into the Abenaki and Sable sub-basins. Development of the. Scotian Basin commenced during the middle Early Jurassic, with extensive deposition during the ...
GEOPHYSICS, VOL. 61, NO. 2 (MARCH-APRIL 1996); P. 422-436, 13 FIGS.
Permeability prediction with artificial neural network modeling in the Venture gas field, offshore eastern Canada
Zehui Huang*, John Shimeld*, Mark Williamson*, and John Katsubet
Canada) are organized into training and supervising data sets for BP -ANN modeling. Data from a fifth well in the same field are retained as an independent data set for testing. When applied to this test data, the trained BP -ANN produces permeability values that compare well with measured values in the cored intervals. Permeability profiles calculated with the trained BP -ANN exhibit numerous low permeability horizons that are correlatable between the wells at Venture. These horizons likely represent important, intra-reservoir barriers to fluid migration that are significant for future reservoir production plans at Venture. For discussion, we also derive predictive equations using conventional statistical methods (i.e., MLR, and MNLR) with the same data set used for BP-ANN modeling. These examples highlight the efficacy of BPANNs as a means of obtaining multivariate, nonlinear models for difficult problems such as permeability estimation.
Estimating permeability from well log information in uncored borehole intervals is an important yet difficult task encountered in many earth science disciplines. Most commonly, permeability is estimated from various well log curves using either empirical relationships or some form of multiple linear regression (MLR). More sophisticated, multiple nonlinear regression (MNLR) techniques are not as common because of difficulties associated with choosing an appropriate mathematical model and with analyzing the sensitivity of the chosen model to the various input variables. However, the recent development of a class of nonlinear optimization techniques known as artificial neural networks (ANNs) does much to overcome these difficulties. We use a back-propagation ANN (BP-ANN) to model the interrelationships between spatial position, six different well logs, and permeability. Data from four wells in the Venture gas field (offshore eastern
factor influencing permeability. Usually, crossplots are used to derive simple empirical relationships between porosity (n) and permeability (k) from core measurements, which are then used to convert well log derived porosity into permeability. Nelson (1994) has provided an in depth discussion and summary of porosity/permeability relationships in sedimentary rocks. Use of such relationships, though, is valid only for unconsolidated sand and homogeneous lithologies (Beard and Weyl, 1973; Bos, 1982), which are rarely present in most practical applications. In more sophisticated approaches, permeability is recognized as a complex function of several interrelated factors such as lithology, pore fluid composition, and porosity. Many well logging tools are designed to be sensitive to these same factors (Asquith and Gibson, 1982; Ahmed et al., 1989 ). Therefore
Petrophysicists, hydrogeologists, and reservoir engineers must obtain estimates of permeability to be able to predict fluid migration pathways and volumes. Sometimes direct permeability measurements are used, but these are usually expensive and, typically, are difficult to obtain. Even when they are available, direct permeability measurements are usually unevenly and sparsely distributed. In practice, the only source of high resolution data to aid in permeability estimation is provided by well logging. However, because of the complexity of the problem, no universally applicable method of estimating permeability from well logs exists (Nelson, 1994). Porosity, which can reliably be estimated from well logs under most circumstances, is often assumed to be the strongest
Manuscript received by the Editor March 25, 1994; revised manuscript received February 27, 1995. *Geological Survey of Canada Atlantic, P.O. Box 1006, Dartmouth, NS, Canada B2Y 4A2. $Geological Survey of Canada, 601 Booth St., Ottawa, Ontario, Canada KIA 0E8. Printed with permission of the Canadian Government. 422
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some, albeit complex and difficult to express, relationship exists between permeability and well logs. To explore this relationship, MLR can be applied to data from cored intervals, and the resulting predictive equation (i.e., log(k) = A 0 + Al X X1 ) is then applied in uncored intervals (Wendt et al., 1986). However, the assumption that log (k) varies linearly with each input variable can be too restrictive in many practical applications. A broad range of MNLR techniques can be used to remove the linearity assumption. However, the accuracy of prediction is extremely dependent on the functional form (or mathematical model). Artificial neural networks (ANNs) comprise a class of nonlinear optimization techniques that does not require a priori selection of the mathematical model. Instead, the relationships among input and output variables are determined automatically as a result of the ANN algorithm. Therefore, ANNs are an ideal choice for handling problems whose exact numerical relationship among variables are unknown or difficult to discover, and the techniques have seen rapid development over the last decade. Recent application of ANNs to geological problems has yielded promising results. ANNs have been used, for example, in mineral and lithology recognition from well logs (Baldwin et al., 1990, Rogers et al., 1992), mineral reserve estimation (Wu and Zhou, 1992), and seismic data processing (Pulli and Dysart, 1990; Wang, 1992). Osborne (1992) presented an example that uses an ANN for permeability estimation, in which porosity measurements and spatial position were used as input variables. We present an example of ANN permeability estimation using spatial position and six well logs, rather than measured porosity, as input variables. Our example is from the Venture gas field, in the Scotian Basin, offshore eastern Canada (Figure 1).
