Advances in Water Resources 96 (2016) 354–372

Contents lists available at ScienceDirect

Advances in Water Resources

journal homepage: www.elsevier.com/locate/advwatres

A general gridding, discretization, and coarsening methodology for

modeling flow in porous formations with discrete geological features

M. Karimi-Fard

∗, L.J. Durlofsky

Department of Energy Resources Engineering, Stanford University, Stanford, CA 94305-2220, United States

a r t i c l e i n f o

Article history:

Received 8 March 2016

Revised 30 July 2016

Accepted 30 July 2016

Available online 1 August 2016

Keywords:

Mesh generation

Fractures

Finite-volume method

Discrete feature modeling

Upscaling

Dual-continuum model

a b s t r a c t

A comprehensive framework for modeling flow in porous media containing thin, discrete features, which

could be high-permeability fractures or low-permeability deformation bands, is presented. The key steps

of the methodology are mesh generation, fine-grid discretization, upscaling, and coarse-grid discretiza-

tion. Our specialized gridding technique combines a set of intersecting triangulated surfaces by con-

structing approximate intersections using existing edges. This procedure creates a conforming mesh of

all surfaces, which defines the internal boundaries for the volumetric mesh. The flow equations are dis-

cretized on this conforming fine mesh using an optimized two-point flux finite-volume approximation.

The resulting discrete model is represented by a list of control-volumes with associated positions and

pore-volumes, and a list of cell-to-cell connections with associated transmissibilities. Coarse models are

then constructed by the aggregation of fine-grid cells, and the transmissibilities between adjacent coarse

cells are obtained using flow-based upscaling procedures. Through appropriate computation of fracture-

matrix transmissibilities, a dual-continuum representation is obtained on the coarse scale in regions with

connected fracture networks. The fine and coarse discrete models generated within the framework are

compatible with any connectivity-based simulator. The applicability of the methodology is illustrated for

several two- and three-dimensional examples. In particular, we consider gas production from naturally

fractured low-permeability formations, and transport through complex fracture networks. In all cases,

highly accurate solutions are obtained with significant model reduction.

© 2016 Elsevier Ltd. All rights reserved.

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1. Introduction

The presence of discrete features such as fractures, faults and

deformation bands within subsurface formations can have a major

impact on fluid flow. The detailed understanding of flow in such

systems is of interest in a variety of engineering fields. Environ-

mental applications include aquifer management, hazardous waste

disposal, and CO 2 sequestration. In the energy sector, oil and gas

recovery, and the exploitation of geothermal reservoirs, often in-

volve flow in fractured systems.

For any of these applications, specialized modeling tools – from

characterization of the geology to the construction of the flow sim-

ulation model – are needed to generate accurate predictions. The

geological characterization of fractured formations is very challeng-

ing and is not the focus of this work. Here we assume that an ex-

plicit and deterministic representation of the fracture distribution

is available (if an ensemble of such distributions is provided, un-

certainty in flow predictions can be quantified). Our objective is to

∗ Corresponding author.

E-mail address: karimi@stanford.edu (M. Karimi-Fard).

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http://dx.doi.org/10.1016/j.advwatres.2016.07.019

0309-1708/© 2016 Elsevier Ltd. All rights reserved.

ntegrate all of the geological data into an efficient flow simulation

odel. More specifically, in this paper we present a comprehen-

ive methodology that includes grid construction, finite-volume-

ased numerical discretization, and flow-based model coarsening

upscaling).

The construction of a flow simulation model from a geological

escription can be decomposed into several steps. First, the geol-

gy is represented geometrically on a grid. As we are interested

n the explicit representation of fractures and faults, a flexible un-

tructured approach is adopted. The geometrical representation of

racture networks is itself a challenge. A brief overview of exist-

ng techniques will be presented when we describe our specialized

pproach. Then, the flow equations are discretized on the unstruc-

ured grid, on which the geological features are represented explic-

tly. A variety of techniques have been applied for flow simulation

n porous media with discrete fractures using both finite-element

nd finite-volume methods. Within the finite-element framework,

he standard Galerkin formulation ( Baca et al., 1984; Juanes et al.,

0 02; Karimi-Fard and Firoozabadi, 20 03; Kim and Deo, 20 0 0 ),

he mixed finite-element method ( Erhel et al., 2009; Hoteit and

iroozabadi, 2008; Ma et al., 2006; Martin et al., 2005 ) and the

M. Karimi-Fard, L.J. Durlofsky / Advances in Water Resources 96 (2016) 354–372 355

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iscontinuous Galerkin method ( Eikemo et al., 2009; Hoteit and

iroozabadi, 2005 ) have been used to simulate single-phase and

ultiphase flow in discrete fracture models. Within the finite-

olume framework, formulations have been presented by, e.g.,

ogdanov et al. (2003b), Granet et al. (2001), Karimi-Fard et al.

2004), Monteagudo and Firoozabadi (2004), Nœtinger (2015), and

eichenberger et al. (2006) . A hybrid approach combining the

nite-element method for the pressure equation and the finite-

olume method for transport has also been investigated ( Geiger

t al., 2009; Matthäi et al., 2007; Nick and Matthäi, 2011 ).

As noted earlier, in this work we apply a finite-volume-based

ethodology. We proceed in this way for several reasons. In addi-

ion to its simplicity, the finite-volume approximation is very flexi-

le in terms of control-volume shape. This enables us to apply the

ame discretization concept at the coarse level as at the fine level.

he finite-volume technique used in this work is an enhanced ver-

ion of the discrete feature model (DFM) originally developed by

arimi-Fard et al. (2004) . The method applies a two-point flux ap-

roximation and entails a grid optimization step that improves grid

rthogonality by shifting the locations of the pressure unknowns.

Depending on the application, if a large number of features are

odeled explicitly, the resulting discrete model could be too large

or practical flow simulations. Simplified models based on effec-

ive properties have been used to address this problem. Many of

hese techniques were developed for highly-fractured formations

ith high fracture connectivity. Such techniques are based on the

oncept of dual-continuum models, and they typically entail ideal-

zed representations of the fracture distribution ( Barenblatt et al.,

960; Cai et al., 2015; Gilman and Kazemi, 1988; Kazemi et al.,

976; Pruess and Narasimhan, 1985; Warren and Root, 1963 ). The

uid exchange between the matrix and fracture is modeled using a

ransfer function. This approach has been generalized by introduc-

ng the concept of multi-rate-mass-transfer (MRMT) to account for

ore flow physics and formation heterogeneity ( Babey et al., 2015;

i Donato et al., 2007; Geiger et al., 2013; Haggerty and Gorelick,

995 ).

We are interested here, however, in the construction of coarse

odels from realistic (nonidealized), geometrically complex frac-

ure distributions. Equivalent absolute tensor-permeabilities for

uch systems have been computed using analytical ( Oda, 1985;

now, 1969 ) and numerical methods ( Bogdanov et al., 2003a;

oudina et al., 1998; Landereau et al., 2001; Lang et al., 2014;

œtinger and Jarrige, 2012; Sævik et al., 2014 ). These procedures

re for single (rather than dual) continuum representations. Single-

ontinuum models of fractured formations are appropriate in two

ypes of systems — when the matrix contribution to flow is neg-

igible (very low matrix porosity and permeability), or when the

ractures are sparse and not extensively connected. For many (if

ot most) problems of practical interest, however, these assump-

ions are not valid. In such cases, a dual-continuum approach,

uch as that applied by Bourbiaux et al. (1998) , Karimi-Fard et al.

20 06) , Ding et al. (20 06) , Gong et al. (20 08) , and Matthäi and

ick (2009) , is required to accurately represent the equivalent (up-

caled) system.

In this work we present a general and systematic methodology

o construct flow simulation models for complex geological forma-

ions. In contrast to existing treatments, our approach is general in

hat there are no assumptions or restrictions on the fracture distri-

ution, degree of fracture connectivity, or matrix properties. Thus,

he same general approach is applicable for a wide range of sys-

ems, from highly fractured formations to heterogeneous porous

edia with few or no fractures. This flexibility is very important

or practical problems, as different parts of a formation can have

uite different characteristics. Our new methodology can also ac-

ount for sealing or low-permeability features, such as deformation

ands, within the same general framework.

This paper proceeds as follows. We first present a specialized

ridding procedure for the geometrical representation of discrete

eological features such as faults and fractures. Then, a finite-

olume discretization technique that is applicable for such mod-

ls is presented, along with a grid optimization procedure that re-

uces the error in the two-point flux approximation. Model coars-

ning, which entails aggregation of fine-grid cells and the flow-

ased computation of coarse-scale flow parameters, is then de-

cribed. Fine and coarse-scale numerical results for several chal-

enging two and three-dimensional problems, which involve natu-

al and engineered fractures, are then presented. A two-phase flow

xample is considered in the Appendix.

We note that earlier versions of some of the procedures within

ur overall methodology have been presented in earlier papers or

onference proceedings. Of the main components of the frame-

ork, only the discretization has been presented previously in full

etail. The general gridding procedure has not been presented, and

he aggregation-based upscaling has only been discussed within

ore limited settings. The examples presented in this work are all

ew and highlight the capabilities of the full methodology.

. Geometrical representation of a set of intersecting surfaces

The generation of grids honoring complex internal geometrical

eatures is a challenging task that has been investigated in many

reas of computational physics. Our focus in this section is on

he geometrical representation of geological structures defined by

hree-dimensional surfaces; e.g., faults and fractures. We wish to

ely as much as possible on existing gridding tools. The majority

f available tools are designed to represent a given model as ac-

urately as possible. This is an important characteristic in many

ngineering applications, though with geological models the exact

ocation of discrete features is often somewhat uncertain. In fact,

ractical models are commonly generated using geostatistical ap-

roaches ( Golder Associates, 2012 ). This motivates the use of pro-

edures that do not exactly honor all aspects of the initial geomet-

ical description.

