Applied Mathematical Modelling 56 (2018) 137–157

Contents lists available at ScienceDirect

Applied Mathematical Modelling

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

Complex analytical solutions for flow in hydraulically

fractured hydrocarbon reservoirs with and without natural

fractures

Arnaud van Harmelen, Ruud Weijermars ∗

Harold Vance Department of Petroleum Engineering, Texas A&M University, 3116 TAMU, College Station, TX 77843-3116, USA

a r t i c l e i n f o

Article history:

Received 28 April 2017

Revised 13 November 2017

Accepted 22 November 2017

Keywords:

Hydraulic fracture flow

Natural fracture flow

Reservoir drainage contours

Streamlines

a b s t r a c t

Reservoir drainage towards producer wells in a hydraulically and naturally fractured reser-

voir is visualized by using an analytical streamline simulator that plots streamlines, time-

of-flight contours and drainage contours based on complex potentials. A new analytical

expression is derived to model the flow through natural fractures with enhanced hydraulic

conductivity. Synthetic examples show that in an otherwise homogeneous reservoir even

a small number of natural fractures may severely affect streamline patterns and distort

the drainage contours. Multiple parallel natural fractures result in a drainage region that is

narrower in the direction normal to the natural fractures while the drainage reach is larger

in the natural fracture direction. Reservoirs with numerous natural fractures are shown to

be characterized by more tortuous drainage patterns than reservoirs without natural frac-

tures. Finally, the analytical flow model for naturally fractured reservoirs is applied to a

natural analog of flow into hydraulic fractures. The tendency of the injected fluid to stay

confined to the fracture network as opposed to matrix flow is entirely controlled by the

hydraulic conductivity contrast between the fracture network and the matrix.

© 2017 Elsevier Inc. All rights reserved.

1. Introduction

In the hydrocarbon industry, economic development of shale reservoirs with low permeability is commonly achieved

by hydraulically fracturing such reservoirs. When natural fractures are present in the reservoir they may support the flow

of reservoir fluids into to the producer wellbore. Maximizing a producer well’s performance therefore requires optimal hy-

draulic fracture planning [1–3] , which in turn necessitates adequate flow models for hydraulic and natural fractures. In this

study, we visualize fluid drainage by hydraulic and natural fractures by employing analytical methods, in contrast to various

semi-analytical and numerical solution methods commonly used [4–6] .

Potential flow theory, which provides closed-form analytical solutions of the Laplace equation, is at the heart of this study

and is applied to derive models of hydraulic and natural fractures for flow simulation. Hydraulic fractures in reservoirs are

directly connected to a producer wellbore and therefore aid in draining reservoir fluids. Such fractures were already modeled

analytically in an earlier study [7] . Natural fractures, on the other hand, are conduits or cracks that either expedite or impede

the flow of reservoir fluids. When such fractures are mineralized they may form impermeable barriers inside the reservoir,

∗ Corresponding author. E-mail address: r.weijermars@tamu.edu (R. Weijermars).

https://doi.org/10.1016/j.apm.2017.11.027

0307-904X/© 2017 Elsevier Inc. All rights reserved.

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138 A. van Harmelen, R. Weijermars / Applied Mathematical Modelling 56 (2018) 137–157

which has been modeled in one of our prior studies [8] . However, the modeling of flow acceleration through multiple cracks

with enhanced hydraulic conductivity has not been modeled before by any analytical method.

The main purpose of our present study is to derive an efficient 2D analytical description for permeable fractures to enable

rapid modeling of flow diversion in reservoirs, made up of a porous medium containing multiple natural fracture systems.

In potential theory one can obtain new solutions by superposing existing solutions. A well-known example of superposition

in fluid mechanics, aerodynamics and electromagnetism is the singularity doublet (or point dipole), which is obtained by

superposing a point source and a point sink in a limiting process [9–11] . With a different approach one can transform an

infinite amount of point sinks into an interval sink [12,13] , which is how we previously modeled a hydraulic fracture [14] .

Similarly, an infinite number of singularity doublets can be transformed into a line doublet [8] , which can approximate high

and low conductivity zones [15] . In this paper, we present a new analytical model of a natural fracture, which we created

by superposing an infinite amount of line doublets.

Although solutions from potential flow theory are obtained only after certain restricting assumptions (discussed in

Section 2 ), the fact that the solutions are closed-form formulae implies that the computational cost of visualizing these

solutions is low. Consequently, densely clustered streamline tracking is achievable at low costs, while enabling uniquely

accurate tracking of the time-of-flight-contours (TOFCs) and drainage contours of reservoir fluids. Another benefit of the

analytical solutions is that velocity fields and pressure fields are obtainable at high resolution. The accuracy of the analytical

models has been verified by matching results to those of a numerical simulator [14] .

This paper is structured as follows. Basic assumptions of fluid flow, a discussion of hydraulic and natural fracture models,

as well as a description of our visualization method and its benefits can be found in Section 2 . Section 3 is dedicated to basic

illustrations of the analytical elements presented in this study. In Section 4 , we combine analytical natural and hydraulic

fracture elements to visualize the drainage of a hydrocarbon reservoir. Lastly, in Section 5 , we illustrate an application of our

analytical natural fracture element by modeling the flow through a complex natural fracture network from a polished rock

slab. The fracture network in the slab shows fluid injection paths and particle flow through the adjacent matrix. Applying

the flow reversal principle, the same slab may serve as an analog for drainage by a natural fracture network in a subsurface

reservoir.

2. Methodology

This section addresses basic assumptions on the reservoir and the fluids inside it, and is followed by a brief review of

previously developed hydraulic and natural fracture models. Next, we explain our visualization method, based on poten-

tial theory for fluid flow [16,17] and the analytical element method [18–26] . Lastly, we highlight the benefits of analytical

models.

2.1. Fluid flow model assumptions

Potential theory and conformal mappings [27] have been applied in various fields to model idealized flow in a viscous

continuum [9–11,28] and form the foundation for each analytical element in this paper. Previous studies have advocated the

modeling of Darcy flow in porous media by potential methods [25,29–35] . The main assumptions behind modeling Darcy

flow with potential theory concern both the reservoir and the reservoir fluids.

The reservoir model in our study is comprised of a porous matrix assumed to be homogeneous, incompressible and con-

fined to a relatively thin layer with large lateral extent. The layer is considered to be a part of a larger reservoir whose

boundaries lie far away from the flow region studied so that boundary effects can be neglected. The assumption of a reser-

voir comprised of a homogeneous matrix implies constant reservoir porosity (the percentage of the reservoir pore space

available for fluid flow) and permeability (the ease with which fluids can traverse through the reservoir). The only hetero-

geneities in our reservoir model are the fractures, which may enhance fluid flow locally. Vertical pressure gradients such

as gravity are neglected, justified by our assumption of a relatively thin reservoir space. With these reservoir assumptions

we can limit the reservoir model to two-dimensional fluid flow, even though fluid flow in three dimensions can also be

described with the analytical element method [19,26] .

The fluid flow we consider is irrotational, incompressible, immiscible and isothermal. Consequently, the fluids’ viscosity

and density are constant. We also assume negligible capillary pressure and disregard any wettability and relative perme-

ability effects of the reservoir fluids. The initial pressure of the reservoir, P 0 , will after flow disturbance by a change agent

�P(z) , become:

P (z) = P 0 + �P (z) . [ Pa ] (1a) When the reservoir fluid develops a pressure gradient, P ( z ) is mainly governed by the reservoir properties (porosity, per-

meability, and height) and reservoir fluid property (viscosity). Assuming positive strength for injectors and negative strength

for producers (see also Appendix A.1 ), we can express P ( z ) as

�P (z) = −φ(z) μk

. [ Pa ] (1b)

A. van Harmelen, R. Weijermars / Applied Mathematical Modelling 56 (2018) 137–157 139

Fig. 1. Sketch of reservoir flow and analytical elements. Illustrated are (1) a vertical wellbore normal to the plane of flow, (2) a hydraulic fracture normal

to the plane of view, (3) a 2D far-field flow confined to the reservoir layer, and (4) a natural fracture, also normal to the plane of view.

Table 1

Reservoir and fluid properties.

