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Generalized Automatic Global Predictive Optimization and Forecasting*October 2024 Also See: Our new optimization page describing our new optimizer SensorOpt is coming soon. We're finishing up our probabilistic workflows to quantify uncertainty in optimizations, predictions, and forecasts. For now, see this SensorOpt log file, Spe5s.txt, for the solution to the deterministic example problem given on our old optimization page (below) to determine the optimal production strategy for SPE51, obtained in a few minutes as opposed to about an hour to make several optimization runs for the previous solution: It maximizes NPV to US $1.9438 billion at 11081 days and recovers 100% of the oil in place. Note that the solution gives an optimal gas flood producing bhp of 3755 psia, the same as the previous solution. This is not a coincidence. It places the reservoir almost entirely in the first-contact miscible region during the optimal gas flood (first-contact fluid saturation pressure = 3874 psia, drawdown is about 80 psia). Also see this SensorOpt log file for the same problem, but with the added constraint that total runtime is less than or equal to 9000 days: Spe5c.txt. The unconstrained problem gives the solution that absolutely maximizes NPV and oil recovery regardless of the time taken to achieve it, and is mostly of academic interest. The constrained optimized solution maximizes NPV to $1.9148 billion. at 8983.6 days, recovers 98.1% of the oil in place, and gives a 12.56% average annual net return on investment (ANROI is the yield on top of inflation, assuming everytthing inflates at the same rate). The unconstrained solution gives an ANROI of 8.5%. We can easily and accurately determine the optimal production strategy for our reservoirs, including the optimal well and fracture spacings that are part of it, using simple models incorporating our reservoir and fluid properties. Numerical error (dispersion) is eliminated through the use of fine gridding or proper upscaling. See our SPE3 and SPE8 pages demonstrating the critical need for proper upscaling in our coarse grid reservoir models. In these spe5 optimization problems, we have thus far assumed that the originally given grid and properties are properly upscaled to reproduce the correct fine-scale solution. Generalized Automatic Global Predictive Optimization and Forecasting* (2013) Our associate Dr. Curtis Whitson and his team developed workflow integration, automation, and iterative optimization software called ECIAM® (Tieto). ECIAM® can integrate any number of linked batch applications to create automated workflows, to which various available iterative optimization methods can be applied. Any program inputs can be automatically and iteratively adjusted to maximize or minimize any benefit function that can be defined from the workflow outputs. They have solved the generalized automatic optimization problem, which naturally leads to the greatest advances in reservoir modeling in decades, when applied to reservoir simulation. Using our reservoir simulator Sensor, we have created automatic workflows in ECIAM® for:
The iterative optimization problems are very similar and differ only in the definition of the benefit function to be minimized or maximized, and identity of the adjusted variables, and the (probabilistic) consideration of uncertainties. Workflows for characterization, upscaling, and history matching are essentially the same. They minimize uncertainty by minimizing a mismatch function. Global predictive optimization adjusts operational variables to maximize an overall value function such as cumulative BOE or NPV, making most instantaneous optimizations obsolete, other than those designed to achieve the globally optimal solution. The probabilistic workflows require as input an arbitrary number of equally-probable realizations of the uncertainties (the number of realizations required must be determined by the user as those that give reproducible probabilistic results). Sensor can represent some of those uncertainties through its Uncertain Inputs features, in which each execution of the program automatically creates an equally-probable realization of the specified uncertainties in the input data. Our programs MakeSpx and SensorPx are used to make an arbitrary number of runs representing the uncertainties and compute the probabilistic results from them, within each iteration of the optimization. Our first example illustrates the use of SensorCast for automated deterministic predictive optimization (considering no uncertainties in the input data other than the optimized variables). It is based on the reservoir and fluid description from the Fifth SPE Comparative Solution Project1. Results from a number of miscible flood (extended black oil) and fully compositional simulators were compared for three given production/injection