Computational and Applied Mathematics Vol. 22, N. 3, pp. 359 395, 2003 Copyright 2003 SBMAC.scielo.r/cam Steam injection into ater-saturated porous rock J. BRUINING 1, D. MARCHESIN 2 and C.J. VAN DUIJN 3 1 Dietz Laoratory, Centre of Technical Geoscience, Mijnoustraat 120 2628 RX Delft, The Netherlands E-mail: J.Bruining@mp.tudelft.nl 2 Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110 22460-320 Rio de Janeiro, RJ, Brazil E-mail: marchesi@impa.r 3 Technische Universiteit Eindhoven, Den Dolech 2 5600 MB Eindhoven, The Netherlands E-mail: c.j.v.duijn@tue.nl Astract. We formulate conservation las governing steam injection in a linear porous medium containing ater. Heat losses to the outside are neglected. We find a complete and systematic description of all solutions of the Riemann prolem for the injection of a mixture of steam and ater into a ater-saturated porous medium. For amient pressure, there are three kinds of solutions, depending on injection and reservoir conditions. We sho that the solution is unique for each initial data. Mathematical suject classification: 76S05, 35L60, 35L67. Key ords: porous medium, steamflood, travelling aves, multiphase flo. Introduction Steam injection is an effective technique to restore groundater aquifers contaminated ith non-aqueous phase liquids NAPL s such as hydrocaron fuels and halogenated hydrocarons [15]. It is also one of the most effective methods to #558/02. Received: 28/XII/02. Accepted: 18/VIII/03. This ork as supported in part y: CNPq under Grant 301532/2003-6, FINEP under CT- PETRO Grant 21.01.0248.00; NWO under Felloship Grant R 75-389, The Netherlands; IMA Univ. of Minnesota, and therefore y NSF, Dietz Laoratory TU Delft, The Netherlands, IMPA Brazil.
360 STEAM INJECTION INTO WATER-SATURATED POROUS ROCK recover oil from medium to heavy oil reservoirs [13]. The main feature of steam injection is the steam condensation front SCF, hich marks the oundary eteen the upstream zone at oiling temperature and the donstream liquid zone elo the oiling temperature. Depending on the situation there may exist an isothermal steam-ater shock at the oiling temperature HISW instead of the SCF. The main result of this ork is a complete and systematic classification of the structure of all possile cases of Riemann solutions. As a first step e have ignored the presence of NAPL s in our model. The model has also applications outside the use of steam for oil recovery or pollutant product recovery, for example in chemical engineering. There is an extensive literature on models of steam drive. Their main focus is the internal structure of the steam condensation front and they are revieed in [4], [5]. In this article e limit ourselves to the simple case of steam displacing ater. Our aim is to investigate a unique ell posed solution of the Riemann prolem for all possile values of the model parameters, providing mathematical validation of our model. This is the first step toards solving the full prolem of groundater NAPL removal. In Section 1, the physical model is presented. It is descried mathematically y alance equations of mass and thermal energy, hich are reritten into a form suitale for analysis. Section 2 presents the asic aves arising in the model; the main concern is to identify their speeds, so as to e ale to find the order in hich they may appear in a linear steam injection experiment. In Section 3, e see that for certain values of initial and oundary data, some of these speeds coincide, giving rise to ifurcation and structural change in the Riemann solution. All solutions of the Riemann prolem are in Section 4. Section 5 verifies that the SCF satisfies Lax s shock inequalities, ut not strictly. Section 6 summarizes our results and conclusions. Appendix A descries notation and values for the physical quantities appearing in the model.
J. BRUINING, D. MARCHESIN and C.J. VAN DUIJN 361 1 Physical and mathematical model 1.1 Physical model We consider linear steam displacement in a homogeneous reservoir of constant permeaility and porosity. The reservoir is initially saturated ith ater. The pressure gradients p /, p g / driving the fluids are small ith respect to the prevailing system pressure p divided y the length of the reservoir. In particular, ithin the short steam condensation zone pressure variations are negligile. Hence e disregard the effect of pressure variation on the density of the fluids and on their thermodynamic properties. The reservoir is horizontal, so gravitational effects vanish. A steam-ater mixture is injected at constant rate u inj and constant steam/ater injection ratio. Transverse heat losses are disregarded. We neglect capillary forces after steam reakthrough at the production end of the reservoir to avoid prolems ith the capillary end effect, hich is outside the present scope of our interest. The effects of temperature on the fluid properties, e.g. ater viscosity µ, steam viscosity µ g, ater density ρ and steam density ρ g are taken into account. Darcy s La determines the fluid motion. The temperature dependence of heat capacities and of the evaporation heat are also taken into account. Capillary pressure as ell as an effective longitudinal heat conduction term are included. We have chosen to descrie condensation in terms of a steam mass condensation rate equation. The mass condensation rate q is alays positive hen the temperature drops elo the oiling temperature T as long as not all steam has condensed, that is S < 1. The stated conditions can e considered representative of steam injection in the susurface for remediation of contaminated sites. As steam is injected the reservoir is heated. Depending on the proportions of steam and ater in the injected mixture, e can distinguish three regimes, hich differ in the structure of Riemann solutions. When pure steam is injected, there ill e a decrease of the steam saturation in the hot zone aay from the injection point, descried as a rarefaction ave, and then a SCF to the cold ater, descried as a shock ith a concentrated source term. This is called situation I. As the ater injection
362 STEAM INJECTION INTO WATER-SATURATED POROUS ROCK rate is increased further a zone in hich the steam saturation is constant ill develop preceding the rarefaction ave until the steam saturation in the hot zone is constant. After this constant state there is a SCF and a cold ater region. This ill e called situation II. Finally hen the ater steam injection ratio is increased further, the steam ank ill not e fast enough to reach the cooling front separating the hot and cold ater zones; thus there is no SCF. This and higher ratios originate in situation III. In all regimes, there is a hot zone and a cool zone, hose oundary moves ith constant speed, as shon in Fig. 1. t A HOT ZONE steam/ ater B C COLD ZONE ater Figure 1 B: Condensation front or cooling front. Each of the enthalpies per unit volume H T, H r T, H g T [J/m 3 ]is defined ith respect to the enthalpy at the initial reservoir temperature T 0 at the standard state. This means that they are all zero at the initial temperature T 0. The enthalpy of steam is sudivided in a sensile part Hg s T and a latent part Hg lt 0, i.e. H g T = Hg st + H g lt 0. The sensile heat Hg st 0 is zero at the initial reservoir temperature. The evaporation heat or the latent heat per unit mass at the initial reservoir temperature T 0 is denoted y 0 = T 0 = H l g T 0 /ρ g T 0. 1 In general T is the evaporation heat per unit mass at temperature T. The enthalpies as a function of temperature are summarized in Appendix A for convenience. x
J. BRUINING, D. MARCHESIN and C.J. VAN DUIJN 363 We assume Darcy s la for to-phase flo, ater and steam respectively, ithout gravity terms: u = kk r µ p, u g = kk rg µ g p g. 2 The liquid ater viscosity and the steam viscosity are temperature-dependent functions see Appendix A. As discussed in [4], the ater mass source term is taken as q T T S 1 for T T, 0 S 1; q = 3 0 otherise. This term is motivated y the idea that the condensation rate is determined y a driving force hich is proportional to its departure from equilirium S = 1 and T = T see also reference [11]. The value of q is considered very large. 1.2 The model equations The mass alance equation of liquid ater and steam read as follos: ϕρ S ϕρ g S g + ρ u + ρ gu g = q, 4 = q. 5 The rock porosity ϕ is assumed to e constant. We include longitudinal heat conduction, ut neglect heat losses to the surrounding rock, in the energy alance equation given elo. By our assumption of almost constant pressure e ignore adiaatic compression and decompression effects. Thus the energy alance is See reference [2], Tale 10.4-1: Hr + ϕs H + ϕs g H g + u H + u g H g = Here κ is the composite conductivity of the rock fluid system [1]: κ T. 6 κ = κ r + ϕ S κ + S g κ g. 7 Equations 4, 5, and 6 are the asic governing equations for the flo.