Although the problem of computing the exact intersection be-

ween a set of surfaces is mathematically well posed, there are

ractical problems depending on the relative positions of the sur-

aces. These problematic configurations are well known and have

een documented ( Holm et al., 2006; Reichenberger et al., 2006 ).

hey can be summarized into situations where surfaces are in

close” proximity and/or are “slightly” overlapping. An exact rep-

esentation of these configurations often leads to excessive degrees

f local grid refinement. This issue can become intractable for large

ystems with many of these problematic configurations. An ex-

mple of exact meshing for a fracture network can be found in

oudina et al. (1998) , where the fractures are represented by pla-

ar polygons. This type of technique can, however, only be applied

o cases without problematic configurations (or to systems where

he problematic configurations have been discarded using the fea-

ure rejection algorithm proposed by Hyman et al. (2014) ).

A way to overcome the geometrical difficulties associated with

roblematic configurations, without removing them entirely from

he model, is to introduce some changes in the geometry of the

urfaces. The basic assumption with this treatment is that these

hanges will not have a significant impact on flow quantities of in-

erest. We expect this to be the case when the fracture connectiv-

ty is maintained, and this is accomplished with our method. We

ow review some of the relevant grid generation techniques, after

hich we describe our approach in detail.

.1. Review of relevant gridding techniques

One of the earliest studies on the gridding of a frac-

ure network using geometrical simplification was presented by

356 M. Karimi-Fard, L.J. Durlofsky / Advances in Water Resources 96 (2016) 354–372

Fig. 1. Examples of triangulated surfaces.

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Maryška et al. (2005) . Their technique is similar to the exact

approach of Koudina et al. (1998) except that, before construct-

ing the triangulation for each polygon, the segments represent-

ing the intersections are simplified to avoid gridding issues. This

technique still requires the construction of exact intersections be-

fore performing the simplifications. More recently, several meth-

ods ( Mallison et al., 2010; Mustapha et al., 2011; Mustapha and

Mustapha, 2007 ) have been proposed which do not require the

construction of exact intersections. These methods essentially use

a “background” grid to simplify the model.

In the method presented by Mustapha and Mustapha (2007) ,

a fracture network is constructed by first identifying the borders

and the intersections of the fractures on a relatively fine three-

dimensional Cartesian grid. This step provides a set of cubes defin-

ing the contours of the fractures and their intersections. The cen-

ters of these cubes are connected and projected on the plane defin-

ing the fractures. This provides an approximate contour which is

used for triangulation. By constraining the shape of the fractures

and their intersections by the Cartesian grid, the method automati-

cally avoids connections with small angles. This technique was im-

proved and extended to represent not only the fracture network

but also the matrix volume in Mustapha et al. (2011) .

A similar technique was proposed by Mallison et al. (2010) . In

this procedure the background mesh is an adaptive structured tri-

angular or tetrahedral grid. Local refinement is performed on the

background mesh near the positions of the fractures, which are

represented implicitly by a distance map. Then, an explicit approxi-

mation of the fractures is obtained by using the background mesh.

Smoothing is performed to honor the actual position of the frac-

tures. Mesh refinement in the vicinity of the fractures is the main

advantage of this method over that presented by Mustapha et al.

(2011) . We note that this approach was applied only for vertical

fractures, though extension to more general fracture distributions

is feasible.

2.2. Description of the new gridding technique

The gridding procedure implemented here is a relatively simple

but quite general approach that uses the geometrical simplification

concepts applied in Mustapha and Mustapha (2007) , Mallison et al.

(2010) , and Mustapha et al. (2011) . As mentioned before, we wish

to rely as much as possible on existing tools, and the proposed

methodology focuses only on the difficulties associated with inter-

section calculations. Simply put, our method takes a set of pre-

triangulated surfaces and combines them into a single conforming

triangulated surface representing all input surfaces. A triangulation

is the most flexible representation of a surface, so any shape or

orientation can be treated. In addition, the surfaces can be nonpla-

nar and/or nonconvex. We typically use the Delaunay triangulation

technique implemented by Shewchuk (2002) , though other tech-

niques could also be used. Because these surfaces are meshed in-

ependently (intersections have not yet been considered), we can

btain high-quality meshes. Fig. 1 depicts examples of the types of

riangulated surfaces that can be handled by our approach.

The methodology can be described by considering only two tri-

ngulated surfaces, and it will be illustrated using surfaces S 1 and

2 depicted in Fig. 2 . Surface S 1 is a triangulated disk and S 2 is

nonplanar surface. The objective is to combine these two sur-

aces into a single conforming triangulated grid. Problems arise

hen the two surfaces are intersected. In our treatment, instead

f calculating the exact intersection, an approximate intersection

s constructed using a set of connected edges from surface S 1 . The

pproach proceeds in two steps, designated “merging” and “cor-

ection.” It is important to note that during the merging process,

urface S 1 is kept unchanged, while nodes and triangles of surface

2 are modified. The two steps proceed as follows:

Merging step: First, we identify all nodes of surface S 2 within

characteristic distance h from surface S 1 ( Fig. 2 a). The distance

is similar to the resolution of the background grid used for

implification in Mustapha and Mustapha (2007) , Mallison et al.

2010) , and Mustapha et al. (2011) , except here the surfaces are

nalyzed relative to each other. These nodes are relocated to the

losest nodes of surface S 1 ( Fig. 2 b). The edges of S 2 connecting

hese relocated nodes represent an approximation of the intersec-

ion ( Fig. 2 e). It is evident in Fig. 2 e that this intersection is not

ecessarily conformal, and thus requires some correction.

Correction step: The objective of this step is to render the inter-

ection conformal. The intersection is conformal if all edges defin-

ng it are shared by both surfaces. As we can see in Fig. 2 e, some

dges are shared and do not require any additional processing. For

ach nonconforming edge of surface S 2 , the following correction

s performed. We first identify the two nodes defining the edge,

nd these nodes are then connected using edges from surface S 1 .

n this case only two edges are needed as shown in Fig. 2 f. The

onconforming triangle ( Fig. 2 g) from S 2 associated with this non-

onforming edge is subdivided into two conforming triangles using

he new edges ( Fig. 2 h). The resulting intersection is now confor-

al. In general, more than two edges may be required to construct

he intersection. The correction procedure identifies the set of con-

orming edges by finding the shortest path on surface S 1 connect-

ng the end points of the original nonconforming edge.

The application of this merging–and–correction technique for

ultiple surfaces is straightforward. The model is constructed in-

rementally by adding the surfaces one by one. The already-

ombined surfaces are considered as a single triangulated surface

new S 1 ) and the new surface (new S 2 ) is added following the

ame procedure. This is repeated until all surfaces are combined.

uring this procedure we keep track of all triangles and the cor-

esponding geological feature so the appropriate properties can be

ssigned. We note that each feature can have its own properties.

his means, for example, that different fracture cells can have dif-

erent permeabilities.

M. Karimi-Fard, L.J. Durlofsky / Advances in Water Resources 96 (2016) 354–372 357

Fig. 2. Construction of the approximate intersection between two surfaces S 1 and S 2 : (a,b,c) perspective view of the surfaces, (d,e,f) top view of the surfaces, (g) noncon-

forming triangle of surface S 2 created after relocating nodes from surface S 2 to surface S 1 , and (h) subdivision of nonconforming triangle to create two conforming triangles.

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Fig. 3. Merging of 30 randomly distributed surfaces. The surfaces are composed of

10 disks, 10 rectangles, and 10 nonplanar surfaces.

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There are of course a number of algorithmic details, and we

ow discuss a few of the more important treatments. In gen-

ral, some amount of “cleanup” will be required after the grid is

onstructed. For instance, during the construction of the surfaces,

ome of the triangles can degenerate, and these must be removed

a degenerated triangle is obtained when two or three nodes of

he same triangle are co-located). In addition, after relocating the

odes, two triangles (which are defined on each of the two sur-

aces) can sometimes now share the same three nodes. In this

ase the two triangles have become identical, so one copy must

e eliminated.

We typically use similar uniform mesh resolution for the in-

ut surfaces. The average length of the edges is used to define the

implification distance h . The distance h provides the tolerance on

proximity,” meaning that all surfaces or parts of surfaces within

his distance are collapsed. An extension (and improvement) of this

reatment would be to automatically mesh the surfaces with adap-

ive mesh refinement close to the intersections, localizing the im-

act of the simplification distance h .

Because the intersections are constructed using edges from a

igh-quality triangulation, we automatically avoid having connec-

ions with small angles. Finally, it is important to note that our

echnique will only change the triangulation close to the intersec-

ions. The rest of the surfaces are unchanged and identical to the

nput surfaces. Consequently, the changes in total fracture area are

enerally small. These can be reduced to a specified tolerance by

efining the fracture mesh.

Fig. 3 depicts the application of the gridding procedure to a

omplex fracture network composed of thirty randomly distributed

riangulated surfaces, which include planar and nonplanar surfaces.

n close view ( Fig. 3 , bottom) we can see how the intersections are

onstructed and how the triangulation is altered near the intersec-

ions due to the merging procedure.

The procedure described thus far provides the gridding of all of

he (intersecting) internal geometrical features. For the volumet-

ic mesh, which resolves the bulk of the domain, we rely on ex-

sting software. As the combined surfaces are now represented by

conforming triangulation, they can be readily used as internal

oundaries for a standard mesh generator. In addition, the exter-

al boundary of the domain can also be represented as triangu-

ated surfaces. In this work we use the software Tetgen, developed

y Si (2004) , to construct a volumetric tetrahedral mesh.

