Appl. sci. Res. Vol. A I

CALCULATIONS ON THE FLOW OF HETEROGENEOUS MIXTURES THROUGH

POROUS MEDIA *) by H. C . BR INKMAN

Laboratory N.V. De Bataafsche Petroleum Maatschappij

Summary M u s k a t and M e r e s 1) have fo rmulated basic d i f ferent ia l equat ions

govern ing the mot ion of heterogeneous f luid mix tures through porous media. They obta ined a solut ion of these equat ions by numer ica l in tegrat ion for the prob lem of a co lumn in i t ia l ly fi l led wi th l iquid saturated wi th gas, which is closed at one end and kept at a constant low pressure at the other 1).

Buck ley and Leveret t2 ) obta ined an ana lyt ica l so lut ion for the prob lem of a co lumn in i t ia l ly fi l led wi th l iquid which is f looded wi th a second immisc ib le l iquid. They found a solut ion in which the saturat ion is a th ree-va lued funct ion of the coord inate a long the column.

In our paper in the f irst place a discussion is given of Buck ley and Leve- re t t ' s solut ion. I t appeared that the t rue solut ion which conta ins a discont i - nu i ty may be der ived f rom the three-va lued solut ion by a discussion of the integra l re lat ion which represents the to ta l l iquid recovery f rom the column. This discussion bears a formal resemblance to that occurr ing in the theory of van der Waa l ' s equat ion of state.

The second prob lem t reated in our paper is that of a vert ica l co lumn in i t ia l ly fi l led wi th l iquid saturated wi th gas under a h igh pressure, which is opened at its lower end. For th is p rob lem we succeeded in f inding an ana ly t i ca l so lut ion for low values of the pressure gradient . Here again the paradox of a th ree-va lued solut ion occurred and led to a d i scont inu i ty in the saturat ion .

A general a rgument is g iven to the effect that th ree-va lued solut ions are unavo idab le for a theoret ica l t reatment based on M u s k a t 's equat ions of prob lems which lead to d iscont inu i t ies in the saturat ion .

w 1. Introduction. The equations o/ [low. The flow of a homoge- neous liquid or gas through a porous material may be described by Darcy's law:

k v = - - - grad p (1)

*) Paper presented at the VIIth Congres of Applied Mechanics, London 1948.

Appl. sei. Res. A i 2l*

- - 3 3 3 -

334 H.C. BR INKMAN

where: v is the mean rate of flow, ~ the viscosity, grad p the pressure gradient, and k the permeability of the porous material.

Relation (1) is a semi-empirical law which may be regarded as an extension of Poiseuille's law by the introduction of the empirical constant k.

M u s k a t and M e r e s i) have further generalized (1) in order to make it applicable to the flow of heterogeneous liquid-gas mixtu- res. They introduced two equations analogous to (1) :

kL v t = - - - - grad p (2a)

~t

kG v~ -- grad p (2b)

where the indices l and g refer to the liquid and the gas phase respectively, while L and G, called the relative permeabilities, are two factors which were chosen so as to obtain a fit between the experimental results and the equations (2a) and (2b).

It should be emphasized that this is an entirely formal procedure; (2a) and (2b) may be regarded as definitions of L and G. The experi- mental work of Wyckof f and Botset3) on the flow of heterogeneous mixtures, indicated that (2a) and (2b) may be very useful for calculations, because of the fact that L and G appeared to have very simple properties, In a good approximation L and G proved to be functions of the saturation only. The saturation S is defined as the volumefraction of the pores filled with liquid.

At first sight this property of L and G seems to be rather surpri- sing. L and G might be expected to be functions of many of the pro- perties of the fluids and of the porous.material, e.g. grad p, ~j/~, k and the interfacial tension. We may be sure that W y c k o f f and B o t s e t's result only holds for a limited range of these variables.

However, as a detailed knowledge of the properties of L and G is lacking, we based our calculations on W y c k o f f and 13 o t s e t's experiments. Their curves for L and G are reproduced in fig. I. This figure shows that the L and G curves are very dissimilar. The G curve shows an inflection point while the L curve does not. This is probably caused by the different wetting properties of the liquid and the gas. The liquid will wet the walls of the pores.

