_.introducing a correction factor c:c=_x qnew i(4)
7th step consists in: calculating new liquid rates (for each producing well the novel rate iscalculated by adding/subtracting the same amounts of water rate, _q±, calculated for the injectorwells connected to it), imposing that the constraints on the total liquid volume (y) are satisfiedthrough the correction factor c1:c1= _y qnew i(5)
8th step consists in: making Matlab write a text file to be included in the 3DSL dataset,summarizing all the new rates and time information.9th step consists in: running the reallocation procedure, following the eight steps listed above,for as many iterations as initially fixed (well rates and time are updated at every _t).Both _t and the set of parameters chosen at the 5th step affect the final results.For all the examples shown here, to signed e was allotted the average field efficiency value, inorder to have a case sensitive value. At the same time, _ was put equal to 2 for all the exam inedcases, so that the weighting function would be nonlinear and the most significant changes in injection rates would occur far-off from the average efficiency, while only small changes would take place around the mean value (Figure 1).For the examples shown below, parameter sensitivities were performed in order to get the bestset for the application of the Matlab code. RESERVOIR MODELS DESCRIPTION
In this paper two synthetic models and a real one are presented.
The numbers of cells for the synthetic models are 20×20×1 for the first case, and 123×54×1for the second case. A uniform discretization of the volume was chosen to accurately describe the phases present in both models are water and oil, which is supposed to be dead oil. Both water and oil have the same viscosity, equal to 1 cp. Oil and water density are, respectively, 780 and1029kg/m3. Datum pressure is equal to 30 bars. For both models, water and liquid constraints are set equal, to let the displacement occur by voidage
replacement, and correspond to 80.000rm3/day and to 150.000rm3/day, respectively. For both models the reallocation procedure starts at day 730;the number of steps, whose duration is 730 days each, is equal to 10.Similarly a real field was tested. The model was available from ECLIPSE; a translation work was done to obtain a model written for 3DSL. The two models were then checked to verify the coherence of the main parameters. Once the match was gained, 3DSL model was used to apply the procedure. The real model, as shown in Figure 4(a, b), is divided into three regions by the presence of faults
Figure 4. (a) Faults divide model three into three regions and (b) model three has three different regions.
Figure 5. Permeability (a) and porosity map (b). Model three.
The number of cells is 68×71×23. The grid cells are not uniform. Permeability and porosity arehighly heterogeneous and range from 0 to 1315mD and from 4 to 27%, respectively (Figure 5(a, b)).The other model properties are summarized in Tables I and II.Table II. Model three: main parameters.Swi (three regions) 0.13; 0.29; 0.06 RESULTS
The three models examined were all used for the three steps workflow related before.First of all, the match and coherence between the two models were always obtained. At thispoint no relevant discrepancies were observed between the two sets of results for the
three models.Figure 6(a, b) shows, respectively, oil and water cumulative production from the third model only,as an example.
At a second stage, Matlab code was applied to each 3DSL model, starting from a certain moment in time, fixed at day 730 for all the three examples. Ten iterations were run, each of them lasting730 days, and the total time for the procedure to run was fixed so as to be compared with the base case ‘do nothing’ simulation run.
As for the frequency of the injection update, the longer the time step is, the higher the rates injected/produced and the major the changes concerning the values of well pairs injection efficiencies during the time steps. For the present work the well pairs injection efficiencies were assumed to remain constant throughout the single time step. Shorter time steps would maybe more suitable to this hypothesis but, on the other hand, they would imply a much higher computational time. In a real field application it might be useful to try with medium time steps (i.e. 6 months),eventually shortening them if, by analyzing the real-time field data, an intervention would appear to be needed.Each model was then compared with its respective optimized case, as shown in Figures 7–9(a, b).The figures again show, respectively, oil and water cumulative production from the field for the three models in sequence. For all the models the generated methodology was shown to give good results and to carry advantages with respect to the mere base case.The two synthetic cases showed good results both in terms of a relevant improvement in oil recovery and of a significant reduction in water production (the results are summarized in Tables III–V). Figure 10(a, b) shows the displacement of model two at the end of the simulation,for the base case and for the reallocated rates case, respectively.
The real case, which at the time of this research was a pure forecast example, since the reservoir was not producing, so no production history was available, showed to gain a certain amount of oil production, but in parallel a higher water production.At the third stage the final rates observed from the 3DSL ‘optimized’ case were input in ECLIPSE, to double check the effectiveness of the optimization for the specific case. Figure 11(a, b) P. A. C. MARESCALCO
Figure 6. Match ECLIPSE versus 3DSL: oil (a) and water cumulative production (b). Model three.
Figure 7. Base case versus reallocated rates case: oil (a) and water cumulative production (b). Model one.
Figure 8. Base case versus reallocated rates case: oil (a) and water cumulative production (b). Model two.
RECOVERY THROUGH REAL-TIME WATER FLOODING OPTIMIZATION
Figure 9. Base case versus reallocated rates case: oil (a) and water cumulative production (b). Model three shows oil and water cumulative production for the second synthetic model, obtained with ECLIPSE before and after the reallocation procedure.For the synthetic models, whose ECLIPSE results downstream of Matlab procedure are as good as expected, the methodology proved to be valid.
Figure 10. Oil displaced at the end of the simulation run: base case (a) and reallocated rates case (b). Model two.
Figure 11. Reallocated rates case versus base case, ECLIPSE check simulation run, oil(a) and water cumulative production (b). Model two As for the real case, the results at this stage are not as good as expected. In the next paragrapha few comments on this will be remarked.
DISCUSSION AND CONCLUSIONS
Looking at the results, it can be generally said that the Matlab code written for this study has added value to the reallocation procedure, making it reliable and attractive for its