Mathematical algorithm development and parametric ...

Mathematical algorithm development and parametric studies with the GEOFRAC three-dimensional stochastic model of natural rock fracture systems

Violeta M. Ivanova, Rita Sousa, Brian Murrihy, and Herbert H. Einstein, Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA.

Corresponding author: Violeta M. Ivanova, 77 Massachusetts Avenue, NE48-308, Massachusetts Institute of Technology, Cambridge, MA 02139. ([email protected])

Key points •

Spatial stochastic model represents natural rock fracture systems

•

New mathematical algorithms model fracture intensity and connectivity

•

Parametric study shows how fracture intensity and size affect connectivity

Index terms •

5104 Fracture and flow

•

3265 Stochastic processes

•

3252 Spatial analysis

•

3275 Uncertainty quantification

•

0545 Modeling

Keywords Fractures, stochastic, model, spatial, connectivity

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Abstract This paper presents results from research conducted at MIT during 2010-2012 on modeling of natural rock fracture systems with the GEOFRAC three-dimensional stochastic model. Following a background summary of discrete fracture network models and a brief introduction of GEOFRAC, the paper provides a thorough description of the newly developed mathematical and computer algorithms for fracture intensity, aperture, and intersection representation, which have been implemented in MATLAB. The new methods optimize, in particular, the representation of fracture intensity in terms of cumulative fracture area per unit volume, P32, via the PoissonVoronoi tessellation of planes into polygonal fracture shapes. In addition, fracture apertures now can be represented probabilistically or deterministically, whereas the newly implemented intersection algorithms allow for computing discrete pathways of interconnected fractures. In conclusion, results from a statistical parametric study, which was conducted with the enhanced GEOFRAC model and the new MATLAB-based Monte Carlo simulation program FRACSIM, demonstrate how fracture intensity, size, and orientations influence fracture connectivity.

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1. Introduction and Background 1.1

Fracture pattern models Geologists and engineers have long recognized the need to spatially characterize rock

fractures (sometimes also called joints) in terms of their orientation, spacing, size, shape, and aperture, and to measure these characteristics [Solomon, 1911; Stini, 1925]. A logical progression from describing individual characteristics is to use aggregate characterization, which is also referred to as “joint system models” or “discrete fracture network (DFN) models”. Dershowitz and Einstein [1988] provide a review of the development of these models, starting from largely deterministic ones [e.g., Irmay, 1957; Snow, 1968] to fully stochastic ones. Important steps in the development of stochastic models were the Baecher disk model [Baecher et al., 1977] and models based on Poisson plane - line processes, such as those by Priest and Hudson [1976] and Veneziano [1978]. Dershowitz [1984] further explored the Poisson plane-line approach and later implemented many of the then existing fracture characterization models into FRACMAN [e.g., see Dershowitz et al. 1993], which has since evolved into the most widely used DFN software. More recent developments in the DFN domain address the inclusion of additional fracture characteristics and especially attempt to make more realistic representations of particular geologies [e.g., La Pointe, 2012]. Several reviews of DFN models also include their application to flow modeling [e.g., National Research Council, 1996; Jing and Hudson, 2002]. Dershowitz [1984] introduced several expressions for characterizing fractured rock masses: most importantly, P32, the total fracture area per volume; as well as a number of connectivity measures. Fracture connectivity indicates if interconnected fracture patterns exist; these are important with regard to flow or instabilities in a rock mass. Fracture connectivity depends on the intensity of fractures: naturally, the more fractures and the larger they are, i.e., 3

the higher the fracture intensity, the more likely they are to intersect and form discrete pathways for flow. Dershowitz and Herda [1992] review the possible fracture intensity measures in 1D (e.g., along a scan line), 2D (e.g., on a trace plane), and 3D (within a rock volume). Fracture intensity P32, the total area of fractures per unit volume, is a 3D measure that is independent of either the size of the modeling volume or the size and shape of individual fractures. These properties of P32 allow us to represent fracture intensity in a modeling volume of any shape and size as well as to decouple the modeling of fracture intensity from the methods of generation of individual fracture shapes and sizes. The GEOFRAC mathematical and computer model, which originates from the MIT work on DFN and further builds on the aforementioned ongoing research process, takes advantage of these properties of P32. After a brief introduction of the original model below, this paper will focus on GEOFRAC’s new developments and applications. 1.2