364 STEAM INJECTION INTO WATER-SATURATED POROUS ROCK Equations 4 and 5 are comined ith the heat alance equation 6, here e also use separation in sensile and latent quantities, to otain: Hr + ϕs H + ϕs g Hg s + u H + u g Hg s κ T = ϕsg H l g ug H l g = ϕsg ρ g 0 ug ρ g 0 = 0 ϕsg ρ g + ug ρ g. Using Eq. 5, this yields Hr + ϕs H + ϕs g Hg s + = q 0 + κ T. u H + u g H s g 8 Let us define the fractional flo functions for ater and steam: f = The capillary pressure k r /µ k r /µ + k rg /µ g, f g = k rg /µ g k r /µ + k rg /µ g. 9 P c = P c S = p g p 10 hich is given y Equation 83, is a strictly monotone decreasing function; it appears in the definition of the capillary diffusion coefficient : = f kk rg µ g dp c ds 0. 11 We notice that vanishes precisely at ater saturations S = S c and S = 1. Using Darcy s la 2 in the asence of gravitational effects and the definition of P c given in Eq. 10, one can easily sho from Eqs. 2 and 11 that: here u = uf S, u g = uf g S g, 12 u = u + u g 13
J. BRUINING, D. MARCHESIN and C.J. VAN DUIJN 365 is the total or Darcy velocity and acts as a saturation-dependent capillary diffusion coefficient. Sustituting 12 into Equations 4, 5 and 8 leads to ϕ ρ S ϕ ρ g S g + ρ uf + ρ g uf g = q + = q + Hr + ϕh S + ϕhg s S g + = q 0 + H H s g S ρ S ρ g S g u H f + Hg s f g + κ T., 14, 15 16 The governing system of equations is 14 16. As to initial conditions, e assume that the reservoir is filled ith ater at saturation S x, t = 0 = S 0 = 1 ith constant temperature Tx,t = 0 = T 0. As to oundary conditions, the total injection rate u inj is specified and constant see Appendix A. The constant steam-ater injection ratio is specified in terms of the ater saturation S inj at the injection side. Lemma 1. q = 0. In a region here the temperature is constant and noncritical, Proof. If the temperature is constant, the enthalpies are constant, so Eq. 8 ecomes ϕh S + ϕh s g S g u + H + H g s u g = q 0. 17 We regroup Eq. 17 and use the mass alance equations 4 and 5. Since the temperature is constant the densities are constant too, so Eq. 17 ecomes H ϕ S + u + Hg s ϕ S g + u g = q 0, H q H s g q = q 0 H, H s 18 g 0 q = 0. ρ ρ g ρ ρ g
366 STEAM INJECTION INTO WATER-SATURATED POROUS ROCK The term in parenthesis in Eq. 18 is minus the enthalpy per unit mass required to convert ater into steam and is therefore non-zero. Consequently e must have that q = 0. Summarizing, e can say that if the temperature is constant in space and time then there is no source term. Remark 1. It is easy to see that the source term q vanishes in regions here either i the temperature is constant, ii the gas saturation is zero, iii the ater saturation is zero. 2 The hyperolic frameork By ignoring capillary pressure and heat conduction diffusive effects, e are in the frameork of first order hyperolic conservation las; this frameork is useful to study the asic aves of the model. Throughout this section e assume that all fluids are in thermodynamic equilirium. Equations 14 and 15, the mass alance equation of liquid ater and steam comined ith Darcy s la read as follos: ϕρ S ϕρ g S g + ρ uf + ρ guf g = q, 19 = q. 20 When e add these equations, e otain the total ater conservation: ϕ ρ S + ρ g S g + uρ f + ρ g f g = 0. 21 Eq. 16 ecomes Hr + ϕh S + ϕhg s S g + uh f + Hg s f g = q 0, 22 or equivalently, as in Eq. 6 Hr + ϕh S + ϕh g S g + uh f + H g f g = 0. 23 Eqs. 21 and 22 ill e used for most of the analysis in this section.
J. BRUINING, D. MARCHESIN and C.J. VAN DUIJN 367 Remark 2. Notice that all speeds defined y Equations 21 and 22 are proportional to u. Thus e can choose any speed to parameterize all the other ones. Let us consider all regions here the mass transfer term vanishes. The mass transfer can vanish ecause of several reasons. Based on these reasons, e classify the regions in the folloing tale. Because the mass source term vanishes Eq. 3, e have the folloing zones in the reservoir: S \T T = T T<T S < 1 hot steam zone xxxxxxxxxxxxx S = 1 hot ater zone cold ater zone Tale 1 Classification according to mass source term. We call hot steam-ater region, or hot region, the hot steam zone together ith hot ater zone, here T = T. We call liquid ater region the hot ater zone together ith the cold ater zone. These regions overlap on the hot ater zone. Remark 3. There is no cold steam zone in Tale 1 ecause at thermodynamical equilirium steam cannot exist at a temperature loer than T. As e ill see, a configuration composed y sequential zones of hot steam, hot ater and cold ater is possile, counting aay from the injection point. At the first interface S = 1 is reached, hile at the second one T = T 0 is reached. A configuration containing only the hot steam zone and the cold ater zone is possile if e interpose the so called SCF, here oth saturation and temperature change aruptly. Lemma 2. The source term in the hot region and in the liquid ater region is zero. That the source term is zero in the hot region follos from Remark 1. That the source term is zero in the liquid region follos from the existence of a single phase and consequent asence of mass and energy transfer eteen phases.
