Issue 
Sci. Tech. Energ. Transition
Volume 78, 2023



Article Number  1  
Number of page(s)  9  
DOI  https://doi.org/10.2516/stet/2022024  
Published online  30 January 2023 
Regular Article
An application of a multiindex, timefractional differential equation to evaluate heterogeneous, fractured rocks
^{1}
Raghavan Group, Inc., PO Box 52756, Tulsa, OK 74152, USA
^{2}
Kappa Engineering, 1500 City West Blvd, Suite 741, Houston, TX 77042, USA
^{*} Corresponding author: raghavan.raj@gmail.com
Received:
7
February
2022
Accepted:
6
December
2022
A multiindex, distributed fractional differential equation is derived and solved in terms of the Laplace transformation. Potential applications of the proposed model include the study of fluid flow in heterogeneous rocks, the examination of bimodal fluid exchange between mobileimmobile regions in groundwater systems, the incorporation of the existence of liesegang bands in fractured rocks, and addressing the influences of faulted and other skin regions at interfaces, among others. Asymptotic solutions that reveal the structure of the resulting solutions are presented; in addition, they provide for ensuring the accuracy of the numerical computations. Fractional flux laws based on Continuous Time Random Walks (CTRW) serve as a linchpin to account for complex geological considerations that arise in the flow of fluids in heterogeneous rocks. Results are intended to be applied at the Theis scale when combined with geological/geophysical models and production statistics to all aspects of subsurface flow: production of geothermal and hydrocarbon fluids, injection of fluids into aquifers, geologic sequestration and hazardous waste disposal. Results may be extended to study the role of complex wellbores such as horizontal and fractured wells and more complex geological considerations such as faulted systems.
Key words: Fractional diffusion / Fractured rocks / Geologic sequestration / Hazardous waste disposal / Geothermal energy / Bimodal fluid exchange
© The Author(s), published by EDP Sciences, 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Nomenclature
C : Compressibility [LT^{2}/M]
f(s): Function defined in equation (15)
K_{ν} (z): Modified Bessel function of the Second Kind of order ν
: Effective permeability, see equation (2)
I_{ν} (z): Modified Bessel function of the First Kind of order ν
α : Exponent; see equation (1)
: “Diffusivity” [L^{2}/T^{α}]; see equation (11)
λ : Dimensionless transfer coefficient; see equation (17)
: Effective mobility []; see equation (2)
ω : Ratio of fracture to total volumes; see equation (12)
Subscripts
Superscripts
1 Introduction
TimeFractional Flow Equations (tFFEs) are particularly useful in the evaluation of systems of complex geology to incorporate the internal architecture and internal imprint involving barriers, intercalations and the like Chu et al. (2017, 2018), Ingle et al. (2020), Raghavan and Chen (2018). Such blockages result in interconnected highconductivity pathways interspersed with low permeability blockages. tFFEs represent models incorporating subdiffusive propagation of the pressure fronts at the Theis scale and result in temporally nonDarcy, nonstationary flow to account for the complex circuitry. As these models are based on motions that depend on a Continuous Time Random Walks (CTRW) (or a waitingtime), such models yield powerlaw declines in flux (Raghavan and Chen, 2017a; Xia et al., 2021). Examples of subdiffusive behaviors are reported in a number of sources in the earth sciences (Bisdom et al., 2016; Caine et al., 1996; Chu et al., 2017, 2018; Cortis and Knudby, 2006; Doe, 1991; Evans, 1988; Jourde et al., 2002; Knudby et al., 2002; Mitchell and Faulkner, 2009; Noetinger and Estebenet, 2000; Noetinger et al., 2001, 2016; Raghavan and Chen, 2018; Savage and Brodsky, 2011; Scholz et al., 1993; Suzuki et al., 2016; Xia et al., 2021). This paper examines applications of distributive fractional differential equations in a manner similar to those discussed in Luchko (2011) in twoporosity rocks with high and low permeability along the lines in Barenblatt et al. (1960). Our choice is dictated because of the model’s ubiquitousness and its ability to apply to situations beyond fractured rocks such as the MobileIMmobilezones (MIM) model of Silva et al. (2009) which is similar to the Multi Rate Mass Transfer (MRMT) model discussed in Haggerty et al. (2000); see Xia et al. (2021). In MIM models the volume of the region with higher connectivity is much larger than the one with lower connectivity. Solutions applicable to the evaluation of single and multiple (interference, tomographic) well tests are presented. The motivation for this study is to evaluate the consequences of finite speeds of propagation of the transient in complex systems of the form where conventional representations of heterogeneity through lognormal representations of permeability are inadequate (Xia et al., 2021). Two exponents that reflect the heterogeneity in permeability and denoted by the symbol α_{j} are used to study the rate of propagation of pressure fronts. These exponents reflect the random walk dimension of fractal objects (Metzler et al., 1994).