Porosity Reservoir permeability Reservoir height Viscosity

n = 0.2 k m = 10 mD (9.87 × 10 −15 m 2 ) h = 1 m μ= 1 cP (0.001 Pa s)

In expression (1b) , μ is the viscosity of the fluid with unit [Pa s], k the permeability, of the relevant reservoir section,with unit [m 2 ], and ϕ( z ) is the potential function [m 2 s −1 ] (see expression (2) ). The assumed reservoir properties are givenin Table 1 .

Natural fractures are modeled as high permeability zones, scaled by the ratio of the matrix permeability ( k m ) and fracture

permeability ( k f ); only high conductivity fractures are considered k f > k m , due to a certain aperture that translates to an

increased fracture permeability with respective to the matrix. Note that streamlines converge into the fracture when k f > k m .

Comparing similar flows in domains with otherwise uniform hydraulic conductivity, the pressure gradient becomes steeper

when the permeability is smaller. When the fluid crosses a domain boundary from the matrix into a fracture with higher

permeability an abrupt pressure drop occurs. Pressures in adjacent points at either side of the permeability boundary are

inversely proportional:

P (z) m · k m = P (z) f · k f ⇒ P (z) m /P (z) f = k f / k m . [ dimensionless ] (1c)The velocities at either side of the boundary will scale accordingly.

2.2. Hydraulic and natural fracture modeling

Hydraulic fracking enhances the reservoir fluid flow towards producer wells and is a common necessity in low permeabil-

ity reservoirs in order to render the reservoir economic. After fracking, the resulting geometric properties of the hydraulic

fractures can be inferred from seismic data, for example, as demonstrated in a previous study [7] . Even though properties of

hydraulic and natural fractures may be difficult to ascertain precisely, numerous models are available to incorporate fracture

properties in numerical reservoir simulators.

The dual porosity model [4,5] is one of the earliest examples of a fracture modeling tool for reservoir simulation pur-

poses. Since the dual porosity model’s inception decades ago various other models, as well as more advanced dual-porosity

based models, have become available [36,37] . Models coupling fluid flow and poro-elastic response predict not only hy-

draulic fracture growth and its interaction with natural fractures during the fracking process [38–42] , but can also estimate

fracture conductivity [43–46] . Most of these fractured reservoir models are semi-analytical or numerical models and there-

fore require numeric discretization and upscaling. Closed-form analytical streamline models, however, are scarce. We have

therefore developed a fully analytical natural fracture model and illustrate how an already known analytical element [12] can

be used to simulate the flow acceleration by natural fractures with enhanced hydraulic conductivity.

A principle sketch of fluid flow in a reservoir is shown in Fig. 1 and includes a single vertical wellbore (represented by

a dot), a hydraulic fracture connected to the wellbore (straight solid line), a natural fracture in the neighborhood of the

hydraulic fracture (double dashed lines) and a far-field aquifer flow entering the flow field on the left. All flow occurs in the

2D plane of view, which means the 3D flow volume has identical solutions at every depth in the reservoir layer viewed from

140 A. van Harmelen, R. Weijermars / Applied Mathematical Modelling 56 (2018) 137–157

above. In what follows, we show systematic examples of the flow impacts for each of the superposed reservoir attributes

shown in Fig. 1 .

2.3. Potential theory and visualization algorithm

The principal tool used for each analytical element we model is the complex potential �( z ), which links the stream

function ( ψ) and potential function ( φ):

�(z) = φ(x, y ) + iψ(x, y ) . [ m 2 s −1 ] (2) The two dimensional velocity components ( v x and v y ) can be extracted from expression (2) by using v x = ∂φ/ ∂x = ∂ψ / ∂y

and v y = ∂φ/ ∂y = −∂ψ / ∂x . The corresponding velocity field V ( z ) can then be expressed by the velocity potential:

V (z) = d�(z) dz

= v x − i v y . [m s −1 ] (3)

More details on complex potentials, stream functions, potential functions, Pólya vector fields and their relationships can

be found elsewhere [14,15,27,28,47,48] .

The complex potential and velocity fields as well as the corresponding pressure field and the propagation of streamlines,

time-of-flight contours, and drainage contours are programmed in Matlab. The streamline tracing method we use is based on

a simple first order Eulerian displacement scheme, hence only the velocity vector components in the x - and y -direction are

required. These velocity vector components ( v x and v y ) can be obtained from the velocity field V ( z ) as shown in expression

(3) .

In order to start tracing a streamline, the initial position z 0 (in complex coordinates with unit [m]) as well as the initial

time t 0 (unit in [s]) from which the tracing starts must be chosen (see [15] for a non-dimensional approach). While we

usually use t 0 = 0, different values may be necessary whenever the velocity field is time-dependent and the streamlinepatterns change over time as a consequence [31–33] . Lastly the size of the time step, �t , needs to be specified. The position

of a tracer element at time t 1 , denoted by z 1 ( t 1 ), can then be calculated:

z 1 ( t 1 ) = z 0 ( t 0 ) + V̄ ( z 0 ( t 0 )) · �t. [ m ] (4) The term V̄ ( z 0 ( t 0 )) is the complex conjugate of V with V = v x − iv y , and expression (4) gives the velocity of the traced

particle at time t 0 and location z 0 , for which the velocity field V ( z ) is used. Generalizing our tracing scheme above, the

location of tracer element at time t j can be found by evaluating

z j ( t j ) = z j−1 ( t j−1 ) + V̄ ( z j−1 ( t j−1 )) · �t. [ m ] (5) In Matlab we evaluate expression (5) by taking expression (3) into account:

z j ( t j ) = z j−1 ( t j−1 ) + [real

{V ( z j−1 ( t j−1 ))

}+ 1 i · imag

{−V ( z j−1 ( t j−1 ))

}]· �t. [ m ] (6)

Although the time step �t is arbitrarily chosen, a small value is particularly required for stronger point sources/sinks in

order to maintain smooth streamlines and TOFCs.

2.4. Benefits of analytical streamline modeling

The analytical formula V ( z ), the simplicity of the first order scheme in expression (6) and the constant value of the time

step �t combined, is what ensures low computational costs of tracking a tracer element during a time step. Together with

the computational processing power of modern computers these fast algorithms enable a swift calculation of a vast amount

of tracer elements. Tracing such dense clusters of streamlines results in smooth visualization of time-of-flight contours, i.e.

the position of all streamline tracers at a certain time t j . Even with all the simplifications of the model and its basic tracing

algorithm, the high accuracy of the analytical models has been verified by matching results to those of a numerical simulator

[14] .

Additionally, the combination of modern computational processing power and the availability of an analytical formula

for the complex potential �( z ) and the velocity field V ( z ) enable high resolution figures of the corresponding pressure field

and velocity field. The resolution of these fields is, in fact, infinite due to the fact that the analytical formulae enable one to

zoom in on any region as accurate as required.

Lastly, by expressing the analytical velocity field by V ( z ) = v x – iv y , any flow stagnation points (i.e. points of zero velocity)can be quickly identified by solving for both v x = 0 and v y = 0, as long as V ( z ) has not become too intricate. However, evenif the analytical velocity field expression becomes too intricate to find the stagnation points analytically, they can still be

found by turning to the infinite resolution of the velocity field and computing time-of-flight contours [14] .

3. Fundamental visualizations of analytical elements

Basic flow patterns of the four analytical elements modeled in this study ( Fig. 1 ) are visualized in this section. First,

the point source and interval source are discussed. Next, the far-field flow is illustrated and superposed with the two prior

A. van Harmelen, R. Weijermars / Applied Mathematical Modelling 56 (2018) 137–157 141

Fig. 2. Vertical wellbore with a hydraulic fracture. Streamlines (blue) and drainage contours (red). Contour spacing is 1 year, total time-of-flight is 10 years,

time step �t = 0.1 day. Other properties are listed in Table 2 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

analytical elements. The new analytical element, representing a natural fracture, is the last element visualized and is su-

perimposed on the three preceding analytical elements. Properties of all analytical elements are chosen for visualization

purposes only and do not need to reflect realistic reservoir values.

3.1. Single well with hydraulic fracture and far-field flow

Reservoir drainage by a single vertical wellbore with one hydraulic fracture ( Fig. 1 ) can be modeled by a point sink and

an interval sink respectively (see Appendix A ). By applying the flow reversal principle [14] , we can obtain the drainage

contours by superposing a point source and interval source.

