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Figure 4.11: Average porosity prediction error was decreased from 8% to 4% with the combined use of the output functional links kh and k/g. — Figure 4 11: Average porosity prediction error was decreased from 8% to 4% with the combined use of the output functional links kh and k/g.

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passed between neurons over the connecting links as shown in Figure 1.1. Figure 1.1: Three-layer neural network. Inputs are in the first layer and the outputs are in the last layer of the network. The middle layer provides the connecting link between the input and output layers. the processing elements that are called neurons which are grouped into layers. Signals are

Figure 1.2: The simulation model is discretized into a grid and the simulations are run. The properties for k, @ and h are varied and simulations done that give different production profiles for each combination of properties.

Figure 1.3: Differences between the numerical simulations and the historical field performance at specific times are input to the network for training. The varied properties (k, @ and h) for each simulation run are used as the output data in the training process. An ANN is constructed so that the desired outputs can be obtained. numerical simulations (Figure 1.3). in mapping the disparities between field performance and the results generated by

Figure 1.4: A new data set that consists of small disparities between the numerical model and the historical field performance is input to the expert system developed. The output from this expert system is a combination of k, @ and h that are representative of the reservoir. Figure 1.4: A new data set that consists of small disparities between the numerical model disparities between the numerical model and the field performance (Figure 1.4).

Figure 1.5: The ANN predicted properties (k, @ and h) obtained from the expert system are input to the simulation model. This numerical simulation should give an improved history match. network has been successful there should be an improved history match.

Figure 2.1: The 16x16 grid block system used to formulate the training data for network development. The grid shows the initial fluid distribution in the reservoir. This is a multi- phase reservoir with one homogeneous property region and one producing well.

Figure 2.2: The 16x16 grid block system used to formulate the training data for network development. The grid shows the initial fluid distribution for the multi-phase reservoir with four homogeneous regions and one producing well.

Figure 2.3: Transmissibility in different regions of the reservoir. The Channel, Region 2, has the least permeability and transmissibility in the reservoir as a result of the limited communication across the channel.

Figure 2.4: The 16x16 grid block system used to formulate the training data for network development. The grid shows the initial fluid distribution for the multi-phase reservoir with four homogeneous regions and three producing wells.

Figure 4.1: Average training errors for each property using different data structures and different numbers of neurons in the hidden layers of a 4-layer neural network. was larger than 25, no improvements were observed.

Figure 4.3: Structure of the back-propagation network (BPN) used in the case study of a single-region, square homogeneous property reservoir with one producer. The delta values for each case are at specific times, as a result, there are multiple delta values for each case. neurons in each of the layers as shown in Figure 4.3. algorithm. In this case, it was determined that the network should have 4, 20, 25, 25 and 3

Table 4.4: Randomly generated properties for 35 cases used in the reservoir simulations for generating training data for tt reservoir with four regions.

Table 4.5: Two reservoir properties and the differences between field performance and the base case simulation were used as inputs. One reservoir property was predicted. The output functional links were used as forcing functions in training the network. Table 4.6: One reservoir property and the differences between field performance and the base case simulation were used as inputs. Two reservoir properties were predicted. The output functional links were used as forcing functions in training the network.

Figure 4.4: Reservoir property prediction errors using two properties as input data while the third property was predicted.

Figure 4.5: Reservoir property prediction errors using one property as input data while the other two properties were predicted.

Figure 4.6: Average training errors with and without time as an input in the training data using a 4-layer neural network and 25 neurons in the hidden layers. Time used as an input decreased the average training errors.

Table 4.7: Inputs used to train the network. These inputs are at specific times for each case

the exception of the porosity predictions that were not significantly affected.

Figure 4.8: Average training errors with and without the input functional links: time(yrs) and Aqo*time(yrs). The training errors for permeability, porosity and thickness were decreased with the use of these input functional links. functional links improved the average training and testing errors (Figure 4.8 & 4.9).

Figure 4.9: Average testing errors with and without the input functional links: time(yrs) and Aqo*time(yrs). The testing errors for permeability, porosity and thickness were decreased with the use of these input functional links.

Figure 4.12: Average thickness prediction error was decreased from 33% to 20% with the combined use of the output functional links kh and k/g.

Figure 4.13: Permeability prediction errors with and without the output functional links: region’s area/region’s permeability and closest and furthest distance from the well to the boundaries/region’s permeability. Both functional links decreased the prediction errors. chosen that resulted in lower average prediction errors of 13%

Figure 4.14: Porosity prediction errors with and without the output functional links: region’s area/region’s permeability and closest and furthest distance from the well to the boundaries/region’s permeability.