As is the case in many fields, reservoirs and seals at Venture are not fully cored. The continuous permeability profiles derived from well logs help interpreters identify and rank various reservoir and sealing units and also aid in correlating these units between wells. Since reservoirs at Venture are overpressured (Mudford and Best, 1989), present-day permeability information is important for further understanding of mechanisms that contribute to overpressure generation. Also at Venture, very few permeability measurements exist within shale horizons, which introduces large uncertainties in the assessment of overpressure generation and maintenance. Again, this is a typical situation in many exploration areas (Katsube et al., 1991), and there is a strong need for accurate methods of high resolution permeability estimation. Following a brief geological summary of the Venture gas field and a brief introduction to ANN techniques, we explain the data sets and preprocessing methods. Then we describe ANN modeling and the results, with a comparison to conventional statistical MLR and MNLR analysis. This comparison is not intended as a mathematically rigorous test of each method's accuracy. Rather, the comparison is presented to highlight the operational strengths and weaknesses of ANNs relative to the other available methods of permeability estimation. Permeability profiles established using the ANN model are examined, together with a short discussion of the geological implications. Geological summary of the Venture gas field Located offshore eastern Canada in the Scotian Basin, the Venture gas field contains hydrocarbon reserves estimated by COGLA (Canada Oil and Gas Lands Administration) at 1.38 tcf of gas and 30 million STP bbl of condensate (Jansa and Urrea, 1990). As Figure 1 shows, the Scotian Basin is divided
FIG. 1. Location of wells in the Venture Gas Field, Scotian Shelf, Offshore Nova Scotia (from Katsube et al., 1991).
Permeability Prediction with ANN
into the Abenaki and Sable sub-basins. Development of the Scotian Basin commenced during the middle Early Jurassic, with extensive deposition during the Mesozoic and Cenozoic. A detailed description of the regional geology and depositional history of the Scotian Basin is provided in Jansa and Wade (1975) and Grant et al. (1986). At Venture, hydrocarbons occur within the lower Missisauga and upper Mic Mac Formations. This zone is greater than 4500 m in depth and is overpressured (Grant et al., 1986). The Mic Mac Formation (Middle to Upper Jurassic) consists of mixed carbonate and clastic facies deposited in a marine environment. The Missisauga Formation (Lower Cretaceous) is characterized by fluvio-deltaic sediments, representing a
period of widespread regression. The gas-bearing sequence in the Venture field is composed of thick sandstone units interbedded with thin shales and limestones. Sandstone to shale ratios within the overpressured region are close to one. Shales at the top of Sandstone Numbers 2A, 2C, 3, 4, and 10 [following stratigraphic terminology used by Mobil (Jansa and Urrea, 1990)] may be laterally continuous and act as seals within the system. This is illustrated by the composite pressure versus depth plot constructed by Mudford and Best (1989) (Figure 2).
