Determination of porosity and permeability in reservoir intervals by artificial neural network modelling, offshore eastern Canada

Transcription

1 Determination of porosity and permeability in reservoir intervals by artificial neural network modelling, offshore eastern Canada Zehui Huang* and Mark A. Williamson Geological Survey of Canada (Atlantic), P.O. Box 1006, Dartmouth, NS, Canada B2Y 4A2 ABSTRACT: A quantitative integration of porosity/permeability measurements and well log data from the major reservoir intervals throughout the basin is carried out using a back-propagation artificial neural network (BP-ANN), modified with the Marquardt algorithm. After data preprocessing and training/ supervising example preparation, a model for the relationship among porosity, permeability and well log responses was established with the BP-ANN technique. The BP-ANN model was then used to construct profiles of porosity and permeability in both cored and uncored wells for the Avalon, Hibernia and Jeanne d Arc formations from well logs. The BP-ANN derived porosity and permeability curves provide a basis for further reservoir studies, such as inter-well permeable units recognition and correlation, and basin-wide reservoir quality evaluation. KEYWORDS: well logs, core analysis data, artificial neural network modelling, porosity (rock) and permeability (rock) INTRODUCTION The oil prone Jeanne d Arc Basin, offshore eastern Canada (Fig. 1), contains 15 significant hydrocarbon discoveries, including the Hibernia and Terra Nova oil fields. The sedimentology and diagenesis of the Hibernia Formation in the Hibernia oil field was examined by Brown et al. (1989). Hurley et al. (1992) reported the reservoir geology and geophysics of the same area, and Salem (1993) estimated physical properties for the Jeanne d Arc Formation in the Hibernia and Terra Nova areas using well logs on the basis of theoretical physical models and empirical relationships. Unfortunately, there are no integrated basin-wide studies on the porosity and permeability of the major reservoirs that quantitatively integrate information from both core and well logs. Such integrated studies are desirable because, (1) the number of core measurements are limited and unable to provide a complete picture of the reservoir intervals, and (2) porosity and permeability (k) (especially the latter) estimated from well logs just based on theoretical physical models or empirical equations from other basins tend to be erroneous because of the unique geological characteristics of each basin s reservoirs. The present study develops a quantitative model for predicting porosity and permeability from well logs, which are important to understanding this basin s petroleum system. Such an understanding will contribute to future exploration, reservoir modelling, reservoir development and management. The detailed quantitative integration of core and well log data presented in this paper is done by using artificial neural network (ANN) modelling techniques. An efficient back-propagation artificial neural network (BP-ANN) modified with the Marquardt algorithm was used. Although we model the relationships among well logs, porosity and permeability, *Present address: Geomath Suite 1125, 200 Westlake Park Blvd, Houston, TX 7707, USA. Petroleum Geoscience, Vol , pp our emphasis has been on estimation of permeability, because conventionally permeability is indirectly converted from porosity values estimated from well logs, using a specific porosity and permeability relationship. Fig. 1. The Jeanne d Arc Basin. Wells with both core measurements and well logs are shown in solid dots, otherwise in open circles /97/$ EAGE/Geological Society, London

2 246 Zehui Huang and M. A. Williamson Fig. 2. (a) Jurassic and Cretaceous lithostratigraphy of the Jeanne d Arc Basin, with reservoirs indicated (after McAlpine 1990). The portion marked by vertical lines in the formation column indicate stratigraphic gaps in a relative sense. (b) A summary on the Avalon, Hibernia and Jeanne d Arc formations, based on work by Brown et al. (1989), McAlpine (1990), Sinclair et al. (1994) and Enachescu et al. (1996). After careful data preprocessing and training/supervising example preparation, a model for the relationship among rock porosity, permeability and well log responses was established with the BP-ANN. The trained BP-ANN was used to estimate profiles of porosity and permeability from well logs for the Avalon, the Hibernia and the Jeanne d Arc formations. The model is also useful for other Mesozoic reservoir intervals (such as the Ben Nevis and Eastern Shoals formations). The porosity and permeability predicted from well logs using the BP-ANN provide a sound basis for detailed study of major reservoir intervals. GEOLOGICAL BACKGROUND The Jeanne d Arc Basin was formed as a result of Mesozoic extensional rift tectonics (McAlpine 1990), and is the deepest of the rift basins in the Grand Banks area (Enachescu 1987). The basin is divided into three structural stratigraphic segments (Fig. 1). South of the Egret Fault, uppermost Jurassic and Lower Cretaceous rocks are virtually absent because of erosion and depositional thinning (Enachescu 1987). Between the Egret Fault and the Hibernia to Ben Nevis trans-basin fault zone, this section thickens dramatically. North of the trans-basin fault zone, a thickened middle Cretaceous sequence has been preserved. Figure 2a shows the Jurassic and Cretaceous stratigraphy of this basin. The Avalon, Hibernia and Jeanne d Arc formations contain important reservoirs, which are mainly charged with hydrocarbons from the early Kimmeridgian Egret Member source rock (Fowler & McAlpine 1994). Figure 2b provides a summary of the three formations. THE DATABASE Data available for the study include digital well logs from the GSC Atlantic s database, and core data (porosity and permeability measurements) provided by Canada Newfoundland Offshore Petroleum Board. The occurrence of the Avalon, Hibernia and Jeanne d Arc formations, and the availability of core data and well logs are listed in Table 1. Although we focus on the Avalon, Hibernia and Jeanne d Arc formations, in order that our model is applicable to other formations and to make use of as much available information as possible, some of the data from the Ben Nevis, Eastern Shoals and Catalina formations were also used (Table 2). Permeability measurements are horizontal permeability to air. The number of porosity/permeability measurements vary from well to well and formation to formation (Table 1). Core data from the Avalon and Hibernia formations are dominated by wells in the Hibernia area. Those from the Jeanne d Arc Formation are mainly from Terra Nova wells. For the Avalon Formation, about 40% of the core data are from the Hibernia