368 STEAM INJECTION INTO WATER-SATURATED POROUS ROCK 2.1 The hot region This region starts ith the hot steam zone, here steam is injected at oiling temperature T. We claim that the Darcy velocity u given y Eq. 13 in the hot region is independent of position. To prove this fact e use equations 19, 20. As the temperature is T, the source term such as given in Eq. 3 vanishes and the densities are constant. We can divide Eqs. 19, 20 y the densities and add the resulting equations and otain our claim. Since the Darcy velocity u is a constant in space in the hot region and since in this ork e also take that u inj is constant in time, the temperature T and the Darcy velocity u are constant in this region. Thus Eq. 22 is satisfied trivially, and oth Eqs. 19 and 20 reduce to any of the to equivalent forms of the Buckley-Leverett prolem for steam and ater that follos: ϕ S + u f = 0, = 0. 24 This equation governs propagation in the hot steam zone, as long as steam and ater are oth present. The classical Oleĭnik construction [10], or equivalently, the fractional flo theory [12] descrie aves in this zone. We ill denote y vs the speed of propagation of saturation aves in the hot steam zone. It is otained from Eq. 24 as the characteristic speed: ϕ S g v s = v s S ; u = u ϕ + u f g f S S, 25 here T = T and e use the nomenclature f S = f S,T. A particular Buckley-Leverett shock for 24 turns out to play a relevant role, separating a mixture of steam and ater from pure ater, oth at oiling temperature. We call it the hot isothermal steam-ater shock or HISW shock eteen the state S,T,u containing steam and the + state 1,T,u containing ater at oiling temperature. It has speed vg, given y v g, = v g, S ; u = u ϕ f g S g S g = u ϕ 1 f S. 26 1 S Notice that, ecause f g S g = 0 = 0 and f g / S g S g = 0 = 0 from Eq. 9 and from the quadratic ehavior of the steam relative permeailities in the
J. BRUINING, D. MARCHESIN and C.J. VAN DUIJN 369 saturation given in Eq. 82, ith n g = 2 e otain: Similarly, for S S c from Eqs. 82 and 9, v g, S = 1; u = 0. 27 v g, = u > 0. 28 ϕ1 S Remark 4. It is easy to verify that vs in Eq. 25 is monotonously increasing in S hen S is less than Sinfl, the inflection ascissa of f, and monotonously decreasing hen S is larger than Sinfl. 2.2 Liquid ater region We recall that the liquid ater zone consists of the hot region, hich is also part of the hot region examined in Section 2.1, and of the cold ater zone. In the liquid ater zone there is no steam, so there is no mass transfer eteen steam and ater. So q = 0. Also, in the liquid region S = 1, so Eqs. 21 and 22 reduce to 2.2.1 Cooling contact discontinuity ϕ ρ + uρ uh Hr + ϕh + = 0, 29 = 0. 30 We ill assume that ρ and C p are essentially constant in the pressure and temperature region of interest. A more complete discussion can e found in [5]. Let us consider a temperature discontinuity from T to T 0, ith speed v,0 in the liquid ater eteen the hot left or upstream state S = 1,T = T,u and the cold right or donstream state S = 1,T = T 0,u 0. For such a cooling contact discontinuity, from Eqs. 29 and 30 one can otain the folloing Rankine-Hugoniot relation, here e denote y u and u 0 the Darcy velocities at the discontinuity sides corresponding to T and T 0 : v,0 = u0 ρ 0 u ρ ϕρ 0 ρ = u 0 H 0 u H Hr 0 + ϕh0 H r + 31 ϕh,
370 STEAM INJECTION INTO WATER-SATURATED POROUS ROCK here H = H T, H r = H r T, ρ = ρ T. 32 We recall that our convention is that enthalpies vanish at T 0 ; then the Rankine- Hugoniot condition can e reritten as v,0 = u0 ρ 0 u ρ ϕρ 0 ρ = u Hr +. 33 ϕh From the second equality in Eq. 33, e otain that H u = Hr + ϕh Hr ρ /ρ0 + u 0, 34 ϕh hich expresses the conservation of ater mass. From the last term in Eq. 33 and from Eq. 34: v,0 = H Hr ρ /ρ0 + u 0. 35 ϕh Remark 5. Notice that the dependence of ρ on temperature is often small. If ρ ere independent of temperature constant, then Eq. 34 ould imply that u = u 0. Remark 6. Since all speeds in this prolem scale ith u, and u inj is constant in time, u 0 and u are constant in time. Remark 7. In the hot ater zone, oth S = 1 and T = T,soq = 0. Since the temperature is constant, so is ρ, thus Eq. 29 says that u is a constant, hich has already een called u in Section 2.1. Equation 30 says that the characteristic speed of temperature aves in the hot ater zone is v = C p T C p r T + ϕc p T u. 36 This is the propagation speed of small temperature perturations near T = T in the hot ater zone.
J. BRUINING, D. MARCHESIN and C.J. VAN DUIJN 371 Remark 8. Under the assumptions that ρ and C p are constant in pressure and temperature, the characteristic speeds 36 evaluated at T, and evaluated at T 0 coincide ith the discontinuity speed 31. In gas dynamics, discontinuities ith this coincidence property are called contact discontinuities. Hence the name e gave to this ave. 2.3 Steam condensation front This is a discontinuity joining a state containing steam and ater at temperature T to pure ater at temperature T 0, a state +; that is, it separates the hot steam zone from the cold ater zone. It satisfies the folloing Rankine-Hugoniot conditions ith speed v SCF for Eqs. 21, 23 eteen states S,T,u and S 0 = 1,T0,u 0. From the ater alance 21 e otain: u ρ f + f g ρ g ϕv SCF ρ S + S g ρ g = u 0 ρ f + f g ρ g + ϕv SCF ρ S + S g ρ g + 37 and from the energy alance 23 e otain: u H f + H g f g v SCF H r + ϕh S + ϕh g S g = u 0 H f + H g f g + v SCF H r + ϕh S g + ϕh g S g +. 38 As no steam exists on the right of the SCF, e can say that S 0 = f 0 = 1 and Sg 0 = f g 0 = 0 and thus Eq. 37 ecomes u ρ f + ρ g f g ϕv SCF ρ S + ρ g S g = ρ 0 u 0 ϕv SCF. 39 Under the same conditions e otain for the heat alance equation 38: u H f + H g f g v SCF H r + ϕh S + ϕh g S g = u 0 H 0 vscf Hr 0 + 40 ϕh0 = 0. The RHS term of Eq. 40 vanishes in the asence of steam ecause of our convention for enthalpies, as far as rock and ater are concerned. For the SCF
372 STEAM INJECTION INTO WATER-SATURATED POROUS ROCK velocity it follos from Eqs. 39, 40 that ρ u 0 = u g f ρ 0 g + ρ f ρ 0 ρ ϕv SCF g S ρ 0 g + ρ S ρ 0 1, 41 H v SCF = u f + H g Bf g Hr + ϕh S +, 42 ϕhb g S g here e used the nomenclature that follos from Eq. 1: H B g H gt = H s g T + H l g = H g + 0 ρ 0 g. 43 Because Sg = 1 S, f g = 1 f and f depends only on the ater saturation in the constant temperature steam zone, e oserve that u 0 depends only on the ater saturation and the Darcy velocity at the left of the SCF as ell as on the velocity of the SCF. From Eqs. 41 and 42, e can rite u 0 in terms of u : u 0 u = ρ ρ 0 ϕ f + ρ g ρ ρ 0 f ρ 0 g S + ρ g S ρ 0 g 1 H f + H g Bf g Hr + ϕh S +. ϕhb g S g 44 Eqs. 42 and 44 represent the speeds v SCF and u 0 in terms of u. Eq. 44 easily [ ] allos to read u in terms of u 0 see Figure 2. We can use the expression u of 0 given y Eq. 44 in Eq. 42 to otain v SCF in terms of u 0 : u [ u v SCF = u 0 0 ] 1 H f + H g Bf g u Hr + ϕh S +. 45 ϕhb g S g Finally, e replace H B g in Eq. 45 y its definition given in Eq. 43. Remark 9. In principle, + states ith temperature T different from T 0 could e considered, ut ecause C p as assumed to e constant such condensation discontinuities do not appear in the Riemann solution.
