Evaluating Flow in Fractal-Fracture Networks: Effect of Variable Aperture

While fractal models are often employed for describing the geometry of fracture networks, a constant aperture is mostly assigned to all the fractures when such models are flow simulated. In nature however, almost all fracture networks exhibit variable aperture values and it is this fracture aperture that controls the conductivity of individual fractures as described by the well-known cubic-law. It would therefore be of practical interest to investigate flow patterns in a fractal-fracture network where the apertures scale in accordance to their position in the hierarchy of the fractal. A set of synthetic fractal-fracture networks and two well-connected natural fracture maps that belong to the same fractal system are used for this purpose. A set of dominant sub-networks are generated from a given fractal-fracture map by systematically removing the smaller fracture segments with narrow apertures. The connectivity values of the fractal-fracture networks and their respective dominant sub-networks are then computed. Although a large number of fractures with smaller aperture are eliminated, no significant decrease is seen in the connectivity of the dominant sub-networks. A streamline simulator based on Darcy's law is used for flow simulating the fracture networks, which are conceptualized as two-dimensional fracture continuum models. A single high porosity value is assigned to all the fractures. The permeability assigned to fractures within the continuum model is based on their aperture values and there is nearly no matrix porosity and permeability. The recovery profiles and time-of-flight plots for each network and its dominant sub-networks at different time steps are compared. The results from both the synthetic networks and the natural data show that there is no significant decrease in fluid recovery in the dominant sub-networks compared to their respective parent fractal-fracture networks. It may therefore be concluded that in the case of such hierarchical fractal-fracture systems with scaled aperture, the smaller fractures do not significantly contribute to connectivity or fluid flow. In terms of decision making, this result will aid geoscientists and engineers in identifying only those fractures that ultimately matter in evaluating the flow recovery, thus building models that are computationally less expensive while being geologically realistic.

Influence of Aperture Distribution on Flow in Fractal-Fracture Networks

<p>While fractal models are often employed for describing the geometry of fracture networks, a constant aperture is mostly assigned to all the fractures when such models are flow simulated. While network geometry controls connectivity, it is fracture aperture that controls the conductivity of individual fractures as described by the well-known cubic-law. It would therefore be of practical interest to investigate flow patterns in a fractal-fracture network where the apertures also scale as a power-law in accordance to their position in the hierarchy of the fractal. A set of synthetic fractal-fracture networks and two well-connected natural fracture maps that belong to the same fractal system are used for this purpose. The former, with connectivity above the percolation threshold, are generated by spatially locating the fractured and un-fractured blocks in a deterministic and random manner. A set of sub-networks are generated from a given fractal-fracture map by systematically removing the smaller fracture segments. A streamline simulator based on Darcy's law is used for flow simulating the fracture networks, which are conceptualized as two-dimensional fracture continuum models. Porosity and permeability are assigned to a fracture within the continuum model based on its aperture value and there is nearly no matrix porosity or permeability. The recovery profiles and time-of-flight values for each network and its dominant sub-networks at different time steps are compared.</p><p>The results from both the synthetic networks and the natural maps show that there is no significant decrease in recovery in the dominant sub-networks of a given fractal-fracture network. It may therefore be concluded that in the case of such hierarchical fractal-fracture systems with scaled aperture, the smaller fractures do not significantly contribute to the fluid flow.</p><p><strong>Key-words: </strong>Fractal-fracture; Connectivity; Aperture; Dominant Sub-networks; Streamline Simulator; Recovery</p>

Clustering, Connectivity and Flow Responses of Deterministic Fractal-Fracture Networks

It is well known that fracture networks display self-similarity in many cases and the connectivity and flow behavior of such networks are influenced by their respective fractal dimensions. In the past, the concept of lacunarity, a parameter that quantifies spatial clustering, has been implemented by one of the authors in order to demonstrate that a set of seven nested natural fracture maps belonging to a single fractal system, but of different visual appearances, have different clustering attributes. Any scale-dependency in the clustering of fractures will also likely have significant implications for flow processes that depend on fracture connectivity. It is therefore important to address the question as to whether the fractal dimension alone serves as a reasonable proxy for the connectivity of a fractal-fracture network and hence, its flow response or, if it is the lacunarity, a measure of scale-dependent clustering, that may be used instead. The present study attempts to address this issue by exploring possible relationships between the fractal dimension, lacunarity and connectivity of fractal-fracture networks. It also endeavors to study the relationship between lacunarity and fluid flow in such fractal-fracture networks. A set of deterministic fractal-fracture models generated at different iterations and, that have the same theoretical fractal dimension are used for this purpose. The results indicate that such deterministic synthetic fractal-fracture networks with the same theoretical fractal dimension have differences in their connectivity and that the latter is fairly correlated with lacunarity. Additionally, the flow simulation results imply that lacunarity influences flow patterns in fracture networks. Therefore, it may be concluded that at least in synthetic fractal-fracture networks, rather than fractal dimension, it is the lacunarity or scale-dependent clustering attribute that controls the connectivity and hence the flow behavior.

