Spe 102266-pa-p

Analytical Solution of NonisothermalBuckley-Leverett Flow Including Tracers

Deniz Sumnu-Dindoruk, SPE, Shell E&P, Unconventional Oil, and Birol Dindoruk, SPE, Shell International E&P

Summary

Mass balances for two immiscible fluids and tracer and convectiveheat balance form a system of three equations (nonisothermalBuckley-Leverett Problem with tracers). Tracer component is con-sidered to investigate the propagation of a tracer to track the flood(or to track a miscible inert contaminant introduced during drill-ing). We have solved the resulting nonisothermal two-phase con-vective flow equation in porous media analytically, including atracer component (i.e., cold or hot waterflooding with and withouttracer). Method of characteristics (MOC) is used as a solutiontechnique after transforming the balance equations in a form thatcan be solved easily with two Welge tangents. Our solution tech-nique is valid for both radial- and linear-flow models.

In practice, these solutions can be used• To investigate the convective flow behavior around the wells

(i.e., sudden fluid losses, convective near-well tracer propagation,analyzing pressure transients).

• To interpret formation-testing-tool responses by detecting thelocation of the thermal front or to estimate the temperature-builduptime that is needed for Horner-type analysis (for the identificationof the formation temperature).

• To calculate the location of the cold or hot water front (ther-mal water) while injecting cold- or hot-water. This will yield thelimit of the maximum temperature disturbance around the well.

• To test/scale relative thermal effects of various systemsagainst one another.

• To test the accuracy of simulators and provide bench-mark solutions.

• To interpret relevant laboratory experiments quickly.Such solutions helped us to understand the depth of influence

of the temperature variations and their influence on the transportproperties, in both radial and linear systems. Solutions analyticallyproved that the thermal front propagates much slower than theflood front. This explains why isothermal black-oil simulators stillwork although the injected-water temperature is not always equalto the reservoir temperature.

Furthermore, we have checked and verified our results againsta commercial thermal simulator and investigated the impact ofnumerical diffusion on the thermal front as well as conductivity.This part of the work revealed that the temperature front is moreprone to numerical diffusion.

Introduction

Cooling resulting from sudden fluid losses (spurt loss) during drill-ing or hot fluid/water (with some additives) treatments around thewellbore is a quite common intervention for the near-wellboreregion. In addition, waterflooding is the most common secondary-recovery mechanism. Short-time behavior of such systems isdominated by the convective terms as defined by the underlyingequations. As many engineers observe, most waterflood modelscan be history matched with isothermal simulators, contrary towhat one might expect (because water has high heat capacity).Such black-oil simulations work because changes in the tempera-

ture and heat capacity of water per unit volume of injection are notenough to extract an excessive amount of heat (in the case ofcold-water injection) from the matrix and the in-situ fluids. Theonly way to extract and/or inject heat is to take the advantage ofphase transformation (i.e., latent heat as in the case of steam in-jection). As described later in the paper, the presence of the real-istic heat losses as the front propagates away from the injectionwell becomes dominant (as convective flow velocity decays radi-ally), further tapering off the thermal gradients induced by theinjected water. Therefore, the impact of cold water can be felt at alimited volumetric region only around the well.

Another observation that can be made during the backflowphase of the subsurface sampling is that contaminants and waterpropagate deeper into the formation than the first-order (convec-tive) thermal effects. The resulting problem is analogous to theBuckley-Leverett (1942) problem, in which the solution can beconstructed by use of a series of tangents similar to Welge’s(Welge et al. 1962; Johns and Dindoruk 1991; Dindoruk 1992;Johns 1992; Hovdan 1989; Bratvold 1989). The late-time tempera-ture behavior, however, is dominated by conduction (Platenkamp1985; Roux et al. 1980; Hashem 1990). Such late-time behavior isalso demonstrated for the solutions presented here by use of acommercial simulator (STARS Manual 2004).

The classes of problems solved here are somewhat differ-ent from the early solutions for nonisothermal displacement inporous media as described by Marx and Langenheim (1959), Lau-werier (1955), and Rubinshtein (1959). The main differences/similarities are

• The classical steam-injection models of the Marx-Langenheim type consider a piston-like displacement and track thesteam-zone growth in the presence of heat losses to the overburdenand underburden, and they are single-phase models (i.e., no rela-tive permeability or fractional flow is incorporated).

• The class of problems solved here are based on threecoupled-balance equations as shown by Eqs. 1 through 3 in thenext section. Both Marx and Langenheim type problem and theclass of problems solved here take the temperature profile as a stepfunction. The temperature zone in the proposed solutions, how-ever, is not aligned with the leading saturation front. This is mainlyowing to losing/gaining heat into/from the porous media from thehot/cold water injection (lack of latent heat).

• The systems of equations also considered here show the cou-pling of tracers with heat injection, and a formal solution is offeredfor the problem in the space of dependent variables.

A thorough overview of these problems and later more analo-gous solutions as proposed by Shutler and Boberg (1972), Myhilland Stegemeier (1978), and others can be found in the monographby Prats (1985).