J. BRUINING, D. MARCHESIN and C.J. VAN DUIJN 373 u 0 u I II III S S 1 * S Figure 2 Speed u versus S for fixed T 0, u 0, otained from Eq. 44. The curve is almost horizontal at S ;ifρ g and Hg B could e neglected, and ρ ere independent of temperature, tangency of the left and right curves at S ould e exact using Eq. 47. 2.4 Cold ater zone In the cold ater zone, S = 1, so q = 0. Since T = T 0 is constant, so is ρ. Thus Eq. 29 says that u is a constant that has een called u 0. Equation 30 says that the characteristic speed of temperature aves in the cold ater zone is v 0 = 3 Wave ifurcation analysis C p T 0 C p r T 0 + ϕc p T 0 u0. 46 Let us consider the situation here the hot steam zone is folloed y a cold ater zone. For such a situation to occur, there must e a steam condensation discontinuity in eteen. Let us first examine the critical case hen the speed of the condensation discontinuity is the same as the characteristic speed in the cold ater zone. 3.1 The hot-cold ifurcation Because speed equality of different aves typically represents resonance and generates ifurcations, let us consider the case hen the SCF speed is so large that it equals the cooling contact discontinuity speed. We expect this ifurcation
374 STEAM INJECTION INTO WATER-SATURATED POROUS ROCK to represent the oundary eteen configurations containing either SCF shocks or cooling discontinuities. Equating the cooling discontinuity speed v,0 from Eq. 33 or equivalently from 35 ith v SCF given y Eq. 45. Using Eq. 26, e conclude that e have the folloing remarkale speed equalities. Theorem 1. Fix T 0 and T or equivalently T 0 and the reservoir pressure. Consider the folloing three shocks: HISW shock eteen S,T,u, 1,T,u, cooling shock eteen 1,T,u, 1,T 0,u 0, SCF eteen S,T,u, S 0,T0,u 0, ith speeds vg, v0 and vscf respectively. If any to of their ave speeds coincide at a certain S = S, then their three speeds coincide at this S. Proof. The proof consists of three parts. The velocities are given in Eqs. 26, 35 and 42. 1 Assume that at S = S e have v g, = v,0. From the equality in speeds, Eqs. 26 and Eqs. 33 e have for S g = 1 S : u ϕ f g S g = u ϕ 1 f 1 S = u H Hr +. 47 ϕh Multiplying numerator and denominator of the second fraction in Eq. 47 y H and sutracting the results to the corresponding terms in the third fraction, e otain: u ϕ f g S g = u H f Hr +. 48 ϕh S Multiplying numerator and denominator of the first fraction in Eq. 48 y Hg B and adding the results to the corresponding terms in the second equation, e otain: u ϕ f g S g H = u f + H g Bf g Hr + ϕh S +. 49 ϕhb g S g From Eqs. 26 and 42, e see that v g, = vscf.
J. BRUINING, D. MARCHESIN and C.J. VAN DUIJN 375 2 Performing the aove calculation in reverse order, e can prove that v g, = v SCF implies v g, = vscf = v,0. 3 Assume that at S e have v,0 = vscf. From Eqs. 35 and 42, and assuming that ρ = ρ0, i.e. the density of ater is independent of temperature, v,0 = u H H r + ϕh H = u f + H g Bf g Hr + ϕh S +. 50 ϕhb g S g Sustituting f = 1 f g, S = 1 S g the last fraction in Eq. 50 e otain: in the numerator and denominator of v,0 = u H H r + ϕh H = u H f g + H g Bf g Hr + ϕh ϕh S g +. 51 ϕhb g S g Sutracting the numerator and denominator of the first fraction from the corresponding terms in the last fraction e otain: v,0 = u ϕ f g H + H B g S g H + H B g, 52 or, from Eq. 26, v,0 = v g,, and the proof is complete. The speed vg, is the Buckley-Leverett speed of propagation of a hot steam shock from S to S = 1 pure hot ater, or no steam governed y Eq. 24. Thus, each pair of states of this one-parameter family of discontinuities S,T,u, 1,T 0,u 0 acts as an organizing center in the space of all solutions of the Riemann prolem; the first memer S,T,u of each such pair is denoted y. This family of discontinuities is parameterized y u 0 for instance, as explained in Remark 2. Theorem 1 provides information related to the structure at the left of the temperature discontinuities. In Figure 3, S corresponds to the saturation of a state. See also Figure 4. The state separates to different configurations; in one of them, there is a hot steam zone and a cold ater zone separated y a SCF, hile in the other there is a hot steam zone, a hot ater zone and a cold ater zone.
