Number density of configurational microstates in steady-state, two-phase flow in model porous media.

Transcription

1 Number density of configurational microstates in steady-state, two-phase flow in model porous media. Marios Valavanides University of West Attica, Athens, Greece Entropy 2018 : From Physics to Information Science and Geometry Barcelona, Spain, May, 2018 Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 1

2 Abstract / Overview Steady-state two-phase flow in porous media is a process whereby a wetting phase displaces a nonwetting phase within a network of pores. The process is critical in many industrial, energy, environmental and biological applications. It is an off-equilibrium stationary process (maintained in dynamic equilibrium at the expense of continuous energy supply to the system). Energy efficiency of the process ( non-wetting phase flow rate per kw spent on pumps ) depends on the rate of global entropy production. It comprises two components: A thermal entropy component, Q/T, in the continuum mechanics scale) and a configurational entropy (a Boltzmann Gibbs entropy component, klnw), due to the existence of a canonical ensemble of flow configurations, physically admissible to the externally imposed flow conditions (PAC ensemble). Implementing combinatorics it was possible to deliver analytical expression for counting the number of microstates, lnw, by contriving an appropriate mixing scheme over the PAC ensemble. Nevertheless, evaluatιon of the B-G analytical expression demands knowledge of the entire PAC ensemble. These PACs are detected by a true-to-mechanism stochastic scale-up model (DeProF). The phase space is scanned on a voxel-partitioning scheme. We deliver the equivalence between the virtual number of microstates estimated by the voxel-partitioning numerical scheme and the actual number of microstates over the continuum phase space of the actual flow. Indicative results for a Berea sandstone model pore network are furnished. Open problems related to the structure of the PAC ensemble, and the proper implementation of the ergodic principle in estimating the macroscopic average flow configuration are discussed. Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 2

3 References 1. Valavanides, Oil & Gas Science Technology67(5) (2012) 2. Valavanides et al., J Pet.Sci.Eng147(2016) 3. Valavanides, Daras, Entropy18(054) (2016) 4. Valavanides, Transp. In Porous Media123(1) (2018) 5. Valavanides, M.S., Oil & Gas Science 73 (6) (2018) ImproDeProF project Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 3

4 Applications of 2φFPM Oil Industry Soil remediation Enhanced Oil Recovery (EOR) Secondary & Tertiary oil displacement in reservoirs to recover trapped oil (50% original oil in place) Use of displacing media: CO2, water + liquid, WAG, polymers, nitrogen, foams, in-place combustion gas etc CO 2 sequestration Problem Typical DNAPL migration processes [from Kamon et al Engineering Geology 70 (2003)] Remedy In-situ DNAPL flushing process [from Khan et al J Env Management 71 (2004)] Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 4

5 The phenomenology of immiscible 2-ph flow in p.m. Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 5

6 Statement of the SS2φFPM Problem Steady-State Two-phase Flow in Porous Media A ~ q~ w q~ w q~ o q~ o Homogeneous Porous medium ~ P i z ~ o w θ Fractional Flow Theory ~ i P i o w q~ ~ i ~ i k i ~ U kr, A ~ ~ z i Capillary number Ca ~ ~ wu w ~ ow k Flow rate ratio i r k i r Ca, r;, 0 A o o r q~ ~ w U w, 0 R q~, ~ U x pm Viscosity ratio ~ ~ o w Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 6

7 Flow Regimes during Steady-State 2-Ph Flow in PM Experimental Study (1). (Avraam & Payatakes, JFM, 293, , 1995) Large Ganglion Dynamics (LGD) Drop Traffic Flow (DTF) Small Ganglion Dynamics (SGD) Connected Pathway Flow (CPF) Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 7

8 Flow Regimes during Steady-State 2-Ph Flow in PM Experimental Study (2) (Avraam & Payatakes, JFM, 293, , 1995) Small Ganglion Dynamics (SGD) Drop Traffic Flow (DTF) Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 8

