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PROCEEDINGS, Thirty-Eighth Workshop on Geothermal Reservoir Engineering Stanford University, Stanford, California, February 11-13, 2013 SGP-TR-198

CONNECTIVITY INDEX AND CONNECTIVITY FIELD TOWARDS FLUID FLOW IN FRACTURE-BASED GEOTHERMAL RESERVOIRS

Younes Fadakar Alghalandis, Chaoshui Xu, Peter A. Dowd

School of Civil, Environmental and Mining Engineering, The University of Adelaide Adelaide, SA, 5005, Australia

e-mail: younes.fadakar@adelaide.edu.au

ABSTRACT

Connectivity measures, such as the connectivity index and the connectivity field, are useful for determining preferential flow directions and flow pathways through fracture networks. However, the current implementation of these measures does not consider the hydraulic properties of the fracture network, which is the issue addressed in this work. We demonstrate that Darcy’s law can be incorporated into the evaluation of these measures using the persistence and aperture properties of fractures in the fracture network. We show that this incorporation can help determine more reliable and accurate flow pathways in the fracture network for three forms of aperture distributions: (i) constant aperture for each fracture cluster (pathway), (ii) variable aperture for each cluster and (iii) variable aperture for each fracture. We also introduce a new concept for the classification of pathways based on the reliability of their assessment, which enhances the understanding of the flow behaviour of fracture-based reservoirs as a result of fracture network expansion processes such as hydraulic stimulation. KEYWORDS stochastic modelling, connectivity measures, flow directions, flow pathways, finite difference

INTRODUCTION

The connectivity of a fracture network is a measure of the potential for fluid flow through the network (Robinson 1983; Renshaw 1996; Sims et al. 2005). It is particularly useful in enhanced geothermal systems (EGS) where the pathways for flow are generated by stimulating the propagation of fractures in such a way as to achieve a connected network (Hanano 2004, Fadakar-A et al. 2013a). Fracture connectivity measures include the Connectivity Index (Xu et al. 2006) and the Connectivity Field (Fadakar-A et al. 2013b). The Connectivity Index (CI) is a probabilistic measure which quantifies the likelihood

of an established connection between two subspaces, so-called the supports, (e.g., injection and production wells) through the fracture network. The Connectivity Field (CF) determines and characterises possible pathways through the fracture network. In addition, CF provides a ranking system for possible pathways in the fracture network with respect to the quality, or strength, of the connectivity. The analysis of CI and CF using stochastic modelling (e.g., Discrete Fracture Network (DFN) method) is a very effective way to assess the main direction of flow in a fracture network (Xu et al. 2006; Fadakar-A et al. 2012; Fadakar-A et al. 2013b). CF and CI are defined by extending the elementary definition of connectivity (Allard 1993; Pardo-Iguzquiza & Dowd 2003; Xu et al. 2006; Renard & Allard 2011; Fadakar-A et al. 2012) which is simply

an indicator function )( yxC 1 defined between

two points (supports) x and y. The indicator measure takes the value 0 or 1, corresponding respectively to two connected or disconnected supports. In practical applications, however, the connectivity measure should incorporate additional parameters, especially for cases where the measure is to be used to evaluate flow through fracture networks in which the physical properties of fractures are critical. In the present work we propose the incorporation of two important fracture properties in the computation of CI and CF measures: the size (length in 2D) and aperture (Renshaw 1996). The proposed method provides a more accurate and more realistic means of assessing the flow characteristics of fracture networks. The fracture size and aperture are incorporated in a manner analogous to Darcy’s law, resulting in weighted forms of the two measures. In addition, we present a comparative study between the weighted forms of CI and CF, using a finite difference method (FD) for the modelling of fluid flow through fracture networks in two dimensions. The results show that significant improvement can be achieved in the accuracy of flow directions in

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fracture networks by means of weighted connectivity measures.

