# A State-of-the-Art Literature Review on Capacitance Resistance Models for Reservoir Characterization and Performance Forecasting

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## Abstract

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## 1. Introduction

- Confirm the presence of sealing or leaking faults, as well as high permeability flow paths (e.g., channels, natural fractures);
- Quantify communication between neighboring reservoirs, and reservoir compartmentalization;
- Determine sweep efficiency of producers;
- Optimize injected fluid allocation during secondary and tertiary recovery.

## 2. Capacitance Resistance Models

#### 2.1. Reservoir Control Volumes

#### 2.1.1. CRMT: Single Tank Representation

#### 2.1.2. CRMP: Producer Based Representation

#### 2.1.3. CRMIP: Injector-Producer Pair Based Representation

#### 2.1.4. CRM-Block: Blocks in Series Representation

#### 2.1.5. Multilayer CRM: Blocks in Parallel Representation

#### 2.2. CRM Parameters Physical Meaning

#### 2.2.1. Connectivities

#### Aquifer–Producer Connectivity

#### Comparisons of CRM Interwell Connectivities with Streamline Allocation Factors

#### Connectivity Interpretation within a Flood Management Perspective

#### 2.2.2. Time Constants

#### 2.3. CRM for Primary Production

#### 2.4. CRM History Matching

- Holanda et al. [33] considered ${\mathbf{C}}_{\mathbf{e}}$ as a diagonal matrix considering each diagonal element proportional to ${\mathbf{q}}_{\mathbf{obs}}$
^{2}. This is equivalent to assume that the errors are independent and proportional to ${\mathbf{q}}_{\mathbf{obs}}$, as a simplifying assumption. In this case, the objective function becomes the relative squared error instead of the absolute squared error. Additionally, a minimum value is set to the ${\mathbf{q}}_{\mathbf{obs}}$^{2}diagonal elements of ${\mathbf{C}}_{\mathbf{e}}$ to avoid overfitting lower rates because when ${\mathbf{q}}_{\mathbf{obs}}$ approaches zero, a high relative error may correspond to an acceptable absolute error. Their results indicated that this approach improved the quality of the history matching. - Holanda et al. [53] applied weights to the diagonal elements of the previous formulation [33]. These weights were defined by heuristic rules aiming to select the most representative data to forecast production with a decline model. Such heuristic rules can be adjusted to improve the probabilistic calibration in large datasets.

#### 2.4.1. Dimensionality Reduction

- Define a spatial window of active injector–producer pairs based on the interwell distance and reservoir heterogeneity, ${f}_{ij}=0$ for wells outside the spatial window [23];
- Define a maximum number of nearest injectors that could affect a producer well [57];
- Assign the same ${\tau}_{j}$ to all layers in the ML-CRM [30];
- Instead of applying the CRM-block representation, i.e., blocks in series, use a first-order tank with a time delay [26];
- Assign a single productivity index per producer in the CRMIP representation [60].

#### 2.4.2. Alternative CRM Formulations

#### Matching Cumulative Production: The Integrated Capacitance Resistance Model (ICRM)

#### Unmeasured BHP Variations: The Segmented CRM

#### Changes in Well Status: The Compensated CRM

#### 2.5. CRM Sensitivity to Data Quality and Uncertainty Analysis

- the amplitude and frequency of uncorrelated variations in the input signals (injection rates and producers’ BHP) because the most relevant dynamic aspects of the system must be observed in the output signals (production rates);
- the amount of data available for history matching, i.e., sampling frequency (e.g., whether production data are reported daily or monthly) and length of the history matching window;
- the properties of the reservoir system, such as permeability distribution, fluid saturation and total compressibility.

## 3. Fractional Flow Models

#### 3.1. Buckley–Leverett Adapted to CRM

#### 3.2. Semi-Empirical Power-Law Fractional Flow Model

#### 3.3. Koval Fractional Flow Model

## 4. CRM Enhanced Oil Recovery

## 5. CRM and Geomechanical Effects

## 6. CRM Field Development Optimization

#### 6.1. Well Control

#### 6.2. Well Placement

## 7. CRM in a Control Systems Perspective

## 8. CRM and Geology

- relate the connectivity decrease from south, where fluvial sands predominate, to the north, where more mudstone exists,
- validate predictions of connectivity proposed by Gardner et al. [101] in their geological cross-sections.