that are randomized initially. However, there are no interconnections among nodes of the same layer. For a BP-ANN, the basis function for nodes in the hidden and output layers should be continuous and monotonically increasing, with asymptotically fixed values as the input approaches plus or minus infinity (i.e., the function is S-shaped) (Kosko, 1992). The basis function used in this study is the sigmoidal function, which takes the form
The training of a BP-ANN is a supervised learning process. Two data sets are involved: a training dataset and a supervising dataset. The learning process involves sending the input values forward through the network and then computing the difference between the calculated output and the corresponding desired output from the training dataset. This error information is propagated backwards through the ANN, and the weights are adjusted through application of the following equation to reduce the error (Rumelhart et al., 1986):
ANN techniques and the algorithm ANNs are, broadly speaking, a class of numerical optimization algorithms originally inspired by studies of the brain and nervous system. In the most general case, ANNs function as nonlinear dynamic systems that learn to recognize patterns after undergoing a training process. The training process involves presenting the ANN with a series of input values and desired output values (training patterns). In effect, the ANN implicitly defines its own, usually very complicated, predictive function as a result of the training process. Once trained, the ANN can be applied to new input data, allowing prediction of the output values. ANNs have two major components: nodes (also called units or neurons) and connections, which are weighted links between the nodes. Each node (except those receiving input values) applies an arbitrarily chosen function (known as a basis, activation, or transform function) to the weighted sum of results passed from other connected nodes. In this manner,
input values are passed through the network topology and transformed into one or more output values. The output values are then compared to the desired values (e.g., measured permeability) and, in accordance with the particular learning algorithm chosen, the connection weights are adjusted. This process iterates until the error between the output values and desired values is minimized. More details about general ANN theory can be found in many recent publications (e.g., Stephen, 1990; Hertz et al., 1991). One of the most popular types of ANN learning algorithms, and the one used for this study, is called back propagation (BP-ANN) (Rumelhart et al., 1986). A BP-ANN always has an input layer, which contains one node for each input variable, an output layer, which contains one node for each output value, and at least one hidden layer with an arbitrary number of nodes. This topology is illustrated in Figure 3. In a BP-ANN, nodes of adjacent layers are fully interconnected by weights
where a and 13 are assumed constants, called the learning rate and momentum factor, respectively, E is the error function, w is the weight vector, and t is the iteration number. For this project, we used a variation of equation (2), known as the "quick-prop" algorithm (Fahlman, 1988). The quickprop algorithm introduces an adaptive estimation of the momentum factor by using gradient information on the current and previous steps, with an assumption that the error versus weight curve for each weight can be approximated by a parabola. Quickprop speeds up the training process in comparison with normal backpropagation, and is also more robust during the learning stage (Fahlman, 1988). In this improved algorithm, the equation for weight adjustment is ©w(t) _ —a a
a E + r3E/aw(taEl)w(taE/aw(t) 1^w(t — 1). (3)
Weight adjustment happens in numerous epochs (iterations). As training carries on, the error based on the training data set gets asymptotically smaller. Theoretically, given
enough nodes and epochs, the error based on the training dataset will approach zero. However, this is undesirable since the ANN will then be fitting random noise in the input data and the ANN will be "overtrained." For this reason, the ANN is applied to the supervising data set at each epoch. The error based on the supervising data set will reach a minimum and then slowly increase as the ANN becomes overtrained. The training process stops when either the error based on the supervising dataset is below an assigned level, or an arbitrarily set number of iterations has been reached. The weights that yield the lowest error based on the supervising data set are regarded as the best generalization and are saved. These weights are then used when the BP-ANN is applied to a new data set for predicting the output values.
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PREPROCESSING OF THE DATA Permeability (specifically, permeability to air) and porosity measurements are available from Sandstone numbers 2A to 13 (approximately 4400 to 5800 m below sea level) penetrated by five wells in the Venture field (B-13, B-43, B-52, H-22, and C-62). The measurements were made by Core LaboratoriesCanada Ltd. The samples, provided by Mobil Oil Canada Ltd., were not cleaned prior to analysis.
Permeability and porosity values, published in Katsube et al. (1991), from shale units intersected by the same wells were included in the database. Data from the first four wells were organized into training and supervising data sets for ANN modeling, while data from Venture C-62 were retained for later use as a totally independent data set to test the trained ANN model. Spatial distribution of the available laboratory measurements is uneven in the first four wells, as shown by Figure 4. In densely sampled intervals, measurements are sometimes only a
• B-43 o B-52
x H-22 HYDROSTATIC PRESSURE
2AAND 2B SS `•
3 SS LU
0 4 SS TO 8 SS
m 5.0 0 LU
10 SS TO 18 SS 5.6 x
FIG. 2. Composite plot of fluid pressure versus depth from Venture wells B-13, B-43, B-52, and H-22 (from Mudford and Best, 1989).
