The reservoir model developed in the previous two sections will now be applied to history-matching production data from a well in a volumetric, internal gas-drive reservoir. Actual oil production and instantaneous gas-oil ratios for the first 3 years of the life of the well are plotted in Fig. 12.1. The data for the problem were obtained from personnel at the University of Kansas and are used here by permission.14
The well is located in a reservoir that is sandstone and produced from two zones, separated by a thin shale section, approximately 1 ft to 2 ft in thickness. The reservoir is classified as a stratigraphic trap. The two producing zones decrease in thickness and permeability in directions where it is believed that a pinch-out occurs. Permeability and porosity decrease to unproductive limits both above and below the producing formation. The initial reservoir pressure was 620 psia. The average porosity and initial water saturation values were 21.5% and 37%, respectively. The area drained by the well is 40 ac. Average thicknesses and absolute permeabilities were reported to be 17 ft and 9.6 md for zone 1 and 14 ft and 7.2 md for zone 2. Laboratory data for fluid viscosities, formation volume factors, solution gas-oil ratio, oil relative permeability, and the gas-to-oil permeability ratio are plotted in Figs. 12.2 to 12.6.
Figure 12.2 Oil and gas viscosity for Schilthuis history-matching problem.
Figure 12.3 Oil and gas formation volume factor for Schilthuis history-matching problem.
Figure 12.4 Solution gas-oil ratio for Schilthuis history-matching problem.
Figure 12.5 Oil relative permeability for Schilthuis history-matching problem.
Figure 12.6 Permeability ratio for Schilthuis history-matching problem.
Table 12.1 Excel Functions Used to Calculate Fluid Property Data for the Schilthuis History-Matching Problem
Table 12.2 Excel Functions Used to Calculate Oil Relative Permeability Curve in the Schilthuis History-Matching Problem.
Solution Procedure
One of the first steps in attempting to perform the history match is to convert the fluid property data provided in Figs. 12.2 to 12.4 to a more usable form. This is done by simply regressing the data to create Excel functions in Microsoft Excel’s Visual Basic Editor for each parameter—for example, oil and gas viscosity as a function of pressure. The resulting functions can be found in the program listing in Table 12.1. The two permeability relationships also need to be regressed for use in the example.
Both the relative permeability to oil and the permeability ratio can be expressed as functions of gas saturation. These Excel functions are structured in a different manner from the fluid property equations. The constants for the regressed equations are placed in an array, shown in Tables 12.2 and 12.3, and a regression is performed between each set of points. These relationships are handled this way to facilitate modifications to the equations used in the program if necessary.
Table 12.3 Excel Functions Used to Calculate Gas to Oil Permeability Ratio Curve in the Schilthuis History-Matching Problem
With the data of Figs. 12.2 to 12.6 expressed in equation format, the history match is now ready to be executed. An example of the Excel worksheet is shown in Table 12.4.
Table 12.4 Excel Worksheet Used in the Schilthuis History-Matching Problem
The Excel sheet is laid out with the reservoir variables such as initial pressure, wellhead flowing pressure, reservoir area, and so on shown at the top of the page. Directly below are the zone-specific data of height, initial hydrocarbon in place, and porosity. Those values are totaled to provide a value for the entire well. Below are the reservoir properties at each average well pressure in 10-psi increments. To the right are the actual production values for the wells, and a comparison between actual and calculated is shown in the graph in the bottom corner. The equations for each of the cells are shown in Table 12.5.
The sheet requires a user-specified delNpguess value in order to determine the reservoir properties at a given pressure. Once those values are determined, the sheet automatically calculates a new delNp and an Np for that pressure. The delNpguess will need to be iterated until it is equal to delNp. To aid in this, the check column was created. At the end of the check column is a cell with the sum of the check values. Using Excel’s built-in solver tool, that cell can be iteratively solved for a minimum value by adjusting the delNpguess for each pressure increment. This allows the user to rapidly solve the set of equations in the Schilthuis balance and get the result at that set of conditions.
