Modeling of 1D Anomalous Diffusion in Fractured Nanoporous ... · contains a time fractional derivative, while super-diffusion is described by a space-fractional derivative. As a

$Page 1: Modeling of 1D Anomalous Diffusion in Fractured Nanoporous ... · contains a time fractional derivative, while super-diffusion is described by a space-fractional derivative. As a$
D o s s i e rCharacterisation and Modeling of Low Permeability Media and Nanoporous Materials

Caractérisation et modélisation de milieux à faible perméabilité et de matériaux nanoporeux

Modeling of 1D Anomalous Diffusion in Fractured

Nanoporous Media

Ali Albinali*, Ralf Holy, Hulya Sarak and Erdal Ozkan

Colorado School of Mines, Petroleum Engineering Department, 1600 Arapahoe Street, Golden, CO 80401 - USAe-mail: [email protected] - [email protected] - [email protected] - [email protected]

* Corresponding author

Abstract — Fractured nanoporous reservoirs include multi-scale and discontinuous fractures coupledwith a complex nanoporous matrix. Such systems cannot be described by the conventional dual-porosity(or multi-porosity) idealizations due to the presence of different flow mechanisms at multiple scales.More detailed modeling approaches, such as Discrete Fracture Network (DFN) models, similarlysuffer from the extensive data requirements dictated by the intricacy of the flow scales, whicheventually deter the utility of these models. This paper discusses the utility and construction of 1Danalytical and numerical anomalous diffusion models for heterogeneous, nanoporous media, whichis commonly encountered in oil and gas production from tight, unconventional reservoirs withfractured horizontal wells. A fractional form of Darcy’s law, which incorporates the non-local andhereditary nature of flow, is coupled with the classical mass conservation equation to derive afractional diffusion equation in space and time. Results show excellent agreement with establishedsolutions under asymptotic conditions and are consistent with the physical intuitions.

Résumé—Modélisation de la diffusion anormale en 1D dans des milieux nanoporeux fracturés—Des réservoirs nanoporeux fracturés comprennent des fractures à échelles multiples et discontinuescouplées à une matrice nanoporeuse complexe. Ces systèmes ne peuvent pas être décrits par lesmodélisations conventionnelles à double porosité (ou multi-porosité) du fait de la présence dedifférents mécanismes de flux à multiples échelles. Des approches de modélisation plus détaillées,telles que les modèles de Réseau de Fracture Discret (DFN, Discrete Fracture Network) souffrent demême des exigences extensives en matière de données, dictées par la complexité des échelles deflux, qui nuisent éventuellement à l’utilité de ces modèles. Le présent article traite de l’utilité et de laconstruction de modèles analytiques et numériques de diffusion anormaux en 1D pour des milieuxhétérogènes nanoporeux, qui se rencontrent communément dans la production de pétrole et de gaz,pour les réservoirs étanches, non conventionnels avec des puits horizontaux fracturés. Une formefractionnée de la loi de Darcy, qui comprend la nature non-locale et héréditaire du flux, est reliée àl’équation de conservation de la masse classique afin d’obtenir une équation de diffusion fractionnéedans l’espace et le temps. Les résultats montrent une parfaite concordance avec des solutions établiesdans des conditions asymptotiques et ils sont conformes aux intuitions physiques.

Oil & Gas Science and Technology – Rev. IFP Energies nouvelles (2016) 71, 56� A. Albinali et al., published by IFP Energies nouvelles, 2016DOI: 10.2516/ogst/2016008

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

http://ogst.ifpenergiesnouvelles.fr/

http://ogst.ifpenergiesnouvelles.fr/

http://ifpenergiesnouvelles.fr/

http://dx.doi.org/10.2516/ogst/2016008

NOMENCLATURE

A Area (L2)B Formation volume factor (L3/L3)ct Total compressibility (LT2/M)D Diffusion coefficient (L2/T)df Fractal dimension

ds Spectral dimensiondw Fractal dimension of the walkh Height (L)I0 Modified Bessel functions of order zero

K Diffusion Coefficient (L2/T)K0 Modified Bessel functions of order zeroK1 Modified Bessel function of order onek Permeability (L2)

ka Permeability (L2/T1�a)ka,b Permeability (Lb�1/T1�a)L Length (L)p Pressure (M/L/T2)

q Flow rate (L3/T)q̂m Matrix flux rate per unit rock volumeR Radial distance (L)r Spherical distance (L)rm Radius of matrix sphere (L)

s Laplace parametert Time (T)u Velocity (L/T)XF Fracture half-length (L)

GREEK

a Time fractional exponentb Space fractional exponentH Anomalous diffusion coefficienth Diffusion index

k Mobility ratiol Fluid viscosity (M/L/T)r Shape factor/ Porosity (L3/L3)

ϑ Bias factor for preferential diffusion directionx Storativity ratio

SUBSCRIPTS

i Coordinate

m Matrixf Natural fractures

F Hydraulic fracture

D Dimensionless

SUPERSCRIPTS

n Time step

ABBREVIATIONS

ADMNF Anomalous Diffusion for Matrix and NaturalFractures

CTRW Continuous-Time Random Walk

MSD Mean-Square DisplacementNFR Naturally Fractured ReservoirsTLM Tri-Linear Model

INTRODUCTION

Fluid flow in naturally fractured nanoporous reservoirs havebeen primarily modeled using approaches that treat the flowdomain as continuum and the flow parameters as representedby their statistical averages. In these models, rock matrix andnatural fractures are generally perceived as low permeabilitystorage medium and highly conductive pathways, respec-tively, that exchange fluids under certain domain and flowconditions. Darcy’s law is the governing fluid flow equationwith the mathematical models utilizing a normal diffusionprocess in the matrix and natural fracture regions — trans-port is considered to be random and following a Gaussian-distributed probability-density function.

Normal diffusion constitutes that a particle’s mean squaredisplacement rh i is related to time linearly; that is:

r2� � ¼ Dt ð1Þ

whereD is the diffusion coefficient. The resulting diffusivityequation used in conventional petroleum engineering fluidflow models is:

@

@tp r; tð Þ ¼ Dr2p r; tð Þ ð2Þ

Several conceptual models have been developed forNaturally Fractured Reservoirs (NFR): (i) a single mediumwith enhanced permeability due to natural fractures, (ii)dual-porosity medium where tight rock matrix feeds intoconductive natural fractures, (iii) matrix with a discrete frac-ture network where each fracture is individually character-ized, and (iv) fractal media where properties are scaled

Page 2 of 25 Oil & Gas Science and Technology – Rev. IFP Energies nouvelles (2016) 71, 56

with distance to some reference point in the domain. Thecurrent models are appropriate for conventional reservoirs,which have permeabilities ranging from tens to hundredsof milli-Darcy (1 Darcy � 10�12 m2). However, the existingapproaches are not adequate in describing flow in nanopor-ous fractured reservoirs due to the extreme petrophysicalcomplexity and velocity-field heterogeneity. Productionrates decline drastically after a short period of time whenthe fluid stored in the natural fractures is produced and thelow-permeability matrix cannot feed into the fractures atrates matching the fracture depletion (Nelson, 2001).

In tight, unconventional reservoirs, such as shale-gas andtight-oil plays, matrix and natural fracture permeabilities aredownscaled significantly. The permeability of the rockmatrix is in the nano- to micro-Darcy range and the poreradii range from a few to no more than hundreds of nano-meters. Natural fractures at different sizes, conductivities,and orientations constitute the discontinuities in these sys-tems due to their large contrast to the matrix conductivity.They may be the product of source-rock maturation, tectonicevents, burial, uplifting, or the changes in the stress field.Generally, the aperture of these fractures is larger than theopening of the matrix pores and flow channels. Additionally,these fractures can be randomly distributed as data from welllogs, core samples and outcrops confirm. Another level ofcomplexity is presented in source rocks due to the presenceof organic material (kerogen) with different pore structureand permeability than the inorganic matrix. (At low pres-sures, existence of organic material may give rise to thecontribution of desorption and molecular diffusion to flow;however, in this paper, such low pressures will not be con-sidered.) This complex environment includes multiple scales(Fig. 1), which create preferential flow paths, differences inpressure profiles, varying fluid compositions, and complexflow regimes (Camacho-Velázquez et al., 2012; Ozkan,2013).

