Steam injection into water-saturated porous rock

 

 

J. Bruining^I; D. Marchesin^II; C.J. Vn Duijn^III

^IDietz Laboratory, Centre of Technical Geoscience, Mijnbouwstraat 120, 2628 RX Delft, The Netherlands, E-mail: J.Bruining@mp.tudelft.nl
^IIInstituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, RJ, Brazil, E-mail: marchesi@impa.br
^IIITechnische Universiteit Eindhoven, Den Dolech 2, 5600 MB Eindhoven, The Netherlands, E-mail: c.j.v.duijn@tue.nl

 

 

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ABSTRACT

We formulate conservation laws governing steam injection in a linear porous medium containing water. Heat losses to the outside are neglected. We find a complete and systematic description of all solutions of the Riemann problem for the injection of a mixture of steam and water into a water-saturated porous medium. For ambient pressure, there are three kinds of solutions, depending on injection and reservoir conditions. We show that the solution is unique for each initial data.

Mathematical subject classification: 76S05, 35L60, 35L67.

Key words: porous medium, steamflood, travelling waves, multiphase flow.

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Introduction

Steam injection is an effective technique to restore groundwater aquifers contaminated with non-aqueous phase liquids (NAPL's) such as hydrocarbon fuels and halogenated hydrocarbons [15]. It is also one of the most effective methods to recover oil from medium to heavy oil reservoirs [13]. The main feature of steam injection is the steam condensation front (SCF), which marks the boundary between the upstream zone at boiling temperature and the downstream liquid zone below the boiling temperature. Depending on the situation there may exist an isothermal steam-water shock at the boiling temperature (HISW) instead of the SCF. The main result of this work is a complete and systematic classification of the structure of all possible cases of Riemann solutions. As a first step we 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 reviewed in [4], [5].

In this article we limit ourselves to the simple case of steam displacing water. Our aim is to investigate a unique well posed solution of the Riemann problem for all possible values of the model parameters, providing mathematical validation of our model. This is the first step towards solving the full problem of groundwater NAPL removal.

In Section 1, the physical model is presented. It is described mathematically by balance equations of mass and thermal energy, which are rewritten into a form suitable for analysis.

Section 2 presents the basic waves arising in the model; the main concern is to identify their speeds, so as to be able to find the order in which they may appear in a linear steam injection experiment. In Section 3, we see that for certain values of initial and boundary data, some of these speeds coincide, giving rise to bifurcation and structural change in the Riemann solution. All solutions of the Riemann problem are in Section 4. Section 5 verifies that the SCF satisfies Lax's shock inequalities, but not strictly. Section 6 summarizes our results and conclusions.

Appendix A describes notation and values for the physical quantities appearing in the model.

 

1 Physical and mathematical model

1.1 Physical model

We consider linear steam displacement in a homogeneous reservoir of constant permeability and porosity. The reservoir is initially saturated with water. The pressure gradients ¶p_w/¶x, ¶p_g/ ¶x driving the fluids are small with respect to the prevailing system pressure p divided by the length of the reservoir. In particular, within the short steam condensation zone pressure variations are negligible. Hence we 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-water mixture is injected at constant rate u^inj and constant steam/water injection ratio. Transverse heat losses are disregarded. We neglect capillary forces after steam breakthrough at the production end of the reservoir to avoid problems with the capillary end effect, which is outside the present scope of our interest.

The effects of temperature on the fluid properties, e.g. water viscosity m_w, steam viscosity m_g, water density r_w and steam density r_g are taken into account. Darcy's Law determines the fluid motion. The temperature dependence of heat capacities and of the evaporation heat are also taken into account. Capillary pressure as well as an effective longitudinal heat conduction term are included.

We have chosen to describe condensation in terms of a steam mass condensation rate equation. The mass condensation rate q is always positive when the temperature drops below the boiling temperature T^b as long as not all steam has condensed, that is S_w < 1.

The stated conditions can be considered representative of steam injection in the subsurface for remediation of contaminated sites. As steam is injected the reservoir is heated. Depending on the proportions of steam and water in the injected mixture, we can distinguish three regimes, which differ in the structure of Riemann solutions. When pure steam is injected, there will be a decrease of the steam saturation in the hot zone away from the injection point, described as a rarefaction wave, and then a SCF to the cold water, described as a shock with a concentrated source term. This is called situation I. As the water injection rate is increased further a zone in which the steam saturation is constant will develop preceding the rarefaction wave until the steam saturation in the hot zone is constant. After this constant state there is a SCF and a cold water region. This will be called situation II. Finally when the water-steam injection ratio is increased further, the steam bank will not be fast enough to reach the cooling front separating the hot and cold water 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, whose boundary moves with constant speed, as shown in Fig. 1.