Modeling of fracture systems with GEOFRAC The GEOFRAC mathematical and computer model represents a rock fracture system as a

network of interconnected convex polygons, which are created as members of sets via a sequence of stochastic processes in a 3D modeling volume. These fracture sets may be hierarchically related to, or independent of, one another, as defined by their geologic setting. The model’s spatial stochastic processes, which were first implemented computationally by Ivanova [1995] and further enhanced by Ivanova [1998] and Meyer [1999] (see also [Meyer et al., 1999]), are based on a theoretical model by Veneziano [1978] who proposed Poisson planes, tessellated by Poisson lines into polygons, as a way to reproduce systems of rock fractures. Figure 1 shows the stochastic processes of the original GEOFRAC model while the brief summary below provides the basis for the new developments. For a more detailed description of the early GEOFRAC model, see [Ivanova and Einstein, 2004]. 4

•

The primary stochastic process consists of a Poisson plane network and represents the main stress field orientation and global fracture intensity.

•

The secondary stochastic process consists of a Poisson line tessellation of the planes, followed by marking of the created polygons as fractured or intact rock, and represents fracture intensity variation by size and location.

•

The tertiary stochastic process consists of random polygon translation and rotation, and represents local variations of fracture positions and orientations.

To measure fracture intensity, GEOFRAC uses cumulative fracture area per rock volume, P32, which was introduced above. The mathematical expressions that relate P32 and the expected area of fracture sizes, E[A], to the Poisson plane and line processes that create the polygonal fracture shapes were derived on the basis of theoretical work by Miles [1964, 1970] and Veneziano [1978] and on additional geometric properties established via computer simulations by Ivanova [1995] with the eponymous C++ program that implemented GEOFRAC. In the original model, P32 and E[A] also depend on marking the polygons by shape and size in the secondary process. Specifically, GEOFRAC represents fracture intensity P32 as

P32 = γµ and the expected area E[A] of polygon-fractures as

E [ A] =

C Aπ

λ2

(2)

Above, µ is the intensity of the Poisson plane process; λ is the intensity of the Poisson line tessellation; and γ and CA are empirical coefficients, established via extensive simulations with GEOFRAC, which reflect how polygons created by the Poisson line tessellation are either marked as fractures or discarded, according to their shape and size.

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Figure 1. Stochastic processes of fracture set generation in the original GEOFRAC model. (a) Primary process: Poisson planes are generated in a 3D modeling volume. (b) Secondary process: Poisson lines tessellate the planes into polygons, which are then marked as fractures by their shape and size. (c) Tertiary process: random translation and rotation, respectively, change the percent of coplanar and parallel fractures.

Fracture intensity variation can be further refined by zone marking, i.e., by conditioning the marking probability on polygon locations or rock properties. For example, the fracture system in Figure 2 consists of two sets, the intensity of which varies in space in relation to each other.

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The GEOFRAC model was further enhanced with specific algorithms to represent fracture systems related to folds [Ivanova, 1998] and fault zones [Meyer, 1999]. These algorithms were applied in case studies [Meyer et al., 1999; Ivanova and Einstein, 2004], which demonstrated the model’s capability to represent fracture orientations related to general stress directions and local geologic structures and fracture intensity as a function of rock properties.

Figure 2. Fracture system modeling with GEOFRAC. (a) An independent fracture set of large fractures with intensity defined by rock properties. (b) A dependent fracture set of smaller fractures with intensity defined by the proximity to fractures of Set 1. (c) A fracture system composed of Set 1 and 2 Set 2.

1.3

Room for improvement of the original GEOFRAC model Some algorithms that generated P32 in the original model were very inefficient.

Especially, the Poisson line tessellation (PLT) creates a large percent of polygons that cannot represent fractures due to their shapes: triangles and quadrangles that are too elongated and/or include sharp angles; therefore they must be discarded during the simulations. Empirically, it

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was established that no more than 40% of the area of a 2D region could be tessellated with PLT into polygons that have fracture-like shapes. In simulations conducted to study the properties of the PLT, over 80% of polygons routinely were discarded due to their shapes [Ivanova, 1998]. Our research focus in the 1990s was on aggregate characterization of the geometry of fracture systems via stochastic processes related to underlying fracturing in nature. However, while this initial research included modeling of fracture areas, it considered neither fracture apertures, nor rock properties such as permeability that control flow and transport. Finally, we implemented the original GEOFRAC model in C++ for a UNIX environment prevalent at MIT in the 1990s. Over the years, maintaining the old code became unsustainable. These and other considerations led to the new developments, which we report next.

