#
Proxy Model Development for the Optimization of Water Alternating CO_{2} Gas for Enhanced Oil Recovery

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}injection is one of the most common methods employed due to a high recovery potential and environmental benefits. To assess the feasibility of CO

_{2}-EOR projects, a reservoir design study must be conducted before optimization is performed. Some studies have demonstrated the advantages of employing proxy models to perform this task in terms of saving huge amounts of computer memory space and time. In this study, proxy models were developed to solve a multi-objective optimization problem using NSGA-II (Non-dominated Sorting Genetic Algorithm II) in two selected reservoir models. The study was performed for a CO

_{2}-WAG (Water Alternating Gas) application, where gas and water injection rates and half-cycle lengths were assessed to maximize the oil recovery and CO

_{2}stored in the reservoir. One model represents a simple geological model (the Egg Model), while the other represents a complex model (the Gullfaks Model). In this study, the good performance of the proxy models generated accurate results that could be improved by increasing the amount of sampling and segmenting the behavior of the reservoir model (depending on the complexity of the reservoir model). The developed proxies have an average error of less than 2% (compared with simulation results) and are concluded to be robust based on the blind test results. It has also been found that to reach the maximum oil recovery using CO

_{2}-WAG, the maximum gas injection rate with the minimum water injection rate is required. However, this configuration may result in a reduction in the total CO

_{2}stored in the reservoir.

## 1. Introduction

_{2}emissions per barrel of oil produced due to the power from shore [6]. Preplanning during exploration is important, but most existing fields were not preplanned as well as this field. Considering the decline in production in most existing fields, an alternative approach to reach the climate targets and meet the energy demands is CO

_{2}Enhanced Oil Recovery (EOR). CO

_{2}-EOR is considered as one of the answers to satisfy the demand in carbon capture, utilization and storage (CCUS), while retaining the profit [7].

_{2}-EOR [8]. Several considerations, for instance, the injection design, rate, cycle, composition, and existing field constraints, are accounted for in this phase. In the context of CO

_{2}-EOR injection design, water-alternating-gas (WAG) has been proven as one of the best designs [9]. CO

_{2}-WAG increases the microscopic displacement efficiency and yields a better mobility control to contact the unswept zones compared to continuous flooding. To design the parameters of the process, optimization needs to be performed. Optimization is a computationally expensive study, as numerous runs are required to solve the problem. Running optimization using existing reservoir simulators demands extensive amounts of time and memory space to solve the problem.

_{2}-WAG as one of the most common Enhanced Oil Recovery (EOR) methods. A proxy model is built as a reservoir model substitute, and the total oil produced and CO

_{2}stored will be maximized as the objective function of the optimization problem. This study will be performed on two geological models, where one acts as a simple model, while the other represents the complexity that is expected in a real field model. Several points that are studied in this research are:

- Development of a proxy model for simple and complex reservoir models that involves sampling using experiments, proxy buildings, and proxy robustness assessments.
- Analysis of the problem and complexity encountered when building a proxy of a complex reservoir model, in comparison with the results of a simple reservoir model proxy development.
- Solving the optimization problem with the generated proxy, both for the simple and complex reservoir models, using an optimization algorithm.

## 2. Basic Theory

#### 2.1. Previous Study on Proxy Modeling

_{2}-WAG to mimic the grid behaviors (pressure and saturation) and well-based behaviors (production rates of oil, gas, and water). Based on the study results, the constructed proxy model learned the preferred pressure, saturation, and rate behavior, in which one year was used as the timestep interval for reporting frequency.

_{2}sequestration study. The study showed that the coarser model required fewer runs for training purposes than the fine grid model. In alignment with that, the cascading procedure illustrated significant errors when observed in the last time step.

_{2}-WAG optimization study using proxy models. Two different machine learning (ML) techniques, namely an ANN [15] and a hybrid support vector [16], were implemented. These developed ML-based proxy models were able to learn the reservoir behavior (oil and water rate) and were then coupled with nature-inspired algorithms for the CO

_{2}-WAG optimization study. Besides that, a study also demonstrated how an ANN could be employed to build a proxy model for waterflooding optimization in a fractured reservoir model [17]. Thereafter, this study was extended to improve the methodology as discussed in [18].