 

 

Each of the enthalpies per unit volume H_w(T), H_r(T), H_g(T) ([J/m³]) is defined with respect to the enthalpy at the initial reservoir temperature T⁰ at the standard state. This means that they are all zero at the initial temperature T⁰. The enthalpy of steam is subdivided in a sensible part (T) and a latent part (T⁰), i.e. H_g(T) = (T) + (T⁰). The sensible heat (T⁰) is zero at the initial reservoir temperature. The evaporation heat or the latent heat per unit mass at the initial reservoir temperature T⁰ is denoted by

In general L(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.

We assume Darcy's law for two-phase flow, water and steam respectively, without gravity terms:

The liquid water viscosity and the steam viscosity are temperature-dependent functions (see Appendix A).

As discussed in [4], the water mass source term is taken as

This term is motivated by the idea that the condensation rate is determined by a ''driving force'' which is proportional to its departure from equilibrium S_w = 1 and T = T^b ( see also reference [11]). The value of q_b is considered very large.

1.2 The model equations

The mass balance equation of liquid water and steam read as follows:

The rock porosity j is assumed to be constant. We include longitudinal heat conduction, but neglect heat losses to the surrounding rock, in the energy balance equation given below. By our assumption of almost constant pressure we ignore adiabatic compression and decompression effects. Thus the energy balance is (See reference [2], Table 10.4-1):

Here k is the composite conductivity of the rock-fluid system [1]:

Equations (4), (5), and (6) are the basic governing equations for the flow.

Equations (4) and (5) are combined with the heat balance equation (6), where we also use separation in sensible and latent quantities, to obtain:

Using Eq. (5), this yields

Let us define the fractional flow functions for water and steam:

The capillary pressure

which is given by Equation (83), is a strictly monotone decreasing function; it appears in the definition of the capillary diffusion coefficient W:

We notice that W vanishes precisely at water saturations S_w = S_wc and S_w = 1.

Using Darcy's law (2) (in the absence of gravitational effects) and the definition of P_c given in Eq. 10, one can easily show from Eqs. (2) and (11) that:

where

is the total or Darcy velocity and W acts as a saturation-dependent capillary diffusion coefficient.

Substituting (12) into Equations (4), (5) and (8) leads to

The governing system of equations is (14)-(16).

As to initial conditions, we assume that the reservoir is filled with water at saturation S_w(x, t = 0) = = 1 with constant temperature T(x, t = 0) = T⁰. As to boundary conditions, the total injection rate u^inj is specified and constant (see Appendix A). The constant steam-water injection ratio is specified in terms of the water saturation at the injection side.

Lemma 1. In a region where the temperature is constant (and noncritical), q = 0.

Proof. If the temperature is constant, the enthalpies are constant, so Eq. (8) becomes

We regroup Eq. (17) and use the mass balance equations (4) and (5). Since the temperature is constant the densities are constant too, so Eq. (17) becomes

The term in parenthesis in Eq. (18) is minus the enthalpy per unit mass required to convert water into steam and is therefore non-zero. Consequently we must have that q = 0. Summarizing, we 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 where either (i) the temperature is constant, (ii) the gas saturation is zero, (iii) the water saturation is zero.

 

2 The hyperbolic framework

By ignoring capillary pressure and heat conduction diffusive effects, we are in the framework of first order hyperbolic conservation laws; this framework is useful to study the basic waves of the model. Throughout this section we assume that all fluids are in thermodynamic equilibrium. Equations (14) and (15), the mass balance equation of liquid water and steam combined with Darcy's law read as follows:

When we add these equations, we obtain the total water conservation:

Eq. (16) becomes

or equivalently, as in Eq. (6)

Eqs. (21) and (22) will be used for most of the analysis in this section.

Remark 2. Notice that all speeds defined by Equations (21) and (22) are proportional to u. Thus we can choose any speed to parameterize all the other ones.

Let us consider all regions where the mass transfer term vanishes. The mass transfer can vanish because of several reasons. Based on these reasons, we classify the regions in the following table. Because the mass source term vanishes (Eq. (3)), we have the following zones in the reservoir:

 

 

We call ''hot steam-water region'', or ''hot region'', the hot steam zone together with hot water zone, where T = T^b. We call ''liquid water region'' the hot water zone together with the cold water zone.

These regions overlap on the hot water zone.

Remark 3. There is no ''cold steam zone'' in Table 1 because at thermodynamical equilibrium steam cannot exist at a temperature lower than T^b.

As we will see, a configuration composed by sequential zones of hot steam, hot water and cold water is possible, counting away from the injection point. At the first interface S_w = 1 is reached, while at the second one T = T⁰ is reached.

A configuration containing only the hot steam zone and the cold water zone is possible if we interpose the so called SCF, where both saturation and temperature change abruptly.

Lemma 2. The source term in the hot region and in the liquid water region is zero. That the source term is zero in the hot region follows from Remark 1. That the source term is zero in the liquid region follows from the existence of a single phase and consequent absence of mass and energy transfer between phases.

