Fractional derivatives and their applications in reservoir engineering problems: A review

Journal of Petroleum Science and Engineering 157 (2017) 312–327

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Fractional derivatives and their applications in reservoir engineering problems: A review Abiola D. Obembe a, Hasan Y. Al-Yousef a, M. Enamul Hossain a, b, *, Sidqi A. Abu-Khamsin a a b

College of Petroleum Engineering and Geosciences, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia Memorial University of Newfoundland, St. John's, NL A1B 3X5, Canada

A R T I C L E I N F O

A B S T R A C T

Keywords: Fractional derivatives Anomalous diffusion Anomalous heat transport Darcy law

The use of fractional derivatives as a mathematical tool towards the development of more robust mathematical models in certain areas of reservoir engineering is gaining attention in both industry and academia alike. However, the main ideology and rationale for employing these fractional-based models are not fully understood. This expository paper carefully analyzes the literature on fractional calculus devoted to reservoir engineering applications. The review covers issues related to its history, definition, applications, and advantage over the classic reservoir engineering based approach. The potentials and challenges/limitation of some of the available approaches are highlighted where appropriate. Finally, some future potential research areas are proposed to fill the gap of the current state-of-the art in the area. This study contributes to the fundamental understanding and further development of fluid flow models that accurately describe the local and global phenomena.

1. Introduction Fractional calculus is a branch of mathematics once thought of has esoteric in nature. In general terms, fractional calculus can be described as a generalization of the classic differentiation and integration to arbitrary order. Ever since its inception, several scholars/mathematicians have contributed to its development. Mathematician such as Liouville, Riemann, Leibniz, Weyl, Abel, Grünwald, Letnikov, Caputo, Riesz, Hadamard, Marchaud, Machado, Chen, Pichaghchi, Katugampola, Coimbra, Davidson, Erd
elyi, Kober, and, Feller come to mind. The Caputo fractional derivative definition has appeared frequently in the porous media literature and is defined by (Caputo, 1967; Samko et al., 1993):

d γ f ðtÞ 1 t m1γ m 0 ∫ ðt  t 0 Þ ¼ f ðt Þdt 0 ; m  1  γ < m: dtγ Γðm  1Þ 0

(1)

where γ is the fractional exponent, t ' is a dummy variable, m is some positive integer bounded by the inequality expressed in Eq. (1), and Γð:Þ denotes the standard Gamma function which is an extension of the factorial function n!. In general, ΓðnÞ ¼ ðn  1Þ!, if n is a positive integer. From Eq. (1), It is observed that the Caputo fractional derivative is given by the convolution of a power law kernel and the ordinary derivative of the function. This way, the fractional derivative of a function f ðtÞ at any time t, is evaluated with a weighted mean of df ðt 0 Þ=dt 0 in the time

interval ½0; t. The fractional exponent term influences the degree to which the past events affect the future events. Caputo and Plastino (2004) argued that such definition demonstrates a feedback system. The Caputo interpretation of the time fractional derivative provides an efficient tool for describing a power law frequency variability of the coefficients by a simple convolution (Zhong et al., 2013). Over the past decade, widespread applications of fractional calculus have been demonstrated in several disciplines such as in dynamical systems theory (i.e. Ergodic theory, plasma physics), stochastic theory (i.e. Reaction-diffusion flow modelling, turbulence), fitting of experimental data, and in characterizing disordered systems (i.e. porous materials, gels etc.) as illustrated in Fig. 1. Literature shows a plethora of works demonstrating the application of the fractional calculus concept in reservoir engineering discipline. For example, there are studies related to modelling the long-tailed nonFickian solute transport in fractured porous rock (Wyss, 1986; Adams and Gelhar, 1992; Becker and Shapiro, 2000; Benson et al., 2000; Berkowitz et al., 2002; Herrick et al., 2002; Levy and Berkowitz, 2003; Reimus et al., 2003; Schumer et al., 2003; Zhou and Selim, 2003; Dentz et al., 2004; Berkowitz et al., 2006; Zhang et al., 2009; Liu et al., 2014; Wu et al., 2015), fluid flow in naturally fractured nano-porous media (Raghavan, 2012; Raghavan and Chen, 2013, 2016; Ozcan, 2014), anomalous heat transport in heterogeneous media (Berkowitz and Scher, 1997, 1998; Berkowitz et al., 2000; Zaslavsky, 2002; Li et al., 2005; Chowdhury and

* Corresponding author. Memorial University of Newfoundland, St. John’s, NL A1B 3X5, Canada. E-mail addresses: [email protected] (A.D. Obembe), [email protected] (H.Y. Al-Yousef), [email protected] (M.E. Hossain), [email protected] (S.A. Abu-Khamsin). http://dx.doi.org/10.1016/j.petrol.2017.07.035 Received 28 August 2016; Received in revised form 11 April 2017; Accepted 12 July 2017 Available online 18 July 2017 0920-4105/© 2017 Elsevier B.V. All rights reserved.

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Fig. 1. Fractional Modelling, a new Meta-Discipline.

Christov, 2010; Fomin et al., 2011), modelling fluid flow in thermally altered zones in porous media (Hossain and Abu-khamsin 2011b; Hossain and Abu-khamsin 2012; Hossain et al., 2015), and fluid flow evolution in some laboratory experiments (Caputo, 1998a, 2000a; Caputo and Plastino, 2004; Martino et al., 2006). It has also been established by numerous scholars that fractional differential equations result from the long-time and long-space limit of the Continuous Time Random Walk (CTRW) formulation (Hilfer and Anton, 1995; Barkai et al., 2000). Fractional diffusion equations have been successfully applied to describe anomalous diffusion behavior, and the power law or heavy tail decay processes observed in some complex systems (Chechkin et al., 2003; Mainardi et al., 2007, 2008; Chen et al., 2010; Obembe et al., 2016b). This paper presents a concise but critical review on the progress made regarding the application of fractional derivatives in reservoir engineering, and to tailor how the future trends in fractional derivatives may be of benefit to effective reservoir management. Emphasis would be given to transient testing applications, anomalous heat transport, and evolution of laboratory experiments. Another vital issue with the literature on fractional derivatives in petroleum engineering literature is the definition of the effective diffusivity (Hossain et al., 2015; Awotunde et al., 2016). A very educative review on memory approach or formalism in a porous medium was the work by Hossain and Islam (2006). The authors demonstrated how different researchers related the fluid memory to various fluid property; i.e. the stress, density, and free energies, etc. The main motivation for this work is in two folds: (i) to establish a proper frame work for scholars and researchers so that they can apply fractional derivatives wisely in reservoir engineering discipline where and when appropriate, (ii) to enlighten or increase the awareness of fractional calculus (i.e. fractional derivative based-mathematical models) in reservoir engineering discipline and highlight their advantages over conventional continuum flow models toward reservoir simulation and management.

2012; Li and Zeng, 2015). In its broadest sense, fractional calculus can be described as a generalization of the classical calculus to real or complex orders i.e. the theory of integrals and derivatives of arbitrarily real or even complex order (Ishteva, 2005). This way, many of the well-known basic properties of classic calculus are preserved. Historically, the concept of fractional calculus can be traced back to Leibniz's letter to L'Hopital in 1695 (Fig. 2), where the derivative of a non-integer order (1/2) was mentioned (Das, 2011). The main motivation of fractional calculus to real-world applications is that certain processes in science and engineering disciplines are not solely constrained to standard calculus. There has been ever increasing evidence that better fits for data obtained in many facets of science, engineering, and finance can be better obtained or captured by fractional models (Sprouse, 2010). In general, the physical interpretation of the integer-order derivatives or integrals are well understood for example; the velocity and the acceleration of a body which imply the first and second derivative of the position of the body as a function of time, the integral of a function also imply area under the curve. On the other hand, the physical interpretation of the fractional derivative is not fully understood but it is expected to have a broader meaning. Recent studies devoted to the physical interpretation of the Fractional operators can be found in references (Nigmatullin, 1992; Li and Chen, 2004; Nigmatullin and Le Mehaute, 2005; Heymans and Podlubny, 2006; Machado, 2009; Guti
errez et al., 2010; Yang, 2012). The literature on the fractional calculus concept shows that there are many definitions of the fractional operators which do not coincide in general. This debacle is understood to be associated with the different approaches employed by different mathematicians to preserve certain properties of the classical integerorder derivative. 3. Applications of fractional derivatives in reservoir engineering discipline In this section, we present a detailed overview of the reported applications of fractional derivatives in the reservoir engineering literature. This section is comprised of four sub-sections each presenting an extensive overview of the reservoir engineering problem, and how the application of fractional derivatives leads to more accurate and reliable flow models.