Characterising Fluid Flow through a Fracture

Darcy’s Law

The fluid flow through a single fracture can be modelled effectively using Darcy’s law (Priest 1993; Diodato 1994; CFCFF 1996; Mohais et al. 2011a; Mohais et al. 2011b). To do so, the following assumptions are made: (i) each fracture is comprised of two parallel flat and smooth plates, and (ii) a fracture is not a void area but is filled with a porous medium with a specified equivalent permeability. As a result, Darcy’s law for flow through a single fracture can be expressed as

L

kQ

eq

PA (1)

where keq (m

2) is the fracture permeability, A (m

2) is

the cross-sectional area to flow, ΔP (Pa) is the

pressure drop at two fracture end-points, μ (Pa.s) is

the viscosity of the fluid (e.g., water), and Q (m3.s

-1)

is the total flow-rate. For fractures in two-dimensions, the area A can be replaced by the aperture and the distance L (m) by the persistence (length) of the fracture. Thus, if the pressure head loss (ΔP) can be evaluated for each fracture in the network, the fluid flow within the fracture can be directly evaluated. In the next section, we demonstrate the use of a finite difference method to compute the pressure heads for nodes (i.e., intersection points between fractures) in two dimensions, which will be used later to evaluate the flow direction model using CI and weighted CI.

Modelling Flow Pressure Heads

In this work a finite difference method detailed in Priest (1993) was adapted to model the flow through two-dimensional fracture networks. An overview of the framework is shown in Fig. 1. The boundary heads are given at the points (circle markers) shown in Fig. 1(a), (b) and (c); and they can be easily set to satisfy the required pressure head conditions. These boundary references can be assigned to the polygon of the study region or to a circle surrounding the study area as shown in Fig. 1(b) and (c). The latter provides a generic and effective solution for studying the fluid flow in any direction in the fracture network regardless of the complexity of fracture network boundaries. The values shown are pressure values ranging from 0 to 1 and are solely for the purpose of demonstration. The flow direction can be analysed under any required rotations of the reference boundary points. After setting the boundary node locations and then the boundary conditions, the

fracture network is analysed for fracture connections. As a result, the pathways shown in Fig. 1(d) are obtained. During the modelling all isolated and partially isolated fractures are eliminated from the network to facilitate further evaluations of the determined pathways. Furthermore, all connected fractures to the pathways are trimmed to remove dead-end pathways. Figure 1(e) shows the final pathways for the presented example. The intersections of fractures with boundary edges generate secondary boundary (outer) nodes, which are assigned head values according to their distance from the initial defined boundary points. Intersections inside the region define inner nodes for which the hydraulic pressure heads are unknown and have to be solved. As a result of this fracture network analysis, a system of equations (AX=B) is set up for all inner nodes as follows.

]4,3,2[,

1

1

n

C

HC

Hn

i

ij

n

i

iij

j (2)

where Hj is the pressure head at node j and Cij is the conductance (Priest 1993) for pathway between nodes i and j which is defined as

Lv

bagCij

12

3

(3)

in which g (9.8 m.s

-2) is the acceleration due to

gravity, a (m) is the aperture of the fracture, b (m) is the third dimension of the fracture (equal to 1 for two-dimensional cases), v (10

-6 m

2.s

-1 for water) is

the kinematic viscosity, and L (m) is the length of fracture. In the formula for Hj, on the assumption that no more than two fractures may pass through a single point, the number n is bounded between 2 and 4 corresponding to the number of pathways departing from a node (i.e., neighbouring edges). Additional assumptions are (i) the rock material is impermeable, inert and incompressible, (ii) the fluid remains continuous and incompressible and (iii) the fracture surfaces are smooth and parallel (Priest 1993). The total flow-rate for a fracture can be found using Q = CΔH. For the example given in Fig. 1(f) the numbers of inner and outer nodes are 55 and 20, respectively, which results in a system of 55 equations. After solving the equations, a further post-processing stage is used to generate a smooth pressure spectrum for pathways between every two adjacent nodes as shown in Fig. 1(f).

