mathematical geoscience.pdf

Chapter 7Groundwater Flow

Groundwater is water which is stored in the soil and rock beneath the surface of theEarth. It forms a fundamental constituent reservoir of the hydrological system, and itis important because of its massive and long lived storage capacity. It is the resourcewhich provides drinking and irrigation water for crops, and increasingly in recentdecades it has become an unwilling recipient of toxic industrial and agriculturalwaste. For all these reasons, the movement of groundwater is an important subjectof study.

Soil consists of very small grains of organic and inorganic matter, ranging in sizefrom millimetres to microns. Differently sized particles have different names. Partic-ularly, we distinguish clay particles (size <2 microns) from silt particles (2–60 mi-crons) and sand (60 microns to 1 mm). Coarser particles still are termed gravel.

Viewed at the large scale, soil thus forms a continuum which is granular at thesmall scale, and which contains a certain fraction of pore space, as shown in Fig. 7.1.The volume fraction of the soil (or sediment, or rock) which is occupied by the porespace (or void space, or voidage) is called the porosity, and is commonly denotedby the symbol φ; sometimes other symbols are used, for example n, as in Chap. 5.

As we described in Chap. 6, soils are formed by the weathering of rocks, andare specifically referred to as soils when they contain organic matter formed by therotting of plants and animals. There are two main types of rock: igneous, formed bythe crystallisation of molten lava, and sedimentary, formed by the cementation ofsediments under conditions of great temperature and pressure as they are buried atdepth.1 Sedimentary rocks, such as sandstone, chalk, shale, thus have their porositybuilt in, because of the pre-existing granular structure. With increasing pressure,the grains are compacted, thus reducing their porosity, and eventually intergranularcements bond the grains into a rock. Sediment compaction is described in Sect. 7.11.

Igneous rock tends to be porous also, for a different reason. It is typically thecase for any rock that it is fractured. Most simply, rock at the surface of the Earth

1There are also metamorphic rocks, which form from pre-existing rocks through chemical changesinduced by burial at high temperatures and pressures; for example, marble is a metamorphic formof limestone.

A. Fowler, Mathematical Geoscience, Interdisciplinary Applied Mathematics 36,DOI 10.1007/978-0-85729-721-1_7, © Springer-Verlag London Limited 2011

387

388 7 Groundwater Flow

Fig. 7.1 A granular porousmedium

is subjected to enormous tectonic stresses, which cause folding and fracturing ofrock. Thus, even if the rock matrix itself is not porous, there are commonly faultsand fractures within the rock which act as channels through which fluids may flow,and which act on the large scale as an effective porosity. If the matrix is porous atthe grain scale also, then one refers to the rock as having a dual porosity, and thecorresponding flow models are called double porosity models.

In the subsurface, whether it be soil, underlying regolith, a sedimentary basin,or oceanic lithosphere, the pore space contains liquid. At sufficient depth, the porespace will be saturated with fluid, normally water. At greater depths, other fluidsmay be present. For example, oil may be found in the pore space of the rocks ofsedimentary basins. In the near surface, both air and water will be present in thepore space, and this (unsaturated) region is called the unsaturated zone, or the vadosezone. The surface separating the two is called the piezometric surface, the phreaticsurface, or more simply the water table. Commonly it lies tens of metres below theground surface.

7.1 Darcy’s Law

Groundwater is fed by surface rainfall, and as with surface water it moves under apressure gradient driven by the slope of the piezometric surface. In order to char-acterise the flow of a liquid in a porous medium, we must therefore relate the flowrate to the pressure gradient. An idealised case is to consider that the pores consistof uniform cylindrical tubes of radius a; initially we will suppose that these are allaligned in one direction. If a is small enough that the flow in the tubes is laminar(this will be the case if the associated Reynolds number is �1000), then Poiseuille

flow in each tube leads to a volume flux in each tube of q = πa4

8μ|∇p|, where μ is

7.1 Darcy’s Law 389

the liquid viscosity, and ∇p is the pressure gradient along the tube. A more realisticporous medium is isotropic, which is to say that if the pores have this tubular shape,the tubules will be arranged randomly, and form an interconnected network. How-ever, between nodes of this network, Poiseuille flow will still be appropriate, and anappropriate generalisation is to suppose that the volume flux vector is given by

q ≈ − a4

μX∇p, (7.1)

where the approximation takes account of small interactions at the nodes; the nu-merical tortuosity factor X � 1 takes some account of the arrangement of the pipes.