3 Porosity and permeability in reservoir intervals 247 Table 1. Occurrence of the Avalon, Hibernia and Jeanne d Arc formations and availability of well log and porosity/permeability measurements in the Jeanne d Arc Basin s exploration wells* Avalon Fm Hibernia Fm Jeanne d Arc Fm Well Name Occurrence logs k & p Occurrence logs k & p Occurrence logs k & p Amethyst F-20 Y Y Y Y Y Y Archer K-19 Y Y Y Y Y Y Ben Nevis I-45 Y Y Y Y Beothuk M-05 Y Y Y Y Y Y Bonne Bay C-73 Y Y Y Y Y Y 18 E. Rankin H-21 Y Y Y Y 5 Egret K-36 Y Y Y Y Egret N-46 Y Y Y Y Flying Foam I-13 Y Y Y Y Fortune G-57 Y Y Y Y 102 Y Y Gambo N-70 Y Y Y Y Y Y Hebron I-13 Y Y Y Y 33 Y Y 57 Hibernia B-08 Y Y Y Y 94 Y Y Hibernia B-27 Y Y 162 Y Y 151 Y Y Hibernia C-96 Y Y 61 Y Y 139 Y Y Hibernia G-55 Y Y 26 Hibernia I-46 Y Y 889 Hibernia J-34 Y Y 280 Hibernia K-14 Y Y 214 Y Y 206 Hibernia K-18 Y Y 41 Y 173 Y Y Hibernia O-35 Y Y Y Y Y Y Hibernia P-15 Y Y Y Y Y 22 Mara M-54 Y Y Y Y Mercury K-76 Y Y Nautilus C-92 Y Y 36 Y Y Y N. Dana I-43 Y Y N. Ben Nevis M-61 Y Y 254 N. Ben Nevis P-93 Y Y 42 Y Y N. Trinity H-71 Y Y Y 45 Panther P-52 Y Y Port au Port J-97 Y Y Y Rankin M-36 Y Y S. Brook N-30 Y Y S. Mara C-13 Y Y 55 Y Y 8 Y Y S. Tempest G-88 Y Y Springdale M-29 Y Y Y Y Y Y Terra Nova C-09 Y Y Y Y Y Y 277 Terra Nova E-79 Y Y Y Y Y Y 279 Terra Nova H-99 Y Y Y Y Y Y 128 Terra Nova I-97 Y Y Y Y Y Y 78 Terra Nova K-07 Y Y Y Y Y Y 82 Terra Nova K-08 Y Y Y Y Y Y Terra Nova K-17 Y Y Y Y Y Y 12 Terra Nova K-18 Y Y Y Y Y Y 197 Trave E-87 Y Y Y Y Voyager J-18 Y Y Y Y Y Y W. Ben Nevis B-75 Y Y Y Y Y Y 22 W. Flying Foam L-23 Y Y Whiterose A-90 Y Y Whiterose E-09 Y Y Y Y Y Y Whiterose J-49 Y Y 64 Y Y Whiterose L-61 Y Y 85 Whiterose N-22 Y Y Y Y Y Y *Y=yes. Number of permeability and porosity measurements. I-46 well. In addition, the measurements are dominated by high porosity and permeability values (Fig. 3), especially in the Hibernia and Jeanne d Arc formations, as a result of preferential sampling of sand-rich intervals. Permeability data from the Avalon Formation are dominated by permeability values ranging from md, as compared to a range of md for the Hibernia and Jeanne d Arc formations. The sampling interval of digital well log data is 0.2 m. The well logs used in this study are the gamma-ray, density, sonic and neutron porosity logs, which mainly respond to variations in the lithology and porosity. For most wells, a complete suite of the four well logs exists. As shown in Table 1, compared to the well log data coverage, the amount of core measurements is small, which itself highlights the need to integrate core data with well logs. The lithostratigraphic scheme, top