2. Fractional calculus: Historical background In this section, the basic information of the concept of fractional calculus is presented, and in no way, should this be considered as an elaborate effort to disseminate the whole literature devoted to fractional calculus. The information presented is sufficient for the casual reader to grasp the key concept of fractional calculus. For detailed information interested readers can refer to text on fractional calculus in the literature (Oldham and Spanier, 1974; Torokhti and Howlett, 1974; Ross, 1975; Podlubny, 1998; Cohen and Henle, 2005; Kilbas et al., 2006; Goshaw, 2007; Sabatier et al., 2007; Abbas et al., 2012; Malinowska and Torres,

3.1. Transient testing in fractured porous media A large percentage of reservoir rocks in the world fall under the classification of naturally fractured or naturally fissured rocks with widespread applications in groundwater, hydro-thermal reservoirs, and petro313

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Fig. 2. History of Fractional calculus.

beyond the scope of this review. Interested readers can refer to the excellent review article by Sahimi and Yortsos (1990b) and text by Hardy and Beier (1994). All fractional calculus formulations employ a non-local flux constitutive law to relate the volumetric flux to the pressure gradient. The non-local flux relationship implies that the observed/predicted flux is influenced by other factors other than the pressure gradient at the desired location at any instant in time. Numerous studies exist in the literature towards the justification of such modifications (Klafter et al., 1997; Molz et al., 2002; Fomin and Chugunov, 2011; Major et al., 2011; Meerschaert, 2012; Benson et al., 2013). Applications of the non-local flux constitutive equations can be found in the literature devoted to; fluid flow in porous media of fractal geometry, in naturally fractured unconventional shale reservoir, and nano-porous and porous materials (Mandelbrot, 1983; Nigmatullin, 1984a; Sahimi and Yortsos, 1990a; Dassas and Duby, 1995; Avnir et al., 1998; Falconer, 2004; Cloot and Botha, 2006; Ochoa-Tapia et al., 2007; Caputo and Cametti, 2008; Camacho-Velazquez et al., 2011; Atangana and Botha, 2012; Raghavan, 2012; Atangana and Bildik, 2013; Atangana and Botha, 2013; Razminia et al., 2015; Le Mehaute, 1984; Nigmatullin, 1984b). In such systems, the classic diffusion approach based on the random Brownian motion of the particles is inappropriate. The normal diffusion model(s) consider the mean square displacement of the diffusing particle to be a linear function of time (Obembe et al., 2016a). More so, the normal diffusion models have been shown to be related to random walks, where only the previous location controls the subsequent particle location. In a nutshell, the classic random walk problems have a mean waiting time that is finite (Raghavan, 2011). The literature demonstrates that anomalous diffusion model may be more appropriate to describe fluid flow in a naturally fractured and disordered nano-porous media (Medeiros et al., 2010; Ozcan et al., 2014; Albinali and Ozkan, 2016). Equation (2) presents a more comprehensive relationship between the mean square variance and time (Ben-Avraham and Havlin, 2000):

thermal reservoirs. By definition, a fractured rock is a porous medium that is intersected by a network of interconnected fractures (Chen, 2000). Fractured rocks are usually recognized to be anisotropic and heterogeneous systems, and can be characterized by models that allow for the rock petro-physical properties (i.e. porosity and permeability) to vary rapidly and discontinuously over the reservoir domain. The literature reveals different approaches for describing transport through such as the double-porosity realization, dual porosity/permeability, triple-porosity realization (Barenblatt et al., 1960; Kazemi, 1969; Zimmerman et al., 1993; Alahmadi, 2010), and the discrete-fracture-and-matrix (DFM) models (Kim and Deo, 2000; Dershowitz et al., 2004; Karimi-Fard et al., 2004). For instance, the double porosity realization, considers the fractures (secondary continuum) and the matrix block (primary continuum) as two different but overlapping continua, interacting through the matrix-fracture interface (Moench, 1984). The porosity and permeability are usually considered to be much greater in the fractures than in the matrix blocks. Therefore, the macroscopic description of fluid flow in naturally fractured systems requires different orders of magnitude representative elementary volumes (REVs) (Civan, 2011). The DFM are usually more accurate than other approaches in predicting single and multi-phase flow in fractured porous media. However, they can sometimes be more difficult to implement for field cases (Ilyasov et al., 2010). In general, DFM models restrict fluid flow to fractures and regard the surrounding rock as impermeable. The DFM models entail explicitly specifying the geometry of random complex and non-orthogonally intersecting fractures which sometimes may not be known due to lack of sufficient information. The challenge with modelling flow in fractured heterogeneous systems lies in the spatial heterogeneity of flow properties. Heterogeneous porous media comprise of various transport units each with different types and order of magnitude rates. A detailed overview of transport units and transport in heterogeneous porous media was presented by Civan (2010). Pressure transient test offers the most direct technique to estimate permeability at the inter-well length scale (Hardy and Beier, 1994). The classic pressure transient models are formulated with the assumption of laterally non-heterogeneous reservoirs. Unfortunately, observations from the core, log, and outcrop data prove otherwise. Thus, the conventional approaches may not work in all situations. To improve these limitations, fractal models and fractional diffusion models have been proposed. The two mentioned approaches include the conventional homogenous reservoir solution as a special case. Discussion on the fractal models is

σ 2r  Dt γ

8 < γ ¼ 1 Normal Diffusion where γ > 1 Superdiffusion : γ < 1 Subdiffusion

(2)

where σ 2r is the variance. Scenarios for which γ ¼ 1 as shown by Eq. (2) are termed as normal diffusion and the corresponding classical diffusivity models are

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appropriate. Implicitly, solutions to such equations are obtained based on the knowledge of the current state of the system. On the other hand, scenarios for which γ≠1, require more information other than the current state of the system (i.e. the history of the process). The dual-porosity idealizations (de Swaan, 1976; Bello, 2009; Bello and Wattenbarger, 2010; Du et al., 2010; Ozkan et al., 2010; Brohi et al., 2011; Li et al., 2011; Samandarli et al., 2011), and triple porosity idealizations (AlAhmadi and Wattenbarger, 2011; Tivayanonda et al., 2012; Zhao et al., 2013; Obembe and Hossain, 2015) have been employed in the literature with varying degree of success. A point worth mentioning is that even such formulations are inadequate, in that there is a lack of a clear scale separation in unconventional reservoirs. In fact it was recently acknowledged that the dual- porosity idealization is a first order approximation for describing the transport network in shale reservoirs (Kuchuk and Biryukov, 2014, 2015). Fractional derivative concept have also been employed for pressure transient testing in scenarios where the geology is described as complex (Thomas et al., 2005). Accordingly, Thomas et al. (2005) noted a pressure drop signature of the form:

Δp  Δt

α

⇀

u ¼ λγ

⇀

u ¼ λγ ΓðγÞ

  γ  t b p ðω; tÞ  pðω; 0Þ Eγ  ηω

(5)

⇀

u¼

∞ X n¼0

zn Γð1 þ γnÞ

(8)