- Kaviani and Jensen [63] found the direction of maximum connectivities was the same as the orientation of the stacked tidal channels in the Senlac Field.
- Yin et al. [15] integrated 4D seismic-based results with CRM evaluations to delineate faults and large-scale conduits and detected a sub-seismic fault, which assisted the history matching of a finite-difference reservoir model for the Norne field (Norway).

## 9. CRM Field Applications

#### 9.1. Primary Recovery

#### 9.2. Secondary Recovery

#### 9.2.1. Evolving Waterflood

#### 9.2.2. Mature Waterflood

#### 9.3. Tertiary Recovery

## 10. Other Reduced Complexity Models

## 11. Unresolved Issues and Suggestions for Future Research

#### 11.1. Gas Content of Reservoir Fluids

#### 11.2. Rate Measurements

#### 11.3. Well-Orientation and Completion Type

- The reverse-CM, wherein the injection rates are history matched while the production rates serve as input variables,
- Subtracting the homogeneous connectivity calculated by the multiwell productivity index (MPI) approach to obtain a ‘geometry adjusted’ connectivity, as was done for Figure 11.

#### 11.4. Time-Varying Behavior of the CRM Parameters

#### 11.5. CRM Coupling with Fractional Flow Models and Well Control Optimization

## 12. Conclusions

- Several aspects must be considered in the design of CRMs to ensure that their applications are fit for purpose. For example, control volume schemes (that is, CRMT, CRMP, CRMIP, CRM-Block, ML-CRM), fractional flow models (that is, Buckley–Leverett based, semi-empirical power-law, and Koval models), optimization algorithms for the history matching and well-control optimization, dimensionality reduction techniques, and data quality and availability.
- The physical meaning of interwell connectivities, time constants, and productivity indices are well understood. For this reason, diagnostic plots from these parameters (that is, connectivity maps, flow capacity plots, and compartmentalization plots) add value to the geological analysis, quickly providing insights into flow patterns and flood efficiencies.
- If the model parameters are considered constant (linear time-invariant system), there is a general matrix structure and solution to all CRM control volume schemes presented in this paper.
- Although CRMs started with mature fields undergoing waterflood, these models were extended to primary recovery, enhanced oil recovery (that is, CO${}_{2}$ flooding, WAG, SWAG, polymer flooding, hot waterflooding), and prebreakthrough scenario in waterflooded fields.
- CRMs are a fast tool for well control optimization in fields with many wells; usually, only production and injection flow rates, producers’ BHP, and well locations are required to obtain the models.
- As an output of the CRM framework, interwell connectivity maps can assist the geological analyses by (1) corroborating results from tracer tests and 4D seismic; (2) determining main directions of flow and the presence of sealing faults, fracture swarms, and high permeability channels; and (3) delineating sand bodies. In addition, CRMs allow for quantifying the drainage volumes associated with each producer.
- Over the last decade, CRMs have also provided valuable insights into the development of other types of reduced-physics models for reservoir simulations.
- Naturally, there will always be room for innovative CRM developments that can provide practical solutions for improving robustness in reservoir characterization, production forecast, and optimization. Currently, the main opportunities exist in (1) improvement in production data quality; (2) understanding model limitations and modeling time-varying behaviors; and (3) more consistent coupling of CRM and fractional flow models.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\mathbf{a}$ | linear inequality constraint matrix |