Pressure-Transient Behavior of Double-Porosity Reservoirs with Transient Interporosity Transfer with Fractal Matrix Blocks

Summary Double-porosity/naturally fractured reservoir models have traditionally been used to represent the flow and pressure behavior for highly fractured carbonate reservoirs. Given that unconventional reservoirs such as shale-oil/gas reservoirs might not be considered to be multiporosity media, the use of the traditional/classical “double-porosity” models might not be adequate (or appropriate). The recent development of anomalous diffusion models has opened the possibility of adapting double-porosity models to estimate reservoir (and related) parameters for unconventional reservoirs. The primary objective of this work is to develop and demonstrate analytical reservoir models that provide (possible) physical explanations for the anomalous diffusion phenomenon. The models considering anomalous diffusion in reservoirs with Euclidean shape are developed using a convolved (i.e., time-dependent) version of Darcy's law. The use of these models can yield a power-law (straight-line) behavior for the pressure and/or rate performance, similar to the fractal reservoir models. The main advantage of using anomalous diffusion models compared with models considering fractal geometry is the reduction from two parameters (i.e., the fractal dimension and the conductivity index) to only one parameter (i.e., the anomalous diffusion exponent). However, the anomalous diffusion exponent does not provide information regarding the geometry or spatial distribution of the reservoir properties. To provide an alternative explanation for the anomalous diffusion phenomenon in petroleum reservoirs, we have developed double-porosity models considering matrix blocks with fractal geometry and fracture networks with either radial or fractal fracture networks. The flows inside the matrix blocks and the fractal fracture network assume that Darcy’s law is valid in its space-dependent (fractal) form, whereas the classical version of Darcy’s law is assumed for the radial-fracture-network case. The transient interporosity transfer is modeled using the classical convolution schemes given in the literature. We have defined the matrix blocks to be “infinite-acting” to represent the nano/micropermeability of shale reservoir. For the system defined by a fractal fracture network and infinite-acting fractal matrix blocks, we have investigated the influence of the fractal parameters (both matrix and fracture network) in the pressure- and rate-transient performance behaviors. We have defined the flow periods that can be observed in these sorts of systems and we have developed analytical solutions for pressure-transient analysis. We demonstrate that the use of the convolved version of Darcy’s law results in a model very similar to the diffusivity equation for double-porosity systems (which incorporates transient interporosity flow). In performing this work, we establish the following observations/conclusions derived from our new solutions: We find that the assumption of a well producing at variable rate (time-dependent inner-boundary condition) has a more-significant effect on the pressure (and derivative) functions and obscures the effects of the properties of the reservoir. We demonstrate that the anomalous-diffusion-phenomena model proposed for unconventional reservoirs can be directly related to the multiporosity concept model. Pressure and pressure-derivative responses can be used in the diagnosis of flow periods and in the evaluation/estimation of reservoir parameters in unconventional reservoirs.

Clustering and Connectivity of Fractal-Fracture Networks: Are they related?

<p>It well known that fracture networks display self-similarity in many cases and the connectivity and flow behavior of such networks are influenced by their respective fractal dimensions. One of the authors have previously implemented the concept of lacunarity, a parameter that quantifies spatial clustering, to demonstrate that a set of 7 nested natural fracture maps belonging to a single fractal system, but different visual appearances have different clustering attributes. Any scale-dependency in the clustering of fractures will also likely have significant implications for flow processes that depend upon fracture connectivity. It is therefore important to address the question as to whether the fractal dimension serves as a reasonable proxy  for the connectivity of a fractal-fracture network or is it the lacunarity parameter that may be used instead. The present study attempts to address this issue by studying the clustering behavior (lacunarity) and connectivity of fractal-fracture patterns. We compare the set of 7 nested fracture maps mentioned earlier which belong to a single fractal system, in terms of their lacunarity and connectivity values. The results indicate that while the maps that have the same fractal dimension have almost similar connectivity values, there exist subtle differences such that both the connectivity and clustering change systematically with the scale at which the networks are mapped. It is further noted that there appears to be an exact correlation between clustering and connectivity values. Therefore, it may be concluded that rather than fractal dimension, it is the lacunarity or scale-dependent clustering attribute that control connectivity in fracture networks.</p>