" # Ka;b ∂a ∂b p ∂b p  ð1  χÞ χ b ∂xb μ ∂t a ∂ðxÞ

(9)

where 0 < a; b  1, and χð0  χ  1Þ is the skewness parameter which controls the bias of the dispersion (Huang et al., 2008), and Ka;b is a phenomenological coefficient with the dimension of [ L2Ta Lb1 ]. The operators

b ∂a ∂ p ∂t a , ∂xb

and

∂b p ∂ðxÞb

stand for the time fractional de-

rivatives, left and right space fractional derivatives respectively. Eq. (9) allows for diverging jump length variances (“long jumps”) and unequal forward and backward jump probabilities at the particle scale. Till date, to the authors knowledge, the generalized constitutive flux law described by Eq. (9) has not yet been explored in the literature. Albeit, simplified variants of Eq. (9) have been applied with varying degree of success in the literature. Accordingly, as pointed out by Holy and Ozkan (2016), such a generalized flux constitutive equation allows for incorporation of the complexities associated with heterogeneous systems and other multiscale flow mechanism. The main advantage of this approach is that it does not require the matrix and fracture petro-physical properties as one would require in the dual-porosity, or triple porosity models. For sake of clarity, time fractional derivative order, a, accounts for presence of flow hindrances to flow, thus describing the degree of subdiffusion. The space fractional derivative order, b, accounts for flow buffers i.e. super-diffusion. Although, Sprouse (2010) pointed out that with a time fractional derivative order, b > 1, a super-diffusive transport can be simulated. Therefore, incorporating Eq. (9) into the continuity equation results in a generalized fractional diffusion equation for the 1D linear flow of a slightly compressible fluid as follows:

where ω is the Fourier mode, γ is an exponent, η is the diffusivity, and Eγ is the Mittag-Leffler function defined as:

Eγ ðzÞ ¼

∂1γ h ⇀ i ∇p x ; t ∂t1γ

where λγ ¼ Kγ =μ and γ < 1. For the case of γ ¼ 1, Eq. (7) reduces to Darcy equation, implying λγ¼1 ¼ K=μ. A generalized constitutive non-local and temporal flux law describing anomalous diffusion in disordered fractured media is described by (Baeumer et al., 2005):

where α is a constant. Eq. (3) implies a power law relationship as opposed to the classical signature which is an exponential decay (Theis, 1935). A power law behavior describes a slower rate of decay, with the contribution of the history of the process playing a huge role. Equations (4) and (5) predict the pressure distribution due to an instantaneous source for the normal diffusion equation, and for the CTRW model (Raghavan, 2011).

(4)

(7)

Equation (7) can also be expressed as follows:

(3)

  b p ðω; tÞ  pðω; 0Þexp ηω2 t

 d t 1 ⇀ ∫ ∇p x ; t 0 dt' dt 0 ðt  t 0 Þ1γ

(6)

Raghavan (2011) concluded that due to the typical nature of the rock fabric in naturally occurring geologic media, transport occurs across disordered structures, rough interfaces, obstacles, cracks, traps, and crevices. He noted that employing fractional derivatives offer a natural way to capture such effects. Furthermore, he presented a flux law of the form:

( " #) ∂ Ka;b ∂a ∂b pðx; tÞ ∂b pðx; tÞ ∂pðx; tÞ  ð1  χÞ χ ¼ ϕcl b ∂x μ ∂t a ∂xb ∂t ∂ðxÞ

Table 1 Summary of some non-local constitutive flux laws presented in the literature. Researchers

Time fractional order Space fractional order Skewness (χ) of of differentiation differentiation

Interpretation of time fractional operator/Sub-diffusion

Interpretation of space fractional operator/Super-diffusion

Dimension of the phenomenological coefficient

Le Mehaute and Crepy (1983) Nigmatullin (1986, 1984a) Compte and Jou (1996) Raghavan (2011) Raghavan and Chen (2013) Ozcan et al. (2014) Chen and Raghavan (2015) Raghavan and Chen (2016) Holy and Ozkan (2016) Albinali and Ozkan (2016)

γ1 γ1 γ1 1γ 1γ 1γ 1γ 1γ 1γ 1γ

NCa/Yes NC/Yes NC/Yes Caputo sense/Yes Caputo sense/Yes Caputo sense/Yes Caputo sense/Yes Caputo sense/Yes Caputo sense/Yes Caputo sense/Yes

NAb/No NA/No NA/No NA/No NA/No NA/No Caputo sense/Yes NA/No Caputo sense/Yes NA/No

L2 T γ1 L2 T γ1 L2 T γ1 L2 T 1γ L2 T 1γ L2 T 1γ L2 T 1γ Lβ1 L2 T 1γ L2 T 1γ Lβ1 L2 T 1γ

a b

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 β 1 β 1

Not clear. Not applicable.

315

(10)

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each with its own temperature. The energy equation is a mathematical statement of conservation of energy in porous media. Employing the subscript ‘m’ for the solid matrix, and ‘f’ for the fluid, the energy equation for the solid phase and fluid phase is presented as follows (Civan, 2010):

where the different arguments for a and b employed in the constitutive flux equations i.e. Eq. (9) in the literature devoted to porous media are presented in Table 1. From a reservoir engineering perspective, Table 1 enables the readers to reflect on how different researchers have attempted to relate the generalized flux relationship to different observed pressure behavior or trends. In conclusion, the fractional diffusion equations obtained by employing various forms of constitutive flux laws capture the anomalous flow behaviors that might occur due to long range dependence and/or trapping events observed in porous media not accounted for by normal (Fickian) diffusion models (Chen and Raghavan, 2015). The major advantage of the anomalous diffusion model over the conventional Darcy-based models for transient testing in naturally fractured porous media or nano-porous media is that these formulations do not require a detailed description of the intrinsic petro-physical properties of the matrix and fracture as well as their spatial distribution. This stems from the fact that the parameters employed in the anomalous diffusion formulations are phenomenological and are dynamic in nature. Interested readers may refer to the work by Holy and Ozkan (2016) for further details. A similar alternative involves the use of fractal fractional diffusion (FFD) models (Chang and Yortsos, 1990; Acuna and Yortsos, 1995; Razminia et al., 2014, 2015) which are beyond the scope of this paper.

   ∂ ρf ϕf ⇀ ¼ ∇: ρf uf þ qmf þ qsource=sink ∂t

(12)

   ∂ ∂ ∂ ∂Tf ⇀ ρf Cpf uTf ¼ κf ϕ þ q_ þ qT ρf Cpf ϕTf þ ∂t ∂x ∂x ∂x

(14)

  q_m ¼ q_ ¼ Asf h Tf  Tm

(15)

In obtaining Eqs. (13) and (14), it is assumed that heat conduction flux is governed by Fourier's law: ⇀ Jc

¼ κ∇T

(16)

with the tensor κ given by:

Accurate modelling of heat transport in fractured porous media is essential to applications such as; geothermal reservoirs, thermal enhanced oil recovery and reservoir management (i.e. detection of water influx in the wellbore) to name a few. Numerous analytical and numerical simulation models have been presented in the literature to tackle this complex phenomenon, see work by Obembe et al. (2016a) for a complete review of such models. In this section, we begin by discussing the mathematical equations describing non-isothermal flow in naturally fractured rocks, and then present an alternative formulation based on the concept of fractional derivatives. Due to the much larger permeability magnitude in the fractures, it is usually considered that fluid flows first into the fractures from the matrix block, and then into adjacent blocks or rather still remains in the fractures (Douglas and Arbogast, 1990). Mathematically, the equations describing non-isothermal fluid flow in naturally fractured rocks involve a combination of the continuity equation, a constitutive flux relationship (i.e. equation describing fluid flow in the fractured porous media, conduction flux), an equation of state, empirical correlations for fluid properties and the energy conservation equation(s). Due to the more rapid nature of fluid flow in the fracture system, it is sufficient to consider the external sources and sinks interact only with the fracture system (Chen, 2000). Therefore, a mathematical statement for the flow through each matrix block, and the flow in the fractures are given by:

(11)

(13)

where qT q_m , and q_ denote the fluid heat source term, the amount of heat added to the matrix and the fluid phase respectively defined by (Civan, 2010):

3.2. Heat transport in fractured porous media

 ∂ðρϕÞ ⇀ ¼ ∇: ρu  qmf ∂t

   ∂

∂ ∂Tm ð1  ϕÞκm þ q_m ρm Cpm ð1  ϕÞTm ¼ ∂t ∂x ∂x

κ s ¼ ks I

(17)