2 Problem statement, flux law, and differential equations
We consider the classic problem of the flow of a slightly compressible, constant viscosity fluid to a cylindrical wellbore subject to the idealization in Barenblatt et al. (1960) and Warren and Root (1963). Initially the pressure, p_{i}, is assumed to be constant everywhere. Fluid compressibility and viscosity are denoted by the symbols c and μ, respectively. As noted in the Introduction, flow in the porous rock of constant porosity, ϕ, and thickness, h, is governed by the CTRW process through two exponents α_{f} and α_{m} where the subscripts f and m stand for the fissure and matrix systems, respectively, in which case the volume of the region corresponding to m is assumed to be much larger than that of f. The opposite would be true for the MIM model. This nomenclature is used merely for convenience and because of familiarity; for example, in the mobileimmobile context the subscripts f and m represent the mobile and immobile regions, respectively. The center of the well is assumed to be the origin with the system assumed to be infinite in its areal extent and as is usual we work in the cylindrical coordinate system through the Laplace transformation. Subdiffusive flow is represented through a fractional derivative defined by Caputo (1967) that may be deduced through the fractional engine discussed in Metzler and Klafter (2000). Detailed and elaborate citations that include theoretical considerations and experimental observations of the fractional flux law may be found in Raghavan and Chen (2020). This flux law is also particularly suitable for considering problems in 2D such as those considered in Beier (1994). The well is assumed to penetrate the rock completely and we consider constantrateproduction at a rate q; wellbore storage effects are not considered.
2.1 The Flux law
The flux law under a CTRW derived in Montroll and Weiss (1965), Noetinger and Estebenet (2000) and Henry et al. (2010) used in this study is given by(1)where the symbol p( x , t) represents fluid pressure,(2)and where ∂^{ α } f(t)/∂t ^{ α } is the temporal fractional derivative defined in the Caputo (1967) sense, namely(3)with Γ(⋅) being the Gamma function. Equation (1) implies that under CTRW or subdiffusive flow, the flux at time, t, is not directly obtainable from the instantaneous pressure distribution, p( x , t). The flux law presumes an independent, identically distributed random variable for the step length, Δx, and a waiting time function, ψ(t), for the time between jumps. Many such time functions have been discussed; see, for example, Feller (1971), Kenkre et al.(1973) and Hilfer and Anton (1995). Expressions similar to equation (1) have been used to describe diffusion in fractal objects, contaminant transport, heat transmission, flow in porous media and for biophysical applications. The exponent α in the expression of the fractional derivative follows a probability density function, , that exhibits powerlaw behavior at long enough times for α < 1 given by Schneider and Wyss (1989):(4)
Here d is the number of dimensions and agrees with the expressions in Carslaw and Jaeger (1959) for d ≤ 3 when α = 1.
2.2 The differential equations
On combining the equation for the conservation of mass with the flux law and the equation of state for a slightly compressible liquid, the Barenblatt et al. (1960) expressions that govern flow in a fractured rock under CTRW are:(5)for the fissure system and for the matrix system by(6)or, in other words,(7)Here Δp is the pressure drop, p_{i} − p(r, t), σ is the shape factor, ℓ is the reference length and the subscript D reflects nondimensionalization of the space variable with respect to ℓ.
On noting that the Laplace transformation of the Caputo fractional operator, , in terms of the Laplace variable, s, is given by(8)Equations (5) and (7), respectively, in terms of the Laplace transformation, become(9)and(10)where , the diffusivity of the fracture system is based on fracture pore volume; that is,(11)with(12)The bar symbol denotes the Laplace transformation. On now combining equations (9) and (10), we have(13)with(14)and(15)
Here(16) (17)and(18)The variables ω and λ, the storativity ratio and dimensionless transfer coefficient, respectively, are the wellknown standard variables used to describe naturally fissured rocks; in this case λ is defined for subdiffusive flow.