From a fundamental fluid mechanics perspective, flow innanoporous unconventional reservoirs takes place on anassembly of short and long flow paths — compared to thetotal medium — in addition to the heterogeneous distribu-tion of these channels. Under these conditions, continuummechanics, which is a major assumption of Darcy’s law, isinapplicable (Fig. 2). Diffusive flows at slower rates takeplace in the matrix and advection has a small contributionto flow due to the diminutive proportion of micropores.Advection, however, is the dominant transport mechanismin natural fractures. Where fractures form a network (contin-uum), advection in fractures dictates the global pressure dis-tribution, which, in turn, governs the local diffusion in thematrix. Therefore, a non-local, hereditary transport resultsin the system. In light of this discussion, unconventionalNFR are conceived as disordered media and flow and

transport are described by anomalous diffusion andfractional calculus in this paper.

Several mathematical assumptions can lead to the formu-lation of anomalous diffusion, which range from assumingthat diffusion follows a power-law as a function of distance,fluid particles follow a Continuous-Time Random Walk(CTRW) behavior, or that the observation and correlationscales are different (Raghavan, 2011). Anomalous diffusionrelates the Mean-Square Displacement (MSD) of a particleto time by the following relation:

r2� � � ta; where

a ¼ 1Normal diffusion

a 6¼ 1Anomalous diffusion

a > 1 Super � diffusion

a < 1 Sub� diffusion:

8>>>>><>>>>>:

ð3Þ

This becomes useful in representing fluid flow in disor-dered, heterogeneous porous structures, specifically at vary-ing scales. In the CTRW model, the transitioning time (orwaiting time) of a particle between two jumps is describedby a probability density function. Longer waiting timescan be associated to an increased flux impediment in thesystem due to obstructions such as discontinuous fractures.As a consequence the particle MSD will grow slower thanin the normal diffusion case. A laboratory study demonstrat-ing the potential of the CTRWmodel to determine the degreeof subdiffusion in a heterogeneous porous media ispresented in Berkowitz et al. (2000). Similarly, it is possibleto define stochastic particle displacement models resulting inMSD growing faster than in the normal case, hence leadingto super-diffusion. The probability density function describ-ing particle displacement is heavy-tailed allowing for longerparticle jumps. One such model was presented by Redner(1989) for a stratified porous medium with layers of differentconductivity.

Physically, sub- and super-diffusion imply that the localflows are temporally and spatially convolved and cannotbe described in terms of instantaneous, local gradients.The diffusion equation describing subdiffusive flowcontains a time fractional derivative, while super-diffusionis described by a space-fractional derivative. As a conse-quence, the solutions fall in the framework of fractionalcalculus.

Previous attempts of using fractional calculus haveutilized the concept of fractals to scale the propertiesspatially. For instance, O’Shaughnessy and Procaccia(1985) have modified the diffusion equation as:

@p r; tð Þ@t

¼ 1

rD�1

@

@rK rð ÞrD�1 @p r; tð Þ

@r

� �ð4Þ

Oil & Gas Science and Technology – Rev. IFP Energies nouvelles (2016) 71, 56 Page 3 of 25

and defined the diffusion coefficient K in a fractal geometryby:

K rð Þ ¼ Kr�H ð5Þ

In Equations (4) and (5), r represents the distance to somereference point in the domain,H being the anomalous diffu-sion coefficient, and D is the fractal dimension. The solutioninvolves a steady-state conductivity, (K); that is assignedspatially and does not take into account the temporal orspatial dependence.

The aim of this paper is to discuss the construction andutility of 1D anomalous diffusion models for oil and gas pro-duction from tight, unconventional reservoirs with fracturedhorizontal wells. Both analytical and numerical solutions ofthe diffusion equation of fractional form are presented. In theanalytical solution, only time-fractional anomalous diffusionis considered because of the difficulties in satisfying the

boundary conditions when space-fractional derivatives areinvolved. The numerical model takes into account both thetemporal and spatial fractional derivatives as the derivativescan be approximated numerically. The focus of the analyticalsolution is to document and discuss the versatility of anoma-lous diffusion to consider various scales and connectivity ofnatural fractures. Numerical solution addresses two issues:handling no-flow boundaries and demonstrating thecombined effect of space- and time-fractional diffusion inpressure-distributions in the reservoir.

1 ANALYTICAL APPROACH

The objective in this section is to apply anomalous diffusionto cover a wider range of heterogeneities in unconventionalreservoirs. The approach taken here utilizes the dual-porosity idealization of fractured reservoirs but considersadditional complexities: (i) natural fractures are discreteand finite in length (do not form a continuum); anomalousdiffusion describes flow in the union of the matrix-fracturesystem, (ii) there is a set of conductive and globally con-nected fractures where normal diffusion prevails, while thematrix includes another set of small, discontinuous fractureswhich gives rise to anomalous diffusion, and (iii) there aretwo scales of fractures; flow in globally connected fracturescan be described by anomalous diffusion due to tortuouspath, varying shape, nonuniform aperture, roughness,cementation, etc., and the diffusion in the matrix is alsoanomalous because of the existence of discontinuousmicrofractures.

For convenience, the analytical solution is derived in twosteps. First, a transfer function is derived to describe fluidtransfer from matrix to global fractures for a radial-flowsystem (vertical well at the center of a cylindrical reservoir)shown in Figure 3. Here, the porous medium is assumed toconsist of spherical matrix elements of uniform radiusenveloped by natural fractures that form a connected

km scale m scale cm scale μm scalea) b) c) d)

Figure 1

Scale heterogeneity in unconventional NFR (Russian, 2013).

Figure 2

Knudsen number and expected flow mechanisms (modifiedfrom Roy et al., 2003).


network for flow. (The extension of the formulation to othermatrix shapes is straightforward.) Pressure is uniform at thesurface of the matrix elements and flux from matrix elementsto the natural fractures is distributed instantaneously anduniformly over one-half the volume of natural fractures(de Swaan, 1976; Kazemi, 1969). The transfer function isderived by considering the conditions in items (i) and (ii)above.

In the second step, the solution is extended to the tri-linearflow model, developed by Ozkan et al. (2009) for multi-fractured horizontal wells in unconventional reservoirs.The tri-linear model couples flow between three contiguousregions: outer reservoir, inner reservoir and hydraulicfracture (Fig. 4). The inner reservoir is considered as adual-porosity region while the outer reservoir and thehydraulic fracture are considered homogeneous (single-porosity) regions. Anomalous diffusion considerations areinvoked into the tri-linear flow model by using the appropri-ate transfer function derived in the first step for the dual-porosity inner reservoir. In this paper, we only discuss thederivation of the transfer function with anomalous diffusion.Details of the solutions presented in this section can be foundin Albinali (2016).

1.1 Transfer Function for Anomalous Diffusion

A CTRW process (Montroll and Weiss, 1965) may be usedto define the constitutive (flux) relation of time-fractionalanomalous diffusion in porous media. The process consistsof probability functions that describe jump lengths and wait-ing times, and will provide the MSD in the sought powerform given in Equation (3). The resulting diffusion equationwill include index/indices that can be used to describe thetransport behavior and the physical structure. The followingequation, proposed by Metzler et al. (1994), was utilized inthis paper:

@a

@tap r; tð Þ ¼ 1

rds�1

@

@rrds�1 @

@rp r; tð Þ

� �ð6Þ

where the diffusion exponent a is related to the diffusionindex h by:

a ¼ 2

2þ hð7Þ

Note here that setting h = 0, or a = 1, corresponds tonormal diffusion and we recover the standard Darcy’s law.Following Raghavan (2011) and Raghavan and Chen(2013), we can write the convective flux as:

u x; tð Þ ¼ � kal

@1�a

@t1�a

@p

@x

� �ð8Þ

and the transient diffusion equation can be stated as:

p x; tð Þ ¼ @1�a

@t1�a

kal

@2

@x2p x; tð Þ

� �ð9Þ

Figure 4

Flow regions in the tri-linear flow model.

Figure 3

Representation of the utilized cylindrical dual-porosity system(Ozkan, 2011).