2.1 The hot region

This region starts with the hot steam zone, where steam is injected at boiling temperature T^b. We claim that the Darcy velocity u (given by Eq. (13)) in the hot region is independent of position. To prove this fact we use equations (19), (20). As the temperature is T^b, the source term (such as given in Eq. (3)) vanishes and the densities are constant. We can divide Eqs. (19), (20) by the densities and add the resulting equations and obtain our claim.

Since the Darcy velocity u is a constant in space in the hot region and since in this work we also take that u^inj is constant in time, the temperature T^b and the Darcy velocity u^b are constant in this region. Thus Eq. (22) is satisfied trivially, and both Eqs. (19) and (20) reduce to any of the two equivalent forms of the Buckley-Leverett problem for steam and water that follows:

This equation governs propagation in the hot steam zone, as long as steam and water are both present. The classical Olenik construction [10], or equivalently, the fractional flow theory [12] describe waves in this zone.

We will denote by the speed of propagation of saturation waves in the hot steam zone. It is obtained from Eq. (24) as the characteristic speed:

where T = T^b and we use the nomenclature (S_w) = f_w(S_w, T_b).

A particular Buckley-Leverett shock for (24) turns out to play a relevant role, separating a mixture of steam and water from pure water, both at boiling temperature. We call it the hot isothermal steam-water shock or HISW shock between the (–) state (, T^b, u^b) containing steam and the (+) state (1, T^b, u^b) containing water at boiling temperature. It has speed given by

Notice that, because f_g(S_g = 0) = 0 and (¶f_g/¶S_g)(S_g = 0) = 0 from Eq. (9) and from the quadratic behavior of the steam relative permeabilities in the saturation given in Eq. (82), with n_g = 2 we obtain:

Similarly, for < S_wc from Eqs. (82) and (9),

Remark 4. It is easy to verify that in Eq. (25) is monotonously increasing in when is less than S^infl, the inflection abscissa of f_w, and monotonously decreasing when is larger than S^infl.

2.2 Liquid water region

We recall that the liquid water zone consists of the hot region, which is also part of the hot region examined in Section 2.1, and of the cold water zone.

In the liquid water zone there is no steam, so there is no mass transfer between steam and water. So q = 0. Also, in the liquid region S_w = 1, so Eqs. (21) and (22) reduce to

2.2.1 Cooling contact discontinuity

We will assume that r_w and are essentially constant in the pressure and temperature region of interest. A more complete discussion can be found in [5].

Let us consider a temperature discontinuity from T^b to T⁰, with speed in the liquid water between the hot left (or upstream) state (S_w = 1, T = T^b, u^b) and the cold right (or downstream) state (S_w = 1, T = T⁰, u⁰). For such a cooling contact discontinuity, from Eqs. (29) and (30) one can obtain the following Rankine-Hugoniot relation, where we denote by u^b and u⁰ the Darcy velocities at the discontinuity sides corresponding to T^b and T⁰:

where

We recall that our convention is that enthalpies vanish at T⁰; then the Rankine-Hugoniot condition can be rewritten as

From the second equality in Eq. (33), we obtain that

which expresses the conservation of water mass.

From the last term in Eq. (33) and from Eq. (34):

Remark 5. Notice that the dependence of r_w on temperature is often small. If r_w were independent of temperature (constant), then Eq. (34) would imply that u^b = u⁰.

Remark 6. Since all speeds in this problem scale with u, and u^inj is constant in time, u⁰ and u^b are constant in time.

Remark 7. In the hot water zone, both S_w = 1 and T = T^b, so q = 0. Since the temperature is constant, so is r_w, thus Eq. (29) says that u is a constant, which has already been called u^b in Section 2.1. Equation (30) says that the characteristic speed (of temperature waves) in the hot water zone is

This is the propagation speed of small temperature perturbations near T = T^b in the hot water zone.

Remark 8. Under the assumptions that r_w and are constant in pressure and temperature, the characteristic speeds (36) evaluated at T^b, and evaluated at T⁰ coincide with the discontinuity speed (31). In gas dynamics, discontinuities with this coincidence property are called contact discontinuities. Hence the name we gave to this wave.

2.3 Steam condensation front

This is a discontinuity joining a state (–) containing steam and water at temperature T^b to pure water at temperature T⁰, a state (+); that is, it separates the hot steam zone from the cold water zone. It satisfies the following Rankine-Hugoniot conditions with speed v^SCF for Eqs. (21), (23) between states (, T^b, u^b) and ( = 1, T⁰, u⁰). From the water balance (21) we obtain:

and from the energy balance (23) we obtain:

As no steam exists on the right of the SCF, we can say that = = 1 and = = 0 and thus Eq. (37) becomes

Under the same conditions we obtain for the heat balance equation (38):

The RHS term of Eq. (40) vanishes in the absence of steam because of our convention for enthalpies, as far as rock and water are concerned. For the SCF velocity it follows from Eqs. (39), (40) that

where we used the nomenclature that follows from Eq. (1):

Because = 1 – , = 1 – and depends only on the water saturation in the constant temperature steam zone, we observe that u⁰ depends only on the water saturation and the Darcy velocity at the left of the SCF as well as on the velocity of the SCF.

