1. 1 JB/SWICA 01-01-01 MODELING FLOW AND SOLUTE TRANSPORT IN THE SUBSURFACE by JACOB BEAR SHORT COURSE ON Copyright © 2002 by Jacob Bear, Haifa Israel. All Rights Reserved. To use, copy, modify, and distribute these documents for any purpose is prohibited, except by written permission from Jacob Bear. Lectures presented at the Instituto de Geologia, UNAM, Mexico City, Mexico, December 6--8, 2003 Professor Emeritus, Technion—Israel Institute of Technology, Haifa, Israel Part 8: 301- 363

2. Adsorption. Ion exchange. Chemical interactions among various species. Interphase transfers, (e.g., dissolution). Decay and growth phenomena. Biotransformations. In addition, we may have Injection/withdrawal of fluids from the considered domain by wells. MASS BALANCE EQUATION. CONTINUUM MODEL: Firset: WHAT MAY HAPPEN TO A CONTAMINANT ALONG ITS PATH: the contaminant may appear or disappear. In the mathematical model, such phenomena are expressed as SOURCES and SINKS,, per unit volume of porous medium. Two groups of sources (sink = - source): 01-01-01JB/SWICA 301

3. Sources of the considered chemical species that result from phenomena that occur at (microscopic) points within the phase. Radioactive decay, chemical reactions,   = strength of such a (macroscopic) source.   = rate of generation of the  -species per unit mass of the phase.  = mass density of the latter. Chemical reactions called homogeneous reactions'. Sources of a considered chemical species that result from the transfer of the latter into the considered phase (or out of it for a sink), across the (microscopic) interphase boundaries: Adsorption, precipitation, dissolution, evaporation, diffusive transfer. `heterogeneous reactions' = rate of transfer of a  -chemical species from the  -phase to all other  -phases, across their common (microscopic) interfaces, per unit volume of porous medium. 01-01-01JB/SWICA 302

4. MASS BALANCE EQUATION. SINGLE COMPARTMENT First: A single component---as an introduction to mass balance. A very simple model: A single cell (compartment) model Aquifer Inflow, I, Outflow, Q, in volume (e.g., m 3 ), per unit time (e.g., year) Concentration in gr / m 3 (ppm). First, WITHOUT SOURCES Sometimes, we consider a number of adjacent cells, in a, two-, or three-dimensional configuration 01-01-01JB/SWICA 303

5. Start by considering a mass balance for an aquIfer cell of finite volume U o, during a finite time interval,  t. The “cell” or “compartment” is "well mixed”, i.e., all fluid and porous medium properties, are uniform within U o. There are no fluxes of the fluid or the solute within U o The cell may be fully or partially saturated with water Mass balance (unsaturated; with sources): The mass of a  -component within the cell is expressed by  S U o c. To be discussed later. denotes the rate (= mass per unit time) at which the  -component moves from the solid and from all other fluid phases into the considered fluid phase across their common microscopic boundaries, per unit volume of porous medium.   denotes the rate of production of  within the fluid phase (e.g., by chemical reactions), per unit mass of fluid of density . Within the cell---fluid saturation S. 30401-01-01 JB/SWICA

6. INFLOW MINUS OUTFLOW = ADDED STORAGE mass  S U o c| t+  t -  S U o c| t =  t(Q in c in - Q out c + f  U o +  S  U o   ) I Mass balance within U o: For a point in time, we divide by  t and let  t 0, obtaining And per unit volume of porous medium: 30501-01-01 JB/SWICA

7. For a saturated domain, with Q in = Q out, and without sources: The analytical solution of this equation for the conditions: t = 0: c = c o, and t 0, c in = 0, The flushing of a number of pore volumes may be required in order to reduce the initial concentration in the cell to a desired level, say MCL. Without adsorption is 30601-01-01 JB/SWICA

8. ADSORPTION Adsorption (the opposite of desorption) is the phenomenon of accumulation of a chemical species present in a liquid (adsorbate) that occupies the void space, or part of it, on the solid matrix (adsorbent) at the liquid-solid interface. The primary driving force for adsorption may be a consequence of the lyophobic (= solute disliking) nature of the solute relative to the solvent that hosts it. Other forces: electrical attraction, van der Waals attraction, chemisorption. Adsorption may be affected by the presence of other species in the liquid phase Within the void space, part of the contaminant is in the fluid (say water), and another part is on the solid. 307 01-01-01 JB/SWICA