To understand the behavior of the nonisothermal aspects oftwo-phase immiscible-fluid flow, we have solved the nonisother-mal Buckley-Leverett problem including an inert tracer and com-pared our results with a commercial simulator. Another reason forsuch comparison is to be able to gauge the heat-flow model againstthe analytical solutions once the correct assumptions are imple-mented correctly.

Cold-water injection and hot-water injection in 1D have beeninvestigated by many authors (Hovdan 1989; Bratvold 1989;Barkve 1989; Lake 1989; Pope 1980; Isaacson 1980). Amongthose, Platenkamp (1985) focused on the reservoir-cooling issueresulting from water injection for North Sea reservoirs. WhilePlatenkamp’s focus was on the pressure-transient aspects as well

Copyright© 2008 Society of Petroleum Engineers

This paper (SPE 102266) was accepted for presentation at the 2006 SPE Annual TechnicalConference and Exhibition, San Antonio, Texas, 24–27 September, and revised for publi-cation. Original manuscript received for review 27 June 2006. Revised manuscript receivedfor review 9 December 2007. Paper peer approved 23 December 2007.

555June 2008 SPE Reservoir Evaluation & Engineering

as injectivity aspects of the flow, this analysis required a specifictemperature profile. In that study, use of temperature profile as astep function was found to yield sufficiently accurate results forthe routine analysis of the falloff surveys. The location of thetemperature front is obtained by use of a simple convective heatbalance assuming piston-like water-saturation profile around thewells. While the outcome of the convective heat balance is in linewith our study, the location of the temperature front is a functionof the fractional-flow function as well as the temperature depen-dence of the fractional-flow function.

In the literature, while most of the focus was on the conductiveheat-transfer aspect of the heat flow, there are a number of ana-lytical solutions (Hovdan 1989; Bratvold 1989; Barkve 1989) forthe convection-dominant version of the problem. These authorsinvestigated the mathematical nature of the problem as well as theuniqueness of the solutions. The outcome of their work was fun-damentally the same, while solutions were obtained with slightlydifferent dependent variables. The solutions were limited mostlyto cold-water injection, however. In this study, we investigate amore general version of the nonisothermal Buckley-Leverett prob-lem, including a passive tracer that may be used to track the floodfronts indirectly.

Statement of the ProblemThe governing mass- and energy-balance equations in dimension-less form are given by the following.

For water mass balance (Bratvold 1989),

�Sw

�tD+

�fw

�Sw

�Sw

�xD+

�fw

�TD

�TD

�xD= 0. . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

For tracer (nonabsorbing) balance,

��cDSw�

�tD+

��cD fw�

�xD= 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

For energy balance:

�TD

�tD+ g

�TD

�xD= 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

where dimensionless independent variables are defined by

xD =x

L, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

tD =qt

�AL, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5)

where xD is dimensionless distance and tD is dimensionless time.In Eqs. 1 through 3, dimensionless dependent variables are defined as

TD =T − Tw

Ti − Tw, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

where cD is dimensionless tracer concentration and TD is di-mensionless temperature. Water saturation is defined as Sw and

fractional-flow function of water is fw. Coefficient g in Eq. 3 isdefined by

g =fw + �

Sw + �, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

where � and � are dimensionless functions of products of ther-mal properties of rock and fluid, as well as porosity. They aredefined by

� =�ocvo

�w cvw − �ocvo, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)

and

� =�ocvo +

1 − �

��rcvr

�wcvw − �ocvo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

Introduction of the g term with � and � leads to scaling of theproblem in terms of thermal properties (Bratvold 1989) becausethermal properties appear only in the g term. Derivation of Eqs. 1,2, and 3 is based on the following assumptions:

• 1D homogeneous porous medium• Negligible effect on fluid properties because of change in

pressure over the displacement length• Convective heat flow only (i.e., conductive effects and heat

loss to overburden and underburden are ignored)• Incompressible flow• No phase change over the displacement length• Fractional-flow functions of saturation and temperature only• Noninteracting and nonadsorbing tracer (Formulation includ-

ing the effects of adsorption is shown in the Appendix and does notcomplicate the methodology presented here.)

• Convective two-phase porous-media flow (i.e., flow-relateddissipative effects such as dispersion/diffusion capillary pressure)

The major limiting assumption above—but necessary for therigorous analytical solutions—is the convective heat-flow assump-tion (i.e., no conduction). The problem proposed here, however, isthe limiting case of little or no conduction; however, that helps usto understand the physics of nonisothermal flow with tracers in aporous medium. Inclusion of conductive heat flow is possible if theequations are solved numerically or semianalytically. The equationsystem (Eqs. 1 through 3) can be solved analytically the subject toRiemann-type boundary conditions (Fig. 1), with the method ofcharacteristics. A direct attempt to solve the system posed by Eqs.1 through 3 generates an eigenvalue problem with a matrix thatrequires calculation of the temperature derivative of the fractionalflow of water. As shown in the Appendix, however, the problemcan be reformulated in terms of a different set of dependent vari-ables, leading to a diagonal coefficient matrix following: the meth-odology proposed by Isaacson (1980) for polymer flooding, amodified version of the methodology by Hovdan (1989) for coldwaterflooding, and the methodology proposed by Dindoruk (1992)for compositional gas-injection problems. As explained in the Ap-pendix, left eigenvectors are used to diagonalize the coefficientmatrix. Left eigenvectors are useful in terms of reducing the equa-tion system to a more explicit form to inspect contact discontinu-ities and recognize Riemann invariants.

