Optimum Operating Conditions for Two-Phase Flows in Pore Networks: Conceptual /Numerical Justification Based on the MEP principle (43) M.S. Valavanides(1) and T. Daras(2) (1) Dept. of Civil Eng., TEI Athens, Greece,
[email protected] (2) School of Environmental Eng., TU Crete, Greece,
[email protected] MaxEnt 2014 34th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering Amboise, France, September 2014 2014 MaxEnt 2014, Amboise, France, 9/2014
M.S.Valavanides “Optimum Operarting Conds. for 2ph Flow in Pore Network – MEP Justification”
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Abstract The mechanistic model DeProF considers steady-state two-phase flow in porous media as a composition of three flow patterns: connected-oil pathway flow, ganglion dynamics and drop traffic flow. The key difference is the degree of disconnection of the non-wetting phase which affects the relative magnitude of the rate of energy dissipation caused by capillary effects, compared to that caused by viscous stresses. An appropriate mesoscopic scale analysis leads to the determination of all the internal flow arrangements of the basic flow patterns that are compatible to the externally imposed flow conditions. The observed macroscopic flow is an average over the canonical ensemble of the flow arrangements. Extensive DeProF simulations revealed that there exist a continuous line [a locus, r*(Ca)] in the domain of the process operational variables -the capillary number, Ca, and the oil-water flowrate ratio, r- on which the efficiency of the process (oil produced per kW dissipated in pumps) attains local maxima. Such maxima have been experimentally identified. Subsequently, the existence of the locally optimum operating conditions could be rationally justified by the following conceptual inference. Steady state two-phase flow in porous media is an off-equilibrium process. The rate of global entropy production (a measure of the process spontaneity) is the sum of two components: the rate of mechanical energy dissipation at constant temperature (thermal entropy), Q/T, and a Boltzmann-type statistical-entropy production component, kDeProF lnΦ, directly related to the number of different physically admissible internal flow arrangements, Φ, associated with every flow condition (configurational entropy). By applying the MEP principle we may infer that optimum operation of the process is met on a locus of conditions whereby the process total entropy production rate takes maximum values. To reduce the falsifiability of that inference, one needs to provide numerical evidence. To do so, it is necessary to deliver: (a) an efficient analytical/numerical scheme, to evaluate the number Φ of the different flow arrangements; (b) an expression for the constant kDeProF in the Boltzmann-entropy expression. Combinatorial considerations provided the analytical background to evaluate the number of different micro-arrangements of the flow per physically admissible solution. Τhe limiting procedure based on Stirling’s approximation has been applied to downscale the excessively large computational effort associated with the numerical handling of operations between large factorial numbers. Still, an appropriate application of the Boltzmann principle needs to be implemented, to deliver an expression for the constant kDeProF pertaining to the sought process.
MaxEnt 2014, Amboise, France, 9/2014
M.S.Valavanides “Optimum Operarting Conds. for 2ph Flow in Pore Network – MEP Justification”
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Scope Immiscible, two-phase flow in porous media is the core process in many industrial applications (recovery of hydrocarbons, soil remediation, protection of aquifers, etc.) A variety of phenomena take place across scales from below the pore scale to the fracture or field scale (hierarchical system). An inherent characteristic of ss2φfpm is the existence of optimum operating conditions (OOC) in terms of efficiency. Scope is to provide a theoretical justification -of the existence of OOC- based on statistical thermodynamics, i.e. the maximum entropy production (MEP) principle MaxEnt 2014, Amboise, France, 9/2014
M.S.Valavanides “Optimum Operarting Conds. for 2ph Flow in Pore Network – MEP Justification”
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Outline The examined process Phenomenology & essentials
The DeProF model essentials (in brief) True-to-mechanism modeling essentials
Predictions of the DeProF model for SS 2φ-f-pm dp/dz, Rel-Perm, intrinsic flow arrangement /variables
Current issues, Open Problems, Ideas & Challenges in 2Φfpm Retrospective examination of rel-perm. diagrams Universal Operational Efficiency Map for SS2φfpm Processes Normative Characterization of 2φ flows in pm Justification of phenomenology based on Statistical Thermodynamics Principles Implementation of different rheology (compressible flows, emulsions, USS, other) Technical Applications (reconsideration of API/RP40 standard, field geometries etc.) MaxEnt 2014, Amboise, France, 9/2014
M.S.Valavanides “Optimum Operarting Conds. for 2ph Flow in Pore Network – MEP Justification”
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References 1. 2. 3. 4. 5. 6. 7.