9. ADSORPTION ISOTHERM is an expression that relates the quantity of the adsorbed species on the solid to the quantity of the same species in the liquid, at a given temperature, under equilibrium conditions. Under equilibrium conditions. Under non-equilibrium conditions. All reactions always take time. However, sometimes, equilibrium is reached relatively quickly. Under certain conditions, equilibrium may be reached after a very long time. Thus, reactions and adsorption: F may be measured in kg/kg, or in moles/kg, while the concentration in the liquid, c, is measured in kg/liter, or in moles per liter. 308 01-01-01 JB/SWICA

10. For m = 1, and replacing the symbol b by the more commonly used symbol K d LINEAR ISOTHERM EXAMPLES OF COMMON ISOTHERMS Fruenndlich (1926), suggested the nonlinear isotherm m < 1 means that as F increases, it becomes more difficult for additional quantities of the adsorbate to be adsorbed. The opposite situation is described by m>1. K d (= F/c) expresses the affinity of the chemical species for the solid phase. It gives, at every instant, the mass of the species on the solid, per unit mass of the latter, per unit concentration of the species in the liquid phase, say, in a unit volume of porous medium. F c KdKd 309 01-01-01 JB/SWICA

11. K d expresses the distribution between fluid and the solid. gr/liter gr/kg liter/kg Expressing partitioning between the fluid and the solid. 310 01-01-01 JB/SWICA

12. EQUILIBRIUM ? Usually, the time characterizing adsorption may be sufficiently short, relative to the times characterizing advection and diffusion in the liquid within the void space, so that equilibrium may be assumed to prevail. ADSORPTION ISOTHERMS: NON-EQUILIBRIUM CONDITIONS: Langmuir (in Adamson, 1967): Lapidus and Amundson (1952): Langmuir (Hendricks, 1972): 311 01-01-01 JB/SWICA

13. Example: For  b = 1.8 kg/liter, K d = 10 liter/kg, c = 3 mgr/liter, F = 30 gr/kg. Hence, in the liquid we have 0.9 mgr/liter of porous medium, while on the solid we have 54 gr/ liter of porous medium. MOST OF THE CONTAMINANT IS ON THE SOLID. Total mass of  per unit volume of porous medium. For mass  c of  in the liquid, we have  b K d c on solid. Langmuir (1915, 1918) suggested a nonlinear equilibrium isotherm Lindstrom et al. (1971) and van Genuchten (1974), presented the Linear isotherm: Why the name “retardation” factor? 312 01-01-01 JB/SWICA

14. In general, Recall: All coefficients are determined experimentally, for a particular chemical species (adsorbate) and solid (adsorbant) When ADSORPTION (equilibrium, but) IS NOT COMPLETELY REVERSIBLE: KdKd F c c max K dd F in At every instant, whether we are on The adsorption or desorption isotherm depends on the prevailing situation and on what happens in the fluid (rising/decreasing concentration). As long as c in the fluid is rising, we use K d in the retardation factor. When c drops, we have to use K dd. NOTE: coefficients 313 01-01-01 JB/SWICA

15. Let us return to the SINGLE CELL, this time WITH ADSORPTION. Adsorption F=K d c Assume equilibrium, with a linear isotherm. We start from the same balance equation for the species in the fluid developed earlier: Asssume no   -type sources, and full saturation: 314 01-01-01 JB/SWICA

16. Write the mass balance for the species ON THE SOLID: We have two (unknown) variables in this (single) equation: c and By summing up the two equations: Or: Single varaible, c. Retardation factor 315 01-01-01 JB/SWICA

17. For Q in = Q out The analytical solution of this equation for the conditions: t = 0: c = c o, and t 0, c in = 0, MCL Longer cleanup time because of adsorption. 316 01-01-01 JB/SWICA

18. Since R d > 1, the time required for reducing the concentration in the cell to a desired level is longer with adsorption than without it. The flushing of the  -contaminant is slower than that of the host liquid. We say that the movement of the contaminant is retarded, relative to the liquid. The coefficient R d is, therefore, referred to as retardation coefficient, or retardation factor. Thus, three interpretations of R d ; Gives the partitioning between  in fluid and on solid. Indicates the reduction of velocity of the contaminant. relative to the fluid. The modified residence time is: R d  U o /Q in. Retards the flushing of the soil. 317 01-01-01 JB/SWICA