Solution Construction. Solution of the equation system defined inEqs. 1 through 3 and subject to the boundary and initial conditionsgiven in Fig. 1 is constructed by use of the method of character-istics. The details of the solution of systems of equations arising inmultiphase transport in porous media is explained by Dindoruk(1992) and Johns (1992). Solution of Eqs. 1 through 3 consists ofshocks, expansion waves, and zone of constant states. The fullsolution is pieced together by use of those components subject toa set of fractional-flow curves shown in Fig. 2. The fractional flowshown in Fig. 2 is generated by use of the relative permeabilitycoefficients shown in Table 1 (Eqs. 10 and 11).

Fig. 1—Boundary and initial conditions (Riemann problem).

556 June 2008 SPE Reservoir Evaluation & Engineering

A sketch of the solution in a 2D state space (TD and g—as perthe transformation described in the Appendix) is shown in Fig. 3.Addition of the tracer just lifts part of the solution path in the cD

dimension, as shown in Fig. 4. Figs. 3 and 4 show the behavior forcold-water injection. For simplicity, we will start describing solu-tion construction by use of Fig. 3:

• Solution starts with a leading shock from Initial Condition ato Point b as dictated by the tangent drawn to the fractional-flowcurve for the reservoir (T�Ti).

• Solution continues with an expansion wave from the Shock

Point b to the Equal Eigenvalue Point c, where g�Sw��fw

�Sw�

TD�1.

The solution of this equation will yield the upstream saturationpoint (S*w) on the fractional-flow curve for the reservoir. This isgeometrically a tangent construction from (−�, −�) to the frac-tional-flow curve of the reservoir.

• Next, the corresponding saturation point on the fractional-flow curve of the injection point (well) needs to be determined.This can be calculated by use of g�S**w ��TD�0�g�S*w��TD�1. BecauseS*w is already known from the previous step, S**w is the only un-known in this equation, and it will give us the location of thelanding point (Point d) on the fractional-flow curve of the injectionpoint. In other words, this point is the point in which the interme-diate eigenvalues of the two extreme temperatures are equal.

• Finally, the trailing segment of the solution will be from thelanding point on the fractional-flow curve of the injection point tothe inlet-contact discontinuity (Point e, 1−Sor with characteristicspeed of zero).

Additon of tracer “lifts” part of the solution path along theconcentration axis, as shown in Fig. 4. Because the tracer is anoninteracting tracer, it will not impact the other variables, and the

location of the tracer can be obtained from�fw

�Sw�

TD�1�

fw

Sw. Geo-

metrically, this is equivalent to drawing a tangent from the ini-tial condition, (0,0), to the fractional-flow curve for the reservoir.The difference between the cases without tracer vs. with tracer isthat the solution-path segment, (m–n), is in the third (concentra-tion) dimension. Corresponding solution construction by use offractional-flow domain is shown in Fig. 1. Saturation profile interms of by use of eigenvalue segments is shown in Fig. 5.

The composition path is somewhat different for hot-water in-jection. In that case, the composition path starts with a leadingshock from (Swc, 0) to the fractional-flow curve for the reservoirand continues until the equal eigenvalue point on the fractional-flow curve for the injection point (well). That point is defined bydrawing a tangent to the fractional-flow curve of the injection point

from (−�,−�). The equation for this tangent is g�Sw��fw

�Sw�

TD�0.

Again, the solution of this equation will yield the upstream satu-ration point (S*w) on the fractional-flow curve for the fractional-flow curve for the injection point. Next, the corresponding satu-ration point on the fractional-flow curve of the reservoir (well) canbe determined by use of g�S**w �|TD�1�g�S*w�|TD�0. Again, S**w is theonly unknown in this equation. Then, the rest of the solution is thesame as that for the cold-water injection.

Corresponding solution construction by use of fractional-flowdomain is shown in Fig. 6. Saturation profiles in terms of use ofeigenvalue segments is shown in Fig. 7.

Sample Solutions. Analytical and numerical solutions are ob-tained for both cold waterflooding and hot waterflooding. Numeri-

Fig. 3—Solution path in TD and g state space.

Fig. 4—Updated solution path in TD, g, and cD state space. a-b-c�-d�-e� path is for nonisothermal Buckley-Leverett withouttracer, and a-b-m-n-c-d-e path is for nonisothermal Buckley-Leverett with tracer.