From Eqs. (41) and (42), we can write u⁰ in terms of u^b:

Eqs. (42) and (44) represent the speeds v^SCF and u⁰ in terms of u^b. Eq. (44) easily allows to read u^b in terms of u⁰ (see Figure 2). We can use the expression of given by Eq. (44) in Eq. (42) to obtain v^SCF in terms of u⁰:

 

 

Finally, we replace in Eq. (45) by its definition given in Eq. (43).

Remark 9. In principle, (+) states with temperature T different from T⁰ could be considered, but because was assumed to be constant such condensation discontinuities do not appear in the Riemann solution.

2.4 Cold water zone

In the cold water zone, S_w = 1, so q = 0. Since T = T⁰ is constant, so is r_w. Thus Eq. (29) says that u is a constant that has been called u⁰. Equation (30) says that the characteristic speed (of temperature waves) in the cold water zone is

 

3 Wave bifurcation analysis

Let us consider the situation where the hot steam zone is followed by a cold water zone. For such a situation to occur, there must be a steam condensation discontinuity in between. Let us first examine the critical case (*) when the speed of the condensation discontinuity is the same as the characteristic speed in the cold water zone.

3.1 The hot-cold bifurcation

Because speed equality of different waves typically represents resonance and generates bifurcations, let us consider the case when the SCF speed is so large that it equals the cooling contact discontinuity speed. We expect this bifurcation to represent the boundary between configurations containing either SCF shocks or cooling discontinuities. Equating the cooling discontinuity speed (from Eq. (33) or equivalently from (35)) with v^SCF given by Eq. (45). Using Eq. (26), we conclude that we have the following remarkable speed equalities.

Theorem 1. Fix T⁰ and T^b (or equivalently T⁰ and the reservoir pressure). Consider the following three shocks: HISW shock between (, T^b, u^b), (1, T^b, u^b), cooling shock between (1, T^b, u^b), (1, T⁰, u⁰), SCF between (, T^b, u^b), (,T⁰,u⁰), with speeds , and v^SCF respectively. If any two of their wave speeds coincide at a certain = 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_* we have = .

From the equality in speeds, Eqs. (26) and Eqs. (33) we have for = 1 – S_*:

Multiplying numerator and denominator of the second fraction in Eq. (47) by and subtracting the results to the corresponding terms in the third fraction, we obtain:

Multiplying numerator and denominator of the first fraction in Eq. (48) by and adding the results to the corresponding terms in the second equation, we obtain:

From Eqs. (26) and (42), we see that = v^SCF.

(2) Performing the above calculation in reverse order, we can prove that = v^SCF implies = v^SCF = .

(3) Assume that at S_* we have = v^SCF.

From Eqs. (35) and (42), and assuming that = , i.e. the density of water is independent of temperature,

Substituting = 1 – , = 1 – in the numerator and denominator of the last fraction in Eq. (50) we obtain:

Subtracting the numerator and denominator of the first fraction from the corresponding terms in the last fraction we obtain:

or, from Eq. (26), = , and the proof is complete.

The speed is the Buckley-Leverett speed of propagation of a hot steam shock from to S_w = 1 (pure hot water, or no steam) governed by Eq. (24). Thus, each pair of states of this one-parameter family of discontinuities (S_*, T^b, u^b), (1, T⁰, u⁰) acts as an organizing center in the space of all solutions of the Riemann problem; the first member (S_*, T^b, u^b) of each such pair is denoted by *. This family of discontinuities is parameterized by u⁰ 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 two different configurations; in one of them, there is a hot steam zone and a cold water zone separated by a SCF, while in the other there is a hot steam zone, a hot water zone and a cold water zone.

 

 

 

 

3.2 The steam-water bifurcation

Let us fix T⁰ and T^b (or equivalently T⁰ and the reservoir pressure). Let us now examine the critical case (†) when the speed of the steam condensation discontinuity is so high that it becomes the same as the characteristic speed of saturation waves in the hot region given in Eq. (25), so the SCF overtakes the cooling discontinuity. One can expect that the SCF cannot exist with higher speed. At S_f, the SCF becomes a left-contact. At such a state (S, T, u) = (S_†, T^b, u^b), we have:

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^b cancels out. Subsequently we can use Equation (44) to calculate the downstream velocity u⁰ in terms of v^SCF. Equivalently, we can use Eqs. (44), (45) to obtain u^b and v^SCF in terms of u⁰.