In general, three parameters are used to describe a fractalstructure; a spectral dimension ds, fractal dimension df andfractal dimension of the walk dw (Mandelbort, 1983). Theseparameters are related by:

ds ¼ 2dfdw

ð10Þ

There is no straightforward relationship between the frac-tal dimension and the diffusion index h and additionalinformation is needed to compute the fractal dimension(Chang and Yortsos, 1990; de Swaan et al., 2012). An exam-ple is given in Camacho-Velázquez et al. (2008) where theCTRW concept was used following the fractional diffusionequation by Metzler et al. (1994). The authors introduced:

df � h� 1 ð11Þ

and

1� b2þ h

ð12Þ

in their formulations and included data from transient andboundary dominated flow tests. Using asymptotic approxi-mations of dimensionless equations and plotting productionversus time they, were able to calculate the slopes that corre-spond to a and m and compute df. Also, they showed how torelate Equations (11) and (12) to the parameters in conven-tional reservoir engineering models via plotting. An over-view about the CTRW and diffusion on fractals can befound in Metzler and Klafter (2000). Another approach tomodel diffusion on fractals was presented in Barker (1988)and Chang and Yortsos (1990) by considering a fractalgeometry and applying the classical conservation of masslaw to obtain a diffusivity equation. The resulting modelsyield isotropic properties that scale with distance, similarto Equation (4) presented by O’Shaughnessy and Procaccia(1985).

In the analytical part of this work, we consider anomalousdiffusion independently for both domains of the dual-porosity idealization by assigning two diffusion exponentsam and af for matrix and natural fractures, respectively.Deriving the equations in this manner gives flexibility inchoosing normal or anomalous diffusion when extendingthe solution into other reservoirs. For example, in reservoirswith high matrix permeability and discontinuous fractures,the solution can be provided as normal diffusion for thematrix and anomalous diffusion for the natural fractures.Another example is for reservoirs with homogeneous andwell-connected natural fractures and heterogeneous tightmatrix where one can apply normal diffusion for the naturalfractures and anomalous diffusion for the matrix.

Starting with the flow in the matrix spheres, we incorpo-rate Equation (6) into the conservation of mass equation toget the continuity equation for a slightly compressible fluidas:

1

r2@

@rr2kml

@1�am

@t1�am

@pm@r

� �¼ /ctð Þm

@pm@t

ð13Þ

Note here that am is the diffusion exponent pertaining tothe matrix spheres. Similarly, flow in the natural fracturesis governed by:

1

R

@

@RRkfl

@1�af

@t1�af

@pf@R

� �þ q̂m ¼ /ctð Þf

@pf@t

ð14Þ

where q̂m represents the flux from matrix to natural fracturesand is given by:

q̂m ¼ � 4pr2m� km

l@1�am

@t1�am

@pm@r

� �r¼rm

" #=

4pr2mhf2

� �ð15Þ

The solution of the above system of equations is obtainedin the Laplace transform domain and leads to the definitionof the following dual-porosity transfer function similar tothose defined by Barenblatt et al. (1960) and Kazemi (1969):

f sð Þ ¼ 2kmhfDkf rmD

gfX 2

F

� �af �am

saf �am�1

� rmDffiffiffiffiffiffibm

pTanh

ffiffiffiffiffiffibm

prmD

� � 1

" #þ gf

X 2F

� �af �1

saf �1 ð16Þ

The details of the derivation of Equation (16) is given inAppendix A. We use the transfer function for the dual-porosity inner region of the tri-linear model (Ozkan et al.,2009) to obtain the results discussed in the next section.An alternative procedure is given in Noetinger and Gautier(1998) and Noetinger et al. (2001). Their work also includesdiscretization of the developed fluid flow equation and theCTRW algorithm over the porous medium.

1.2 Verification of the Analytical Solution and Discussionof Results

Synthetic data in Table 1 are used to verify the results of theanalytical solution (ADMNF) against the Tri-Linear Model(TLM). The models are first compared under homogeneousreservoir conditions by setting diffusion exponents of theanalytical solution to unity since the tri-linear model consti-tutes normal diffusion. Also, the dual-porosity functions inboth models were set to unity to represent a homogeneous


reservoir. Figure 5 shows an excellent agreementbetween the bottomhole pressures obtained from the twomodels.

Next, the models are compared under dual-porosity ideal-ization. The diffusion exponents of the analytical solutionare again set to unity. Several runs were conducted under dif-ferent values of matrix thickness. Transmissivity (k) andstorativity (x) ratios of de Swaan (1976) and Kazemi(1969) are used to label the tri-linear model runs. Transmis-sivity is a ratio of the matrix flow capacity to that of the

natural fractures, i.e. smaller values indicate larger matrixslabs. Storativity is a measure of the fluid storage in naturalfractures to that in the matrix. These parameters are definedas:

k ¼ rX 2F

kmkf

ð17Þ

and

x ¼ /ctð Þf/ctð Þf þ /ctð Þm

ð18Þ

where r is a shape factor for the matrix blocks and, for thespherical geometry used in this work, r ¼ 12 (Kazemiet al., 1976). Solid lines correspond to the tri-linear modeland circles correspond to the analytical solution in Figure 6.The results show excellent agreement under the dual-porosity idealization.

After verifying the analytical solution for asymptoticcases, we use it to investigate the effects of anomalous diffu-sion. Figure 7 considers normal diffusion in fractures(af ¼ 1Þ and anomalous diffusion in matrix for the data inTable 1. In Figure 7, the case for af ¼ am ¼ 1 correspondsto the conventional dual-porosity model. For the values ofam 6¼ 1, Figure 7 displays the effect of different levels ofthe matrix heterogeneity. Because matrix contribution comesin at later times (after depleting the network of naturalfractures), Figure 7 focuses on the bottomhole pressure atlate times.

In the second case shown in Figure 8, for a fixed value ofam ¼ 1, a range of the diffusion exponent of the natural

1.E+2

1.E+3

1.E+4

1.E+5

1.E+6

1.E+7

1.E+8

1E-2 1E-1 1E+0 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7

Pre

ssur

e D

rop,

Pa

Time, hour

ADMNF-H TLM -H

Figure 5

Verification for homogeneous reservoir. TLM-H versus devel-oped solution (ADMNF-H).

TABLE 1

Data used for analytical solution verification

Formation thickness, h, m 76.2

Wellbore radius, rw, m 0.0762

Horizontal well length, Lh, m 853.44

Number of hydraulic fractures, nF 15

Distance between hydraulicfractures, dF, m

60.96

Distance to boundary parallel towell (1/2 well spacing), xe, m

76.2

Inner reservoir size, ye, m 30.48

Viscosity, l, Pa�s 3E-4

Hydraulic fracture porosity, /F,fraction

0.38

Hydraulic fracture permeability,kF, m

24.93E-11

Hydraulic fracture totalcompressibility, ctF, Pa

�11.45E-8

Hydraulic fracture half-length,xF, m

76.2

Hydraulic fracture width, wF, m 0.003

Matrix permeability, km, m2 9.87E-24

Matrix porosity, /m, fraction 0.05

Matrix compressibility, ctm, Pa�1 1.45E-9

Matrix radius, rm, m 3.05

Natural fractures permeability,kf, m

29.87E-13

Natural fractures porosity, /f,fraction

0.35

Natural fractures compressibility,ctf, Pa

�17.98E-5

Constant flow rate, q, m3/s 2.76E-4


fractures, af , from 1 to 0.1 is considered. Values less thanone indicate subdiffusion in the fracture network.As expected, higher-pressure drops are observed as flowdeviates from normal diffusion (smaller af values). Thiscan also be interpreted that flow becomes hindered as flowdeviates from normal diffusion due to increased heterogene-ity of the velocity field.

In Figures 9 and 10, we consider some cases of mixed dif-fusion. Case a, shown in Figure 9, considers two combina-tions of normal diffusion in natural fractures with normaland subdiffusive flows in the matrix. The discrepancybetween the bottomhole pressures for normal and subdiffu-sive matrix flows at late times (after the fracture system isdepleted) indicates that the analytical model proposed in thiswork extends the conventional dual-porosity approach tomore complex matrix heterogeneities.

Case b shown in Figure 10 is similar to Case a in Figure 9except for subdiffusion, instead of normal diffusion, infractures. For practical purposes, the difference between theresults for normal and anomalous diffusion is not discerniblein Figure 10. This result should be expected based on thereduced conductivity of fractures in the case of anomalousdiffusion; that is, even though the matrix flow capacityincreases when am increases from 0.1 to 1, the limited flowcapacity of fractures due to anomalous diffusion dictateshow much fluid can be transferred from matrix to fractures.

To emphasize the significance of the results shown inFigures 7–10, it must be noted that the classical dual-porosity approach is only capable of representing the hetero-geneity caused by the contrast between the volume-averagedproperties of the matrix and fracture media. As discussed inthe Introduction, when the heterogeneity reaches the limit ofcontinuum mechanics, flow phenomena cannot be repre-sented in terms of the average properties of the medium.Figures 7–10 indicate that the new dual-porosity solutionprovides a convenient means of covering a wider range ofheterogeneity encountered in unconventional reservoirs.