3 Solution
As is well known, the principal advantage of expressing the differential equation in the form noted is that if f(s) were set to unity, then the expression is identical to that for flow in a porous rock for subdiffusive systems and, further, if α_{f} were equal to unity then the differential equation reduces to that of a homogeneous rock considered in the literature in great detail. Thus it is immediately obvious that the solution for flow to a well in the form of a cylindrical system is given as (Theis, 1935)(19)where, Δp_{f} (r, t) = [p_{i} − p_{f} (r, t)] is the magnitude of the drop in pressure below the initial pressure, p_{i}. The symbols I_{0} (x) and K_{0} (x) represent the modified Bessel functions of the first and second kind of order 0, respectively. The kernel, u, is given by(20)Equation (19), of course, presumes that the initial condition is(21)The simple consequence of using in the kernels is that the solutions in Raghavan and Chen (2020) that were derived for the singleporosity case may be used directly to write expressions for the pressure distributions when pressures are governed primarily by the fissures (at early times) and as a result of both the fissures and matrix (total system) at later times. For example, we note that for early times (s → +∞), the faster leg is dominant reflecting the fracture pore volume and we have(22)at intermediate times(23)the lowertransmissivity, slowerregion plays a significant role, and under such circumstances we have(24)and at long enough times (s → 0+), the faster region again dominates but reflects the total pore volume and(25)The approximation given in equation (24) results in pressure distributions that are yet to be reported in the literature. As we see both exponents, α_{f} and α_{m} play a role in the transitional period from one behaving like a system governed only by the fast leg, equation (22), to one governed by both regions, equation (25). As we will see the responses are distinctly different from those of the classical case. This type of coupling is not evident in Xia et al. (2021); the two legs do not operate in parallel. We refer to the three approximations given in equations (22), (24) and (25) as Flow Regime 1, Flow Regime 2 and Flow Regime 3, respectively.
We now seek a solution to equation (19) in an infinitely large system subject to the following boundary conditions:(26)and(27)Equation (27) specifies the condition for a well of finite radius, r_{w}, which we take to be the reference length; thus the dimensionless distance r_{D} is given as(28)The pressure distribution, , subject to the above conditions may be obtained in a straight forward manner to be(29)If we are to understand the structure of the solutions, however, we need to approximate equation (29) by considering a linesource well; that is, assume x K_{1} (x) → 1 as x → 0 and in this case we may derive solutions for the three f(s) approximations considered.
3.1 Solutions for the situation xK_{1} (x) as x → 0
Under this assumption, we may write equation (29) as(30)To generalize and explore approximate solutions subject to the linesource approximation, we define dimensionless pressure, p_{Df} (r_{D}, t_{Df}), and dimensionless time, t_{Df}, respectively, by(31)and(32)With the asymptotic representation of K_{0} (z) as z → + 0 given by Carslaw and Jaeger (1959) (33)where γ is Euler’s constant, and the relationship in Oberhettinger and Badii (1973) given by(34)where ψ(·) is the Digamma function and L is the Laplace transformation, the dimensionless pressures for the three approximate expressions of f(s), equations (22), (24) and (25) are, respectively, given by(35)for early times,(36)for intermediate times, and(37)for long times.
As noted earlier the structure of the solution during Flow Regime 2 is significantly different from that reported in the literature; StreltsovaAdams (1978) now both α_{f} and α_{m} influence pressure. In addition, we see that during Flow Regime 3 two time scales, namely, those involving the fracture volume and the total volume affect the response with the fracture volume being, perhaps, more dominant. As expected, all three expressions given above reduce to the classical case for α_{f} = 1 and α_{m} = 1. Also note that as ω → 1, the role of Flow Regime 2 diminishes. The derivative or logarithmic derivative, , corresponding to equations (35), (36) and (37) are, respectively,(38) (39)and(40)Again for α_{f} and α_{m} = 1, these results correspond to the semilogarithmic approximation of the Theis (1935) solution, namely the Cooper and Jacob (1946) approximation. In concluding this discussion it must be clear that two requirements are to be satisfied. The value of the Laplace variable, s, needs to such that those required of the approximations of f(s) hold and those required for s to be small enough for the logarithmic approximation to apply also hold.