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Fig 1: Framework for studying directional flow in two-dimensional fracture networks: (a) four pressure head values set in the corners providing simple left-to-right pressure gradient, (b) four pressure head values set on main axial directions (c) 24 pressure head values set on a circle calculated via the Hamming filter, (d) marking isolated/partially isolated fractures after intersections analysis, (e) extracted and trimmed final pathways, and (f) pressure head solution for the boundary setting as of (c).

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Modelling flow for a particular direction can be done by rotating the reference boundary points against the centre of the study region which, in this example, is the geometrical centroid. We developed a model to extract directional flow information (Fig. 2), which honours simultaneously the orientation of the fractures (β), their lengths and apertures, and pressure head values obtained by the method described above. The model is defined as

)(max dfmax (4)

in which

)()()( ,1,1,1 half kmn πππ (5)

where )(, qpπ is a projection operator to the range

[p,q] applied on lengths of fractures (l), apertures (a)

and head values (h) of two end-points for each

fracture, and is the nearest integer function. For

the fracture network in the demonstration example (Fig. 1) the most dominant direction of flow is determined to be in the NW-SE direction, as shown in Fig. 2. The second major direction is NE-SW. In the rose diagrams shown in Fig. 2, two bin sizes, 15 and 10 degrees, were used to analyse the binning effect (Wells 2000). It appears that they are consistent from an overall orientation perspective. The full directional coverage shown in Fig. 2 on the right emphasises the dominancy of the two main flow directions in the fracture network: ~135 and ~45 degrees. A simplified flowchart of the proposed framework is shown in Fig. 3. In our implementation we used Graph Theory algorithms to facilitate the nodal representation of fracture intersections, their neighbourhoods, degrees of connection and edge (path) weighting.

Fig 2: The resulting orientations for flow are shown as rose diagrams: grey-filled for 15 degree bins and solid-black line for 10 degree bins. The boundary conditions were as in Fig. 1(c). For full directional coverage the boundary references were rotated by 15 degree steps; the resulting orientations are shown on the right.

Fig 3: The framework for a finite difference method used to model the flow through fracture networks in two dimensions. The bottom section confined by dashed lines is repeated for any direction to obtain full coverage of orientations in the inlet pressure heads.

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Fig 4: (left) Directional SCI verifies homogeneity for the isotropic fracture network. (right) The anisotropy in the anisotropic fracture networks was clearly depicted by Directional SCI.

Fig 5: The variation in the length of pathway and also in the aperture of the connected fractures in the pathway affects the heat exchange process in the EGS reservoir. Using CI the two pathways are reported as [1, 1] but using WCI they are reported

as }]1..0[,|),{( Rbaba depending on the lengths and apertures of fractures forming the pathways.

CI AND CF FOR FLOW MODELLING

Preferential Flow Directions using CI

Directional stationary CI (SCI) can reveal any anisotropy in the connectivity affected by fracture orientations of the fracture network (Xu et al. 2006; Fadakar-A et al. 2012). For example, Fig. 4 shows the CI curves for two isotropic and anisotropic fracture networks. The anisotropy is due to preferred orientations in the fracture network and is specified by setting higher values for the concentration

parameter κ (≥10) in the von Mises-Fisher distribution function. In the right subplot of Fig. 4, higher CI values suggest that the direction of higher fracture connectivity is a preferred flow direction, that is, NE-SW.

Incorporating Length and Aperture in CI: WCI

Incorporating the length and aperture of fractures forming the pathways between two connected supports (cells) increases the reliability of CI as a measure for flow assessment as the new measure is more closely related to realistic flow conditions as discussed in the previous sections. The example shown in Fig. 5 demonstrates the problem with CI, which does not take into account the length and

aperture variation of fractures forming the connection pathway; these features are the most influential factors in the response of the network (Renshaw 1996) to applications such as flow and heat extraction in geothermal energy systems (Mohais et al. 2011b). Figure 6 shows the shortest path length (SPL) measure computed simultaneously during the evaluation of CI for every pair of supports by means of Dijkstra’s algorithm from Graph Theory. It is apparent that realistic flow evaluation through CI requires consideration of the length and aperture of fractures in the pathways connecting the two cells (wells). The idea is implemented using the newly proposed Weighted CI (WCI). It has also been reported that fracture apertures are positively correlated with fracture lengths in a fracture network (Koike & Ichikawa 2006). That is, in the simulation and modelling, a wider aperture for longer fractures can be assumed. Further complexity can be incorporated by using Marked Point Processes in the relationship between length and aperture (Xu et al. 2007). In the demonstration in Fig. 6, the variation of SPL for different distances h depicts a curve that is more or less linear for low h values. The fluctuation in the SPL curve for higher h values is, however, difficult to explain.