4 248 Zehui Huang and M. A. Williamson Table 2. Wells where both core and well log data are available for the Ben Nevis, Eastern Shoals and Catalina formations* Ben Nevis Fm Eastern Shoals Fm Catalina Fm Well Name Occurrence logs k & p Occurrence logs k & p Occurrence logs k & p Hebron I-13 Y Y 71 Y Y 71 Hibernia G-55 Y Y 28 Hibernia I-46 Y Y 622 Hibernia J-34 Y Y 155 Hibernia K-14 Y Y 26 N. Ben Nevis M-61 Y Y 52 N. Ben Nevis P-93 Y Y 67 S. Mara C-13 Y Y 113 Terra Nova I-97 Y Y 18 Terra Nova K-07 Y Y 42 W. Ben Nevis B-75 Y Y 372 Whiterose L-61 Y Y 31 *Y=yes. Number of permeability and porosity measurements. and thickness of these formations used in this study follows McAlpine (1989). MODELLING OF POROSITY, PERMEABILITY AND WELL LOGS The modelling approach In this study we have taken an advanced non-linear modelling approach, the artificial neural network (ANN), to integrate core data with well log data. Previously, we used a back-propagation artificial neural network (BP-ANN) to establish a satisfactory model for the relationship between well logs and rock permeability in the Sable Subbasin offshore eastern Canada (Huang et al. 1996). A comparison with conventional statistical methods (multiple linear/non-linear regression analysis) has highlighted the promise of BP-ANNs as a method for obtaining multivariate, nonlinear models for difficult problems such as permeability estimation (Huang et al. 1996). The main advantages of a BP-ANN approach include: (1) accommodation of non-linearity in the data; and (2) freedom from a priori selection of mathematical models (equations). The complicated relationship among porosity, permeability and well logs can be established through a supervised training process using training and supervising examples. The training examples train the BP-ANN model and improve its performance, while the supervising examples monitor whether the BP-ANN has reached the optimal status and prevent it from being overtrained. Ideally, the two example sets should be statistically similar. With our problem, an example consists of four well log readings (the pattern, which is referred to as the input value) followed by permeability and porosity values (the actual answer, or definition to the pattern, which is referred to as the desired output value). One major drawback of a BP-ANN approach is the time needed for the training process. The potential of convergence to a local minimum is a another pitfall. The training time is long because the ordinary BP-ANN algorithm and some of its modified speedy versions are usually slow to converge when modelling a complicated problem which has many variables in both input and output and a lot of examples. In addition, a number of trial runs are needed to determine the optimal network architecture. To avoid converging to local minima, for a particular network architecture, several training runs are necessary, which also takes a large amount of time. To improve the efficiency of BP-ANN modelling, we use a fast BP-ANN modified with the Marquardt algorithm (Hagan & Menhaj 1994). Explanations of the BP-ANN algorithm and its Marquardt modification are presented in the Appendix. Details of ANNs and BP-ANNs can be found elsewhere (Rumelhart et al. 1986; Hertz et al. 1991). Baldwin et al. (1990), Rogers et al. (1992), Osborne (1992) reported examples of ANN applications in geosciences. Data pre-processing The purpose of the data pre-processing is to construct reliable and comprehensive training/supervising examples. Our experience indicates that data pre-processing is one of the important steps applying the BP-ANN modelling approach to geological problems. For the efficient integration of core data with well log data, we need to consider the following problems. (1) For some wells, depth shifts (or depth mis-matching) exist between the depth of well logs and coring depth. Matching well log readings with core measurements from an incorrect depth gives the pattern an erroneous definition and could result in wrong training/supervising examples. With many erroneous examples, the ANN will not be properly trained. (2) The vertical resolution of well logs is not equivalent to that of core sample measurements. The core plug porosity and permeability values are only representative of a small rock mass, while a single well log reading is a composite of petrophysical properties within a radius of centimeters to meters, depending on tool design. Low porosity and permeability values measured on a shaly sample from a thin shale interbed in thick sandstones (or, vice versa, high values from a thin sandstone interbed in thick shales) may not be resolved by well logs at all. Therefore, it should be avoided to combine core measurements from a thin interbed beyond well log s resolution with well log readings for making an training or supervising example. (3) Poor hole conditions can cause well logs measuring a larger volume of drilling fluid and mud cake than normal hole conditions. Use of caliper logs can assist in correcting poor hole conditions. However, the caliper log is not widely available in this basin, which prevents us from using this log for this correction.