4 Computational results
The principal goal is to illustrate the nature of responses for subdiffusive flows in distributed timefractional flow systems and illustrate the consequences of subdiffusive flow by elucidating the roles of the exponents, α_{f} and α_{m}. The interaction between the exponents, α_{f} and α_{m}, produces a wide range of responses and we document the range of expectations. Both pressure and derivative responses of the rigorous solution are compared with the asymptotic expressions derived here in order to discern the structure of the solutions. Our focus is on Flow Regime 2. The roles played by the constants λ and ω are similar to those for the classical solutions and are not addressed in any detail as they are well documented in Warren and Root (1963) and Kazemi (1969). All solutions of equation (19) and associated boundary conditions, equations (26) and (27), are computed in the usual way by the Stehfest algorithm (1970a, 1970b). The examples presented provide evidence for both the viability of the model and the accuracy of the computations.
Figure 1 presents pressure responses at the wellbore, p_{wD}(t_{Df}), for the simplest system one may envisage, α_{f }= 1 with α_{m} being the parameter of interest; three values of α_{m} are considered. The complement of these results, namely the corresponding derivative curves, , are shown in Figure 2. In both figures responses for the classical case are included to serve as a backdrop. For α_{f }= 1 as noted in equations (35) and (37) straight lines with slopes of 1.151 on semilog coordinates will be evident for Flow Regimes 1 and 3, respectively, and this point is confirmed in Figure 1. During intermediate times, Flow Regime 2, the solution for α_{m }= 1 is flat as suggested in equation (36) whereas the solutions for α_{m} < 1 are distinctly different displaying broad sweeping patterns. As shown in Figure 2, the magnitude of the responses becomes muted as α_{m} decreases; further, the duration for which Flow Regime 2 exists increases. Both these observations essentially reflect the reduction in effective transmissivity for lower values of α_{m}.
Figure 1
The well response for production at a constant rate; dimensionless pressure, p_{wD}. The influence of the exponents for the faster and slower regions is demonstrated. The faster region is assumed to obey classical diffusion; that is, α_{f} = 1; the slower system assumes subdiffusive flow and three values of the exponent α_{m}: 1, 0.8 and 0.6. 
Figure 2
Logarithmic derivative responses corresponding to the results of Figure 1. The shapes of the responses for α_{m} < 1 reflect the consequences of the lower effective transmissivity. 
The results in Figure 3 consider the situation when α_{f} < 1. Results for both dimensionless pressure, p_{wD}(t_{Df}), and its derivative, , are shown on loglog coordinates to account for the existence of the term in equations (35) and (37). The circles and the squares display the duration for which Flow Regimes 1 and 3, respectively, exist. It is striking that the transitional period like that shown in Figure 2 during which Flow Regime 2 would be evident is entirely nonexistent. Conditions for which Flow Regime 2 would exist are not readily apparent for tFFEs describing subdiffusive flow in twoporosity rocks when α_{f} is much different from α_{m}. Most important to note here is that the time of transition from Flow Regime 1 to Flow Regime 3 is short and powerlaw responses with slopes reflecting the exponent of t_{Df}, namely, (1 − α_{f})/α_{f}, are not evident. This result is markedly different from what we see when we consider Cartesian coordinates; see Raghavan (2012) who showed that at early times the well response to a fractured well in Cartesian coordinates is given by(41) Chu et al. (2017, 2018) have used equation (41) to identify subdiffusive systems in convincing, compelling ways.
Figure 3
The well response for production at a constant rate is considered; the top and bottom curves, respectively, are the dimensionless pressure responses, p_{wD}, and the corresponding logarithmic derivatives, . The circles and squares are the analytical responses corresponding to Flow Regimes 1 and 3, respectively. Flow Regime 2 is essentially absent. 
Figure 4 explores the transitional period further and two situations where α_{f} → α_{m} are considered. Results that are complements of the ones discussed in Figures 1 and 3 are considered here. The top curve represents an α_{f} of 0.75 and α_{m} of 0.6 and this response may be compared to the derivative curve in Figure 3. Unlike that situation, here Flow Regime 2 dominates the responses shown for several log cycles; the other two Flow Regimes are not evident. The responses for the bottom curve may be compared to the situation where α_{f} and α_{m} are equal in Figure 2. Although muted, here again, we note the existence of a ushaped derivative curve with a rather flat bottom; the circles corresponding to Flow Regime 2 do yield (1 − α_{f})/α_{f} because the logarithmic term in equation (39) is a constant as the αs are equal. Finally, Flow Regime 3 prevails.