Fig. 2—Path construction on the fractional-flow curves for coldwaterflooding.


cal solutions are obtained by use of a commercial simulator(STARS Manual 2004). Input data are given in Tables 1 through 3.Thermal properties and densities shown in Table 2 are used incombination with the fractional-flow function shown below:

fw�Sw, T � =1

1 +�w

�o�T �

kro

krw

. . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)

The fractional-flow function is a function of saturation resultingfrom relative permeabilities and a function of temperature throughviscosities. The method described here, however, is general andcan take any combination of these dependencies (i.e., relative per-meabilities as a function of temperature). For the solutions pre-sented here, Corey-type relative permeabilities are used:

krw = krwo �S�nw

kro = kroo �1 − S�no

S =Sw − Swr

1 − Sor − Swr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)

The input parameters for the relative permeability functions areshown in Table 1, and the fluid viscosities are shown in Table 3.

Cold Waterflooding. For this case, we have assumed injectionof cold water in which the oil viscosity increases within the tran-sition zone behind the temperature front. For the example caseshown in Fig. 8, we have assumed 0.99-cp cold water displacing2-cp oil, while oil viscosity increases to 8 cp behind the thermalfront. Solution starts from downstream to upstream with a Buck-ley-Leverett-type shock front (a→b) on the fractional-flow func-tion for the reservoir. Then it continues with an expansion wave(b→c). The expansion wave on the fractional-flow function for thereservoir is followed by a trailing shock (c→d), owing to tempera-ture change. That shock is followed by a zone of constant state(d→d), owing to the differences in the speeds of propagation be-tween the trailing shock front and the expansion wave, (d→e), onthe fractional-flow curve for the injection point (well). As seen inFig. 8, the temperature front causes c→d shock and, as expected,it is also aligned with it. The tracer shock, (m→n), does not in-terfere with the structure of the temperature and saturation profiles.The location of the tracer shock is a function of pore volumesinjected and the amount of initial water saturation within the tran-sition zone. Tracer concentration moves slower than the leadingshock because of the absence of tracer in the initial water satura-tion. One of the main observations that can be made is that thetemperature shock moves slower than the injected water front, and

Fig. 6—Path construction on the fractional-flow curves forhot waterflooding.

Fig. 7—Solution-path construction by use of the eigenvalues forhot waterflooding.

Fig. 5—Solution-path construction by use of the eigenvalues forcold waterflooding.


the speed of propagation is a function of the thermal heat (mass)capacity ratio (�/�) of the porous medium as well as the flowproperties of residing and invading fluids.

Although it is easy to solve the equation system posed by Eqs.1 through 3 by use of a conservative fine-difference scheme (re-sults not shown here), one of our main motivations is to implementthe assumptions cited here and solve the same/near-identical prob-lem by use of a commercial simulator and compare the accuracy ofthe solution. In Fig 8, we have used 1,000 gridblocks in the di-rection of flow, and the numerical results agreed with the analyti-cal results extremely well. Among all the dependent variables, thetracer front exhibited a higher level of numerical dispersion (Lantz1971) because of its self-sharpening nature, and it moved fasterthan the temperature front. Resulting from the entry and exit to thetwo-phase-flow region, the leading saturation shock did not exhibithigh levels of dispersion with respect to TD and cD fronts. Level ofnumerical dispersion is a function of numerical Peclet number andis a function of the derivative of the fractional-flow function(Lantz 1971). Further discussion on numerical dispersion in thecontext of 1D conservation equations can be found in Orr (2007).

Hot Waterflooding. For the hot-waterflooding case, we haveassumed injection of hot water in which the oil viscosity decreaseswithin the transition zone behind the thermal front. For the ex-ample case shown in Fig. 9, we have assumed that 0.69-cp hotwater displacing 8-cp oil while oil viscosity decreases to 2 cpbehind the thermal front (Table 3). Corresponding solution con-struction by use of fractional-flow domain is shown in Fig. 6.Saturation profiles by use of eigenvalue segments are shown inFig. 7 Solution starts from downstream to upstream with a Buck-ley-Leverett-type shock front (a→b) on the fractional-flow func-tion for the reservoir. Then it continues with an expansion wave(b→c). An expansion wave on the fractional-flow function for thereservoir is followed by a zone of constant state (c→c) because thelanding point of the trailing shock, (c→d), on the fractional-flowfunction for injection point (well) travels slower for the samesaturation. Again, the trailing shock, (c→d), is caused—and isaligned—by the temperature shock. The tracer shock, (m→n),does not interfere with the structure of the temperature and satu-ration profiles. Similar to the cold-water-injection case, the loca-tion of the tracer shock is a function of pore volumes injected andthe amount of initial water saturation within the transition zone.Like the cold-water injection case, tracer concentration movesslower than the leading shock because of the absence of tracer inthe initial water saturation. The temperature shock moves slowerthan the injected water front, and the speed of propagation is afunction of the thermal heat (mass) capacity ratio, (�/�), of theporous medium as well as the flow properties of the residing andinvading fluids.

The numerical solutions are also compared with the analyticalsolution in Fig. 9 at tD�0.25 (pore volumes injected). In thenumerical solutions, 1,000 gridblocks are used in the direction offlow, and the numerical results agreed with the analytical resultsextremely well. Behavior of the numerical solution and the nu-merical quality with respect to the analytical solution is the sameas the cold-waterflooding case and therefore will not be elaboratedonce more.