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Fig 6: CI and SPL vs. h; larger h values result in lower CI values and higher SPL values. The fluctuation on the right of the SPL curve can be explained as an edge effect caused by the geometrical boundaries of the fracture networks. This computation was conducted on 60 realizations on a grid of 25×25 and for each h value 100 samples were taken.

Fig 7: Variations in the length and aperture of the pathways affect the flow direction depending on the configuration established by the interconnection between fractures and the support locations. (B) is the response of CI in the evaluation of flow direction of a system shown in (A) while (C) to (F) are assessments by means of WCI. (G) shows possible resulting flow directions.

Fig 8: Flow (velocity factor) through fractures using (top) Lattice Boltzmann method and (bottom) Finite Element method for (A) equal aperture, (B) variable aperture per pathway, and (C) variable aperture per fracture. FEM was done using the COMSOL software package.

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Fig 9: The framework and the code developed in this research have been checked extensively by applying it to various typical networks. As shown on the right the finite difference (FD), CI and WCI methods have correctly determined the flow direction for a particular case shown on the left.

One observation is that at locations close to the edges of the study region there is less complexity in the pattern of intersecting fractures. Thus, in some cases it is possible to have a shorter SPL than at smaller distances. The SPL curve explains the complexity of fracture intersections in the study region as a function of distance h. We propose a weighting factor (w) based on the length and aperture of fractures forming the pathways between two supports as

m

i

m

ii

i

i l

a

lw

1 1

1)1

( (6)

where a is the aperture and l is the length of fractures in pathways connecting two supports for each connectivity analysis. The connectivity indices are weighted by w resulting in the WCI measure. Figure 7 helps explain the improvement of using WCI over CI for modelling flow directions. For comparison, we conducted fluid flow modelling using two well-known numerical methods: Lattice Boltzmann (LBM; Xu et al. 2005) and Finite Elements methods (FEM). Three typical cases including: (i) equal aperture for pathways, (ii) variable aperture per pathway and (iii) variable aperture per fracture were prepared as matrices for LBM and also the same structures of fractures were prepared as vector files for FEM using COMSOL software. As demonstrated in Fig. 8 the effects of variations in the length and aperture are significant in the ultimate preferred pathway for fluid flow in each case.

Comparison of the results

A full coverage of orientation, that is, from 0 to 180 degrees with a step size 10 degrees was used for a comparative study of flow directions using the three methods: CI, WCI and FD. The distance h for CI and WCI was set to 0.2 (relative to the 1×1 extent of the study region) and the support size was set to 0.025. The number of sampling pairs of point with distance h in the study region was 1,500 per realization. The

resulting orientations were normalised to [0.2, 1] for the purpose of comparing the results from the three modelling methods. Simple fracture networks were created to test the consistency and accuracy of the methods one of which is shown in Fig. 9. Figure 10 shows the results from applications of CI, WCI and FD to 70 realizations generated using the DFN approach with the following model settings: the locations of fractures were simulated by a homogeneous Poisson point process with point density of λ=120; the orientations of fractures were derived from a von Mises-Fisher distribution (κ=1); fracture lengths were generated from an exponential distribution with λ=0.1. The FD rose diagrams are shown in blue the CI in red and the WCI in green. WCI consistently shows better directional correlation with FD compared to CI, apart from a few cases where the results of both CI and WCI are less similar to FD (bottom row figures in Fig. 10). Taking FD as the correct solution, Fig. 11 shows the errors incurred when using CI and WCI for the assessment of flow directions.