scenarios. The Scenarios are described in detail in Appendix 1 of the referenceibid. The first Scenario is 2 years of depletion with producing BHP=1000, followed by a WAG flood with 1 year gas and water injection cycles. The second Scenario is a primary WAG flood with 3 month gas and water injection cycles with higher maximum rates. The third Scenario is 1 year of depletion using BHP=1000, followed by a 1 year waterflood, followed by a WAG flood using the same 3 month cycle and WAG rates as Scenario 2 except that maximum gas injection rate is increased from 20000 mcf/d to 30000 mcf/d. Scenarios 1 and 3 use a minimum producing BHP of 1000 psia during all production periods, but Scenario 2 limits producing BHP to 3000 psia during WAG cycling, to better operate in the miscible region. The Sensor compositional model datafiles for these three scenarios are spe5a.dat, spe5b.dat, and spe5c.dat. These datafiles do not include Sensor's options for treatment of oil and gas relative permeability and Pcgo as functions of gas/oil interfacial tension, as miscibility is approached, which may make a significant difference in results. We used the optional maximum trapped gas saturation of .20 for Krg hysteresis. In Scenarios 1 and 3, we used both injector and producer for depletion during the depletion periods (for a better determination of optimal strategy). The problem specification includes a lab-measured minimum miscibility pressure (MMP) of 3000 psia, which is used in miscible flood simulators as a condition of miscibility. Miscibility in reservoirs is generally not achieved at MMP measured in a laboratory slim-tube experiment, because of the compositional mixing in the reservoir due to heterogeneity and multiphase flow and capillary and gravity forces and thermodynamics, which prevents the very gradual 1D enrichment that is required to achieve MMP well below the pressure required for first-contact miscibility (the FCM pressure) in the lab experiment (see our MMP page). Optimal minimum (producer) miscible flood pressure is generally slightly lower than the first-contact miscible (FCM) pressure. An FCM pressure exists for a given fluid/solvent if there exists a closed P-x phase envelope (a hubble point, dew point, or critical point exists for all mixtures) as the maximum saturation pressure of any mixture. Sensor prints the phase envelope for the given oil / injected gas mixture in all output files, showing that the FCM pressure for the given oil and injected gas compositions is 3874.8 psia at a composition of (.2 oil, .8 solvent). Scenario 1 (spe5a.dat) terminates at 6317.9 days when the field watercut limit is exceeded, with oil and gas recoveries of 70.2% and 34.8%, cumulative produced oil, gas, and water are 29398 mstb, 40117 mmscf, and 4411 mstb, respectively. Cumulative water and gas injection are 35040 mstb and 32015 mmscf, respectively. HC average pressure at end of run is 3286 psia. Scenario 2 (spe5b.dat) terminates at 6186 days when the field GOR limit is exceeded, with oil and gas recoveries of 78.4% and -1.4%, cumulative produced oil, gas, and water are 32824 mstb, 59655 mmscf, and 73 mstb, respectively. Cumulative water and gas injection are 29738 mstb and 59976 mmscf, respectively. HC average pressure at end of run is 3811 psia. Scenario 3 (spe5c.dat) terminates at 3531 days when the field GOR limit is exceeded, with oil and gas recoveries of 72.0 and 17.6%, cumulative produced oil, gas, and water are 30166 mstb, 44469 mmscf, and 2585 mstb, respectively. Cumulative water and gas injection are 31845 mstb and 40357 mmscf, respectively. HC average pressure at end of run is 3501 psia. To improve accuracy of the results, we added Sensor's options for treatment of oil and gas relperms and capillary pressure as functions of gas/oil interfacial tension (KRIFT and TENSION). If we assume a lab reference IFT for the air/water system used to determine Pcgo, then the Pcgo scaling performed by our TENSION option scales the curve down to negligible values at immiscible and miscible reservoir conditions. So in the following datafiles we use the KRIFT option to treat oil and gas relperms as functions of IFT, but omit the TENSION option for Pcgo scaling. The KRIFT option without the following PC label automatically zeroes Pcgo. The Sensor datafiles for the 3 Scenarios using KRIFT are spe5akrift.dat, spe5bkrift.dat, and spe5ckrift.dat. Results are: Scenario 1 (spe5akrift.dat) terminates at 6357.4 days when the field watercut limit is exceeded, with oil and gas recoveries of 70.3% and 34.7%, cumulative produced oil, gas, and water are 29430 mstb, 40574 mmscf, and 4553 mstb, respectively. Cumulative water and gas injection are 35040 mstb and 32489 mmscf, respectively. HC average pressure at end of run is 3218 psia. Scenario 2 (spe5bkrift.dat) terminates at 6478.8 days when the field GOR limit is exceeded, with oil and gas recoveries of 79.3% and 