Solution in Radial Coordinates. In reality, we are more inter-ested in the radial-flow geometry than the linear geometry. Be-cause the MOC solutions are similar, it is possible to perform asimple coordinate transformation as defined by Welge (Welge etal. 1962) and, later, by others in a more generalized form (Johnsand Dindoruk 1991; Johns 1992). The following dimensionless

variables will transform the solutions for linear geometry into ra-dial coordinates without repeating the solution process:

rD =r

L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)

is for the radial distance, and

tD =qt

�L2�h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13)

is for the dimensionless time in radial coordinates. Here, L indi-cates some characteristic distance in the radial domain. Therefore,relabeling the linear distance xD as rD

2 (xD≡rD2 ) in Figs. 8 and 9

yields the analytical solutions for radial coordinates, as shown inFigs. 10 and 11. In this case the dimensionless distance is equiva-lent to the dimensionless radial distance squared. The comparativedetails of radial vs. linear dimensionless variables are shown byJohns (1992). Numerical implementation of the radial flow re-quires a somewhat different approach, however. For that, the do-main needs to be gridded appropriately by use of a radial-gridsystem (Fig. 12). In addition, to capture the flow around the wellaccurately, a logarithmically spaced grid system is needed. Nu-merical solutions shown in Figs. 10 and 11 are generated by use of1,000 grid cells. As in the case of linear geometry, the numericalsolutions agree well with the analytical solutions.

Numerical Experiments. In this section, we will focus on limitednumerical experiments to understand the impact of some of the

Fig. 9—Comparison of analytically constructed profiles for lin-ear flow (method of characteristics) vs. numerical solution forhot waterflooding at tD=0.25 pore volumes of injection. Nx=1,000for the numerical solution.

Fig. 8—Comparison of analytically constructed profiles for lin-ear flow (method of characteristics) vs. numerical solution forcold waterflooding at tD=0.25 pore volumes of injection.Nx=1,000 for the numerical solution.


variables that we did not consider to set up the problem for theanalytical solutions.

Grid Sensitivity. For the sake of brevity, we will show only twoof the runs we have performed: fine- and coarse-grid runs. Fine-grid runs are performed by use of 1,000 grids as before, andcoarse-grid runs are performed by use of 10 grids. In addition towhat we show here, we performed simulation runs with finer grids(i.e., 10,000), and the results were similar to those shown here with1,000 gridblocks. Comparison of the numerical results for bothcold-water and hot-water-injection cases are shown in Figs. 13and 14. As can be seen in these figures, the temperature profileand the tracer profiles are more prone to the numerical dispersionthan is the fluid saturation. In addition to the differences in thenumerical Peclet number (Lantz 1971), one of the reasons forhigher levels of dissipation of the temperature profile is that tem-perature has more room to dissipate for all the phases and the rock,and it is also a self-sharpening wave (Jeffery 1976).

Tracer dispersion is somewhat analogous to temperature; how-ever, it moves faster and its dissipation is also related to the dis-sipation of Sw because it resides in the water phase only. Thedissipation characteristics of the tracer front, however, are similarto a semishock, in which one side of the shock resides on thefractional-flow curve. In explicit solutions, dimensionless timestepsize must be less than dimensionless grid size for single-phaseflow and much less than grid size if the saturation derivative of thefractional-flow function is greater than 1, as it will be for theS-shaped fractional-flow curve appropriate to two-phase flow in aporous medium, particularly when the injected fluid has a viscositythat is less than that of the oil displaced. As a result, the effects of

numerical dispersion can be reduced by reducing grid size (andtherefore timestep size), but they cannot be eliminated entirely forthe finite-difference scheme. The limitation on timestep size is aversion of the Courant-Friedrichs-Lewy condition (Courant et al.1967), which states that the finite-difference scheme of a simple

one-point explicit scheme is unstable if�ptD

xD1 for all equations

and characteristic speeds [�p, eigenvalues (Orr 2007)].Effect of Thermal Conduction/Overburden/Underburden Heat

Loss. Next, we have examined the impact of thermal conduction,which had to be neglected to solve the system of equations ana-lytically. The values used in the numerical investigation are givenin Table 4. The conductivity of underburden and overburden are22.47 and 22.79 Btu/day-ft-ºF, respectively. The density and con-ductivity products are 38.68 and 35.44 Btu/ft3-ºF for overburdenand underburden, respectively.

Numerical results including the impact of thermal conductionshown in Figs. 15 and 16 are obtained with Cartesian grids forlinear displacement. In the same figures, numerical solution with-out the effects of conduction is also superimposed for comparisonpurposes. Overall, the thermal front dissipates at tD�0.25. Thelocation of the tracer front and the saturation front (and the profile)do not change appreciably, however. In this study, the main focuswas the convection-dominant displacement, and a methodical ap-proach to form benchmark solutions was presented for the solutionof saturation, temperature, and tracer equations. Although we didnot investigate the impact of heat conduction thoroughly, the im-pact of heat conduction appears as significant on the basis of thecases that we have considered for this work. Further work is

Fig. 11—Comparison of analytically constructed profiles for ra-dial flow (method of characteristics) vs. numerical solution forhot waterflooding at tD=0.25 pore volumes of injection. Nx=1,000for the numerical solution.