Permeability Prediction with ANN
few centimeters apart. Most of the measurements were collected using samples from Sandstone Numbers 3, 5, and 6. Fewer are from Sandstone numbers 2, 7, 8, and 11. Sandstone numbers 13 and 14 only have seven and one measurement(s), respectively. There are no data available for Sandstone numbers 4, 8, 9, 10, and 12. Spontaneous potential, gamma ray, density, sonic, neutron porosity, and density correction logs are available for all the wells at Venture. These logs were used, along with the geographic coordinates and depth of the measured data, for permeability prediction. Although resistivity logs are also available at Venture, they were not used since resistivity measurements are more sensitive to fluid composition than porosity when several types of fluid (water, gas and oil) are present. As with any attempt at combining well log and core data, ANN modeling has to address the following problems: 1) Shifts between recorded well log depths and sample depths are possible for a number of reasons. While every attempt is made to remove these depth shifts, undetected depth shifts could cause significant errors in permeability predictions. 2) The spatial scale of well log measurements is not equivalent to that of sample measurements (Worthington, 1991; Enderlin et al., 1991). Well-log measurements are more spatially averaged than core data. Permeability measured from core is representative of only a small rock mass, while a single well log reading is a composite result of petrophysical properties within a radius of cm to m (depending on which tool is being used). Small scale heterogeneities in permeability between core samples a few centimeters apart may not be resolved by well logs at all.
3) Because of the strong heterogeneous and anisotropic nature of permeability in most natural rocks, it is often difficult to define a characteristic volume that is suitable for numerical calculations. We must keep in mind that a measured value can serve only as an estimate of permeability over a small interval. 4) Errors in well log data are caused by poor borehole conditions. Washout, caving, abnormal mudcake, etc., are all capable of adversely affecting well log responses. Typically though, these conditions can be identified by an experienced interpreter. If it is not possible to correct the well log measurements for the effects of poor borehole conditions, then intervals of poor borehole conditions can be flagged in the final interpretation. To address the possibility of depth shifts, we plotted laboratory measured permeability and porosity against several well logs that respond primarily to lithology and porosity. In sections with sufficient data, such a plot can indicate whether depth shifts exist between the well logs and core samples. In most sections from the first four wells, porosity and permeability measurements are consistent with the well log readings (see examples in Figure 5), although sections with sparse data are difficult to judge. Overall, we are reasonably confident that problems caused by depth shifts in the first four wells are minor. However, in Venture C-62 there is a systematic depth shift between well log depths and core sample depths. The shift is about 5 m and was corrected. The second and third issues, relating to spatial scaling, are difficult to address. In an attempt to match the spatial scale of well log measurements with sample measurements, we averaged laboratory measured permeability values that are less than 0.6 m apart and have a corresponding porosity difference
FIG. 3. The structure of the BP-ANN used for modeling. W h stands for weights connecting the input and hidden layers, and Wh o `
for those connecting the hidden and output layers.
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of less than 3%. While these criteria are arbitrary, the general intent is to reduce the significance attached to any one laboratory measurement. Thus, the precision of laboratory measurements was reduced to more closely match that of the well log measurements. There are 300 permeability measurements (raw data) from Venture B-52, B-43, H-22, and B-13. After averaging, the number of data points was reduced to 213, of which 73 were averaged values. Figure 6 displays the population distribution and cumulative probability of both the raw data and the mixture of averaged and unaveraged permeability measurements. The population contains few permeability values that are less than 0.1 millidarcies or that are greater than
1000 millidarcies [equivalently, log(k) < —1 and log(k) > 3) (Figure 6)]. The lack of low permeability values is caused mainly by preferential sampling in sandstone and siltstone intervals rather than shale intervals. Table 1 explains how we have combined and organized the permeability measurements, well-log data, and spatial position into training, supervising, and testing data sets, and comments on their purpose in this study. Because the number of permeability measurements available is limited, the desired output values in the supervising data set comprised the 73 averaged permeability measurements. The well-log input values corresponding to these measurements consisted of the spatially closest, one-point readings, rather than the 5-point averaged
FIG. 4. Composite plot of laboratory permeability measurements in Venture B-13, Venture B-43, Venture B-52, and Venture H-22.
