D-optimal design is a 3-level experimental model, so it considers interactions and quadratic effects in the response surface. The model screens the less effective variables, combines the important parameters and selects the best outcome (Okenyi and Omeke 2012). D-optimal design minimizes the overall variance of the regression coefficients by maximizing the determinant. The number of simulation runs is higher when using this type of model and increases multiple folds with increasing number of factors. For N number of factors, the total runs are:
D-optimal model allows factors to have multiple levels. D-optimal design is a rigorous design based on quadratic regression. It includes square terms and linear terms. We enabled a Computer Modeling Group (CMG) workflow to use D-optimal design to implement response surface modeling, and the RSM polynomial fit will be of higher order, i.e., linear + quadratic + interaction parameters terms are used to generate a proxy model to validate the simulation results.
The workflow for the RSM is:
- Define the objective functions, i.e., oil recovery and HCPV of CO2 injected.
- Evaluate uncertainty factors and their distribution affecting the objective functions.
- Analyze the parameters (heavy hitters) that will highly influence the response of the optimizing parameter (one parameter at a time, OPAAT, analysis).
- Use design of experiments (D-optimal design) to generate simulation cases.
- Run numerical simulator for all simulation cases.
- Generate proxy model for each objective function.
For this study, it was desired to achieve a proxy model considering the effect of interaction parameters. An acceptable R-square of 0.85 is defined in the engine. Once the initial accuracy is achieved, more experiments are generated to achieve a higher R-square value. The RSM approach will first try to fit a linear relationship between the objective function and each critical parameter. If a parameter has a nonlinear relationship with the objective functions, a quadratic term (x2) will define the relation. If modifying 2 parameters at the same time has a stronger effect than the sum of their individual linear or quadratic effects, a cross-term (xy) will define the relationship with the objective function.
It must be noted that the RSM approach only considers completion parameters to generate a proxy model. It means that only those parameters which are relevant to stimulation operations are considered for this study. Formation properties are not evaluated in this study as these properties cannot be changed and each reservoir has its characteristic set of defined formation and petrophysical properties.
Therefore, the importance and usefulness of the proxy model is only after a reservoir model is built for the particular formation and then for operations. There is a need to carry out sensitivity analysis to have confidence in predicted recoveries from a formation. Stress-dependent permeability and adsorption properties defined for the base model are kept constant and not considered for the proxy model approach. Table 8 provides the variable range between maximum and minimum values for the uncertainty parameters used for generating a proxy model.
Table 8 Uncertainty parameters with maximum, minimum and base values for the RSM approach.
Oil recovery: RSM
The input is defined to produce a proxy model, which considers interaction and quadratic terms. The reduced quadratic model is utilized to create a tornado chart to indicate the significance of each uncertainty and also improve the proxy model. The polynomial fit consists of linear terms, quadratic terms and parameter interaction terms.
Each term has its own statistical significance affecting the objective functions. The reduced quadratic model initially generates a proxy model consisting of all quadratic terms and interaction terms. The model then removes the statistically insignificant terms from the proxy equation. Figure 12 summarizes the effect estimate of uncertainty parameters on oil recovery generated by the RSM engine.
Fig. 12 Effect estimate of each uncertainty parameters and interaction of parameters on oil recovery (%).
It can be observed that the RSM approach also follows a similar trend to that observed in OPAAT analysis. The result shows that the lateral distance between the injector and producer hydraulic fractures is the most critical parameter to optimize oil recovery in a CO2-EOR scenario. It is followed by the injection pressure and the distance between wells. Several interaction parameters have more critical effect on oil recovery than individual uncertainty. The significant advantage of the RSM approach over the OPAAT analysis is that the RSM has the ability to consider the effect of interaction and the quadratic effect of uncertainties, which is not achieved through the OPAAT.