11.0%, cumulative produced oil, gas, and water are 33191 mstb, 64812 mmscf, and 192 mstb, respectively. Cumulative water and gas injection are 31558 mstb and 62252 mmscf, respectively. HC average pressure at end of run is 3664 psia. Scenario 3 (spe5ckrift.dat) terminates at 3376.2 days when the field GOR limit is exceeded, with oil and gas recoveries of 71.5 and 22.0%, cumulative produced oil, gas, and water are 29946 mstb, 42947 mmscf, and 21918 mstb, respectively. Cumulative water and gas injection are 31280 mstb and 37822 mmscf, respectively. HC average pressure at end of run is 3501 psia. Adding the KRIFT option made the most difference in gas production in Scenarios 2 and 3 (8.6% more in Scenario 2). The economic model parameters specified in Example 1 below were used in the above datafiles, while retaining the originally specified economic limits. Reported NPV at end of run for the last 3 KRIFT runs are $1.40575E9 at 6357.4 days, $1.57631E9 at 6478.8 days, and $1.43252E9 at 3376.2 days, respectively. Example 1 - Extension of the SPE5 problem to determine the optimal production/injection strategy Determine the optimal production/injection scenario for the SPE5 problem. Allow for a possible primary depletion period, possibly followed by a waterflood, possibly followed by a WAG flood, possibly followed by blowdown. Use the same maximum production rate for the producer (12000 stb oil/d) along with the same minimum producing bhp of 1000 psia for depletion, waterflooding, and blowdown. Assume that the maximum well injection rates given in any period of the original Scenarios represent fixed well or facility limits. Instead of the specified limiting GOR and WCUT values for shutting in the field, use the following economic model variables to determine the optimal scenario by maximizing Net Present Value:
Solution We first define the optimization variables and their initial values and minimum and maximum limiting values for optimization in the ECIAM Optimizer. A link is created for each variable to its specification in one of the data or output files. The initial values are based on original specifications or on our best guesses of what we think may be optimal: Variable Description Min Max Initial TD depletion end time, d 10 7300 365 DTWF waterflood time, d 10 7300 365 DTWAG WAG flood time, d 10 10000 6500 DTWAGW water inj. cycle time, d 1.0 500 91.25 DTWAGG gas inj. cycle time, d 1.0 500 91.25 DTBD blowdown time 365 7300 1370 BHPWFI wf injector max bhp 3000 6500 6000 (assume frac P > 6500) BHPWAGP wag producer min bhp 1000 4000 3000 BHPWAGI wag injector max bhp 4000 6000 4500 The set limits allow any of the production phases to be eliminated or to be included down to their minimums and up to their maximum time limits. The optimized values of these variables represent the optimal operating strategy. The Sensor data file prior to optimization using the initial values of the defined variables is spe5d0.dat. spe5d0.out reports that the NPV at end of run for the initial guess is $1.5766E9 at 8600 days (about the same as Scenario 2 but 3 years later), and the fractional return on investment ROI is 5.485. NPV = cumulative npv of production - cumulative npv of investment, $ at t=0 ROI = cumulative npv of production / cumulative npv of investment, fraction Injected fluid costs are accounted for as investment expenses. We note that miscible recovery is not achieved in much of the bottom 2 layers, due to gravity override of the oil by the injected gas. So we make a test run changing the injection well perforation from layer 1 to layer 3. NPV increases to $1.61823E9 at 8600 days (about $42 million higher), and ROI increases by 27% to 6.916. We then copy this modified datafile (spe5di.dat) to Sensor.dat in our ECIAM project directory data folder. This is an example of manual predictive optimization, using engineering judgment. An alternative is to define as additional optimization variables the injector (and producer) completion layer ranges (as K1 K2 in the well data) for all or for each of the production phases, but for this example we will consider the completed layers as fixed in the bottom layer for both wells in all production phases. The above variables are linked directly to their specifications in the Sensor.dat file (the DT* variables are defined in comments). Other "auxilliary" variables representing end-time specifications in the Sensor datafile are defined in the optimizer as specified functions of the optimization variables: TW end of waterfloodng. = TD + DTWF TWAG end of WAG flooding = TD + DTWF + DTWAG TEND end of blowdown = TD + DTWF + DTWAG + DTBD The auxilliary variables are automatically calculated and set in the Sensor.dat datafile by the optimizer at the beginning of each optimization iteration. The recurrent well data in Sensor.dat (spe5di.dat) containing all of the defined variables is: WELL We define the objective