19. RADIOACTIVE AND OTHER FIRST ORDER DECAY Radioactive and certain other decay phenomena A PRODUCTS, may be expressed as a first-order rate law, in the form Further discussion later = first order rate constant for the radioactive decay, N = number of atoms of the radioactive material. We could have used molar concentrations instead of N. In words: the rate of decrease (because of the minus sign) in N is proportional to N. Integrating the above expression from N = N o at t = 0, to any t, gives Half-life = time required for the concentration (or the number N) to be reduced by 2, 318 01-01-01 JB/SWICA

20. In principle, no equilibrium can be reached until the radioactive material has completely disappeared. When a  -species (concentration c) in a fluid  -phase undergoes radioactive decay within a porous medium domain, with or without adsorption, the source term, expressing the rate of appearance of the species in a macroscopic mass balance equation, is given by: Actually, disappearance For decay, or degradation of a  -species in a fluid  -phase: k  = degradation rate constant for the species in the fluid phase. In the fluid, and on the solid: …per unit volume of porous medium. 319 01-01-01 JB/SWICA

21. We continue the discussion of a SINGLE CELL, this time WITH ADSORPTION, AND DECAY. Adsorption F=K d c Assume equilibrium, with a linear adsorption isotherm. We start from the same balance equation for the species in the fluid developed earlier: The contaminant undergoes a first order (e.g., radioactive) decay:   = - c,   s = - F. ? 320 01-01-01 JB/SWICA

22. On the solid: By summing the two equation, the mass balance for the porous medium: With , and Q in = Q out 321 01-01-01 JB/SWICA

23. For c in = 0, For unsaturated flow, we cannot divide by  and by R d (  ). …and more about sources: = 0 322 01-01-01 JB/SWICA

24. AQUEOUS PHASE--GASEOUS PHASE PARTITIONING The gas-liquid interface may be visualized (modeled) as composed of two ‘boundary layers’, or `films': a `gas film', and a `liquid film', on the respective sides of the interface. The transfer of a chemical species from a gaseous phase (in which it exists as a component) to be dissolved in the liquid phase, and vice versa, may be visualized as: (1) transport of the species from the bulk gaseous phase to the surface of a `gas film'. (2) Diffusion through the gas film. (3) Diffusion through the `liquid film'. (4) Transport into the bulk liquid. A characteristic time is associated with each step. Diffusion through the liquid phase is, usually, the `rate limiting step'. (AIR-WATER PARTITIONING) 323 01-01-01 JB/SWICA

25. HENRY’s LAW SOLUBILITY OF A GAS IN WATER (e.g., air in water): (…but air is composed of many components…) = Henry's law constant (in atm) = mole fraction of gas dissolved in liquid = partial gas pressure (in atm) g X p For the simultaneous dissolution of a number of gases: 324 01-01-01 JB/SWICA

26. HENRY’s LAW describes aqueous phase—gaseous phase partitioning of a  –species: Henry’s coefficient. Henry’s coefficient can be expressed as: S  w = solubility of  = mass of  dissolved per unit volume of w-phase, R = universal gas constant (82.06 cm  atm mole  deg  ), and T is the absolute temperature, and P   vapor pressure of the pure component at that temperature. 325 01-01-01 JB/SWICA

27. We continue the discussion of a SINGLE CELL, this time WITH ADSORPTION, AND VOLATILIZATION. Adsorption F=K d c Assume equilibrium, with a linear adsorption isotherm. We start from the same balance equation for the species in the fluid, in the gas, and on the solid, as developed earlier: Both water at a constant saturation S w, and a gas at saturation S g occupy the void space. The  -contaminant is a volatile one, partitioned between the liquid, the solid, and the gas (at constant pressure) Henry’s law: 326 01-01-01 JB/SWICA

28. In gas Per unit volume of porous medium: In liquid On solid where is another Retardation Factor For a volatile adsorbing species. The mass balance equation: +??? Adsorption? The component is partitioned between the gas, the water, and the solid 327 01-01-01 JB/SWICA

29. The solution, under the same conditions as those of the case with adsorption only (constant pressure and saturation): The effective residence time for a gas at saturation S g, is now S g  U o R v /Q in. …and now something about chemical reactions. Why is this part of a discussion on SOURCES? 328 01-01-01 JB/SWICA