#### 2.2. Possibilities for Proxy Modeling Improvement

_{2}and H

_{2}property estimations [19,20] and substitutes for reservoir models as detailed in the previous section. In the domain of reservoir and production engineering, proxy models can provide an alternative to overcome the current computational limitations in reservoir modeling (tremendous run time and memory consumption). In this aspect, most studies have worked with only one geological model and there are few studies focusing on the feasibility of this idea begin applied in different geological models (considering geological uncertainty) [21,22]. Some studies have also mentioned that the proxy would reflect the complexity of the reservoir model, yet quantitative results are to be found [14]. Therefore, an investigation was performed to check whether a proxy model is a feasible substitute for the reservoir models studied in this paper. It is worth noting that the proxy models are generally case specific.

#### 2.3. Workflow

- 1
- Determining the study objective:A proxy model only learns from the given sets of data and information, and this creates a limitation for the model, making it case specific. Different proxies need to be built for different study objectives, leading to sampling, proxy input–output combinations, study limitations, and the requirement of algorithms to solve the problem.
- 2
- Data sampling:After determining the study objectives, proxy scale, and input data for the proxy, data sampling can be performed. Data sampling is usually performed by running the reservoir model. The results to be learned are then sampled for the proxy learning dataset. Data sampling can be performed using available statistical sampling.
- 3
- Data management:After obtaining the sampling plan, reservoir model runs will be performed. Many data points can be obtained, yet not all of them will be used for proxy model training.
- 4
- Designing and building the proxy model:Using available machine learning or deep learning models, a proxy model can be built. The proxy model will approximate the numerical reservoir model. It should mimic the nonlinearity in responses of the numerical model. The complexity of the proxy model itself reflects the complexity of the reservoir model. Studies on proxy modeling have used approximately the same workflow shown in Figure 1.

#### 2.4. Artificial Neural Network (ANN)

#### 2.5. NSGA-II (Non-Dominated Sorting Genetic Algorithm II)

- 1
- Elitist principle;
- 2
- Explicit diversity preserving mechanism;
- 3
- Emphasis on the non-dominated solutions.

## 3. Materials and Methods

#### 3.1. Reservoir Model

_{2}-WAG design using the same optimization algorithm. An overview of the complexity difference between the models is listed in Table 1.

^{3}after initializing the fluid model.

#### 3.2. Fluid Model

_{2}-EOR by Al-Adasani [27] modeled with the Peng–Robinson equation. Constant composition expansion (CCE), differential liberation expansion (DL), multi-stage separator, slim tube, swelling, and multi-contact test data are available. These PVT test data are sufficient for fluid modeling in the CO

_{2}-EOR study. EOS PR78 was used in this study.

_{20+}and regressing on conventional PVT test data (CCE, DL, multi-stage separator test). After that, lumping was performed into seven components. Regression was then performed based on the available advanced PVT tests (slim tube and swelling test). The matched PVT results are shown in Figure 4.

#### 3.3. Depletion Scenario

^{3}for the Egg model and 17.2 million sm

^{3}for the Gullfaks model). Both will be depleted for eight years, supported with water injection to keep the reservoirs undersaturated. Equal injection rates and production rate limits were applied to both models. All wells are active (producing and injecting), starting in 2013 and continuing to operate until 2021. Table 2 shows the depletion study details and results obtained from each model.

_{2}-WAG. Figure 5 shows the results for both reservoirs. A higher recovery is seen when CO

_{2}-WAG is applied (3 months half-cycle, 6000 sm

^{3}/day field water injection rate, and 0.9 Msm

^{3}/day field gas injection rate) compared to waterflooding (3000 sm

^{3}/day field injection rate).

#### 3.4. Optimization Study

#### Objective Function

_{2}stored in the reservoir. Based on this, two objective functions need to be defined. The parameters that will be studied are the half-cycle length (HC), the field gas injection rate (q

_{g}), and the field water injection rate (q

_{w}), with several segmentations and ranges. These are formulated as follows:

## 4. Proxy Development Workflow

#### 4.1. Design of Experiments

- Half-cycle length:Sampling was performed for 3, 6, 9, and 12 month half-cycle lengths. This means that the proxy will not predict the CO
_{2}-WAG behavior out of the half-cycle length correctly.