A | drainage area of the control volume, ft${}^{2}$ |

$\mathbf{A}$ | state matrix |

${\mathbf{a}}_{\mathbf{eq}}$ | linear equality constraint matrix |

$\mathbf{b}$ | linear inequality constraint vector |

B | number of blocks |

$\mathbf{B}$ | input matrix |

${\mathbf{b}}_{\mathbf{eq}}$ | linear equality constraint vector |

C | CRM number |

$\mathbf{C}$ | output matrix |

${\mathbf{C}}_{\mathbf{e}}$ | error covariance matrix |

${c}_{t}$ | total compressibility, psi${}^{-1}$ |

$\mathbf{D}$ | feedforward matrix |

E | effective oil-solvent viscosity ratio |

f | interwell connectivity |

F | cumulative flow capacity |

${f}^{\prime}$ | fraction of injected flow rate allocated to each layer |

${f}_{o}$ | fractional flow of oil |

${f}_{PLT}$ | fraction of production flow rate coming from each layer |

${f}_{w}$ | fractional flow of water |

$\mathbf{G}(s)$ | transfer function |

H | heterogeneity factor |

$\mathbf{I}$ | identity matrix |

J | productivity index, bbl/(day × psi) |

k | permability, md |

${K}_{val}$ | Koval factor |

L | ratio of sampled data points to number of parameters |

${\mathbf{l}}_{\mathbf{b}}$ | lower bound vector |

m | oil relative permeability exponent |

M | end-point mobility ratio |

n | water relative permeability exponent |

${N}_{ft}$ | number of time steps until end of forecasting window |

${N}_{inj}$ | number of injectors |

${N}_{L}$ | number of layers |

${N}_{p}$ | cumulative liquid production, bbl |

${N}_{par}$ | number of parameters |

${N}_{prod}$ | number of producers |

${N}_{t}$ | number of time steps until end of history matching window |

$NPV$ | net present value |

p | pressure, psi |

$\overline{p}$ | average pressure of the control volume, psi |

${p}_{wf}$ or BHP | producer bottomhole pressure, psi |

q | liquid production rate, bbl/day |

$\mathbf{q}$ | liquid production rates vector, bbl/day |

${q}_{BHP}$ | contribution of unknown BHP variations to flow rate, bbl/day |

${q}_{p}$ | liquid production rate disregarding crossflow, bbl/day |

${Q}_{c}$ | crossflow between layers, bbl/day |

r | discount rate per period |

s | Laplace variable |

S | normalized average water saturation |

${S}_{or}$ | residual oil saturation |

${S}_{w}$ | water saturation |

${S}_{wr}$ | irreducible water saturation |

t | time, day |

${t}_{D}$ | dimensionless time |

T | transmissibility, bbl/(day × psi) |

${T}_{s}$ | segmented time, day |

$\mathbf{u}$ | input vector |

${\mathbf{u}}_{\mathbf{b}}$ | upper bound vector |

$\mathbf{U}(s)$ | input vector in Laplace space |

${\mathbf{U}}_{t}$ | well trajectories matrix for optimization |

${V}_{p}$ | pore volume, bbl |

w | injection rate, bbl/day |

${w}^{*}$ | effective water injected in the control volume, bbl/day |

W | cumulative water injected, bbl |

${W}^{*}$ | effective cumulative water injected in the control volume, bbl/day |

$\mathbf{x}$ | state vector |

${x}_{D}$ | dimensionless distance |

$\mathbf{y}$ | output vector |

$\mathbf{Y}(s)$ | output vector in Laplace space |

z | history matching objective function |

Greek Letters | |

$\alpha $ | power-law coefficient for semi-empirical fractional flow model |

$\beta $ | power-law exponent for semi-empirical fractional flow model |

${\lambda}_{vj}$ | interwell connectivity between virtual injector and producer |

$\mu $ | viscosity, cp |

$\rho $ | density, lbm/bbl |

$\tau $ | time constant, day |

${\tau}_{p}$ | time constant for primary production, day |

$\varphi $ | porosity |

$\Phi $ | cumulative storage capacity |

$\mathbf{\chi}$ | parameters vector |

$\omega $ | price |

Subscripts and Superscripts | |

a | aquifer |

b | b-th block |

$cap$ | capacity of the surface facilities |

i | i-th injector |

$in$ | input |

j | j-th producer |

k | k-th time step |

$max$ | maximum |

$min$ | minimum |

o | oil |

$\mathbf{obs}$ | observed data |

$out$ | output |

$\mathbf{pred}$ | predicted data |

s | solvent |

v | v-th producer is shut-in |

w | water |

$\alpha $ | $\alpha $-th layer |

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**Figure 1.**Types of reservoir models (adapted from Gildin and King [1]).