κ f ¼ kf I þ ρf Cpf Dm

(18)

where Dm is the mechanical dispersion term. Assuming the fluid flow in the subsurface is sufficiently slow, then the assumption of Local Thermal Equilibrium (LTE) i.e. Tf ¼ Tm ¼ T, is valid and thus adding Eqs. (13) and (14) results in:

   i ∂ ∂ h ∂ ∂T ⇀ ρf Cpf uT ¼ κe þ qT ρCp b T þ ∂t ∂x ∂x ∂x

(19)

where the bulk storage capacity and effective diffusion-dispersion tensor are defined by:



ρCp

 b

¼ ρm Cpm ð1  ϕÞTm þ ρf Cpf ϕ;

κe ¼ κ f ϕ þ ð1  ϕÞκ m

(20)

Eq. (19) is solved to obtain the temperature distribution in the porous media. Note Eq. (19) is the simplified form of the classic (Fourier- based) energy equation. On the other hand, the unequal temperature formulation i.e. Eqs. (13) and (14) which implies Local thermal non-equilibrium, is appropriate for the following situations; a huge difference exists between the rate of heat conduction through the solid matrix and the rate of heat transfer in the fluid (Emmanuel and Berkowitz, 2007), or when the solid phase thermal diffusivity is negligible compared to heat transfer between the fluid and solid, or when the fluid flow velocity is high (Nield and Bejan 2013). Anomalous heat transport has been observed in fractured media comprising of poorly connected fractured network (Luchko and Punzi, 2011) and in other systems where anomalous diffusion occurs (Gurtin and Pipkin, 1968; Nunziato, 1971; Zanette and Alemany, 1995; Berkowitz et al., 2002; Dentz and Berkowitz, 2003; Cortis et al., 2004; Li et al., 2005; Chowdhury and Christov, 2010; Ilyasov et al., 2010; Suzuki et al., 2015). Generally, Eqs. (13), (14) and (19) are sufficient under the notion of standard diffusion and Fourier-based heat transport. Similarly, extending the CTRW approach a non-Fourier heat transport model has been presented in the literature (Emmanuel and Berkowitz, 2007; Geiger and Emmanuel, 2010; Augustin et al., 2014). For the sub-diffusion flow problem, employing a probability density function ψ, with first moment τ, and Laplace transform of the form (Emmanuel and Berkowitz, 2007; Luchko and Punzi, 2011):

where the fluid velocities in the matrix block and fractures are determined from Darcy constitutive equation. The contribution of the matrix to flow into the fracture is incorporated through the extra source term qmf . Two techniques have been successfully applied to model the matrix-fracture fluid transfer term. One approach involves using shape factors (Warren and Root, 1963; Kazemi, 1969), while the second approach involves imposing boundary conditions explicitly on the matrix blocks (Barenblatt et al., 1960). For further details, please refer to references above. For sake of completeness, modelling heat transport in porous media involves considering the porous media has two phases (solid and fluid) 316

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Fig. 3. Pressure profile for different values of γ after, (a) 10 min (b) 20 min, and (c) 40 min.

Fig. 4. Temperature profile for different values of γ after, (a) 10 min (b) 20 min (c) 40 min.

ψðsÞ ¼ 1  ðτsÞγ ; for s→∞

with the dyadic symbol ‘:’, representing a tensor product. and

(21)

where s is the Laplace space variable, results in the fractional FokkerPlanck equation (fFPe) of the form below for the energy equation under LTE (Emmanuel and Berkowitz, 2007):

i   i ∂1γ h  ∂ h ⇀ ρCp b T ¼ 1γ ∇: κ γe ∇T  ρf Cpf uγ :∇T ∂t ∂t





b eff ðsÞ vψ ϕρf Cpf :∇ T b f ðθ; sÞ b f ðθ; sÞ  Tf ;init ðθÞ ¼  M ϕρf Cpf s T  κ ψ ϕρf Cpf b f ðθ; sÞ : ∇∇ T

(22)



ρCp uT

 in

   ⇀ ¼ κ e;γ ∇T  ρf Cρf uγ :∇T ; x ¼ 0; t > 0

b MðsÞ ¼ t1 s

1þ

P ¼ Pi ; 0  x  L; t ¼ 0 ;

(26)

Q ¼ Qinj ; x ¼ 0; t > 0

(27)

P ¼ Pout ; x ¼ L ; t > 0 ;

(28)

ρCp

b MðsÞ   P wi

(32)

i sþWi

Table 2 Rock and fluid properties values for numerical computation.

The literature to date under the assumption of Local Thermal NonEquilibrium (LTNE) to the author's knowledge is still unclear from a fractional derivative perspective. The CTRW models or ‘master equations’ for both the LTE and LTNE energy equations in Laplace space are presented in Eqs. (29) and (30) for sake of completeness.



(31)

where t1 , wi , and Wi parameters represent the characteristic time for transitions from one site to another, trapping rate and release rate respectively. Interested readers may refer to the work by Emmanuel and Berkowitz (2007) for full details. In summary, in naturally fractured rocks with a high degree of spatial heterogeneity in the flow field or thermal dispersivities, the heat transport behavior will tend towards anomalous behavior, and the classic Fourier-type heat equation may not be adequate.

(24)

(25)

b ðsÞ ψ 1ψ b ðsÞ

b eff ðsÞ ¼ M

(23)

∂T ¼ 0; x ¼ L ; t > 0 ; ∂x

(30)

b b eff ðsÞ are the memory function and effective memory where MðsÞ, and M function given by

where the coefficient κ γe has a different physical interpretation from κ e in Eq. (18). Furthermore, the flow velocity is obtained from a generalized anomalous flux equation of the form presented in Eq. (7) or any of its variants. The pressure is also obtained by solving the time fractional diffusion equation of the form of Eq. (10). Figs. 3 and 4 were obtained employing Eq. (22), and the input data of Table 2 to illustrate the behavior of anomalous diffusion and heat transport in a porous medium. The following initial and boundary conditions are applied:

T ¼ Ti ; 0  x  L; t ¼ 0 ;



h  

  b b ðθ; sÞ  κ ψ ρCp b ðθ; sÞ  Tinit ðθÞ ¼  MðsÞ vψ ϕρf Cpf :∇ T sT b b i b ðθ; sÞ : ∇∇ T (29)

317

Fluid and rock properties

Fluid and rock properties

Pi ¼34.023 atm (500 psia) To ¼298.15 K Tinlet ¼393.15 K Pref ¼1 atm (14.7 psia) Δt ¼10 s βf ¼ 9 104 K1 Pout ¼ 34.023 atm (500 psia) Nx ¼ 80 Diamter ¼0.0381 m (1.5 inches) kw;γ ¼0.6 W-s1γ /(m-K) ks;γ ¼2.5 W-s1γ /(m-K) tol ¼ 1  1010

ϕref ¼0.36 ρs ¼ 2650 kg/m3 Tt ¼2400 s (40 min) ρref ¼997 kg/m3 μref ¼0.00089 kg/ms cw ¼4.556105 atm1 (3.1106 psi1) cμp ¼-2.73577 105 atm1 (1.86106 psi1) cμT ¼ 2.7102 K1 Cpw ¼4186 J/kg/K Cps ¼820 J/Kg/K βϕ ¼9105 K1 L ¼0.1524 m (6 inches)