2 NUMERICAL APPROACH

As noted in Introduction, the analytical difficulty in handlingno-flow boundaries in space-fractional anomalous diffusionmakes the numerical treatment of the problem desirable.In addition, the numerical solution has the potential toextend the utility of the anomalous diffusion model to othercases of practical interest such as multiphase flow. In thissection, we present an implicit finite-difference scheme tomodel linear anomalous diffusion for a single phase, slightlycompressible fluid flowing in a finite reservoir initially at

1.E+4

1.E+5

1.E+6

1.E+7

1E+4 1E+5 1E+6 1E+7

Pre

ssur

e D

rop,

Pa

Time, hour

αm = 1

αm = 0.9

αm = 0.7

αm = 0.4

αm = 0.2

αm = 0.1

Figure 7

Results for subdiffusive flow in matrix.

1.E+2

1.E+3

1.E+4

1.E+5

1.E+6

1.E+7

1.E+8

1.E+9

1E-2 1E-1 1E+0 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7

Pre

ssur

e D

rop,

Pa

Time, hour

αf = 1

αf = 0.9

αf = 0.7

αf = 0.4

αf = 0.2

αf = 0.1

Figure 8

Results for subdiffusive flow in natural fractures.

1.E+1

1.E+2

1.E+3

1.E+4

1.E+5

1.E+6

1.E+7

1E-2 1E-1 1E+0 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7

Pre

ssur

e D

rop,

Pa

Time, hour

λ = 20, ω = 5.41E-4 λ = 50, ω = 2.16E-4 λ = 250, ω = 4.33E-5 λ = 500, ω = 2.16E-5

ADMNF1 ADMNF2 ADMNF3 ADMNF4

Figure 6

Verification under dual-porosity idealization.


uniform pressure, with a constant-rate boundary at x = 0, anda no-flux boundary at x = L (Fig. 11).

The fractional flux law incorporating space and timenon-locality is defined by Chen and Raghavan (2015) as:

uðx; tÞ ¼ � ka;bl

@1�a

@t1�a

@b

@xbpðx; tÞ

!ð19Þ

where 0 < a < 1 and 0 < b < 1 are the fractional derivativeexponents in time and space respectively, l is the fluid vis-cosity, and ka;b is the phenomenological coefficient withdimensions L1þbT 1�a. Note that, in the asymptotic case ofa ¼ b ¼ 1, the flux law reverts back to the well-knownDarcy’s law with ka;b in dimensions of L2. For a < 1 thediffusion is expected to slow down (subdiffusion) while inthe case of b < 1 particle motion should accelerate(superdiffusion).

The classical mass conservation equation in 1D is givenby:

� o@x

u x; tð ÞB

� �¼ /ct

B

@p x; tð Þ@t

ð20Þ

Combining Equations (19) and (20), and introducing theinitial and boundary conditions, the following initialboundary value problem is obtained:

@

@x

ka;blB

@1�a

@t1�a

@bp x; tð Þ@xb

! !

¼ /ctB

@p x; tð Þ@t

for 0 < x < L; t > 0; ð21Þ

p x; 0ð Þ ¼ pinitial for 0 � x � L; ð22Þ

u 0; tð Þ ¼ � ka;bl

@1�a

@t1�a

@bp 0; tð Þ@xb

!

¼ q 0; tð ÞA

for t � 0 constant rate boundaryð Þ;ð23Þ

Figure 11

Schematic of the linear system examined. The boundaries x = 0and x = L correspond to the constant-rate (q = constant) and no-flux (q = 0) boundaries respectively.

1.E+0

1.E+1

1.E+2

1.E+3

1.E+4

1.E+5

1.E+6

1.E+7

1.E+8

1E-2 1E-1 1E+0 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7

Pre

ssur

e D

rop,

Pa

Time, hour

αf = 0.1, αm = 1

αf = 0.1, αm = 0.1

Figure 10

Case b for mixed diffusion process.

1.E+2

1.E+3

1.E+4

1.E+5

1.E+6

1.E+7

1E-2 1E-1 1E+0 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7

Pre

ssur

e D

rop,

Pa

Time, hour

αf = 1, αm = 1

αf = 1, αm = 0.1

Figure 9

Case a for mixed diffusion process.


u L; tð Þ ¼ � ka;bl

@1�a

@t1�a

@bp L; tð Þ@xb

!

¼ 0 for t � 0 no� flux boundaryð Þ ð24Þ

The spatial domain [0, L] is discretized into a block-centered grid of Imax blocks and uniform block lengthDx = L/Imax. The grid block centers are labeled with theindex xi where i = 1, . . ., Imax as shown in Figure 12. Thetime domain [0,T] is discretized into N + 1 time steps withuniform time increment Dt = T/N and the time steps arelabeled with the index tn = nDt, n = 0, . . ., N (Fig. 13).The numerical approximations of the function p(xi, tn) aredenoted by pni .

2.1 Approximation of Fractional Derivatives

The time and space integer derivatives of the mass conserva-tion (Eq. 20) are approximated by the forward and centraldifferences, respectively, leading to:

o@x

u x; tð ÞB

� �¼ /ct

B

@p x; tð Þ@t

¼> � 1

�x

u

B

h iiþ1

2

� u

B

h ii�1

2

� �

¼ /ctB

Pnþ1i � Pn

i

�tð25Þ

or after substituting the flux (Eq. 19):

1

�x

ka;blB

@1�a

@t1�a

@bp x; tð Þ@xb

!" #iþ1

2

0@

� ka;blB

@1�a

@t1�a

@bp x; tð Þ@xb

!" #i�1

2

!¼ /ct

B

Pnþ1i � Pn

i

�tð26Þ

where the subscripts i 12 denote the grid block interfaces

and the superscripts nþ 1 and n denote the simulation timesteps.

Time-Fractional Derivative

In Equation (26), the time fractional derivatives in the fluxterms at grid block interfaces (xi 1

2) are defined in the Caputo

(1967) sense, allowing the use of integer-order initial condi-tions. The left-sided Caputo derivative is defined as:

C0DC

at f tð Þ ¼ 1

C p� að ÞZ t

0

@pf sð Þ@tp

t � sð Þp�1�a ds ð27Þ

where p� 1 < a < p, p being the smallest integer larger than a,and C(x) is the gamma function. Substituting Equation (27)

with 0 < a < 1 into the time fractional derivatives ofEquation (26) yields:

@1�a

@t1�a

@bp xi12; tn

� �@xb

0@

1A ¼ C

0DC1�atnþ1

@b xi12; tn

� �@xb

0@

1A

¼ 1

C 1� 1� að Þð ÞZ tnþ1

t0¼0

@

@t

@bp xi12; tn

� �@xb

0@

1A

� tnþ1 � sð Þ� 1�að Þds n ¼ 0; . . . ;N � 1 ð28Þ

Using the approach presented in Murio (2008), afterrewriting Equation (28) as a summation of integrals overuniform time intervals, Dt, and evaluating the time derivativein each integral term by the first order forward approxima-tion, the following expression is obtained:

@1�a

@t1�a

@bp xi12; tn

� �@xb

0@

1A ¼ ra;�t

Xnþ1

k¼1

x að Þk

�@bp xi1

2; tnþ2�k

� �@xb

�@bp xi1

2; tnþ1�k

� �@xb

0@

1A ð29Þ

where

ra;�t ¼ 1

C 1þ að Þ1

�t1�a ð30Þ

and

x að Þk ¼ ka � k � 1ð Þa; x að Þ

1 ¼ 1 ð31Þ

The complete derivation of Equation (29) is presented inAppendix B and further details are presented in Holy (2016).

Figure 12

Spatial discretization of the examined system.


Equation (29) can be further rearranged to yield:

@1�a

@t1�a

@bp xi12; tn

� �@xb

0@

1A ¼

ra;�t

@bp xi12; tnþ1

� �@xb

þXnk¼1

�x að Þk þ x að Þ

kþ1

� � @bp xi12; tnþ1�k

� �@xb

� x að Þnþ1

@bp xi12; t0

� �@xb

266666666664

377777777775

ð32Þ

Two important observations can be made from the frac-tional time derivative approximation in Equation (32). First,the evaluation of the derivative at tnþ1 requires the space-fractional pressure derivatives at all previous time steps fromt0 to tn. Second, for a? 1,x að Þ

k ? 1 for k �1 and ra;�t? 1,and hence Equation (32) requires the evaluation of the spacefractional derivative at tnþ1 and t0 only. Because the systemis initially at rest, @bp xi1=2; t0

� @xb ¼ 0, and, thus, the

equation reduces to evaluating the spatial derivative at tnþ1

in the asymptotic case a = 1. Hence, Equation (32) revertsto the classical implicit formulation for normal diffusion.