The point source velocity field can be expressed as (see Appendix A.1 ):

V (z) = q 1 (t) 2 πhn

1

z − z c , [m s −1 ] (7)

where q 1 ( t ) is the volumetric flow rate in [m 3 s −1 ], z c is the location of the vertical wellbore, h the height of the reservoir

[m], and n the reservoir porosity [% or fraction]. The velocity field description of an interval source is given by (see Appendix

A.2 ):

V (z) = q 2 (t) 2 πhnL

e −iβ{

log ( e −iβ (z − z c ) + 0 . 5 L ) − log ( e −iβ (z − z c ) − 0 . 5 L ) }. [m s −1 ] (8)

The volumetric flow rate, location of the interval source, height of the reservoir, and the reservoir porosity in expression

(8) are respectively denoted by q 2 ( t ), z c , h , and n . The length of the interval source is given by L (with unit [m]) and the

angle of orientation is given by β (see Fig. A4 ). In order to later understand superposed flow patterns in a naturally fractured reservoir containing vertical wellbores,

hydraulic fractures and a natural aquifer drive, we first illustrate fluid drainage due to only a single wellbore and a single

hydraulic fracture. Fig. 2 shows such drainage and flow pattern due to a vertical wellbore with a hydraulic fracture. Applying

the Eulerian particle tracing algorithm of expression (6) to expressions (7) and (8) , allows the construction of streamlines

and drainage contours ( Fig. 2 ) by employing the flow reversal principle and superposing a point source and interval source

of equal strength. The inner- and outermost time-of-flight contours correspond respectively to t = 1 year and t = 10 year.While the innermost drainage contour exhibits a shape typical for a combined interval source and point source drainage

pattern (see Appendix A ), the drainage contours of later stages are more circular.

Fluid flow in any reservoir can be influenced by a natural drive from an aquifer located far from the reservoir. The far-

field flow is an analytical element that represents such a natural drive. Because the aquifer is located far from the reservoir,

the resulting far-field flow is unidirectional (angle α) and has a constant velocity ( U ∞ ). Denoting the reservoir porosity againby n , leads to the far-field velocity field:

V (z) = U ∞ n

e −iα. [m s −1 ] (9)

Before including natural fractures in our reservoir model, we first visualize the impact of a far-field flow if superposed

on the fluid flow pattern of Fig. 2 . Fig. 3 (a) visualizes a far-field flow propagating through the reservoir (see Table 2 for

properties). Because the streamlines are not progressing towards any hydraulic fracture or wellbore (hence colored green),

they remain unperturbed and uniform throughout the reservoir. The far-field flow in Fig. 3 (a) travels from left to the right

with a uniform velocity of U ∞ = 0.5 m/year, the total travelled distance in a 10 year time period equals 25 m due to the

142 A. van Harmelen, R. Weijermars / Applied Mathematical Modelling 56 (2018) 137–157

Fig. 3. Far-field flow in a reservoir. Contour spacing is 1 year, total time-of-flight is 10 years, time step �t = 0.1 day. Properties of fracture, wellbore and far-field flow are given in Table 2 . (a): Only far-field flow in the reservoir. (b): Far-field flow superposed on the flow due to the hydraulic fracture and

vertical wellbore as in Fig. 2 . Drainage contours (red) show drained area over time. Stagnation point marked by black square. (For interpretation of the

references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 4. Natural fracture model. L and W are the length and width; z c is the center; z a1 , z a2 , z a3 , and z b2 are the corners; β is the line doublet angle (see

Appendix A ), while γ is the rotation angle of the natural fracture. Unidirectional arrows denote direction of flow through the fracture domain.

Table 2

Properties of analytical elements in Figs. 2, 3 , and 5 .

Location Strength Angle Length Width Height Porosity

Vertical wellbore z c = 2.5 + 0 ∗1i q ( t ) = −1.84 × 10 −7 m 3 /s N/A N/A N/A h = 1 m n = 0.2 Hydraulic fracture z c = 0 + 0 ∗1i q ( t ) = −1.84 × 10 −7 m 3 /s β = 0 ° L = 5 m N/A h = 1 m n = 0.2 Far-field flow N/A U ∞ = 0.5 m/year α = 0 ° N/A N/A N/A n = 0.2 Natural fracture z c = – 6 – 5 ∗1i υ = 3.68 × 10 −6 m 4 /s β = 120 ° L = 10 m W = 0.5 m h = 1 m n = 0.2

γ = 60 °

porosity value n = 0.2. In order to visualize the impact of a far-field flow on the drainage pattern in Fig. 2 , only the oildrained by the well and the hydraulic fracture is traced with drainage contours in Fig. 3 (b). Oil left of the producer can

be produced (blue streamlines), while most of the oil above, below and to the right of the vertical wellbore is pushed

downstream, past the wellbore (green streamlines). In contrast to the more smooth and rounded edge of the drainage con-

tours in Fig. 2 , the contours in Fig. 3 (b) have a sharper edge (pointing left). The presence of a far-field flow also leads to

the occurrence of a stagnation point not far to the right of the vertical wellbore, indicated by a blue and green stream-

line meeting each other head on and marked by a black square on the red envelope where time-of-flight contour bungle

( Fig. 3 (b)).

3.2. Impact of single natural fracture on fluid flow

The next element we introduce in the reservoir is our analytical element to describe flow through a highly permeable

natural fracture. Since a conductive natural fracture enhances the fluid velocity locally, there should only be a local effect

on fluid velocity. The reservoir fluids should also be able to flow through all sides of the natural fracture model ( Fig. 4 ),

although there is a preferred flow direction between two opposite sides (indicated with blue arrows). The other two sides

of the fracture are indicated by dashed lines. In Section 4 , we illustrate various results of flow through a reservoir with

A. van Harmelen, R. Weijermars / Applied Mathematical Modelling 56 (2018) 137–157 143

Fig. 5. Natural fracture. Drainage contour spacing is 1 year, total drainage time is 10 years, time step �t = 0.1 day. Properties of fractures, wellbore and far-field flow are given in Table 2 . (a) Full field view of reservoir. Contours show total drained area over time. (b) Zoom of the fracture area. Stagnation

point marked by black square.

multiple natural fractures. Here, we first illustrate the flow distortion of Fig. 3 (b) when a highly conductive natural fracture

is present.

Crucial in the derivation of this natural fracture model is the superposition of an infinite amount of line doublets (see

Appendix B ), which was inspired by a previous paper [15] where we superposed four line doublets to approximate local

areas of high and low conductivity. The derivation of the natural fracture model can be found in Appendix B . The final

velocity field expression reads:

V (z) = −i · υ · e −iγ

2 πhn · L · W [log ( −e −iγ ( z − z a 2 )) − log (−e −iγ (z − z a 1 ) )

+ log (−e −iγ (z − z b1 )) − log (−e −iγ (z − z b2 ) ].

[m s −1 ] (10)

In Fig. 4 and expression (10) , υ is the natural fracture strength [m 4 s −1 ] and the natural fracture’s corners are given byz a1 = z c – exp( i γ ) ·(0.5 L + 0.5 W ·exp( i β)), z a2 = z c – exp( i γ ) ·(0.5 L – 0.5 W ·exp( i β)), z b1 = z c – exp( i γ ) ·(– 0.5 L + 0.5 W ·exp( i β)),and z b2 = z c – exp( i γ ) ·(– 0.5 L – 0.5 W ·exp( i β)).

Fig. 5 (a) illustrates the impact of a single natural fracture on the streamline pattern and drainage contours of

Fig. 3 (b) (properties given in Table 2 ). Our principal sub-conclusion is that the drainage area perpendicular to the hydraulic

fracture is narrower due to the presence of the natural fracture . Further note that the two streamlines that meet each other

head on in Fig. 3 (b) no longer do so ( Fig. 5 (a)), meaning that the location of the stagnation point (black square) has shifted

due to the presence of the natural fracture. The streamlines close to the hydraulic and natural fracture are densely spaced

and difficult to distinguish. Fig. 5 (b) therefore shows a zoom-in of the immediate area around the hydraulic and natural

fracture. Many of the streamlines near the natural fracture propagate towards it and, once inside, rush through the natural

fracture. Note that streamlines do not necessarily traverse the entire length of the fracture, as they may exit at another side

( Fig. 5 (b)). Specific streamlines have been marked by an arrowhead for easier visual tracing. The tightly spaced streamlines

inside the natural fracture indicate that streamline jetting occurs, i.e. the velocity inside the natural fracture is much larger

than outside of it. Although the streamline jetting effect implies higher velocities inside the natural fracture, the streamlines

do not travel instantaneously through the natural fracture due to its finite strength υ , as scaled by the permeability contrastbetween the natural fracture and the matrix (see Section 2.1 ). Also evident is that the natural fracture model does exactly

what was intended: there is a preferred direction of flow, but all sides of the natural fracture are permeable.