376 STEAM INJECTION INTO WATER-SATURATED POROUS ROCK speed v * v SCF hot steam ater zone cold ater zone v g, v, 0 hot ater zone s s ater saturation * 1 s Figure 3 Schematic ifurcation diagram near S for fixed u 0, T 0 versus S, the saturation on the left of the HISW shock or of the SCF. 3.2 The steam-ater ifurcation Let us fix T 0 and T or equivalently T 0 and the reservoir pressure. Let us no examine the critical case hen the speed of the steam condensation discontinuity is so high that it ecomes the same as the characteristic speed of saturation aves in the hot region given in Eq. 25, so the SCF overtakes the cooling discontinuity. One can expect that the SCF cannot exist ith higher speed. At S, the SCF ecomes a left-contact. At such a state S,T,u = S,T,u,ehave: v s S ; u u ϕ f S S = v SCF. 53 It is easy to find numerically or graphically see Fig. 5 the solution of Eq. 53 using Eq. 42 and solve for S. Notice that u cancels out. Susequently e can use Equation 44 to calculate the donstream velocity u 0 in terms of v SCF. Equivalently, e can use Eqs. 44, 45 to otain u and v SCF in terms of u 0. Remark 10. If Figure 5 ere dran to scale for the actual data nothing ould e visile. The draing is for illustrative purposes. In the numerical example studied in detail, e find the folloing values: S = 1.32449, f = 0.00471, S infl = 0.884747, S = 0.308042, S = 0.991435, and S = 0.983167.
J. BRUINING, D. MARCHESIN and C.J. VAN DUIJN 377 f 1 f * S * S * 1 S Figure 4 Finding S from second equality in Eq. 47, y a solid line ith slope ϕh /H r + ϕh ; dashed ounding lines through 0, 0 and S, 0. f f 1 f S, f S S S * S infl 1 S Figure 5 Graphical solution of Eqs. 53 and 42 for steam-ater ifurcation. See Eq. 54. The solid line represents the SCF shock. Equating v SCF given y Eq. 53 ith Eq. 42, making f g = 1 f, S g = 1 S, e otain f S S = f S f S S, 54
378 STEAM INJECTION INTO WATER-SATURATED POROUS ROCK here S = H r /ϕ + H g B Hg B H, f = H B g H B g H, 55 and all quantities are evaluated at the oiling temperature T. The physics of ater at normal pressure dictates that at the oiling temperature, Hg B <H, thus S < 0 and f < 0. 56 Remark 11. Since for steam ater S,T satisfies the inequalities 56, there is another solution point S,f for Eq. 53 closer to 1,1 as shon in Fig. 5. Hoever, it does not play any role in the Riemann solution of the current prolem ecause it exceeds S, according to Remark 10. The contact ifurcation S separates different ave structures in the steamater zone as can e seen in Fig. 6. speed v SCF T constant + rarefaction v s v SCF constant I II III s s * v g, Figure 6 Structure of the steam-ater zone elo solid curve marked y vs,vscf T, v SCF, vg, given in Eqs. 25, 42 ith S =S, Eq. 42 ith S <S inj <S, and Eq. 26 respectively. The figure is not dran to scale. In Figures 11 and 2, e sho the characteristic speed v SCF and u for each S, at temperature T for fixed u 0. As e shall see in Section 4.2, the diagram in Fig. 11 determines the structure of the Riemann solution in the steam zone. 1 s inj
J. BRUINING, D. MARCHESIN and C.J. VAN DUIJN 379 Remark 12. By inspection of Figure 5, e see that slightly aove S there exist S + and slightly elo S there exist S, such that in the limit as S + = S e have v SCF S + = v SCF S larger than v SCF S, satisfying as ell S + S v SCF S + = u ϕ f S + f S S v SCF S = u ϕ f S f Sutracting these to equations, dividing y S + S and taking the limit as S +,S S e recover that u ϕ f S S = v SCF, 59 and otain that S maximizes v SCF, as illustrated in Figures 3, 6 and 11. An analogous argument holds at S. Remark 13. The SCF shocks are represented in Figure 5 as segments ith slope v SCF /u eteen S,f and S, f for 0 S S. We see that as S increases v SCF decreases and the shock amplitude S S increases. Remark 14. We have shon that v SCF /u has an extremum at S ; the Figure 5 shos that this slope has an extremum at S. Thus v SCF /u also has an extremum at S. 3.3 Waves in the liquid ater region Because the initial reservoir temperature is T 0, the liquid ater region must alays contain a cold ater zone at temperature T 0 far aay from the place here hot steam is injected. If the liquid ater region receives ater at temperature T from the steam zone, the liquid ater region consists of a hot liquid ater zone at temperature T and a cold ater zone at temperature T 0, separated y a cooling discontinuity that moves ith speed v,0 given y Eq. 35. This cooling discontinuity exists provided v,0 >v g, from Eq. 26 i.e. the hot isothermal steam-ater shock velocity HISW. In this case the HISW shock at hich
380 STEAM INJECTION INTO WATER-SATURATED POROUS ROCK the steam saturation ecomes zero and the cooling shock here the temperature jumps to the amient temperature are separated. See region S>S in Fig. 3. On the other hand, if v,0 >v g, ere to e violated, there ould e no hot ater zone and no cooling discontinuity. See region S<S in Fig. 3, here there is a steam condensation front instead of a cooling discontinuity. 3.4 Waves in the hot steam zone The aves in this zone can e found y a pure Buckley-Leverett or Oleĭnik analysis of Eq. 24, ith one caveat. In the sequence of zones starting at the injection ell, the first zone is a steam zone, and the last one is a cold ater zone, ith heat flo governed y the system 29 30. The cold ater zone is reached either via a steam condensation shock or via a cooling shock. In the latter case, if there is no other shock eteen the steam zone and the cold ater zone, all aves in the steam zone must have speeds that do not exceed the cooling shock speed v,0 given y Eq. 34. In particular, if there is a HISW shock ith speed given y vg, in Eq. 26, e must have v g, v,0 saturation rarefaction ave ith speed given y vs. Similarly, if there is a in Eq. 25, e must have vs v,0, the cooling contact discontinuity velocity. Because of Theorem 1, e see that the restrictions aove are satisfied precisely for saturation S in the hot ater zone ith values eteen [S, 1], see Figure 11. For steam ater at the conditions considered in this ork, one can verify that the steam ater ifurcation ater saturation S is smaller than S in Fig. 5. Because S <S, there are no Buckley-Leverett shocks eteen [S c,s ]. This is so ecause elo S there are no shocks as rarefaction ave velocities increase monotonically from S c to S.AtS the velocity is equal to the SCF velocity. Beteen S and S there are no shocks as the rarefaction ave velocities are larger than the SCF velocity. Another case of interest occurs if pure steam is injected, i.e. S = S c, the connate ater saturation. In this case, a saturation rarefaction ave starts at x = 0 in the steam zone. A mixture of steam and ater can also e injected. As long as S inj <S,atx = 0 there is a constant state folloed y a saturation rarefaction ave. Of course, in the Buckley-Leverett solution for the steam zone the rarefaction ave containing a saturation value S satisfies the geometric