5 Porosity and permeability in reservoir intervals 249 Fig. 3. Histograms of porosity and permeability measurements for the Avalon Formation (a, b), the Hibernia Formation (c, d) and the Jeanne d Arc Formation (e, f). To remove possible depth shifts, we plotted core measurements alongside four well logs (Fig. 4). In sections with sufficient core data, such plots help to determine whether depth shifts exist between the well logs and core samples. Examination of these plots for possible depth shifts were performed with the recognition that the petroleum industry preferentially take cores and samples from sand-prone intervals rather than shaly intervals, and knowledge of the nature of general relationships between lithologies and well log responses and those among porosity, permeability and lithologies. Examples of such knowledge applicable to siliciclastic sequences are: (1) shaly intervals have higher gamma ray, lower density, higher sonic and higher neutron porosity readings than sandy intervals; (2) shaly intervals should have lower permeability than sandy intervals; (3) if the lithology does not vary significantly, decrease in density, and increase in sonic travel time and neutron porosity may relate to increase in porosity and hence permeability. However, in sections with sparse measurements it is difficult to judge whether shifts between well log depth and core depth exist. To alleviate the second problem, we avoid taking isolated porosity/permeability values that lack a distinctive and reasonable well log response for construction of examples. Most of the training examples are from intervals that have at least three laboratory measurements with the maximum consecutive intersample distance being less than 1 m and the difference between

6 250 Zehui Huang and M. A. Williamson Fig. 4. An example of correcting depth shift between well logs and core measurements (Hibernia Formation, Hibernia C-96). (a) Evidence of depth shifts are as the follows. Relatively uniform and high permeability values in zone 1 correspond to a shaly interval. A few low permeability values in the uppermost part of zone 2 fall in a sandstone zone. The high permeability values in lower part of zone 2 are in a shaly interval. Permeability values in zone 3 imply a fairly homogeneous sandy interval, yet the gamma log shows a shaly layer near the top. Permeability values and well logs in zone 4 exhibit similar contradictions. (b) The depth shifts were corrected by moving core A downward 4.5 m and moving core B downward 6 m. the maximum and the minimum of logarithmic permeability values (log 10 k) being less than one. A further criterion for supervising example construction is that the interval sampled must be thicker than 1 m. Such intervals are relatively homogeneous and are expected to be unambiguously resolved by the well logs. Since measurements on shale-prone intervals are less than those on sand-prone intervals, some special training/supervising examples for low porosity and permeability values (e.g. k<0.1 md) are taken at locations with distinctive and reasonable well log responses regardless of the sample number and thickness requirements. Thus, measurements representing shale-prone intervals are preserved in the example sets. Figure 5 shows how we construct examples after depth shift correction. Although much effort was invested to solve the problems of depth shift and difference in vertical resolution between core and well log data, problems were only

7 Porosity and permeability in reservoir intervals 251 Fig. 5. An example of constructing training and supervising examples after depth shift correction (Jeanne d Arc Formation, Terra Nova K-07). For each of the intervals marked by a vertical bar, the averaged well log readings of N depth points are combined with the average of the porosity/permeability values to become a training example. If the interval is thicker than 1 m, the well log readings at one randomly selected depth point are removed, and the averaged well log readings of N 1 points are combined with the averaged measurements to form a supervising example. Thus, the training and supervising examples are similar, but not the same. Two types of isolated measurements were recognized. The first has distinctive well log responses, and the second does not. For the first type (special points), three-point averaged well log readings were combined with the measured values to form a training example and small random noise was added to it to produce a supervising example. The measurements belonging to the second type were abandoned, as they are unresolvable by well logs. Fig. 6. Histograms of permeability values for the (a) training and (b) supervising example sets. lessened, rather than completely solved. Averaging well log readings in relatively thick intervals in example construction (Fig. 5) may alleviate the influence from any poor hole conditions, depending on the depth span of poor hole conditions. The examples are taken from the Ben Nevis, Avalon, Catalina, Hibernia and Jeanne d Arc formations. The data in the Eastern Shoals Formation were excluded from the example extraction process to serve as a relatively independent testing data set for the trained BP-ANN. As a result of the example extraction criteria outline above, some wells with sparse core measurements contributed little to example construction. The data from such wells are also qualified as independent testing sets. The number of training and supervising examples thus constructed are respectively, 497 and 333. The number of examples is smaller than that of original core measurements. The number is reduced because some scattered individual measurements are not taken as examples, and in relatively homogeneous intervals, several, or in rare cases even tens of measurements, are represented by their averages to form the examples. Fewer, but more representative, examples can reduce the requirement for computer memory and increase the efficiency of modelling. An additional benefit of abandoning and grouping measurements is that in both the training and supervising example sets, the number of high and low porosity/ permeability values become closer than in the raw data (Fig. 6). This makes the examples less dominated by those from sandstone intervals. The last step of data preprocessing is to normalize the well log readings and porosity values to a 0 to 1 range, and the log 10 k values to a 1 to 1 range, so that they are compatible

