Here a brief summary of the performance of the two feedback approaches developed in this work are given. The comparison is made based on the results obtained in Sections 4.3 for RHC strategy and 5.2.2.1 for SOC since same cases of uncertainty and reservoir systems were considered.
It can be seen from Table 5-20 that for the four cases, loses incurred by RHC approach are higher. In fact, an unacceptable loss of 15.21% resulted in Case IV as a result of implementing this feedback technique whereas the loss is only 2.09% for the same case by employing SOC approach. Based on these results, RHC can be said to be sensitive to model/system mismatch. The sensitivity of the formulated CV through SOC principle is however very minimal in comparison.
Table 5-20: Comparison between SOC and RHC Methods
NPV ($) Loss (%)
Cases BM SOC RHC OC SOC RHC
I 182,775.00 182,297.70 182,274.70 182,775.00 - - II 159,723.50 159,428.90 159,320 159,096.40 0.19 0.25 IIII 239,271.50 228,116.4 222,286.10 220,918.20 4.66 7.10 IV 487,520.10 477,309.60 413,365.02 333,904.70 2.09 15.21
5.5 Conclusions
In this chapter, a novel method of data-driven SOC was developed where the target CV is the gradient of the objective function with respect to control. The method does not require the gradient information (explicit expression of the gradient) but is computed based on data through finite difference scheme. The concept was first developed for static optimization (both unconstrained and constrained cases) which was tested on a hypothetical case. A wonderful performance was seen with the method which is far better than local SOC. Some important points were observed in the cause of implementation of the method, some of which are
1. The more the number of reference points, the better the performance, although this has a detrimental effect on the computational time. On the other hand, the use of multiple neighbourhood points does not contribute to the superior performance of the method; this is because CV functions are only computed at reference points.
2. Using central difference scheme produced the best performance than forward and backward differences.
3. The methodology was also tested for situations where the disturbance is completely unknown. Here, variables ranking based on separable and monotonicity rules were employed to deal with the situation. Again, a tremendous performance was recorded with a loss as low as 0.00007789.
4. Application of the method to constrained scenario has also yielded excellent results with performance exactly as that of NCO approximation method which requires explicit expression of the NCO. A zero loss was achievable in this case.
The method was then extended to solve dynamic optimization problems with particular focus on waterflooding process. Implementation of the method was done on both simplistic and more realistic reservoir sizes. The feedback benefits of SOC in counteracting uncertainties in rock and fluid properties were
realised through various case studies. The following can be concluded from the study:
1. In the absence of system/model mismatch the OC approach was seen to have a better performance than SOC as expected. The difference is not significant however; the loss recorded by SOC was only 0.26% for the simplistic reservoir size and 0.11% for the real case.
2. With introduction of uncertainty of various forms and degree into the system which includes uncertainty in permeability, porosity, size, geometry, structure and shape of relative permeability curves, the developed feedback approach performed extremely well.
3. The relative performance of SOC method was observed to increase with the degree of uncertainty considered in the system. For instance, when uncertainty was considered in permeability only, gain achieved is in the range of 0.26% - 1.03%. Introducing more mismatches simultaneously in the form of reservoir size, geometry and structure, the gain was seen to shift up to 24.03% - 30.04%. Comparing this with the BM case where all properties were assumed to be known a priori, losses of only 0.54% - 2.09% were incurred by SOC as against 24.44% - 31.51% by OC.
4. In most of the cases studied, the shape of the injection trajectories found by SOC approach resemble those of the BM despite the presence of uncertainties, a situation that led to finding optimum oil and water production profiles, hence close to optimal NPVs.
5. Uncertainty is not considered in the formulation of the CVs due to complexity of oil reservoir, the robustness of the CVs is therefore entirely due to the feedback nature of the SOC strategy. With introduction of uncertainties in the CV formulation, the performance of the technique can be improved further.
6. In summary the designed CVs can be regarded as simple and robust, therefore are insensitive to uncertainties. This was also confirmed through sensitivity analyses on the CVs and individual measurements.
6 Optimal Multivariable Feedback Control for Reservoir
Waterflooding
6.1 Introduction
In Chapter 5, SOC methodology for dynamic systems was developed and applied to waterflooding process. Impressive results were reported for various geological uncertainties considered. However, only systems with one manipulative variable (one degree of freedom) were considered. In the present chapter, the methodology was extended to optimize waterflooding process of higher degrees of freedom, because real oil and gas fields consist of several production and injection wells in operation and hence multivariate problems are encountered.
As was in Section 5.2.2, gradients of the objective function with respect to controls (CVs) were obtained based on a nominal model through regression. These CVs were then applied to reservoirs with different degrees of uncertainties in properties ranging from permeability, shape of relative permeability curves, and size, geometry and structure of reservoir.
The CVs were found to be robust in the presence of all the above uncertainties with performance similar to case where the reservoir properties were assumed to be known a priori. With the application of the CVs to the nominal model, only a negligible loss was incurred. Furthermore, implementation of the CVs to cases with model/system mismatch leads to a gain of up to 95% over an open-loop solution.