Figure 14 shows the weights �x að Þk þ x að Þ

kþ1

� �attributed

to the past space-fractional derivatives as a function of awhen computing solutions for tnþ1 ¼ 50 based in Equation(32). As can be seen, a values closer to one result in deriva-tives putting more weight on past states while a valuestowards zero make the derivative more local, with theasymptotic case of a = 0 reverting to the first derivative.It should also be noted that the weight coefficients naturallyadd up to zero for any 0 � a � 1 and any number of priortime steps tn. Hence, the approximation presented inEquations (29) and (32) is not a truncated series.

Space-Fractional Derivative

The space-fractional derivatives at the grid block interfacesare defined as weighted two-sided Caputo derivatives based

on the symmetric Caputo derivative presented by Klimekand Lupa (2011). The space-derivative at the interface atxi+1/2 and time tn+1 is:

@bp xiþ12; tnþ1

� �@xb

¼ #C0D

bxiþ1

2

� 1� #ð ÞCxiþ1

2

Db

Lð33Þ

where 0 < b < 1, 0 < ϑ < 1 is the weighting factor allowing toset a bias on the preferred diffusion direction on either sideof the point of interest, xiþ1=2. In Equation (33), C

0DCbxiþ1=2

and Cxiþ1=2

Db

Lare the left and right-sided Caputo derivatives,

respectively, as shown in Figure 15.The left-sided Caputo derivative (from the left boundary

at x = 0 to the interface at xi+1/2) is defined as:

C0D

bxiþ1

2

¼ 1

C 1� bð ÞZ x

iþ12

0

@p n; tnþ1ð Þ@n

� xiþ12� n

� ��bdn ð34Þ

The right-sided Caputo derivative (from the point at xi+1/2to the right boundary at x ¼ L) is defined as in Kilbas et al.(2006):

Cxiþ1

2

Db

L¼ �1

C 1� bð ÞZ L

xiþ1

2

@P n; tnþ1ð Þ@n

n� xiþ12

� ��bdn

ð35Þ

Both derivatives are discretized using the approachpresented in Zhang et al. (2007), leading to the followingapproximations:

C0D

bxiþ1

2

¼ rb;�x

Xim¼1

x bð Þm Pnþ1

iþ2�m � Pnþ1iþ1�m

� ð36Þ

and

Cxiþ1

2

Db

L¼ �rb;�x

XImax�i

m¼1

x bð Þm Pnþ1

iþm � Pnþ1i�1þm

� ð37Þ

where

rb;�x ¼ 1

C 2� bð Þ1

�xbð38Þ

and

x bð Þm ¼ m1�b � m� 1ð Þ1�b; x bð Þ

1 ¼ 1 ð39Þ

The detailed derivation of Equations (34) and (35) isgiven in Appendix C and Holy (2016).

Figure 13

Temporal discretization of the examined system.


Substituting the left and right-sided Caputo approxima-tions, Equations (34) and (35), into Equation (33), thefinite-difference approximation to the space-fractionalderivative at xi+1/2 becomes:

@bp xiþ12; tnþ1

� �@xb

¼ rb;�x

#Xim¼1

x bð Þm Pnþ1


�

þ 1� #ð ÞXImax�i

m¼1

x bð Þm Pnþ1


�

8>>>><>>>>:

9>>>>=>>>>;

ð40Þ

In a similar manner, the finite-difference approximation tothe space fractional derivative at the interface at xi�1=2 andtime tnþ1 becomes:

@bp xi�12; tnþ1

� �@xb

¼ rb;�x

#Xi�1

m¼1

x bð Þm Pnþ1

iþ1�m � Pnþ1i�m

�

þ 1� #ð ÞXImaxþ1�i

m¼1

x bð Þm Pnþ1

i�1þm � Pnþ1i�2þm

�

8>>>><>>>>:

9>>>>=>>>>;ð41Þ

Figure 14

Temporal dependence of the time-fractional derivative as a function of a.


Two important observations can be made from the space-fractional derivative approximations: first, the evaluation ofthe two-sided derivative at time tnþ1 requires pressures atevery grid block, leading to a fully populated Imax � Imaxð Þiteration matrix; second, for b ! 1; x bð Þ

m ! 0 for m > 1,and rb;�x ! 1=�x, the approximations in Equations (40)and (41) reduce, respectively, to:

@bp xiþ12; tnþ1

� �@xb

¼ 1

�x# Pnþ1

iþ1 � Pnþ1i

�

þ 1� #ð Þ Pnþ1iþ1 � Pnþ1

i

� g ¼ Pnþ1iþ1 � Pnþ1

i

�xð42Þ

and

@bp xi�12; tnþ1

� �@xb

¼ 1

�x# Pnþ1

i � Pnþ1i�1

�

þ 1� #ð Þ Pnþ1i � Pnþ1

i�1

� g ¼ Pnþ1i � Pnþ1

i�1

�xð43Þ

which correspond to the classic central difference approxima-tionsof thefirstderivativesatgridblock interfacesatxiþ1=2 andxi�1=2 in case of normal diffusion. In addition, as for the timefractional derivative, theweight coefficients attributed to eachgrid block naturally add up to zero for any 0 < b < 1 and anynumberofgridblocks.Hence, theapproximationspresentedinEquations (40) and (41) are not truncated series.

Substituting the time- and space-fractional derivativeapproximations, Equations (29), (40) and (41), into Equation(26), an implicit finite-difference scheme for the anomalousdiffusion equation is obtained at time tnþ1:

1

�x

ka;blB

� �nþ1

iþ12

ra;�t

rb;�x

#Xim¼1

x bð Þm Pnþ1


�

þ 1� #ð ÞXImax�i

m¼1

x bð Þm Pnþ1


�

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

�@bp xiþ1

2; tn

� �@xb

þXnþ1

k¼2

x að Þk

@bp xiþ12; tnþ2�k

� �@xb

�@bp xiþ1

2; tnþ1�k

� �@xb

0@

1A

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

� 1

�x

ka;blB

� �nþ1

i�12

ra;�t

rb;�x

#Xi�1

m¼1

x bð Þm Pnþ1

iþ1�m � Pnþ1i�m

�

þ 1� #ð ÞXImaxþ1�i

m¼1

x bð Þm Pnþ1

i�1þm � Pnþ1i�2þm

�

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

�@bp xi�1

2; tn

� �@xb

þXnþ1

k¼2

x að Þk

@bp xi�12; tnþ2�k

� �@xb

�@bp xi�1

2; tnþ1�k

� �@xb

0@

1A

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

¼ /ctB

Pnþ1i � Pn

i

�tð44Þ

Figure 15

Left and right side contributions to the space fractional deriva-tive computed at xi+1/2.


Treatment of Boundaries

The rate q at the constant flux boundary (x = 0) is specifiedexplicitly in grid block 1 by:

u

B

h i1�1

2

¼ q

BAfor tn � 0 ð45Þ

At the no-flux boundary (x ¼ L), the flux is zero andhence for grid block Imax is:

u

B

h iImaxþ1

2

¼ 0 for tn � 0 ð46Þ

2.2 Verification of the Numerical Solution

The finite-difference scheme presented in the previoussection is used to model the flow of a slightly compressiblefluid towards a hydraulic fracture in a complex porous mediaat constant production rate within one-fourth of the fracturedrainage area. The objective is to qualitatively assess the

influence of the exponents, a and b, and verify the solution.For simplicity, the fluid and rock properties are treatedas constants and the synthetic data used is presented inTable 2.

TABLE 2

Fluid and rock properties used for verification of the numerical solution

Formation thickness, h, m 76.2

Fracture half-length, xf, m 76.2

Distance to no-flux boundary, m 76.2

Phenomenological coefficient,ka,b, m

1+bd1-a4.935E-17

Porosity, /, fraction 0.2

Total compressibility, ct, Pa�1 5.801E-9

Viscosity, l, Pa.s 3E-4

Constant flowrate, q, m3/s 2.76E-5

Initial pressure, p, MPa 34.474

Figure 16

Pressure profiles in the studied system at times t = 10, 20, 50, 200 days as a function of a with b = 1. The case a = 1 corresponds to normaldiffusion while a < 1 corresponds to subdiffusion.