Permeability Prediction with ANN
Permeability and porosity measurements (which are shown as solid diamonds and dots, respectively) with the neutron porosity, gamma ray, density and sonic logs.
Huang et al.
FIG. 6. Population distribution and cumulative probability of measured permeability (A, B for the raw data and C, D for the mixture
of averaged and unaveraged values).
Table 1. Description of training, supervising, and testing data sets (k = permeability; Well A = Venture B -52, B = Venture B -43, C = Venture H -22, D = Venture B-13, E = Venture C-62). Desired output
213 k measurements (both averaged and unaveraged) from wells A to D.
the BP-ANN compares the desired output and feeds the error backwards to adjust the weight.
73 k measurements (averaged) from wells A to D.
253 k measurements (both averaged and unaveraged) from Well E.
5-point averaged readings of 6 well logs at the corresponding depth and the location data from wells A to D. 1-point reading of 6 well logs at the corresponding depth and the location data from wells A to D. 6 continuous well logs and the location data from Well E.
the BP-ANN monitors the evolution of the error on this dataset to determine the optimum state of the training process. the trained BP-ANN are feed with inputs from Well E. The k data will be plotted alongside the computed k curve only for comparison.
Permeability Prediction with ANN
readings used in the training data set. Thus the two data sets were not identical, but their population distributions were similar. To make the inputs vary within the domain of the basis functions (i.e., the sigmoidal function), each input was linearly scaled to the interval [0, 1]. A logarithmic transformation was applied to the permeability values, and these results were then linearly scaled to the interval [-0.5, 0.5] which is the output range of the BP-ANN we used. PERMEABILITY PREDICTION: ANNS IN COMPARISON WITH OTHER APPROACHES
ANN permeability modeling In our problem, the BP-ANN is given nine nodes (three for the spatial position and six for well logs) in the input layer and one node (permeability) in the output layer. The number of nodes in the hidden layer of the BP-ANN was determined through a series of trials with networks of differing architectures. We found that a network with 12 nodes in the hidden layer (see Figure 3) had the least root-mean-squared error (RMSErr) based on the supervising data set compared to networks with fewer or more nodes in the hidden layer. Actually, finding the optimum architecture for a BP-ANN is one of the most time-consuming steps in network training. In another study being carried out, we have introduced the dynamic node creation scheme by Ash (1989). This scheme has eased the architecture-defining problem. Figure 7 displays the evolution of RMSErr during BP-ANN training, for both the training and supervising data sets. Typically, RMSErr decreases rapidly in the first 5000 training epochs. The RMSErr for the training data set constantly decreases as training carries on, although the reduction becomes very slight in the later stages (epoch > 125 000).
However, the RMSErr curve for the supervising data set reaches a minimum at approximately the 298 000th epoch and increases slightly, or remains virtually unchanged, afterwards. Thus, the BP-ANN weights at the 298 000th epoch were taken as the best generalization state. With the weights saved at the best generalization state, we obtained ANN predicted permeability values from the input values for the supervising data set. Figure 8a is a crossplot of measured permeability against the BP-ANN predicted permeability with inputs from the supervising dataset. The slope of the best-fit line, determined using the reduced matrix axis (RMA) method (Davis, 1986), is 45.3 0 and has a correlation coefficient (R) of 0.8. Here the slope of the RMA line indicates the fundamental goodness of the fit, and R indicates the scatter. These two values in Figure 8a indicate that the best trained BP-ANN can make an almost one-to-one prediction for the supervising dataset and the scatter is quite small. Scatter around the best-fit line may be caused by problems related to resolution, noise in the well log reading, local borehole conditions, and to a lesser extent, errors in the permeability measurements themselves. MLR and MNLR permeability modeling For comparison, we modeled exactly the same training dataset using the MLR approach from Davis (1986). We set the nine input values (scaled well log responses, etc.) as the independent variables and the transformed and scaled permeability as the dependent variable. Using the linear predictive equation (i.e., log(k) = A 0 + 1,y 1 A X X1 ) derived from the training data set, we calculated permeability from the nine input values from the supervising data set. Figure 8b is a crossplot showing the measured permeability in the supervising data set and the calculated values using the linear predictive equation. The slope of the RMA is 28.0° with
G N L .1.