30. l RATE OF REACTION l Reaction rate of species A and of species C, resp Reactions described by a stoichiometric equation : Reactant Product When reactions are reversible, we use. Rate of a reaction expresses the decrease in the concentration of a reactant, or the increase in that of a product, per unit time. A B + C Example: 329 01-01-01 JB/SWICA

31. The minus sign means disappearance of the species: one molecule of the reactant A disappears for one molecule of the product,C, produced. A B + C Recall that always, a derivative is the rate of increase It is more common to define the rate of reaction by: in which [A] (equivalently, n A = c A /M A, with M A denoting the molar mass of A) represents the molar concentration of A. We note that 330 01-01-01 JB/SWICA

32. Stoichiometric equation expresses a chemical reaction: with the rate of the reaction (moles per second in a given volume of solution), in which  ’s,  = A, B, …,P, Q,.... are the stoichiometric coefficients which describe the relative number of moles of each reactant and those of each product that participate in the considered reaction. We have a single rate for the entire reaction. A + B C, Rate law rate constant of the reaction 331 01-01-01 JB/SWICA Reactants to products.

33. The rate law for a the general reaction The rate expressed by this equation is said to be of  -order in A,  -order in B, etc FIRST AND HIGHER ORDER RACTIONS first-order rate law. first-order rate constant (dims. T -1 ). second-order rate law second order rate constant (dims. M -1 T -1 ). 332 01-01-01 JB/SWICA

34. When a considered species participates in chemical reactions which cause its concentration within the fluid phase to increase (or decrease), we express the strength of the source (= rate of production) of that species in the (macroscopic) balance equation of the latter, in the form the SOURCE in the balance equation of that chemical species: expresses the mass of the  -species added to the fluid  -phase by the j-th chemical reaction, per unit volume of fluid phase, per unit time. 333 01-01-01 JB/SWICA

35. 334 01-01-01 JB/SWICA Rate of reaction Again:

36. MASS BALANCE of a  -chemical species. The continuum approach. FUNDAMENTAL BALANCE EQUATION Consider a  -component in a fluid  -phase (liquid, or gas) that occupies the entire void space, or part of it, at a    S  ). Start: single fluid phase, and a single component, Hence, no subscript and no superscript. Objective: MACROSCOPIC EQUATION. by averaging the microscopic equation, directly, by recalling the structure of a balance equation. The mass balance equation for a chemical species at the microscopic level: Advective flux Diffusive flux Source 335 01-01-01 JB/SWICA

37. By averaging: THE MACROSCOIC MASS BALANCE EQUATION: Recall: Advective + Dispersive fluxes 336 01-01-01 JB/SWICA

38. c  = average concentration of the component in the fluid-phase, V  = average fluid velocity. J*  = dispersive flux = -D.grad c  J    = diffusive flux =- D.grad c . = rate of mass transferred from  -phase to all  -phases, per unit volume of porous medium, across common microscopic interfaces.        = Rate of production of  per unit volume of porous medium. Physical interpretation of terms:  c = Mass of  per unit volume of porous medium. = Rate of increase of  c = Total flux of  per unit area of porous medium (by advection, dispersion and diffusion). 337 01-01-01 JB/SWICA

39. Physical interpretation of terms:  c = Mass of  per unit volume of porous medium. = Rate of increase of  c = Total flux of  per unit area of porous medium (by advection, dispersion and diffusion). THE DIRECT APPROACH = rate of mass transferred from  -phase to all  -phases, per unit volume of porous medium, across common microscopic interfaces.        = Rate of production of  per unit volume of porous medium. Leads to (the same balance equation: Recall the interpretation of minus Divergence of a flux. 338 01-01-01 JB/SWICA

40. in which we replace the fluxes and the sources by appropriate expressions. What about PUMPING AND ARTIFICIAL RECHARGE through wells as A SOURCE AND A SINK? Wells as boundary conditions or as sources/sinks. It is a source when the  -component is being produced in the domain. In a sink (= negative source), the component disappears from the domain at the concentration prevailing at the sink's location at the time of withdrawal. FOR A CONTINUOUS ACCRETION OR WITHDRAWAL: = R ext (x, t) c R (x, t). = - Q ext (x, t) c (x, t) 339 01-01-01 JB/SWICA