4. Multiple natural fractures in a reservoir system

The next step is to examine the new analytical element of natural fractures in two different scenarios with far-field flow:

(1) natural far-field drive due to an aquifer confined between the upper and lower boundaries of the reservoir, (2) man-made

imposed far-field flow due to the presence of a wellbore, with consequent drainage of fluid from the reservoir.

4.1. Far-field flow due to aquifer drive in naturally fractured reservoir

Fig. 6 visualizes four different cases (properties in Table 3 ) of a far-field flow with two superposed natural fractures: (a)

two parallel natural fractures aligned with far-field flow, (b) two parallel natural fractures misaligned with far-field flow,

(c) far-field flow with two obliquely superposed natural fractures that are mutually perpendicular (non-crossing), (d) far-

field flow with two obliquely superposed natural fractures, that are mutually crossing and orthogonal. Fig. 6 shows that

each natural fracture significantly alters the far-field flow streamline pattern and the time-of-flight contours. Where an

undisturbed far-field flow progresses with straight time-of-flight contours ( Fig. 3 (a)), the presence of the natural fractures

results in elongated time-of-flight contours at the tips of the natural fractures. Flow between two fractures ( Fig. 6 (a) and

(b)) is slowed down, as can be seen by the reduced time-of-flight contour spacing. Our principal sub-conclusion is that in an

144 A. van Harmelen, R. Weijermars / Applied Mathematical Modelling 56 (2018) 137–157

Fig. 6. Far-field flow superposed with various natural fracture configurations. Total time-of-flight is 10 years, time contour spacing is 1 year, and �t = 0.1 day. (a) Parallel natural fractures aligned with far-field flow. (b) Parallel natural fractures misaligned with far-field flow (c) orthogonal natural fractures

with far-field flow. (d) Orthogonal crossing natural fractures with far-field flow.

Table 3

Properties of analytical elements in Fig. 6 .

Location Strength Angle Length Width Height Porosity

Far-field flow N/A U ∞ = 0.3 m/year α = 0 ° N/A N/A N/A n = 0.2 Natural fractures (6a) z c1 = 0 – 5 ∗1i υ1 = 3.68 × 10 −7 m 4 /s β1 = 90 ° γ 1 = 0 ° L 1 = 5 m W 1 = 0.5 m h 1 = 1 m n 1 = 0.2

z c2 = 0 + 5 ∗1i υ2 = 3.68 × 10 −7 m 4 /s β2 = 90 ° γ 2 = 0 ° L 2 = 5 m W 2 = 0.5 m h 2 = 1 m n 2 = 0.2 Natural fractures (6b) z c1 = 0 – 5 ∗1i υ1 = 3.68 × 10 −7 m 4 /s β1 = 90 ° γ 1 = 30 ° L 1 = 5 m W 1 = 0.5 m h 1 = 1 m n 1 = 0.2

z c2 = 0 + 5 ∗1i υ2 = 3.68 × 10 −7 m 4 /s β2 = 90 ° γ 2 = 30 ° L 2 = 5 m W 2 = 0.5 m h 2 = 1 m n 2 = 0.2 ‘Natural fractures (6c) z c1 = –4.5 – 2 ∗1i υ1 = 3.68 × 10 −7 m 4 /s β1 = 90 ° γ 1 = −60 ° L 1 = 5 m W 1 = 0.5 m h 1 = 1 m n 1 = 0.2

z c2 = 1.5 + 1 ∗1i υ2 = 3.68 × 10 −7 m 4 /s β2 = 90 ° γ 2 = 30 ° L 2 = 5 m W 2 = 0.5 m h 2 = 1 m n 2 = 0.2 Natural fractures (6d) z c1 = 0 + 0 ∗1i υ1 = 3.68 × 10 −7 m 4 /s β1 = 90 ° γ 1 = 45 ° L 1 = 10 m W 1 = 0.5 m h 1 = 1 m n 1 = 0.2

z c2 = 0 + 0 ∗1i υ2 = 3.68 × 10 −7 m 4 /s β2 = 90 ° γ 2 = −45 ° L 2 = 10 m W 2 = 0.5 m h 2 = 1 m n 2 = 0.2

otherwise homogeneous reservoir, far-field flow drainage contours may become severely distorted due to the presence of natural

fractures .

4.2. Natural fracture interference on drainage area around vertical producer well

While the drainage region of a vertical wellbore is ideally circular ( Fig. A2 ), unknown natural figures may lead to unex-

pected drainage regions. Fig. 7 shows streamlines and drainage contours of a producer well with adjacent three different

natural fracture systems: (a) single pair of parallel natural fractures (unidirectional) next to a vertical wellbore, (b) fracture

system of multiple parallel natural fractures (unidirectional) surrounding a vertical wellbore, (c) fracture system (bidirec-

tional) near a vertical wellbore. Table 4 lists the properties of the fractures and the wellbore, and we further assume poros-

ity n = 0.2, height h = 1 m, fracture width W = 0.5 m and fracture length L = 5 m. Additionally, all fractures have strengthυ = 3.68 × 10 −7 m 4 /s and the wellbore strength is q(t) = 3.68 × 10 −7 m 3 /s.

Fig. 7 (a) highlights how even two relatively short natural fractures can already distort the streamline pattern near the

wellbore, as well as the drainage contour shape. Fig. 7 (b) and (c) show that natural fractures will lead to elongated drainage

contours in the direction of the natural fractures. Our principal sub-conclusion is that a parallel set of multiple natural

A. van Harmelen, R. Weijermars / Applied Mathematical Modelling 56 (2018) 137–157 145

Fig. 7. Vertical wellbore superposed with different natural fracture systems. Total time-of-flight is 10 years, time contour spacing is 1 year, and �t = 0.1 day. (a) Unidirectional natural fracture system next to a vertical wellbore. (b) Unidirectional natural fracture system surrounding a vertical wellbore.

(c) Bidirectional fracture.

Table 4

Properties of analytical elements in Fig. 7 .

Location Angle

Vertical wellbore z c = 0 + 0 ∗1i N/A Natural fractures (6a) z c1 = –0.5 – 4.5 ∗1i β1 = 90 ° γ 1 = −135 °

z c2 = 4.5 + 0.5 ∗1i β2 = 90 ° γ 2 = 45 °Natural fractures (6b) z c1 = –0.5 – 4.5 ∗1i β1 = 90 ° γ 1 = −135 °

z c2 = 4.5 + 0.5 ∗1i β2 = 90 ° γ 2 = 45 °z c3 = –6 – 2 ∗1i β3 = 90 ° γ 3 = −135 °z c4 = 2 + 6 ∗1i β4 = 90 ° γ 4 = 45 °

Natural fractures (6c) z c1 = –4.5 – 2 ∗1i β1 = 90 ° γ 1 = −135 °z c2 = 1.5 + 1 ∗1i β2 = 90 ° γ 2 = 45 °z c3 = 0 + 5 ∗1i β3 = 90 ° γ 3 = 90 °

146 A. van Harmelen, R. Weijermars / Applied Mathematical Modelling 56 (2018) 137–157

Fig. 8. Drainage distortion due to natural fractures. Total time-of-flight is 10 years, time contour spacing is 1 year, and �t = 0.1 day. (a) Overview of analytical elements and flow direction inside natural fractures. (b) Streamlines and drainage contours in hydraulic and naturally fractured reservoir. (c)

Streamlines and contours in reservoir without natural fractures.

Table 5

Properties of analytical elements in Fig. 8 .