2. New developments for the GEOFRAC fracture system model New advances of GEOFRAC include programming its mathematical algorithms in MATLAB to take advantage of built-in routines that are optimized for runtime and memory storage; implementing the Poisson-Voronoi Tessellation in the secondary stochastic process to improve the efficiency of fracture size and shape generation; and developing algorithms to represent fracture apertures and intersections in order to assess flow through interconnected fractures. Since MATLAB is available on all computer platforms and widely used for engineering computations, this also facilitates potential applications of GEOFRAC by others. 2.1

The new MATLAB code of GEOFRAC

Coding GEOFRAC’s original and new algorithms in MATLAB made the numerical model execute much more efficiently than the old C++ code. MATLAB® by MathWorks, Inc., is an

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algorithm-development environment that enables extensive computations via thousands of builtin routines that are optimized for memory storage and runtime; e.g., see [Attaway, 2012]. In our MATLAB code a fracture set is stored in a structure cell array: a data type that provides the means to store hierarchical data in a single entity. Figure 3 illustrates the FractureSet structure and its five fields (data containers); every field stores an array of data about the polygons in the fracture set, such as vertex and center coordinates, area, radius, and aperture.

Figure 3. The FractureSet structure cell array in the new MATLAB-based GEOFRAC. The FractureSet structure contains five fields (data containers): Polygon – an array of m 3xn matrices, where m is the number of fractures and every 3xn matrix represents a fracture-polygon that is defined by n vertices; Area – an mx1 array that contains the respective areas of polygons in Polygon; C – an array of m 1x3 vectors that contain the coordinates of polygon centers; Re – an mx1 array that contains the equivalent radii of the polygons (an equivalent radius is the radius of a circle with area equal to that of the polygon); H – an mx1 array that contains the average apertures of the polygon-fractures. In the example above, there is only one fracture of pentagonal shape, therefore Polygon contains one 3x5 matrix, C contains one 1x3 vector, and C, Re, and Area are 1x1 arrays. 9

2.2

Poisson - Voronoi Tessellation

The Poisson-Voronoi Tessellation (PVT), which is a Voronoi tessellation based on a Poisson point process, was implemented for the secondary stochastic process in GEOFRAC. The PVT is constructed following three steps, which are illustrated in Figure 4. 1. A Poisson Point Process with intensity ρ creates a set of points, P, on a plane. 2. A Delaunay Triangulation on the Poisson point set P divides the plane into triangles. 3. A Voronoi Tessellation, created by the intersections of the orthogonal bisectors of the triangular segments of the Delaunay Tessellation, divides the plane into polygons.

Figure 4. Creating fracture-polygons via the Voronoi tessellation and Delaunay triangulation. (a) Schematic detail. (b) GEOFRAC simulation of the tessellation.

In depth presentation of the Delauney triangulation and the Voronoi tessellation (also called “Voronoi Diagram” and “Dirichlet Tessellation”) can be found in [Møller, 1994] and [Okabe et al., 2000]. The statistical properties of the population of polygons created by the Voronoi tessellation are known from theoretical derivations (e.g., see [Miles, 1970]) and simulations of

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millions of Voronoi “cells” [Meijering, 1953; Gilbert, 1962; Hinde and Miles, 1980; Tanemura, 2003). Specifically, the expected number of a polygon’s vertices, E[N], and the expected area and standard deviation, E[A] and σA, respectively, of polygon areas created by a Voronoi tessellation based on a Poisson point process of intensity ρ can be computed as follows:

E[N ] = 6 E[A] =

1

(3)

ρ

(

)

σ A = E  A 2  − E [ A ] = 2

0.529

ρ

The properties of the polygons created by the Poisson-Voronoi tessellation make it very suitable for reproducing fracture-like shapes. For example, mathematically, it is known that the Delauney Triangulation maximizes the minimum angle of triangles, hence “skinny” triangles with very acute angles are unlikely; this leads to Voronoi polygons (also called “cells”) that tend to be “round”, i.e., they tend to be equilateral and without very sharp angles. E[N] = 6 in the equation above indicates that polygons tend to be hexagons. (For comparison, the Poisson Line Tessellation from the earlier version of GEOFRAC produces polygons with E[N] = 4, over 60% of which are too elongated or otherwise geometrically unsuitable for modeling natural fractures.) Also, the number of Voronoi triangles is very small compared to polygons with more vertices and those few triangles tend to be equilateral; for a detailed study, see [Tanemura, 2003]. The geometric properties of the polygons created by the Poisson-Voronoi tessellation allow us to make the assumption that all polygons are good for representing fractures, since the number of “bad” polygons is negligible compared to those with fracture-like shapes. This eliminates the need for marking of polygons by shape or size in the secondary process of GEOFRAC. Hence, in Eq. 1, the coefficient γ=1, i.e., the entire Poisson planes could be tessellated into fracture-like shapes via the Poisson-Voronoi tessellation. (Note that polygons still