Although our objective here is to represent all surfaces explic-

tly with a conforming mesh, it is important to note that there

re alternative approaches based on implicit representations of the

urfaces. These techniques bypass some of the gridding difficul-

ies and account for the features within the discretization scheme.

inite-element approaches along these lines include the extended

nite-element method ( Berrone et al., 2013a; 2013b; 2014; Fuma-

alli and Scotti, 2013 ) and the hybrid mortar method ( Benedetto

t al., 2016; Pichot et al., 2012 ). Another approach is the hierarchi-

al fracture modeling procedure ( Hajibeygi et al., 2011; Lee et al.,

001; Li and Lee, 2008; Moinfar et al., 2011 ), where a finite-volume

ormulation is used and the discrete fractures are “embedded” into

Cartesian grid. These approaches simplify the gridding step but

hey require specialized discretization techniques. By contrast, with

ur explicit, conforming-grid representation, many discretization

echniques are directly applicable. This is a key motivation for us-

ng this type of approach.

358 M. Karimi-Fard, L.J. Durlofsky / Advances in Water Resources 96 (2016) 354–372

Fig. 4. Three-dimensional illustration of a DFM grid. The polyhedrons representing

the matrix are separated by three connected fracture polygons.

Fig. 5. Definition of connection angles θ for three configurations: (a) two adjacent

matrix control-volumes, (b) adjacent matrix and fracture control-volumes, (c,d) con-

nected fracture control-volumes. Points M and F are the locations of pressure nodes

within the matrix and fracture control-volumes, and n represents the normal to the

interface. For fractures, n is specified to be inside the plane of the fracture, and nor-

mal to the intersection line ( L ).

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3. Flow model and discretization technique

In this work we consider flow through a porous formation,

which is described by a mass conservation equation for each com-

ponent and Darcy’s law for each phase. For a single-phase fluid in

the absence of gravity we have:

∂(φρ)

∂t + ∇ · (ρu ) = q, (1)

u = − k

μ∇p, (2)

where the fluid density and viscosity are designated by ρ and

μ. The flow is defined by Darcy velocity u and pressure p , and

the source/sink term is denoted by q . The rock properties are

the porosity φ and isotropic permeability k . We assume that the

same flow equations are valid inside the fractures, with corre-

sponding porosity and permeability. In addition, an aperture must

be prescribed for the fractures. These flow equations can be di-

rectly extended to multiphase, multicomponent cases. The basic

discretization, however, can be described through consideration of

Eqs. (1) and (2) .

The discretization technique will be presented and imple-

mented for general polyhedral grids. This allows us to not only ap-

ply it to grids generated by our methodology, which are mostly 3D

tetrahedral grids or 2.5D prismatic grids, but also to other types

of (conforming) grids that may be generated through use of other

software packages. In particular, the discretization allows for hy-

brid grids where different types of elements are combined within

the same grid. The discretization technique used in this work is an

extension of the finite-volume-based discrete feature model (DFM)

presented by Karimi-Fard et al. (2004) . In the geometrical repre-

sentation, the matrix part of the model can be discretized using

any type of polyhedron. The fracture network is identified by poly-

gons defining the polyhedrons. Fig. 4 depicts an example of such a

grid. Although not represented explicitly in the grid, an aperture is

associated to each fracture element in addition to the porosity and

the permeability. This is necessary to enable the correct calculation

of fracture and matrix pore volume.

The key quantity of interest when applying a finite-volume ap-

proximation for Eqs. (1) and (2) is the mass flow rate between two

adjacent control-volumes. When applying a two-point flux approx-

imation, the discrete form of the mass flow rate Q ij between adja-

cent control-volumes i and j is expressed as:

Q i j = λT i j (p i − p j ) , (3)

where λ = ρ/μ is the interface mobility. The transmissibility T ij is

defined by Karimi-Fard et al. (2004) as

T i j =

αi α j ∑

αn , with α =

kA

D

, (4)

n

here α is a sub-transmissibility. For each control-volume we

efine as many sub-transmissibilities as there are neighbors.

he sub-transmissibility α for a given connection as defined in

q. (4) involves A , the area of the interface between the two

onnected control-volumes, D , the distance from the pressure

ode to the interface along the line connecting the two pres-

ure nodes, and k , the control-volume permeability. For matrix-

atrix connections and matrix-fracture connections, two half-

ransmissibilities are used and T ij is calculated by harmonic averag-

ng. For fracture-fracture connections we can have more than two

ub-transmissibilities when there are intersections. The expression

or T ij in Eq. (4) accounts for all contributions, as shown in Karimi-

ard et al. (2004) . We note finally that, in the case of multiphase

ow, the mobility λ is modified to include relative permeability

ffects. In either multiphase or single-phase flow cases, the treat-

ent of λ is handled by the finite-volume simulator; i.e., it is not

reated within our methodology.

The use of two-point flux approximations technically requires

n orthogonal grid (with anisotropic permeability, the require-

ent is k-orthogonality). A grid is orthogonal when, for all adja-

ent control-volumes, the line connecting the two pressure “nodes”

s orthogonal to the shared interface. This criterion is automati-

ally satisfied by construction when using a perpendicular bisector

PEBI) grid, but it is not the case for the highly-flexible polyhe-

ral grids considered here. The orthogonality of a connection de-

ends on the position of the pressure nodes, and it can be evalu-

ted by the angle between the normal to the interface and the line

onnecting the two pressure nodes. If this angle is zero, the con-

ection is orthogonal. If all connection angles are zero, the grid is

rthogonal. We now describe our approach for the quantification

nd enhancement of grid orthogonality based on the optimization

echnique presented in Karimi-Fard (2008) .

In Fig. 5 , the connection angle, θ , is defined for different con-

gurations. Points M and F are the locations of the pressure nodes

ithin the matrix and fracture control-volumes. Note that these

an be significantly removed from the surface/volume centroid.

he definition of θ is straightforward for matrix-matrix ( Fig. 5 a)

nd matrix-fracture connections ( Fig. 5 b). For fracture-fracture

M. Karimi-Fard, L.J. Durlofsky / Advances in Water Resources 96 (2016) 354–372 359

c

l

(

c

t

i

t

n

v

p

c

o

s

�

�

w

θ

n

f

c

c

i

c

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s

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t

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t

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c

d

t

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o

a

t

o

c

i

a

r

i

v

p

f

m

u

g

t

m

t

t

i

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a

q

a

i

Fig. 6. Two-dimensional illustration of the pattern search optimization procedure

for a single control-volume. In this case a position that renders all connections or-

thogonal exists.

o

f

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t

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e

t

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f

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p

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2

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onnections, we first project the fracture pressure nodes onto the

ine L that defines the intersection ( Fig. 5 c). Then, the barycenter

G ) of all projected pressure nodes is calculated ( Fig. 5 d). Pressure

ontinuity is imposed at the barycenter ( G ), and a different connec-

ion angle is evaluated for each fracture control-volume, as shown

n Fig. 5 d.

By shifting the pressure nodes it is possible to improve the or-

hogonality of the grid and to eventually make the connections

ear orthogonal. As a node is connected to all surrounding control-

olumes, improving the orthogonality of one connection will im-

act the orthogonality of the other connections. To handle this

oupling, we use an iterative optimization technique. The quality

f a node is computed as the average of all connection angles as-

ociated with it. For control-volume i , we thus define the quantity

i as:

i =

1

n

n ∑

j=1

θi j , (5)

here n is the number of neighbors of control-volume i and

ij is the connection angle with neighbor j . For an orthogo-

al grid, �i is zero for all control-volumes. We note that dif-

erent expressions could be used to define �i . For instance one

ould apply a weighted average to diminish the impact of some

onnections.

The objective of the optimization is to minimize all �i by mov-

ng the pressure nodes. A new location for a pressure node is ac-

eptable (feasible) if it remains inside the corresponding control-

olume. For a matrix control-volume the node must remain in-

ide the polyhedron defining the matrix, and for a fracture control-

olume the node must remain on the fracture plane and inside

he polygon defining the fracture. We apply a pattern search algo-

ithm (direct search method), as described by Kolda et al. (2003) ,

o perform the optimization. This family of methods involves a sys-

ematic search and is straightforward to implement. The geometri-

al constraints needed in the optimization can be handled without

ifficulty.

Fig. 6 illustrates several steps of the search algorithm for a

wo-dimensional configuration on a single control-volume. For this

ase, a pressure-node location that renders all connections orthog-

nal does exist ( Fig. 6 , top), but in general this is not the case

nd the objective is to minimize the nonorthogonality. The op-

imization procedure starts with an initial guess ( Fig. 6 a). The

bjective function ( �i ) is evaluated at all points on a stencil

entered around the initial guess point, and the pressure node

s moved to the best location ( Fig. 6 b). If no improvement is

chieved at any of the stencil points ( Fig. 6 c), the stencil size is

educed ( Fig. 6 d) and the search again proceeds. The procedure

s repeated until the stencil size reaches a (specified) threshold

alue.

Because of the cell-to-cell coupling noted above, rather than ap-

ly multiple optimization steps within a control-volume, we per-

orm a single optimization step in each control-volume and then

ove to the next control-volume. The optimization then contin-

es, looping on all cells, with a particular stencil size. Once the

rid orthogonality stops improving, the stencil size is reduced and

he optimization is continued. We proceed in this manner until the

inimum stencil size is reached. We typically use the barycen-

er of each control-volume as the initial guess for the optimiza-

ion algorithm. Any other location is possible as long as the node

s inside the control-volume. It is important to reiterate that this

ptimization is performed only once, in the pre-processing step,

nd that it does not affect the efficiency of the simulations. The

uantities directly impacted by the optimization procedure are

ll the distances to the faces of the control-volumes (distance D

n Eq. (4) ).

Another way to achieve orthogonality is by changing the shape

f the control-volumes. This option is not considered in this work

or practical reasons. In general, when a grid is provided for a given

eological model, each control-volume is already prescribed a set

f effective properties. If the shapes of the control-volumes are

hanged, these properties would need to be recomputed. By con-

rast, our approach involves only shifts of the pressure nodes – we

etain the original grid and associated properties.