i 11 100000
7. Root mean squared error (RMSErr) evolution of the training data and supervising data during BP-ANN training.
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a correlation coefficient of 0.65, which indicates that, in a general sense, the predictive equation from MLR analysis provides results that are worse than those obtained using the BP-ANN (Figure 8a). This is especially true at the data extremes. The reason for comparing BP-ANN and MLR results is because MLR is a conventionally used method of permeability estimation (Allen, 1979; Wendt et al., 1986). However, mathematically, this comparison is unfair because we have compared a nonlinear optimization technique to a linear one (even
though we use log(k) in both methods). This example merely serves to emphasize that permeability estimation from well logs is a complex, highly nonlinear problem. For a fairer comparison, it is necessary to compare the BP-ANN results with another nonlinear optimization technique. Again with the same training data set, we used a LevenbergMarquardt procedure, which is a commonly used MNLR method. Details about the Levenberg-Marquardt algorithm are given by Flannery et al. (1986), while Entyre (1992) provides a geological example where the algorithm is used to
FIG. 8. Crossplot of measured permeability against predicted permeability from a BP-ANN model (a), from linear equation (log(k) = A o + X Z9 I A, X X,) resulted from MLR analysis (b), from a nonlinear equation (log(k) = Ap + fl XA') resulted from MNLR analysis, with parameters being estimated with the conventional Levenberg-Marquardt routine (c), and from a same form of nonlinear equation resulted from MNLR analysis, but with parameters being estimated with a robust Levenberg-Marquardt version using Tukey's biweight scheme to weight each X vector (d). Here, kp is the predicted permeability and k m the measured permeability. The straight line is the RMA line.
Permeability Prediction with ANN
predict petrophysical parameters from well logs. As a trial, we used two nonlinear function forms: the first being log(k) = A 0 + H 1 XA , , and the second being log(k) = A 0 + X^9 1
With each function form, we used both a conventional and a robust weighting scheme to estimate the parameters (e.g., A 0 , A 1 , ... A 9 in the first function form). The conventional weighting scheme treats each independent variable vector X on an equal basis, while the robust scheme assigns varying weights using a weighting function [e.g., Tukey's biweight or Huber 1.5 robust weighting, see Entyre (1992)]. MNLR analysis achieved convergence only with the first function form. With the second function form, the conventional weighting scheme failed to converge and the robust version converged only occasionally with poor results. Therefore, we will present and discuss only the results from the first function form. Figure 8c is a crossplot showing the measured permeability in the supervising set and the calculated values using the nonlinear predictive equation (log(k) = A 0 + [I i9 1 XAi) from MNLR, with parameters being estimated with the conventional Levenberg-Marquardt routine. The slope of the RMA is 28.1° with a correlation coefficient of 0.66. The results are marginally better than those from MLR analysis (see Figure 8b), but compare poorly with the BP-ANN results. Figure 8d is a crossplot showing the measured permeability and the calculated values using the same form of nonlinear predictive equation from MNLR, with parameters being estimated with a robust Levenberg-Marquardt version, in which Tukey's biweight scheme is used as the weighting function. The slope of the RMA is 32.1° with a correlation coefficient of 0.67. Again, these results are only slightly better than the results from MLR analysis (see Figure 8b), but are still worse than the BP-ANN modeling results (Figure 8a). We also used a Huber 1.5 robust weighting function. The result (not displayed) is similar to what is shown in Figure 8c. From the point of view that we have used identical input variables, and both BP-ANN and MNLR are nonlinear opti 0