Figure 4
Derivative responses illustrating interactions between α_{f} and α_{m}. The intent is to demonstrate the influence on well responses as α_{f} → α_{m}. The circles denote Flow Regime 2 and the squares represent Flow Regime 3. 
The examples considered above are a small set of the cases we have considered and form a representative set.
5 Discussion and concluding remarks
A multiindex, distributed fractional differential equation is developed to examine flow in rocks that may be characterized by twoconductivity systems, one a heterogeneous system consisting of dead ends, intercalations and the like wherein fluid movement occurs rapidly and the second, a slower forcing system which is also presumed to be heterogeneous. The interfaces between these regions are not necessarily distinct and may contain entities such as mineral deposits, skin regions, etc. The net effect is that pressure propagation is delayed and other complex phenomena in the form of flow reversals etc. need to be addressed; Raghavan (2004). A twoindex timefractional, subdiffusive system offers a potential path to address such situations. The timefractional equation follows the outlines of the Barenblatt et al. (1960) model as it is a convenient vehicle for presenting the essential elements we desire; furthermore, it is both ubiquitous and well understood. This model opens up a wealth of options for examining nonlocal phenomena where diffusion is central; it is intended for situations where the geology is complex in fields such as the evaluation of geothermal systems, planning of contaminant transport, ascertaining waste disposal options and the like. Unlike conventional formulations, Gaussian statistics is not obeyed. Situations where this is the case are noted in the Introduction. Unlike DFN formulations, wherein the rules of Gaussian statistics are observed and details of smaller subunits are needed to predict performance; fractional diffusion assumes that such details may be dispensed with. It has occurred to us that incorporation of fractional diffusion within the subunits would be a useful exploration.
The principal contribution of this work is that subdiffusive features are introduced in a rigorous way for the first time. Three specific flow periods may be identified during transient flow and the intermediate flow period may be absent when α_{f} is much different from α_{m}. When evident, the characteristics of this period may be used to identify subdiffusive flow as it is much different from the classical case. The duration of this period increases as the magnitude of α_{m} decreases all other things being constant and as ω → 1.
Potential applications of tFFE models include fractured rocks where heterogeneity such as variations in permeability, variations and/or the width of fissure apertures are dominant, existence of liesegang bands (mineral deposits abutting fissures) and complex interconnectedness (Belayneh et al., 2006; Cacas et al., 2001; Kang et al., 2014; Reiss, 1980; Sharp et al., 1996), lowerconductivity skin regions in sandstones (Fu et al., 1994, solute retention, Zhang et al., 2020) and “water pileup” in bimodal transport that results in blockages at low permeability regions and consequent early arrivals in regions of higher permeability (Xia et al., 2021). Nonlocal pressure fronts of the kind discussed here may have to be incorporated into field studies even if other information such as solute transport suggests otherwise (Zhang et al., 2020). As the model is developed in terms of sources and sinks and the Laplace transformation, numerical implementation is straight forward and extensions to other well configurations and boundary conditions is also straight forward (Raghavan and Chen, 2017b). We do emphasize that tests must be supplemented with information on reservoir architecture (geological model, maps, logs, cores, etc.) for a complete analysis.
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All Figures
Figure 1
The well response for production at a constant rate; dimensionless pressure, p_{wD}. The influence of the exponents for the faster and slower regions is demonstrated. The faster region is assumed to obey classical diffusion; that is, α_{f} = 1; the slower system assumes subdiffusive flow and three values of the exponent α_{m}: 1, 0.8 and 0.6. 

In the text 
Figure 2
Logarithmic derivative responses corresponding to the results of Figure 1. The shapes of the responses for α_{m} < 1 reflect the consequences of the lower effective transmissivity. 

In the text 
Figure 3
The well response for production at a constant rate is considered; the top and bottom curves, respectively, are the dimensionless pressure responses, p_{wD}, and the corresponding logarithmic derivatives, . The circles and squares are the analytical responses corresponding to Flow Regimes 1 and 3, respectively. Flow Regime 2 is essentially absent. 

In the text 
Figure 4
Derivative responses illustrating interactions between α_{f} and α_{m}. The intent is to demonstrate the influence on well responses as α_{f} → α_{m}. The circles denote Flow Regime 2 and the squares represent Flow Regime 3. 

In the text 
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