**Figure 3.**More than 260 public-domain documents concerning Capacitance resistance models (CRMs) or their applications have appeared since 2006. Source: Google Scholar. 2016–18* indicates publications through 29 September 2018.

**Figure 4.**Reservoir control volumes for CRM representations: (

**a**) single tank (CRMT); (

**b**) producer based (CRMP); (

**c**) injector–producer pair based (CRMIP); (

**d**) blocks in series (CRM-block); (

**e**) multi-layer or blocks in parallel (ML-CRM).

**Figure 5.**(

**a**) CRM response to a sequence of step injection signals for several values of interwell connectivity; (

**b**) physical meaning of time constants: percent of stationary response achieved at a specific dimensionless time.

**Figure 6.**(

**a**) example of modified Brooks and Corey [71] relative permeability model; (

**b**) Buckley–Leverett prediction of the flood-front advance; (

**c**) water-cut sensitivity to parameters in Equation (29); the title of each subplot indicates which parameter is changing with values shown in the legends (base case: $w=1$ bbl/day, ${V}_{p}=1$ bbl, ${S}_{wr}=0.2$, ${S}_{or}=0.2$, $M=0.33$, $m=3$, $n=2$; observation: w and ${V}_{p}$ are normalized for the base case).

**Figure 7.**(

**a**) example of history matching the late time water-oil ratio (WOR) with the power-law relations (these four producers are in the reservoir shown in Figure 9a). Water-cut sensitivity to parameters of the semi-empirical fractional flow model: (

**b**) ${\alpha}_{j}$, and (

**c**) ${\beta}_{j}$.

**Figure 8.**(

**a**) water-oil ratio (WOR) resulting from history matching the early and late time water-cut with the Koval fractional flow model (these four producers are in the reservoir shown in Figure 9a). Water-cut sensitivity to parameters of the Koval fractional flow model: (

**b**) ${V}_{p}$, and (

**c**) ${K}_{val}$.

**Figure 9.**(

**a**) fluvial environment reservoir based on the SPE-10 model, previously described in [25]; (

**b**) flow capacity plot for four producers. ‘PROD5’ is the most efficient producer in terms of sweep efficiency while ‘PROD3’ is the least efficient one, which can potentially improve through EOR processes.

**Figure 11.**(

**a**) Marsden Field well (red triangles are injectors and black squares are producers) and sandbody (red lines) positions defined by geological interpretation. (

**b**) Expanded view showing connectivities to a producer, P50, near the edge of a sandbody. Connectivities are normalized by subtracting the homogeneous connectivities from them; ${\lambda}^{\prime}={\lambda}_{meas}-{\lambda}_{homog}$, where $\lambda $ is the CRM gain (modified from [43]).

**Figure 12.**Matching field data of four wells with primary-CRM (modified from [11]).

**Figure 13.**(

**a**) stabilized flow contributions from each well; (

**b**) diagnostic plot shows lack of compartmentalization [11].

**Figure 14.**CRMP analysis of P-1 well’s prebreakthrough (

**a**), and post-breakthrough (

**b**) scenarios [49].

**Figure 15.**(

**a**) CRMT match of the total liquid rate, (

**b**) CRMT match with different fractional-flow models [99].

**Table 1.**Some of the optimization algorithms used for capacitance resistance model (CRM) history matching.

Reference | Algorithm | Highlights |
---|---|---|

Kang et al. [56] | Gradient projection method within a Bayesian inversion framework. | Converted Equation (19) into a equality constraint. Analytical formulation for gradient computation based on sensitivity of the model response to its parameters. Each iteration takes the direction of the projected gradient that satisfies the constraints. |

Holanda et al. [52] | Sequential quadratic programming (SQP), numerical gradient computation, BFGS approximation for the Hessian matrix. | Even though gradient-based formulations may be fast and straightforward to implement, they also rely on a proper choice of initial guess to avoid convergence to a local minima. |