J. BRUINING, D. MARCHESIN and C.J. VAN DUIJN 381 compatiility condition: u inj 4 Construction of the Riemann solution ϕ f S S v SCF. 60 Here e descrie a systematic ay of constructing the Riemann solution through a ave curve. Then e summarize the resulting Riemann solution. u u 0 I II III S S 1 * S Figure 7 Speed u 0 for fixed T, u inj, otained from Eq. 7 and Fig. 2. 4.1 The ave curve Let us fix the initial state of the reservoir as S 0 = 1, T = T 0, u = u 0, hich is necessarily the rightmost constant state in the Riemann solution. It turns out to e convenient for the discussion to imagine that an aritrary value for u 0 has een specified. Let us decrease the injection saturation S inj from S = 1toS = S c ; in our case this corresponds to changing S at the oiling temperature, the ater saturation at the HISW shock ith speed given in Eq. 26. For each S inj,e construct the sequence of elementary aves and constant states ith decreasing speeds from right to left, that is from S 0 to S. There is a constant state to the left of S, so there is no other ave to the left of S, the steam shock. For each S, e mark its ave speed, forming the solid curves in Fig. 11. This sequence, parameterized y S, is called a ackard ave curve from S = 1,T 0,u 0. It comines all information needed for descriing the structure of the Riemann solution, and for verifying necessary speed relations for the admissiility of the
382 STEAM INJECTION INTO WATER-SATURATED POROUS ROCK Figure 8 To cases of steam injection: at connate ater saturation S c or at a lo ater saturation. With injection at S c there is a rarefaction ave and a SCF. With injection of lo ater saturation aove S c there is a constant state efore the rarefaction. shocks involved. On rarefaction segments of this ackard ave curve, the characteristic speed decreases, hile on shock segments of this ave curve, the shock speed increases [8], [14]. We refer to Fig. 11. When S inj lies eteen S and 1, the ave ith fastest possile speed is the cooling shock in the liquid ater region see Section 3.4, so the ackard ave curve corresponds to such a shock ave, ith speed given y v,0 in Eq. 35. There is a hot steam-ater region and a cold ater zone. In the hot steam-ater region, generically there is a Buckley-Leverett rarefaction-shock and a constant state. The hot steam-ater region terminates ith the HISW shock. The analysis from no on relies on the fact that for steam ater in the actual reservoir, S infl <S.ForS inj ithin S,S, the ave ith sloest speed is the SCF, ith speed v SCF. At the left of the SCF, characteristic speed vs exceeds
J. BRUINING, D. MARCHESIN and C.J. VAN DUIJN 383 Figure 9 Steam injection at intermediate ater saturation. We get a constant state upstream and the steam condensation front. The SCF moves faster than in Figure 8. v SCF, so there can e no rarefactions nor shocks to the left of the SCF. Thus, for S inj in such range, the solution from left to right consists of a constant state of steam and ater at temperature T,aSCF jumping from S inj constant cold ater zone. Recall that S <S infl. Therefore, from S inj to 1.0, and a ithin S c,s, the sloest speed is a hot steam ater rarefaction speed v s, hich is smaller than vscf v s and v SCF coincide at the left state S, so there are rarefactions to the left of the SCF. Thus the solution consists of a constant state ith saturation S inj previous case this constant state disappears if S inj from S inj to S,aSCF from S to 1, and a constant cold ater zone. as in the = S c, a rarefaction ave Let us explain ho the Darcy velocity u is constructed for each value of S inj. If S inj lies in the interval S, 1, u is given y Eq. 34, and the speed v,0 of the cooling shock is given y Eq. 35 Case III. As explained in Section 2.1, the velocity u is constant in the hot region and equals u. If S inj lies in the
384 STEAM INJECTION INTO WATER-SATURATED POROUS ROCK Figure 10 Steam injection at a high ater saturation. We get a constant state upstream, the Buckley-Leverett-saturation shock and the cooling discontinuity. These to aves have distinct speeds; they are oth faster than the SCF in Figure 9. interval S and S, u is given y Eq. 44. The speed of the SCF is given y Eq. 45, Case II. If S inj lies in the interval S c,s Case I, since u has to match at the oundary of Cases I and II, u is given y Eq. 45 ith S replaced y S. Thus u is independent of S inj in Case I. A summary of the ehavior of u S inj is presented in Fig. 2. Here u S inj is the value of u calculated in the previous paragraphs for a fixed u 0. We are ready to aandon the assumption that u 0 is knon. This is impractical, since normally one specifies u inj rather than u 0. We take advantage of the fact that all speeds are proportional to find the actual speed in the cold ater zone
J. BRUINING, D. MARCHESIN and C.J. VAN DUIJN 385 speed v s v SCF v, 0 v, 0 I II III S S S S c * 1 III S infl S Figure 11 Wave speed diagram for left states S,T,u reachale from the right state 1,T 0,u 0 ; for this right state the curve marked v SCF is the Rankine-Hugoniot curve projected onto the S,vplane. The velocity of the SCF is v SCF Eq. 42, vs is the hot region saturation characteristic speed Eq. 25, and v,0 is the velocity of the cooling contact discontinuity Eq. 35. u 0 S inj for specified uinj, as follos: u S inj u 0 = uinj u 0 S inj. 61 From this equation e recover u 0 S inj, essentially y inverting the variale represented in the ordinate in Fig. 2. Thus e otain Fig. 7. 4.2 Summary of the Riemann solution The solution consists of three parts, viz. a hot region A at constant oiling temperature, an infinitesimally thin cooling front B or discontinuity, here all possile steam condensation occurs, and a cold liquid ater region donstream C. See Figure 1. As e have seen, the nature of the solution changes and there are three possile cases I, II, and III, depending on the injected steam quality S inj g = 1 S inj. Cases II and III are separated y the hot-cold ifurcation, hile Cases I and II are separated y the steam-ater ifurcation.