12.3.2.2 Discussion of History-Matching Results
When the program is executed using the original data given in the problem statement as input, the oil production rate and R, or instantaneous GOR, values obtained result in the plots shown in Fig. 12.7. Notice that the calculated oil production rates, shown in Fig. 12.7(a), begin higher than the actual rates and decrease faster with time or with a greater slope. The calculated instantaneous GOR values are compared with the actual GOR values in Fig. 12.7(b) and found to be much lower than the actual values.
At this point, it is necessary to ask how the calculated instantaneous GOR values could be raised in order for them to match the actual values. An examination of Eq. (10.33) suggests that R is a function of fluid property data and the ratio of gas-to-oil permeabilities. To calculate higher values for R, either the fluid property data or the permeability ratio data must be modified. Because fluid property data are readily and accurately obtained and the permeabilities could change significantly in the reservoir owing to different rock environments, it seems justified to modify the permeability ratio data. It is often the case when conducting a history match that an engineer will find differences between laboratory-measured permeability ratios and field-measured permeability ratios. Mueller, Warren, and West showed that one of the main reasons for the discrepancy between laboratory kg/ko values and field-measured values can be explained by the unequal stages of depletion in the reservoir.15 For the same reason, field instantaneous GOR values seldom show the slight decline predicted in the early stages of depletion and, conversely, usually show a rise in gas-oil ratio at an earlier stage of depletion than the prediction. Whereas the theoretical predictions assume a negligible (actually zero) pressure drawdown, so that the saturations are therefore uniform throughout the reservoir, actual well pressure drawdowns will deplete the reservoir in the vicinity of the wellbore in advance of areas further removed. In development programs, some wells are often completed years before other wells, and depletion is naturally further advanced in the area of the older wells, which will have gas-oil ratios considerably higher than the newer wells. And even when all wells are completed within a short period, when the formation thickness varies and all wells produce at the same rate, the reservoir will be depleted faster when the formation is thinner. Finally, when the reservoir comprises two or more strata of different specific permeabilities, even if their relative permeability characteristics are the same, the strata with higher permeabilities will be depleted before those with lower permeabilities. Since all these effects are minimized in high-capacity formations, closer agreement between field and laboratory data can be expected for higher capacity formations. On the other hand, high-capacity formations tend to favor gravity segregation. When gravity segregation occurs and advantage is taken of it by shutting in the high-ratio wells or working over wells to reduce their ratios, the field-measured kg/ko values will be lower than the laboratory values. Thus the laboratory kg/ko values may apply at every point in a reservoir without gravity segregation, and yet the field kg/ko values will be higher owing to the unequal depletion of the various portions of the reservoir.
Table 12.5 Equations Used in the Schilthuis History-Matching Problem
Figure 12.7 Schilthuis history match using original data.
The following procedure is used to generate new permeability ratio values from the actual production data:
1. Plot the actual R values versus time and determine a relationship between R and time.
2. Choose a pressure and determine the fluid property data at that pressure. From the chosen pressure and the output data in Table 12.4, find the time that corresponds with the chosen pressure.
3. From the relationship found in step 1, calculate R for the time found in step 2.
4. With the value of R found in step 3 and the fluid property data found in step 2, rearrange Eq. (10.33) and calculate a value of the permeability ratio.
5. From the pressure chosen in step 2 and from the Np values calculated from the chosen pressure, calculate the value of the gas saturation that corresponds with the calculated value of the permeability ratio.
6. Repeat steps 2 through 5 for several pressures. The result will be a new permeability ratio–gas saturation relationship.
In Excel, the solution resembles Table 12.6.
Table 12.6 Excel Worksheet Illustrating the Calculation of the New Permeability Ratio
The reader should realize that in steps 2 and 5 the original permeability ratio was used to generate the data of Table 12.4. This suggests that the new permeability ratio–gas saturation relationship could be in error because it is based on the data of Table 12.4 and that, to have a more correct relationship, it might be necessary to repeat the procedure. The quality of the history match obtained with the new permeability ratio values will dictate whether this iterative procedure should be used in generating the new permeability ratio–gas saturation relationship. The new permeability ratios determined from the previous six-step procedure are plotted with the original permeability ratios in Fig. 12.8.