Fig. 12—Well configuration and grid system used for the simu-lation of radial-flow cases (Nr =1,000 by use of logarithmicallyspaced grids).

Fig. 13—Numerical simulation of cold waterflooding with atracer by use of fine (Nx=1,000) and coarse (Nx=10) grids attD=0.25. Solid lines are the analytical solutions by use of themethod of characteristics.

Fig. 10—Comparison of analytically constructed profiles for ra-dial flow (method of characteristics) vs. numerical solution forcold waterflooding at tD=0.25 pore volumes of injection.Nx=1,000 for the numerical solution.


needed to quantify the time scales that second-order effects (con-duction, heat losses to underburden and overburden) make impor-tant. Impact of conduction is discussed within the context ofsteamflood by Prats (1985). Although the results are not shownhere, we have also included the effects of overburden and under-burden heat losses. The inclusion of those effects did not changethe results significantly for the saturation and tracer profiles (forthe time scale considered). The temperature profile changed asmall amount, however.

Sensitivity to Rock/Fluid Thermal Properties. We have brieflystudied the impact of mass-based heat capacity of the rock andfluid system. In this analysis, fractional-flow functions of the sys-tem were not changed. The relevant parameters for the heat-capacity sensitivity study are shown in Table 5. In Table 5, wehave tried to capture a wide range of realistic properties, but it ispossible to widen the ranges considered here even further. On thebasis of the scaling shown in Eq. 3, the thermal front will propa-gate as dictated by the final values of � and � as well as thefractional-flow function of the system. Because the fractional flowis kept the same in all cases, we can isolate the impact of � and �.In all cases, the slow shocks seen in the saturation profiles corre-spond to temperature shocks, and the temperature shock movessignificantly slower than the leading saturation front. The propa-gation behavior of the thermal front is similar in the case of hot-water injection. In fact, the speeds of the thermal fronts are thesame because we have considered the same fractional-flow curves,as explained in the Sample Solutions section.

Between the parameters � and �, � shows more variability withrespect to � because of the contribution of the porosity term. It is

possible, therefore, to correlate the location of the temperaturefront with respect to � alone for a given oil/water system. Becausefinding the location of the temperature front can be obtained easily,as explained in the Solution Construction section, by drawing atangent to the fractional-flow curve from (−�, −�), there is no needto develop a generalized correlation for the speed of the tempera-ture front. It is still possible, however, to see the speed of thetemperature front decrease as � increases (Fig. 17). Higher �means that the proportion of the heat transferred to the rock ishigher, and, therefore, the temperature front will lag behind thesaturation front even more. In this figure, � varied between 0.48and 0.83 for the given set of fractional-flow function described bythe parameters given in Tables 2 and 3.

ConclusionsWe have solved the nonisothermal Buckley-Leverett problem bothfor hot- and cold-water injection including an inert tracer andcompared analytical solutions with numerical solutions. The pri-mary conclusions are1. Simplified analytical solution of the nonisothermal Buckley-

Leverett problem with tracer (both hot- and cold-water injec-tion) is shown to be equivalent to three tangents drawn on thefractional-flow function of the system.

2. The temperature front propagates much slower than the satura-tion and tracer fronts and dissipates significantly when conduc-tive terms are considered. The mobility changes induced by thetemperature front, therefore, cannot be mimicked in black-oilsimulators by modifying the mobility by use of the standardinteracting tracer options.

3. By use of the radial transformation of Welge (Welge et al.1962), radial solutions are constructed. The temperature frontpropagates (in distance) further in radial coordinates because ofquadratic dependence of the volume on radial distance.

4. The temperature front slows down as more heat is transferred tothe rock matrix (i.e., low porosity, high �).

5. Radial solution of the problem is compared with the numericalsolutions and shows the temperature front is more prone tonumerical dispersion.

6. Numerical solutions agree well with the analytical solutions.

Fig. 15—Numerical simulation of cold waterflooding with atracer by use of fine grids (Nx=1,000) at tD=0.25 including theeffects of conduction. Solid lines are the analytical solutions byuse of the method of characteristics.

Fig. 16—Numerical simulation of hot waterflooding with a tracerby use of fine grids (Nx=1,000) at tD=0.25 including the effects ofconduction. Solid lines are the analytical solutions by use of themethod of characteristics.

Fig. 14—Numerical simulation of hot waterflooding with a tracerby use of fine (Nx=1,000) and coarse (Nx=10) grids at tD=0.25.Solid lines are the analytical solutions by use of the method ofcharacteristics.


Nomenclaturea, b � parameters for Langmuir-type adsorption function (Eq.

