Modeling of 1D Anomalous Diffusion In Fractured Nanoporous Media

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1 LowPerm2015 Colorado School of Mines Low Permeability Media and Nanoporous Materials from Characterisation to Modelling: Can We Do It Better? IFPEN / Rueil-Malmaison June 2015 CSM Modeling of 1D Anomalous Diffusion In Fractured Nanoporous Media A. Albinali, R. Holy, H. Sarak, & E. Ozkan Colorado School of Mines UREP UNCONVENTIONAL RESERVOIR ENGINEERING PROJECT Colorado School of Mines 1 CSM

2 Introduction Flow in Fractured Nanoporous Media Multiple flow mechanisms at different scales. Advection is the fastest with minimal contribution to total flow due to small proportion of the micro-pores and fractures In nano-pores, much slower diffusive processes occur due to concentration gradient, osmotic pressure, etc. Local diffusion is a function of the global pressure distribution due to advective flow. 2

3 Introduction Our objective is not to create a replica of the flow system We want to predict the performance of the system Definition of transport in porous media is phenomenological Conventional perceptions of porous media flow (in petroleum engineering) are not sufficient and practical for heterogeneous nanoporous reservoirs Conventional Perceptions Porous medium is a continuum Saturation, pressure, etc., are volume-averaged quantities Fluxes are related to gradients through empirical coefficients Constitutive relationships Darcy s law, relative permeability, macroscopic capillary pressure, etc. Conventional perceptions require (and permit) macroscopic averaging of intrinsic properties. 3

4 Introduction 80% of the pores in the Barnett core have a pore size less than 5 nanometers (Bruner et al., 2011) At what scale should the flow in unconventional reservoirs be perceived? Knudsen Number: Kn = λ Λ = No-Slip Conditions Continuum Flow Slip Conditions Slip Flow Mean Free Path of Fluid Molecules Macroscopic-Average Pore-Diameter Transitional Flow Free-Molecular Flow Darcy Flow (No-Slip Flow) Gas-Flow in Nanopores (Slip Flow ) 4 0ç Kn Non-Darcy Flow Darcy Flow Macro-scale pores Fast-Evolving Processes Fluctuations negligible Averaging & Upscaling Domain-scale modeling Bulk properties Non-Darcy Flow Nano-scale pores Slow-Evolving Processes Fluctuations significant Pore-scale characterization Pore-scale modeling Intrinsic properties Knè

5 Introduction Fundamental Problem of Traditional Modeling Traditional models of transport in porous media rely on continuum methods Continuum methods assume small or finite spatial correlations of process variables. Requirement of small or finite spatial correlations of process variables places restrictions on the rock properties and process parameters. 5

6 Introduction Fundamental Problem for Unconventional Reservoirs In unconventional reservoirs, continuum assumption become dubious because of ü various types of petrophysical heterogeneity, ü strong scale dependency of phase behavior and non-local gradient of the mean process variables. ü the cascade of scales of fracture networks, which is suggestive of fractal geometry Our problem is the macroscopic description of unstable processes in the absence of length scale cutoffs. Transport in these systems has long-range correlations and it cannot normally be described by classical methods. 6

7 Flow in Fractured Nanoporous Media Consider (characterize) the heterogeneity of the velocity field instead of the petrophysical heterogeneity of the flow domain Anomalous (Fractional) Diffusion Normal Diffusion: The probability density function in space, evolving in time, is of Gaussian type. The mean square displacement of a particle is a linear function of time: σ r 2 ~ Dt Anomalous Diffusion: In heterogeneous media, especially where the continuum assumption does not hold, The mean square displacement of a particle is not a linear function of time σ r 2 ~ Dt α α = 1 Normal Diffusion α > 1 Superdiffusion α < 1 Subdiffusion. 7

8 Flow in Fractured Nanoporous Media Anomalous (Fractional) Diffusion Physical Implications: Flux is temporally & spatially convolved Flux cannot be defined in terms of instantaneous & local gradients Physical meaning of the phenomenological coefficient of flux is unclear Mathematical Implications: Anomalous diffusion includes space and time fractional derivatives Solutions fall in the framework of fractional calculus Mathematical statements of the physical boundary conditions are not straightforward 8

9 Flow in Fractured Nanoporous Media 1D anomalous diffusion model Single-phase production from tight, unconventional reservoirs with fractured horizontal wells Analytical Only time-fractional (subdiffusion); problems with no-flow boundaries Extension to dual-porosity formulation Testing the concepts Numerical General anomalous diffusion solution Single porosity, single phase Treatment of boundary conditions Potential to extend to multi-phase flow 9

10 Analytical Approach Dual porosity idealization Dual Porosity Media Fractures Matrix Block w f L z z y w f L x L y x w f 10

11 Analytical Approach Time-Fractional Anomalous Diffusion u(x, t) = k μ 1 α t 1 α. p x 0 Dual-porosity idealization utilized to describe: Anomalous diffusion in the union of matrix+natural fractures Anomalous diffusion in matrix coupled with normal diffusion in the natural fractures Anomalous diffusion in matrix and natural fractures (independent) Solution verified for α = 1 (normal diffusion) with Homogenous reservoir solution Dual-porosity medium solution 11

12 Analytical Approach Subdiffusive Flow in Matrix 12

13 Analytical Approach Subdiffusive Flow in Fracture Network 1.E+5 1.E+4 1.E+3 p, psi 1.E+2 1.E+1 1.E+0 αf = 1 αf = 0.9 αf = 0.7 αf = 0.4 αf = 0.2 αf = E- 1 1.E- 2 1E- 2 1E- 1 1E+0 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 1E+7 t, hrs 13

14 Numerical Approach Space- and Time-Fractional Anomalous Diffusion Numerical formulation Treatment of boundary conditions Pressure distributions in the reservoir Solution verified for normal diffusion with hydraulic fracture solution 14

15 Numerical Approach 15

16 Numerical Approach

17 Numerical Approach

18 Numerical Approach

19 Numerical Approach

20 Numerical Approach

21 Numerical Approach

22 Numerical Approach β = 1 α = 1 Subdiffusion Superdiffusion

23 Conclusions Anomalous diffusion provides a means of modeling flow and transport in unconventional reservoirs The real problem problem is the physical interpretation of the phenomenological coefficients This relates to characterization and upscaling problems

24 Conclusions The Real Problem: Constitutive Phenomenological Relation (Flux): Normal Diffusion Anomalous (Fractional) Diffusion v = k p µ x v α,β = k α,β µ Fractional derivative γ C ξ γ = 1 Γ 1 γ ( ) ξ 0 ( ξ ξ ) γ C 1 α β p t 1 α x β 0 < α, β < 1 ξ d ξ How do you measure a phenomenological coefficient which is defined by a non-local, memory dependent flux law? Time-fractional diffusion k α = α 2n tα 1 k 24