Location Strength Length Width

Vertical wellbore z c = 2.5 + 0 ∗1i q ( t ) = −9.20 × 10 −8 m 3 /s N/A N/A Hydraulic fracture z c = 0 + 0 ∗1i q ( t ) = −3.68 × 10 −7 m 3 /s L = 5 m N/A Natural fracture (1) z c = –7 – 10 ∗1i υ = 3.68 × 10 −7 m 4 /s L = 8 m W = 0.5 m Natural fracture (2) z c = 1 – 6.5 ∗1i υ = 1.38 × 10 −7 m 4 /s L = 3 m W = 0.5 m Natural fracture (3) z c = 9 – 6 ∗1i υ = 2.30 × 10 −7 m 4 /s L = 5 m W = 0.5 m Natural fracture (4) z c = –11 – 3 ∗1i υ = 2.76 × 10 −7 m 4 /s L = 6 m W = 0.5 m Natural fracture (5) z c = –3.76 – 1.51 ∗1i υ = 1.84 × 10 −7 m 4 /s L = 4 m W = 0.5 m Natural fracture (6) z c = 5 – 2.5 ∗1i υ = 0.92 × 10 −7 m 4 /s L = 2 m W = 0.5 m Natural fracture (7) z c = 11 + 0 ∗1i υ = 1.84 × 10 −7 m 4 /s L = 4 m W = 0.5 m Natural fracture (8) z c = –0.74 + 1.51 ∗1i υ = 1.84 × 10 −7 m 4 /s L = 4 m W = 0.5 m Natural fracture (9) z c = –4 + 4 ∗1i υ = 2.76 × 10 −7 m 4 /s L = 6 m W = 0.5 m Natural fracture (10) z c = 8 + 5 ∗1i υ = 1.84 × 10 −7 m 4 /s L = 4 m W = 0.5 m Natural fracture (11) z c = 5 + 10 ∗1i υ = 3.22 × 10 −7 m 4 /s L = 7 m W = 0.5 m Natural fracture (12) z c = –5 + 12 ∗1i υ = 3.22 × 10 −7 m 4 /s L = 7 m W = 0.5 m Natural fracture (13) z c = 3 + 15 ∗1i υ = 2.30 × 10 −7 m 4 /s L = 5 m W = 0.5 m

fractures will tend to narrow the drainage region normal to the natural fractures while extending the drainage reach in the

direction of the natural fractures ( Fig. 7 (b)).

4.3. Hydraulic fractured vertical wellbore in a subparallel natural fracture system

We now expand on the results of basic natural fracture systems ( Sections 4.1 and 4.2 ), by considering a larger natural

fracture system with a single preferred angular orientation as is often observed in naturally fractured reservoirs ( Fig. 8 (a)).

The reservoir considered contains a vertical wellbore (black dot), a single hydraulic fracture (straight black line) with rotation

angle β = 0 °, and a system of thirteen natural fractures ( Fig. 8 (a)). Natural fracture numbers (5) and (8) are connected atthe tip of a hydraulic fracture ( Fig. 8 (a)), facilitating flow into the hydraulic fracture. The principal direction of flow inside

each natural fracture is indicated with blue arrows ( Fig. 8 (a)). For fractures (1)–(7) the rotation angle γ = 45 °, whereas forfractures (8)–(13) γ = −135 °. For all natural fractures β = 135 °. For all analytical elements we assume a depth of h = 1 m,and porosity of n = 0.2. Further properties of all analytical elements are given in Table 5 .

Fig. 8 (b) reveals the complex drainage pattern after flow superposition of the vertical wellbore, the hydraulic fracture and

the thirteen natural fractures. Specific streamlines have been marked with an arrowhead to facilitate easier visual tracing of

their flow pattern. While the streamlines still move from all directions towards the hydraulic fracture and vertical wellbore,

the contours in Fig. 8 (b) illustrate how natural fractures can distort the drainage area as compared to the same reservoir

without natural fractures ( Fig. 8 (c)). In Fig. 2 where the hydraulic fracture and vertical wellbore had equal strengths, drainage

contours at early times are pear-shaped. In contrast, the strength of the vertical wellbore in Fig. 8 (c) only drains 25% of the

amount drained by the hydraulic fracture ( Table 5 ), leading to more elliptical drainage contours even early on in the drainage

process. Another important insight is that drainage optimization of a naturally fractured reservoir by multiple producers

requires high-resolution streamline visualization, which has now become possible with the analytical element derived in

our study.

Our principal sub-conclusion is that natural fractures may severely distort the drainage contour pattern around a single

producer well. Although natural fractures do not necessarily lead to increased drainage (total drainage surface/volume for

A. van Harmelen, R. Weijermars / Applied Mathematical Modelling 56 (2018) 137–157 147

Fig. 9. Orthogonal photograph of polished rock slab with injection veins. (a) Filled fracture veins with interpreted directions of the original largest ( σ 1 )

and intermediate ( σ 2 ) principal stress axes. Major veins open first normal to σ 1 and then normal to σ 2 , which likely swapped with σ 1 after hydraulic

loading of the main veins. (b) Interpreted principal fracture network (yellow lines) used for flow model of fracture network in Fig. 10 . (For interpretation

of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 8 (b) and (c) are identical for identical time contours), the drainage pattern with natural fractures ( Fig. 8 (b)) is more

tortuous than when natural fractures are absent ( Fig. 8 (c)).

5. Fluid injection into hydraulic fractures

During hydraulic fracturing operations to stimulate the connectivity between the reservoir and the wells in a shale field,

a progressive network of fractures is created by sequentially isolating wellbore sections in a frac stage. In the Eagle Ford

formation, Texas, each stage of about 300 ft long has 6 perforation clusters (where a shotgun pierces the cement casing)

spaced 50 ft apart, with set of 6 perforations concentrated in a 2 ft stretch of the wellbore. Each stage will thus create 6

main fractures and over the total length of the 10,500 ft long horizontal wellbore, up to 35 stages will result in an array

of 210 transversal fractures. Unfortunately, the detailed geometry of the fractures created by hydraulic fracturing is poorly

known, as fracture diagnostic tools such as micro-seismic monitoring devices have poor resolution.

This study uses, as an analog for man-made hydraulic fractures in deep rock formations, a natural fracture network

created in a Proterozoic rock ( Fig. 9 (a) and (b)) from the Aravalli Supergroup [49–51] , extracted near the villages of Bidasar-

Charwas, Churu district in the state of Rajasthan, India. The rock was invaded by hydrothermal veins which hydraulically

fractured the rock under high pressure before being exhumed by tectonic uplift and erosion; a polished slab from the rock

face was imaged in a stonework shop, Bryan, Texas. Although the precise natural pressure responsible for the injection of the

hydraulic veins is unknown, the pressure has exceeded the strength of the rock and was large enough to open the fractures

at several km burial depth, thus being in the order of 100 MPa. The fluid was injected into the fractures as well as into a

pervasive system of micro-cracks connected to the main fracture. Based upon the splaying and provenance of the fractures,

we assume veins propagated from the top downward in the images of Fig. 9 (a) and (b).

The slab of Fig. 9 (a) and (b) may serve as a natural analog for flow into hydraulic fractures in shale reservoirs, with the

limitation that shale and our analog rock may have different elastic moduli, different petrophysics and grain size. Nonethe-

less, we postulate the injection patterns of the hydrothermal veins (preserved after cooling and subsequent exhumation)

provide a useful analog for fluid injection when hydraulic fracturing is applied to hydrocarbon wells. The flow in the main

fractures and matrix is modeled in Fig. 10 (a) and (b). Our simulation does not account for the creation of the fractures, but

assumes these have already developed and are subsequently flushed by the injection fluid.

The first case , modeled in Fig 10 (a), assumed the natural fractures have a hydraulic conductivity moderately higher than

that of the matrix (scaled by k f / k m ∼ 1.68). Such a limited permeability contrast results in fluid flowing faster through thefour or five main veins, as indicated by the time-of-flight contours of Fig. 10 (a), while the matrix still shows major fluid pen-

etration but with lagging time-of-flight. A second case ( Fig. 10 (b)) assumed the hydraulic conductivity of the natural fractures

is significantly larger than that of the matrix ( k f / k m ∼ 7.78). The increased permeability contrast ( Fig. 10 (b)) is achieved byincreasing the strength of all natural fractures used in the first case ( Fig. 10 (a)) by a factor of ten. In contrast to the progres-

sive flow of fluid through the matrix in Fig. 10 (a), large matrix regions remain un-impregnated (white space) in Fig. 10 (b),

because most of the injected fluid finds its way downward through the fractures. We emphasize the permeability contrasts,

specified for Fig. 10 (a) and (b), are estimations, as the inputs in our fracture flow model are in terms of relative strengths.