To relate this to macroscopic variables, and in particular the porosity φ, we ob-serve that φ ∼ a2/d2

p , where dp is a representative particle or grain size so that

q/d2p ∼ −(

φ2d2p

μX)∇p. We define the volume flux per unit area (having units of veloc-

ity) as the discharge u. Darcy’s law then relates this to an applied pressure gradientby the relation

u = − k

μ∇p, (7.2)

where k is an empirically determined parameter called the permeability, having unitsof length squared. The discussion above suggests that we can write

k = d2pφ2

X; (7.3)

the numerical factor X may typically be of the order of 103.To check whether the pore flow is indeed laminar, we calculate the (particle)

Reynolds number for the porous flow. If v is the (average) fluid velocity in the porespace, then

v = uφ

. (7.4)

If a is the pore radius, then we define a particle Reynolds number based on grainsize as

Rep = 2ρva

μ∼ ρ|u|dp

μ√

φ, (7.5)

since φ ∼ a/dp . Suppose (7.3) gives the permeability, and we use the gravitationalpressure gradient ρg to define (via Darcy’s law) a velocity scale2; then

Rep ∼ φ3/2

X

(ρ√

gdp dp

μ

)2

∼ 10[dp]3, (7.6)

2This scale is thus the hydraulic conductivity, defined below in (7.9).


where dp = [dp] mm, and using φ3/2/X = 10−3, g = 10 m s−2, μ/ρ = 10−6 m2 s−2.Thus the flow is laminar for d < 5 mm, corresponding to a gravel. Only for free flowthrough very coarse gravel could the flow become turbulent, but for water percola-tion in rocks and soils, we invariably have slow, laminar flow.

In other situations, and notably for forced gas stream flow in fluidised beds or inpacked catalyst reactor beds, the flow can be rapid and turbulent. In this case, thePoiseuille flow balance −∇p = μu/k can be replaced by the Ergun equation

−∇p = ρ|u|uk′ ; (7.7)

more generally, the right hand side will be a sum of the two (laminar and turbulent)interfacial resistances. The Ergun equation reflects the fact that turbulent flow in apipe is resisted by Reynolds stresses, which are generated by the fluctuation of theinertial terms in the momentum equation. Just as for the laminar case, the parame-ter k′, having units of length, depends both on the grain size dp and on φ. Evidently,we will have

k′ = dpE(φ), (7.8)

with the numerical factor E → 0 as φ → 0.

7.1.1 Hydraulic Conductivity

Another measure of flow rate in porous soil or rock relates specifically to the passageof water through a porous medium under gravity. For free flow, the pressure gradientdownwards due to gravity is just ρg, where ρ is the density of water and g is thegravitational acceleration; thus the water flux per unit area in this case is just

K = kρg

μ, (7.9)

and this quantity is called the hydraulic conductivity. It has units of velocity. A hy-draulic conductivity of K = 10−5 m s−1 (about 300 m y−1) corresponds to a perme-ability of k = 10−12 m2, this latter unit also being called the darcy.

7.1.2 Homogenisation

The ‘derivation’ of Darcy’s law can be carried out in a more formal way using themethod of homogenisation. This is essentially an application of the method of mul-tiple (space) scales to problems with microstructure. Usually (for analytic reasons)one assumes that the microstructure is periodic, although this is probably not strictlynecessary (so long as local averages can be defined).

7.1 Darcy’s Law 391

Consider the Stokes flow equations for a viscous fluid in a medium of macro-scopic length l, subject to a pressure gradient of order �p/l. If the microscopic(e.g., grain size) length scale is dp , and ε = dp/l, then if we scale velocity withd2p�p/lμ (appropriate for local Poiseuille-type flow), length with l, and pressure

with �p, the Navier–Stokes equations can be written in the dimensionless form

∇.u = 0,

0 = −∇p + ε2∇2u,(7.10)

together with the no-slip boundary condition,

u = 0 on S : f (x/ε) = 0, (7.11)

where S is the interfacial surface. We put x = εξ and seek solutions in the form

u = u(0)(x, ξ) + εu(1)(x, ξ) . . . ,

p = p(0)(x, ξ) + εp(1)(x, ξ) . . . .(7.12)

Expanding the equations in powers of ε and equating terms leads to p(0) = p(0)(x),and u(0) satisfies