M u s k a t and M e r e s J) have applied (2a) and (2b) in order to

CALCULATIONS ON THE FLOW O~ HETEROGENEOUS MIXTURES 335

treat the following problem : A column consisting of porous material is initially filled with liquid, saturated with gas under a high pressure The pressure at one end of the column is then lowered and kept at a constant low value. A mixture of liquid and gas will leave the column at this end and the course of pressure and saturation is calculated by numerical methods. A reaHnsight in the consequences of the assump- tion of (2a) and (2b) can only be reached by obtaining analytical solutions for some problems and by studying their properties. It was therefore thought worth while to investigate some problems which allow of an analytical solution. The fundamental problem how to derive (2a) and (2b) from a detailed study of the flow of two phases

,I re/a b~ l permea3i//g/e.s

o ! ---',5

Fig. 1.

through a pore system, is not discussed here. The basic equations for our calculations are obtained by writing

down the equations of continuity for the liquid and the gas phase. The equation for the gas is complicated by the fact that the gas may dissolve into the liquid. As a result we have to distinguish between the pure liquid and a liquid phase which is a mixture of pure liquid and dissolved gas.

We introduce: [ - - the porosity of the medium; S - - the satura- tion; M - - the mass of the gas dissolved in unit volume of the liquid phase; N - - the mass of the pure liquid in unit volume of the liquid phase; Q~ the density of the gas phase. The quantities M, N and 0~ are functions of the pressure p.

The equations of continuity read:

O(NS) div (Nvl) = - -1 Ot (3a)

336 H.c. BRINKMAN

div (~.v~) + div (My,) = --la(MS)at laEq, (lot--S)] (3b) Substitution of v t and v, from (2a)and (2b) leads to two partial

differential equations for the saturation S and the pressure p. These equations are non-linear. Nevertheless analytical solutions exist for nontrivial cases.

w 2. The /low o/ two immiscible liquids. A very simple case is obtained for the problem of two immiscible, incompressible liquids, indicated as a and b. In this case the mass of dissolved material M = 0, while the liquid densities of both phases are constant. We discuss the one-dimensional case of a column initially filled with liquid a into which liquid b is introduced at one end. The saturation S is defined as the fraction of the pore volume filled with liquid a.

In the equations of flow (2a) and (2b) relative permeabilities A and B instead of L and G have to be introduced. A and B are functions of S like L and G. However, depending on the wetting properties of a and b, the exact course of A and B may differ considerably from that of L and G.

Equations (3a) and (3b) then become, after substitution of (2a) and (2b):

0 (kA op) aS (4a)

a apl as (4b)

A solution of these equations was given by B u c k 1 e y and L e- v e r e t t 2). We will extend the discussion of the general solution of the equations, as this will give an introduction into the more com- plicated problem of the next section.

Adding (4a) and (4b) gives

or

a /'A ~ ) ap [ _ o (s)

op c (6)

ax A/n,, + Bin,,

CALCULATIONS ON THE FLOW OF HETEROGENEOUS MIXTURES 337

where ck is total rate of flow of both phases together. Substitution of (6) into (4a) gives

Introduct ion of a function of the saturation called F and defined by

A/~la F(S) =

A/n~ + B/no yields

dF OS / OS (7)

dS Ox -- kc Ot

The approximate course of the functions F and dF/dS has been sketched in fig. 2.

l

0 --.,_s I O

Fig. 2.

/ --'.S

The general solution of (7) is:

s = x +- [ - ds t (8)

where v/is an arbitrary function, to be determined from the initial conditions.

Let the column extend from x = 0 to x = X, and assume that initially S = [ for 0 < x < X, while at x = X liquid b is intro- duced into the column at the rate ck. Hence at t = 0, S is a function of x as indicated by the solid lines in fig. 3. According to (8) the argu 2 m ent of ~p will vary with t but at a different rate for different

Appl. sci. Res, A ! 22

338 H. C. BRINKMAN

values of S, in consequence of the presence of the factor dF/dS." As a result according to equation (8) after a certain time t the course of S will have the shape as indicated in fig. 3 by the dotted curve. This course of S is paradoxical, as it is a three-valued function for a certain region of values of x. In this region the physical meaning of the solution is not yet cleaI'.