mization techniques, our comparison seems fair. However, the biggest factors affecting the MNLR performance are the selected mathematical model and its sensitivity to the chosen input variables. This becomes especially important when there are large covariances between the input variables. One would comment that, had we used a different function form that is able to handle the nonlinearity in the data better than the one we chose or, had we focused more effort on input variable selection, MNLR would have likely performed much better. The problem is: what is the exact form of this better function? 1 Also, determining the covariance matrix of the input variables a in a nonlinear parameter space is difficult and time-consuming. The purpose of our example, however, is not to say that MNLR is incapable of producing results comparable to the BP-ANN. Instead, we hope to clearly illustate the operational strengths and weaknesses of each method given exactly the same dataset. Permeability prediction with other methods As we have mentioned earlier, one common method of estimating permeability is to build a porosity-permeability relationship from core measurements, and then apply this relationship to porosity values derived from well logs in uncored intervals. Alternatively, a published and fairly well accepted porosity-permeability relationship can be used (e.g.,
the modified Kozeny-Carman equations (Ungerer et al., 1990) which takes a nonlinear form). Both these methods were applied to the supervising dataset constructed for the BP-ANN, MLR, and MNLR attempts. With the first method, we obained a linear predictive equation from the 213 permeability values and the corresponding porosity measurements (Figure 9a). At depths corresponding to the 73 averaged permeability values in the supervising data set, the readings from both the density log and the neutron porosity log were used to calculate apparent porosity. Porosity was derived from the density log using 2.65 and 1.0 g/cm 3 as the matrix and fluid density, respectively. Then, apparent porosity was assumed to be the average of the porosity values derived from the density log and the neutron porosity log. Finally, shale volume (and hence the bulk volume of bound water) was calculated using the gamma-ray log to convert apparent porosity to effective porosity.
FIG. 9. (a) Permeability-porosity relationships built from core measurements. (b) Crossplot of measured permeability against predicted permeability using the permeability-porosity relationship in (a). The straight line is the RMA line.
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The crossplot of measured against predicted permeability from well-log-derived porosity (Figure 9b) using the linear relationships shown in Figure 9a illustrates that the first method performs rather poorly at Venture. In Figure 9b, the RMA correlation coefficient is 0.52 and the slope is 51°. The second method, using the modified Kozeny-Carman relationship, was also unsatisfactory. The relationship was applied using the same log-derived porosity values as before, and taking 96 000 m 2 /m 3 as the specific surface area for fine sandstone. As Figure 10 shows, the correlation coefficient and slope of the RMA line (R = 0.49, slope = 48°) are as poor as the first method. The main problem with these two approaches is that less information is included in the calculations than with the BP-ANN, MLR, and MNLR approaches, even though the Kozeny-Carman equation is a nonlinear function. Other problems include the use of a single matrix density and fluid density when calculating porosity, and a single specific surface area, which is a function of grain size, in calculating permeability. In reality, these parameters vary with lithology and fluid type. Permeability prediction using the ANN in Venture wells We applied the trained BP-ANN to the data set from the Venture C-62 well as a completely independent test of its ability to estimate permeability from well logs. Predicted permeability profiles from the trained BP-ANN were derived with well log data from the interval between 4788 and 5230 m. Figure 11 shows both the predicted permeability and the measured values. Some of the measured permeability values were averaged according to the criteria as mentioned before. Except for a few intervals, the good agreement between predicted and measured values is obvious. This agreement is impressive considering the fact that the data from Venture C-62 were not incorporated into the training and supervising sets.
Although the systematic shift of 5 m between well log and core sample depths has been removed when plotting Figure 11,
FIG. 10. Crossplot of measured permeability against predicted permeability using the modified Kozeny-Carman relationship (Ungerer et al., 1990). The straight line is the RMA line.