Weber [19], Lasdon et al. [57] | GAMS/CONOPT (gradient-based, local search), automatic computation of first and second partial derivatives. | The objective function is based on the mismatch for a one step ahead prediction from the measured data. The problem is solved in a sequence of four steps that include defining a suitable initial guess, determining injector–producer pairs with zero gains and excluding outliers. A global optimization algorithm capable of identifying local minima has demonstrated the occurrence of multiple local solutions in several examples. |

Wang et al. [58] | Stochastic simplex approximate gradient (StoSAG). | For an example of a heterogenoues reservoir with five injectors and four producers, the StoSAG demonstrated convergence with less iterations and to a smaller value of objective function than with the projected gradient and ensemble Kalman filter methods. |

Mamghaderi et al. [29], Mamghaderi and Pourafshary [34] | Genetic algorithms (global optimization) | Genetic algorithm is applied for the history matching of the ML-CRM and justified by the significant increase in the number of parameters compared to other CRM representations (Table 2). |

Jafroodi and Zhang [31], Zhang et al. [35] | Ensemble Kalman filter (EnKF). | Model parameters are sequentially updated as more data is gathered. Thus, it is possible to track and analyze the time-varying behavior of the parameters. Multiple models are obtained providing insight in the uncertainty of production forecasts and estimated parameters. Model constraints have not been explicitly considered. |

**Table 2.**Dimension of the history matching problem for several CRM representations without dimensionality reduction and without additional parameters for the primary production term. * The number of parameters for the ML-CRM was estimated based on Equations (12) and (13), assuming no data available from production logging tools or smart completions (i.e., unknown ${f}_{PLT,j\alpha}$ and ${f}_{i\alpha}^{\prime}$) and occurrence of crossflow between layers.

Model | Dimension | |
---|---|---|

Constant BHP | Varying BHP | |

CRMT | 2 | 3 |

CRMP | $\left({N}_{inj}+1\right){N}_{prod}$ | $\left({N}_{inj}+2\right){N}_{prod}$ |

CRMIP | $3{N}_{inj}{N}_{prod}$ | $4{N}_{inj}{N}_{prod}$ |

CRMT-Block | $B+1$ | $B+2$ |

CRMIP-Block | $(B+2){N}_{inj}{N}_{prod}$ | $(B+3){N}_{inj}{N}_{prod}$ |

ML-CRM * | ${N}_{L}({N}_{prod}({N}_{inj}+{N}_{t}+2)+{N}_{inj})$ | ${N}_{L}({N}_{prod}({N}_{inj}+{N}_{t}+3)+{N}_{inj})$ |

EOR Process | Reference (s) | Highlights |
---|---|---|

CO${}_{2}$ flooding | Sayarpour [12] | Proposed a logistic equation to mimic the increase in oil rates due to mobilizing residual oil during CO${}_{2}$ injection, while accounting for the fact that oil remaining in the reservoir is a finite resource. However, this logistic equation is independent of the CO${}_{2}$ injection rate, which is assumed to be constant, and four parameters must be history matched for each slug of CO${}_{2}$ injection, which might be impractical. The history matched data was obtained from a compositional reservoir simulator. |

Eshraghi et al. [14] | Application of the CRMP with the semi-empirical power-law fractional flow model and heuristic optimization algorithms for miscible CO${}_{2}$ flooding cases with data from a grid-based compositional reservoir model. | |

Water alternating gas (WAG) | Sayarpour [12] | Applied the CRMT and CRMP with the semi-empirical power-law fractional flow model to a pilot WAG injection in the McElroy field (Permian Basin, West Texas). |