Valavanides, Constantinides & Payatakes, TiPM (1998) Valavanides & Payatakes, AWR, 24, 385-407 (2001) Valavanides & Payatakes, CMWR XIII, ISBN 9058091236, 239-243 (2000) Valavanides & Payatakes, SPE78516 (2002) Valavanides & Payatakes, SCA2003-18 (2003) Valavanides & Payatakes, SPE88713 (2004) Valavanides, SPE135429 ATCE2010 (2010)
The ImproDeProF Project http://users.teiath.gr/marval/ArchIII/ImproDeProF.html 8. Valavanides, Oil & Gas Science Technology 67(5) 2012 Review & state-of-the-art 9. Valavanides, SCA2014-047 10. Tsakiroglou et al., SCA2014-041 11. Valavanides et al., RFP 2014, Sept. 2014, Amsterdam, NL.
MaxEnt 2014, Amboise, France, 9/2014
M.S.Valavanides “Optimum Operarting Conds. for 2ph Flow in Pore Network – MEP Justification”
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The examined process: Immiscible, stationary (a.k.a. “steady-state”) two-phase flow in porous media (or networks)
Applications Oil & H/C industry (upstream/downstream), soil remediation processes, Reactors, PEM fuel cells etc.
Phenomenology & essentials tbd
MaxEnt 2014, Amboise, France, 9/2014
M.S.Valavanides “Optimum Operarting Conds. for 2ph Flow in Pore Network – MEP Justification”
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Applications of 2φ 2φFPM
Enhanced Oil Recovery (EOR) Secondary & Tertiary oil displacement in reservoirs to recover trapped oil ( ~50% of original oil in place) Use of displacing media: CO2, water + liquid polymers (and combinations), nitrogen, foams, in-place combustion gas etc.
MaxEnt 2014, Amboise, France, 9/2014
M.S.Valavanides “Optimum Operarting Conds. for 2ph Flow in Pore Network – MEP Justification”
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Applications of 2φ 2φFPM Soil remediation
Problem
Remedy
Typical DNAPL migration processes [from Kamon et al Engineering Geology 70 (2003)]
In-situ soil flushing process [from Khan et al J Env Management 71 (2004)]
MaxEnt 2014, Amboise, France, 9/2014
M.S.Valavanides “Optimum Operarting Conds. for 2ph Flow in Pore Network – MEP Justification”
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Statement of the SS2φFPM Problem Stationary Two-phase Flow in Porous Media ~ A
~ qw
~ qw
~ qo
~ qo
(− ∆~P i )
Homogeneous Porous medium
Fractional Flow Theory
θ
∆~z
o
(
~ ~ k − ∆Pi ~ U i = ~ k ir µi ∆~z
)
i = o, w
(
i i In Conventional Fractional Flow Theory k r = k r Si , xpm
But in reality MaxEnt 2014, Amboise, France, 9/2014
w
)
(
k ir = k ir Ca , r; κ, θ 0A , θ 0R , x pm
)
M.S.Valavanides “Optimum Operarting Conds. for 2ph Flow in Pore Network – MEP Justification”
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Flow Regimes during Steady-State Two-Phase Flow in Porous Media. Experimental Study (1). (Avraam & Payatakes, JFM, 293, 207-236, 1995)
Large Ganglion Dynamics (LGD)
Small Ganglion Dynamics (SGD) MaxEnt 2014, Amboise, France, 9/2014
Drop Traffic Flow (DTF)
Connected Pathway Flow (CPF)
M.S.Valavanides “Optimum Operarting Conds. for 2ph Flow in Pore Network – MEP Justification”
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The DeProF model essentials
MaxEnt 2014, Amboise, France, 9/2014
M.S.Valavanides “Optimum Operarting Conds. for 2ph Flow in Pore Network – MEP Justification”
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Evolution of DeProF theory for steady-state 2phase flow in porous media (SS2φFPM) Time- & scale-wise evolution of research leading to the development of the DeProF theory Theoretical and semi-analytical models
Monte-Carlo simulation of the fate of solitary oil ganglia during immiscible µ-displacement in p.m.