Sensitivity on Time-Fractional Exponent a

Figure 16 shows the impact of the time-fractional exponent aon the pressure distribution in the reservoir at different times,where b ¼ 1, T ¼ 200 days, N ¼ 200, �t ¼ TN ¼ 1 day,and the number of grid blocks, Imax = 250. The case for a ¼ 1corresponds to normal diffusionwhile the cases for a < 1 cor-respond to subdiffusion,which is clearly shownby the smallerdrainage distances at t ¼ 200 days. In cases for a ¼ 1 anda ¼ 0:99, the pressure disturbance has clearly reached theouter boundary while transient flow is still dominant fora < 0:4. In addition, pressure drawdowns at the fracture faceare more significant when a < 1 (Fig. 17), which is in agree-ment with the responses obtained by the analytical solution.

Sensitivity on Space-Fractional Exponent b

Figure 18 shows the impact of the space fractional exponentb on the pressure distribution in the reservoir at different

Figure 17

Pressure drop versus time at the fracture face as a function of awith b = 1. The case a = 1 corresponds to normal diffusionwhile a < 1 corresponds to subdiffusion.

Figure 18

Pressure profiles in the studied system at times t = 10, 20, 50, 200 days as a function of b with a =1 and #= 0. The case b = 1 corresponds tonormal diffusion while b < 1 corresponds to superdiffusion.


times, where a ¼ 1, T ¼ 200 days, N ¼ 200,�t ¼ TN ¼ 1,the number of grid blocks, Imax = 250, and the weighting fac-tor, # ¼ 0 (for right-sided derivative contribution only).Note that in case of # ¼ 0, the iteration matrix is reducedto an upper triangular matrix. The case b ¼ 1, correspondsto normal diffusion while the cases b < 1 correspond tosuperdiffusion. This is shown through the larger drainagedistances as well as smaller pressure drawdowns at the frac-ture face (Fig. 19).

From a purely qualitative point of view, the implementedfinite difference scheme seems to be well suited for handlingthe imposed boundary conditions although neither extensivetesting nor stability and convergence analysis have been per-formed at this stage. Another important point to address isthe selection of the weighting factor #. Figure 20 showsthe impact of using the right-sided (# ¼ 0), symmetric(# ¼ 0:5) and left-sided (# ¼ 1) space fractional derivativeat t = 10 days for different values of b, keeping all otherparameters unchanged. For b ¼ 1, the normal diffusion caseis obtained and is unaffected by the weighting factor. Forb < 1, although the trend of superdiffusion is observed inall three cases, the cases for # ¼ 0:5 and # ¼ 1 seem to bephysically irrelevant due to the concave downward shapeof the pressure profiles.

CONCLUSION

A modified flux law incorporating spatial and temporalheterogeneity of the velocity field has been coupled withthe classic mass conservation equation to model anomalousdiffusion in fractured nanoporous media. The model has

been used to simulate the complex flow behaviors aroundhydraulically fractured horizontal wells in unconventionalreservoirs. The resulting partial differential equationincludes non-integer time and space derivatives, which aresolved by means of fractional calculus. In the analytical solu-tion only the time dependency of flux has been studied due

Figure 19

Pressure drop versus time at the fracture face as a function of bwith a = 1 and #= 0. The case b = 1 corresponds to normal dif-fusion while b < 1 corresponds to superdiffusion.

Figure 20

Pressure profiles in the studied system at t = 200 days as a func-tion of b and #.


to the difficulty of handling no-flux boundary conditionswhen space fractional derivatives are introduced. This draw-back is overcome in the numerical approach.

Sensitivities run on the time-fractional exponent a are inagreement in the analytical and numerical solutions showinglarger pressure drops near the flux boundary and smallerdrainage distances for smaller values of a. Such cases seemto be representative of porous media dominated by tightmatrix and disconnected natural fractures leading to subdif-fusion. The space-fractional exponent b has the oppositeeffect. Decreasing b values lead to smaller pressure dropsat the flux boundary and larger drainage distances at a partic-ular time. Such cases might represent porous media domi-nated by a well-connected natural fracture network leadingto superdiffusion.

REFERENCES

Albinali A. (Forthcoming 2016) Analytical Modeling of FracturedNano-Porous Reservoirs, PhD Dissertation, Colorado School ofMines.

Barenblatt G.E., Zheltov I.P., Kochina I.N. (1960) Basic conceptsin the theory of seepage of homogeneous liquids in fissured rocks,Journal of Applied Mathematics and Mechanics 24, 5, 1286-1303.

Barker J. (1988) A generalized radial flow model for hydraulic testsin fractured rock, Water Resources Research 24, 10, 1796-1804.

Berkowitz B., Scher H., Silliman S.E. (2000) Anomalous transportin laboratory-scale, heterogeneous porous media, Water ResourcesResearch 36, 1, 149-158.

Camacho-Velázquez R., Fuentes-cruz G., Vásquez-Cruz M. (2008)Decline-Curve Analysis of Fractured Reservoirs with FractalGeometry, SPE Reservoir Evaluation & Engineering 11, 3,606-619.

Camacho-Velázquez R., Vásquez-Cruz M.A., Fuentes-Cruz G.(2012) Recent Advances in Dynamic Modeling of Naturally Frac-tured Reservoirs, SPE Latin American and Caribbean PetroleumEngineering Conference, Mexico City, Mexico, April 16-18.

Caputo M. (1967) Linear Models of Dissipation whose Q is almostFrequency Independent-II, Geophysical Journal 13, 5, 529-539.

Chang J., Yortsos Y.C. (1990) Pressure transient analysis of fractalreservoirs, SPE Formation Evaluation 5, 1, 31-38.

Chen C., Raghavan R. (2015) Transient flow in a linear reservoirfor space-time fractional diffusion, Journal of Petroleum Scienceand Engineering 128, 194-202.

de Swaan A. (1976) Analytic Solutions for Determining NaturallyFractured Reservoir Properties by Well Testing, Society of Petro-leum Engineers Journal 16, 3, 117-122.

de Swaan A., Camacho-Velázquez R., Vásquez-Cruz M. (2012)Interference Tests Analysis in Fractured Formations with a Time-fractional Equation, SPE Latin American and Caribbean PetroleumEngineering Conference, Mexico City, Mexico, April 16-18.

Holy R. (Forthcoming 2016) Numerical Investigation of 1D Ano-malous Diffusion in Fractured Nanoporous Reservoirs, PhD Dis-sertation, Colorado School of Mines.

Kazemi H. (1969) Pressure Transient Analysis of Naturally Frac-tured Reservoirs with Uniform Fracture Distribution, Society ofPetroleum Engineers Journal 9, 4, 451-462.

Kazemi H., Merrill Jr L.S., Porterfield K.L., Zeman P.R. (1976)Numerical Simulation of Water-Oil Flow in Naturally FracturedReservoirs, Society of Petroleum Engineers Journal 16, 6, 17-326.

Kilbas A.A., Srivastava H.M., Trujillo J.J. (2006) Fractional Inte-grals and Fractional Derivatives, in Theory and Applications ofFractional Differential Equations, Elsevier, Amsterdam.

Klimek M., Lupa M. (2011) On Reflection Symmetry in FractionalMechanics, Scientific Research of the Institute of Mathematics andComputer Science 10, 1, 109-121.

Mandelbort B.B. (1983) The Fractal Geometry of Nature, W.H.Freeman and Company, New York.

Metzler R., Glockle W.G., Nonnenmacher T.F. (1994) FractionalModel Equation for Anomalous Diffusion, Physica A: StatisticalMechanics and its Applications 211, 1, 13-24.

Metzler R., Klafter J. (2000) The RandomWalk’s Guide to Anoma-lous Diffusion: a Fractional Dynamics Approach, Physics reports339, 1, 1-77.

Montroll E.W., Weiss G.H. (1965) Random Walks on Lattices. II,Journal of Mathematical Physics 6, 2, 167-181.

Murio D.A. (2008) Implicit finite difference approximation for timefractional diffusion equations, Computers and Mathematics withApplications 56, 4, 1138-1145.

Nelson R.A. (2001) Geologic Analysis of Naturally FracturedReservoirs, 2nd ed., Gulf Professional Publishing, Woburn.