function in the ECIAM Optimization window by creating a link to the value of NPV reported at end of run in the Run Summary of the Sensor.out file and assign it as the objective function to be maximized. Also reported are the values of the maximum NPV achieved during the run and the time it is achieved. That time is the optimal end time as reported by Sensor, if it occurs before TEND. If not, the (exactly) optimal end time for any given scenario is easily determined by increasing TEND until NPV begins to decrease. After setting up the above (9 optimization, 3 auxiliary, and 1 objective function) variables in the ECIAM Optimization and Linkz windows (taking about 20 minutes), we are ready to optimize. The general solution method to find a global maximum, which is demonstrated here as a manual process but can be automated, is to keep executing the same or different 'solvers' available in ECIAM in series, and by trying different starting points until we're unable to significantly improve NPV. This is simply automation and improvement of the method that engineers have always used in reservoir modeling to make optimizations between discrete alternatives, for example as examined in the original SPE5 paper for the three given production scenarios using various reservoir simulators. We select the Reflection optimizer (or "solver", a proprietary Petrostreamz/Tieto implementation of the Simplex method) in the ECIAM Optimization window, and allow for 500 maximum iterations. The number of required iterations and convergence is highly dependent on the number of optimization variables (and the optimization method used). We then select "Run" in the Optimizer window and the optimizer begins to iterate. The optimization terminates after 500 iterations with an optimized NPV of $1.89303E+9 at 10992 days, an increase of $172 million or 17%, 6 years later. Total workflow execution time is about 12 minutes. Spe5_opt1.xlsx gives the iterative history of the optimization. The best found solution is repeated at the end as iteration no. 501.. The optimized datafile Sensor.dat is copied to spe5_opt1.dat. The values of the optimized variables are: TD 170.3299861 DTWF 238.0863996 DTWAG 9988.348595 DTWAGW 35.82821765 DTWAGG 495.3740961 DTBD 595.5213266 BHPWFI 4671.993396 BHPWAGP 3349.564431 BHPWAGI 4483.139139 TW 408.4163857 TWAG 10396.76498 TEND 10992.28631 NPV 1893030000 Since DTWAG has reached its maximum limit, we reset its maximum value in the optimizer to 12000 days. We also reset minimum DTWAG to 6000, and continue optimization allowing another 500 iterations. NPV is increased by over $500 million to 1.94581E+9 at 12052.2 days (spe5d_opt2.dat, spe5d_opt2.out). This solution is near-optimal (near a local or global optimum). NPV for this solution is within 0.1% of the best solution that we have found for this problem upon further optimization, as described below. TD 136.50915184286688 DTWF 599.9401793463742 DTWAG 10659.776837735773 DTWAGW 11.928941344948099 DTWAGG 477.7340847527599 DTBD 655.9837457117383 BHPWFI 4767.714517037251 BHPWAGP 3755.2291830134295 BHPWAGI 4558.572982510338 TW 736.4493311892411 TWAG 11396.226168925015 TEND 12052.209914636755 NPV 1945810000 To check for possible further improvement, we switch to the NOMAD optimizer in ECIAM, and allow another 500 iterations, which take another 14 minutes to complete. NOMAD works well for improving the accuracy of near optimal solutions, and also by itself on most cases. NPV improves by almost a million dollars to 1.946674E9 at 11736.1 days (spe5d_opt3.dat, spe5d_opt3.out): TD 136.51121714380835 DTWF 599.9427757247006 DTWAG 10294.159347921015 DTWAGW 11.929684912891672 DTWAGG 477.7286329125976 DTBD 705.4821524795834 BHPWFI 4767.722027222493 BHPWAGP 3755.222388083925 BHPWAGI 4558.572267254601 TW 736.453992868509 TWAG 11030.613340789525 TEND 11736.09549326911 NPV 1946673753.4 An optimal solution is indicated mathematically by dNPV/dx = 0 for all optimization variables x. No solution found by the optimizer will be exactly optimal. When the optimizer converges in less than the maximum allowable iterations, its solution is very close to a local or global optimum When it does not converge but enough iterations are allowed, the optimizer will usually find and 'bounce around' a local or global optimum but without satisfying convergence tolerances. This tends to increasingly happen as the number of optimization variables increases, and as their co-dependencies become stronger. Although the optimization variables are completely independent, their optimal values are very highly dependent on each other (with the exception of optimal water and gas flood operating pressures, which are mainly dependent on phase behavior). This makes the NPV response surface very complex with multiple local optima and the optimization extremely difficult or impossible to perform manually when there are more than just a few