small random shifts in depth between the two values still exist. Because of the scale and resolution issues previously described, a crossplot of measured against predicted permeability (not shown), with each pair taken according to the depth match as shown on Figure 11, does not show good correlation. However, when the results are binned into 1.0 m windows and the closest matches are crossplotted, the RMA correlation coefficient is 0.85 and the slope is 37.5° (Figure 12). Although Figure 12 shows less scatter around the RMA line in comparison with Figure 8a (the supervising data set), the reduced slope of the RMA line in Figure 12 reveals that the BP-ANN underestimated high permeabilities and overestimated low permeabilities in the testing dataset. The principal reason why some of the extreme points do not match may be that, in the training and supervising data sets, there are only a few measured permeability values that are less than 0.1 millidarcies or greater than 1000 millidarcies. Also, the population distribution of the Venture C-62 well is somewhat different from that of the training and supervising sets. Therefore, during training, the BP-ANN did not receive enough information to acquire the ability to predict very low and very high permeability values at some intervals. Nevertheless, given generally good performance of the trained BP-ANN as shown in Figure 8a and Figure 12, we calculated continuous permeability curves for the other four Venture wells. Figure 13 displays these permeability curves, plotted along with the measured permeability, from 4400 m (near the top of Unit 2) to 5800 m. Since the training and supervising data are from these four wells, the measured permeability and predicted ones display better agreement than what was observed in Figure 11. It is worth noting that, in the BP-ANN permeability model, permeability values are constrained to fall within the interval between 10 -5 millidarcies and 10 5 millidarcies. The maximum value does not occur in the predicted permeability curves, however, the minimum value appears over several intervals. For example, at Venture B-13, calculated permeability values are 10 -5 millidarcies around 4400 m (top of Unit 2), but the actual permeability in that interval could be lower. With the aid of gamma ray logs (not shown), we successfully correlated high permeability zones (sandstone and siltstone intervals) and low permeability zones (shale and limestone intervals) between the Venture wells. Correlation of the BPANN permeability curves reveals the relative importance of various seals in terms of their thickness, thickness variation, lateral continuity, and the predicted permeability magnitude. This level of detail and accuracy, as shown by the measured values plotted alongside, would not be possible if we had used only unevenly distributed permeability measurements (preferentially sampled from sandy intervals). Figure 13 is provided to show the distribution of low and high permeability zones within the reservoir intervals. Such information serves as a useful platform, in conjunction with pressure and flow test data, for detailed analysis of producing zone potential and connectivity. Examination of Figure 13 indicates that over this reservoir interval, the important seals appear at the top of No. 2, No. 2A, No. 3, No. 4, No. 6, No. 10, No. 13, and No. 16 sandstones. These shale and limestone seals may have significantly influenced the development of overpressure in the reservoir.
Permeability Prediction with ANN
The BP-ANN permeability model has a number of operational advantages over conventional methods, including MLR and MNLR analysis. In searching for complicated interrelationships among geological/geophysical properties and processes, the main advantages of an ANN approach over conventional methods, apart from its ability to handle nonlinearity, are that no a priori choice of the underlying mathemat-
ical model is necessary, and little effort is required to determine the sensitivity of the model to the chosen input variables. The main drawback of BP-ANN modeling is the amount of time required to train the network. Further ANN experiments with other learning algorithms are needed for more efficient use of this new technique. The database used in this study is rather small. Therefore, our trained ANN is data and area specific. When new data are available, the ANN must be retrained with the updated data
FIG. 11. Continuous predicted permeability (curve), with the trained BP-ANN, and measured permeability values (crosses) at Venture C-62.
Huang et al.
FIG. 12. Crossplot of measured permeability against the pre-
dicted ones with the trained BP-ANN at Venture C-62. The straight line is the RMA line.
sets. Inclusion of data with sufficient geographic, environmental, diagenetic, and geochronological diversity, will lead to a more widely applicable ANN permeability model. The correlation of predicted permeability curves in four Venture wells helped us identify major intrareservoir seals, which may have controlled the development and maintenance of overpressures, and will influence present-day gas production in this field. The important seals are shale and limestone intervals located immediately above the No. 2, No. 2A, No. 3, No. 4, No. 6, No. 10, No. 13, and No. 16 sandstones. ACKNOWLEDGMENTS Z. Huang was supported by a Canadian Government Laboratory Visiting Fellowship. The authors wish to thank the staff of the Canada-Nova Scotia Offshore Petroleum Board for their assistance during data collection. Early versions of the manuscript were read by Al Grant and Matthew Salisbury of Atlantic Geoscience Centre. Constructive suggestions and comments from two anonymous reviewers are appreciated. In this study, we used a publicly available neural network software package called Nevprop, which was developed by the Centre for Biomedical Modeling Research at the University of Nevada. GSC Contribution No. 66394. REFERENCES
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Permeability Prediction with ANN
Fig. 13. Predicted permeability curve and measured permeability values at Venture B-52, Venture B-43, Venture H-22 and Venture B-13 and the inter-well correlation.