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3.3. Memory formalism for laboratory evolution experiments The Darcy's law is the most widely employed constitutive equation relating the volumetric flux to the applied pressure or potential gradient in a porous medium (Darcy, 1856). Many researchers have proposed different extensions to the classic Darcy's equation by accounting for slip, inertia, non-Darcy flow etc. (Brinkman, 1949; Sposito, 1980; Whitaker, 1986; Kim et al., 2001; Auriault et al., 2010; Adler et al., 2013; Bear, 2013). Further, Darcy's law is not a diffusion model but rather a constitutive equation that relates the fluid potential/pressure drop to the volumetric flux in porous media as mentioned earlier. Rather, it is the substitution of Darcy's law into the continuity equation that results in a diffusion model or alternatively the diffusivity equation as referred to in Well-Testing niche. It has been reported in the literature (Caputo, 1998a; Martino et al., 2006; Di Giuseppe et al., 2010) that under certain conditions the rock permeability may vary locally because of different factors. For instance, fluids can react chemically with the porous medium, solid particles embedded with the reservoir fluids may deposit or get attached along the pore throats, rock compaction, mineralization, precipitation, grain re-arrangement, fines migration, clay swelling and temperature variations within the porous media due to injection of hotter fluids. In such cases, Darcy's equation may not be appropriate to describe fluid flow process because the permeability evolution is not known priori. The memory formalism was developed to provide an alternative approach to incorporate the hereditary nature of porous rocks as well as the fluid rheology into the classic constitutive equations employing the fractional calculus concept (Caputo, 1998a, 2000a, 2003; Caputo and Plastino, 2004; Di Giuseppe et al., 2010; Zhong et al., 2013). Caputo and Plastino (2004) modified Darcy's equation to account for the reduction in permeability with time via introducing the time fractional derivative operator to reflect the system memory. The main advantage of the memory formalism in describing flux of the fluid is that it allows the use of more than two parameters instead of one parameter K in the classic Darcy's equation. The memory parameters introduced may then be estimated utilizing experimental data. It should be recognized that the equations and parameters resulting from the memory formalism are phenomenological and must be calibrated against experimental data. The generalized memory formalism (i.e. non-local constitutive equations) describing the variation of rock and fluid properties with time (i.e. the local phenomena) may be described by the following system of equations based on the literature by researchers (Caputo, 1998a, 1998b, 2000a; Di Giuseppe et al., 2010):





   a þ b ∂m2=∂t m2 p ¼ f þ g ∂m2=∂t m2 fmðx; tÞ  m0 g    e þ h ∂n1=∂t n1 q ¼  c þ d ∂n2=∂t n2 ∇p

∇:q þ

∂m ¼0 ∂t

Fig. 5. Flowchart illustrating memory formalism approach for fluid flow evolution.

combined in such a way that either the flux evolution (q) or the pressure evolution (p) at certain location(s) along the porous medium can be obtained. This may entail deriving analytical solution employing the Laplace transform method or via the numerical solution of the resulting equation(s). The inverse problem of obtaining the memory parameters is tackled by curve fitting the simulated data (i.e. output from the memory formalism) with the observed data derived from the laboratory experiments, or other efficient minimization techniques. Iaffaldano et al. (2005) carried out experimental investigations to understand the permeability reduction observed during the diffusion of water in sand layers. Based on their results they concluded that the reduction in permeability observed was a result of grain rearrangement and compaction. Based on the data observed from five experiments with sand and water the average values of the memory parameters obtained _ of 108 m2s0.5. It is worth are n≈ 0.5, and a pseudo-permeability (k) stating that estimate of pseudo-permeability obtained is out of range of typical permeability of sand obtained from Darcy's equation which is around 1013 to 109 m2 (Bear, 1972, 2013) which is not unusual due to the different dimensions of both variables. Caputo (2000a, 2000b, 1998a) proposed a modified form of Darcy's law refer to Eq. (36), by introducing the time fractional derivative to account for the local permeability changes in the porous media. Subsequently, he demonstrated a method to determine the two parameters defining his memory diffusion model.

(33)

(34)

q ¼ η

(35)

  ∂γ ∂p ∂t γ ∂x

(36)

Caputo and Plastino (2003) proposed a modified constitutive relation refer to Eq. (37), to better describe the diffusion process of fluids in porous media. The authors modified the classic Darcy's equation by incorporating an additional space fractional derivative of pressure. The authors argued that the memory term (i.e. space fractional derivative term) captures the effect of the medium previously affected by the fluid. It is worth mentioning that the time-memory is suitable for accounting for local phenomena, and the space memory captures the variations in space.

where

0  n1 < 1; 0  n2 < 1 and 0  m1 ; 0  m2 < 1 From a practical point of view (i.e. to reduce the number of parameters) it is advisable to allow n1 ¼ n2 ¼ m1 ¼ m2 ¼ n. The coefficients a, b, c, d, e, f, g, h and n are memory parameters. The models in the literature on memory formalism in laboratory experiments are variants of Eqs. (33)–(35) and are presented herein. Fig. 5 depicts a flowchart illustrating the memory formalism algorithm employed in typical laboratory evolution experiments conducted by researchers (Caputo, 1998a, 1998b, 2000a; Di Giuseppe et al., 2010). The flowchart depicted in Fig. 5 demonstrates that in arriving at the memory parameters, the governing equations (Eqn. (33) to Eqn. (35)) are

q ¼ a1

∂1þγ ∂ p þ a2 p ∂x ∂x1þγ

(37)

In an attempt to better represent fluid flow through porous media,

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modified material-balance equation (MBE) by including a stress-strain formulation for both the porous rock and fluid. The authors claim that the proposed MBE can also be applied to fractured formations with dynamic features. Furthermore, an improvement of 5% in oil recovery was observed from the new MBE over the classic MBE. However, the proposed MBE requires accurate rock and fluid compressibility data obtained from laboratory measurements or from reliable correlations. Hossain et al. (2008a) proposed an extension of the classic flow diffusivity equation by incorporating fluid and rock memory. This was derived by introducing the Caputo fractional derivative to the classic Darcy's law. The authors believed the introduction was necessary to account for the variation of fluid and formation properties with time. The proposed model is a nonlinear integro-differential equation refer to Eqs. (40) and (41). Furthermore, they proposed an explicit finite difference scheme to solve the resulting nonlinear equation.

Caputo (1998b) combined the modified Darcy's equation (Eq. (36)) and a modified constitutive relationship between pressure and density (i.e. Hook's law, see the work by K€ ornig and Müller (1989)) to incorporate the rheology of the fluid as follows:

a5 ρ þ a6

  ∂n1 ρ ∂n2 p ¼ A a7 p þ a8 n n ∂t 1 ∂t 2

(38)

where the coefficient A has dimension of L2 T2, the memory parameters a5 and a7 are positive and dimensionless, while the other memory parameters a6 , and a8 are positive and have dimensions of sn1 , and sn2 respectively and 0 < n1 < 1; 0 < n2 < 1. Table 3 presents a summary of some of the memory flow models in the literature were associated with modifications to the classic Darcy's equation or Fick's equation and one or more constitutive equation.

1 ∂η ∂p ∂ ϕct ∂p Z þ cf Z þ Z ¼ η ∂x ∂x ∂x η ∂t

4. The Hossain memory model Every reservoir formation is unique in its properties and fluid content(s). However, these rock attributes and the saturation of the stored fluid(s) are strongly related to the depositional environment, migration path, in-situ pressure, temperature, geologic events and subsequent rock compaction. A detailed description of the pore network and the adequate description of the rheological behavior of the complex reservoir fluid(s) are fundamental to the accuracy of any fluid flow model developed to describe flow through reservoir rocks. Hossain et al. (2008a) attempted to address the issue of incorporating the time dependency of rock and fluid properties explicitly into the existing fluid flow models. The authors introduced a memory model to capture the time evolution of rock and fluid properties with time employing the fractional calculus concept. It was pointed out that a memory model may be better suited to describing the inherent changes in rock properties (i.e. porosity, and permeability) and fluid properties (i.e. density, and viscosity) with time and space. Accordingly, Darcy's equation was modified to introduce the notion of rock and fluid memory according to the constitutive relationship below (Hossain et al., 2008a; Hossain, 2016):

  ∂1γ ∂p u ¼ η 1γ ∂x ∂t

⇀

Z¼





∂2 p ∂ξ∂x

∂ξ (41)

ΓðγÞ

Hossain et al. (2008b) proposed a stress-strain relationship applicable to non-Newtonian fluid flow in porous media. They included for temperature variations, surface tension, pressure variations and fluid and rock memory. The authors investigated the effect of memory on the stress-strain curve assuming a homogeneous, isotropic porous media. They concluded that the stress strain behavior was a strong function of time, distance, and the memory parameter. The proposed stress-strain model equations are represented below:

τT ¼

kΔpAyz ΓðγÞ  μ20 ηρ0 ϕyI t

I ¼ ∫0

(39)

ðt  ξÞ k

ð1γÞ





∂σ ΔT ∂T αD Ma



 ∂ux E eRT  ∂y

(42)