A-12)A � area open to flow, ft2

cD � dimensionless tracer concentrationcDtD

� partial derivative of dimensionless concentration withrespect to dimensionless time

cDxD� partial derivative of dimensionless concentration with

respect to dimensionless distanceci � injected tracer concentration, ppm

cvo � heat capacity of oil, BTU/lbm-oFcvr � heat capacity of rock, BTU/lbm-oFcvw � heat capacity of water, BTU/lbm-oF

D � dimensionless adsorption term in Eq. A-13D� � adsorption term as described by Eq. A-12fS � partial derivative of fractional flow of water with

respect to water saturation, fS =�fw

�SwfTD

� partial derivative of fractional flow of water with

respect to dimensionless temperature, fTD=

�fw

�TDfw � fractional flow of water

g � dimensionless ratio function defined by g=fw+�

Sw+�

gtD� partial derivative of g with respect to dimensionless

timegxD

� partial derivative of g with respect to dimensionlessdistance

h � dimensionless ratio function defined by h=fw

Swh� � modified dimensionless ratio function defined by h�=

fw

Sw+dD

dcD

.

I � identity matrixJ � coefficient matrix (Eq. A-1)

kro � relative permeability of oilko

ro � endpoint relative permeability of oilkrw � relative permeability of waterko

rw � endpoint relative permeability of waterL � distance, ft

no � relative permeability exponent of oilnw � relative permeability exponent of waterNr � Number of grids in r-directionNx � Number of grids in x-direction

q � injection rate, ft3/dayr � radial distance, ft

rD � dimensionless radial distance

S � scaled water saturation defined by S=Sw−Swr

1−Sor−Swr, fraction

Sor � residual oil saturation, fractionStD

� partial derivative of water saturation with respect todimensionless time

Sw � water saturation, fractionSw,lead� water saturation of the leading (fast) shock, fraction

Swi � initial water saturation, fractionSwr � residual water saturation, fractionSxD

� partial derivative of water saturation with respect todimensionless distance

Sw* � water saturation obtained from g�Sw�=�fw

�Sw�

TD=1

Sw** � water saturation obtained from g�S**w �|TD=0=g�S*w�|TD=1

t � time, daysFig. 17—Characteristic propagation speed of TD front vs. �.


tD � dimensionless timeTD � dimensionless temperature

TDtD� partial derivative of dimensionless temperature with

respect to dimensionless timeTDxD

� partial derivative of dimensionless temperature withrespect to dimensionless distance

Ti � initial (reservoir) temperatureTw � inner-boundary temperature

x � linear distancexD � dimensionless distanceX� � eigenvectors

� � dimensionless property function defined by �=�ocvo

�wcvw−�ocvo

� � dimensionless property function defined by

�=�ocvo+

1−�

��rcvr

�wcvw−�ocvo

� � eigenvalues�o � viscosity of oil, cp�w � viscosity of water, cp�o � density of oil, lbm/ft3

�r � density of rock, lbm/ft3

�w � density of water, lbm/ft3

� � porosity, fraction

AcknowledgmentAuthors thank Shell E&P for granting permission to publish thismanuscript.

ReferencesBarkve, T. 1989. The Riemann Problem for Nonstrictly Hyperbolic System

Modeling Nonisothermal, Two-Phase Flow in a Porous Medium. SIAMJ. of Applied Mathematics. 49 (3): 784–798. DOI: 10.1137/0149045.

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Courant, R., Friedrichs, K.O., and Lewy, H. 1967. On the Partial Differ-ence Equations of Mathematical Physics. IBM J. of Research and De-velopment Development 11 (2): 215–234.

Dindoruk, B. 1992. Analytical Theory of Multicomponent Multiphase Dis-placement in Porous Media. PhD dissertation, Stanford, California:Stanford University.

Fayers, F.J. 1962. Some Theoretical Results Concerning the Displacementof a Viscous Oil by a Hot Fluid In a Porous Medium. J. of FluidMechanics. 13: 65–76. DOI: 10.1017/S002211206200049X.

Hashem, M. 1990. Saturation Evaluation Following Water Flooding. En-gineering thesis, Stanford, California: Stanford University.

Hovdan, M. 1989. Water Injection—Incompressible Analytical SolutionWith Temperature Effects. Technical Report MH–1/86. In Norwegian.Stavanger: Statoil.

Isaacson, E. 1980. Global Solution of a Riemann Problem for Non StrictlyHyperbolic System of Conservation Laws Arising in Enhanced OilRecovery. New York City: The Rockefeller University.

Jeffery, A. 1976. Quasilinear Hyperbolic Systems and Waves. London:Pitman Publishing.

Johns, R.T. 1992. Analytical Theory of Multicomponent Gas Drives WithTwo-Phase Mass Transfer. PhD dissertation, Stanford, California:Stanford University.

Johns, R.T. and Dindoruk, B. 1991. Theory of Three-Component, Two-Phase Flow in Radial Systems. In Scale-Up of Miscible Flood Pro-cesses. Final report, Contract No. DE-FG21-89MC26253-5, US DOE,Washington, DC.