 

 

Remark 10. If Figure 5 were drawn to scale for the actual data nothing would be visible. The drawing is for illustrative purposes. In the numerical example studied in detail, we find the following values: S^† = –1.32449, f^† = –0.00471, S^infl = 0.884747, S_† = 0.308042, S_†† = 0.991435, and S_* = 0.983167.

Equating v^SCF given by Eq. (53) with Eq. (42), making = 1 – , S_g = 1 – S_w, we obtain

where

and all quantities are evaluated at the boiling temperature T^b. The physics of water at normal pressure dictates that at the boiling temperature, < , thus S^† < 0 and

Remark 11. Since for steam-water (S^†, T^†) satisfies the inequalities (56), there is another solution point (S_††, ) for Eq. (53) closer to (1,1) as shown in Fig. 5. However, it does not play any role in the Riemann solution of the current problem because it exceeds S_*, according to Remark 10.

The contact bifurcation S_† separates different wave structures in the steam-water zone as can be seen in Fig. 6.

 

 

In Figures 11 and 2, we show the characteristic speed v^SCF and u^b for each , at temperature T^b for fixed u⁰. As we shall see in Section 4.2, the diagram in Fig. 11 determines the structure of the Riemann solution in the steam zone.

 

 

 

 

 

 

 

 

 

 

Remark 12. By inspection of Figure 5, we see that slightly above S_† there exist S₊ and slightly below S_† there exist S_–, such that in the limit as S₊ = S_– we have v^SCF(S₊) = v^SCF(S_–) larger than v^SCF(S_†), satisfying as well

Subtracting these two equations, dividing by (S₊ – S_–) and taking the limit as S₊, S_– ® S_† we recover that

and obtain 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 with slope (v^SCF/u^b) between (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 shown that (v^SCF/u^b) has an extremum at S_†; the Figure 5 shows that this slope has an extremum at S_††. Thus (v^SCF/u^b) also has an extremum at S_††.

3.3 Waves in the liquid water region

Because the initial reservoir temperature is T⁰, the liquid water region must always contain a cold water zone at temperature T⁰ far away from the place where hot steam is injected. If the liquid water region receives water at temperature T^b from the steam zone, the liquid water region consists of a hot liquid water zone at temperature T^b and a cold water zone at temperature T⁰, separated by a cooling discontinuity that moves with speed given by Eq. (35). This cooling discontinuity exists provided > from Eq. (26) i.e. the hot isothermal steam-water shock velocity (HISW). In this case the HISW shock at which the steam saturation becomes zero and the cooling shock where the temperature jumps to the ambient temperature are separated. See region S > S_* in Fig. 3.

On the other hand, if > were to be violated, there would be no hot water zone and no cooling discontinuity. See region S < S_* in Fig. 3, where there is a steam condensation front instead of a cooling discontinuity.

3.4 Waves in the hot steam zone

The waves in this zone can be found by a pure Buckley-Leverett or Olenik analysis of Eq. (24), with one caveat. In the sequence of zones starting at the injection well, the first zone is a steam zone, and the last one is a cold water zone, with heat flow governed by the system (29)-(30). The cold water zone is reached either via a steam condensation shock or via a cooling shock. In the latter case, if there is no other shock between the steam zone and the cold water zone, all waves in the steam zone must have speeds that do not exceed the cooling shock speed given by Eq. (34). In particular, if there is a HISW shock with speed given by in Eq. (26), we must have < . Similarly, if there is a saturation rarefaction wave with speed given by in Eq. (25), we must have < , the cooling contact discontinuity velocity.

Because of Theorem 1, we see that the restrictions above are satisfied precisely for saturation in the hot water zone with values between [S_*, 1], see Figure 11. For steam-water at the conditions considered in this work, one can verify that the steam water bifurcation water saturation S_† is smaller than S_* in Fig. 5. Because S_† < S_*, there are no Buckley-Leverett shocks between [S_wc, S_*]. This is so because below S_† there are no shocks as rarefaction wave velocities increase monotonically from S_wc to S_†. At S_† the velocity is equal to the SCF velocity. Between S_† and S_* there are no shocks as the rarefaction wave velocities are larger than the SCF velocity.

Another case of interest occurs if pure steam is injected, i.e. S_w = S_wc, the connate water saturation. In this case, a saturation rarefaction wave starts at x = 0 in the steam zone. A mixture of steam and water can also be injected. As long as < S_†, at x = 0 there is a constant state followed by a saturation rarefaction wave. Of course, in the Buckley-Leverett solution for the steam zone the rarefaction wave containing a saturation value S_w satisfies the geometric compatibility condition:

 

4 Construction of the Riemann solution

Here we describe a systematic way of constructing the Riemann solution through a wave curve. Then we summarize the resulting Riemann solution.