Noetinger B., Gautier Y. (1998) Use of the Fourier-Laplace trans-form and of diagrammatical methods to interpret pumping tests inheterogeneous reservoirs, Advances in Water Resources 21, 7,581-590.

Noetinger B., Estebenet T., Landereau P. (2001) A direct determi-nation of the transient exchange term of fractured media using acontinuous time random walk method, Transport in porous media44, 3, 539-557.

Ozkan E., Brown M., Raghavan R., Kazemi H. (2009) Comparisonof Fractured Horizontal-Well Performance in Conventional andUnconventional Reservoirs, SPE Western Regional Meeting, SanJose, California, March 24-26.

Ozkan E. (2011) On Non-Darcy Flow in Porous Media: ModelingGas Slippage in Nano-pores, SIAMMathematical & ComputationalIssues in the Geosciences Meeting, Long Beach, California, March21-24.

Ozkan E. (2013) A Discourse on Unconventional Reservoir Engi-neering – The State of the Art after a Decade, UnconventionalReservoir Engineering Project Consortium Meeting at ColoradoSchool of Mines, Golden, Colorado, Nov. 8-11.

O’Shaughnessy B., Procaccia I. (1985) Diffusion on Fractals, Phys-ical Review A 32, 5, 3073-3083.

Raghavan R. (2011) Fraction Derivative: Application to TransientFlow, Journal of Petroleum Science and Engineering 80, 1, 7-13.

Raghavan R., Chen C. (2013) Fractured-Well Performance underAnomalous Diffusion, SPE Reservoir Evaluation & Engineering16, 3, 237-254.

Redner S. (1989) Superdiffusive Transport Due to Random Veloc-ity Fields, Physica D: Nonlinear Phenomena 38, 1-3, 287-290.


Roy S., Raju R., Chuang H.F., Cruden B.A., Meyyappan M. (2003)Modeling gas flow through microchannels and nanopores, Journalof Applied Physics 93, 8, 4870-4879.

Russian A. (2013) Anomalous Dynamics of Darcy Flow and Diffu-sion through Heterogeneous Media, PhD Dissertation, UniversitatPolitècnica de Catalunya.

Stehfest H. (1970) Numerical Inversion of Laplace Transforms,Communications of the ACM 13, 1, 47-49.

Zhang X., Lv M., Crawford J., Young I.M. (2007) The impact ofboundary on the fractional advection-dispersion equation for solutetransport in soil: Defining the fractional dispersive flux with theCaputo derivatives, Advances in Water Resources 30, 5,1205-1217.

Manuscript submitted in May 2015

Manuscript accepted in May 2016

Published online in August 2016

Cite this article as: A. Albinali, R. Holy, H. Sarak and E. Ozkan (2016). Modeling of 1D Anomalous Diffusion in FracturedNanoporous Media, Oil Gas Sci. Technol 71, 56.


APPENDIX

Appendix A: Derivation of the Analytical Solution

The mass balance equation for spherical control volume is

� 1

r2@

@rr2qov� ¼ @

@tqo/mð Þ ðA:1Þ

The time-dependent Darcy equation given as equation governs the flux m. The right side of equation can be written as

@

@tqo/mð Þ ¼ qo

@

@p

@p

@t/mð Þ þ /m

@

@p

@p

@tqoð Þ ¼ @p

@tqo

@/m

@pþ /m

@qo@p

� �ðA:2Þ

We have

cm ¼ 1

/m

@/m

@pðA:3Þ

and

co ¼ 1

qo

@qo@p

ðA:4Þ

We define

ct ¼ co þ cm ðA:5Þ

Now

@p

@tqo

@/m

@pþ /m

@qo@p

� �¼ @p

@tqocm/m þ /mqoco½ ¼ @p

@tqocm/m þ /mqoco½ ¼ @p

@tqoct/mð Þ ðA:6Þ

Then

1

r2@

@rr2qo

kml

@1�am

@t1�am

@pm@r

� �¼ @p

@tqoct/mð Þ ðA:7Þ

For a slightly compressible fluid, the equation can be reduced to

1

r2@

@rr2kml

@1�am

@t1�am

@pm@r

� �¼ @p

@tct/mð Þ ðA:8Þ

Defining dimensionless variables as

pD ¼ kf h

qBl�p ðA:9Þ

rD ¼ r

X FðA:10Þ

and

tD ¼ gfX 2

F

t ðA:11Þ


with

gf ¼kf

l /ctð Þf ðA:12Þ

Taking @am�1

@tam�1 of both sides, we can write the matrix flow equation in dimensionless form as

2

rD

@pmD@rD

þ @2pmD@r2D

¼ X 2F

gm

� �gfX 2

F

� �am @ampmD@tamD

ðA:13Þ

Introducing

wmD rD; tDð Þ ¼ pmD rD; tDð ÞrD ðA:14Þ

bm ¼ X 2F

gm

� �~gfX 2

F

� �am

sam ðA:15Þ

and applying Laplace transformation, we get

d2�wmD

dr2D� bm�wmD ¼ 0 ðA:16Þ

which have a general solution of

�wmD ¼ Aexp �ffiffiffiffiffiffibm

prD

� �þ Bexp


prD

� �ðA:17Þ

Boundary condition

�pm r ¼ 0; tð Þ ¼ finite ðA:18Þ

in dimensionless form is

pmD rD ¼ 0; tDð Þ ¼ finite ðA:19Þ

and in terms of wmD is

wmD rD ¼ 0; tDð Þ ¼ 0 ðA:20Þ

This yields

A ¼ �B ðA:21Þ

and we can write

�wmD ¼ B expffiffiffiffiffiffibm

prD

� �� exp �


prD

� �h iðA:22Þ

or equally

�wmD ¼ 2Bsinhffiffiffiffiffiffibm

prD

� �ðA:23Þ


Reverting the expression to �pmD

�pmD ¼ 1

rD�wmD ¼ 2B

sinhffiffiffiffiffiffibm

prD

� rD

" #ðA:24Þ

The second boundary condition is

�pmð Þr¼rm¼ �pf

� �R¼rm

ðA:25Þ

in dimensionless form and Laplace domain is

pmDð ÞrD¼rmD¼ pfD

� �RD¼rmD

ðA:26Þ

and

�pmDð ÞrD¼rmD¼ �pfD�

RD¼rmDðA:27Þ

respectively. The solution for the matrix is given as

�pmDð Þ ¼ �pfD�

rmD

rmDsinh


prmD

� " #

sinhffiffiffiffiffiffibm

prD

� rD

" #ðA:28Þ

For the flow in the natural fractures, we repeat steps of (A.1) to (A.8) and we need to include the flux term from the matrixregion to get

1

R

@

@RRkfl

@1�af

@t1�af

@pf@R

� �þ q̂m ¼ /ctð Þf

@pf@t

ðA:29Þ

Matrix elements supply natural fractures by

q̂m ¼ � 4pr2m� km

l@1�am

@t1�am

@pm@r

� �r¼rm

" #=

4pr2mhf2

� �ðA:30Þ

Taking @af �1

@taf �1 of both sides, putting the equation in dimensionless form and applying Laplace transform we get

1

RD

@

@RDRD

d�pfDdRD

� �� 2kamX F

hfDkaf

gfX 2

F

� �af�am

saf �an d�pmDdrD

� �rD¼rmD

� gfX 2

F

� �af �1

saf �pfD ¼ 0 ðA:31Þ

We can substitute for d�pmDdrD

� �rD¼rmD

from previous step and defining

f sð Þ ¼ 2kamhfDkaf rmD

gfX 2

F

� �af�am

saf �am�1 rmDffiffiffiffiffiffibm

pTanh


prmD

� � 1

" #þ gf

X 2F

� �af �1

saf �1 ¼ 0 ðA:32Þ

and

u ¼ sf sð Þ ðA:33Þ

we arrive at

1

RD

@

@RDRD

@�pfD@RD

� �� u�pfD ¼ 0 ðA:34Þ


This is a modified Bessel’s equation of order zero with general the solution

�pfD ¼ C1I0ffiffiffiu

pRD

� þ C2K0ffiffiffiu

pRD

� ðA:35Þ

The outer boundary condition is

�pf R ! 1; sð Þ ¼ 0 ðA:36Þ

which leads to

C1 ¼ 0 ðA:37Þ

since

limx!1 I0 xð Þð Þ ¼ 1 ðA:38Þ

At the wellbore

@�pfD@RD

� �R¼RwD

¼ � 1

sðA:39Þ

which leads to

ffiffiffiu

pC2K1

ffiffiffiu

pRwD

� ¼ � 1

sðA:40Þ

Solving for C2 and substitute in the equation, we get

�pfD ¼ � 1

sffiffiffiu

p K0ffiffiffiu

pRDð Þ

K1ffiffiffiu

pRwDð Þ ðA:41Þ

The solution can be inverted back to the time domain using Stehfest (1970) algorithm.