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AppendixEqs. 1 through 3 can be written in matrix form as:

�StD

TDtD

cDtD

� + �fs fTD

0

0 g 0

0 0 h��

SxD

TDxD

cDxD

� = �0

0

0� , . . . . . . . . . (A-1)

where h [water particle velocity as Isaacson (1980)] is defined by

h =fw

Sw. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-2)

This can also be rewritten symbolically as

�I�Y�TD− �J�Y�xD

= 0�, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-3)

where [I] is the identity matrix, [J] is the coefficient matrix, and Yis the vector of dependent variables Sw, TD, and cD. The eigenvalueproblem posed by Eq. A-1 is

�J − �I�X� = 0�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-4)

The eigenvalues for this system are

�1 =�fw

�Sw, �2 = g, �3 =

fw

Sw, . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-5)

and the corresponding eigenvectors are

X� 1 = �1

0

0�, X� 2 =�

�fw

�Sw

g −�fw

�TD

0� , X� 3 = �

0

0

1�. . . . . . . . . . (A-6)


Left eigenvectors can be extracted by use of

�X1 X2 X3��fs fTD

0

0 g 0

0 0 h�

−��X1 X2 X3� = 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-7)

for �1=�fw

�Sw, yielding the left eigenvector

X� 1 = � �fw

�Sw− g,

�fw

�TD, 0�. . . . . . . . . . . . . . . . . . . . . . . . . (A-8)

Multiplying both sides of Eq. A-1 with the left eigenvector de-scribed by Eq. A-8 leads to new set of dependent variables de-fined by

�g

�tD+ fs

�g

�xD= 0

�TD

�tD+ g

�TD

�xD= 0

�cD

�tD+ h

�cD

�xD= 0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-9)

or in matrix form,

�gtD

TDtD

cDtD

� + �fs 0 0

0 g 0

0 0 h��

gxD

TDxD

cDxD

� = �0

0

0� . . . . . . . . . . (A-10)

The differences between the new system of equations and thesystem posed by Eq. A-1 are (1) a new dependent variable isintroduced in place of water saturation, (2) the coefficient matrix,[J], is diagonal, and (3) the temperature derivative of the frac-tional-flow function is not needed. The eigenvalues of the systemdefined by Eq. A-10, however, are the same as the system definedby Eq. A-1, and they are

�1 =�fw

�Sw, �2 = g, �3 =

fw

Sw. . . . . . . . . . . . . . . . . . . . . . . (A-11)

One of the main advantages of the new formulation originatesfrom the structure of the coefficient matrix [J]. The eigenvectorsdefined by the eigenvalues are all constant. That means that g isconstant along constant TD, or TD is constant along constant g, andsimilarly, cD lifts the solution only in the space defined by theother two dependent variables (TD and g), as shown in Fig. 4.

Inclusion of noninteracting adsorption term for tracer transportwill not complicate the solutions or the equation system. Adsorptioncan be represented by use of a Langmuir-type adsorption function:

D� =ac

1 + bc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-12)

The transport equation for the tracer (Eq. 2) becomes:

��cDSw + D�

�tD+

��cD fw�

�xD= 0, . . . . . . . . . . . . . . . . . . . . . . . . . (A-13)

where D is the dimensionless adsorption function. Simplificationof Eq. A-13 will yield

�cD

�tD+� fw

Sw +dD

dcD

� �cD

�xD= 0, . . . . . . . . . . . . . . . . . . . . . . . (A-14)

h� =fw

Sw +dD

dcD

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-15)

h� term is a modified version of the h term including a term relatedto adsorption, as shown in Eq. A-9 and as defined by Eq. A-2.In this case (the case with adsorption), the propagation speed ofthe tracer can be obtained by drawing a tangent from (−h(0),0)to the fractional-flow curve as explained in the section on solu-tion construction.

SI Metric Conversion Factors°F (°F−32)/1.8 � °Cft2 × 9.290 304* E−02 � m2

ft3 × 2.831 685 E−02 � m3

lbm × 4.535 924 E−01 � kg

*Conversion factor is exact.

Deniz Sumnu-Dindoruk is a staff reservoir engineer and teamleader at Shell Exploration and Production Company, Uncon-ventional Oil in Houston where she works in using unconven-tional thermal recovery techniques to produce heavy oil andbitumen reservoirs. She holds BS and MS degrees in petroleumengineering from Middle East Technical University, Turkey anda PhD degree in petroleum engineering from Stanford Univer-sity. Birol Dindoruk is a principal technical expert in reservoirengineering working for Shell International E&P since 1997, andadjunct faculty at the University of Houston, department ofchemical engineering. He is a global consultant for fluid prop-erties (PVT), miscible/immiscible gas injection, EOR and simu-lation. Dindoruk holds a PhD degree in petroleum engineeringfrom Stanford University and an MBA degree from the Univer-sity of Houston. He is a recipient of SPE Cedric K. FergusonMedal in 1994. Dindoruk was one of the Co-Executive Editors ofSPE Journal of Reservoir Evaluation and Engineering (2004–2006) and is currently the Editor in Chief for Journal of Petro-leum Engineering Science and Technology.


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