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SensorPx Example 1 (input data files are spe1*.mspx, spe1*.dat) If you have not made Makespx the default program to open .mspx files and SensorPx the default program to open .spx files, do so as indicated in Section 5.1 after copying your data files to the work directory. This example is based on spe1, and is a black oil case of gas injection in a 10 by 10 by 3 grid. We attempt to quantify uncertainty in oil recovery (and all other output variables) due to specified uncertainties in the entire input permeability and porosity fields (900 total unknowns), and then attempt an optimization of well completion. The questions we wish to answer are: 1. How accurately can we quantify (forecast) production in stochastic (Monte-Carlo) modeling, as a function of the number of scenarios considered, for the given set of uncertainties? 2. Can we sufficiently quantify uncertainty in this example to determine how to best complete the injector (vertically), given its areal location and the uncertainties in the input reservoir description? Consider completion options (a) as originally specified in layer 1 (for which it is assumed we have log and core data), (b) bottom layer only, (c) all layers, and (d) bottom two layers only. The Sensor data files that are input to Makespx and that specify completion options a, b, c, and d are spe1a.dat, spe1b.dat, spe1c.dat, and spe1d.dat, respectively. These data files use the Sensor Uncertain Inputs features described below in order to generate any desired number of cases having equally probable values and combinations of the uncertain descriptive reservoir variables that are randomly populated from their input probability distributions.
Upon completion of the above, the SensorPx output file spe1a.log will open in Notepad, and binary output files spe1a.p10, spe1a.p50, and spe1a.p90 will be written. The results of all Sensor runs made will be in the results subdirectory of the work directory, named casen.out and fortn.61, n=nadd+1,nadd+ncases, where ncases is the total number of Sensor cases specified to be run under CASES, and nadd is the total number of existing results files added with any ADD specifications, in the .mspx data file. The output files of any Sensor cases that fail to run to completion are written to the badruns subdirectory, and corresponding error messages will appear in the output SensorPx .log file. The first Makespx input data file considered in this example is spe1a1k.mspx. The data file specifies that SensorPx will compute P10, P50, and P90 results for 1000 Sensor executions of the Sensor data file spe1a.dat in 8 simultaneous sets of (1000 / 8 =) 125 sequential runs: 10 50 90 8 CASES spe1a.dat 1000 END By default, exceedance probabilistic values are computed. The CUM option can be used to instead compute cumulative probabilistic values: Exceedance Probability: There is at least an x% chance that the value of the variable will be greater than or equal to its exceedance Px value (it is greater than or equal to that value in at least x% of runs). The exceedance P10 value of a variable is a high number, and P90 is a low number. Cumulative Probability: There is at least an x% chance that the value of the variable will be less than or equal to its cumulative Px value (it is less than or equal to that value in at least x% of runs). The cumulative P10 value of a variable is a low number, and P90 is a high number. The only differences between the spe1a.dat Sensor case and the original spe1.dat case are the Uncertain Inputs specified in spe1a.dat for the permeability and porosity distributions, and the change of the BHP of the injector from 10000 to 9000, to avoid the effects of a negative oil compressibility pvt data error. The originally specified porosity and permeability values, constant by layer, are assumed to represent the mean values. The value of r specified in the data shown below is the ratio of the standard deviation to the mean, for a normally distributed variable. In general (applying to both normal and log-normal distributions of a variable X), the input value of r is defined as r = (Xmean – X-1sd) / Xmean (5.1) where X-1sd is the value of the variable at one standard deviation below the mean (r was defined by Dykstra and Parsons as the ‘coefficient of permeability variation’ for log-normal distributions of permeability). Each execution of the Sensor data file spe1a.dat will generate an equally probable combination of the uncertainties that are populated according to their specified distributions, using a random number generator. The well block properties are assumed to be less uncertain (they have a much smaller entered value of r) because of the assumption of existing log and core data where they have been completed as originally specified:
C i1 i2 j1 j2 k1 k2 mean r min max DISTRIBUTE KX KY 1 10 1 10 1 1 500 .5 0 10000 DISTRIBUTE KZ 1 10 1 10 1 1 50 .5 0 10000 DISTRIBUTE KX KY 1 10 1 10 2 2 50 .5 0 10000 DISTRIBUTE KZ 1 10 1 10 2 2 50 .5 0 10000 DISTRIBUTE KX KY 1 10 1 10 3 3 200 .5 0 10000 DISTRIBUTE KZ 1 10 1 10 3 3 19.23 .5 0 10000 DISTRIBUTE POROS 1 10 1 10 1 3 .3043 .2 0 1 C ASSUME WELL BLOCK PROPERTIES ARE LESS UNCERTAIN DISTRIBUTE KX KY 1 1 1 1 1 1 500 .01 0 10000 DISTRIBUTE KZ 1 1 1 1 1 1 50 .01 0 10000 DISTRIBUTE KX KY 10 10 10 10 3 3 50 .01 0 10000 DISTRIBUTE KZ 10 10 10 10 3 3 50 .01 0 10000 DISTRIBUTE POROS 1 1 1 1 1 1 .3043 .01 0 10000 DISTRIBUTE POROS 10 10 10 10 3 3 .3043 .01 0 10000
For the first POROS distribution specified above (normal by default), applying to all but the well block porosities, the standard deviation is (r*mean=) 0.06043, and 99.74% of a large number of properly populated porosity values should lie between 3 standard deviations of the mean, which is between 0.1230 and 0.4856. We wish to examine the accuracy of our probabilistic estimates as a function of the number of cases considered. Since pore volume (porosity) is treated as uncertain, and for the sake of simplicity, we will focus only on cumulative oil recovery here, rather than on fractional recovery. In a more complex example we might examine the variance of all probabilistic cumulative production and injection values. Cumulatives are recommended, rather than rates, which are averages over specific timesteps – rates are reported at the end of each timestep, but in any analysis of them, rates should be taken from the rate of change of the cumulatives, (Cn+1-Cn)/(tn+1-tn), and should be considered to apply at the middle of each common reported time period (which may include more than one timestep in any given Sensor case, since timestepping will generally vary by case). Ideally, we would consider all cumulative production/injection figures and include an economic model to compute Px,y,z results of net present value (NPV). That can easily be done with a separate economics package, and an integrated economics model is planned as a future enhancement of SensorPx.
Figure 5.1 shows the results of 3 executions of spe1a1k.mspx to compute P10, P50, and P90 forecasts for 1000 runs of Sensor case spe1a.dat. The large deviations in Px,y,z results for the 3 sets of runs indicates than many more runs are needed to accurately quantify uncertainty in cumulative oil production.
Figure 5.1
Figures 5.2 through 5.5 show the results of 3 executions of spe1a10k.mspx, spe1b10k.mspx, spe1c10k.mspx, and spe1d10k.mspx to compute P10, P50, and P90 forecasts for 10000 runs of Sensor cases spe1a.dat, spe1b.dat, spe1c.dat, and spe1d.dat, respectively. The good reproducibility of Px,y,z results for the 3 sets of runs in all cases indicates that accuracy may be sufficient in order to differentiate between the probable performance of the completion options a, b, c, and d. That of course depends on how strongly the results are affected by the operational options. We can make reliable optimizations only if the variations in probabilistic results due to operational alternatives are significantly greater than the estimated errors in those results. We can estimate those errors by examining the reproducibility of probabilistic results computed from the results of multiple sets of cases having different random realizations of the unknowns.
Figure 5.2 (option a)
Figure 5.3 (option b)
Figure 5.4 (option c)
Figure 5.5 (option d)
Figure 5.6 Comparison of P10, P50, and P90 cumulative oil production for options a,b,c,d
Figure 5.6 compares probabilistic cumulative oil production for the last set of 10000 runs made for each of the 4 options (the gold curves in Figures 5.2 – 5.5). It appears that option c offers slightly better probabilistic performance than option d (completion in layers 2 and 3), which is the second best option, and that the originally specified top layer completion is the worst possible choice. But the differences between option c and d probabilistic results are small, and are possibly within the margin of error, based on these sets of 10000 runs. In order to verify that case c is optimal, and to examine the further increase in accuracy of predictions and our ability to make optimizations regarding operational options having less effect on results than these options do, 3 sets of 100000 runs were made comparing options c and d (Makespx data files are spe1c100k.mspx, spe1d100k.mspx). The results are shown in Figure 5.7 and confirm our conclusions that option c, completion of the injector in all layers, is most likely to be the optimal choice. The probabilistic results of the 3 sets of runs for case c and for case d are graphically indistinguishable, while all case c results clearly exceed those for case d by a small margin. End-of-run Px,y,z figures are also given in Table 5.1. The Px,y,z figures at end-of-run for case c differ by a maximum of 0.3%, while case d values vary up to a maximum of 0.4%, and the case c values are all higher than the case d values, by an average of about 1%. The end-of-run P90 figure averages indicate that the minimum cumulative oil production in the highest producing (best) 90% of the cases c runs exceeds that in the case d runs by about 1%, which is a significant figure of about 0.5 million barrels.