- 2.
- Gas injection rate and water injection rate:The gas injection and water injection rates are sampled for each possible half-cycle length equally, since the probability for half-cycle length is equal for each possible value. More samples can be added if the proxy cannot learn from the given number of samples. The distribution of this parameter probability is uniform. Parameters are not dominating the others. Hence, the usual LHS can be performed.

#### 4.2. Data Preparation

_{2}PR. These variables are available as output results of a simulation run. In addition, the half-cycle length, timestep, gas injection rate, and water injection rate are used as the proxy inputs. Normalization is performed to help the convergence of the ANN. Maximum–minimum normalization will be performed as:

#### 4.3. Proxy Building

_{2}PR and FOPR proxies. For the Gullfaks model, a sharp production change is identified after 1 month, since the injection fluid (water or CO

_{2}) was changed, which can be observed as a delay compared to the Egg model. This results in proxy segmentations based on the injection phase.

#### 4.4. Proxy Training and Validation

#### 4.5. Blind Test

#### 4.6. Optimization Study

## 5. Results

#### 5.1. Proxy Structure Results

- 1.
- Egg Model:
- -
- The base: both proxies (FOPR and FCO
_{2}PR) are segmented based on injection phase. - -
- Timestep split: the proxy is then split again into 1st year, 2nd year–5th year, and 6th year–10th year.

- 2.
- Gullfaks Model:
- -
- The base: both proxies (FOPR and FCO
_{2}PR) are segmented based on injection phase. - -
- Half-cycle split: the base is then segmented based on low half-cycle lengths (3 and 6 months) and high half-cycle lengths (9 and 12 months).
- -
- Timestep split: the proxy is then split again into 1st year, 1st year–2.5th year, 2.5th year–5th year, 5th year–7.5th year, and 7.5th year–10th year.

#### 5.2. Proxy Training, Validation, and Testing Results

_{2}PR. This is due to the error obtained at early times (first year), as the difference in small value leads to a significant error calculated in APRE term. Nevertheless, FCO

_{2}PT (Field CO

_{2}Produced Total) should be the focus, as it will be used in the optimization study. The proxy and reservoir model results (selected randomly) are shown in Figure 7 for the Egg Model and Figure 8 for the Gullfaks Model.

_{2}PR maximum error. Observing the data for every timestep indicates that the error is due to the early time errors, the same as observed in the training validation phase.

_{2}PT errors are plotted in Figure 9 for the Egg Model and Figure 10 for the Gullfaks Model. The blind-tested proxy is then ready to be used as a proxy to represent the reservoir model. An optimization study was then performed with this proxy model.

#### 5.3. Optimization Study Results

## 6. Discussion

#### 6.1. Data Complexity

^{3}/d gas injection rate, and 3000 sm

^{3}/d water injection rate. This case is shown until 1800 days only. The other case is for a 360 day half-cycle length, 2 Msm

^{3}/d gas injection rate, and 9000 sm

^{3}/d water injection rate. These cases were run as training cases for both models, as one of Latin Hypercube edges.

_{2}PR behavior, a sharp increase in CO

_{2}production can be seen for the 90 day half-cycle. However, in the case of the 360 day half-cycle for the Gullfaks model, there is a smooth transition rather than a sharp increase or drop in production behavior. This different behavior will be learned using the same proxy model if it is not segmented based on half-cycle length. From FOPR plots, higher production rates can be noticed compared to the base case shown in Figure 5.

#### 6.2. Number of Samples Needed

#### 6.3. Proxy Model Development Process

#### 6.4. Proxy Robustness

#### 6.5. Optimization Results

_{2}stored in the reservoir will reduce. For the Egg model, this behavior is almost linear. However, for the Gullfaks model, it is not anywhere near linear. All statistics for the optimization process are tabulated in Table 8.

## 7. Conclusions

_{2}process. The proxy models were proven to be powerful modeling tools and can efficiently tackle the problems associated with conventional reservoir simulations, such as runtime and memory needed to store the results. In this context, the proxy models were built to mimic the oil production rate and CO

_{2}production rate, where the Egg model was the simple reservoir model and the Gullfaks model was the complex reservoir model in this study. Several proxy segments were used to construct these proxy models. Each segment was made using an ANN. The robustness (as a whole proxy) was tested using APRE, both during training and validation and blind testing. The results can be improved by changing the ANN, the assessment of the proxy, or even the optimization algorithm. In addition, the following points can be highlighted.