Finally, the discrete form of the combined Eqs. (1) and (2) for

ach control-volume i , using a backward-Euler approximation of

he time derivative, can be written as:

φn +1 i

ρn +1 i

− φn i ρn

i

t v i +

∑

j

Q i j = q i v i , (6)

here v i is the volume of control-volume i, n and n + 1 indicate

ime level, and t is the time step. The sum is over all adjacent

ontrol-volumes, with Q ij as defined in Eq. (3) . As mentioned be-

ore, this equation can be extended to the multiphase, multicom-

onent case without any additional geometrical considerations.

In the case of a highly-distorted grid and/or highly-anisotropic

ermeability, a multipoint flux approximation will be required

Aavatsmark et al., 1998; Edwards, 20 0 0; Mlacnik and Durlofsky,

006 ). In the context of discrete fracture modeling, Sandve et al.

2012) have extended the method proposed by Karimi-Fard et al.

2004) to account for full-tensor permeability. A similar technique

as presented more recently by Ahmed et al. (2015a ; 2015b ). In

his work we only consider isotropic porous media, where the use

f a two-point flux approximation combined with pressure-node

ptimization is acceptable.

At this point, using the gridding and discretization techniques

resented thus far, a fine-grid discrete feature model can be con-

tructed and simulated. Examples of flow simulations using such

360 M. Karimi-Fard, L.J. Durlofsky / Advances in Water Resources 96 (2016) 354–372

Fig. 7. Geometrical partitioning. Seed distribution (left) and corresponding coarse

grid (right).

Fig. 8. Geological partitioning. Explicit representation of a fracture within the

coarse-grid model.

u

c

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f

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t

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s

s

s

models will be presented later. We now describe how these de-

tailed fine-scale models can be accurately coarsened to enable

much more efficient flow predictions.

4. Model coarsening procedure

When there are a large number of discrete features, the un-

structured fine-grid model can become very large, making flow

simulations computationally expensive. The coarsening technique

we now describe enables substantial reductions in the number

of cells in the model, thus leading to considerable speedup. Our

coarsening procedure has two main parts – the construction of the

coarse grid, and the computation of flow properties on this grid.

We now describe these procedures in turn.

4.1. Coarse-grid construction

The coarse grid is constructed by partitioning the fine grid. As

a consequence, a coarse-scale control-volume is represented by an

aggregation of a set of fine-grid cells. We apply three approaches

to partition the fine grid. These procedures, described below, are

referred to as geometrical partitioning, geological partitioning, and

flow-based partitioning.

Geometrical partitioning: Geometrical partitioning can be used

to control the level of resolution in different parts of the domain.

We use a technique based on discrete Voronoi tessellation. This

technique was presented by Karimi-Fard and Durlofsky (2012b ) in

the context of heterogeneous reservoirs, and it is adapted here to

account for discrete geological features such as faults and frac-

tures. This approach has several advantages, most notably the fact

that the only input data required are a set of nodes (or seeds)

within the domain. No connectivity or mesh information is speci-

fied. The discrete Voronoi tessellation is then constructed by asso-

ciating each fine-grid cell to its closest seed. The number of blocks

in the coarse model is equal to the number of seeds. Considering

T = { T j } , the list of all tessellations, the fine-grid cells within a

given tessellation T j are identified through the application of

T j = { x i ∈ X : ‖ x i − s j ‖ ≤ ‖ x i − s k ‖

with s j , s k ∈ S and j � = k } . (7)

Here X represents all fine-grid cells, S denotes all seeds, x i desig-

nates the location of the fine-grid cell i (specifically the location

of the pressure node within the cell), s j represents the location of

seed j , and ‖ . ‖ is the Euclidean norm. As the tessellations are con-

structed using a set of fine-grid cells, their shapes are generally

irregular due to the irregularity of the underlying fine-grid mesh.

Fig. 7 depicts three examples of coarse-grid construction using

discrete Voronoi tessellation. As shown in Fig. 7 (top), a uniform

distribution of seeds will provide a Cartesian coarse grid, while a

random distribution of seeds ( Fig. 7 , middle) leads to an unstruc-

tured coarse grid. The technique can also be applied to a surface as

shown in Fig. 7 (bottom). Local mesh refinement can be achieved

by providing a denser distribution of seeds in some parts of the

model. This technique is very flexible and can be applied to do-

mains of any shape. The only requirement is an underlying fine-

grid mesh, which is the starting point of our overall procedure.

Geological partitioning: Geometrical partitioning is generally

combined with geological partitioning. The objective is to repre-

sent explicitly a given geological feature at the coarse level. Fig. 8

shows an example of geological partitioning, where a fracture is

represented explicitly in the coarse-grid model. This partitioning

allows us to conserve the pore-volume and the surface area of a

feature at different coarsening levels, which can be important for

simulation accuracy. In this paper we use the geological partition-

ing mainly to represent fractures and faults, but it can also be

sed to capture other geological features such as high-permeability

hannels or layers in a stratigraphic model.

Flow-based partitioning: In general, by introducing flow informa-

ion into the coarsening procedure, it is possible to achieve higher

ccuracy with fewer degrees of freedom. This partitioning is per-

ormed by first computing the pressure solution for a given flow

onfiguration on the fine grid, and then using iso-pressure surfaces

o partition the fine grid into a set of coarse blocks through ag-

lomeration. This idea was used by Karimi-Fard et al. (2006) to

onstruct the coarse grid around a fracture network, and it was

hown to provide a high level of accuracy in the coarse-scale pres-

ure solution. A similar idea was used in Karimi-Fard and Durlof-

ky (2012a ) to construct flow-based coarse grids in the near-well

M. Karimi-Fard, L.J. Durlofsky / Advances in Water Resources 96 (2016) 354–372 361

Fig. 9. Two-dimensional schematic of space partitioning procedure and the corre-

sponding graph representation. The first type of partitioning (lower left) is appro-

priate for high-permeability features, while the second type of partitioning (lower

right) is required for sealing or low-permeability features.

r

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4

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t

a

I

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M

t

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k

h

u

i

S

t

l

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o

a

fl

c

F

v

a

c

a

m

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v

T

w

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i

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e

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p

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a

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b

T

V

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t

T

w

a

H

c

egion. This partitioning technique can be combined with geomet-

ical and geological partitioning.

.2. Partitioning and cell-to-cell connectivity in the coarse model

An important aspect of our coarsening procedure is its general-

ty in terms of the way in which the coarse model is partitioned

nd in how the coarse-scale cell-to-cell transmissibilities are com-

uted. We now describe some key points regarding the space par-

itioning (coarse-scale transmissibility computation is discussed in

ection 4.3 ). Fig. 9 (top) illustrates a two-dimensional model con-

aining four features within a rectangular matrix block. In this ex-

mple, geometrical and geological partitioning are both applied.

n the images on the left in Fig. 9 , the features are assumed to

e high-permeability fractures. For this model, the partitioning is

hown in Fig. 9 (left). Here the matrix is first partitioned into two

oarse blocks (geometrical partitioning), and then clusters of iso-

ated features within each matrix block are partitioned into sep-

rate feature control-volumes (geological partitioning). The final

oarse model is composed of five control-volumes – matrix cells

1 and M

2 , and feature cells F 1 , F 2 , and F 3 . The connections be-

ween these cells are represented using a graph or a connectiv-

ty list ( Fig. 9 , bottom). Note that there are two fracture control-

olumes, F 1 and F 2 , within the left block. Since F 1 and F 2 do not

ntersect, they are treated as separate control-volumes.

Because the matrix and features are represented separately, this

odel can be seen as a general dual-continuum model. In fact, a

ey advantage of our framework is that this dual-continuum be-

avior emerges naturally at the coarse scale. By associating vol-

me and porosity to each vertex of the graph and a transmissibil-

ty to each edge, a discrete model, analogous to that presented in

ection 3 for the fine-grid model, is obtained. Note that we define

wo types of connections, indicated in Fig. 9 (bottom) by paral-

el lines (for connections between two feature cells or two matrix

ells) and dotted lines (for connections between matrix and fea-

ure cells). The computation of transmissibility for these two types

f connections will be based on different flow solutions, which is

ppropriate given the different physics governing the two types of

ow.

As is evident in Fig. 9 (left), a feature, or a cluster of features,

an cut completely through a matrix block (for instance, feature

3 and matrix M

2 ). In this example, such divided matrix control-

olumes are still treated as a single control-volume. This is clearly

simplification, but it typically provides accurate flow results for

ases with high-permeability features such as open fractures. In

ddition it allows for a considerable reduction in the number of

atrix control-volumes. It is important to note that this treatment

s used in standard dual-porosity models.

In the case of low-permeability features, such as (partially) seal-

ng faults or deformation bands, subdivision of the matrix control-

olumes is required to model the pressure jump across the feature.

his is illustrated in the second type of partitioning ( Fig. 9 , right),

here the matrix region M

2 is subdivided into three disconnected

atrix subregions ( M

21 , M

22 and M

23 ). The corresponding model

s now able to capture pressure discontinuities.

For highly faulted models, the additional matrix partitioning

ay generate a large number of matrix control-volumes, and some

ype of clustering may be required. A technique based on effec-

ive matrix properties, along the lines of that proposed by Odling

t al. (2004) , could be useful in this context. When applying our

ethodology, if the subdivision of the matrix blocks does not lead

o an excessive number of cells, it is performed even with high-

ermeability fractures, as the corresponding coarse model will be

ore accurate. In the next section, we describe how the flow in-

eractions between the different subregions are computed.

