2. Background

2.1 Discrete Feature Network Modeling
2.1.1 DFN Approach
2.1.2 Integration of Geological and Hydraulic Data
2.2 Forward Modeling
2.3 Neural Network Theory

2. BACKGROUND

One of the key advantages of the discrete feature network (DFN) approach is that it leverages existing discrete feature data to support reservoir engineering. Although large quantities of expensive data such as FMI logs, flow logs, and outcrop maps are collected as part of typical reservoir characterization projects, it is less common for this data to be used to directly support reservoir development. The software developed in this report provides methods to extract more value out of geological, hydrological, and geophysical data by using that data to support the implementation of discrete feature network models. The use of these DFN models such as that illustrated in Figure 2-1 for reservoir development is the focus of the research project, "Fractured Reservoir Discrete Feature Network Technologies", (Dershowitz et al., 1997).

This section describes three technologies which underlie the data analysis procedures developed in Sections 3 through 6:

• DFN Modeling (Section 2.1),
• Forward Modeling (Section 2.2), and
• Neural Networks (Section 2.3).

This provides the background and context for the data analysis procedures.

2.1 Discrete Feature Network Modeling

The discrete feature network modeling approach is one of three basic approaches available to approximate reservoir formations for flow modeling and analysis (Figure 2-2). The DFN approach has significant advantages, and disadvantages when compared to the alternatives. This section describes features of the DFN approach which are relevant to the DFN data analysis procedures which are the primary focus of this report.

2.1.1 DFN Approach

Discrete feature network (DFN) modeling is an approach in which the discrete features which control flow at a specific scale are modeled explicitly. Thus, at the scale of an individual well, a DFN model would model the fractures intersecting the well, where these provide the primary conduit for oil production (Figure 2-3a). At the scale of a 100 m grid cell, it might be necessary to model fractures on the order of 5 to 100 m (Figure 2-3b). At the reservoir scale, a DFN model would need to explicitly model faults, conductive strata, and bedding planes of 100m to 100 km scales (Figure 2-3c,d).

In addition to these major discrete features, DFN models include the "background" permeability due to rock matrix and smaller, less significant discrete features either as continuum elements (Figure 2-4a), approximate dual porosity coupling terms (Figure 2-4b), or background discrete fracturing (Figure 2-4cFigures/2-4.GIF).

DFN models applied at scales from 10m to 100 km are illustrated in Figure 2-5.

Due to its focus on the discrete features which control flow at each scale, the DFN approach is particularly well suited to oil reservoir applications, which must simultaneously deal with processes at a wide range of scales, from the well bore (0.1m scale) to the field scale. In addition, because the DFN approach explicitly accounts for the geological features which control flow, it is able to directly incorporate geological data which is frequently ignored in many conventional continuum approaches.

Discrete feature models are frequently implemented using elements of order lower than that of the problem being solved. Thus, three-dimensional reservoirs are modeled using networks of two dimensional elements representing discrete fractures (Figure 2-6a), or one-dimensional elements representing flow conduits (Figure 2-6b). This can provide significant computational and theoretical advantages over models which represent these features using continuum elements (Figure 2-6c). However, all three of the models in Figure 2-5 are DFN models, since they explicitly represent key discrete features explicitly.

2.1.1.1 Fracture Generation

The generation of geologically and hydraulically realistic patterns of discrete flow conduits and flow barriers is the key to the DFN approach. DFN fracture generation extrapolates field measurements to produce a model which reproduces the key hydraulic features of the oil field. Figure 2-7 illustrates examples of discrete features which can be modeled by the DFN approach.

The type of fracture generation used depends on the amount of data available to describe a particular feature. DFN models generate the following classes of discrete features to match the level of data available:

· deterministic (know completely), in which case they are entered to the DFN model directly,

· conditioned (known partially), in which case the fractures are generated from statistical populations, conditioned to match the known properties such as the location of borehole intersections or geophysical features,

· stochastic (known statistically), in which case the fractures are generated from a statistical description of their spatial structure, intensity, size, orientation, and hydraulic properties.