Figure 12.8 First iteration of permeability ratio for history-matching problem.
It is now necessary to regress the new permeability ratio–gas saturation relationship and input the new data into the Excel worksheet before the calculation can be executed again to obtain a new history match. When this is done, the calculation yields the results plotted in Fig. 12.9.
The new permeability ratio data has significantly improved the match of the instantaneous GOR values, as can be seen in Figure 12.9(b). However, the oil production rates are still not a good match. In fact, the new permeability ratio data have yielded a steeper slope for the calculated oil rates, as shown in Figure 12.9(a), than what is observed in Figure 12.7(a) from the original data. A look at the calculation scheme helps explain the effect of the new permeability ratio data.
Figure 12.9 History match after modifying permeability ratio values.
Because the new values of instantaneous GOR were calculated with the new permeability ratio data, which in turn were determined by using Eq. (10.33) and the actual GOR values, it should be expected that the calculated GOR values would match the actual GOR values. The flow rate calculation, which involves Eq. (8.45), does not use the permeability ratio, so the magnitude of the flow rates would not be expected to be affected by the new permeability ratio data. However, the time calculation does involve Np, which is a function of the permeability ratio in the Schilthuis material balance calculation. Therefore, the rate at which the flow rates decline will be altered with the new permeability ratio data.
To obtain a more accurate match of oil production rates, it is necessary to modify additional data. This raises the question, what other data can be justifiably changed? It was argued that it was not justifiable to modify the fluid property data. However, the fluid property data and/or equations should be carefully checked for possible errors. In this case, the equations were checked by calculating values of Bo, Bg, Rso, μo, and μg at several pressures and comparing them with the original data. The fluid property equations were found to be correct and accurate. Other assumed reservoir properties that could be in error include the zone thicknesses and absolute permeabilities. The thicknesses are determined from logging and coring operations from which an isopach map is created. Absolute permeabilities are measured from a small sample of a core taken from a limited number of locations in the reservoir. The number of coring locations is limited largely because of the costs involved in performing the coring operations. Although the actual measurement of both the thickness and permeability from coring material is highly accurate, errors are introduced when one tries to extrapolate the measured information to the entire drainage area of a particular well. For instance, when constructing the isopach map for the zone thickness, you need to make assumptions regarding the continuity of the zone in between coring locations. These assumptions may or may not be correct. Because of the possible errors introduced in determining average values for the thickness and permeability for the well-drainage area, varying these parameters and observing the effect of our history match is justified. In the remainder of this section, the effect of changing these parameters on the history-matching process is examined. Table 12.7 contains a summary of the cases that are discussed.
In case 3, the thicknesses of both zones were adjusted to determine the effect on the history match. Since the calculated flow rates are higher than the actual flow rates, the thicknesses were reduced. Figure 12.10 shows the effect on oil producing rate and instantaneous GOR when the thicknesses are reduced by about 20%.
By reducing the thicknesses, the calculated oil production rates are shifted downward, as shown in Figure 12.10(a).
This yields a good match with the early data but not with the later data, because the calculated values decline at a much more rapid rate than the actual data. The calculated instantaneous GOR values still closely match the actual GOR values. These observations can be supported by noting that the zone thickness enters into the calculation scheme in two places. One is in the calculation for N, the initial oil in place, which is performed by using Eq. (12.3). Then N is multiplied by each of the ΔNp/N values determined in the Schilthuis balance. The second place the thickness is used is in the flow equation, Eq. (8.45), which is used to calculate qo. The instantaneous GOR values are not affected because neither the calculation for N nor the calculation for qo is used in the calculation for instantaneous GOR or R. However, the oil flow rate is directly proportional to the thickness, so as the thickness is reduced, the flow rate is also reduced. At first glance, it appears that the decline rate of the flow rate would be altered. But upon further study, it is found that although the flow rate is obviously a function of the thickness, the time is not. To calculate the time, an incremental ΔNp is divided by the flow rate corresponding with that incremental production. Since both Np and qo are directly proportional to the thickness, the thickness cancels out, thereby making the time independent of the thickness. In summary, the net result of reducing the thickness is as follows: (1) the magnitude of the oil flow rate is reduced, (2) the slopes of the oil production and instantaneous GOR curves are not altered, and (3) the instantaneous GOR values are not altered.