41. When sources exist at N isolated points, x m, m = 1, 2,... N, the source term in the mass balance equation takes the form where R ext m represents the rate of injection (in terms of volume added per unit time) at point x m at time t, of fluid at the known concentration, c R m, and  (x - x m ) denotes the Dirac delta function (or the Dirac distribution, or the unit impulse) defined formally by For withdrawal To be inserted into the mass balance equation. 340 01-01-01 JB/SWICA

42. Mass balance equation for a single component in a single gluid: Decay, Chem. React. ? ? Two unknowns in a single equation. f Mass balance for the component on the solid surface Sum up the two equations, neglecting the source: 341 01-01-01 JB/SWICA

43. We have, eliminated the terms that represent the rate of transfer of the component from the fluid phase to the solid (and vice versa). adsorption decay In indicial notation pumping recharge Radioactive …..=….. 342 01-01-01 JB/SWICA

44. The mass balance equation for the fluid: For a constant density and a non-deformable porous medium: Under the same conditions, for a component: 343 01-01-01 JB/SWICA

45. Another form. Note that the pumping rate does not appear in this equation. Some more about RETARDATION Simple case: No external sources or sinks exist. The solid's density is constant, and a nondeformable solid matrix. No degradation or decay phenomena take place. The considered component adsorbs to the solid under conditions of equilibrium, following a linear isotherm, with Then, the mass balance equation: +… 344 01-01-01 JB/SWICA

46. For comparison, the same conditions, but without adsorption, i.e., K d = 0. The two equations are similar, except that in the former equation, the average fluid velocity is replaced by V/R d and the coefficient of hydrodynamic dispersion is replaced by D h /R d. CONCLUSION: Under the assumption of equilibrium adsorption described by a linear isotherm, the effect of adsorption is to retard the advance of the component (as part of it is adsorbed to the solid). 345 01-01-01 JB/SWICA

47. In 1-d: A semi-infinite column The concentration c 0 is maintained at x = 0, and A constant flow takes place through the column The point c = 0.5 advances at a speed V/R d, and that Curve is steeper, indicating a smaller coefficient of hydrodynamic dispersion. Approx. 346 01-01-01 JB/SWICA

48. VOLATILE CMPONENT A gas, g, and a liquid, l, occupy the void space (at saturations S g, and S l ) The considered contaminant is a VOC (= volatile organic compound,  ) present as a vapor component in the gas; No `free product' (i.e., liquid VOC) is present in the void space Concentration c  g, in the gaseous phase (and no water vapor). Concentration c  l in the aqueous liquid (and no dissolved gas). Concentration F  as an adsorbate on the (rigid) solid. Chemical equilibrium prevails among all phases. Adsorption F  = K d c  l. The partitioning between the gas and the liquid obeys Henry's law: component may undergo first order decay, with rate coefficients g, l and s, in the gas, liquid, and solid phases, respectively. 347 01-01-01 JB/SWICA

49. The balance equation for the mass of VOC in the gas: The balance equation for the mass of VOC in the liquid: The balance equation for the mass of VOC on the solid Balance equation for the mass of the component in the porous medium: Gas extraction: 348 01-01-01 JB/SWICA

50. Another form: Another Retardation factor: With partitioning between the liquid, the gas, and the solid. 349 01-01-01 JB/SWICA

51. EULERIAN-LAGRANGIAN FORMULATION Lagrangian formulation, viz., from the point of view of an observer that moves with the considered quantity, as it travels. Definition of total (hydrodynamic) derivative::: 350 01-01-01 JB/SWICA Very useful in numerical techniques of solution

52. To obtain a unique solution of a balance equation in a given porous medium domain, we have to specify: COMPLETE TRANSPORT MODEL Boundary conditions Initial conditions GENERAL BOUNDARY CONDITION : For any extensive quantity, in the absence of sources/sinks on the boundary (e.g., in heat flow with phase change), there is no jump in the total flux of that quantity across the boundary. For the mass of contaminant, in a multi-phase system: jump in (.. ) 351 01-01-01 JB/SWICA

53. Boundary with a body of fluid. e.g., a large lake or a river. for the gaseous phase in the soil, the atmospheric air above ground surface WELL MIXED WATER BODY. The condition of no-jump in the normal component of the total flux Contradiction??? Consider specific cases: 352 01-01-01 JB/SWICA