Figure 5.7 P10, P50, P90 cumulative oil production for 3 sets of 100000 runs, cases c and d
The maximum deviations observed at end-of-run in our 3 sets of 10000 runs for cases c and d (shown in Figs. 5.4 and 5.5 and also from Table 5.1) are about 1.3% and 1.1%, respectively. So with an order of magnitude increase in the number of runs, from 10000 to 100000, we achieved only about a 3-fold increase in the accuracy of our probabilistic forecasts, with a corresponding approximate 3-fold increase in our ability to distinguish between the relative performance of operational options. We saw approximately the same increase in apparent accuracy (based on variation in results of only a few sets of runs) of probabilistic results when going from 1000 to 10000 runs. Table 5.1 gives data and output file names along with their predicted P10, P50, and P90 values of cumulative oil recovery at end of run (10 years) for all executions of Makespx/SensorPx in this example. These tabular end-of-run results are taken from the “FINAL FIELD RESULTS” table in the SensorPx output .log files. An example from the SensorPx output file spe1a1ka.log from our first execution of spe1a1k.mspx / runs.bat / runspx.bat is shown in Table 5.2 (your results will differ). Table 5.1 End-of-Run Px,y,z CUMOIL values for all Example 1 MakeSpx/SensorPx Executions
Table 5.2 FINAL FIELD RESULTS VARIABLE TIME PX PY PZ (CASE) (CASE) (CASE) ------------------------------------------------------------------------ QOIL 0.3650000E+04 0.1088708E+05 0.8912476E+04 0.7593289E+04 36 384 518 QWAT 0.3650000E+04 0.0000000E+00 0.0000000E+00 0.0000000E+00 0 0 0 QGAS 0.3650000E+04 0.8936689E+05 0.8111063E+05 0.7603425E+05 454 132 420 QWI 0.3650000E+04 0.0000000E+00 0.0000000E+00 0.0000000E+00 0 0 0 QGI 0.3650000E+04 0.1000000E+06 0.1000000E+06 0.1000000E+06 183 1000 256 WCUT 0.3650000E+04 0.0000000E+00 0.0000000E+00 0.0000000E+00 0 0 0 GOR 0.3650000E+04 0.1197446E+05 0.9151039E+04 0.7008436E+04 523 312 271 CUMOIL 0.3650000E+04 0.5039073E+05 0.4666276E+05 0.4213393E+05 927 233 650 CUMWAT 0.3650000E+04 0.0000000E+00 0.0000000E+00 0.0000000E+00 0 0 0 CUMGAS 0.3650000E+04 0.1970040E+06 0.1707808E+06 0.1419334E+06 120 848 525 CUMWI 0.3650000E+04 0.0000000E+00 0.0000000E+00 0.0000000E+00 0 0 0 CUMGI 0.3650000E+04 0.3602181E+06 0.3496366E+06 0.3307256E+06 196 225 397 OILREC 0.3650000E+04 0.1796322E+02 0.1634342E+02 0.1486434E+02 121 635 650 GASREC 0.3650000E+04 -0.4398073E+02 -0.4905497E+02 -0.5189258E+02 102 104 268 PAVGHC 0.3650000E+04 0.8354913E+04 0.8259162E+04 0.8124512E+04 502 41 793 PAVG 0.3650000E+04 0.8354880E+04 0.8259124E+04 0.8124478E+04 502 41 793 QGLIFT 0.3650000E+04 0.0000000E+00 0.0000000E+00 0.0000000E+00 0 0 0 CUMGLIFT 0.3650000E+04 0.0000000E+00 0.0000000E+00 0.0000000E+00 0 0 0 The case number from which the computed Px,y,z results were taken is given below each reported result. Note that except for the two virtually identical average pressures, no two Px, Py, or Pz results are from the same case. There is no such thing as a Px, Py, or Pz case – only the probabilistic results have any significant meaning, and none of the individual cases indicated can possibly represent any probabilistic behavior or description, beyond the singular Px, Py, or Pz result that they predict at the reported time(s). However, we may be able to find many cases that match P50 oil, gas, and water production/injection results fairly well, and by tuning the Uncertain Inputs to match the p50 results to historical production, we can maximize the rate at which history matches can be found stochastically. Those found cases will have very different descriptions, as demonstrated in Example 4. Large numbers of history matches can then quantify uncertainty in predictions that are made from them. |
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