Properties of the natural fractures and far-field flow, including the assigned strengths, are listed in a supplementary table

(online version). Scaling rules for the permeability contrast in general are discussed in some further detail in Section 6.1 .

Permeability scaling for the specific example of Fig. 10 (a) and (b) is detailed in Appendix C .

Some important conclusions may be drawn from the simple analog study of Figs. 9 and 10 . First, hydraulic fracturing of

a matrix with relatively low permeability contrast may require excessive injection fluid as the matrix absorbs a significant

amount of the injected fluid ( Fig. 10 (a)). Any hydrocarbons will be effectively swept away from the injected well, which

requires longer flow-back and may cause permanent reservoir damage. A larger permeability contrast warrants the matrix

near the fractures is not or insignificantly invaded by the injected fluid. In our simulation, constant injection causes fluid

148 A. van Harmelen, R. Weijermars / Applied Mathematical Modelling 56 (2018) 137–157

Fig. 10. The complex natural fracture system of the prototype slab ( Fig. 9 ). Streamlines are colored magenta; time-of –flight contours are visualized using

rainbow colors ranging from dark blue (streamline progression after year 1) to red (progression after year 10). time-of-flight spacing is 1 year, total time-

of-flight is 10 years, and �t = 0.1 day. (a) Low permeability contrast ( k f / k m ∼1.68) between natural fractures and matrix. (b) High permeability contrast ( k f / k m ∼7.78) between natural fractures and matrix. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

to spurt into the matrix from the fracture tips ( Fig. 10 (b)), because a constant flux is maintained. However, the continuous

pressure increase to maintain the flux generally reaches a limit in real fracturing operations. Our principal sub-conclusions

are that (1) natural fractures may distort the fluid propagation front, but when the permeability contrast with the matrix

remains small ( k f / k m ∼1.68), the matrix is still massively invaded by the injection fluid ( Fig. 10 (a)), and (2) a relativelyhigh natural fracture conductivity compared to matrix ( k f / k m ∼ 7.78) leaves large matrix areas between the fractures barelytouched by fluid flow ( Fig. 10 (b)).

6. Discussion

This study offers an analytical 2D model for single phase flow in fractured porous media. The modeling of flow acceler-

ation through multiple cracks with enhanced hydraulic conductivity has not been modeled before by any analytical method.

Previous analytical models were limited to flow through a single channel or slit [52,53] . Such descriptions only apply to sin-

gle fracture “channels”, whereas our model can be used to model an infinite amount of natural fractures, accounting for any

mutual flow interference. More recently, numerical solution schedules have been proposed, with variable benchmark results

[6] . Although no benchmarking against the numerical test cases is attempted here, our present study provides a basis for

such benchmarking in a future study.

The models presented here make assumptions about fracture strength and width, as well as about the direction of flow

in the fractures. These are critical inputs in our model and some nuances for each of these inputs are highlighted below.

6.1. Fracture strength

Using mostly synthetic examples of flow in fractured porous media, we allocate to each fracture (segment) a particular,

a priori strength. In the examples of Section 4.1 ( Fig. 4 , Table 3 ), we make no attempt to differentiate strengths of the

fractures. By giving all examples equal strengths, the effect of different fracture orientations is uniquely highlighted. The

purpose is to show the model is capable of simulating single-phase flow in fractured porous media. However, the physical

relevance of what we model will be determined by accurate knowledge of the input parameters from the prototype reservoir

[54] . In practice, we often do not know the detailed fracture properties. For example, fracture diagnostics in the hydrocarbon

industry has revealed that we currently still lack devices to tell us precisely what are the physical attributes of either natural

or hydraulic fractures. We may know their length and maybe variation in aperture, using fracture propagation models to

arrive at an average fracture width, but effective permeability may need to be estimated by inverse means (i.e., well testing).

In order to aid translation of model results using fracture strength υ to practical applications, it is useful to scale thefracture strength using standard fracture properties like length, L f , width, W f , depth, D f , and permeability, k f . A starting point

is Darcy’s Law:

V f = −κ f L f

�P

μ. [m s −1 ] (11)

with V f denoting flow rate in the fracture of the fluid with viscosity, μ, and a pressure differential �P . The discharge rateof the fracture, q f , is given by

q f = −κ f L f

�P

μ( W f . D f ) . [ m

3 s −1 ] (12)

A. van Harmelen, R. Weijermars / Applied Mathematical Modelling 56 (2018) 137–157 149

The connection with the fracture strength, υ , is

υ f = −q f L f = −( W f . D f ) �P

μκ f . [ m

4 s −1 ] (13)

The permeability of a fracture in the model of Fig. 10 can be estimated by substituting in Eq. (13) an arbitrary viscosity

μ = 1 cP, unit fracture depth D f = 1 m and fracture width W f = 0.005 m, and using Eq. (1b ) with the specific complexpotential expression of the natural fracture given in Eq. (B12) to obtain from the model the pressure differential inside a

segment of the fracture. Similarly, the pressure differential is measured in the model outside the fracture, in the adjacent

matrix, to solve for k m . Then the permeability contrast k f / k m is solved. We designed our method with one important feature

in mind: the ability to use pressure measurements, from the field to adjust the model strength inputs in a feed-back loop

when all other properties are known until the model pressure matches the measured or separately modeled pressures in

the real fractured reservoir. This will be future work as a natural extension of the current paper.

Alternatively, a quick estimation of the permeability contrast is obtained by comparing the relative velocities inside and

outside the fracture zone, using Eq. (11) to define the permeability ratio and assuming L f / L m = 1: κ f k m

= V f V m

�P m �P f

. [ dimensionless ] (14)

If the boundary condition in the model would be a constant pressure differential between the top and bottom of the

model space (so that �P m = �P f ), Eq. (14) simplifies to: V f κm = V m κ f . Using Eq. (14) , with velocity measurements in ourmodel as summarized in Appendix C , we obtained a permeability contrast of k f / k m ∼ 1.68 for each fracture in the lowpermeability case of Fig. 10 (a), and k f / k m ∼ 7.78 for each fracture in the high permeability contrast case of Fig. 10 (b).In our model, the far-field velocity arrived at the upper boundary with a uniform speed, but then velocity differentials

develop inside the model, most likely accompanied by some differences between �P m and �P f . When �P m / �P f > 1, which

is likely in natural cases, our inferred permeability contrast estimations based on the measured model velocities will be

proportionally larger.

6.2. Fracture width

There is no limitation to using any smaller fracture widths in Eq. (10) . In the examples of Sections 1 –4 , relatively large

widths were taken in order to let streamlines, as they slip into the fracture(s) at a certain angle, still be visible in the

flow visualization images (see Figs. 5 (b) and 6 (d)). Natural fracture zones can have widths of 0.5 m or more. In fact, much

larger widths (up to km scale) occur in fractured zones along strike-slip faults [55 , 56] , which may have either reduced or

increased permeability depending on trans-tensional or trans-pressional far-field stress and any clay mineral alterations in

such zones. Admittedly, man-made hydraulic fractures in hard rocks will have much narrower apertures (widths), which is

why our model in Section 5 uses a width of 0.005 m (see the supplementary online table with input data for Fig. 10 ).

6.3. Direction of flow

The direction of flow inside a fracture is captured by the model’s design (see Appendix B ). The principal direction of

flow inside the fracture is indicated in Fig. B2 , i.e. orthogonal to the width W of the fracture. We control at which of the

two ‘width sides’ streamlines should enter, either by rotating the fracture 180 ° or by multiplying the strength with −1.However, the actual resulting direction of flow near and inside the natural fracture is strongly dependent on the superposed

flow, which can be represented by a range of analytical elements (source flow, far-field flow, etc.). From our experiments,

it became clear that we can model any fractured flow constellation, one of the critical questions being what is the flow

direction in the fracture for a given superposed flow? Because our study focuses on fractures with permeability higher than

the matrix, the fractures act as acceleration conduits in the direction of the far-field flow pressure gradient, which is how

we determined the flow polarity of the fractures. There is an important coupling between the assumed flow polarity and

fracture strength: if the assigned flow polarity is "wrong", this could be corrected by a sign reversal in the fracture strength.