4.1 The wave curve

Let us fix the initial state of the reservoir as = 1, T = T⁰, u = u⁰, which is necessarily the rightmost constant state in the Riemann solution. It turns out to be convenient for the discussion to imagine that an arbitrary value for u₀ has been specified. Let us decrease the injection saturation (at the boiling temperature) from S_w = 1 to S_w = S_wc; in our case this corresponds to changing , the water saturation at the HISW shock with speed given in Eq. (26). For each , we construct the sequence of elementary waves (and constant states) with decreasing speeds from right to left, that is from to . There is a constant state to the left of , so there is no other wave to the left of , the steam shock. For each , we mark its wave speed, forming the solid curves in Fig. 11. This sequence, parameterized by , is called a backward wave curve from (S_w = 1, T⁰, u⁰). It combines all information needed for describing the structure of the Riemann solution, and for verifying necessary speed relations for the admissibility of the shocks involved. On rarefaction segments of this backward wave curve, the characteristic speed decreases, while on shock segments of this wave curve, the shock speed increases [8], [14]. We refer to Fig. 11.

When lies between S_* and 1, the wave with fastest possible speed is the cooling shock in the liquid water region (see Section 3.4), so the backward wave curve corresponds to such a shock wave, with speed given by in Eq. (35). There is a hot steam-water region and a cold water zone. In the hot steam-water region, generically there is a Buckley-Leverett rarefaction-shock and a constant state. The hot steam-water region terminates with the HISW shock.

The analysis from now on relies on the fact that for steam-water in the actual reservoir, S^infl < S_*. For within (S_†, S_*) , the wave with slowest speed is the SCF, with speed v^SCF. At the left of the SCF, characteristic speed exceeds v^SCF, so there can be no rarefactions nor shocks to the left of the SCF. Thus, for in such range, the solution (from left to right) consists of a constant state of steam and water at temperature T^b, a SCF jumping from to 1.0, and a constant cold water zone.

Recall that S_† < S^infl. Therefore, from within (S_wc, S_†), the slowest speed is a hot steam-water rarefaction speed , which is smaller than v^SCF ( 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 with saturation as in the previous case (this constant state disappears if = S_wc), a rarefaction wave from to S_†, a SCF from S_† to 1, and a constant cold water zone.

Let us explain how the Darcy velocity u^b is constructed for each value of . If lies in the interval (S_*, 1), u^b is given by Eq. (34), and the speed of the cooling shock is given by Eq. (35) (Case III). (As explained in Section 2.1, the velocity u is constant in the hot region and equals u^b.) If lies in the interval S_† and S_*, u^b is given by Eq. (44). The speed of the SCF is given by Eq. (45), (Case II).

If lies in the interval (S_wc, S_†) (Case I, since u^b has to match at the boundary of Cases I and II, u^b is given by Eq. (45) with S_w replaced by S_†. Thus u^b is independent of in Case I.

A summary of the behavior of u^b() is presented in Fig. 2. Here u^b() is the value of u^b calculated in the previous paragraphs for a fixed u⁰.

We are ready to abandon the assumption that u⁰ is known. This is impractical, since normally one specifies u^inj rather than u⁰. We take advantage of the fact that all speeds are proportional to find the actual speed in the cold water zone u⁰() for specified u^inj, as follows:

From this equation we recover u⁰(), essentially by inverting the variable represented in the ordinate in Fig. 2. Thus we obtain Fig. 7.

4.2 Summary of the Riemann solution

The solution consists of three parts, viz. a hot region (A) at constant boiling temperature, an infinitesimally thin cooling front (B) or discontinuity, where all possible steam condensation occurs, and a cold liquid water region downstream (C). See Figure 1.

As we have seen, the nature of the solution changes and there are three possible cases (I), (II), and (III), depending on the injected steam quality = 1 – . Cases (II) and (III) are separated by the hot-cold bifurcation, while Cases (I) and (II) are separated by the steam-water bifurcation.

Case (I) occurs when the saturation wave velocity (; u^inj) < v^SCF (see Eq. (25)); it consists of a sequence of a constant state at the injection end, a rarefaction wave in the hot steam zone (A) ending with saturation S_† at the SCF (B) with speed v^SCF defined by saturations S_† and S_w = 1, and a cold water constant state in (C). The constant state in (A) disappears if the injection saturation is S_wc, that is, pure steam is injected. See Fig. 8. The rarefaction disappears for (; u^inj) = v^SCF, as in this case = S_†.

Case (II) occurs for (, u^inj) < v^SCF < (; u^inj); see Eqs. (25), (26) and (27). This case consists of a hot constant steam-water state in (A), the SCF (B) with speed v^SCF defined by left and right saturations and 1 (see Eq. (45)), and a constant cold water state in (C). See Fig. 9.