8 252 Zehui Huang and M. A. Williamson Fig. 7. A three-layered BP-ANN used for modelling in this study. W 1 denotes the weights linking the nodes in the input and hidden layers, and W 2 are the weights connecting the nodes in the hidden and output layers. with the output range of the transform function used in the BP-ANN (see Appendix). BP-ANN modelling The BP-ANN (Fig. 7) is given four nodes in the input layer for four well logs, and two nodes in the output layer for predicting porosity and permeability. We used one hidden layer in the BP-ANN, and the number of nodes in the hidden layer was determined through a series of trial runs with networks of a differing number of hidden nodes. The number of hidden nodes was varied between eight and 12. With each network architecture we executed five trial runs. The performance of the BP-ANN was measured by its minimum RMSErr (root of mean squared error) of the supervising examples. The best trained BP-ANN (i.e. that with the lowest minimum RMSErr) of the five runs for each architecture was used to make comparisons between varying architectures. We expect that the trained BP-ANN should be able to predict permeability and porosity from well logs with an averaged error less than 0.5 logarithmic permeability (md), i.e. one half order of magnitude, and 0.03 porosity units in fraction. This allows for the errors in the data associated with depth shift, vertical resolution and poor hole conditions. This expectation was set as the error goal. We found that the lowest number of hidden nodes needed to reach the error goal is 10. With an increased number of hidden nodes, the RMSErr can only be slightly reduced. Therefore, the BP-ANN with 10 hidden nodes was considered as the most suitable architecture, and the weights which yield the lowest RMSErr for the supervising set with this architecture are used as the optimal model representing the relationship among rock porosity, permeability and well logs. The BP-ANN modified by the Marquardt algorithm greatly improved the modelling efficiency. For a practical comparison, we trained an adaptive variable learning rate version of the BP-ANN (Vogl et al. 1988) with the same architecture (i.e. four input nodes, 10 hidden and two output nodes) and the same training/supervising example set, on the same computing platform. The Marquardt BP-ANN usually converges to a lowest RMSErr for the supervising example set within 600 training epochs, and the time taken in the five runs for the architecture varies from 6 to 46 min. However, the BP-ANN of the variable learning rate version took over 36 h or longer to reach a comparable lowest RMSErr for the same architecture. This shows that the Marquardt version of BP-ANN is at least 47 times more efficient than the variable learning rate version for our problem. Fig. 8. (a) Cross-plot of measured permeability against predicted ones from the trained BP-ANN for the supervising examples. (b) Cross-plot of measured porosity against predicted ones from the trained BP-ANN for the supervising examples. Figures 8a and 8b are cross-plots of measured permeability and porosity against the BP-ANN predictions, respectively, for the supervising example set. The best-fit lines (solid line: Y=k*X+b) were calculated using the reduced major axis (RMA) method (Davis 1986). The slope of the RMA line indicates the general goodness of the fit, and R 2 the scatter. In Fig. 8a, the slope of the best-fit line is nearly 44 and the correlation coefficient (R 2 ) is These two values indicate that the trained BP-ANN can make satisfactory permeability predictions for the supervising examples. In Fig. 8b, the slope of the best-fit line is over 42 and the R 2 is Figure 9 shows the cumulative graphs of the absolute difference between the measured and predicted permeability and porosity, respectively, of the supervising examples. For over 68% of the predicted permeability the absolute differences from the

9 Porosity and permeability in reservoir intervals 253 Fig. 9. Cumulative graphs of the absolute difference between the measured and predicted permeability and porosity, respectively, of the supervising examples. φ m and k m stand for measured porosity and permeability, and φ p and k p the predicted. measured values are less than a half order of magnitude. Only for less than 4% of the predicted values are the differences over one order of magnitude. For over 68% of the predicted porosity, the absolute differences from the measured ones are less than 0.03 (fraction). For less than 5% of the predicted porosity the differences are over 0.06 (fraction). The scatter in Fig. 8 and the 4 5% of the predicted values with large errors (Fig. 9) can be attributed to such factors as uncorrected depth shifts and spurious well log responses caused by poor hole conditions. RESULTS AND APPLICATION Figure 10 shows the predicted permeability and porosity curves using the trained BP-ANN and the original core measurements in the Eastern Shoals Formation at the Hibernia I-46 well, which is a sequence of interbedded sandstone and siltstone. Although the core measurements were not used for constructing examples, the predicted curves and the measurements agree with each other, except for a few measurements. This agreement may be due that the sandstone in the Eastern Shoals Formation was deposited in a environment similar to that of the Avalon Formation. Several factors may explain some discrepancies between the predicted curves and the measurements. First, it can be expected that some high and low measured values from thin interbeds (e.g. <0.15 m) unresolvable by well logs could not be realized with the trained BP-ANN. Errors in laboratory measurements may be a second reason. A third reason may be caused by small depth shifts between the well log and core samples. In spite of some discrepancies (Fig. 10), the general Fig. 10. Predicted permeability and porosity curves and the measured values (crosses) in the Eastern Shoals Formation at Hibernia I-46 well. agreement does indicate that the trained BP-ANN is able to reveal the average characteristics of the reservoirs and provide a reliable basis for reservoir study in this basin. The trained BP-ANN also performed satisfactorily in the Eastern Shoals Formation in other wells. Core data from the