9 The DeProF model essentials Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 9

10 U ~ w Decomposition into Prototype Flows DeProF Externally imposed system parameters: V ~ A ~ V ~ q~ w A ~ S o,cpf =1 S o,dof CPF U ~ o,dof U ~ o,cpf Ca, r;, 0 A, DTF GD 0 R, x pm U ~ o A ~ q~ o DOF=GD&DTF U ~ w,dof CPF GD&DTF V ~ GD V ~ DOF The following variables are introduced: Flow Arrangement Variables (FAV): {S w, β, ω} Prototype Flow Variables: {U, S} U = {U o,cpf, U o,dof, U w,dof } S = {S o,dof, S o,d, S o,g } j DTF cell D / C q~ w uc C D / q~o ~ q i k Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 10

11 The DeProF model Process ~ l, ~ x (Pore network) (Oil & water) q~ o, q~ pm ~ ~ ~ o, w, ow, A, w ( Pumps ) R System param. x pm κ, θ Α, θ R Operational par. Ca, r DeProF mech/stic model algorithm RESULT The macroscopic rheological state equation: x x Ca, r;, A, R, x pm Reduced Macrosocpic Pressure Gradient x P ~ z~ ~ ow ~ k Ca And k ro, k rw Relative Permeabilities! Interstitial physical characteristics of SS 2φ flow in pm S w, β, ω, Flow arrangement variables (FAV) η ο,cpf, η o,g Oil flow rates in CPF & DOF (GD) U ow,dof Flowrate of o/w interfaces f OF, Coefficient of oil fragmentation, ξ ow,d Flowrate of o/w interface through DTF n G Ganglion size distribution Energy utilization factor Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 11 f EU

12 Domain of Physically Admissible Configurations (PAC) (DeProF theory) For each set (Ca, r) of system parameter values the 2φ flow visits a continuum of physically admissible flow configurations represented by (S w,β,ω) the PAC domain (cloud of red balls). The PAC domain is a canonical ensemble r= 2.5 κ=1.45 Ca= A unique set of values for S w, β and ω (black balls) are obtained by averaging over the PAS domain. The volume of the PAC domain (red cloud) is a measure of the process number of degrees of freedom and S w of the process contribution to configurational entropy. Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 12

13 r= 0.1 r= 1.0 Effect of flowrate ratio r=q o /q w on the PAC domain (degrees of freedom) S w S w For any fixed value of Ca and as r gradually increases, the PAC domain progressively swells, extends to a max. size and shrinks to zero (just as 2-ph flow cease to be sustainable). When the PAS domain attains a max. volume, the 2φ flow is as rich as possible in different flow configurations. A systematic behavior is observed for the other o/w systems examined (κ=6 & 3,35) S w S w r= 2.5 Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 13 r= 6.3 κ=1.45 Ca=1.190x10-6 r= 4.0 S w r= 10.0 S w

14 10000 Reduced mechanical power dissipation of the total flow W 3D κ=1.45 Reduced pressure gradient x Ca (x10-6 ) Ca (x10-6 ) W ~ W ~W 1Φ W ~ ~ kμ ~ γ~ Ca 2 ow w x ~ p z~ γ~ ~ k Ca ow Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 14

15 Energy efficiency aspects of SS2φFPM (efficiency = m 3 /s of recovered oil per kw spent in pumps) (as revealed by DeProF model predictions) Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 15

16 Energy utilization factor (f EU = r/w) Critical Flow Conditions (CFCs) f EU = {flow rate of oil} / {mechanical power supplied to the system} 0,15 0,15 r/w 0,10 =1.45 2D r/w 0,10 =1.45 3D ,5 0,5 Ca (x10-6 ) 1,5 2, ,5 0,5 Ca (x10-6 ) 1,5 2,0 For any Ca=const, Locus r*(ca) : f EU (Ca, r*)=max[f EU =(Ca, r)] r*(ca) Critical Flow Conditions (CFCs) Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 16