Case (III) occurs for a typically small region for the cooling contact discontinuity velocity (given in Eq. (35)) with > (, u^inj) (see Eq. (26)), i.e. the hot isothermal steam-water shock velocity. In this case, there is no SCF. In the hot region (A) there is a constant state with steam-water, then another constant state of pure hot water at the same boiling temperature, separated by a Buckley-Leverett shock. Then there is a cooling shock with speed , where the saturation of water is constant (S_w = 1) and the temperature changes from T^b to the reservoir temperature T⁰. (See Fig. 10.)

Figure 6 illustrates the saturation dependence of the various velocities that are the basis of the steam-water 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, we will compare the SCF speed to the left and right characteristic speeds. We will conclude that from the point of view 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/m³] at constant pressure, i.e. (T) is the heat capacity of water and (T) is the heat capacity of steam. In the same way we define the thermal expansivity of water and steam a_w(T) and a_g(T) as minus the temperature derivative of the density divided by the density (see Appendix A).

Eqs. (21)-(22) may be written in quasilinear form as:

We restrict our attention to regions where ¶u/¶x = 0 and q = 0, that is, away from any kind of shocks. Thus, the LHS terms of Eqs. (62), (63) vanish.

We let

Multiplying the RHS of Eq. (62) by -(H_w - ) and of Eq. (63) by (r_w – r_g) and adding leads to a new equation, which will be used instead of Eq. (63):

We let

Thus in regions where ¶u/¶x = 0 and q = 0, that is, away from any kind of shocks (see Remarks 1 and 2), Eqs. (62)-(65) may be written in matrix form as:

Let m be a characteristic speed. Then the determinant of the following matrix must vanish:

Since the matrix above is upper triangular, the characteristic speeds are easily read from the diagonals:

(It is easy to check that A_III never vanishes.)

Now, in the liquid water region on the right of the SCF, S_g = 0, f_g = 0, = 0, the characteristic speeds are

The latter speed has already been calculated in Eq. (46).

On the other hand, in the hot steam zone the characteristic speeds are

The first speed has already been calculated in Eq. (45). The second speed of thermal waves is shown in Figure 11 as a function of S_w at T = T^b.

For a (-) state for the SCF with S_w in (S_*, S_f) we have that the thermal characteristic speed in the steam zone satisfies 0 < v⁰ < v^SCF, and the steamfront velocity satisfies < v^SCF < , so the SCF would be called a 2-shock in Lax's classification scheme. However, Lax's theorem only applies to shocks with small amplitude, while the SCF is a large shock, and only if the governing equations were a system of conservation laws satisfying appropriate technical hypotheses, such as genuine nonlinearity, which is actually violated at the inflection S^infl. Moreover, even the Lax inequalities are violated starting at the steam-water bifurcation; there is no conclusive mathematical evidence that the SCF shock needed to complete the Riemann solution is physically admissible. This is the issue left open.

 

6 Summary and conclusions

A complete and systematic description of all possible solutions of the Riemann problem for the injection of a mixture of steam and water into a water-saturated porous medium, for all possible reservoir temperatures and pressures below the water critical point. For each Riemann data, we found a unique solution.

As determined by the dissipative effects of capillary porous forces combined with the mass source term given in Eq. 3, the internal structure of the SCF is consistent with the Riemann solution in this work. This fact is demonstrated in a companion paper [4].

 

7 Acknowledgments

This work was partially done at IMA, University of Minnesota, and therefore partially funded by 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.

 

REFERENCES

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[2] J. Bear, Dynamics of Fluids in Porous Media, p. 650 and 651, Dover Publications, Inc., New York (1972).         [ Links ]

[3] R.B. Bird, W.E. Stewart and E.N. Lightfoot, Transport Phenomena, John Wiley & Sons (1960).         [ Links ]

[4] R.H. Brooks and A.T. Corey, Properties of porous media affecting fluid flow, J. Irr. Drain. Div., Proc., Am. Soc. Civ. Eng. June (1966), 61-88.         [ Links ]

[5] J. Bruining, D. Marchesin and S. Schecter, Steam condensation waves in water-saturated porous rocks, in preparation, 2002.         [ Links ]

[6] J. Bruining, D. Marchesin and C.J. Duijn, Steam injection into water-saturated porous rock, IMPA preprint 136/2002, 2002.         [ Links ]

[7] L.P. Dake, Fundamentals of Reservoir Engineering, Elsevier Science Publishers, Amsterdam, 1978.         [ Links ]

[8] F.A.L. Dullien, Porous media; fluid transport and pore structure, Academic Press, N.Y. (1979).         [ Links ]

[9] E. Isaacson, D. Marchesin, B. Plohr and J.B. Temple, Multiphase flow models with singular Riemann problems, Mat. Apl. Comput., 11 2 (1992), 147-166.         [ Links ]

[10] M. C. Leverett, Capillary behavior in porous solids, Trans., AIME, 142 (1941), 142-169.         [ Links ]