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could be marked by other criteria, e.g., according to proximity to other features such as fault zones.) Therefore, now:

P32 = µ

(4)

where µ is the intensity of the Poisson plane process and P32 is fracture intensity. In summary, in the new GEOFRAC the mean and standard deviation of polygon areas are a function only of the point intensity ρ (Eq. 3), but do not depend on the size or shape of the region in which the polygons are generated. P32, the cumulative fracture area per unit volume, on the other hand, depends only on µ (Eq. 4), the intensity of Poisson planes, but not on λ. Thus GEOFRAC now allows one to represent P32 independently of the fracture size variation. The MATLAB-based enhanced GEOFRAC model generates a fracture set as follows: 1. The desired mean fracture size E[A] and fracture intensity P32 in a modeling volume V are given as input. E[A] and P32 can be derived from field data; for example, methods for deriving P32 are described by Dershowitz and Herda [1992] and for deriving E[A] by Zhang et al. [2002], Mauldon [2000], and Kulatilake [1993]. 2. In the primary stochastic process, Poisson planes of intensity µ are generated in the volume V, where the intensity of the Poisson plane process is computed as:

µ = P32

(5)

3. In the secondary stochastic process, a Poisson point process with intensity ρ is generated on the planes, which are then divided into polygons by a Voronoi tessellation, where the intensity of the underlying point process is computed as:

ρ=

1 E [ A]

12

(6)

4. In the tertiary stochastic process, polygons are randomly translated and rotated. The orientation of the fractures can be simulated to follow the distribution observed in the field (e.g., uniform or Fisher); for example, Einstein, et al. [1979] discuss possible orientation distributions and how to consider possible biases. Figure 5 illustrates the process of generating a fracture set with the new GEOFRAC model.

Figure 5. Fracture set generation with the new GEOFRAC model. (a) Primary stochastic process: Poisson planes. (b) Secondary stochastic process: Poisson-Voronoi tessellation of the planes. (c) Tertiary stochastic process: random translation and rotation of polygons. 13

2.3

Fracture aperture modeling

Rock fracture aperture plays an important role in rock mechanics and fractured rock hydrogeology. To model fracture aperture distribution in the new version of GEOFRAC, two options are now available: one is deterministic, the other probabilistic. In both cases the aperture of an individual fracture is assumed to be constant along its area. 2.3.1 Deterministic approach Observations of fracture properties from field studies have led some researchers to propose a power-law correlation between fracture trace length and aperture [e.g., Stone, 1984; Vermilye and Scholtz, 1995; Johnston and Mccaffrey, 1996; Marrett, 1996]. Similarly, a new algorithm in GEOFRAC allows one to model fracture aperture according to a power-law function, such that:

h = α 2Reβ

(6)

where Re is a polygon’s equivalent radius (i.e., the radius of a circle with the same area), h is its aperture, and the coefficients α and β depend on the site’s geology. The literature references indicate what coefficients might be used. 2.3.2 Probabilistic approach Others have proposed a lognormal distribution of hydraulic fracture apertures [e.g., Dverstop and Andersson, 1989; Cacas et al., 1990]. A new GEOFRAC algorithm, which uses the inverse transform method [Devroye, 1986], now enables one to model fracture apertures according to a truncated lognormal distribution, such that:

fTR ( h ) =

f ( h) h

max

∫ f ( h ) dh

hmin

14

, hmin ≤ h ≤ hmax

(7)

where hmin and hmax are the lower and upper limit, respectively, and f(h) is defined as follows:

f(h)=

 − ( ln h − µ )2  exp  , 0 ≤ h ≤∞ 2σ 2 hσ 2π   1

(8)

Above, f(h) is the full lognormal distribution of the aperture, h, with parameters µ and σ. By definition, fTR(h) has a value of zero for any h less than hmin or greater than hmax.