Stochastic and conditioned discrete features are generally based on statistical distributions, stochastic spatial processes, and correlation structures. Frequently, these patterns are based on geological information encoded as a 3D spatial field (Figure 2-8).

These spatial patterns can be imported from geological frameworks such as IRAP, StrataModel, and GeoFrame.

2.1.2 Integration of Geological and Hydraulic Data

Because the DFN approach uses geologic and hydraulic data differently from continuum approaches, the use of both DFN and continuum approaches ensures that the value of information obtained from the data will be greater when the two methods are used than if either approach were used in isolation. Figure 2-9 illustrates the manner in which field data is used in the DFN approach to obtain quantitative model parameters.

This report describes software developed to derive the statistical distributions to describe:

• spatial location (Spatial)
• orientation and sets (NeurISIS)
• transmissivity and connectivity (FracDim and Flare)

In addition to these parameters, the compete statistical description of stochastic fractures for fracture generation requires statistical distributions to describe:

• size
• shape
• storativity (porosity)

All of the above parameters can be defined as either independent or correlated distributions.

Table 2-1 summarizes ways in which the DFN approach can increase the value extracted from reservoir geological, geophysical, and hydraulic data by deriving DFN Model parameters.

Table 2-1 Use of Field Data in the Discrete Feature Network Approach

2.2 Forward Modeling

The oil industry has in the past emphasized the "inverse modeling approach". In this approach, models are tweaked to obtain the best possible match to well test or production data. The "inverse approach" favors models with large numbers of adjustable parameters, since these provide the best matches. The "inverse approach" also discourages the direct incorporation of geological information, and particularly discrete feature information, since this constrains the models which can be calibrated.

The Flare analysis which is described in this report is an example of the use of the "forward modeling" approach, which is the complement of the inverse modeling approach. While inverse modeling seeks to identify the single set of model parameters which best matches in situ measurements, forward modeling seeks to establish the range of model parameters which could be consistent with field measurements.

As illustrated in Figure 2-10, forward modeling starts with a prior model for the reservoir. For DFN models, this consists of assumed distributions for:

• fracture spatial structure
• set definitions
• orientation
• size
• hydraulic and storage properties

The model is then implemented based on these assumptions, and several realizations of this model are generated by Monte Carlo simulation. Using these models, a set of simulated field measurements are obtained by simulated FMI logging, geophysics, field mapping, and well testing. Monte Carlo simulation with several realizations is used to reflect the fact that even for known model parameter distributions, there is still considerable uncertainty regarding in situ conditions.

The distributions of simulated measurements are compared against in situ measurements. If a significant proportion of the realizations provide a reasonable match to field measurements, then the assumed DFN model parameters can be considered a possible representation of in situ conditions. The process can then be repeated, modifying the model assumptions consistent with the parameter ranges appropriate to known geological conditions.

One advantage of the forward approach over the inverse approach is that the range of possible reservoir conditions is identified, rather than a single "calibrated" set of parameters. In addition, the parameters of the DFN model are physical parameters such as feature size and orientation, which can be constrained by geological data beyond the purely hydraulic parameters which are generally adjusted in inverse approaches.

2.3 Neural Network Theory

Neural networks are a sophisticated form of non-linear pattern recognition that are used in such diverse areas as stock market analysis, loan application screening, disease diagnosis, and medical expert systems (Eberhart & Dobbins, 1993). They are particularly well-suited for problems in which the input and output variables vary spatially and in scale of importance, are of different mathematical types (e.g., class, ordinal, and continuous variables) and are complexly interrelated. Neural network have found geologic application in a variety of areas including slope stability analysis (Xu & Huang, 1994), rock and soil mechanics (Ellis et al., 1995; Feng, 1995; Lee & Sterling, 1992), fracture network hydrology (La Pointe et al., 1995; Thomas & La Pointe, 1995) and prediction of earthquake intensity and liquefaction (Goh, 1994; Tung et al. 1994).