 ∂c ∂p c ∂k ∂p ∂2 p  þ c 2 dξ ∂ξ ∂ξ k ∂ξ ∂ξ ∂ξ

(43)

Memory models were presented in the works by (Hossain and Abukhamsin 2011a; Hossain and Abu-khamsin 2011b; Hossain and Abukhamsin 2012; Hossain et al., 2015; Hassan and Hossain, 2016) to describe the temperature evolution in oil reservoir during a thermal flood. In those studies, the authors emphasized that the variation of rock and fluid properties with time in a thermal flood can be best captured by utilizing Eq. (39). Recently, Hossain (2016) revealed that the resulting memory-based diffusivity equation predicts a lower pressure drop when compared to the classic Diffusivity equation based on Darcy's equation. The author also claimed that the pressure difference between both models grew more significantly with time. In his own words, he concluded that with continual oil withdrawal from the reservoir, the effect of memory became stronger and may be considered as some form of reservoir-drive mechanism.

where η, is a composite variable defined; η ¼ Ktμ , and the operator ∂t∂ 1γ 1γ

1γ

ð1γÞ

t

∫ 0 ðt  ξÞ

(40)

is the time fractional operator of order 1  γ, and is interpreted in the Caputo sense. Employing Eq. (39) as a constitutive equation describing fluid flow in a porous medium, the author has presented numerous applications of the memory formalism. The author argued that the memory-based models demonstrated superior performance/predictive capabilities in comparison to the Darcy's-based models. Next, a quick review of some of the authors work in the literature is revisited. Hossain and Islam (2009) investigated the influence of time and reservoir rock properties on cumulative oil production. They introduced a

Table 3 Summary of some memory flow models presented in the literature. Author

Caputo (1998b) Caputo (1998a) Caputo (2003) Caputo and Plastino (2003) Caputo and Plastino (2004) Martino et al. (2006) Di Giuseppe et al. (2010) Fomin et al. (2010) Suzuki et al. (2015)

Const. Eq. with memory 2 1 1 1 2 1 2 1 2

Fractional derivative operator Time

Space

Interpretation

Yes Yes No No Yes Yes Yes Yes Yes

No No Yes Yes No No No Yes Yes

Caputo Caputo Caputo Caputo Caputo Caputo Caputo Caputo Caputo

319

Memory parameters

Dependent variables (unknown in fractional equation)

Calibration

6 2 4 2 7 4 4 3 6

p p p p p p u C C; and T

No No No No No Yes Yes Yes Yes

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Table 4 Reservoir rock and fluid parameters and boundary condition input for simulation. Parameter

Value

L W H Δx Kγ Qx0 QyL pi μ ϕ Δt tmax ct

5000 ft. (1524 m) 1000 ft. (304.8 m) 100 ft. (30.48 m) 100 ft. (30.48 m) 50 mD:day 0:1 200 bbl/day (3.68104 m3/day) 0 bbl/day (0 m3/day) 4500 psi (3.17107 Pa) 0.78 cp (7.8104 Pa s) 0.35 0.25 days 30 days 7.0106 psi1 (1.0153109 Pa1)

It is interesting to see the author(s) apply the fractional derivative concept via Eq. (39) for the above fluid flow problems. The conclusions of the author(s) are not surprising in that the resulting fluid problem obtained by incorporating Eq. (39) is totally different, see Eq. (44). Furthermore, employing Eq. (39) as a constitutive equation for fluid flow through porous media implies the typical rock permeability experiments carried out routinely in the laboratory (i.e. based on Darcy's equation) would not apply anymore. However, till date, all routine core permeability measurements are performed employing Darcy's equation. Incorporating Eq. (39) into the continuity equations results in:

       K ∂ ∂p ∂p D1γ ¼ ϕct t 1γ t μ ∂x ∂x ∂t

Fig. 7. Pressure history at control volume one for γ equal 0.7 for the Hossain memory model, Anomalous diffusion model, and the Darcy's-based flow model.

The plots exhibited through Figs. 6–8 reveal that the pressure estimates predicted by the Hossain et al. model differ from the anomalous diffusion models. A close observation of the same plots (i.e. Figs. 6–8), reveals the pressure values estimated from the Hossain et al. model falls in between the anomalous and the Darcy-based models. However, as the magnitude of fractional derivative increases, the difference in pressures predicted by the Hossain et al. model and the anomalous diffusion model decreases. The exhibited trend is not unusual in that as the fractional exponent increases, the contribution of the time multiplier to the transmissibility term or diffusivity term diminishes. Of interest, Fig. 9 reveals that both the flux relations in both the anomalous diffusion model and the Hossain et al. model reduce to Darcy's law for γ ¼ 1. It is worth stressing that the Darcy-based flow model i.e. Eq. (46), predicts higher pressure drops compared to the other two flow models as the magnitude of the fractional exponent reduces. In summary, the analysis indicates that the Hossain memory formalism does not describe sub/super diffusion behavior, and as it stands, as no correlation or link with the fractional diffusion equations describing anomalous transport in porous media which are obtained as a limiting case of the CTRW models.

(44)

Equation (44) is different from the anomalous diffusion equation i.e. Eq. (45) describing fluid flow in porous media.



  2    Kγ ∂p ∂p D1γ ¼ ϕc t t ∂x2 ∂t μ

(45)

For sake of comparison, the simulated pressure evolution at the first control volume in an oil reservoir is demonstrated employing Eqs. (44) and (45) and utilizing Darcy's law as the appropriate constitutive equation in the continuity equation i.e. Eq. (46) with the data listed in Table 4 are illustrated in Figs. 6–9.



K μϕct

 2    ∂p ∂p ¼ ∂x2 ∂t

(46)

Fig. 6. Pressure history at control volume one for γ equal 0.5 for the Hossain memory model, Anomalous diffusion model, and the Darcy's-based flow model.

Fig. 8. Pressure history at control volume one for γ equal 0.9 for the Hossain memory model, Anomalous diffusion model, and the Darcy's-based flow model. 320

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and the time fractional derivative of order 1 2 , is interpreted in the Caputo sense in convolution form as follows (Babak and Azaiez, 2014) =

1

=

d 2 f ðtÞ 1 2

=

dt

1 df 1 df t dξ ¼ pffiffiffiffiffi* ¼ ∫ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi πt dt πðt  ξÞ dξ

(49)

However, the above unified fractional derivative model gave a good match with established transient inter-porosity flow models only during the fracture-dominant and transitional fracture-to-matrix dominant flow regimes. For further details, refer to original manuscript by the authors. 5.2. Heat transport modelling in geothermal reservoirs Observed breakthrough curves obtained from field tests are usually characterized by the appearance of ‘‘anomalous’’ early arrivals and longtail, late-time arrivals of solutes not predicted by classic Fickian based transport models (Dentz et al., 2004). It was argued that the major challenge to modelling geothermal formations lies in the unresolved heterogeneities (i.e. residues) that vary over several orders of magnitude and span a huge range of spatial scales (Eggleston and Rojstaczer, 1998; Sidle et al., 1998). The continuous time random walk (CTRW) formulation has successfully been applied to describe the non-Fickian transport, in such ensemble-averaged sub-regions. Discussion on the CTRW is beyond the scope of the present study, however, the CTRW was developed based on the transition rate approach to transport. For further details on the CTRW models, refer to references and those therein (Kenkre et al., 1973; Klafter and Silbey, 1980; Berkowitz and Scher, 1995; Hughes, 1996; Berkowitz et al., 2002). Interestingly, fractional derivatives result as the limiting case/subset of the CTRW models. The corresponding anomalous solute transport models capture the combined effect of the various mechanisms that result in the broad spectrum of typical local solute transit times in the media.

Fig. 9. Pressure history at control volume one for γ equal 1 for the Hossain memory model, Anomalous diffusion model, and the Darcy's-based flow model.