Appendix B: Finite Difference Approximation to the Time Fractional Derivative

The finite difference approximation Equation (29) to the time fractional derivative Equation (28) is derived as follows:

@1�a

@t1�a

@bp xi12; tn

� �@xb

0@

1A ¼ C

0DC1�atnþ1

@bp xi12; tn

� �@xb

0@

1A; n ¼ 0; . . . ;N � 1

¼ 1

C 1� 1� að Þð ÞZ tnþ1

t0¼0

@

@t

@bp xi12; tn

� �@xb

0@

1A tnþ1 � sð Þ� 1�að Þds ðB:1Þ

Using the approach presented in Murio (2008) this can also be written as the summation of integrals overintervals �t:

@1�a

@t1�a

@bp�xi1

2; tn

@xb

!¼ 1

C 1� 1� að Þð ÞXnþ1

k¼1

Z k�t

k�1ð Þ�t

@

@t

@bp xi12; tn

� �@xb

0@

1A tnþ1 � sð Þ� 1�að Þds ðB:2Þ


Replacing the time derivative term by the first order forward approximation:

@1�a

@t1�a

@bp�xi1

2; tn

@xb

!

ffi 1

C að ÞXnþ1

k¼1

Z k�t

k�1ð Þ�t

@bpp�xi1

2;tk

@xb � @bpp�xi1

2;tk�1

;

@xb

!

�ttnþ1 � sð Þ� 1�að Þds ðB:3Þ

¼ 1

C að ÞXnþ1

k¼1

@bp�xi1

2;tk

@xb � @bp�xi1

2;tk�1

@xb

!

�t

Z k�t

k�1ð Þ�ttnþ1 � sð Þ� 1�að Þds

Integration of the integral term leads to:

@1�a

@t1�a

@bp�xi1

2; tn

@xb

!¼ 1

C að ÞXnþ1

k¼1

@bp�xi1

2;tk

@xb � @bp�xi1

2;tk�1

@xb

!

�t� tnþ1 � sð Þ1�ð1�aÞ

1� ð1� aÞ

" #k�t

k�1ð Þ�t

¼ 1

C að Þ1

a

Xnþ1

k¼1

@bp�xi1

2;tk

@xb � @bp�xi1

2;tk�1

@xb

!

�t� tnþ1 � sð Þa½ k�t

k�1ð Þ�t

¼ 1

C 1þ að ÞXnþ1

k¼1

@bp�xi1

2;tk

@xb � @bp�xi1

2;tk�1

@xb

!

�t

� nþ 1ð Þ�t � k�tð Þaþ nþ 1ð Þ�t � k � 1ð Þ�tð Þa" #

¼ 1

C 1þ að ÞXnþ1

k¼1

@bp�xi1

2;tk

@xb � @bp�xi1

2;tk�1

@xb

!

�t

nþ 1� k þ 1ð Þa� nþ 1� kð Þa

" #�ta

¼ 1

C 1þ að Þ1

�t1�a

Xnþ1

k¼1

@bp xi12; tk

� �@xb

�@bp xi1

2; tk�1

� �@xb

0@

1A n� k þ 2ð Þa

� n� k þ 1ð Þa" #

ðB:4Þ

Or after shifting indices (so that the summation starts at tnþ1), the Caputo approximation becomes:

@1�a

@t1�a

@bp xi12; tn

� �@xb

0@

1A ¼ ra;�t

Xnþ1

k¼1

x að Þk

@bp xi12; tnþ2�k

� �@xb

�@bp xi1

2; tnþ1�k

� �@xb

0@

1A ðB:5Þ

where:

ra;�t ¼ 1

C 1þ að Þ1

�t1�a ðB:6Þ

and

x að Þk ¼ ka � k � 1ð Þa; x að Þ

1 ¼ 1 ðB:7Þ


Appendix C: Finite Difference Approximation to the Space Fractional Derivative

The discretization of the space fractional potentials at grid block interfaces is based on the approach presented in Zhang et al.(2007).

The finite difference approximation Equation (36) to the left-sided space fractional derivative Equation (34) is derived asfollows:

C0DC

bxiþ1

2

¼ 1

C 1� bð ÞZ x

iþ12

0

@P n; tnþ1ð Þ@n

xiþ12� n

� ��bdn; i ¼ 1; . . . ; Imax � 1 ðC:1Þ

The integral can be rewritten as summation of integrals over grid block lengths �x from grid block 1 to i where the firstderivative is approximated by the central difference at grid block interfaces x1þ1=2 to xiþ1=2:

C0DC

bxiþ1

2

ffi 1

C 1� bð ÞXim¼1

Z m�x

m�1ð Þ�x

Pnþ1mþ1 � Pnþ1

m

� �x

xiþ12� n

� ��bdn ðC:2Þ

¼ 1

C 1� bð ÞXim¼1

Pnþ1mþ1 � Pnþ1

m

� �x

�xiþ1

2� n

� �1�b

1� b

264

375m�x

m�1ð Þ�x

¼ 1

C 2� bð ÞXim¼1

Pnþ1mþ1 � Pnþ1

m

� �x

� i�x� m�xð Þ1�b þ i�x� m� 1ð Þ�xð Þ1�bh i

¼ 1

C 2� bð ÞXim¼1

Pnþ1mþ1 � Pnþ1

m

� �x

iþ 1� mð Þ1�b � i� mð Þ1�bh i

�x1�b

¼ 1

C 2� bð Þ1

�xbXim¼1

Pnþ1mþ1 � Pnþ1

m

� iþ 1� mð Þ1�b � i� mð Þ1�b

h iðC:3Þ

After shifting indices in Equation (C.3) (so that the summation starts from node xi) the left sided Caputo approximation inspace becomes:

C0DC

bxiþ1

2

¼ rb;�x

Xim¼1

x bð Þm Pnþ1


� ðC:4Þ

where

rb;�x ¼ 1

C 2� bð Þ1

�xbðC:5Þ

and

x bð Þm ¼ m1�b � m� 1ð Þ1�b; x bð Þ

1 ðC:6Þ

Using the same approach, the finite difference approximation Equation (37) of the right-sided space fractional derivativeEquation (35) is derived as follows:

Cxiþ1

2

DCb

L¼ �1

C 1� bð ÞZ L

xiþ1

2

@P n; tnþ1ð Þ@n

n� xiþ12

� ��bdn; i ¼ 2; . . . ; Imax ðC:7Þ


The integral can be rewritten as summation of integrals over grid block lengths �x from grid block i to Imax where the firstderivative is approximated by the central difference at grid block interfaces xi�1=2 to xImax�1=2:

Cxiþ1

2

DCb

Lffi �1

C 1� bð ÞXImax�i

m¼1

Z iþmð Þ�x

i�1þmð Þ�x

Pnþ1iþm � Pnþ1

i�1þm

� �x

n� xiþ12

� ��bdn ðC:8Þ

¼ �1


m¼1

Pnþ1iþm � Pnþ1

i�1þm

� �x

�xiþ1

2� n

� �1�b

1� b

264

375

iþmð Þ�x

i�1þmð Þ�x

¼ �1


m¼1

Pnþ1iþm � Pnþ1

i�1þm

� �x

iþ mð Þ � ið Þ1�b � i� 1þ mð Þ � ið Þ1�bh i

�x1�b

¼ �1

C 2� bð Þ1

�xbXImax�i

m¼1

Pnþ1iþm � Pnþ1

i�1þm

� m1�b � m� 1ð Þ1�bh i

ðC:9Þ

Or in compact formation:

Cxiþ1

2

DCb

L¼ �rb;�x

XImax�i

m¼1

x bð Þm Pnþ1


� ðC:10Þ

where rb;�x and x bð Þm are defined as in (C.5) and (C.6) respectively.


Documents

Modeling of 1D Anomalous Diffusion in Fractured Nanoporous ... · contains a time fractional derivative, while super-diffusion is described by a space-fractional derivative. As a