In all cases, the simulated flow may still correspond to a natural example, because it is the combination of flow polarity

and fracture strength that controls the system. Measurements from nature on polarity and fracture strength are needed to

further validate dynamic similarity between any model and prototype flow.

7. Conclusions

In this study, we derived a new analytical model based on potential flow theory that emulates flow through natural

fractures. We conclude that streamlines, drainage contours and time-of-flight contours become severely distorted due to the

presence of (multiple) natural fractures in an otherwise homogeneous reservoir ( Fig. 6 ). For a unidirectional or bidirectional

natural fractures system, drainage is supported in the direction of the natural fractures while the drainage is narrower

normal to the natural fractures ( Fig. 7 ). Even though natural fractures do not need to lead to increased drainage, the drainage

pattern is more tortuous than when the fractures are absent ( Fig. 8 ). In a complex natural fracture system ( Fig. 10 ), large

matrix zones remain difficult to reach for frac fluid when natural fracture conductivity is significantly larger than the matrix

150 A. van Harmelen, R. Weijermars / Applied Mathematical Modelling 56 (2018) 137–157

conductivity. However, when the contrast between matrix and natural fracture permeability is relatively small, the matrix is

completely invaded by the injection fluid.

Acknowledgments

We thank Marble Craft, Bryan, Texas for providing access to their Indian facing stones used in the analysis of Fig. 9 . The

samples were photographed June 2015 by Bill Crawford, International Ocean Discovery Program at Texas A&M University.

This study was funded by start-up funds provided to the senior author by the Texas A&M Engineering Experiment Station

(TEES).

Appendix A. Overview of analytical elements

A.1. Point sources and sinks

In analytical reservoir modeling, a point source element represents a vertical injector wellbore in the planar ( x , y ) field

( Fig. A1 (a)), whereas a vertical producer wellbore is modeled by a point sink ( Fig. A1 (b)). The velocity field expression for

such an analytical point element is given by

V (z) = m (t) 2 π(z − z c ) , [m s

−1 ] (A1)

where z c is the location of the wellbore and m ( t ) its strength (for producers m ( t ) > 0 and for injectors m ( t ) < 0).

The strength m ( t ) can be more explicitly expressed [13] by using q ( t ) for the volumetric flow rate of the well (with unit

[m 3 s −1 ]), h for the height of the reservoir [m], and n for the reservoir porosity [dimensionless]:

m (t) = q (t) hn

, [ m 2 s −1 ] (A2)

Streamlines and time-of-flight contours (TOFCs) of a single vertical injector wellbore are obtained with our tracing algo-

rithm (expression (6) ) and visualized in Fig. A2 . Fluid flows equally in all directions because of a constant wellbore flowrate

and a homogeneous reservoir. Streamlines therefore move through the reservoir in strictly radial direction. Consequently,

the isochronous time-of-flight contours are circles.

The streamlines in Fig. A2 represent fluid flowing from the injector wellbore in outward direction, but can also be in-

terpreted as fluid flowing towards a producer wellbore. The TOFCs can therefore be interpreted as fluid front advance or as

Fig. A1. (a) Point source, (b) point sink (adapted from Weijermars and Van Harmelen [7] ).

Fig. A2. Streamlines (blue) and time-of-flight contours (red) of a point source. Parameters: z c = 0 + 0 ∗1i, q ( t ) = 1.84 × 10 −7 m 3 /s, h = 1 m, n = 0.2, and �t = 0.1 day. TOFC spacing is 1 year, total TOF is 10 years. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

A. van Harmelen, R. Weijermars / Applied Mathematical Modelling 56 (2018) 137–157 151

Fig. A3. (a1) Array of point sources, (a2) array of point sinks , (b1) interval source, (b2) interval sink (adapted from Weijermars and Van Harmelen [7] ).

Fig. A4. Interval source element.

drainage contours, respectively, called the flow reversal principle [13] . Hence, the flow reversal principle enables visualiza-

tion of drainage contours by using a point source. The wellbores modeled with expressions (A1) and (A2) are also called

line sources, as they extend in a straight line perpendicular to the plane of flow.

A.2. Interval sources and sinks

Interval sources and sinks can be employed not only to model injector and producer wellbores that coincide with the

plane of flow, but to model hydraulic fractures as well [6,13] . While a closely spaced array of point sources/sinks ( Fig. A3 ,

items a1,2) can approximate an interval source/sink ( Fig. A3 , items b1,2) [7] , obtaining a mathematical description of such

an interval element requires the superposition of an infinite amount point sources/sinks in a horizontal interval of finite

length [11] .

The velocity field description of an interval source is given by (see Weijermars and Van Harmelen [7] for a detailed

derivation):

V (z) = q (t) 2 πhnL

e −iβ{

log ( e −iβ (z − z c ) + 0 . 5 L ) − log ( e −iβ (z − z c ) − 0 . 5 L ) }. [m s −1 ] (A3)

In expression (A3) , q ( t ) is the volumetric flow rate [m 3 s −1 ] of the interval source ( q ( t ) < 0 for an interval sink), z c is thelocation of the center [m], h is the reservoir height [m], n the porosity [dimensionless], L the length [m], and β the rotationangle (see Fig. A4 ).

A hydraulic fracture, modeled with the flow reversal principal as an interval source, is in this study always connected

to a vertical wellbore as it drains reservoir fluid directly into the wellbore. Understanding the flow pattern of a superposed

vertical wellbore and hydraulic fracture requires familiarity with the basic flow pattern of an interval source. Fig. A5 there-

fore visualizes the streamlines and time-of-flight contours of an interval source. Note that the blue arrows will point inward

if the interval source was used to model a hydraulic fracture (due to the flow reversal principle). Because the streamlines

start along the interval source, the TOFCs of Fig. A5 are no longer circles like the TOFCs in Fig. A2 . Even though the shape of

the drainage contours in Figs. A2 and A5 are different, the total drained area is per definition equal because the volumetric

flow rate, height and porosity used are identical.

152 A. van Harmelen, R. Weijermars / Applied Mathematical Modelling 56 (2018) 137–157

Fig. A5. Interval source streamlines (blue) and TOFCs (red). TOFC spacing is 1 year, total TOF is 10 years. Interval source parameters: z c = 0 + 0 ∗1i, q ( t ) = 1.84 × 10 −7 m 3 /s, h = 1 m, n = 0.2, L = 5 m, β = 0 ° and �t = 0.1 day. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. A6. (a) Singularity doublet, (b) multiple singularity doublets, (c) transformed into line doublet (adapted from Weijermars and Van Harmelen [7] ).

Fig. A7. Streamlines of a line doublet with z c = 0 + 0 ∗1i, W = 5 m, β = 0 °, and strength υ = 3.68 × 10 −6 m 4 /s. Porosity n = 0.2 and height h = 1 m.

A.3. Singularity doublets and line doublets

Although singularity doublets and line doublets are not visualized in the main body of this paper, they are paramount to

the derivation of our natural fracture model. The singularity doublet ( Fig. A6(a )) is obtained by superposing a point source-

sink pair and then decreasing their distance and inversely increasing their strength in a limiting process [7] . Superposing

an infinite amount of singularity doublets in a line leads ( Fig. A6(b )) to the construction of a line doublet ( Fig. A6(c )).

The streamlines of a single line doublet are shown in Fig. A7 . The TOFCs are not visualized, because of the closely spaced

streamlines and their recirculation.

The line doublet’s velocity field is given in expression (A4) , where z c is the line doublet’s center [m], W is its total width

[m], υ is the line doublet strength [m 4 s −1 ] and β is its rotation angle. Expression (A4) is identical to expression (C9) inWeijermars and Van Harmelen [7] with z a = z c – 0.5 W · exp( i β), z b = z c + 0.5 W · exp( i β) and m ∗ = υ/(2 πhn ).