Laochamroon- vorapongse [80], Laocham- roonvorapongse et al. [13] | Represented a single injector as two pseudoinjectors at the same location, one only injecting water and the other one only injecting CO${}_{2}$. Different values of interwell connectivities were obtained for these pseudoinjectors, revealing that the flow paths are dependent on the type of injected fluid. Field examples are presented for miscible WAG in a carbonate reservoir in West Texas, and immiscible WAG in a sandstone, deep water, turbidite reservoir. Additionally, the following diagnostic plots supplemented the analysis of surveillance data for WAG processes: reciprocal productivity index plot, modified Hall plot, WOR and GOR plot, and EOR efficiency measure plot. | |

Simultaneous water and gas (SWAG) | Nguyen [47] | Proposed an oil rate model derived from Darcy’s law assuming that water and CO${}_{2}$ are displacing oil in two separate compartments and relative permeability curves are known. Presented several examples of CRM to SWAG injection in comparison to grid-based compositional reservoir models and in the SACROC field (Permian Basin, West Texas). |

Hydrocarbon gas and nitrogen injection | Salazar et al. [81] | Applied a three-phase, four-component fractional flow model to predict production rates of oil, water, hydrocarbon gas and nitrogen gas in a deep naturally fractured reservoir in the South of Mexico. |

Isothermal EOR (solvent flooding, surfactant-polymer flooding, polymer flooding, alkaline surfactant polymer flooding) | Mollaei and Delshad [82] | Even though this work was not focused on CRM, there is an undeniable overlap in the underlying concepts, as the model developed is based on segregated flow (Koval model), material balance, and the flow capacity and storage concept. It assumes that there are two flood fronts displacing the oil to the producers. This model can provide insight for a future research on fractional flow models amenable to CRM in EOR processes. |

Hot waterflooding | Duribe [83] | Coupled CRM with energy balance and saturation equations to account for a time-varying ${J}_{j}(t)$ and, consequently, ${\tau}_{j}(t)$, mainly due to the water saturation increase and oil viscosity reduction. The results were compared with a grid-based thermal reservoir simulator. |

CO${}_{2}$ sequestration${}^{*}$ | Tao [84], Tao and Bryant [85], Tao and Bryant [86] | Application of the CRMP with the semi-empirical power-law fractional flow model for supercritical CO${}_{2}$ injection in an aquifer with data obtained from a grid-based compositional reservoir simulator. The main objective is to define an optimal strategy for each injector that maximizes field CO${}_{2}$ storage (i.e., minimizes CO${}_{2}$ production) under a constant fieldwide injection rate. |

Geothermal reservoirs ${}^{*}$ | Akin [87] | History matching of the CRMIP to infer interwell connectivities and improve the strategy for reinjection of produced water in a geothermal reservoir located in West Anatolia, Turkey. |

Li et al. [28] | Although not explicitly stated, their tanks network model is analogous to the CRM-block. However, in this model, production rates are the input and pressure drawndowns are the output. In addition, a complexity reduction technique is applied and production from some wells are clustered into a single tank. Their framework is applied to the Reykir and Reykjahlid geothermal fields in Iceland. |

_{2}storage and geothermal reservoirs are not EOR processes by definition (because oil is not been produced), these references are included here because they fit in the context of more complex exploitation processes involving chemical interactions and heat transfer.

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**MDPI and ACS Style**

Holanda, R.W.d.; Gildin, E.; Jensen, J.L.; Lake, L.W.; Kabir, C.S. A State-of-the-Art Literature Review on Capacitance Resistance Models for Reservoir Characterization and Performance Forecasting. *Energies* **2018**, *11*, 3368.
https://doi.org/10.3390/en11123368

**AMA Style**

Holanda RWd, Gildin E, Jensen JL, Lake LW, Kabir CS. A State-of-the-Art Literature Review on Capacitance Resistance Models for Reservoir Characterization and Performance Forecasting. *Energies*. 2018; 11(12):3368.
https://doi.org/10.3390/en11123368

**Chicago/Turabian Style**

Holanda, Rafael Wanderley de, Eduardo Gildin, Jerry L. Jensen, Larry W. Lake, and C. Shah Kabir. 2018. "A State-of-the-Art Literature Review on Capacitance Resistance Models for Reservoir Characterization and Performance Forecasting" *Energies* 11, no. 12: 3368.
https://doi.org/10.3390/en11123368