.3. Computation of coarse-grid flow parameters

The coarse-grid connectivity can be viewed in terms of a graph,

s shown in Fig. 9 (bottom). When using a finite-volume formula-

ion at the coarse level, the required flow parameters are the bulk

olume V I and the porosity φI for each coarse block I (which cor-

espond to the vertices of the graph), and the transmissibility T IJ etween two adjacent coarse blocks I and J (edges of the graph).

he coarse volume and porosity are defined by:

I =

∑

i ∈ I v i , φI =

1

V I

∑

i ∈ I v i φi , (8)

here v i and φi are the bulk volume and porosity, respectively,

f fine-grid cell i , which falls within coarse block I . The transmis-

ibility between two adjacent coarse blocks can be computed in

any different ways ( Durlofsky et al., 2012; Farmer, 2002; Renard

nd de Marsily, 1997; Wen and Gómez-Hernández, 1996 ). In this

ork we apply flow-based transmissibility upscaling, as these ap-

roaches can provide better accuracy for highly-heterogeneous sys-

ems ( Chen et al., 2003 ). The basic approach is straightforward:

iven a local fine-scale pressure solution in a region covering the

wo adjacent coarse blocks and their shared interface, the coarse

ransmissibility is defined by:

IJ =

Q IJ

p I − p J , (9)

here the coarse-grid average pressures p I and p J , and the associ-

ted coarse-grid flow rate Q IJ , are computed as follows:

p I =

1

V I

∑

i ∈ I v i p i , p J =

1

V J

∑

j∈ J v j p j ,

Q IJ =

∑

(i ∈ I, j∈ J) Q i j =

∑

(i ∈ I, j∈ J) T i j (p i − p j ) .

(10)

ere p i and p j designate the fine-scale pressures that lie within

oarse blocks I and J, ij indicates the interface between fine-scale

362 M. Karimi-Fard, L.J. Durlofsky / Advances in Water Resources 96 (2016) 354–372

Fig. 10. Two-dimensional illustration of flow-based upscaling procedure. Left: Problem setup showing the feature distribution and the permeability heterogeneity of the

matrix. Middle: Example of pressure field construction for parallel flow. The same set of directional flows are used to construct two pressure fields within the matrix for

two connections with different orientations. Right: Exam ple of pressure fields used to compute the flow exchange between the feature/well and the surrounding medium.

All pressure fields are represented by iso-pressure curves with pressure values varying from high (red) to low (blue).

u

a

s

(

b

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s

w

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a

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d

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t

cells i and j ( ij lies on the interface between I and J ), and T ij is the

transmissibility for this connection.

To actually compute the coarse flow parameters, we must spec-

ify the local fine-grid pressure field to be used in Eq. (10) . These

pressure fields derive from fine-scale solutions of the pressure

equation, and the choice of flow problem depends on the local

flow physics. In the case of heterogeneous porous media without

fractures, a local pressure solution can be generated by impos-

ing a pressure difference on a domain containing the two coarse

blocks, which acts to drive flow through the shared interface. Many

variations of this method have been applied in the past to com-

pute upscaled permeability or transmissibility ( Aarnes et al., 2007;

Chen et al., 2003; Durlofsky, 1991; Durlofsky et al., 2012; Has-

sanpour et al., 2010; Karimi-Fard and Durlofsky, 2012a; Prévost

et al., 2005; Wen et al., 2003 ). Similarly, when computing proper-

ties in a coarse cell containing a well, a well-driven flow problem

is solved. This leads to more of a radial flow pattern in the vicinity

of the well. The associated pressure field is used to compute the

well index, which is the transmissibility between the wellbore and

the coarse well block ( Ding, 1995; Durlofsky et al., 20 0 0; Karimi-

Fard and Durlofsky, 2012a ), along with cell-to-cell transmissibili-

ties. Finally, in the context of fractured porous media, Karimi-Fard

et al. (2006) have shown that transmissibilities between the frac-

ture network and the surrounding matrix can be accurately cap-

tured through use of a flow problem that mimics the transient ex-

change between matrix and fracture.

These findings are used (and adapted) in this work to compute

all of the required coarse-scale transmissibilities. We define two

general types of flow patterns: parallel flow patterns, which are

b

sed for matrix-matrix and feature-feature connections (marked

s solid lines in the graph representation in Fig. 9 , bottom), and

ource-driven flow patterns, used for matrix-feature connections

marked as dotted lines in the graph representation in Fig. 9 ,

ottom). A source-driven flow is also applied to compute coarse

ell indices. We now describe the construction of these pressure

olutions.

Parallel flow: We first focus on the construction of parallel flow

ithin the matrix. To construct the local pressure solution, we

se a modified version of the technique proposed by Karimi-Fard

nd Durlofsky (2012a ). The basic idea is to first generate a set

f linearly independent large-scale steady-state pressure solutions,

esignated p n . We then use the superposition principle to ap-

ropriately combine these solutions to provide, for each connec-

ion, the desired local flow pattern. For three-dimensional prob-

ems, we typically use three global pressure solutions, with each

orresponding to large-scale flow in one of the three coordinate

irections. The original method was developed for heterogeneous

orous media without fractures; here it is extended to also han-

le discrete features. Considering the case of matrix-matrix con-

ections, the following steady-state pressure equation is solved for

he matrix ( ):

∇ · ( k ∇p ) = 0 , with p = p b (x ) on �b & � f , (11)

here �b represents the boundary of the domain, and �f repre-

ents all discrete features, as shown in Fig. 10 (left, top).

Three pressure solutions are generated by setting the value of

he boundary condition p b ( x ) to (in turn) x, y and z . These pressure

oundary conditions are also imposed on the discrete features ( �f ).

M. Karimi-Fard, L.J. Durlofsky / Advances in Water Resources 96 (2016) 354–372 363

T

�

t

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i

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t

T

p

s

t

N

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t

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n

(

o

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t

T

d

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l

p

a

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g

a

c

N

γ

w

N

l

g

w

T

J

G

t

Fig. 11. Two-dimensional illustration of flow-based upscaling procedure for feature-

feature connections: (a) fine and coarse representation of a discrete feature network

(DFN), with cells in the coarse model represented by different colors, (b) pressure

distribution for flow in the x and y directions (blue corresponds to low pressure

and red to high pressure), (c) example of two coarse connections and the associated

combined pressure fields.

c

m

Q

c

t

w

l

F

a

s

m

p

p

b

t

C

(

t

a

t

t

t

a b

his is because, if the boundary conditions are only imposed on

b , the pressure information will be weak within an isolated ma-

rix block surrounded by high-permeability (or low-permeability)

eatures. Such a pressure field can lead to inaccurate transmissibil-

ties, since both Q IJ and | p I − p J | can be very small. By imposing

ressure boundary conditions on all features, a well-defined pres-

ure field is obtained inside all isolated matrix blocks. A different

ay to address this problem would be to remove all features from

he model when constructing the pressure fields for the matrix.

he former approach is adopted here as it leads to a simpler im-

lementation and will be seen to provide accurate results.

Fig. 10 (middle, top) depicts an example of the resulting pres-

ure fields for flow in the x and y directions. The pressure varia-

ion is not precisely linear in x or y due to matrix heterogeneity.

ote also that the pressure field displays clear variation in matrix

egions that are surrounded by fractures. This allows for the com-

utation of meaningful T IJ in these regions.

The reference pressure fields p n can be combined in different

ays to obtain the desired flow pattern. In our previous work

Karimi-Fard and Durlofsky, 2012a ), pressure constraints were used

o define a directional flow pattern, and the reference pressures

ere combined to satisfy these constraints using a least-square ap-

roach. Here, the reference pressures are combined based on their

orresponding local pressure gradients. Specifically, for each trans-

issibility T IJ , the “target” flow direction is defined by the vector

IJ , which is parallel to the line connecting the barycenters of the

adjacent) target cells I and J . We designate ( ∇p n ) IJ as the estimate

f the pressure gradient induced by the pressure field p n over a

egion covering coarse block I and J . The contribution assigned to

ach reference pressure field is based on the angle ϕ

n IJ

between the

arget flow direction n IJ and the direction of the gradient ( ∇p n ) IJ .

his means that, if the gradient is perpendicular to the target flow

irection, the corresponding pressure field will not contribute to

he transmissibility calculation.

A formal way to estimate the local pressure gradient for so-

ution n , ( ∇p n ) IJ , is to use linear regression. Assuming a linear

ressure field, P n IJ

= a n IJ

x + b n IJ

y + c n IJ

z + d n IJ , the coefficients a n

IJ , b n

IJ , c n

IJ ,

nd d n IJ

are obtained by minimizing || P n IJ (x, y, z) − p n (x, y, z) || 2 using

he pressure values at all fine-grid cells ( x i , y i , z i ) over the region

f interest (we typically use all fine cells within blocks I and J ). The

radient of the linear pressure P n IJ , defined by (a n

IJ , b n

IJ , c n

IJ ) , provides

local estimate of ( ∇p n ) IJ . Then, for each p n we compute:

os (ϕ

n IJ

)=

n IJ · (∇p n ) IJ ‖ n IJ ‖ ‖ (∇p n ) IJ ‖

. (12)

ow, the contribution for each pressure field p n is given by:

n IJ =

| cos (ϕ

n IJ

)| ∑ N m =1 | cos

(ϕ

m

IJ

)| , (13)

here N is the number of reference pressure solutions (typically

= 3 for 3D models). Finally, the weight w

n IJ

for each flow prob-

em is obtained by scaling by the magnitude of the corresponding

radient:

n IJ =

γ n IJ

‖ (∇p n ) IJ ‖

. (14)

he local pressure field for the two adjacent control-volumes I and

can now be constructed as:

p =

N ∑

n =1

w

n IJ p

n . (15)

iven this pressure solution, all quantities required for the compu-

ation of T IJ are now available.

It is important to reiterate that, using a simpler flow specifi-

ation such as one generic pressure solution, we may encounter

any regions for which the flow field is too weak (i.e., very small

IJ and | p I − p J | ) to compute accurate T IJ . This is particularly the

ase with highly-heterogeneous systems and distorted cells. The

reatment presented here assures that ∇p is oriented in such a

ay that both Q IJ and | p I − p J | are physically reasonable, which

eads to accurate T IJ .

Two examples of combined pressure fields are shown in

ig. 10 (middle). Zoom-ins in the vicinity of the two target cells

re shown in the bottom two images. It is apparent that, con-

istent with the discussion above, iso-pressure lines are approxi-

ately parallel to the IJ interface, which indicates that the local

ressure fields generated from our procedure lead to appropriate

ressure variation and flow over the regions of interest.