5. Conclusions and outlook In this study, a concise review of the reported applications of fractional derivatives in reservoir engineering problems in the literature is presented. Fractional diffusion equation reflects the L
evy process and provides an efficient means of characterizing the power law phenomena of anomalous diffusion. The integer-order derivative (i.e. classic diffusion equations) is simply a limited case of the fractional-derivative based models. More so, the literature reveals the origin of the anomalous behavior to the occurrence of a broad distribution of heterogeneity length scales and long-range heterogeneity correlations. More important, the parameters of the resulting fractional derivative models have clear physical significance and must be obtained from a data fitting of experimental or field measurements. On the negative side, fractional-based models are computationally expensive and require enormous data storage requirement and are of a phenomenological description which does not necessarily reflect the physical mechanism behind the scenes. The literature survey has revealed the following areas to have potential research elements:

5.3. Variable-order anomalous diffusion models Constant order fractional order fractional diffusion equations have been employed successfully to describe the anomalous diffusion observed in porous media. However, it has been evident in some porous media that the constant order fractional diffusion models are inadequate for characterizing some complex diffusion behavior (Kobelev et al., 2003; Chechkin et al., 2005; Santamaria et al., 2006; Sun et al., 2009). For instance, if the diffusion process is such that the medium structure or the external field changes with time. The variable-order (VO) fractional models have been successfully applied to handle such diffusion behavior (Soon, 2005; Umarov and Steinberg, 2009; Diaz and Coimbra, 2010; Sweilam and Assiri, 2015; Sweilam and AL-Mekhlafi, 2016; Obembe et al., 2017). The theoretical justification of the VO time fractionaldiffusion model was presented by Chechkin et al. (2005) starting from the continuous time random walk (CTRW) scheme. The major limitation of the VO fractional diffusion equation compared with the constant-order counterpart lies in the difficulty of obtaining an analytical solution. Thus, numerical solution represents the only practical way to investigate such evolution diffusion problems. The literature reveals two types of VO fractional differential definitions. The first considers the derivative order has no memory related to past derivative order values (Coimbra, 2003; Sun et al., 2009), the second definition considers the derivative order has memory related to past derivative order values (Lorenzo and Hartley, 2002).

5.1. Transient inter-porosity flow in naturally fractured media Babak and Azaiez (2014) presented a unified fractional derivative approach to describe the fluid flow of slightly compressible fluid in naturally fractured reservoirs. This novel technique was arrived at by the juxtaposition of the flow at the micro scale and macro scale in the matrix blocks and the fractures respectively employing the transient interporosity flow behavior at the interface of both porous media (i.e. matrix blocks and fractures). Accordingly, the fractional derivative approach provides a convenient framework to unify the existing transient inter-porosity models developed for different shapes of blocks in any medium dimensions. The final form of the derived fracture flow model considering a slab type matrix block geometry is given by 1   ∂ψ f ∂ 2ψ f Kf þ τm 1  ∇: ∇ψ f ¼ 0 ∂t μ ∂t 2

=

=

ϕf cf

(47) 5.4. Multi-parameter estimation for model calibration

where

τm ¼

Am BVm pffiffiffiffiffi ϕ cm ηm ; Vm BV m

ηm ¼

Km ϕm cm μ

The incorporation of non-local and temporal flux constitutive equations into fluid flow mathematical models eventually result to the final step of model calibration with experimental, numerical solution, or field data to obtain the memory/fractional diffusion parameters i.e. γ, Kγ for the anomalous diffusion models presented by (Awotunde et al., 2016;

(48)

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Raghavan and Chen, 2016; Obembe et al., 2016b), df , γ, and θ for fractional fractal diffusion models (Chang and Yortsos, 1990; Yao et al., 2008, 2009; Wang et al., 2009; Zheng et al., 2012; Yang et al., 2014). This implies that the above parameters simulate the properties of the porous media. The literature reveals that parameter estimation methods for fractional differential equations is still in its infancy stage as opposed to that of the corresponding integer order partial differential equations. Methods such as; linear least-squares method (Wang et al., 2009; Fomin et al., 2010), a combination of the hybrid simplex search and a particle swarm optimization method (Liu and Burrage, 2011), Levenberg–Marquardt method (Ozcan, 2014; Yu et al., 2015, 2016), and the Bayesian method (Wang and Zabaras, 2005; Ma and Zabaras, 2009; Angelov and Slavtchova-Bojkova, 2012; Fan et al., 2015, 2016).

where I is the identity operator, and other terms are defined in the nomenclature section. Alternatively, Caputo (1998a) presented the mathematical equations governing stress-induced flow in a porous elastic media by considering a stress-strain equation (Biot, 1941), and a constitutive equation relating the stress field, pore pressure, and fluid density. Employing Eq. (36) as the constitutive equation relating the volumetric flux and pore fluid pressure, he arrived at:

5.5. Memory fluid flow model for anisotropic porous media

with the diffusivity term (A) in Eq. (56) is expressed as:

∇:ðσ þ σIÞ þ ½ρϕ þ ð1  ϕÞρs ℘∇z ¼ 0

A

# "  2 2Gð1  νu Þ B2 ð1 þ νu Þ ð1  2νÞ A¼η ð1  2νÞ 9ð1  νu Þðνu  νÞ

Due to the anisotropic nature of naturally occurring fractured porous media, a general extension of the modified Darcy equation was presented in the work by Caputo (1998b) as follows:

  ∂n1 qi ∂n2 pj a1 qi þ a2 n ¼ hij a3 pj þ a4 n ∂t 1 ∂t 2

    ∂γ 2 ∂ σ σ ∇ þ 3p þ 3p ¼ kk kk =B =B ∂t ∂t γ

(54)

(55)

(56)

Refer to nomenclature section for the definition of terms above. A critical assessment of the literature reveals the effects of stress is usually neglected in memory based models (perhaps, for convenience). Therefore, the contribution of stress effects in memory flow equations deserves more investigation moving forward.

(50)

⇀

where, qi are the components of q, pj are the components of, hij is a ⇀

symmetric dimensionless second order tensor, qðx; y; z; tÞ is the flux, pðx; y; z; tÞ is the pore pressures in the medium and 0 < n1 < 1; 0 < n2 < 1. The memory parameter a1 is positive and dimensionless, while other memory parameters a2 , a2 , and a3 have dimensions of sn1 , s, and s1þn2 respectively and are all positive. Eq. (50) may be further simplified by considering the anisotropy to be independent of coordinate system. Therefore, rotating the coordinates to the principal directions of the anisotropy yields (Caputo, 1998b):

a1 qi þ a2

  ∂n1 qi ∂n2 pi a ¼ h p þ a i 3 i 4 ∂t n1 ∂t n2

5.7. Non-Darcy effects In petroleum engineering discipline, certain deviations such as high velocity flow (important in gas reservoirs, or near wellbore flow effects due to completion design) and flow of non-Newtonian fluids lead to deviations from Darcy's law. To account for such deviations, the usual approach involves modifying Darcy's law. One widely employed equation for modelling turbulent flow in gas reservoirs is the Forchheimer empirical equation (Forchheimer, 1901; Barree and Conway, 2004) relating the pressure gradient to the volumetric flow rate as demonstrated by Eq. (57)

(51)

where hi are the absolute values of the diagonal elements of hij .

 5.6. Stress-induced memory flow fluid models

u¼

c1 K ∂p δ μ ∂x

with ¼

(58)

1 1 þ c3 Kβμu ρjuj , and c3 ¼ c1  c2 .