10 254 Zehui Huang and M. A. Williamson Fig. 11. Predicted permeability and porosity curves and the measured values (crosses) in: (a) the Avalon Formation at Whiterose L-61 well, (b) the Hibernia Formation at Hibernia K-14 well, and (c) the Jeanne d Arc Formation at Terra Nova E-79 well. Hibernia Formation in the South Mara C-13 well and that from the Jeanne d Arc Formation in the East Rankin H-21 well were not used in training/supervising examples. The trained BP- ANN still produced porosity/permeability curves that agreed with the core measurements in these two wells. Since the examples are all from Mesozoic strata that represent a variety of clastic depositional environments, we expect that the trained BP-ANN is applicable to most of the Mesozoic strata of the Jeanne d Arc Basin and adjacent areas. We applied the BP-ANN model to estimate permeability and porosity profiles from well logs for the Avalon Formation (40 wells), the Hibernia Formation (36 wells) and the Jeanne d Arc Formation (37 wells). The BP-ANN model was also applied to the Ben Nevis and Catalina formations in a smaller number of wells. In most of the wells where core measurements exist, we observed agreement between the predicted curves and the measurements (Figs 11 and 12). Discrepancies can be explained for similar reasons to the Eastern Shoals Formation (Fig. 10). For example, the reservoir intervals of the Hibernia Formation in the Hibernia K-14 well are mainly of thick sandstones with few thin interbeds, and therefore, the measured values and the predicted curves agree well (Fig. 11a). On the other hand, the reservoir intervals of the Jeanne d Arc Formation in the Terra Nova E-79 well ( m) contain many thin shale and calcareous siltstone interbeds about m thick, and some of the measured low permeability and porosity values in these thin beds were not accurate estimates. Nevertheless, the overall general agreement observed in these wells further increases our confidence in using the BP-ANN model derived curves in wells where only well logs are available to study the reservoir intervals. The continuous permeability and porosity curves reveal detailed vertical variations. Since shaly intervals are not routinely sampled and measured for permeability, predicted permeability values in the shaly intervals allow quantitative examination and evaluation of vertical intra-reservoir flow barriers at well locations. We defined at least five intra-reservoir barriers with permeability less than 0.01 md in the Main Hibernia Zone (MHZ) as defined by Brown et al. (1989). Some of the barriers are over 15 m thick. In the Jeanne d Arc Formation, there are at least eight such barriers. Efficient vertical communication among permeable layers may have to rely on open fractures and faults. With many predicted porosity and permeability curves from the exploration wells in this basin, we are able to examine the distribution of high quality reservoir rocks, for example the

11 Porosity and permeability in reservoir intervals 255 Table 3. Good reservoir* thickness and thickness percentage in the Avalon Formation at well locations Well name Formation thickness Good reservoir thickness Thickness percentage Fig. 12. Predicted permeability and porosity curves and the measured values (crosses) in the Ben Nevis Formation at West Ben Nevis B-75 well. Avalon Formation (Table 3). Table 3 shows that in the Avalon Formation good reservoirs are thick in the Terra Nova area, around the North Ben Nevis wells, and in the west part of the Hibernia Field. CONCLUSIONS (1) We have used an ANN modelling approach to achieve an efficient and reliable quantitative integration of measured porosity/permeability and four well logs that are mainly indicators of lithology and porosity. The process includes, (1) data preprocessing and example preparation, (2) BP-ANN modelling and (3) testing and application of the BP-ANN model. Our experience indicates that, of these stages, the most important is the first one. An in-depth understanding of the core and well log data is very important. The efficiency of BP-ANN training itself has been improved greatly by using a BP-ANN modified by the Marquardt algorithm. (2) Comparison of measured and predicted values for the supervising examples and testing data set indicate that the Amethyst F Archer K Ben Nevis I Beothuk M Bonne Bay C Fortune G Gambo N Hebron I Hibernia B Hibernia B Hibernia C Hibernia G Hibernia I Hibernia J Hibernia K Hibernia K Hibernia O Mara M Mercury K Nautilus C N. Ben Nevis M N. Ben Nevis P S. Mara C Springdale M Terra Nova C Terra Nova E Terra Nova H Terra Nova I Terra Nova K Terra Nova K Terra Nova K Terra Nova K Voyager J W. Ben Nevis B W. Flying Foam L Whiterose A Whiterose E Whiterose J Whiterose L Whiterose N *Those with porosity higher than 10% and permeability higher than 10 md. trained BP-ANN has satisfactorily generalized the relationship between porosity, permeability and well logs. This trained model is useful for predicting porosity and permeability in Mesozoic strata elsewhere in the basin. (3) Continuous BP-ANN derived porosity and permeability curves provide more complete and quantitative portraits of important reservoir intervals. These curves allow basinwide evaluation of reservoir quality and distribution and facilitate detailed examination of vertical variations of permeable and less permeable units. (4) Finally, it should be pointed out that the BP-ANN modelling approach is just a generalized mathematical model of the relationships among rock s porosity, permeability and well log responses, but not a physical one. The BP-ANN modelling is actually a learn-from-data process. It is able to extract information in the training examples. The predictive value of such a model is limited by what is represented by the training examples and the resolution of well log data. The training examples are from strata of various clastic environments located in different parts of the basin, the lithological control on variations in