To determine the effect on the history match of varying the absolute permeabilities, the permeabilities were reduced by about 20% in case 4. Figure 12.11 shows the oil production rates and the instantaneous GOR plots for this new case. The quality of the match of oil production rates has improved, but the quality of the match of the instantaneous GOR values has decreased. Again, if the equations involved are examined, an understanding of how changing the absolute permeabilities has affected the history match can be obtained.
Table 12.7 Description of Cases
Figure 12.10 History match of case 3. Case 3 used the new permeability ratio data and reduced zone thicknesses.
Figure 12.11 History match of case 4. Case 4 used the new permeability ratio data and reduced absolute permeabilities.
Equation (8.45) suggests that the oil flow rate is directly proportional to the effective permeability to oil, ko:
Equation (12.4) shows the relationship between the effective permeability to oil and the absolute permeability, k. Combining Eqs. (8.45) and (12.4), it can be seen that the oil flow rate is directly proportional to the absolute permeability. Therefore, when the absolute permeability is reduced, the oil flow rate is also reduced. Since the time values are a function of qo, the time values are also affected. The magnitude of the instantaneous GOR values is not a function of the absolute permeability, since neither the effective nor the absolute permeabilities are used in the Schilthuis material balance calculation. However, the time values are modified, so the slope of both the oil production rate and the instantaneous GOR curves are altered. This is exactly what should happen in order to obtain a better history match of the oil production values. However, although it has improved the oil production history match, the instantaneous GOR match has been made worse. By reducing the absolute permeabilities, it has been found that (1) the magnitude of the oil flow rates are reduced, (2) the magnitude of the instantaneous GOR values are not changed, and (3) the slopes of both the oil production and instantaneous GOR curves are altered.
By modifying the zone thicknesses and absolute permeabilities, the magnitude of the oil flow rates and the slope of the oil flow rate curve can be modified. Also, while adjusting the oil flow rate, slight changes in the slope of the instantaneous GOR curve are obtained. In case 5, both the zone thickness and the absolute permeability are changed in addition to using the new permeability ratio data. Figure 12.12 contains the history match for case 5. As can be seen in Figure 12.12(a), the calculated oil flow rates are an excellent match to the actual field oil production values. The match of instantaneous GOR values has worsened from cases 2 to 4 but is still much improved over the match in case 1, which was obtained by using the original permeability ratio data.
Figure 12.12 History match of case 5. Case 5 used the new permeability ratio data and modified zone thicknesses and absolute permeabilities.
A second iteration of the permeability ratio values may be necessary, depending on the quality of the final history match that is obtained. This is because the procedure used to obtain the new permeability ratio data involves using the old permeability ratio data. The calculated instantaneous GOR values do not match the actual field GOR values very well, so a second iteration of the permeability ratio values is warranted. Following the procedure of obtaining new permeability ratio data in conjunction with the results of case 5, a second set of new permeability ratios is obtained. This second set is plotted in Figure 12.13, along with the original data and the first set used in cases 2 to 5.
Figure 12.13 Second iteration of permeability ratios for the history-matching problem.
By using the permeability ratio data from the second iteration and by adjusting the zone thicknesses and absolute permeabilities as needed, the results shown in Figure 12.14 are obtained. It can be seen that the quality of the history match for both the oil production rate and the instantaneous GOR values is very good. When a history match is obtained that matches both the oil production and instantaneous GOR curves this well, the model can be used with confidence to predict future production information.
Figure 12.14 History match of case 6. Case 6 used permeability ratio data from a second iteration and modified zone thicknesses and absolute permeabilities.
Table 12.8 Input Data for History-Matching Example.
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