∇ξ .u(0) = 0,

0 = −∇ξp(1) + ∇2

ξ u(0) − ∇xp(0),

(7.13)

equivalent to Stokes’ equations for u(0) with a forcing term −∇xp(0). If wj is the

velocity field which (uniquely) solves

∇ξ .wj = 0,

0 = −∇ξP + ∇2ξ wj + ej ,

(7.14)

with periodic (in ξ ) boundary conditions and u = 0 on f (ξ ) = 0, where ej is theunit vector in the ξj direction, then (since the equation is linear) we have (summingover j )3

u(0) = −∂p(0)

∂xj

wj . (7.15)

We define the average flux

〈u〉 = 1

V

∫V

u(0) dV, (7.16)

3In other words, we employ the summation convention which states that summation is impliedover repeated suffixes, see for example Jeffreys and Jeffreys (1953).


where V is the volume over which S is periodic.4 Averaging (7.15) then gives

〈u〉 = −k∗.∇p, (7.17)

where the (dimensionless) permeability tensor is defined by

k∗ij = ⟨

wji

⟩. (7.18)

Recollecting the scales for velocity, length and pressure, we find that the dimen-sional version of (7.17) is

〈u〉 = − kμ

.∇p, (7.19)

where

k = k∗d2p, (7.20)

so that k∗ is the equivalent in homogenisation theory of the quantity φ2/X in (7.3).

7.1.3 Empirical Measures

While the validity of Darcy’s law can be motivated theoretically, it ultimately relieson experimental measurements for its accuracy. The permeability k has dimensionsof (length)2, which as we have seen is related to the mean ‘grain size’. If we writek = d2

pC, then the number C depends on the pore configuration. For a tubular net-

work (in three dimensions), one finds C ≈ φ2/72π (as long as φ is relatively small).A different and often used relation is that of Carman and Kozeny, which applies topseudo-spherical grains (for example sand grains); this is

C ≈ φ3

180(1 − φ)2 . (7.21)

The factor (1 − φ)2 takes some account of the fact that as φ increases towards one,the resistance to motion becomes negligible. In fact, for media consisting of unce-mented (i.e., separate) grains, there is a critical value of φ beyond which the mediumas a whole will deform like a fluid. Depending on the grain size distribution, thisvalue is about 0.5 to 0.6. When the medium deforms in this way, the descriptionof the intergranular fluid flow can still be taken to be given by Darcy’s law, butthis now constitutes a particular choice of the interactive drag term in a two-phaseflow model. At lower porosities, deformation can still occur, but it is elastic not vis-cous (on short time scales), and given by the theory of consolidation or compaction,which we discuss later.

4Specifically, we take V to be the soil volume, but the integral is only over the pore space volume,where u is defined. In that case, the average 〈u〉 is in fact the Darcy flux (i.e., volume fluid flux perunit area).

7.2 Basic Groundwater Flow 393

Table 7.1 Different grainsize materials and theirtypical permeabilities

k (m2) Material

10−8 gravel

10−10 sand

10−12 fractured igneous rock

10−13 sandstone

10−14 silt

10−18 clay

10−20 granite

In the case of soils or sediments, empirical power laws of the form

C ∼ φm (7.22)

are often used, with much higher values of the exponent (e.g. m = 8). Such be-haviour reflects the (chemically derived) ability of clay-rich soils to retain a highfraction of water, thus making flow difficult. Table 7.1 gives typical values of thepermeability of several common rock and soil types, ranging from coarse graveland sand to finer silt and clay.

An explicit formula of Carman–Kozeny type for the turbulent Ergun equationexpresses the ‘turbulent’ permeability k′, defined in (7.7), as

k′ = φ3dp

175(1 − φ). (7.23)

7.2 Basic Groundwater Flow

Darcy’s equation is supplemented by an equation for the conservation of the fluidphase (or phases, for example in oil recovery, where these may be oil and water).For a single phase, this equation is of the simple conservation form

∂

∂t(ρφ) + ∇.(ρu) = 0, (7.24)

supposing there are no sources or sinks within the medium. In this equation, ρ is thematerial density, that is, mass per unit volume of the fluid. A term φ is not presentin the divergence term, since u has already been written as a volume flux (i.e., the φ

has already been included in it: cf. (7.4)).Eliminating u, we have the parabolic equation

∂

∂t(ρφ) = ∇.

[k

μρ∇p

], (7.25)

and we need a further equation of state (or two) to complete the model. The simplestassumption corresponds to incompressible groundwater flowing through a rigid


porous medium. In this case, ρ and φ are constant, and the governing equationreduces (if also k is constant) to Laplace’s equation

∇2p = 0. (7.26)

This simple equation forms the basis for the following development. Before pur-suing this, we briefly mention one variant, and that is when there is a compressiblepore fluid (e.g., a gas) in a non-deformable medium. Then φ is constant (so k is con-stant), but ρ is determined by pressure and temperature. If we can ignore the effectsof temperature, then we can assume p = p(ρ) with p′(ρ) > 0, and

ρt = k

μφ∇.