Similarly, a set of pressure solutions is needed for the (possi-

ly disconnected) network containing the features. This will enable

he computation of coarse-scale feature-feature transmissibilities.

onsider the fine-scale discrete fracture network shown in Fig. 11 a

left). For a pressure solution corresponding to flow in the x direc-

ion, an appropriate set of boundary conditions is to specify p b = x

t the end points of all fractures. This boundary specification leads

o flow through both connected and disconnected fractures and

hus allows for the computation of all of the required T IJ . The other

wo global flow problems (for 3D systems) correspond to setting

p b = y and p b = z.

For linear fractures of homogeneous permeability, with pressure

t fracture end points specified to be p = x, the global pressure

364 M. Karimi-Fard, L.J. Durlofsky / Advances in Water Resources 96 (2016) 354–372

p

t

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Q

w

o

d

c

a

b

c

m

i

v

(

g

d

i

r

m

5

a

t

O

a

b

m

i

t

2

(

c

g

d

g

u

o

f

p

t

s

s

field in the fractures can be reasonably approximated as p = x . This

can be readily seen (and is exact) for isolated fractures or sim-

ple networks. For more general cases, such as those involving frac-

tures with a high degree of curvature or varying aperture, this ap-

proximation will incur some error, and for such cases it may be

preferable to solve the pressure equation over the discrete frac-

ture network (though we note that this pressure solution will re-

quire some detailed numerical treatments for complex networks).

For the cases considered here, however, this pressure approxima-

tion is reasonable and leads to accurate coarse models.

The coarse-scale representation of the fracture network is

shown in Fig. 11 a (right). The coarse model contains nine control-

volumes, which are indicated by different colors in the figure. Six

of these control-volumes are composed of clusters (two or more)

of fractures, while three correspond to isolated fractures. The p = x

and p = y pressure approximations are illustrated in Fig. 11 b. These

reference pressure “solutions” are then combined using the same

technique as is used for matrix-matrix connections; i.e., we con-

struct a pressure solution with ( ∇p ) IJ approximately oriented with

n IJ . Fig. 11 c depicts the combined pressure field and the associated

flow direction for two coarse connections.

Source-driven flow: A source-driven flow solution is used for the

computation of all feature-matrix transmissibilities. These trans-

missibilities represent exchange between features and the sur-

rounding matrix. In the case of high-permeability fractures within

a low-permeability matrix, the pressure field reaches steady state

very quickly in the fractures, but there is a potentially long tran-

sient in the matrix. Rather than attempt to construct transmissi-

bilities from the solution of a compressible (time-dependent) pres-

sure equation, we use the same approach as in Karimi-Fard et al.

(2006) and instead solve the “pseudo-steady-state” pressure equa-

tion. Pseudo steady state here refers to the late transient period,

when ∂ p / ∂ t becomes constant in space and time.

The pseudo-steady-state pressure equation is given by:

∇ · ( k ∇p ) = φF , with

p = 0 on � f , ∂ p

∂n

= 0 on �b , F = 1 in . (16)

The φF term on the right hand side, with F an arbitrary constant,

derives from the (constant) ∂ p / ∂ t term at pseudo steady state, but

it appears in the same form as a source term in the steady-state

pressure equation. As in Karimi-Fard et al. (2006) , the boundary

conditions are constant pressure for all discrete features and no-

flow on the border of the domain. The only difference here is

that Eq. (16) is solved once globally, while in the previous work

it was solved locally for each coarse cell. An example of this type

of pressure solution is shown in Fig. 10 (right, top). From the

pressure contours, we can see that flow is from the matrix into

the fractures. Using the computed pseudo-steady-state pressure

solution, all feature-matrix transmissibilities are evaluated using

Eqs. (9) and (10) . We note that a pseudo-steady-state pressure so-

lution is also used with low-permeability features. In this case this

solution again provides a pressure field that is appropriate for the

computation of feature-matrix T IJ .

In this paper, well indices (which link the wellbore to the

coarse cells in which it is completed; i.e., open to flow) are com-

puted numerically, as in Karimi-Fard and Durlofsky (2012a ). In this

case we use the pressure field from a well-driven flow problem.

An example of such a pressure field is depicted in Fig. 10 (right,

middle).

We conclude this section with a few technical remarks. For all

of the examples in this paper, the required pressure solutions are

computed globally. Because the pressure equation is linear, this can

be done efficiently even for quite large models. In addition, these

equations are solved only once at the beginning of the coarsening

rocedure. If the fine model is too large for global pressure solu-

ions, our coarsening procedure can still be applied by dividing the

lobal domain into a few (overlapping) subdomains, and then ap-

lying the technique for each subdomain.

For the parallel-flow pressure fields, we typically use three

eneric pressure solutions for three-dimensional problems. It is

mportant to note, however, that the methodology allows for the

se of additional pressure fields. For instance, in the near-well re-

ions, in addition to the three pressure fields we could also in-

orporate the pressure field from a well-driven flow. This would

rovide a smooth transition in the flow problems used to com-

ute transmissibilities (i.e., from radial flow near the well to paral-

el flow away from the well).

Finally, we comment briefly on the form of the coarse model

or cases involving two-phase or multiphase flow. In such cases,

e represent the flux of a given phase between coarse control-

olumes I and J using an expression similar to Eq. (3) . Specifically,

n the absence of gravitational and capillary effects, we write

w

IJ = λw

T IJ (p I − p J ) , Q

n IJ = λn T IJ (p I − p J ) , (17)

here Q

w

IJ and Q

n IJ

are the flow rates for the water and nonaque-

us phases, λw

and λn are the saturation- and possibly pressure-

ependent phase mobilities (handled by the simulator), T IJ is the

oarse-scale transmissibility computed as described above, and p I nd p J are coarse-cell pressures. In our treatment, the phase mo-

ilities in Eq. (17) are taken to be the same between the fine and

oarse scales; the only upscaled parameter is the rock and geo-

etric part of the transmissibility, T IJ . This type of representation

s common in upscaling and has been shown to perform well in a

ariety of settings, including for oil-water flow in fractured systems

Gong et al., 2008; Karimi-Fard et al., 2006 ), assuming sufficient

rid resolution is used. For highly coarsened models, or systems

isplaying other complications such as highly heterogeneous cap-

llary pressure, procedures involving the computation of upscaled

elative permeabilities, which provide upscaled phase mobilities,

ay additionally be required.

. Numerical results

We now apply our modeling workflow to several problems. For

ll cases presented, the fine-grid models are considered to provide

he reference solution against which we compare coarse solutions.

ne might choose to further refine the fine-grid model in order to

ssure that numerical convergence (to within some tolerance) has

een achieved, though this can be time consuming, especially for

ultiphase flow cases, and is not attempted here. The fine models

n this work are taken to be small enough so the reference solu-

ions can be obtained with reasonable computation.

The first problem considered is a 2.5D model requiring only a

D mesh generator. For this case a direct Delaunay triangulation

Shewchuk, 2002 ) is applied and a prismatic grid with one layer is

onstructed. For the other cases, the models are fully 3D and our

eneral gridding procedure is applied. The resulting fine and coarse

iscrete models are compatible with any connectivity-based (i.e.,

enerally unstructured) subsurface flow simulator. In this work, we

se Stanford’s General Purpose Research Simulator (GPRS), devel-

ped by Cao (2002) and Jiang (2007) . The simulator is run using a

ully implicit formulation with a first-order upstream-weighted ap-

roximation of the transported-flux term. Backward Euler time in-

egration is applied. The linear system at each Newton iteration is

olved using a block GMRES iterative solver with constrained pres-

ure residual preconditioning.

M. Karimi-Fard, L.J. Durlofsky / Advances in Water Resources 96 (2016) 354–372 365

Fig. 12. Reservoir configuration including ten hydraulic fractures and one horizontal

well.

5

l

p

T

u

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w

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d

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a

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a

f

w

m

t

W

o

t

C

t

m

s

o

o

e

w

p

p

f

Fig. 13. Three scenarios of activated natural fractures within the SRV (green vertical

lines indicate hydraulic fractures).

Table 1

Gas density and viscosity as a function of pressure.

p (psi) ρ (kg/m

3 ) μ (cp, 10 −3 Pa.s)

14 .7 0 .0058186 0 .0080

264 .7 0 .0801917 0 .0096

514 .7 0 .1545677 0 .0112

1014 .7 0 .3033337 0 .0140

2014 .7 0 .6008412 0 .0189

2514 .7 0 .7494264 0 .0208

3014 .7 0 .8979238 0 .0228

4014 .7 1 .1957555 0 .0268

5014 .7 1 .4942338 0 .0309

9014 .7 2 .5123258 0 .0470

i

p

T

3

3

t

p

A

r

f

g

p

s

s

c

c

a

c

r

.1. Simulation of gas production from a naturally fractured

ow-permeability formation

In this example we consider gas production from low-

ermeability formations, such as tight gas or shale gas reservoirs.

o improve productivity, these formations are typically stimulated

sing hydraulic fracturing. The fracturing process activates pre-

xisting natural fractures, creating the so-called stimulated reser-

oir volume (SRV). To illustrate the capability of our methodology,

e construct a series of synthetic models for systems of this type.

Fig. 12 depicts a formation with ten hydraulic fractures and one

orizontal well. The natural fractures are defined by two sets of

ractures, which are oriented perpendicular and parallel to the hy-

raulic fractures. The average spacing is different for each fracture

et. For parallel natural fractures the spacing is about 10 m, and

or the set perpendicular to the hydraulic fractures the spacing is

bout 15 m. In addition, we randomly perturb the paths of the

ractures to increase the complexity of the model. Three scenarios

re considered in the simulations. For the first scenario, the natural

ractures are fully activated and they provide a fully-connected net-

ork. This is the typical assumption when applying a dual-porosity

odel. For the other two models, we randomly disconnect some of

he fractures. This is intended to represent incomplete stimulation.

e assume that all activated natural fractures are open to flow.