2.4

Fracture intersection algorithms

Fracture connectivity is essential for assessing flow in fractured rocks. For the purpose of computing flow paths, a fracture intersection algorithm was developed, implemented in GEOFRAC, and optimized; the algorithm, which is illustrated in Figure 6, follows these steps: 1. For every polygon, the radius Ri of the sphere that encloses it is computed. 2. For every pair of non-coplanar polygons, the distance Dij between centers is computed. If Ri and Rj are the radii of two spheres, then the spheres intersect only if Ri+Rj < Dij. 3. If two spheres intersect, the intersection, if any, between the polygons, is computed. For step 3 above, we implemented in GEOFRAC the algorithms developed by Locsin [2005] to compute intersections between polygons. (See also [Locsin and Einstein, 2012]). An additional process, called “clean fracture algorithm”, was implemented in the GEOFRAC program to determine and retain only those fractures that might contribute to flow [Sousa, et al. 2012]. Namely, once all intersections between polygons have been determined, the clean fracture algorithm finds and retains only those polygons that intersect either at least two other fractures, or one of the modeling volume boundaries and at least one other fracture.

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Figure 6. Fracture intersection algorithm. (a) The spheres enclosing all polygons are computed and the intersections between spheres, if any, are determined. Two spheres intersect if the distance between their centers is smaller than the sum of their radii. Above the enclosing spheres of polygons 1 and 2 intersect, but neither of them intersects with the enclosing sphere of polygon 3. Therefore polygons 1 and 2 might be intersecting, but neither of them could be intersecting with polygon 3. C1, C2, and C3: center of polygon 1, 2, and 3, respectively. R1, R2, and R3: radius of the enclosing sphere of polygon 1, 2, and 3, respectively. D12: distance between the centers of polygons 1 and 2. D13: distance between the centers of polygons 1 and 3. (b) For every pair of intersecting spheres, such as that of polygons 1 and 2, the intersection between the polygons, if any, is determined.

The purpose of fracture system models is to provide a basis for modeling engineering behavior of rock masses such as flow. However, prior to applying a flow model, fracture connectivity can provide initial, useful information. In the next section, we present results from our parametric study of how the fracture intensity P32 and fracture sizes affect fracture connectivity.

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3. Parametric study of connectivity between fracture polygons To study the relationships between fracture intensity, size, and connectivity, we conducted Monte Carlo simulations to determine the mean fracture connectivity, C, for a range of expected fracture areas, E[A], and intensity, P32. First, we conducted the simulations for random polygon orientations and locations within the modeling volume that were generated according to a uniform probability distribution; thus we studied fracture connectivity for different P32 and E[A] without any bias for fracture orientations. Next, we conducted simulations for the same combinations of P32 and E[A], but with restrictions on the orientations. The statistical methods we applied, the additional computer algorithms we developed, and the results we obtained are presented in Sections 3.1, 3.2, and 3.3 below, respectively. 3.1

Statistical sampling for the connectivity simulations

An infinite number of fracture systems with given fracture intensity P32 and expected area of the polygons E[A] can be generated within a control volume, V. One could evaluate the effect of P32 and E[A] on the mean fracture connectivity by running GEOFRAC thousands of times for all combinations of P32 and E[A] of interest and computing fracture intersections and connectivity. However, such a process would be computationally very costly due to the time needed for every individual simulation. An alternative approach is to execute a minimal number of simulations such that the observed mean connectivity of the sample can be presented within a confidence interval from the true mean connectivity. We apply this method to compute the mean fracture connectivity of a fracture system with intensity P32 and mean fracture size E[A] as follows. Let C and σ C2 be the true mean and the true variance of the mean fracture connectivity for a combination of P32 and E[A]. In n simulations for the given P32 and E[A], one would obtain

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n different values of the mean connectivity, C1, C2, …, Cn. The Ci are independent and identically distributed random variables, every one of which has an expected value E[Ci]=C. Therefore: C1 + C2 + ... + Cn n E C  = C

C=

(9)

C2 VAR(C) = n

σC =

σC

n

where C is the sample mean of fracture connectivity (with “connectivity” defined as the number

( )

of other fractures with which a fracture intersects) and E C , VAR C , and σ C are its expected value, variance, and standard deviation, respectively. In the equations above, the relationships