It should be kept in mind, however, that initially S, considered as a function of x, shows a discontinuity at x ---- X. This discontinuity will move through the column at a certain rate. It will now be proved that the true solution (indicated by crosses in fig. 3) has a disconti- nuity at x = Xl, where the value of x I is a function of the time which has to be determined.

'I . . . . . . . ..-.,1 / .- .... g . , , /

i '%, / j C_ x~ X ~ x

Fig. 3.

The velocity of the discontinuity may be determined by consider- ing two imaginary cross-sections through the column at both sides of the plane of discontinuity x I. The change in liquid content of the region bounded by these cross-sections can be determined in two ways, on the one hand by applying (2a) and (2b) to a calculation of the rate of flow through the cross-sections, on the other hand by calculating the change in liquid content due to the movement of the boundary x 1.

Equating the two results yields

/ (1 - - S) dxl kB C d--t = - - ~b A/~h + B/rl--b" (9)

This is an equation determining the rate of propagation of x 1. However, a much easier way of determining x I is presented by con- sidering the total quantity R of liquid b in the column:

R =/ ) ( I - -s ) dx. (I0) 0

CALCULAT IONS ON THE FLOW OF HETEROGENEOUS MIXTURES 339

In fig. 4 R/] is indicated by the shaded areas a for the three-valued solution and b for the true solution. These shaded areas should be equal. Therefore, the construction of the discontinuity at x I may be performed as indicated in fig. 4c where the two shaded regions should be of equal area.

~s

,5

1

,s ]

Fig. 4.

---* .lr

"---'*X

c?

C

It is easily proved that this construction is equivalent to a solution of equation (9). In fact (9) is a differential equation derived from the equations of flow, while (10) is an integral relation into which the solution for S of the equations of flow has to be substituted. The equivalence of (9) and (10) may be proved by straightforward calcul- ation by differentiating (10) with respect to t.

The paradox discussed in this paragraph bears a formal resem- blance to that occurring in the theory of van der Waals' equation of state. There also the discussion of an integral relation (f Vdp) leads to an analogous geometrical construction.

An analogous paradox occurs in the theory of chromatography.

w 3. A vertical column containing liquid and gas. In this section a more complicated problem is considered, viz. a vertical porous

340 H.C. BRINKMAN

column initially filled with liquid saturated with gas, which column is opened at its lower end. For this problem the equations of flow (2a) and (2b) have to be extended so as to include gravitational forces. The only way in which this can be done simply and consistently is:

kL vl ~ - - - - (grad p + Q:g) (l 1 a)

*h

kG vg -- - - (grad p + egg) (l 1 b)

where 0~ is the liquid density and g the acceleration of gravity. In how far (1 l a) and (1 I b) are a good description of this type of

flow remains to be determined by experiment..The precise meaning of p, for instance, is a very difficult problem which involves a dis- cussion of the interfacial tensions; p as well as S should be considered to be mean values over a small volume element which still contains many pores.

The following assumptions about various physical properties of liquid and gas are made:

Henry's law: M = cp; Boyle's law: eg = cp; and further: N = 0~ = constant.

The constants of Boyle's law and Henry's law are taken to be the same. This is often the case in actual gas-liquid mixtures.

With these simplifying assumptions the introduction of (1 l a) and (llb) into the equations of continuity (3a) and (3b) yields, for our one-dimensional case:

OIL( Op )I ~h] OS (12a) Sx ~x + Ng -- k St

(op + (12b/ Sx p L ~ + Ng' rh] op

- - + ~ \, Sx k St The following boundary conditions, determining the solution of

these equations, are chosen. At its upper end (x = X) the column is assumed to be shut off. Therefore v~ ---- 0 and vg = 0 at x = X. This condition is met by taking S = 0 and sp/ax = --~g. Physically this means that at the upper end of the column an infinitesimal region, where only gas is present, is formed; all liquid proceeding downwards.