These three models are illustrated in Fig. 13 , where the network

f natural fractures is depicted within the SRV. It is evident that

he fracture connectivity is much weaker for the last model (model

), and a standard dual-porosity model may not be adequate in

his case. The models are discretized using a 2D unstructured fine

esh. For each fine model, coarse models are constructed and the

imulation results are compared. A key advantage of our method-

logy is that the same coarsening technique is applied regardless

f the connectivity of the fractures.

We consider the following compressible single-phase flow

quation:

∂(φρ)

∂t = ∇ ·

(ρk

μ∇p

)+ q, (18)

here ρ and μ are respectively the density and viscosity (both

ressure-dependent) of the gas, as defined in Table 1 . Note that the

ressure-dependence of the gas mobility is taken to be of the same

orm on all scales. This treatment is consistent with that described

n Eq. (17) for two-phase flow. The reservoir (matrix) porosity and

ermeability are 5% and 10 −5 md ( 9 . 87 × 10 −21 m

2 ), respectively.

he porosity and permeability of activated natural fractures are

0% and 10 md ( 9 . 87 × 10 −15 m

2 ), and for the hydraulic fractures,

0% and 6 × 10 4 md ( 5 . 92 × 10 −11 m

2 ). The depth of the forma-

ion is 20 0 0 m. The reservoir is fully saturated with gas, and the

ressure at the top of the formation is 50 0 0 psi (3.4 × 10 7 Pa).

ll three scenarios have the same original gas in place. Simulation

esults will be presented in terms of recovery factor.

The gas is produced at constant pressure (500 psi, 3.4 × 10 6 Pa)

rom a horizontal well connected to all hydraulic fractures. We ne-

lect the pressure drop along the horizontal well. Due to the low

ermeability of the matrix rock, a long transient is typically ob-

erved. We first simulate gas production for 20 years for all three

cenarios using the fine-grid models. Then, coarsened models are

onstructed for each case. Due to the geometric complexity asso-

iated with the fracture distribution, the fine-grid models require

large number of cells. For these examples, model A has 184,839

ells, model B has 181,238 cells, and model C has 148,997 cells.

The fine-grid simulation results are summarized in Fig. 14 . The

ecovery factor is normalized by the long-time asymptotic value

366 M. Karimi-Fard, L.J. Durlofsky / Advances in Water Resources 96 (2016) 354–372

Fig. 14. Cumulative gas production for three fine and coarse models (left). Recovery factors normalized by long-time cumulative gas production. Pressure field after one year

of production for the fine models (right).

Fig. 15. Top: example coarse grid for model B with 1869 cells. Bottom: three close

views, with coarse cells identified by different colors.

a

i

d

t

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f

v

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T

a

s

p

u

m

n

T

n

o

s

p

s

f

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t

v

m

fi

m

o

of the cumulative gas production for the fine-grid model. The cu-

mulative gas production clearly varies with the degree of fracture

connectivity. As expected, the least gas recovery is observed for the

scenario with the lowest fracture connectivity (model C). Pressure

fields after 1 year of production are presented in Fig. 14 (right).

Consistent with the production profiles, the depletion is the most

extensive for the case with highly-connected natural fractures

(model A).

We now apply the coarsening technique presented in this pa-

per for each of the three scenarios. Fig. 15 shows an example

of a coarse grid for model B. This grid contains 1869 cells and

is constructed by combining geometrical and geological partition-

ing. For the geometrical partitioning a regular seed distribution is

used, leading to a Cartesian grid. The geological partitioning sep-

rates the fractures from the matrix. Fractures are clearly visible

n the close views ( Fig. 15 , bottom). For this partitioning, to re-

uce the overall number of coarse cells, matrix cells cut by frac-

ures are kept as single control-volumes (first type of partition-

ng in Fig. 9 ). It is also evident in Fig. 15 (bottom) that connected

ractures within a matrix block are combined into a single control-

olume (individual control-volumes are indicated by color).

In Fig. 14 , in addition to the fine-grid results, we also present

he cumulative gas production for the “converged” coarse models.

o achieve these converged results, for each model we constructed

nd ran a series of coarse models with increasing resolution. Re-

ults for the coarsest model that still provided high accuracy are

lotted. For these examples, we achieve highly accurate results by

sing, respectively, 3.7%, 18.8%, and 19.4% of the fine-grid cells for

odels A, B, and C. Interestingly, for model A (highly-connected

atural fractures), a dramatic reduction in model size is possible.

his is presumably due to the relative simplicity of the flow dy-

amics for this case, and is consistent with the fact that this type

f model can be effectively upscaled to a dual-porosity-type repre-

entation. For models B and C, due to the more complicated flow

hysics, larger coarse models are required to accurately capture the

olution.

This example illustrates that the large number of cells needed

or the fine-grid models is essentially dictated by the geometrical

ssues associated with complex fracture distributions. By relaxing

he geometrical constraints on the shapes of coarse-scale control-

olumes, and appropriately computing flow parameters, the coarse

odels are able to provide results very similar to those from the

ne-grid models, while using far fewer control-volumes.

To provide a sense of the rate of convergence of the coarse

odels, in Fig. 16 we plot the recovery factor for several grid res-

lutions (1%, 5% and 10% of the cells in the reference fine-grid

M. Karimi-Fard, L.J. Durlofsky / Advances in Water Resources 96 (2016) 354–372 367

Fig. 16. Impact of model resolution on gas production results.

m

i

m

v

d

g

i

t

5

s

a

m

t

∇

w

t

q

t

c

a

s

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f

a

5

f

t

d

t

d

s

m

(

F

s

t

1

o

(

t

e

c

i

W

m

T

1

c

F

t

w

s

e

(

b

h

i

r

e

(

S

t

m

o

d

g

i

(

i

c

F

t

t

l

s

t

o

5

w

odel) for models B and C. Model accuracy clearly increases with

ncreasing refinement, as would be expected. It is noteworthy that

odels containing only 5–10% of the number of fine-grid cells pro-

ide results of reasonable accuracy. We note finally that models of

ifferent resolution can be constructed efficiently, since the same

lobal pressure solutions can be reused for all of the transmissibil-

ty calculations. This means that “convergence” assessments, of the

ype shown in Fig. 16 , can be readily performed.

.2. Transport through fractured porous media

The next two examples involve flow and transport of a passive

calar. Such models are relevant to pollutant transport in fractured

quifers and to the storage of radioactive waste in fractured for-

ations (though of course many additional complications arise in

hese problems). The governing equations are:

· ( k ∇p ) + q t = 0 , (19)

∂ ( φC )

∂t = ∇ · ( Ck ∇p ) + q, (20)

here C is the transported quantity (e.g., concentration), q t is the

otal source/sink term, q is the source/sink term for the transported

uantity, and all other variables are as defined previously. This is

he simplest setup to test our discretization and coarsening pro-

edure for a transport problem. For the first example we combine

heterogeneous matrix with a few discrete features, while in the

econd example a complex fracture network embedded in a homo-

eneous matrix is considered. In the Appendix, we present results

or a two-phase flow case. All coarse grids are unstructured and

re constructed using the discrete Voronoi tesselation.

.2.1. Heterogeneous system with sparse discrete features

Fig. 17 illustrates the problem setup. Twenty nonplanar vertical

eatures are explicitly represented ( Fig. 17 a). These features have

he same size and shape but different orientations. The flow is

riven by one injection well and one production well. Although

he model is fully three-dimensional, the flow is essentially two-

imensional, which facilitates visualization and comparison of re-

ults. Fig. 17 b shows the matrix permeability map. Matrix per-

eability varies from about 1 md ( 9 . 87 × 10 −16 m

2 ) to 10 0 0 md

9 . 87 × 10 −13 m

2 ). The matrix porosity is kept constant at 25%.

or a particular simulation, all of the discrete features have the

ame properties. For the first set of runs, high-permeability fea-

ures, with permeability of 10 6 md ( 9 . 87 × 10 −10 m

2 ), porosity of

00%, and aperture of 1 mm, are considered. For the second set

f simulations, the features are assumed to have low permeability

10 −6 md, 9 . 87 × 10 −22 m

2 ).

The model is discretized using tetrahedral elements for the ma-

rix and triangular elements for the features. A total of 209,993

lements are used for the fine-grid model. Three unstructured

oarse models are constructed by combining geometrical partition-

ng (with irregular seed distribution) and geological partitioning.

e note that in this case, cut matrix blocks are represented by

ultiple control-volumes (second type of partitioning in Fig. 9 ).

hese models contain 20% (42,017 cells), 10% (21,005 cells), and

% (2117 cells) of the number of fine-grid cells. An unstructured

oarse grid with even fewer cells (524 cells) is shown in Fig. 17 c.

low simulations are performed with constant rate specified for

he injector and constant pressure for the producer. The fine-scale

ell indices are computed as described by Karimi-Fard and Durlof-

ky (2012a ). The coarse-grid results are compared with the refer-

nce fine-grid model using recovery data, well-to-well productivity

Q / p , or well-to-well flow rate divided by the difference in well-

ore pressures), and concentration maps.

Fig. 18 summarizes the simulation results for the case with

igh-permeability features. The well-to-well productivity is a key

ndicator of the accuracy of the coarse-scale single-phase flow pa-

ameters (transmissibilities and well indices), and we can see that

ven for the coarsest model (2117 cells) the error in Q / p is small

0.71%). Similar accuracy is observed in the tracer recovery curves.

light differences in model results are evident in the concentra-

ion maps ( Fig. 18 , right). In order to compare concentrations for

odels with different unstructured grids at different levels of res-

lution, the concentration values were interpolated onto a two-

imensional horizontal plane defined by a fine uniform Cartesian

rid. This plane is positioned in the middle of the model. The

nterpolation was performed using the visualization tool Tecplot

Tecplot 360 TM

, 2013 ). Some smearing of the concentration front

s evident in the coarser models, but the overall flow pattern is

aptured accurately.