- A higher number of samples is needed for a more complex reservoir. For the Egg Model, 68 samples are enough, but for the Gullfaks model, 97 samples are required to develop an appropriate proxy model.
- To obtain the maximum oil recovery for CO
_{2}-WAG study, a maximum gas injection rate with a minimum water injection rate is needed. This will, however, reduce the amount of CO_{2}stored in the reservoir. A linear relationship between water injection rate, total oil produced, and total CO_{2}stored can be observed in the simple reservoir model. The complex reservoir model shows the same trend, but the relationship is not linear. This is because of the reservoir heterogeneity that affects the sweep efficiency and the CO_{2}entrapment.

## 8. Future Recommendations

- The workflow developed here can be employed as a good starting point to develop proxy models, for example, to be used in optimization studies.
- A separate study regarding the amount of sampling and sampling method is recommended as there is no clear guideline on how the sampling must be performed.
- APRE leads to positive and negative errors. This might lead to incorrect overall error values, and hence incorrect proxy robustness assessments. The AAPRE (average absolute percentage relative error) might be more suitable to assess the performance of the proxy, as this will remove the possibility of the negative errors being nullified by other positive value errors.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

_{2}Enhanced Oil Recovery and Storage), a strategic cooperation between NTNU and SINTEF (https://www.ntnu.edu/ceors, accessed on 18 July 2022). We would like to thank Curtis Hays Whitson for guidance in PVT modeling and Whitson AS for the Advanced PVT Course that aided in the PVT modeling in this study.

## Conflicts of Interest

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**Figure 1.**Workflow of proxy modeling [10].

**Figure 3.**Model illustration, faults, well locations, and perforation zones for (

**a**) the Egg Model and (

**b**) the Gullfaks Model.

**Figure 6.**Workflow of proxy building, the optimization study, and the software involved in this study.

**Figure 7.**Proxy results and comparison with the reservoir simulation output, sampled cases show different injection patterns (3 months, 6 months, and 12 months) (Egg Model).

**Figure 8.**Proxy results and comparison with the reservoir simulation output, sampled cases show different injection patterns (3 months, 6 months, and 12 months) (Gullfaks Model).

**Figure 9.**Proxy results and comparison with the reservoir simulation output (Egg Model), blind test.

**Figure 10.**Proxy results and comparison with the reservoir simulation output (Gullfaks Model), blind test.

**Figure 12.**Differences between the Egg Model and the Gullfaks Model regarding rate response to the WAG injection phase (

**left**: 90 day half-cycle,

**right**: 360 day half-cycle).

Parameter | Egg Model | Gullfaks Model |
---|---|---|

Permeability | Channel distribution | Heterogeneous |

Porosity | Homogeneous | Heterogeneous |

Faults | 0 | 14 |

Fluid | From SPE136530 | From SPE136530 |

Transmissibility | No multiplier | Heterogeneous multiplier |

Relative permeability | Sand preset | Sand preset |

Grid system | Cartesian | Cornerpoint |

Grid size | Homogeneous | Heterogeneous |

Initial condition | 320 bar, 120.85 °C, 1850 m | 320 bar, 120.85 °C, 1850 m |

Wells | Three injectors, three producers | Three injectors, three producers |

Perforations | Throughout all layers | Different for each well |

Parameter | Egg Model | Gullfaks Model |
---|---|---|

Injection rate | 2000 sm^{3}/day/well | 2000 sm^{3}/day/well |

Production rate limit | 2000 sm^{3}/day/well | 2000 sm^{3}/day/well |

Start of depletion | 1 January 2013 | 1 January 2013 |

End of depletion | 1 January 2021 | 1 January 2021 |

STOOIP | 16.55 Msm^{3} | 17.24 Msm^{3} |

STOIP (2021) | 8.94 Msm^{3} | 8.90 Msm^{3} |

RF | 46.0% | 48.4% |

P_{avg} (2021) | 254.9 bar | 264.3 bar |

Parameter | Value |
---|---|

Learning function | Adam |

Learning rate | 10^{−6}–10^{−2} |

Number of hidden layers | 1–3 |

Number of nodes | 2–20 |

Activation function | relu |

Batch size | 64 |

Dropout | - |

Training ratio | 75 |

Validation ratio | 25 |

Loss function | MSE |

Epochs | 100, then 1500 |

Parameter | APRE, Egg Model (%) | APRE, Gullfaks Model (%) | ||||
---|---|---|---|---|---|---|