( )

between E C , VAR C , and σ C , on one hand, and the true mean, C

and true variance, σ C2 ,

on the other hand, are based on theoretical derivations, which are presented in detail in classical texts on statistical sampling and data modeling; e.g., see [Bertsimas and Freund, 2004]. When n tends to infinity, the standard deviation σ C of the sample mean will tend to zero. More importantly, for the purpose of optimizing the simulations, the Central Limit Theorem states that when n is moderately large (say, n>30), C is approximately Normally distributed. Therefore, we can use the properties of the Normal distribution to estimate the true mean, C, of the random variable C within a specific confidence interval, as follows. First, in a pilot set of thirty simulations for given P32 and E[A], we compute the mean fracture connectivity, ci, in every simulation, as well as the observed sample mean, c , observed

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sample variance,

2

, and observed sample standard deviation,

, of the mean fracture

connectivity for the thirty simulations in the pilot:

c=

c1 + c2 + ... + cn n

∑ ( c − c) n

s2 =

i=1

, where n=30 in the pilot

n −1

∑( c − c) n

s=

2

i

(10)

2

i

i=1

n −1

Next, we compute a β% confidence interval for the true mean, C, using the observed standard deviation, s, as an approximation of the true standard deviation, σ C2 , and applying the properties of the Normal distribution. (Again, one can refer to [Betsimas and Freund, 2004] for the theory of computing confidence intervals.) The β% confidence interval for C is the interval:

χs χs    c − n , c + n  , where the number χ is that number for which

P(− χ ≤ Z ≤ χ ) = β /100 , where Z is the Standard Normal. For β =90%, χ=1.645. For β =95%, χ=1.960. For β =98%, χ=2.326. For β =99%, χ=2.576.

(11)

By rearranging the equation above, we can compute the number of simulations, N, which we need to run, so that the observed sample mean of fracture connectivity from these simulations will be within a tolerance ±t from the true mean, C, with a confidence of β %. Namely:

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 c − t, c + t  is the confidence interval, where

t=

N=

χs N

, which can be rearranged as

χ 2s2

(12)

t2

Above, the desired β and ±t are given as input, the variance s2 of the sample mean of fracture connectivity is computed from a pilot of thirty simulations, and the coefficient χ for the given % is obtained from statistical tables for the Standard Normal distribution. Finally, we perform N simulations for the given P32 and E[A] and compute the mean fracture connectivity, CN, from these simulations. CN is our estimate for the true mean of fracture connectivity, C, for the given P32 and E[A]. Note that every time one might conduct N simulations, the mean connectivity computed from these simulations would likely be somewhat different. However, we can expect that 95% of the resulting confidence intervals [CN–t, CN+t] would, on average, contain the true mean C of fracture connectivity for that P32 and E[A]. 3.2

Fracture connectivity simulations with FRACSIM and GEOFRAC

A new MATLAB algorithm, called FRACSIM, which implements the statistical methods described in Section 3.1, was developed for the Monte Carlo simulations to compute fracture connectivity with GEOFRAC. Specifically: first, FRACSIM executes thirty simulations (by running GEOFRAC thirty times for a given set of P32 and E[A]) and computes the sample mean,

c , and standard deviation, s, of fracture connectivity in the pilot; next, using the sample standard deviation of mean connectivity, it computes the necessary number, N, of simulations for a given tolerance, t, and a confidence interval; then, it performs the additional simulations needed for the

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confidence interval (by running GEOFRAC again as many times as needed), and computes the mean fracture connectivity, CN; finally, it iterates over the desired ranges of P32 and E[A]. An additional algorithm, called informally GEOFRAC “Lite”, was developed as a modified version of GEOFRAC for use in the parametric study. Since fractures would be orientated and placed randomly in the modeling volume in this study (i.e., once generated, polygons would not retain any “knowledge” of which random plane they came from), in an individual simulation all fractures could be generated from one plane only. Using the PoissonVoronoi Tessellation (described in Section 2.2), for every given E[A], GEOFRAC “Lite” generates polygons on a large enough area in the modeling plane, so that enough polygons are produced for even the largest desired P32. Then, polygons are randomly selected, placed, and oriented into the control volume. This process is repeated polygon by polygon until the cumulative area of fractures in the modeling volume, divided by the volume, V, reaches the desired P32 for the simulation. Thus, we conduct the parametric study with a deterministic P32 and a probabilistically obtained expected area E[A] of polygons created by the Poisson-Voronoi tessellation. FRACSIM and GEOFRAC (and/or GEOFRAC “Lite”) were used in the parametric study for a range of P32 and E[A] to determine their effect on the mean connectivity C. 3.3