variables. It also makes it virtually impossible to replace or sufficiently estimate the simulator-predicted NPV function or its optimization with any proxy model, neural network, or any 'artificial intelligence' or 'machine learning' method. To demonstrate this, NPV results for an unlimited number of equally-probable realizations of this example problem were offered to those claiming proxy or neural-network or any ai/ml model applicability to reservoir engineering. No solutions were offered based on the 100 realizations given nor were any more requested. See the SPE Reservoir technical group discussion of "Artificial Intelligence / Machine Learning Problem Solving Methods" (SPE and Reservoir group memberships are required). Also see our page on Artificial Intelligence. The optimizer may find a local or a global optimum in any given optimization run. A locally optimal solution that may be found can depend on the iterative solution path taken by the optimizer from the assumed initial variable values, which depends on those values and the specified variable limits and the optimization algorithm. So to find a global optimum, we use different starting points for optimization while trying to allow for any reasonable values of the variables. Continuing an optimization with the same or a different optimizer sometimes results in an improved solution. Some optimizers are much better than others. Of the available optimizers in ECIAM, we have found that both the Reflection and the NOMAD solvers work well on this problem and others like it. From the ECIAM Pipe-it Manual,
We can check that our optimized solutions are close to a local or global maxima by perturbing each optimization variable in turn, using perturbations that are large enough to have a significant effect on the objective function. At an optimal solution, all perturbations of all variables plus and minus should result in a decrease in NPV. Test runs were made by manually perturbing all variables by small amounts in the optimizer interface to confirm that spe5_opt3 is very close to an optimal solution, and all perturbations made result in decreasing NPV. For example, changing time of depletion TD to either 130 or 140 days results in significant decrease in NPV. Even the very small water injection cycle time of 11.93 for the WAG flood is found to be very accurately optimal, which is a little surprising given the very long optimal gas injection cycle time of 477.73 days. These tests take only a few minutes in this case. However, since the last NOMAD optimization increased NPV by about a million dollars, we continue with another 500 NOMAD iterations (spe5d_opt4.dat, spe5d_opt4.out). NPV increases by about half a million dollars to 1.947266E9 at 11736.0 days. Another 1000 iterations of NOMAD (spe5d_opt5.dat, spe5d_opt5.out) increases NPV by about $330 thousand dollars to $1.947597E9 at 11735.92 days. The NOMAD optimizations are slowly increasing the accuracy and NPV of the same optimal strategy that was found in the second optimization using the Reflection solver, spe5_opt2. Our spe5d_opt5 solution, taking a total time of about an hour to perform the 5 optimizations, is TD 136.3965841775175 DTWF 600.0111131013296 DTWAG 10294.03529942354 DTWAGW 11.946483704052952 DTWAGG 477.67823466207545 DTBD 705.4771862199365 BHPWFI 4768.00586143727 BHPWAGP 3755.3412987055735 BHPWAGI 4558.521919736671 TWF 736.4076972788471 TWAG 11030.442996702386 TEND 11735.920182922322 NPV 1947597546.3 To see it could possibly be better to completely deplete the reservoir first, we run case spe5d_dep.dat simulating only depletion to 10,000 days. The only optimization variable is the optimal end time, which Sensor reports at the end of spe5d_dep.out as the time of maximum NPV: MAX NPV (PRIOR TO ABEX) = 0.6271021E+09, AT TIME = 0.7197584E+04 DAYS A very important observation is that if production strategy is not optimized and only depletion is considered, NPV of the reservoir is reduced by 67.8%. We then specify 7197.58 days as end time of depletion TD and deactivate it as an optimization variable, and reoptimize using 500 iterations of the Reflection solver, starting with the other optimized variable values from spe5d_opt5. The best found solution (spe5d_dopt1.dat, spe5d_dopt1.out) is NPV = 1930191954.0 at time 18982.5 days. We then apply 500 iterations of NOMAD and NPV improves by about $500,000 to 1.930689E9 at 19031.9 days giving our optimal solution for the case of mandatory depletion to 7197.59 days (spe5d_dopt2.dat, spe5d_dopt2.out). TD 7197.58 (specified) DTWF 512.9597784073119 DTWAG 10602.090610824198 DTWAGW 3.2521265727310475 DTWAGG 483.77406322367017 DTBD 719.2467111404744 BHPWFI 4391.326519799346 BHPWAGP 3689.1731626916003 BHPWAGI 4517.1076610443215 TWF 7710.539778407312 TWAG 18312.63038923151 TEND 19031.877100371985 NPV 1930689086.2 So requiring total depletion first only costs 17 million dollars