17 Extensive DeProF model simulations spanning: 5 orders in Ca: -8<logCa<-4 5 orders in r : -2<logr<2 and 5 viscosity ratio values: κ=μ o /μ w ={0.33, 0.67, 1.00, 1.5, 3.0, 20.0} Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 17

18 -4>logCa>-8 Incr. o/w viscosity ratio, κ=μ o /μ w Water saturation, S w S w κ=0,33 S w κ=7 Sw κ=0 S w κ=1,50 S w κ=3, Connected-oil pathway flow (CPF) vol. fraction, β β κ=0,33 β κ=7 β κ=0 β κ=1,50 β κ=3, Drop traffic flow (DTF) vol. fraction in DOF, ω ω κ=0,33 ω κ=7 ω κ=0 ω κ=1,50 ω κ=3, Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM

19 Energy efficiency (oil output per kw spent), f EU =r/w 0,5 f EU =r/w κ=0,33 0,5 f EU =r/w κ=7 0,5 f EU =r/w κ=0 0,5 f EU =r/w κ=1,50 0,5 f EU =r/w κ=3,00 0,3 0,3 0,3 0,3 0,3 0,1 0,1 0,1 0,1 0, >logCa>-8 Incr. o/w viscosity ratio, κ=μ o /μ w Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 19

20 Open Problem (... & tentative solutions...) Existence of CFC in ss2φfpm Theoretical justification on the basis of statistical thermodynamics (maximum entropy production principle) Valavanides, 2010, SPE Valavanides & Daras, 2016, Entropy 58 Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 20

21 f EU vs N Ω PAS r*(ca) vs r (Ca) r/w 0,10 = D r/w 0,10 = D f EU 5 Operational effciency Ca (x10-6 ) ,5 Volume of PAS, V PAS 0,5 2,0 1,5 κ=6 r*(ca) Ca (x10-6 ) ,5 Volume of PAS, V PAS 0,5 2,0 1,5 κ=1, = =1.45 N Ω PAS degrees of freedom Ca (x10-6 ) ,5 0,5 1,5 2,0 r (Ca) Ca (x10-6 ) ,5 0,5 1,5 2,0 Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 21

22 S Ca r k lnn Ca r SYS, ss2 fpm PAS, Identification of Sources of Entropy S Ca, r S Ca, r S Ca r UNIV SUR SYS, S SUR Ca,r WCa,r Q Ca,r T 0 T 0 Molecular level (bulk & interfaces) thermal entropy Q/T Q: energy released to the environment (or dissipated as heat ) at temperature T Core/Field - level configurational entropy k lnw W: number of microstates freely & equiprobably attained k: Boltzmann-type constant for the particular process Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 22

23 The asapp concept (MEP) 1 T W Ca (x10-6 ) -1 0 logr ln(a D N PAS ) d e m o 40 d ln k e m o D (A D ) = C f d e m o d e m o d e m o Ca (x10-6 ) Volume of PAS, V PAS Ca (x10-6 ) = ,5 logr 2,0 1, ,5 0,15 0, Ca (x10-6 ) f EU =r/w logr 1 2 S SUR Entropy released to the Surroundings S SYS Entropy produced within the System S UNIV Total Entropy produced in the Universe W : reduced rate of mechanical energy dissipation = heat dumped to the surroundings T : absolute temperature k D : bridge from meso-to-macroscopic physics (similar to Boltzmann s const) - not yet estimated! N PAS : number of physically admissible solutions (internal flow arrangements at mesoscopic scale) A D : correlation factor between number of DeProF estimated (N PAS ) and actual number of flow arrangements f EU : Energy utilization factor (oil flow rate per kw of power dissipated in pumps) C f : correlation coefficient Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 23

24 Configurational Entropy (1) Identification of Microstates (a) Identification ofand decomposition intotwo complementary domains: 1) frontal area perpendicular to macroscopic flow for the connected-oil pathway flow (CPF) / balls-in-boxes problem (b) 2) reference volume for the disconnectedoil flow (GD+DTF) / chains-in-barbs problem (c) (c) Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 24