Results from the parametric study of fracture connectivity

In the first part of the parametric study, we applied FRACSIM and GEOFRAC for a range of P32 and E[A] and uniformly distributed fracture orientations; i.e., the generated fracture networks had no preferred orientation of the polygons that were placed in the modeling volume. The mean connectivity CN (where connectivity is defined as the number of polygons, with which a polygon intersects) was computed for the given combinations of P32 and E[A]. The results were used to build contours of CN versus P32 and E[A], which are illustrated in Figure 7 (solid curves). 21

Once the effect of P32 and E[A] on the connectivity CN was determined, the simulations were repeated for the same range of P32 and E[A], but with polygon orientations following a Fisher distribution with K=10 [see definition by Fisher, 1953]. (For comparison, a Fisher p.d.f. with K=0 is essentially the uniform distribution; as K increases, fracture orientations become more narrowly clustered around a mean orientation.) Figure 7 also shows the connectivity contours for networks of fractures with orientations following a Fisher p.d.f. with K=10 (dotted curves). As expected, there is a clear “shift” in connectivity: for the same P32 and E[A], when the orientations are more clustered around a mean orientation (i.e., when fractures are more likely to have similar orientations), as is the case with fractures with orientations from a Fisher p.d.f. with K=10, their connectivity (i.e., the mean number of intersections per fracture) decreases. The connectivity contours in Figure 7 are constructed from data points, which are summarized in Table 1. Every data point corresponds to a Monte Carlo simulation for the given E[A] and P32. The results from the parametric study suggest that fracture connectivity is a non-linear function of both the fracture size, measured by the expected mean area E[A], and the fracture intensity, measured as cumulative fracture area per unit rock volume, P32. As can be expected, connectivity increases as either E[A] or P32 increases; however, E[A] appears to have a greater effect on fracture connectivity than P32 does. Namely, as the fracture size increases, relative to the size of the entire control volume, fractures tend to intersect even if there are very few of them; on the other hand, as E[A] decreases, even at very high P32, fractures do not intersect. The implication for fracture flow is that large fractures are likely to provide dominant pathways for flow, since they are most likely to intersect one another. Small fractures, even if their overall

22

intensity is very high, are less likely to intersect and provide discrete pathways for flow. Small fractures essentially increase the overall rock porosity, rather than forming discrete flow paths. In Figure 7, the region with 2 ≤ C ≤ 5 (for uniform orientations) is shaded for emphasis. Such connectivity and the corresponding P32 and E[A] are possibly associated with the greatest likelihood that interconnected fractures control the flow in the predominant direction(s) of fracture connectivity. If C5, there might be so many fracture paths, that flow would occur along fractures without any specific path(s) being more significant than others.

4. Conclusions This paper presented recent research developments at MIT on fracture system modeling with the GEOFRAC three-dimensional stochastic model. New mathematical algorithms were developed for modeling fracture intensity via the Voronoi tessellation, which improves the process of generating polygonal shapes that represent fractures; for assigning apertures to the generated polygon-fractures, deterministically or probabilistically; and for computing intersections among the polygons. Thus enhanced, the GEOFRAC model optimizes the representation of fracture intensity in terms of cumulative fracture area per unit volume in a region of interest of any size or shape. In addition, the new FRACSIM program was developed for conducting Monte Carlo simulations with the enhanced GEOFRAC. All new algorithms were implemented in MATLAB. The statistical parametric study with GEOFRAC and FRACSIM demonstrates how fracture intensity, size, and orientation might influence fracture connectivity. Namely, connectivity increases as either fracture intensity, measured by the cumulative fracture area per rock volume P32, or fracture size, measured by the expected area of individual fractures E[A], increases, whereas clustering of fracture orientations around a preferred direction reduces connectivity. 23

Figure 7. Estimate for the true mean fracture connectivity C as a function of fracture intensity P32 and expected fracture area E[A] with a tolerance t=0.01 for a confidence interval β=99%. Grid intersections correspond to pairs of P32 and E[A], for which the average number of fracture intersections, CN, was computed from N simulations, where N is a function of t and β . Contours for CN were interpolated between these data points, which are summarized in Table 1. Solid curves are contours for uniformly distributed fracture orientations; dotted curves contours for fracture orientations following a Fisher p.d.f. with K=10. The shaded area denotes the region of 2