Figure 4.16: Prediction errors of k, @ and h using 2, 3, 4 and 5 hidden layers in the network with varying output functional links. Inset: Average prediction errors of the properties k, @ and h. The 4-hidden layer network gave the lowest average prediction errors.

the properties of the reservoir are unknowns. serve as a forcing function in the output. The eigenvalues were all tried as outputs since

where D;, D2, D3, and D4 are the distances from the center of the producing well to the boundaries of the reservoir.

Figure 4.17: Structure of the back-propagation network (BPN) used in the case study of a square homogeneous reservoir with four regions and one producer. The network predicts k, 6 and h for each region. the properties were obtained using this network. each region using 7 inputs and 37 outputs are shown Figure 4.17. Satisfactory results for

Figure 4.20: Actual and predicted thickness values utilizing the final network. The use of the eigenvalue as an output functional link further decreased the prediction errors.

Figure 4.21: Average prediction errors of permeability, porosity and thickness using different numbers of hidden layers and varying functional links in the network. The network with six hidden layers gave the lowest prediction errors for k, @ and h. the network are described in Tables 4.9 & 4.10.

Figure 4.22: Comparison of the base case permeability with the ANN-predicted permeability values using the output functional link permeability thickness. Figures 4.22, 4.23 and 4.24.

Figure 4.25: Comparison between the historical profile and the simulated profile obtained from scenario 1. The graphs show large deviations from the historical profile, the largest being observed in the water production.

Figure 4.27: Comparison between the historical profile and the simulated profile obtained from the ANN-predicted properties. There is a good history match for the oil and gas production as well as the pressure profile. The history match for water production is improved but it is not as close a match as the oil and gas production profiles.

Figure 5.1: Initial fluid distribution and the location of the producing wells in the M4.1 reservoir shown on a curvilinear grid. is the result of case 36 shown in Table 5.1.

Table 5.3: Inputs and outputs used in training the network for the single-region homogeneous property reservoir.

Table 5.4: Final inputs and outputs used in training the network for the single-regior homogeneous property reservoir.

Figure 5.2: Architecture of the neural network used to predict the properties of the single-region M4.1 reservoir.

Figure 5.3: Training errors for k, o and h for 38 simulated cases. The average training error for each property was 15%, 3% and 7% respectively, with an overall average training error of 8%.

Figure 5.4: Testing errors for k, @ and h for 10 simulated cases. The average testing error for each property was 11%, 3% and 8% respectively with an overall average testing error of 7%.

Figure 5.5: Simulated production profiles for the single-region M4.1 reservoir using the ANN-predicted properties compared to the historical production data.

Figure 5.6: The four delineated regions in the M4.1 reservoir Region North of Fault A, the Channel, the East Levee and the W on a curvilinear grid: est Levee respectively. The regions were delineated according to compartmentalization and varying permeability in each region. The Channel, Region 2, was simulated as having the least transmissibility and permeability due to the restricted communication between the | East and West Levees.

‘able 5.5: Randomly generated properties used in the 38 simulation runs for formulating the training data for the reservoir w our regions.

Table 5.6: Randomly generated properties used in the 10 simulation runs for formulating the testing data for the reservoir with fot regions.

Figure 5.8: Training errors for k, @ and h in each region of the reservoir. The overall average training error was 35%.

Figure 5.9: Testing errors for k, 9 and h in each region of the reservoir. The average testing error was 20%.

additional functional links, which can be explored in future studies.

Table 5.7: Predicted properties for each region of the M4.1 reservoir using two training data sets. The training data w formulated using a narrow data range with 38 simulations and a wide data range with 53 simulations. The ANN-predict properties using both networks were similar.

Table 5.8: Randomly generated values for the GOCs and OWCs used in 38 simulations for formulating the training data for the reservoir with four regions.

Table 5.9: Randomly generated values for GOCs and OWCs used in 10 simulations for formulating the testing data for the reservoir with four regions.

Table 5.10: Inputs and outputs to the network used in the prediction of the properties k. 8, h, GOC and OWC.

Table A.1: Rock and fluid data used in the initialization of the artificial neural network (ANN) model.

Table A.2: End point input data used to generate relative permeability curves based on Corey’s Model for the water-oil system.

Figure A.1: Relative permeability to oil and water in a two-phase system.

Figure A.2: Relative permeability to oil and gas in a two-phase system.

Figure A.3: Capillary pressure curve in the oil-water system.

Figure A.4: Capillary pressure curve in the gas-oil system.

Table A.5: Black oil PVT data derived from fluid analysis in the M4.1 reservoir. Table A.6: Black oil PVT data used in the reservoir simulator to design the ANN model.

Table D.1: Rock and fluid data used in the initialization of the M4.1 simulation model.