[11] O. Oleinik, Discontinuous Solutions of Nonlinear Differential Equations, Usp. Mat. Nauk. (N.S.) v.12, pp. 3-73, 1957. English transl. in Amer. Math Soc. Transl. Ser. 2 26, 95-172.         [ Links ]

[12] P.F. Peterson, Diffusion Layer Modeling for Condensation with Multicomponent Non-condensable Gases, Journal of Heat Transfer, Transactions of the ASME, Vol 122 (2000), 716-720.         [ Links ]

[13] G.A. Pope, The application of fractional flow theory to enhanced oil recovery, SPEJ, 191-205, June 1980.         [ Links ]

[14] M. Prats, Thermal Recovery, Monograph Series, SPE, Richardson, TX, 7 (1982).         [ Links ]

[15] S. Schecter, D. Marchesin and B. Plohr, Classification of codimension-one Riemann solutions, Journal of Dynamics and Differential Equations, vol. 13 3 (2001), pp. 523-588.         [ Links ]

[16] A. Shoda, C.Y. Wang and P. Cheng, Simulation of constant pressure steam injection in a porous medium, Int. Comm. Heat Mass Transfer Vol 25 6 (1998), 753-762. (1998).         [ Links ]

[17] W.S. Tortike and S.M. Farouq Ali, Saturated-steam-property functional correlations for fully implicit reservoir simulation, SPERE (November 1989) 471-474.         [ Links ]

[18] R.C. Weast, CRC Handbook of chemistry and physics, 58^th edition, CRC Press, Inc. 1977-1978.         [ Links ]

 

 

Received: 28/XII/02.
Accepted: 18/VIII/03.
This work was supported in part by: CNPq under Grant 301532/2003-6, FINEP under CTPETRO Grant 21.01.0248.00; NWO under Fellowship Grant R 75-389, The Netherlands; IMA (Univ. of Minnesota), and therefore by NSF, Dietz Laboratory (TU Delft, The Netherlands), IMPA (Brazil).

 

 

#558/02.

 

 

Appendix A - Physical quantities; symbols and values

In this Appendix we summarize the values and units of the various quantities used in the computation and empirical expressions for the various parameter functions. For convenience we express the heat capacity of the rock in terms of energy per unit volume of porous medium per unit temperature i.e. the factor 1 – j 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 water

We use reference [16] to obtain all the temperature dependent properties below. The water and steam densities used to obtain the enthalpies are defined at the bottom. First we obtain the boiling point T^b at the given pressure p, i.e.

where = log(p) and p is the pressure in [k Pa]. The evaporation heat [J/kg] is given as a function of the temperature T at which the evaporation occurs. We use atmospheric pressure (p = 101.325 [k Pa]) in our computations, to make the example representative of subsurface contaminant cleaning.

The liquid water enthalpy h_w(T) [J/kg] as a function of temperature is approximated by

The steam enthalpy h_g [J/kg] as a function of temperature is approximated by

For the latent heat [J/kg] or evaporation heat L(T) we obtain

The sensible heat of steam (T) in [J/m³] is given as

We also use the temperature dependent steam viscosity

The temperature dependent water viscosity m_w is approximated by

For the steam density tion of temperature T[K] we use a different expression than [16] because our interest is a steam density at constant pressure, which is not necessarily in equilibrium with liquid water.

where p is the total pressure at which the steam displacement is carried out, R=8.31 [J/mol K] and Z is the Z-factor (see e.g. Dake [6]) and = 0.018 kg/mole is the molar weight of water. For the atmospheric pressures of interest here the Z-factor is close to unity. The liquid water density as a function of the temperature T[K] is given as

A.2 Constitutive relations

We use a porosity j that is representative for unconsolidated sand. The relative permeability functions k_rw and k_rg are considered to be power functions of their respective effective saturations [7], i.e.

S_we = (S_w – S_wc)/(1 – S_wc), S_ge = S_g /(1 – S_wc).

The effective saturations require knowing the connate water saturation S_wc. In all our examples we use a fourth power of the effective saturation for the relative water permeability and a quadratic dependence for the steam relative permeability.

The relative permeability functions k_rw and k_rg are considered to be power functions of their respective saturations [7], i.e.

For the computations we take n_w = 4, n_g = 2. The connate water saturation S_wc is given in the table.

The capillary pressure is of the Brooks-Corey type based on the dimensionless capillary pressure from P_c(S_w = 0.5)/(s_wg) = 0.5. The capillary pressure between steam and water is given by the empirical expression which combines Leverett's approach to non-dimensionalize the capillary pressure [9] with the semi-empirically determined saturation dependence suggested by Brooks [3]:

where g is a parameter that in many cases assumes values between 0.3 and 0.7. We use g = 0.5 and l_s = . Finally s_wg = 0.058 is the water-vapor interfacial tension. We disregard its temperature dependence and use the value at the boiling point (see [17], p. F-45).