
Technology
CRM trains linear correlation between variation of production rates against variation of injection rates with account of bottom-hole pressure history in producers.
against material balance and require current FDP volumetrics, PVT and SCAL models.
The CRM has certain specifics for oil producers, water injectors, gas injectors and field/sector analysis.
CRM – Single-Tank Capacitance Resistance Model
The CRM model is trying
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left( p_e(t) - p_{wf}(t) \right) |
where
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and as such depends on completion/lift settings (defining
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) and how formation pressure is maintained
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The
and constants are related to some primary well and reservoir characteristics: LaTeX Math Block |
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\beta = c_t \, V_\phi |
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\tau = \frac{\beta}{J} = \frac{c_t V_\phi}{J} |
where
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over time.
It keeps declining due to the offtakes:
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where
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The first assumption of CRM is that productivity index of producers stays constant in time:
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p_e[q_1^{\uparrow}(t), q_2^{\uparrow}(t) |
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and maintained by either aquifer or Fluid Injection and in the latter case depends on injection rates
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:
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,q_2^{\downarrow}(t),q_3^{\downarrow}(t) |
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The combination of LaTeX Math Block Reference |
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, LaTeX Math Block Reference |
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and LaTeX Math Block Reference |
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lead to the correlation between production rates, injection rates and bottomhole pressure variation.
The ultimate purpose of CRM is to correlate the long-term (few months or longer) injection rates history with production rates history and BHP history (recorded by PDG).
It is essentially based on the fact that production rate responds to changes in BHP and offset injection.
The major assumptions in CRM model are:
- productivity index of producer stays constant in time
- dynamic drainage volume of producer is finite and constant in time (which is equivalent to PSS flow regime)
- total compressibility within the drainage volume of a given producer stays constant in time
Assumption 2 means that interference between producers is fairly constant in time despite the rate variations and their impact on the dynamic drainage volumes.
Goals | Objectives |
Identify and prioritise production optimisation opportunities | Generate production and formation pressure forecasts based on the bottom-hole pressure and injection rates |
Identify and prioritise redevelopment opportunities | Assess productivity index of producing wells |
Identify and prioritise surveillance candidates | Assess dynamic drainage volume around producing wells |
| Quantify connectivity between injectors and producers |
| Assess water flood efficiency against expectations and / or between wells or well groups |
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Advantages |
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| Limitations |
Fast-track | It only models injector-producer system |
Requires minimum input data (BHP history, fowrate history and FVF) | Requires eventful history of injection rates variations |
Robust procedure (no manual setups) | Requires productivity index of producers to stay constant |
Does not involve full-field 3D dynamic modelling and associated assumptions | Requires the drainage volumes of all producers stay the same throughout the modelling period |
| Only applicable for specific subset of PSS fluid flow regimes |
Technology
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The CRM trains linear correlation between variation of production rates against variation of injection rates with account of bottom-hole pressure history records in producers.
See Capacitance-Resistivity Model @model
Inputs | Outputs |
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Production rate history | Productivity Index for the focus producer |
Bottom-hole pressure history | Drainage volume by the focus produce producer |
Injection rate history | Share of injection going towards the focus producer |
PVT model |
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| CRM is a specific case of MDCV with the following unit-rate transient responses:
| DTR | CTR from offset producers | CTR from offset injectors |
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UTR | LaTeX Math Block |
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| p_{1nn}(t) =J_n^{-1} \left( 1 - \frac{t}{\tau_n} \right) |
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| p_{1nm}(t) = 0 |
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| p_{1nm}(t) = \frac{f_{nm}}{J_n \tau_n} \cdot t |
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Pressure drop | LaTeX Math Block |
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| \delta p_{1nn}(t) = \frac{1}{J_n \tau_n} \cdot t |
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| \delta p_{1nm}(t) = 0 |
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| \delta p_{1nm}(t) = \frac{f_{nm}}{J_n \tau_n} \cdot t |
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Log Der | LaTeX Math Block |
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| p'_{1nn}(t) = \delta p_{1nn}(t) |
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| p'_{1nm}(t) = 0 |
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| p'_{1nm}(t) = \delta p_{1nm}(t) |
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See also CRM as MDCV @model for derivation. |
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See Also
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Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis
[ Capacitance-Resistivity Model @model ]
References
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Arthur Aslanyan, Mathematical aspects of Multiwell Deconvolution and its relation to Capacitance Resistance Model, arxiv.org/abs/2203.01319
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: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:10.2118/177106-MS
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| Application
- Assess current production performance
- current distribution of recovery against expectations
- current status and trends of recovery against expectations
- current status and trends of reservoir depletion against expectations
- current status and trends of water flood efficiency against expectations
- compare performance of different wells or different groups of wells
- Identify and prioritize surveillance opportunities
- Identify and prioritize redevelopment opportunities
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The second assumption is that drainage volume of producers-injectors system is finite and constant in time:
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V_\phi = V_{rocks} \phi = \rm const |
The third assumption is that total formation-fluid compressibility stays constant in time:
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c_t \equiv \frac{1}{V_{\phi}} \cdot \frac{dV_{\phi}}{dp} = \rm const |
which can be easily integrated:
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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,...