386 STEAM INJECTION INTO WATER-SATURATED POROUS ROCK Case I occurs hen the saturation ave velocity vs Sinj ; uinj v SCF see Eq. 25; it consists of a sequence of a constant state at the injection end, a rarefaction ave in the hot steam zone A ending ith saturation S at the SCF B ith speed v SCF defined y saturations S and S = 1, and a cold ater constant state in C. The constant state in A disappears if the injection saturation is S c, that is, pure steam is injected. See Fig. 8. The rarefaction disappears for vs Sinj ; uinj = v SCF, as in this case S inj = S. Case II occurs for vg, Sinj,uinj <v SCF <vs Sinj ; uinj ; see Eqs. 25, 26 and 27. This case consists of a hot constant steam ater state in A, the SCF B ith speed v SCF defined y left and right saturations S inj and 1 see Eq. 45, and a constant cold ater state in C. See Fig. 9. Case III occurs for a typically small region for the cooling contact discontinuity velocity v,0 given in Eq. 35 ith v,0 >v g, Sinj,uinj see Eq. 26, i.e. the hot isothermal steam-ater shock velocity. In this case, there is no SCF. In the hot region A there is a constant state ith steam-ater, then another constant state of pure hot ater at the same oiling temperature, separated y a Buckley-Leverett shock. Then there is a cooling shock ith speed v,0, here the saturation of ater is constant S = 1 and the temperature changes from T to the reservoir temperature T 0. See Fig. 10. Figure 6 illustrates the saturation dependence of the various velocities that are the asis of the steam-ater zone structure, for Cases I, II, III. 5 Lax conditions for the steam condensation front Despite the fact that our system does not satisfy Lax s theorem hypotheses, e ill compare the SCF speed to the left and right characteristic speeds. We ill conclude that from the point of vie of Lax s inequalities, the SCF is a 2-shock or a limit of such shocks. We introduce the heat capacities C p T as the temperature derivatives of the enthalpies [J/m 3 ] at constant pressure, i.e. C p T is the heat capacity of ater and Cg p T is the heat capacity of steam. In the same ay e define the thermal expansivity of ater and steam α T and α g T as minus the temperature derivative of the density divided y the density see Appendix A.
J. BRUINING, D. MARCHESIN and C.J. VAN DUIJN 387 Eqs. 21 22 may e ritten in quasilinear form as: u [ ρ f + ρ g f g = ϕ S ρ α + S g ρ g α g T + ] S f ρ g ρ u[ ρ α + f g ρ g α g + ρ g ρ f / T T ρ ρ g f S ] S, 62 q 0 u [ p C H f + Hg s f r g = ϕ [ f C p + H H s g S ] + u + H H s g f / T T + ϕ + S C p + S gc p g + f gcg p H Hg s f S T ] S. 63 We restrict our attention to regions here u/ = 0 and q = 0, that is, aay from any kind of shocks. Thus, the LHS terms of Eqs. 62, 63 vanish. We let A I = Cp r ϕ + S C p + S gcg p, A II = f C p + f gcg p + H Hg s f / T. 64 Multiplying the RHS of Eq. 62 y H Hg s and of Eq. 63 y ρ ρ g and adding leads to a ne equation, hich ill e used instead of Eq. 63: [ ϕ H Hg s S ρ α + S g ρ g α g + ] T ρ ρ g AI + u [H H sg f ρ α + f g ρ g α g + ρ g ρ f / T 65 We let + ρ ρ g A II ] T = 0. A III = H H s g S ρ α + S g ρ g α g + ρ ρ g AI, 66
388 STEAM INJECTION INTO WATER-SATURATED POROUS ROCK A IV = H Hg s f ρ α + f g ρ g α g + ρ g ρ f / T + ρ ρ g AII. 67 Thus in regions here u/ = 0 and q = 0, that is, aay from any kind of shocks see Remarks 1 and 2, Eqs. 62 65 may e ritten in matrix form as: A + B S = 0. 68 T Let µ e a characteristic speed. Then the determinant of the folloing matrix must vanish: ρg ρ S ρ α + S g ρ g α g µϕ 0 A III 69 ρg ρ f / S f ρ α + f g ρ g α g + ρ g ρ f / T +u. 0 A IV Since the matrix aove is upper triangular, the characteristic speeds are easily read from the diagonals: µ = u ϕ f and µ = u A IV. 70 S ϕ A III It is easy to check that A III never vanishes. No, in the liquid ater region on the right of the SCF, S g = 0, f g = 0, f S = 0, the characteristic speeds are µ = 0 and v 0 = C p T 0 C p r T 0 + ϕc p T 0 u0. 71 The latter speed has already een calculated in Eq. 46. On the other hand, in the hot steam zone the characteristic speeds are v s = u ϕ f, and vt S = u A IV T ϕ A III T. 72 The first speed has already een calculated in Eq. 45. The second speed of thermal aves is shon in Figure 11 as a function of S at T = T.
J. BRUINING, D. MARCHESIN and C.J. VAN DUIJN 389 For a state for the SCF ith S in S,S e have that the thermal characteristic speed in the steam zone satisfies 0 <v 0 <v SCF, and the steamfront velocity satisfies vt <vscf <vs, so the SCF ould e called a 2-shock in Lax s classification scheme. Hoever, Lax s theorem only applies to shocks ith small amplitude, hile the SCF is a large shock, and only if the governing equations ere a system of conservation las satisfying appropriate technical hypotheses, such as genuine nonlinearity, hich is actually violated at the inflection S infl. Moreover, even the Lax inequalities are violated starting at the steam-ater ifurcation; there is no conclusive mathematical evidence that the SCF shock needed to complete the Riemann solution is physically admissile. This is the issue left open. 6 Summary and conclusions A complete and systematic description of all possile solutions of the Riemann prolem for the injection of a mixture of steam and ater into a ater-saturated porous medium, for all possile reservoir temperatures and pressures elo the ater critical point. For each Riemann data, e found a unique solution. As determined y the dissipative effects of capillary porous forces comined ith the mass source term given in Eq. 3, the internal structure of the SCF is consistent ith the Riemann solution in this ork. This fact is demonstrated in a companion paper [4]. 7 Acknoledgments This ork as partially done at IMA, University of Minnesota, and therefore partially funded y NSF. H.B. thanks Shell for the continuous support of the steam drive recovery research at the Delft University of Technology. We also thank Beata Gundelach for careful and expert typesetting of this paper. We thank the referees for suggestions that improved the paper. Appendix A Physical quantities; symols and values In this Appendix e summarize the values and units of the various quantities used in the computation and empirical expressions for the various parameter
390 STEAM INJECTION INTO WATER-SATURATED POROUS ROCK Physical quantity Symol Value Unit Water, steam fractional functions f,f g Eq. 9. [m 3 /m 3 ] Water-steam frac. flo, hot region f, f g [m 3 /m 2 s] Porous rock permeaility k 1.0 10 12 [m 2 ] Water, steam relative permeailities k r,k rg Eq. 82, 82. [-] Pressure p 1.0135 10 5 [Pa] Mass condensation rate q Eq. 3. [kg /m 3 s] Mass condensation rate coefficient q 0.01 [kg /m 3 sk] Steam injection rate u inj 9.52 10 4 [m 3 /m 2 s] Water, steam phase velocity u,u g Eq. 2. [m 3 /m 2 s] Total Darcy velocity u u + u g, Eq. 13. [m 3 /m 2 s] Flo rate in hot region u Eq. 24, 34. [m 3 /m 2 s] Flo rate in cold ater zone u 0 Eq. 31. [m 3 /m 2 s] SCF velocity v SCF Eq. 42. [m/s] Cooling contact disc. speed, hot ater zone v Eq. 36. [m/s] Thermal characteristic speed, cold ater zone v 0 Eq. 46. [m/s] Saturation characteristic speed, hot region vs Eq. 25. [m/s] Hot isothermal steam-ater shock velocity vg, Eq. 26. [m/s] Cooling contact discontinuity velocity v,0 Eq. 35. [m/s] Water, steam heat capacity C,C p g p dh /dt, dhg s/dt [J/m3 K] Effective rock heat capacity Cr p 2.029 10 6 [J/m 3 K] Steam enthalpy H g ρ g T h g T h T 0 [J/m 3 ] Steam sensile heat Hg s Eq. 77. [J/m 3 ] Steam latent heat Hg l ρ g T 0 [J/m 3 ] Rock enthalpy H r ρ r Cr p T T 0 [J/m 3 ] Water enthalpy H ρ T h T h T 0 [J/m 3 ] Water, rock enthalpy at oiling temperature H, H r H T, H r T [J/m 3 ] Steam total, sensile enthalpy at oil. temp. Hg B, H g Eq. 43. [J/m 3 ] Tale 2 Summary of physical input parameters and variales.