in final NPV, but at a very much longer final time (19032 vs. 11736 days, 20 years longer!). If discount factor were significant here, the lost value due to requiring initial depletion of the reservoir would be huge. To compare optimizers and to possibly find a better solution, we try 4 successive applications of 500 maximum iterations of the NOMAD optimizer, with the same spe5di.dat starting point, with maximum DTWAG set to 12000. A slightly different locally-optimal solution is found (in less than an hour) with an NPV of 1944930000, with the main difference being a much shorter depletion period of only 10 days, which is the specified minimum (spe5d_optn4.dat, spe5d_optn4.out): TD 10 DTWF 733.2169148921967 DTWAG 9947.705966472626 DTWAGW 1.0603598713874816 DTWAGG 125.81382192373273 DTBD 661.9549714326859 BHPWFI 4706.688094139099 BHPWAGP 3737.332844734192 BHPWAGI 5689.139604568481 TWF 743.2169148921967 TWAG 10690.922881364822 TEND 11352.877852797508 NPV 1944930000
Both spe5d_opt5 and spe5_dopt2 solutions might be improved slightly but are tightly converged to optimal solutions, which is confirmed with declining NPV upon small perturbations of the variables. In spe5d_dopt2, the extremely small optimal water injection phase time of 3.2521 days (with gas injection time of 483.77 days) for the WAG flood and its strong effect on NPV is very surprising, but accurate. Values of 3.2 and 3.3 days both give reduced NPV values of 1.929770E9 and 1.928870E9, respectively (reductions of about 0.8%). We believe that spe5d_opt5 represents the globally optimal solution. The first-contact miscible pressure for the oil and the specified injected solvent is given by the maximum saturation pressure on the solvent/oil phase diagram printed n the Sensor output file, and is equal to 3874.8 psia. It is interesting that the optimal minimum bottomhole pressure for the producer (3755 psia, first determined accurately in spe5d_opt2) is only just below this fcm pressure, and not even close to the given lab-measured slimtube MMP of 3000 psia. Any (miscible) simulator assuming a reservoir MMP applied to Scenario 2 of the original spe5 case is incorrect. Incorrect application of lab-measured slimtube MMP to field operations is likely the main reason for their failures to achieve high miscible recovery. The final oil saturation map shown below for the optimized case shows that miscible recovery was achieved in most of the reservoir. This is not true in any of the original Spe5 scenarios (counter to statements in the paper). Final oil and gas recoveries in spe5d_opt5 are 100.9% and 17.8%, respectively, and final average reservoir pressure is 1002 psia. Very high miscible recovery is achieved in the confined model, just as might be achieved from interior wells in a pattern miscible flood. This case might be considered to be a model of a quarter-5 spot pattern, with the well rates representing partial well values. To do so rigorously for this example problem, only the well PI values and block sizes on the boundary need adjustment for the partial wells placed along the outer lateral boundaries of half and quarter blocks that are created using our EDGE option. The constant value of the lateral block size can then be added as an optimization variable in order to additionally optimize well spacing for the simultaneous pattern flood. We have long known that simultaneous pattern operation is generally required to optimize recovery from our petroleum reservoirs. SO
TIME = 11735.9 DAYS DATE: 0 0 0 MAP WINDOW 1
The SensorCast workflow is very simple and consists only of the Sensor datafile being executed by Sensor to produce its ascii text output file. The optimization variables are graphically linked to their values in Sensor.dat and the NPV objective function is graphically linked to its reported value at the end of Sensor.out:
Example 1b What is the value of the final blowdown period in the optimized solution of Example 1? Answer: Fort.75 from run spe5d_opt5 gives NPV vs. time output from the economic model, showing that NPV increased by $27.03 million in the final 705.4 days of the run during blowdown.
1. Killough, J. E., "Fifth Comparative Solution Project: Evaluation of Miscible Flood Simulators", SPE 16000 presented at the Ninth SPE Symposium on Reservoir Simulation held in San Antonio, Texas, February 1-4, 1987
* This example problem was first presented in December 2020 in the SPE Connect Reservoir Group Discussion of "Artificial Intelligence / Machine Learning Problem Solving Methods" in which it was proven that no such useful methods exist in reservoir engineering, invalidating very many published claims, and SPE technical sections and courses. A spreadsheet giving NPV results for some number of realizations of the unknowns was given with any number offered, but nobody was able to predict an optimal solution.
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