Refer to Nomenclature section for the definition of variables in Eqs. (57) and (58). The Forchheimer flux relationship i.e. Eq. (58) may be modified to incorporate the effect of memory as follows (Rami, 2015):

(52)

where u ¼ ∂w ∂t , with other variables specified in the nomenclature section. By extension, Eq. (52) may be modified to account for the whole spectrum of diffusion by employing Eq. (9) as the corresponding flux law as compared to Darcy's equation utilized in Eq. (52). This way, Eq. (52) yields:

" # ∂ Kγ;β ∂1γ ∂β p ∂β p ðw  w_ s Þ ¼  ð1   χÞ χ ∂t ∂xβ μ ∂t 1γ ∂ðxÞβ

(57)

where c1 and c2 are conversion factors. Re-arranging Eq. (57) in the form of Darcy's equation yields

Accounting for the effect of rock deformation i.e. stress-strain effect is key to predicting the long-term performance of any deformable porous medium. If we assume that the porous medium is composed of a linear elastic material with small deformations in the fluid phase (w), and a solid phase (w_ s ), Biot (1955) derived a generalized Darcy's equation of the form

∂ 1 ðw  w_ s Þ ¼  Kð∇p  ρ℘∇zÞ ∂t μ

∂p μ ¼ u þ c2 βu ρu2 ∂x c1 K

u¼

  c1 Kγ ∂γ ∂p δ γ μ ∂t ∂x

(59) γ

where the time fractional operator (∂t∂ γ ) is interpreted in the Caputo sense. Furthermore, the author revealed that for δ to remain dimensionless, βuγ (i.e. the modified non-Darcy coefficient) has dimension of ½T γ L1 , with δ re-expressed as follows:

(53)

In Eq. (53), the effect of gravity is assumed negligible. Along with Eq. (53), an equilibrium relationship of the form described in Eq. (54), and a constitutive equation between stress and strain tensors give rise to the resulting system of equations modelling stress induced fluid flow (Chen et al., 2006).

δ¼

1þ

1  1:47 c3 Kγ βuγ ρjuj ; where βuγ ¼ ϕ0:53 Kγ μ

(60)

The applicability of Eq. (59) as a modified Forchheimer flux relationship deserves further verification/validation moving forward. 322

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5.8. Multicomponent diffusion

where ϖ is the thermodynamic correction factor portraying the nonideal behavior. For gaseous mixtures at low to moderate pressures and for thermodynamically ideal liquid mixtures, the thermodynamic factor, ϖ ¼ 1. Furthermore, for this limiting case, Fick and Maxwell-Stefan diffusivity are identical to each other. However, it has been reported that the major drawback of the Maxwell-Stefan diffusion theory is that the diffusion coefficients with the exception of the diffusion of dilute gases, do not correspond to the Fick's diffusion coefficients (Curtiss and Bird, 1999). Further discussion of the Maxwell-Stefan diffusion model is beyond the scope of this research. To the best of our knowledge, no application of fractional derivative has been demonstrated till date on the multicomponent diffusion. Perhaps, this may be worth investigating moving forward. On a final note, the multiple diffusing species must undergo a complex diffusive behavior (e.g. time/space dependent diffusion coefficients) for the use of the fractional derivative (anomalous diffusion theory) to be valid (Periasamy and Verkman, 1998).

The sometimes unrealistic behavior of Fick's theory in relating the effective Fick diffusivity of component i in a multicomponent mixture was illustrated by Krishna and Wesselingh (1997). By carefully analyzing the results from four different diffusion experiments in the literature, the authors enumerated the shortcomings/limitations of Fick's theory. These experiments include: (i) the diffusion in an ideal ternary gas mixture (Duncan and Toor, 1962), (ii) diffusion in mixed ion system (Vinograd and McBain, 1941), (iii) ultrafiltration of an aqueous solution of polyethylene glycol and dextran (van Oers, 1994), and (iv) the transport of nbutane and hydrogen across zeolite membrane (Kapteijn et al., 1995). In addition, they established that the Maxwell–Stefan diffusion provides a robust framework for describing multicomponent diffusion in both fluids and dilute gases (Maxwell, 1866; Stefan, 1871). The Maxwell–Stefan diffusion formulation is a steady state model derived from the premise that the deviation from the equilibrium between the molecular friction and thermodynamic interactions leads to a diffusion flux (Rehfeldt and Stichlmair, 2007). It is argued that the Maxwell-Stefan diffusion formulation is more comprehensive than the “classical” Fick's diffusion theory because it accounts for the possibility of negative diffusion coefficients (Taylor and Krishna, 1993). Interestingly, Krishna and Wesselingh (1997) demonstrated that for a binary system, the Fick diffusivity D can  by the expression below: be related to the Maxwell-Stefan diffusivity D

 D ¼ ϖD

Acknowledgements The authors would like to acknowledge the support provided by King Abdulaziz City for Science and Technology (KACST), through the Science & Technology Unit at King Fahd University of Petroleum and Minerals (KFUPM), for funding this work through project No. 11-OIL1661-04, as part of the National Science, Technology and Innovation Plan (NSTIP).

(61)

Nomenclature

1 A

Am Asf BV B BVm cl cf co C Cp df D  D Dm Eγ ðzÞ G h hij H ⇀ Jc

k K Kγ L n p q qmf Rs s t t1

m2 s2; see Eqn. (56) Total matrix block surface, m2; see Eq. (47) Matrix pore surface available per unit bulk volume, m1; see Eq. (15) Bulk volume of all medium, m3; see Eq. (47) Skempton's coefficient Bulk volume of porous matrices, m3; see Eq. (47) Fluid compressibility; Pa1 Fractured media compressibility; Pa1 Oil compressibility; Pa1 Solute concentration, mol.m3 Specific heat capacity; J kg1 K1 Fractal (Hausdroff) dimension Fick diffusivity in the binary mixture, m2 s1; see Eq. (61) Maxwell-Stefan diffusivity in the binary mixture, m2 s1; see Eq. (61) Mechanical dispersion m1; see Eq. (18) Mittag-Leffler function; see Eq. (6) Shear modulus of the medium, kg s2 m1; see Eq. (56) Film heat transfer coefficient, Wm2 K1; see Eq. (15) Dimensionless diffusivity tensor; see Eqn. (50) Reservoir height, m Diffusion energy flux, W m2; see Eq. (16) Thermal conductivity, W m1 K1 Permeability, m2 Pseudo-permeability m2 s 1-Ɣ Reservoir length, m Dimension Pressure, Pa Fluid mass flow rate in the porous medium, kg m2 s1 Fluid flow from the matrix to fracture, kg m2 s1; see Eq. (11) Solution gas ratio, sm3/sm3 Laplace variable Time, s Characteristic time for transitions from one site to another 323

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b T T

Temperature in Laplace space Temperature, K

⇀

Fluid velocity vector, m s1

⇀ uγ

Anomalous fluid velocity, m/s; see Eq. (21) Drained Poisson ratio of the medium dimensionless; see Eq. (56) Undrained Poisson ratio of the medium dimensionless; see Eq. (56) Total volume of matrix block, m3; see Eq. (29) Deformation in the fluid phase, m; see Eq. (52) Trapping rate s1; see Eq. (25) Deformation in the solid phase, m; see Eq. (52) Release rate s1; see Eq. (25) Reservoir width, m a complex function

u

ν νu Vm w wi w_ s Wi width z

Greek symbol λ Mobility, m s kg1 [ LT/M] χ Skewness parameter, first introduced in Eq. (8) β Space order of fractional differentiation βu Non-Darcy flow coefficient, m1; see Eq. (57) βuγ Modified non-Darcy coefficient, sɣ m1; see Eq. (60) γ Fractional exponent, see Eq. (1) γg Specific gas gravity, fraction ξ Dummy variable; see Eq. (30) ϕ Porosity, fraction θ' Structure parameter θ Spatial position in CTRW formulation σ Stress tensor, kg s2 m1; see Eq. (54) σ ij Component of the stress tensor, kg s2 m1 σ kk hydrostatic part of the stress tensor, σ ij , kg s2 m1; see Eq. (55) Variance; see Eq. (2) σ 2r Dummy variable t' τ First moment of probability density function ψ Γ Standard Gamma function, see Eq. (1) ρ Density, kg m3 ℘ The magnitude of the gravitational acceleration; see Eq. (52) η Ratio of pseudo-permeability of the medium to fluid viscosity, kg1 m3 s 2-Ɣ κf Fluid phase thermal conductivity, W m1 K1, see Eq. (13) κm Solid matrix thermal conductivity, W m1 K1, see Eq. (14) κγe Effective pseudo-thermal conductivity, W s1Ɣm1 K1, see Eq. (22) ψ Probability density function Pseudo pressure, first introduced in Eq. (28) ψf ω Fourier mode ϖ The thermodynamic correction factor for the binary mixture, dimensionless; see Eq. (61) Subscripts b f init m s c

Bulk Fracture Initial Matrix block Solid Continuous phase

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