Min | Average | Max | Min | Average | Max | |

FOPR | −2.307 | −0.129 | 3.227 | −2.364 | 0.042 | 1.443 |

FCO_{2}PR | −9.882 | 0.842 | 10.962 | −2.871 | 1.270 | 11.166 |

FOPT | −2.435 | −0.268 | 2.449 | −2.533 | −0.125 | 1.429 |

FCO_{2}PT | −3.252 | 1.052 | 6.114 | −6.048 | −0.909 | 2.778 |

Parameter | APRE, Egg Model (%) | APRE, Gullfaks Model (%) | ||||
---|---|---|---|---|---|---|

Min | Average | Max | Min | Average | Max | |

FOPR | −1.078 | 0.646 | 2.85 | −0.7 | 0.511 | 3.93 |

FCO_{2}PR | −3.538 | 2.944 | 12.433 | −1.823 | 2.006 | 10.055 |

FOPT | −1.575 | 0.168 | 2.162 | −0.728 | 0.194 | 3.405 |

FCO_{2}PT | −1.783 | 1.74 | 5.81 | −2.191 | −0.071 | 5.612 |

Parameter | Egg Model | Gullfaks Model | ||||
---|---|---|---|---|---|---|

PETREL | Proxy | Relative Error (%) | PETREL | Proxy | Relative Error (%) | |

FOPR * | - | - | 0.26 | - | - | 1.04 |

FOPT (Msm^{3}) | 2.49 | 2.51 | 1.05 | 2.08 | 2.1 | 1.19 |

FCO_{2}PR * | - | - | 1.23 | - | - | 0.45 |

FCO_{2}PT (Msm^{3}) | 78.84 | 79.43 | 0.75 | 95.91 | 94.83 | −1.12 |

_{2}PR are not specified as they comprise a set of data points.

Process | Egg | Gullfaks | |
---|---|---|---|

Sampling | LHS | 68 | 97 |

Sampling time | 9 h 43 m | 12 h 35 m | |

Run time per case | 9 m 25 s | 8 m 45 s | |

Memory space per case | 97 MB | 85 MB | |

Proxy Development | FOPR | 2 ANNs | 8 ANNs |

FCO_{2}PR | 5 ANNs | 9 ANNs | |

Total time needed | 1 h 23 m | 2 h 49 m | |

Proxy Performance | Run time | 8.75 s | 9.16 s |

Memory space | 14 KB | 13 KB |

Egg | Gullfaks | ||
---|---|---|---|

Pareto optima | 40 | 32 | |

Time needed | 2 h 25 m | 2 h 23 m | |

Runs needed | 1030 | 1030 | |

Eclipse equivalent time | 6 d 17 h 39 m | 6 d 5 h 38 m | |

Results | Half-cycle (months) | 3–6 | 3–9 |

Gas rate (Msm^{3}/d) | 1.98–2 | 1.95–2 | |

Water rate (sm^{3}/d) | 3000–9000 | 3000–9000 |

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## Share and Cite

**MDPI and ACS Style**

Matthew, D.A.M.; Jahanbani Ghahfarokhi, A.; Ng, C.S.W.; Nait Amar, M.
Proxy Model Development for the Optimization of Water Alternating CO_{2} Gas for Enhanced Oil Recovery. *Energies* **2023**, *16*, 3337.
https://doi.org/10.3390/en16083337

**AMA Style**

Matthew DAM, Jahanbani Ghahfarokhi A, Ng CSW, Nait Amar M.
Proxy Model Development for the Optimization of Water Alternating CO_{2} Gas for Enhanced Oil Recovery. *Energies*. 2023; 16(8):3337.
https://doi.org/10.3390/en16083337

**Chicago/Turabian Style**

Matthew, D Aqnan Marusaha, Ashkan Jahanbani Ghahfarokhi, Cuthbert Shang Wui Ng, and Menad Nait Amar.
2023. "Proxy Model Development for the Optimization of Water Alternating CO_{2} Gas for Enhanced Oil Recovery" *Energies* 16, no. 8: 3337.
https://doi.org/10.3390/en16083337