25 Configurational Entropy (2) Estimation of Number of Microstates Combinatorics P Ρ Ρ N COP COP COP N! COP Ρ DOF Ρ K DOF1 K CP CP! ballsink CP Ρ N DOF2 COP boxes! N N DTF DTF C N chainsin i N!N C! N! DTF I N max i1 C N C barbs! N! i Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 25

26 Configurational Entropy (3) Estimation of Number of Microstates P Ρ COP Ρ DOF N COP! K K CP CP! N COP! N DTF N N DTF! C! 1 I max N i! i1 lnρ ln K! lnn! lnk N! CP ln COP COP I max N DTF NC! ln NDTF! ln N j! CP j1 lnρ K CP ln K CP N COP ln n! ln N COP n ln n n K N lnk N COP Imax N DTF NC ln NDTF NC NDTFlnNDTF Ni lnni CP Stirling s approximation CP i1 COP Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 26

27 Configurational Entropy (4) Estimation of Number of Microstates Separation of extensive & intensive contributions lnρ K CP M 1 β lnβ 1 βln1 β Imax G G β 1 ωln1 ωωlnωω n ln n i1 i i If β=0 (no CPF), i.e. all oil is disconnected lnρ M 1 Imax G G β 1 ωln1 ωωlnωω n ln n i1 i i And if all oil is disconnected in singlets: lnρ M 1 ωln1 ω ωlnω Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 27

28 Configurational Entropy (5) Estimation of Boltzmann-type constant pending problem! S SYS = k DeProF ln P k DeProF : the Boltzmann s constant(s) for the particular process Generalized formulation as an effective expression for:,, S Ca, r k ln P Ca, r k ln P k ln P SYS DePr of CPF CPF, j DOF DOF, j j1 j1 N PAC Ca, r N Ca r N Ca r k ln 1 ln 1 CPF j j j j j1 PAC NPAC, Imax kdof Ca r 1j 1 j ln1 j j ln j j n i ln n i j1 i1 PAC j Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 28

29 Results (1) ln K CP ln 1 ln 1 Imax G M 1 1 ln1 ln ni ln n i1 G i Capillary number Ca 10 6 Viscosity ratio 1,50 Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 29

30 Results (2) ln K CP ln 1 ln 1 Imax G M 1 1 ln1 ln ni ln n i1 G i Capillary number Ca 10 6 Viscosity ratio 1,50 Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 30

31 Results (3) Imax G ln KCP ln1 ln1 M 1 1 ln1 ln ni ln n i1 G i Capillary number Ca 10 7 Viscosity ratio 1,50 Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 31

32 CONCLUSIONS (1/2) Two-phase flow in p.m. is a process in dynamic equilibrium regulated by oil disconnection and capillarity effects that restrain /inhibit the superficial transport of fluids low-end flow regime) bulk phase viscosities high-end flow regime). Process engineers must always judge where to set the balance between capillarity or viscosity (to increase process efficiency) Two-phase flow in p.m. is inherently rich in interstitial flow configurations (the PAC ensemble) The shape of the PAC ensemble is atypical and restructures (as externally imposed flow conditions change). Latent systematic trend in restructuring (mutation) over the process independent variables (as revealed by accounting the number of microstates per PAC.) Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 32

33 CONCLUSIONS (2/2) Open problems Derivation of the Boltzmann type constants appearing in the analytical expression for the configurational entropy is still an open problem. The DeProF model is based on ergodicity considering that all PACs are equiprobable. An unresolved modeling problem is to properly implement ergodicity considering the PAC ensemble shape characteristics. Implementation of ergodicity should also consider /challenge: a number-of- microstates-weighted PACs-averaging methodology. PACs residing in central part of the PAC cloud would be visited more frequently than PACs residing in the outskirts (?) Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 33

34 Welcome feedback, input & ideas Thank you! Acknowledgements ImproDeProF project Entropy 2018, Barcelona M.S. Valavanides: Number Density of Configurational Microstates in SS 2for SS 2ph Flow in PM 34