Population Balance Eqs
Immiscible µdisplacement and ganglion dynamics in p.m.
Payatakes et al, 1981
Payatakes, 1982
Payatakes & Dias, 1984
Pore-tomesoscale
Meso-tomacroscale
Pore-tomesoscale
Ganglion dynamics &
Network models for 2φFPM.
Theoretical model of collisioncoalescence of oil ganglia.
Network simulation of SS2φFPM
Dias & Payatakes, 1986 (a&b)
Constantinides & Payatakes, 1991
Constantinides & Payatakes, 1996
Valavanides et al, 1998
Mesoscale
Pore-scale
Mesoscale
Mesoscale
Motion of solitary oil ganglia in p.m.
Experimental works
Mechanistic Model of SS2φFPM based on ganglion dynamics (GD)
SS2φ flow in planar & non-planar model networks
Hinkley et al, 1987
Avraam et al, 1994
Num. solution of PBEs
Flow regimes & relperms during SS2φFPM Avraam & Payatakes, 1995
DeProF: mechanistic model of SS2φFPM decomposition in prototype flows Valavanides & Payatakes, 1998, 2000 & 2001 Pore-tomacroscale
DeProF prediction of optimum operating conditions (OOC) for SS2φFPM Valavanides & Payatakes, 2003 Pore-tostatistical thermodynamics scale
Flow regimes & relperms during SS2φFPM / strong wettability
aSaPP: conceptual justification of the existence of OOC in SS2φFPM Valavanides, 2010 Statistical thermodynamics
Reveal of latent experimental evidence on the existence of OOC for SS2φFPM Valavanides, 2011
Avraam & Payatakes, 1999
1980
# pores: 1
1990
10
103
p.m. scale: Pore Study scales: microscale
MaxEnt 2014, Amboise, France, 9/2014
2000
106 network /core mesoscopic
2010
109
+∞ field
macroscopic
statistical thermodynamics
M.S.Valavanides “Optimum Operarting Conds. for 2ph Flow in Pore Network – MEP Justification”
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DeProF A True-to-mechanism Theoretical Model For Stationary Two-Phase Flow in Porous Media (SS2φFPM)
Modeling essentials • Decomposition into 3 Prototype Flows: Connected-oil Pathway Flow & Disconnected Oil Flow= (Ganglion Dynamics + Drop Traffic Flow) • Physicochemical characteristics of oil/water/p.m. • Dynamic wettability (contact angles) • Mobilization and stranding probabilities for disconnected oil • Accounting of unit cell Conductivities for all flow configurations • Implementation of Effective medium theory • Hierarchical modeling / Scale-up pore-to-”core”-to-field scales • Physically admissible solutions & ergodicity principles MaxEnt 2014, Amboise, France, 9/2014
M.S.Valavanides “Optimum Operarting Conds. for 2ph Flow in Pore Network – MEP Justification”
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The “DeProF” DeProF” model Process ~ ~ l, xpm
RESULT System param.
(Pore network)
~ ,µ ~ ~ µ o w, γow, θA, θR
xpm κ, θΑ, θR
(Oil & water)
Operational par.