In addition, at reservoir scale, uncertainties among individual parameters simultaneously affect the flow modeling. In Fig. 12, ‘Maximum’ is the maximum value for the oil recovery among all simulation runs, and similarly, ‘Minimum’ is the minimum oil recovery calculated among all simulation runs. After estimating the effect of uncertainties, the response surface model fits a proxy equation with critical parameters to estimate oil recovery factor without running simulations.
Equation (1) is the proxy equation as a function of uncertainty parameters analyzed by the RSM engine. The generated proxy equation is modeled with 20 uncertain terms. The equation is valid only for the Middle Bakken Formation.
HCPV of CO2 injected: RSM
CO2 injectivity is a critical consideration for CO2 EOR as well as CO2 sequestration. Figure 13 illustrates the effect estimation of uncertainty parameters on hydrocarbon pore volume of CO2 injected. The result shows that the injection pressure is the most critical parameter for both HCPV of CO2 injected and oil recovery.
It is followed by distance between fractures and distance between wells. It is also observed that the injection pressure is combined with several interaction parameters and the effect of these interaction parameters has both negative and positive effects on the amount of CO2 injected. Therefore, it can be concluded that it is critical to evaluate the effect of interaction parameters on the objective functions instead of relying on OPAAT single-parameter analysis.
Fig. 13 Effect estimate of uncertainty parameters on HCPV of CO2 injected.
From the RSM approach, a wider perspective of the simulation results could be analyzed and the range of objective function with dependence on uncertainty could be easily predicted.
For final analysis, a cross-plot was generated from all experimental runs as shown in Fig. 14. The plot defines the CO2 utilization factor for improving oil recovery. In general, a CO2 utilization factor of 3:1 can be observed from this cross-plot. It signifies that every 3% hydrocarbon pore volume of CO2 injected into the Middle Bakken Formation can lead to an incremental oil recovery of 1%.
Fig. 14 Cross-plot of objective functions establishing a relation between the amounts of CO2 injected for every additional percentage of oil recovery.
The research concluded that facilitating oil recovery from tight oil reservoirs by CO2 injection could be greater than primary depletion depending on natural and induced fracture network connectivity. The presence of natural fractures significantly affects the flow migration of CO2 in the reservoir, directly impacting the sweep efficiency. Major conclusions of incorporating various physical processes into reservoir model are:
- Heterogeneity in the reservoir rock and permeability anisotropy are critical for unconventional tight oil formations and need to be considered in reservoir simulation studies. Reservoir heterogeneity and permeability anisotropy significantly affect the oil recovery as well as the CO2 breakthrough time in the production stream.
- Sensitivity analysis of the critical parameters with the OPAAT and RSM approaches provides significant understanding of critical parameters. The lateral distance between the injector hydraulic fracture and producer hydraulic fracture is the most critical parameters that affect oil recovery. Other critical factors are hydraulic fracture permeability of the production well and then the distance (acre spacing) between the injection and production wells. Natural fracture connectivity strongly influences the role of hydraulic fracture permeability and fracture half-length in the reservoir. These parameters become insignificant in the absence of natural and induced fractures in the formation.
- The RSM model estimated injection pressure as the most critical parameter for both the amount of HCPV of CO2 injected and the oil recovery. It is followed by the distance between fractures and the distance between wells. It was also noticed that when the injection pressure was combined with several interaction parameters, the effect of these interaction parameters had negative as well as positive effects on the amount of CO2 injected. It can hence be concluded that it is critical to evaluate the effect of interaction parameters on the objective functions instead of relying on the OPAAT single-parameter analysis.
- A CO2 utilization factor of 3:1 was evaluated using the base reservoir model for improving oil recovery in the Middle Bakken Formation. It signifies that for every 3% HCPV of CO2 injected into the formation, an incremental oil recovery of 1% could be achieved.
The authors greatly appreciate the support from the Computer Modeling Group Ltd. for providing the simulation software, and the support from the Warwick Energy Group and University of Oklahoma to publish this work.
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Wei Tian [email protected]
Edited by Yan-Hua Sun
© The Author(s) 2017
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