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is drainage volume at time moment . LaTeX Math Block |
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\frac{dV_{\phi}}{dp} = c_t \, V_{\phi} \ \cdot |
The change in drainage volume
is leading to formation pressure variation LaTeX Math Block |
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c_t \equiv \frac{1}{V_{\phi}} \cdot \frac{dV_{\phi}}{dp} = \frac{1}{V_{\phi}} \cdot \frac{1}{p_i - p_r(t) } \cdot \Bigg[ \int_0^t q_{\uparrow}(\tau) d\tau - f \int_0^t q_{\downarrow}(\tau) d\tau \Bigg] = \rm const |
The last equation can be rewritten as:
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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
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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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q_{\uparrow}(\tau) = f q_{\downarrow}(\tau) - c_t \, V_\phi \, \bigg[ \frac{d p_{wf}(t)}{d t} + J^{-1} \frac{d q_{\uparrow}}{d t} \bigg] |
which leads to
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.The target function is:
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E[\tau, \beta, f] = \sum_k \big[ q^{\uparrow}(t_k) - \tilde q^{\uparrow}(t_k) \big]^2 \rightarrow \min |
The constraints are:
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\tau \geq 0 , \quad \beta \geq 0, \quad 0 \leq f \leq 1 |
CRMP – Multi-tank Producer-based Capacitance Resistance Model
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q^{\uparrow}_j (t) = \sum_i^{n_i} f_{ij} q^{\downarrow}_i(t) - \tau_j \cdot \frac{ d q^{\uparrow}_j}{ dt } - \beta_j \cdot \frac{d p_j}{dt} |
The target function is:
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E[\tau, \beta, f] = \sum_k \sum_j \big[ q^{\uparrow}_j(t_k) - \tilde q^{\uparrow}_j(t_k) \big]^2 \rightarrow \min |
The constraints are:
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\tau_j \geq 0 , \quad \beta_j \geq 0, \quad f_{ij} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{ij} \leq 1 |
ICRM – Multi-Tank Integrated Capacitance Resistance Model
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Q^{\uparrow}_j (t) = \sum_i^{n_i} f_{ij} Q^{\downarrow}_i(t) - \tau_j \cdot \big[ q^{\uparrow}_j(t) - q^{\uparrow}_j(0) \big] - \beta_j \cdot \big[ p_j(t) - p_j(0) \big] |
The target function is:
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E[\tau, \beta, f] = \sum_k \sum_j \big[ Q^{\uparrow}_j(t_k) - \tilde Q^{\uparrow}_j(t_k) \big]^2 \rightarrow \min |
The constraints are:
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\tau_j \geq 0 , \quad \beta_j \geq 0, \quad f_{ij} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{ij} \leq 1 |
References
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RAFAEL WANDERLEY DE HOLANDA, CAPACITANCE RESISTANCE MODEL IN A CONTROL SYSTEMS FRAMEWORK: A TOOL FOR DESCRIBING AND CONTROLLING WATERFLOODING RESERVOIRS, 2015.pdf
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