Testing of the two permeability estimation methods proposed above in the situation of noise present in the input pressure was performed on a range of simple 2D models. The models exhibit different kinds of permeability heterogeneity: high permeable channels, low permeable zones and partially sealing faults. Each model has two wells – producer and injector – located at the opposite edges of the grid, for which a constant BHP control is set. The initial reservoir pressure for all models is 15 MPa, and the BHP at producer and injector are 10 MPa and 20 MPa respectively.
The exact pressure maps taken from the simulation models were perturbed by adding the Gaussian correlated random noise shown in Figure 3.8. The noise peak values are within ± 1 MPa, and the RMS value is 0.31 MPa. Since both methods #1 and #2 failed to work with the perturbed pressure map, some map smoothing was done prior to running the procedures. The degree of smoothing was taken the same for all three cases.
Figure 3.8 Correlated random noise added to the pressure maps.
The comparative performance of permeability estimation methods #1 and #2 for three simple models is shown in Figure 3.9 (these three models are also considered in the next section, so for consistent referencing they are labelled as Case 1, Case 9, Case 11 on the figure). When examining the different models, the magnitude of the pressure noise introduced should be compared with the magnitude of the total pressure signal. The total signal for each case equals the pressure map from the model minus the initial pressure, i.e. PPinit P15MPa, since it is pressure difference which is inferred from the time-lapse seismic.
Case 1 is a reservoir with a high permeable channel between the wells, which is not connected to them. The RMS of the pressure signal (signal = P – 15 MPa) equals 1.28 MPa, so the noise shown in Figure 3.8 constitutes 28% of the signal in terms of their RMS ratio. The high permeable channel is resolved reasonably well by method #1 using the exact pressure map, although some artefacts appear. Applying the method for the perturbed (and smoothed) pressure converts the channel into a feature that connects the two wells, has relatively small permeability, and inherits its geometry from the streamlines pattern. The method essentially failed here. Method #2 with the exact pressure input is good in resolving the channel geometry, however the overall permeability values become somewhat underestimated. This is likely to be the consequence of the regularisation, since the channel itself does not connect to the wells and has a small impact on the pressure map. Running algorithm #2 with the perturbed pressure input results in a failure to estimate permeability anywhere except for the wells vicinity.
Case 9 is a reservoir with a more complex channel picture, where the channels connect the two wells. The RMS of the pressure signal is 1.81 MPa, so the noise/signal ratio equals 17%. Method #1 with exact pressure resolves the part of channels connecting the wells, but does not resolve the remaining channels since there is almost no fluid flow taking place there, and consequently almost no streamlines. When the same method is run with the perturbed pressure, it also resolves the channel to a certain extent. The channel now takes its geometry from the streamlines bundle. Algorithm #2 with the exact pressure works well in estimating the permeability map. Since the high permeable feature connects the wells in this model, the algorithm resolves the channels even for the perturbed pressure input (middle bottom map). However, as will be shown below, a visually appealing picture does not mean a good reproduction of the historic fluid flow.
Case 11 is a homogeneous model with a sealing fault which baffles approximately 80% of the cross-sectional area between the wells. The RMS of the pressure signal for this model equals 2.49 MPa, which corresponds to 12% noise/signal ratio. Algorithm #1 with exact pressure produced a decent estimation for the part of the reservoir covered by the streamlines. Its application to the perturbed pressure also resulted in a sensible estimate for the lower part of the map. Again, the imprint of the streamlines geometry is clearly seen on the permeability features. The presence of the fault cannot be inferred from the estimated permeability map, instead, the map shows a channel going through the lower part of the map connecting the two wells. Method #2 with the exact input
gives a good quality permeability estimate, and the fault position can be inferred from it. Supplying the perturbed input pressure for the method led to poor permeability estimation, which is only valid in the immediate vicinity of the wells.
Case 1, P noise 28% Case 9, P noise 17% Case 11, P noise 12%
E x ac t p er m ea b ilit y 50 mD 500 mD 0 mD 100 mD 50 mD sealing fault Me th o d # 1 , ex ac t P Me th o d # 1 , p er tu rb ed P Me th o d # 2 , ex ac t P M eth o d # 2 , p er tu rb ed P
Figure 3.9 Testing of permeability estimation methods #1 and #2. Columns: different models. Rows, top to bottom: exact permeability, estimation #1 from the exact pressure, estimation #1 from the perturbed pressure, estimation #2 from the exact pressure, estimation #2 from the perturbed pressure.
Further testing of the permeability maps resulting from the noisy pressure input was conducted for Case 9 which gave the most favourable results above. To do the testing, the maps (rows 3, 5 in the middle column in Figure 3.9) were supplied to the simulation model which was run with the constant rate controls at wells to ensure the same injected and produced volumes as in the original exact model. By doing this, I look at how well the resolved lateral variations of permeability allow reproduction of the historic saturation front propagation, whereas reproduction of the historic well BHP’s could then be achieved by adjusting the average reservoir permeability and the well skin factors. The simulation results are shown in Figure 3.10.
Figure 3.10 Saturation front prediction at step 2 using the permeability maps estimated from the noisy pressure for Case 9. Left: method #1, right: method #2. The maps show the predicted saturation, the solid black line shows the reference exact saturation front.
Method #2 which relies only on the pressure map produced a permeability map which visually resembles the underlying exact permeability, but gives a rather poor prediction of the saturation front. Method #1 also accounts for the saturation when making estimates, and despite the estimated permeability does not look quite similar to the underlying one, the saturation front prediction looks better, although there is space for further improvement. Thus, usage of the saturation information by method #1 results in better saturation forecasts, as opposed to method #2.
From the testing conducted, I can conclude the following. Method #1 with noiseless input works reasonably well, however errors may emerge in the velocity analysis, as was discussed in the previous chapter. A noisy pressure map requires smoothing for the method to work, and the stronger the smoothing applied, the closer the resulting map to the pressure map of a homogeneous model. In the noisy case method #1 can resolve the permeability features which follow the streamlines direction, as we saw for Cases 9 and
11. The map of permeability calculated in this way will however have a strong imprint of the streamlines geometry. The ability of the method to work in the “noisy” circumstances results from the usage of the saturation maps in addition to pressure. For the same reason one may expect better prediction of the saturation fronts compared to method #2. Method #2 is generally good for the exact input pressure taken from a 2D model, although it may somewhat fail in resolving the permeability features located far from the wells and not connected to them (cf. Case 1). For the perturbed input the method works notably worse. It either estimates permeability only close to the wells (cf. Cases 1, 11), or, even if the permeability estimation looks promising (cf. Case 9), it still proves un-sucessful in predicting the saturation fronts.
The levels of noise I considered in this exercise range from 12% to 28%, which is quite low compared to the errors one may expect for the pressure maps inverted from seismic. Method #2, as additional testing revealed, may start failing for the noise being as low as 5%. High sensitivity to the pressure noise could be a consequence of the implementation of this particular algorithm. On the other hand, this could be due to a more fundamental restriction of resolving the permeability map from a pressure map. In the following section I will examine the latter possibility and address the question of the information content of the pressure map.