12 256 Zehui Huang and M. A. Williamson Fig. 13. The training flow chart of a three-layer BP-ANN modified with the Marquardt algorithm (see Appendix for explanation). porosity and permeability should be learned by the BP-ANN model. Variations related to thin bed or lamination that are unresolved by well logs can not be predicted when applying the trained ANN model to well logs. The heterogeneity reflected in the predicted porosity and permeability curves are of the same order of resolution as the well logs. This study has been financially supported by GSC and OERD. Mr W. Chipman of Canada Newfoundland Offshore Petroleum Board is thanked for providing porosity and permeability measurements. Drs Al Grant and Kevin Coflin of Geological Survey of Canada (Atlantic) have made useful suggestions on the first version of this paper. Constructive suggestions and comments from reviewers Brian Moss and Jerry Jensen are greatly appreciated. Using the BP-ANN in Fig. 7 as an example, the input to a node in the hidden layer is: The output of that hidden node is: The input to a node in the output layer is: The output of that output node is: H(i)=f(S 1 (i)). (3) O(i)=f(S 2 (i)). (5) APPENDIX: BACK-PROPAGATION NEURAL NETWORK WITH MARQUARDT ALGORITHM A BP-ANN always has an input layer containing input nodes, an output layer containing output nodes, and at least one hidden layer containing hidden nodes (Fig. 7). In a BP-ANN, nodes of adjacent layers are connected by weights. However, there are no connections among nodes of the same layer. The nodes in the hidden and output layers process the data through application of a transform function (or basis function, activation function). A BP-ANN s transform function should be continuous, S-shaped and monotonically increasing, with asymptotically fixed values as the input approaches plus or minus infinity. Commonly used functions are the log sigmoid function, whose outputs range from 0 to 1, and hyperbolic tangent sigmoid function, whose outputs are from 1 to 1. The latter takes the following form: With the normal BP-ANN algorithm, in one training epoch of a supervised training process, the input values in the training examples are fed forward through the network to produce computed output values. The difference between the computed output and the desired output values for the training examples is fed backward applying a learning rule called the Delta rule (Rumelhart et al. 1986) to update the weights in order to make the calculated outputs approach the desired outputs as closely as possible. Meanwhile, the BP-ANN is presented with all supervising examples, and the error index (e.g. RMSErr) for the supervising examples is recorded to monitor the performance of the BP-ANN. This process stops when the error index reaches the error goal, with the weights in the last epoch being saved, or when a pre-set maximum number of epochs is reached, with the weights which yielded the lowest error index being saved for BP-ANN model application. The Marquardt algorithm is a non-linear regression analysis technique (Marquardt 1963). Although a BP-ANN is not a single function, the Marquardt modification treats the weights as the coefficients of a function, since the transfer function in the hidden and output nodes take a simple