~ qo, ~ qw
Ca, r
DeProF mech/stic model algorithm
The macroscopic rheological state equation:
(
x = x Ca, r; κ, θA , θR , xpm
)
(“Pumps”) Reduced Macrosocpic Pressure Gradient
~ ~ ∂P k x= ~~ ∂z γow Ca And kro, krw Relative Permeabilities! MaxEnt 2014, Amboise, France, 9/2014
Interstitial physical characteristics of SS 2φ flow in pm Sw, β, ω, Flow arrangement variables (FAV) ηο,CPF, ηo,G Oil flow rates in CPF & DOF (GD) Uow,DOF Flowrate of o/w interfaces fOF, Coefficient of oil fragmentation, ξow,D Flowrate of o/w interface through DTF nG Ganglion size distribution fEU Energy utilization factor
M.S.Valavanides “Optimum Operarting Conds. for 2ph Flow in Pore Network – MEP Justification”
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Decomposition into Prototype Flows “DeProF” DeProF”
{
Externally imposed system parameters: Ca , r; κ, θ0A , θ0R , xpm ~ V
~ A
}
~ βV So,CPF=1
~ qw ~ Uw = ~ A
~ U o,CPF
CPF
DTF
~ U o, DOF
GD
So,DOF DOF=GD&DTF
q~ o ~ Uo = ~ A CPF
~ U w ,DOF
~ ~ ω ≡ V GD V DOF
GD&DTF
Prototype Flow Interlocking Condition: (SSFD2φPM)
The following variables are introduced: Flow Arrangement Variables (FAV): {Sw, β, ω} Prototype Flow Variables: {U, S} U = {Uo,CPF, Uo,DOF, Uw,DOF} S = {So,DOF, So,D, So,G}
MaxEnt 2014, Amboise, France, 9/2014
~ ~ ~ ∂Po ∂P w ∂P ~γow Ca = ~ = ~= ~ x ~ ∂z ∂z ∂z k
M.S.Valavanides “Optimum Operarting Conds. for 2ph Flow in Pore Network – MEP Justification”
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DOF modelling - microscopic scale ~ l
(1)
DTF domain
DTF cell q~ wD / C G ~ q uc
j
~ q uc
i
k
~ q oD / C
GD domain
Ganglion Cells
~ u 5G
GD domain reduced cell-conductances: −1 ~ ~ qG µ 1 1 1 b g bjik , n = ~ucb,,Gn ~w3 = A bjik + B n Ca χG u G ∆ p jik , n l n (x )
DTF domain reduced cell-conductances:
g Djik
b : “C,G”, “E,G” or “X,G”
A Djik (U, S ) + D ~ ~ quc µw D = ~ D ~3 = σ m 1 D ∆p jik l B jik (U, S ) + D 2 Ca χ
−1
A bjik , B bn : reduced effects of bulk phases & interphases u Gn (x ) : reduced ganglion velocity, χG :tortuosity of ganglion spine MaxEnt 2014, Amboise, France, 9/2014
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DOF modelling - microscopic scale
(2)
Criterion of Mobilization of Ganglia & Droplets
θ0A
( )
~ 0 Jα θ A
j ~ Jmin, j
~ l
j i
~ ∆L ji
( )
~ 0 Jα θ A
θji
∫
−∞
~ VnD
Configuration II (Droplet in throat)
θ 0R
~ Jdr ,i
i
Ganglion mobilization condition: ~ LGn
( )
~ 0 Jα θ R
i
~ − ∇P
+∞
Configuration I (Droplet invading throat)
~ VnD
( )
~ 0 Jα θ R
Droplet mobilization condition:
[ ( )
( )]
∂~p ~ ~ P(y )dy ≥ 2~γow Jdr,i θ0R − Jmin, j θ0A ∂~z
~ 2 l 2
+∞
∫
−∞
[ ( ) ( )]
∂~p ~ ~ P(y )dy ≥ 2~γow Jα θ0R − Jα θ0A ∂~z ∀ α ∈ {I, II}
∀ j, i = 1,...,5
Ganglia & droplets, as members of a dense population, can move even even at Ca