CALCULATIONS ON THE FLOW OF HETEROGENEOUS MIXTURES 341

At the lower, end (x = 0), where the column is open, a condition determining the rate of flow of one of the phases might be chosen as a boundary condition. Another possibility is to give the value of @lax at x = 0. However, a different condition will prove to be more expedient, as we shall see at a later stage in the discussion.

At first sight one might expect the search for an analytical solu- tion of equations (12a) and (12b) to be a hopeless task. It is a sur- prising fact, however, that such a solution can be found if one further assumption is made, viz. that the pressure gradient @lax is every- where small compared to p/x . The assumption of a small value of 9p/gx means that at any moment the pressure p and therefore the solubility M and the density of the gas Qg can be taken to be constant throughout the column. The value of ap/ox may be kept small in practice by regulating the rate of flow at the lower end of the column. The calculation is further simplified if we assume that the pressure in the column is given as a function of t:

p = po ~ (13)

This means that the rate of pressure decrease ir~ the column has been fixed. This rate, determines the rate at which the column is depleted. It is related to the rate of saturation decline and, therefore, to the rate of flow at the lower end of the column. Relation (13) may, therefore, be accepted as a substitute for the missing boundary condition at the lower end of the column.

With these assumptions we have introducr such simplification as to-make the solution of the equations a relatively easy affair.

Equation (12b) can be integrated with respect to x from an arbi- trary value of x to x =X. As the left hand member of (12b), is a derivative with respect to x, while the right hand member does not depend on x, according to our assumptions, the integration yields an equation from which we obtain the following expression for ap/ax

~p _ (~,/a/k ) (X - - x) - - LNg- - G (~/~g) cpg (14) Ox L + (~7,1~7~) C

This expression for @lax is now introduced into equation (12a). A differential equation for S is obtained:

k st - ax (bp - - a) (15)

342 H.C. BRINKMAN

where H is a function of S" H = 1 + ,hG/,igL, while b and d are constants" b = ~/t cg/~ and d = ~t Ng/~g. The general solution of this equation is:

~(X--x) G. b H +(d--bp) ~- + p exp {--fHdS}.fG exp {fHdS}dS

----- ~v [p exp {--fndS}l (16)

where ~0 [ ] is an arbitrary function and d = ~l[a/k. In order to determine ~0, the boundary conditions have to be intro-

duced into the solution. These boundary conditions are"

fo r t=O :P :Po ; fo r t = Oandx : / :X : S = I; for t > Oandx=X:S=O.

s"

P

- " * X

Fig. 5.

The introduction of these boundary conditions in (16) is not very easy. In order to clarify our procedure an S-x-p space has been represented in fig. 5. The saturation S is a function of x and t, but instead of t the pressure p is introduced as an independent variable by means of (13). In the figure the boundary conditions are drawn as solid lines through which an integral surface should be laid. It is immediately clear that no simple surface can be laid through the three solid lines which intersect at angles of 90 degrees.

An integral surface satisfying conditions a and b is easily found. It is given by

S

P = Po exp {fgdS}. (17) 1

CALCULATIONS ON THE FLOW OF HETEROGENEOUS MIXTURES 343

In this solution S is a function of p only. The integral surface, therefore, consists of straight lines parallel to the x-axis. It is indi- cated in fig. 5.

The construction of a solution also satisfying condition c was found to be possible by laying an integral surface through the vertical line x = X, p = P0 in fig. 5. Substitution of these values of x and p in the solution (16) leads to a definition of ~v:

G (d - - bPo ) -~ + bPo exp {-- f HdS}.f G exp {f HdS} dS =

= ~v {P0 exp {- - fHdS} (18) or introducing:

S Y = P0 exp {-- f HdS} (20)

1

G s s %v(y) = (d - - bPo ) ~ + byfG exp {fHdS} dS.. (21)

1 1

For any value of S a value of y and a value of ~p may be deduced from these two equations. Thus ~v is known as a function of y. Sub- stitution of this function in (16) gives a definite relation between S, p and x.