V (z) = −i υ · e iβ(

iβ)(

iβ) . [m s −1 ] (A4)

2 πhn z − ( z c + 0 . 5 W · e ) z − ( z c − 0 . 5 W · e )

A. van Harmelen, R. Weijermars / Applied Mathematical Modelling 56 (2018) 137–157 153

Fig. B1. Superposition of j line doublets, each of width W and angle β . Flow direction indicated by blue arrows. (For interpretation of the references to

color in this figure legend, the reader is referred to the web version of this article.)

Appendix B. Mathematical derivation of the natural fracture description

B.1. Superposing line doublets

The velocity field description of a natural fracture can be obtained by considering an infinite amount of line doublets.

First we set z c = 0 in expression (A4) :

V (z) = −i · e iβ(

z − 0 . 5 W · e iβ)(

z + 0 . 5 W · e iβ) υ

2 πhn . [m s −1 ] (B1)

Next, we assume that there are j line doublets of identical width W and angle β , with centers spaced evenly along thereal interval [ −0.5 ·L , 0.5 ·L ] ( Fig. B1 ). Fluid flow in Fig. B1 is indicated with blue arrows.

In order to maintain a cumulative uniform strength of υ , the strength of each line doublet has to equal υ/ j . By let-ting x k denote the location of the k th line doublet, the corresponding velocity field is found by applying the principal of

superposition:

V (z) = j−1 ∑ k =0

−i · e iβ(z − x k − 0 . 5 W · e iβ

)(z − x k + 0 . 5 W · e iβ

) υ2 πhn · j . [m s

−1 ] (B2)

B.2. Distance between two line doublets

In order to turn expression (B2) into a Riemann integral we first partition the interval [ −0.5 ·L , 0.5 ·L ] into j intervals thatare defined by the points

ˆ x k = −0 . 5 L + L

j k, 0 ≤ k ≤ j. [ m ] (B3)

Therefore, the center of a line doublet ( x k ) is located at

x k = ˆ x k +1 + ˆ x k

2 = −0 . 5 L + L

j k + L

2 j for 0 ≤ k ≤ j − 1 . [ m ] (B4)

Next, the spacing between two centers ( �x k ) is defined as

�x k = x k +1 − x k = L

j , 0 ≤ k ≤ j − 2 . [ m ] (B5)

B.3. From line doublets to natural fracture

Multiplying expression (B2) with L / L then gives

V (z) = j−1 ∑ k =0

−i · e iβ(z − x k − 0 . 5 W · e iβ

)(z − x k + 0 . 5 W · e iβ

) υ2 πhn · L

L

j . [m s −1 ] (B6)

Splitting this sum results in the expression

V (z) = j−2 ∑ k =0

−i · e iβ(z − x k − 0 . 5 W · e iβ

)(z − x k + 0 . 5 W · e iβ

) υ2 πhn · L �x k

+ −i · e iβ(

z − x j−1 − 0 . 5 W · e iβ)(

z − x j−1 + 0 . 5 W · e iβ) υ

2 πhn · j . [m s −1 ] (B7)

154 A. van Harmelen, R. Weijermars / Applied Mathematical Modelling 56 (2018) 137–157

Fig. B2. Natural fracture model. L and W are the length and width; z c is the center; z a 1 , z a 2 , z a 3 , and z b2 are the corners; β is the doublet angle (Fig. B1 ),

while γ is the rotation angle of the natural fracture. Intended flow direction indicated with blue arrows.

By letting j → ∞ , the latter part of expression (B7) vanishes and we are left with the integral:

V (z) = ∫ 0 . 5 L

−0 . 5 L

−i · e iβ(z − x k − 0 . 5 W · e iβ

)(z − x k + 0 . 5 W · e iβ

) υ2 πhn · L d x k . [m s

−1 ] (B8)

Evaluating this integral results in the velocity field description of a natural fracture located at the origin, with total width

w and total length l , and strength υ:

V (z) = −i · υe iβ

2 πhn · L

[log ( −z + x k − 0 . 5 W · e iβ ) − log (−z + x k + 0 . 5 W · e iβ )

2 · e iβ · 0 . 5 W

]0 . 5 x k = −0 . 5 L

= −i · υ2 πhn · L · W

[log (−z + 0 . 5 L − 0 . 5 W · e iβ ) − log (−z + 0 . 5 L + 0 . 5 W · e iβ )

+ log (−z − 0 . 5 L + 0 . 5 W · e iβ ) − log (−z − 0 . 5 L − 0 . 5 W · e iβ ) ].

[m s −1 ] (B9)

B.4. Natural fracture description

Lastly, letting z c and γ respectively denote the natural fracture’s center and angle of rotation (see Fig. B2 ), we use theconformal mapping f ( z ) = e −i γ ·( z – z c ) to find the velocity field for a natural fracture. This velocity field is found by evalu-ating V ( f ( z )) · f ’( z ):

V (z) = −i · υ · e −iγ

2 πhn · L · W [log ( −e −iγ ( z − z c ) + 0 . 5 L − 0 . 5 W · e iβ ) − log (−e −iγ (z − z c ) + 0 . 5 L + 0 . 5 W · e iβ )

+ log (−e −iγ (z − z c ) − 0 . 5 L + 0 . 5 W · e iβ ) − log (−e −iγ (z − z c ) − 0 . 5 L − 0 . 5 W · e iβ ) ].

[m s −1 ]

(B10)

In the above formula for a natural fracture, L and W are the total length and width, υ is the strength, z c the center,and γ its rotation angle. The angle β is the angle of the original line doublet. By considering the natural fracture’s corners( Fig. B2 ), we can simplify expression (B10) to

V (z) = −i · υ · e −iγ

2 πhn · L · W [log ( −e −iγ ( z − z a 2 ) )

− log (−e −iγ (z − z a 1 )) + log (−e −iγ (z − z b1 )) − log (−e −iγ (z − z b2 )

], [m s −1 ] (B11)

with z a 1 = z c − exp ( iγ ) · ( 0 . 5 L + 0 . 5 W · exp ( iβ) ) , z a 2 = z c − exp ( iγ ) · ( 0 . 5 L − 0 . 5 W · exp ( iβ) ) , z b1 = z c − exp ( iγ ) · ( −0 . 5 L + 0 . 5 W · exp ( iβ) ) , and z b2 = z c − exp ( iγ ) · ( −0 . 5 L − 0 . 5 W · exp ( iβ) ) .

The complex potential function is then obtained by integrating the natural fracture description, resulting in

�(z) = −i · υ · e −iγ

2 πhn · L · W [( z − z a 2 ) log ( −e −iγ ( z − z a 2 ) ) − (z − z a 1 ) log (−e −iγ (z − z a 1 ))

−iγ −iγ ] . [ m 2 s −1 ] (B12)

+(z − z b1 ) log (−e (z − z b1 )) −(z − z b2 ) log (−e (z − z b2 )

A. van Harmelen, R. Weijermars / Applied Mathematical Modelling 56 (2018) 137–157 155

Fig. C1. Vertical scale gives permeability contrasts between matrix and each fracture ( k f / k m ) for (a) low permeability case, and (b) high permeability case,

fracture numbers ( x -axis) correspond to those used in the supplementary online table.

Appendix C. Obtaining the permeability contrast per fracture

As set out in Section 6.1 , model velocities can be used to obtain a quick approximation of the permeability contrast

between fractures and the matrix (see Eq. (14) ). Pressures prevailing at the time of injection vein formation in the specific

rock slab used in our study ( Figs. 9 and 10 ) remain poorly constrained. We used the modeled velocity field to estimate the

permeability contrast with the matrix for each fracture. The velocity varies across the lateral width of each fracture, which

is why we calculated the velocity at a one hundred points inside each fracture and subsequently calculated the average

velocity per fracture. Next we divided the average fracture velocity by the average velocity in the adjacent matrix, assum-

ing �P m / �P f = 1. Fig. C1 plots the inferred permeability contrast for each individual fracture using the outlined procedure,resulting in permeability contrast estimates per fracture of k f / k m ∼ 1.68 in the low permeability case ( Fig. 10 (a)), and apermeability contrast of k f / k m ∼ 7.78 for each fracture in the high permeability case of ( Fig. 10 (b)).

Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.apm.2017.11.027 .

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Complex analytical solutions for flow in hydraulically fractured hydrocarbon reservoirs with and with