There are many types of neural networks, but all share a common architecture consisting of neurons and synapses (Figure 2-11). A neuron is simply a node in the network which uses a non-linear transfer function to convert an input signal (value) to an output signal. Neurons are connected by synapses. A synapse takes the output signal from one neuron, multiplies it by a synaptic weight, and passes the modified signal to an adjacent neuron as input. Depending on the number of incoming and outgoing synapses connected to it, a neuron can be classified into one of three categories:

1. Input neurons have zero incoming synapses and one or more outgoing synapses. They are used to represent input variables, and take the variable value as their output.
2. Output neurons have one or more incoming synapses and zero outgoing synapses. They are used to represent output variables, and produce an output signal which equals the predicted variable value.
3. Hidden neurons have one or more incoming synapses and one or more outgoing synapses. They sit between the input and output neurons and pass signals through the network.

A distinct advantage of neural networks over other classification methods is their ability to learn the relative importance and complex interrelations among input and output variables. By changing the neuron transfer functions, the synaptic weights, or the network connectivity, a neural network can be conditioned to provide the expected response for a given input pattern. Once trained, a neural network can then be used to make predictions for input patterns whose correct classification is unknown.

Fracture conductivity studies may be considered an exercise in discriminant analysis: given a variety of geological and environmental parameters, is a particular fracture likely to be conductive or not? Backpropagation neural networks are well-suited for this purpose. In a backpropagation neural network, the input, hidden, and output nodes are arranged in layers. A single input layer, consisting only of input neurons, is connected to an output layer, consisting only of output neurons, through one or more hidden layers, consisting only of hidden neurons (Figure 2-11). Each neuron in a given layer is connected to all neurons in the preceding and following layers by synapses, which are characterized by their synaptic weight.

As an example, consider a fracture conductivity dataset consisting of the input and output parameters listed in Table 2-2. Of the five input variables, three are continuous, one is of class type, and the remaining one is boolean. The single output parameters is of class type, indicating the fracture set. A backpropagation neural network constructed for this problem would require at least five input nodes, one output node, and perhaps a single hidden layer containing about three hidden nodes (Figure 2-12).

Table 2-2 Input and Output Parameters for Fracture Conductivity Study

In a backpropagation network, the network connectivity and the neuron transfer function are held constant, and network behavior is modified by adjusting synaptic weights. Initial synaptic weights are assigned from a random distribution. The neural network is then presented with a series of training patterns, and an error signal is computed from the difference between the network's output signal and the desired output signal. In an iterative procedure known as back-propagation of errors, the synaptic weights connecting each layer are modified so as to reduce the output error. In this way, the network is trained to successfully classify the training data. Any backpropagation network with one or more hidden layers using a non-linear neuron transfer function is capable of learning complex non-linear mappings (Eberhart and Dobbins, 1990). Once trained, a neural network can be used to make predictions for data sample whose output parameters are unknown (e.g., assignment of additional fractures to sets).

Additional information can be obtained by examining the synaptic weights of a trained neural network. These weights provide a record of the network's classification strategy and of the input parameters most important for classification. Synaptic weights can be viewed graphically using a Hinton diagram (Figure 2-13), or examined quantitatively by computing the neural network's relation factors. Of these, the simplest is relation factor one, which indicates the strength of the output signal produced by a single neuron.

The probabilistic neural network (PNN) (Specht, 1990) is based on a combination of probability theory and Bayesian statistics, and was developed primarily for solving multivariate classification problems (Masters, 1993). The definition of fracture sets can be considered a classical classification problem, with the following special attributes:

• The combination of diverse data types, including orientations, vectors, real, integer, and ordinal values, and classes
• The need to consider both upper and lower hemisphere orientation vectors when defining distances between orientation values
• The bias toward definition of sets with distinctive mean orientations and minimized dispersion
• The need to consider geologists' judgment for definition of prior distribution assumptions

The probabilistic neural network algorithm is designed to provide a classification which minimizes mis-classification of fractures to the wrong set. The classification system which has the minimum "cost" of mis-classification is termed "Bayes Optimal" (Parzan, 1962).