In principle, therefore, we have now solved our problem. When making numerical applications, however, it was found that the solu- tion in this form is not of much use. Both ~v and y appeared to attain enormous values, which cancelled again after substitution in (16). This made impossible satisfactory numerical evaluation of the rela- tion between S, p and x. It was found, however, that these difficul- ties could be avoided by introducing a new function instead of ~v.

A function S 1 of p and S was defined by the following relation: S

P = P0 exp {f HdS}. (22) St

Introduction of (22) into (20) and (21) leads to an expression for ~v: S St

~v[p exp { - - f HdS}] -~ %v[p o exp { - - f ndS}] ---- 1 1

GI s, s, = (d - - bPo ) -H-~ + bp~ exp. {--/ l idS}, .fG, exp {fHdS} dS

where G t = G(S1) and n t ---- H(St).

344 H. C. BRINKMAN

The solution is now obtained in the form :

,~(X - - x) G G, H +(d--bp)-~--(d--bpo )~+

S~ S S

+ bPo exp {-- f HdS} .f G exp {f HdS} dS = 0. (23) I Sj 1

For numerical applications a value of S and S 1 should be chosen and p and x be'calculated from (22) and (23).

In this way the integral surface may be constructed. A cross- section through it for a certain value of p (and, therefore, because of (13), for a certain value of t) is drawn in fig.-6. In this problem

I

,s 1

X ,x Fig. 6.

S again proves to be a many-valued function of x. Another remark- able property of the curve in fig. 6 is that it cannot be extended "beyond a certain S-value, at which S1, as defined by (22), reaches the value 1. For larger Sl-vahies H is not defined and (23) cannot be applied numerically. Now for S l = 1 relation (22) reads:

S P = P0 exp {f HdS} (24)

1 which is equal to (17).

This means that the solutions (17) and (23) intersect at the end- point of the S--x curve of fig. 6.

In fig. 7 both solutions have been drawn. It appears that the solutions (17) and (23) together form a complete solution of our equations meeting all boundary conditions. The integral surface consists of the surface already drawn in fig. 5 together with the surface (23) which meet along a certain curve. Surface (23) contains a fold. A cross-section through this complicated integral surface for a certain value of p is given in fig. 7.

CALCULATIONS ON THE FLOW OF HETEROGENEOUS MIXTURES 345

In fig. 7 S is again a three-valued function of x. This difficulty is overcome in the same way' as discussed in the preceding paragraph. A discontinuity of the S-x curves occurs, which is indicated in fig. 7. Its x-value is determined by equating the shaded areas in fig. 7.

11

J

X ~x

Fig. 7.

The physical meaning of this discontinuity is that in the upper part of the column a regio.n of low saturation (a gascap) is formed, separated by a sharp boundary from the rest of the column. In this lower region the saturation appears to be constant throughout the column at any time.

w 4. The inevitability o/ three-valued solutions. In this section it will be shown that the equations of flow (2a) and (2b) will inevitably lead to three-valued S-x functions.

,.s 1 (-

Fig. 8.

,X

In fig. 8 we have represented by a solid curve a situation where S is a monotonous function of x. Now according to (2a) and (2b) for S -~ 1, v~-+0 and for S~0, v~-+0. This means that for these S-values one of the phases is immobilized. Accordingly the satura- tion remains constant.

346 CALCULATIONS ON THE FLow OF HETEROGENEOUS MIXTURES

Therefore, when e.g. a liquid is introduced into the porous mate- rial at a high x-value, an increase of S will only occur in the middle region of the S-interval. This is sketched in fig. 8. The area under the dotted S-curves increases more and more as more liquid is in- troduced into the column, leading to three-valued S-functions as indicated in fig. 8.

A discontinuous S-x curve which corresponds to the actual situa- tion in the porous material cannot be obtained as a solution of the differential equation describing the rate of change of S.

We conclude that three-valued S-x functions are inevitable with this way of treatment.

Received 16th September, 1948.

REFERENCES

1) Muska t, M. ,and M. W. Meres , Phys ics7 (1936) 346. 2) Buck ley , S .E . ,and M.C. Leveret t, Trans. A.I.M.M.E. 146(1942) 107. 3) Wyckof I , R .D . , and H. G. Botset , Physics 7 (1936) 325.