54.  | fb = 1.0 rigid and stationary solid This is a boundary condition of the third kind (in terms of c). Without advection: 2 nd kind b.c.. (only diffusion) NO DIFFUSION even when ??? The error in the conclusion stems from the assumption that a `well mixed' zone exists on the external side of the boundary. This assumption, in the absence of advection, combined with the sharp boundary approximation, must yield no mass flux by diffusion across it. Recall: there is no diffusion in a well-mixed zone. 353 01-01-01 JB/SWICA

55. To overcome difficulty: introduce a transition zone, boundary layer, or buffer zone. The sum of dispersive and diffusive fluxes through the transition zone is proportional to the average concentration gradient, and that the latter is proportional to the concentration difference c - c 0. The condition of continuity of flux of the contaminant at the boundary:  = a transfer coefficient. The dispersive-diffusive flux through the buffer zone can be expressed, following Fick's law, in the form: For the fluid as a whole: 354 01-01-01 JB/SWICA

56. Boundary condition: In the absence of advection, or when q r.n <<  *, the boundary condition Conclusion: With q r.n >>  *, Boundary condition of the third type. For a one-dimensional case 355 01-01-01 JB/SWICA

57. PHREATIC SURFACE Boundary condition is derived from the requirement of no-jump in the flux normal to the phreatic surface. The component's total flux, relative to the moving phreatic surface: The no-jump condition 356 01-01-01 JB/SWICA

58. With Now insert F = h (x, y, t) - z, to obtain a condition in terms of and c, h. A third type boundary condition for c. Note: 357 01-01-01 JB/SWICA

59. The phreatic surface is the lower boundary of the unsaturated zone The saturated zone is a `well mixed zone' at concentration c| sat. The no-jump (in flux) condition takes the form HOWEVER, USUALLY, GROUNDWATER IN THE SATUURATED ZONE IS CONTINUOUSLY IN MOTION, WITH THE CONCENTRATION VARYING IN TIME. 358 01-01-01 JB/SWICA

60. SEEPAGE FACE Continuity of flux of a component: Stationary seepage face, i.e., u = 0 With This is a boundary condition of the second kind. 359 01-01-01 JB/SWICA

61. We divide the problem domain along the surfaces of discontinuity into sub-domains in each of which no such discontinuities exist. On the boundary of discontinuity, we need one condition for the left domain, and one for the right one. The two models are coupled, and have, to be solved simultaneously. Back to a 360 01-01-01 JB/SWICA

62. INITIAL CONDITIONS The concentration of the considered component, at t = 0: c (x, y, z, 0) = f 3 (x, y, z), where f 3 is a known function. COMPLETE TRANSPORT MODEL FOR A SINGLE COMPONENT Velocity and saturation have to be known---from a flow model. Whenever the density is concentration-dependent, the two problems have to be solved simultaneously. The mathematical statement, or model Equation(s) that describe the configuration of the surface that bounds the porous medium problem domain. A list of the dependent variables (also for flow model). Flux equations for the mass of the considered fluid phase 361 01-01-01 JB/SWICA

63. Partial differential mass balance equations for the relevant fluid phases. Mass and momentum balance equations for the solid, when the latter is considered to be deformable. Partial differential equations that describe mass balances of the considered component Dispersive, and diffusive flux equations Constitutive equations for the fluid phases, for the solid (in case the latter is deformable), and for the component. Expressions for the various external sources and sinks (for mass of fluid phases and mass of the considered component). Statement of initial conditions for each of the relevant state variables. Statement of boundary conditions the relevant extensive quantities— mass of fluid phases, and mass of the considered component. Numerical values, or functional relationships for all the coefficients. 362 01-01-01 JB/SWICA

64. To obtain a closed set of equations, within the framework of a well posed problem, we need an equal number of equations. The next step is to determine the number of primary variables (or degrees of freedom}) of the problem. 363 01-01-01 JB/SWICA End of part 8

65. 30001-01-01 JB/SWICA MODELING FLOW AND SOLUTE TRANSPORT IN THE SUBSURFACE by JACOB BEAR WORKSHOP I Part 8:slides 300-370 Copyright © 2000 by Jacob Bear, Haifa Israel. All Rights Reserved.