The results with low-permeability features are summarized in

ig. 19 . Similar accuracy is observed for this case. From the concen-

ration maps, we can see that the low-permeability features essen-

ially act as barriers. As a result, tracer breakthrough occurs slightly

ater here relative to the case with high-permeability features. Re-

ults between the two cases are not dramatically different since

he features are relatively sparse. It is noteworthy, however, that

ur procedure is applicable for both types of systems.

.2.2. Complex three-dimensional fracture network

We now consider a fully three-dimensional fracture network

ithin a box-shaped formation. The network includes three differ-

368 M. Karimi-Fard, L.J. Durlofsky / Advances in Water Resources 96 (2016) 354–372

Fig. 17. Problem setup: (a) feature distribution and well locations, (b) matrix permeability map, (c) example coarse grid with 524 control-volumes.

Fig. 18. Comparison of simulation results between the coarse-grid models and the

reference fine-grid model for the case with high-permeability features. The concen-

tration maps are shown on a horizontal cross-section after 0.2 pore-volume injec-

tion.

Fig. 19. Comparison of simulation results between the coarse-grid models and the

reference fine-grid model for the case with low-permeability features. The con-

centration maps are shown on a horizontal cross-section after 0.2 pore-volume

injection.

w

o

o

t

ent fracture shapes – disks, rectangles, and nonplanar rectangles.

The objective here is to generate a complex model in order to illus-

trate the capabilities of our workflow. Although not based on real

data, we believe the level of complexity considered in this example

is realistic, and could be representative of the difficulties associated

ith actual geological models. We emphasize that no assumptions

r restrictions are made on the orientation of the fractures.

The model is constructed by randomly combining 50 surfaces

f each type, for a total of 150 fractures. The resulting fracture dis-

ribution is shown in Fig. 20 . The corresponding fine-grid model

M. Karimi-Fard, L.J. Durlofsky / Advances in Water Resources 96 (2016) 354–372 369

Fig. 20. Problem setup with 150 randomly distributed fractures within a porous

matrix block.

c

t

t

i

c

t

p

r

s

c

t

e

p

t

h

t

L

o

n

d

t

Fig. 21. Comparison of tracer recovery and well-to-well productivity for complex

three-dimensional fracture network.

i

c

t

6

i

t

u

s

i

f

b

a

r

s

q

c

i

i

l

c

a

t

f

t

c

i

u

v

v

fi

i

o

i

p

ontains 289,673 elements. The matrix properties are uniform in

his case (permeability of 10 md and porosity of 25%). The frac-

ures have the same properties as the high-permeability features

n the previous example.

Three unstructured coarse-grid models are constructed. These

ontain 57,998 cells (20% of the fine model), 29,124 cells (10% of

he fine model), and 2898 cells (1% of the fine model). As in the

revious example, flow simulations are performed using constant

ate at the injector and constant pressure at the producer. The

imulation results are presented in Figs. 21 and 22 . The recovery

urves show very fast breakthrough, which would be expected for

his highly-connected fracture network. Consistent with previous

xamples, high accuracy is again observed for the tracer recovery

rofiles and well-to-well productivity.

Fig. 22 provides a comparison of the concentration maps af-

er 0.15 pore-volume injection. Each image shows results for both

orizontal ( x − y ) and vertical ( z − y ) cross-sections. The horizon-

al plane is located at 0.5 L z and the vertical plane at 0.5 L x , where

z and L x are the model dimensions in the z and x directions. The

verall flow pattern is captured very accurately with 20% of the

umber of fine-grid cells, and results with 10% of the cells still

isplay reasonable accuracy (part of the discrepancy is likely due

o the interpolation onto a Cartesian grid, as required for visual-

zation). Smearing is apparent when 1% of the number of fine-grid

ells is used ( Fig. 22 , lower right), though even in this case the

racer recovery and Q / p results retain a high degree of accuracy.

. Concluding remarks

In this paper, a comprehensive framework for modeling flow

n fractured formations was presented. The methodology addresses

he three main steps required for the construction of efficient sim-

lation models. First, a specialized gridding technique was pre-

ented to construct the mesh associated with a set of intersect-

ng triangulated surfaces. Next, the flow equations were discretized

or the resulting unstructured grid using a finite-volume approach

ased on an optimized two-point flux approximation. And finally,

systematic coarsening technique was presented to achieve model

eduction. The applicability of the methodology was illustrated for

everal problems involving fractured formations, for which a se-

uence of coarse models was constructed. The high degree of ac-

uracy of these coarse models was demonstrated through compar-

son of simulation results for reference fine-grid models.

A key advantage of our coarsening approach is the flexibility

t enables in the definition of the coarse control-volumes. This al-

ows us to somewhat dissociate geometric complexity from flow

omplexity. This was illustrated by the accurate flow results

chieved for coarse models of highly-fractured systems. In addi-

ion, regardless of the coarsening level, the pore-volume of the

racture network and the surface area between the fractures and

he surrounding matrix are conserved in our methodology. This

ontributes significantly to the accuracy of the models, especially

n transport computations. The coarsening procedure can also be

sed to “clean up” a fine-grid model by agglomerating clusters of

ery small cells. In this case the overall resolution may not change

ery much compared to the original model, but by avoiding arti-

cially small cells the performance of the flow simulator may be

mproved considerably.

The focus of this work was on the computation and evaluation

f single-phase flow parameters at different coarsening levels. This

nformation is directly applicable for multiphase and multicom-

onent flow (cell pore volumes, cell-to-cell transmissibilities and

370 M. Karimi-Fard, L.J. Durlofsky / Advances in Water Resources 96 (2016) 354–372

Fig. 22. Concentration maps for horizontal (left portion of each image) and vertical (right portion of each image) cross-sections after 0.15 pore-volume injection.

Fig. 23. Comparison of water flooding simulation results between the coarse-grid

models and the reference fine-grid model for case with high-permeability features.

d

t

a

1

a

a

S

b

b

E

a

a

well indices are the same for these problems as for single-phase

flow systems), though additional coarse-scale treatments may be

required in some cases. These could include the use of upscaled

relative permeability, mobility, and/or capillary pressure functions,

all of which can be incorporated within our modeling framework.

Another way to approach multiphase multicomponent flow prob-

lems is to use adaptive mesh refinement, which can focus grid res-

olution in critical parts of the flow, such as fronts. Our coarsening

procedure provides the required flexibility for the application of

such a technique, and preliminary results on this topic have been

presented by Karimi-Fard and Durlofsky (2014) . In future work, we

plan to pursue a general formulation that combines the framework

presented here with adaptive mesh refinement.

Unit conversions

1 md = 9 . 869233 × 10 −16 m

2

1 psi = 6.894 × 10 3 Pa

1 cp = 10 −3 Pa.s

Acknowledgments

We thank the industrial affiliates of the Stanford University

Reservoir Simulation Research Consortium (SUPRI-B), Chevron En-

ergy Technology Company and Total S.A. for partial funding of this

work.

Appendix A. Two-phase oil-water example

We now consider a two-phase flow example. The system in-

cludes gravitational but not capillary pressure effects. The govern-

ing equations can be written in terms of a pressure equation and

a saturation equation, which resemble Eqs. (19) and (20) , though

several complications arise. These include the appearance of a

saturation-dependent total mobility term in the pressure equation,

a nonlinear flux function in the saturation equation, and gravita-

tional terms in both equations.

We assess performance for the example in Section 5.2.1 , for the

case with high-permeability fractures. The model is initially satu-

rated with oil and water is injected into the system. We use the

ead oil capability within the simulator GPRS to model the wa-

er flooding process. The density and viscosity of oil are 785 kg/m

3

nd 1 . 2 × 10 −3 Pa.s, and the corresponding quantities for water are

025 kg/m

3 and 3 × 10 −4 Pa.s. We use quadratic relative perme-

bility functions for both phases. This leads to significant spatial

nd temporal variation in the phase mobilities. As discussed in

ection 4.3 , only the rock and geometric part of the transmissi-

ility (single-phase parameter T IJ ) is computed in our aggregation-

ased coarsening. The cell-to-cell flow rates are represented using

q. (17) , and the functional forms of the mobility/relative perme-

bility functions do not change with scale.

Using the same setup as in Section 5.2.1 , a fine-grid model

nd two coarse models are simulated. Fig. 23 summarizes the

M. Karimi-Fard, L.J. Durlofsky / Advances in Water Resources 96 (2016) 354–372 371

s

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(

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t

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A

A

A

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B

B

B

B

B

B

B

C

C

C

D

D

D

D

D

D

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E

E

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K

imulation results for water cut (fraction) at the production well.

ccuracy similar to that for the passive tracer case is observed

or the model with 42,017 cells, though some degradation in ac-

uracy is seen for the 2117-cell model. This reduction in accuracy

s likely due to the additional physics (e.g., saturation-dependent

obilities) in the two-phase flow case. Simulation times are also

eported in the figure. Although the fine model is not very large

209,993 cells), this water flooding simulation is still quite slow

more than 24 hours for just 0.4 pore-volume of injection). This be-

avior, which is typical for models of multiphase flow in fractured

orous media, is due to the small time steps and nonlinear conver-

ence difficulties experienced by the simulator. The aggregation-

ased coarsening approach appears to provide an effective means

f improving simulation efficiency for such systems, as the 42,017-

ell model gives nearly identical results to the reference solution

ith a speedup factor of 12.

This example suggests that our overall methodology is applica-

le to two-phase flow cases, even though only the rock and ge-

metric part of the coarse-scale transmissibility is computed. If

ur intent is to use very coarse models, however, then multiphase

ffective properties may additionally be required. Alternatively,

oarse-model accuracy can be (consistently) improved through use

f mesh refinement in key regions, regardless of the complexity of

he underlying flow physics.

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