Preferential Flow Pathways using CF

The Connectivity Field (CF) is useful for determining and characterising preferential flow pathways in fracture networks (Fadakar-A et al. 2012; Fadakar-A et al. 2013b). CF is defined as

R

CF dvC (7)

where for a two-dimensional grid of size nm covering the region R, the connection measure

}..1,..1,{ njmiCC ij

is calculated as the

indicator values between the thij cell and all other

cells (with support v). This generates a total of nm

sets of connectivity matrices.

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Fig 10: The eight resulting flow directions from 70 realizations of a fracture network using three methods: CI as red, WCI as green and FD as blue rose diagrams. WCI matches the FD roses significantly better than CI.

Fig 11: Histograms of orientation errors for CI and WCI based on 70 realizations of a fracture network model; WCI provides noticeably lower error values compared to CI; error statistics

include the minimum ( e ), mean ( e ), mode ( e~ ) and maximum

( e ). This suggests WCI consistently provides more accurate

results than CI for modelling flow directions.

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Fig 12: The procedure of determining ranked flow pathways using the CF measure.

An extension of CF, the Probabilistic CF (PCF), applies the concept of CF directly to the fracture network model by means of Monte Carlo simulations (Dowd et al. 2007; Xu & Dowd 2010; Fadakar-A et al. 2012). It has been shown that PCF is closely related to an extension of CI: the CI Field (CIF), as follows when written in discrete form:

RR

R

CCICIF

CCFPCF

k

i

k

i

k

i

i

k

kk

1

11

111

111

(8)

where C is a connectivity measure, η is a standardization factor, R is the region of study, and k is the number of realizations from Monte Carlo simulation. The concept of incorporating length and aperture of fractures proposed for CI can also applied to CF, hence the weighted CF, WCF, to enhance its capability in connectivity assessment. In the following section, we demonstrate the use of CF in determining and characterising fracture networks when applied either to a single realization or to a model (further discussions can be found in Fadakar-A et al. (2012). Due to limitations on the size of this

manuscript, the connectivity assessments using WCF will be covered in future publications. An example of CF application for the evaluation of potential pathways in a realization of fracture network is shown in Fig. 12. Pathways are ranked on the basis of the values of CF, representing the strength of the connectivity in the network. In addition, the pathways suggest that NE-SW is the dominant direction for flow in the network. Further potential applications of the proposed ranking system based on CF values could be for the prediction of the final extended fracture network after the stimulation process, i.e., regions with higher CF values are areas for potentially more efficient fracture stimulation. Further information can be extracted from the CF map with regard to regions with high contrasts of CF values shown as small circles in Fig. 12. These areas are the most critical regions in the network that will affect the extent of the fracture stimulation. They are very important areas on which the stimulation process should be focused in order to create an extensive fractured reservoir. From this perspective, CF analysis provides critical information for prediction, planning and connectivity assessment of fracture network reservoirs.

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CONCLUDING REMARKS

Connectivity Index (CI) and Connectivity Field (CF) are two connectivity measures that have been shown to be useful in characterising the connectivity of fracture networks and in providing practical assessments such as flow characteristics for fracture-based geothermal reservoirs (e.g., EGS). In this paper we proposed a method to incorporate the size and aperture of fractures in the two methods. Incorporating these properties makes it possible to link much more realistically the connectivity of fracture networks to fluid flow (using a method analogous to Darcy’s law), which in turn significantly increases the reliability of the connectivity analysis. In particular, we discussed the application of the proposed measures in directional flow analysis with the benefit of increased accuracy and reliability. The rose diagrams resulting from CI and WCI were compared to results from the finite difference method. It was shown that the incorporation of the length and aperture of fractures improves significantly the consistency between the results from the connectivity measure and the finite difference solutions. We also demonstrated the application of CF to the direct determination of flow pathways in fracture networks and to providing a ranking system for pathways that can be used to predict and assess the extent of fracture networks as the result of a multi-stage hydraulic fracture stimulation process. Acknowledgements The work reported here was funded by Australian Research Council Discovery Project grant DP110104766.

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