13 Porosity and permeability in reservoir intervals 257 form. The Marquardt modification on BP-ANN lies in the way of weight updating. While normal BP-ANN is a steepest descent algorithm, the Marquardt version is an approximation to Newton s method. Depending on the variation of a training parameter (µ) during training, the Marquardt BP-ANN is steepest descent in weight updating when µ is large, and is Gauss Newton when µ is small. Hagan & Menhaj (1994) have shown that the Marquardt BP-ANN is much faster than two other quick versions by Vogl et al. (1988) and Nguyen & Widrow (1990). Although the Marquardt BP-ANN increases memory requirements, it is manageable with modern computers if the number of weights is less than a few hundred. Figure 13 shows the flow chart for a three-layer Marquardt BP-ANN. Only training examples (X and Y pairs) are used for illustration. After initialization, it proceeds as follows: (1) Feed forward inputs (X) through the network and calculate the outputs (O), errors (E), and RMSErr1. (2) Considering RMSErr1 as a function of all weights, compute the Jacobian matrix (J(W)), through back-propagation of the errors through the network. (3) Calculate the change in the weights (ΔW). (4) RMSErr2 is computed using W+ΔW. If RMSErr2 is smaller than RMSErr1, reduce µ by Δµ down, update and save the weights, and return to step 1 after checking if the BP-ANN is converged or has reached the maximum number of training epochs (step 5). Otherwise, increase µ by Δµ up, and return to step 3. The most important step is the second one. Assuming that W 1 is a n 1 by m 1 matrix, W 2 is a n 2 by m 2 matrix, and there are q pairs of X and Y vectors, with the length of X and Y being n and m, respectively, the J(W) is: Using the last row of the Jacobian matrix as an example, the terms in the J(W) are calculated as follows: For a supervised training process, step 5 is to be modified. After weight updating, the input values in supervising examples should be fed forward through the BP-ANN and the RMSErr for the supervising examples (denoted as RMSErrS) is calculated. If the RMSErrS is the smallest amongst all calculated, the weights are saved. One of the conditions in step 5 for stopping training should use RMSErrS instead of RMSErr1. REFERENCES BALDWIN, J. L., BATEMAN, R. M. & WHEATLEY, C. L Application of a neural network to the problem of mineral identification from well logs. The Log Analyst, 31, BROWN, D. M., MCALPINE, K. D. & YOLE, R. W Sedimentology and sandstone diagenesis of Hibernia Formation in Hibernia Oil Field, Grand Banks of Newfoundland. AAPG Bulletin, 73, DAVIS, J. C Statistics and Data Analysis in Geology. 2nd edn, Wiley, Chichester. ENACHESCU, M. E The tectonic and structural framework of the northwest Newfoundland continental margin. In: Beaumont, C. & (6)

14 258 Zehui Huang and M. A. Williamson Tankard, A. J. (Eds) Sedimentary Basin and Basin-forming Mechanisms. Canadian Society of Petroleum Geologists Memoirs 12, , HARDING, S. C., SMEE, G. R., BRISCOE, R. G., EMERY, D. J. & HALLSTROM, S. K Structural, tectonic and seismo-stratigraphic study of the Terra Nova oil field, offshore Newfoundland CSEG National Convention Abstract Book, FOWLER, M. G. & MCALPINE, K. D The Egret Member, a prolific Kimmeridgian source rock from offshore Eastern Canada. In: Katz, B. J. (ed.) Petroleum Source Rock Case Studies. Springer, Berlin, HAGAN, M. T. & MENHAJ, M. B Training feedforward networks with the Marquardt algorithm. IEEE Transactions on Neural Networks, 5, HERTZ, J., KROGH, A. & PALMER, R. G Introduction to the Theory of Neural Computation. Addison-Wesley, London. HUANG, Z., SHIMELD, J. & WILLIAMSON, M. A Permeability prediction with computer neural network modeling in the Venture Gas Field, offshore eastern Canada. Geophysics, 61, HURLEY, T. L., KREISA, R. D., TAYLOR, G. G. & YATES, W. R. L The reservoir geology and geophysics of the Hibernia Field, offshore Newfoundland. In: Halbouty, M. T. (ed.) Giant Oil and Gas Fields of the Decade AAPG Memoirs, 54, MARQUARDT, D An algorithm for least squares estimation of non-linear parameters. Journal of the Society of Industrial Application of Mathematics, MCALPINE, K. D Lithostratigraphy of fifty-nine wells, Jeanne d Arc Basin. Geological Survey of Canada Open File Report Mesozoic stratigraphy, sedimentary evolution, and petroleum potential of the Jeanne d Arc Basin, Grand Banks of Newfoundland. Geological Survey of Canada Paper NGUYEN, D. & WIDROW, B Improving the learning speed of 2-layer neural networks by choosing initial values of adaptive weights. In: Proceedings of IJCNN, vol. 3, OSBORNE, D. A Neural networks provide more accurate reservoir permeability. Oil & Gas Journal, 90, ROGERS, S., FANG, J. H., KARR, C. L. & STANLEY, D. A Determination of lithology from well logs using a neural network. AAPG Bulletin, 76, RUMELHART, D. E., HINTON, G. E. & WILLIAMS, R. J Learning internal representations by error propagation. In: Rumelhart, D. E. & McClelland, J. L. (eds) Parallel Distributed Processing; Foundations. MIT Press, SALEM, H. S A preliminary study of the physical properties of the Terra Nova and Hibernia oil fields in the Jeanne d Arc Basin, offshore Newfoundland, Canada. Geological Survey of Canada Open File Report 2686, 67 pp. SINCLAIR, I. K., SHANNON, P. M., WILLIAMS, B. P. J., HARKER, S. D. & MOORE, J. G Tectonic control on sedimentary evolution of three North Atlantic borderland Mesozoic basins. Basin Research, 6, VOGL, T. P., MANGIS, J. K., ZIEGLER, A. K., ZINK, W. T. & ALKON, D. L Accelerating the convergence of the backpropagation method. Bio. Cybern., 59, Received 21 June 1996; revised typescript accepted 26 February 1997.