J. BRUINING, D. MARCHESIN and C.J. VAN DUIJN 391 Physical quantity Symol Value Unit Water, rock enthalpy, reservoir temperature H 0, H r 0 H T 0, H r T 0 [J/m 3 ] Water, steam saturations S,S g Dependent variales. [m 3 /m 3 ] Connate ater saturation S c 0.15 [m 3 /m 3 ] Water injection saturation S inj See Section 4. [m 3 /m 3 ] Hot-cold ifurcation ater saturation S Theorem 1. [m 3 /m 3 ] Steam-ater ifurcation ater saturation S Eq. 53. [m 3 /m 3 ] Steam-ater ifurcation ghost saturation S See Remark 11. [m 3 /m 3 ] Water saturation at inflection S infl Frac. flo infl. sat. [m 3 /m 3 ] Temperature T Dependent variale. [K] Reservoir temperature T 0 293. [K] Boiling point of ater steam T Eq. 73. [K] Steam thermal expansion coefficient α g 1/ρ g ρ g / T p [/K] Water thermal expansion coefficient α 1/ρ ρ / T p [/K] Water, steam thermal conductivity κ, κ g 0.652, 0.0208 [W/mK] Rock, composite thermal conductivity κ r, κ 1.83, Eq. 7. [W/mK] Saturation exponent for P c λ s 0.5, Eq. 83 [-] Water, steam viscosity µ, µ g Eq. 79, Eq. 78. [Pa s] Water, steam densities ρ, ρ g Eq. 81, Eq. 80. [kg/m 3 ] Water, steam, rock densities oiling temp. ρ, ρ g, ρ r Eq. 32 [kg/m 3 ] Water, steam, rock densities reservoir temp. ρ 0, ρ0 g, ρ0 r [kg/m 3 ] Interfacial tension σ g 58 10 3 [N/m] Rock porosity ϕ 0.38 [m 3 /m 3 ] Water evaporation heat at reservoir temperature 0 Eq. 1. [J/kg] Capillary diffusion coefficient Eqs. 11, 12. [m 3 /m 3 ] Tale 2 continuation
392 STEAM INJECTION INTO WATER-SATURATED POROUS ROCK functions. For convenience e express the heat capacity of the rock C p r in terms of energy per unit volume of porous medium per unit temperature i.e. the factor 1 ϕ is already included in the rock density. All other densities are expressed in terms of mass per unit volume of the phase. A.1 Temperature dependent properties of steam and ater We use reference [16] to otain all the temperature dependent properties elo. The ater and steam densities used to otain the enthalpies are defined at the ottom. First e otain the oiling point T at the given pressure p, i.e. T = 280.034 + l14.0856 + l1.38075 + l 0.101806 + 0.019017l, 73 here l = logp and p is the pressure in [k Pa]. The evaporation heat [J/kg] is given as a function of the temperature T at hich the evaporation occurs. We use atmospheric pressure p = 101.325 [k Pa] in our computations, to make the example representative of susurface contaminant cleaning. The liquid ater enthalpy h T [J/kg] as a function of temperature is approximated y h T = 2.36652 10 7 3.66232 10 5 T + 2.26952 10 3 T 2 7.30365T 3 + 1.30241 10 2 T 4 1.22103 10 5 T 5 + 4.70878 10 9 T 6. 74 The steam enthalpy h g [J/kg] as a function of temperature is approximated y h g = 2.20269 10 7 + 3.65317 10 5 T 2.25837 10 3 T 2 + 7.3742T 3 1.33437 10 2 T 4 + 1.26913 10 5 T 5 4.9688 10 9 T 6. 75 For the latent heat h l g [J/kg] or evaporation heat T e otain h l g = 7.1845 10 12 + 1.10486 10 10 T 8.8405 10 7 T 2 + 1.6256 10 5 T 3 121.377T 4 1 2. 76 The sensile heat of steam Hg st in [J/m3 ]isgivenas Hg s T = ρ g hg T h T 0 T 0. 77
J. BRUINING, D. MARCHESIN and C.J. VAN DUIJN 393 We also use the temperature dependent steam viscosity µ g = 5.46807 10 4 + 6.89490 10 6 T 3.39999 10 8 T 2 + 8.29842 10 11 T 3 9.97060 10 14 T 4 + 4.71914 10 17 T 5. 78 The temperature dependent ater viscosity µ is approximated y µ = 0.0123274 + 27.1038 T 2.17342 109 1.86935 1011 +. T 4 T 5 23527.5 1.01425 107 + T 2 T 3 79 For the steam density as a function of temperature T [K] e use a different expression than [16] ecause our interest is a steam density at constant pressure, hich is not necessarily in equilirium ith liquid ater. ρ g T = p M H 2 O 80 ZRT here p is the total pressure at hich the steam displacement is carried out, R=8.31 [J/mol K] and Z is the Z-factor see e.g. Dake [6] and M H2 O = 0.018 kg/mole is the molar eight of ater. For the atmospheric pressures of interest here the Z-factor is close to unity. The liquid ater density as a function of the temperature T [K] is given as ρ T = 3786.31 37.2487T + 0.196246T 2 5.04708 10 4 T 3 + 6.29368 10 7 T 4 3.08480 10 10 T 5. 81 A.2 Constitutive relations We use a porosity ϕ that is representative for unconsolidated sand. The relative permeaility functions k r and k rg are considered to e poer functions of their respective effective saturations [7], i.e. S e = S S c /1 S c, S ge = S g /1 S c. The effective saturations require knoing the connate ater saturation S c.in all our examples e use a fourth poer of the effective saturation for the relative