[ρp′(ρ)∇ρ

], (7.27)

which is a nonlinear diffusion equation for ρ, sometimes called the porous mediumequation. If p ∝ ργ , γ > 0, this is degenerate when ρ = 0, and the solutions displaythe typical feature of finite spreading rate of compactly supported initial data.

7.2.1 Boundary Conditions

The Laplace equation (7.26) in a domain D requires boundary data to be prescribedon the boundary ∂D of the spatial domain. Typical conditions which apply are a noflow through condition at an impermeable boundary, u.n = 0, whence

∂p

∂n= 0 on ∂D, (7.28)

or a permeable surface condition

p = pa on ∂D, (7.29)

where for example pa would be atmospheric pressure at the ground surface. Anotherexample of such a condition would be the prescription of oceanic pressure at theinterface with the oceanic crust.

A more common application of the condition (7.29) is in the consideration offlow in the saturated zone below the water table (which demarcates the upper limitof the saturated zone). At the water table, the pressure is in equilibrium with the airin the unsaturated zone, and (7.29) applies. The water table is a free surface, andan extra kinematic condition is prescribed to locate it. This condition says that thephreatic surface is also a material surface for the underlying groundwater flow, sothat its velocity is equal to the average fluid velocity (not the flux): bearing in mind(7.4), we have

∂F

∂t+ u

φ.∇F = 0 on ∂D, (7.30)

if the free surface ∂D is defined by F(x, t) = 0.


7.2.2 Dupuit Approximation

One of the principally obvious features of mature topography is that it is relativelyflat. A slope of 0.1 is very steep, for example. As a consequence of this, it is typi-cally also the case that gradients of the free groundwater (phreatic) surface are alsosmall, and a consequence of this is that we can make an approximation to the equa-tions of groundwater flow which is analogous to that used in shallow water theoryor the lubrication approximation, i.e., we can take advantage of the large aspect ra-tio of the flow. This approximation is called the Dupuit, or Dupuit–Forchheimer,approximation.

To be specific, suppose that we have to solve

∇2p = 0 in 0 < z < h(x, y, t), (7.31)

where z is the vertical coordinate, z = h is the phreatic surface, and z = 0 isan impermeable basement. We let u denote the horizontal (vector) component ofthe Darcy flux, and w the vertical component. In addition, we now denote by∇ = ( ∂

∂x, ∂

∂y) the horizontal component of the gradient vector. The boundary condi-

tions are then

p = 0, φht + u.∇h = w on z = h,

∂p

∂z+ ρg = 0 on z = 0;

(7.32)

here we take (gauge) pressure measured relative to atmospheric pressure. The con-dition at z = 0 is that of no normal flux, allowing for gravity.

Let us suppose that a horizontal length scale of relevance is l, and that the corre-sponding variation in h is of order d , thus

ε = d

l(7.33)

is the size of the phreatic gradient, and is small. We non-dimensionalise the variablesby scaling as follows:

x, y ∼ l, z ∼ d, p ∼ ρgd,

u ∼ kρgd

μl, w ∼ kρgd2

μl2 , t ∼ φμl2

kρgd.

(7.34)

The choice of scales is motivated by the same ideas as lubrication theory. The pres-sure is nearly hydrostatic, and the flow is nearly horizontal.

The dimensionless equations are

u = −∇p, ε2w = −(pz + 1),

∇.u + wz = 0,(7.35)


with

pz = −1 on z = 0,

p = 0, ht = w + ∇p.∇h on z = h.(7.36)

At leading order as ε → 0, the pressure is hydrostatic:

p = h − z + O(ε2). (7.37)

More precisely, if we put

p = h − z + ε2p1 + · · · , (7.38)

then (7.35) implies

p1zz = −∇2h, (7.39)

with boundary conditions, from (7.36),

p1z = 0 on z = 0,

p1z = −ht + |∇h|2 on z = h.(7.40)

Integrating (7.39) from z = 0 to z = h thus yields the evolution equation for h in theform

ht = ∇.[h∇h], (7.41)

which is a nonlinear diffusion equation of degenerate type when h = 0.This is easily solved numerically, and there are various exact solutions which

are indicated in the exercises. In particular, steady solutions are found by solvingLaplace’s equation for 1

2h2, and there are various kinds of similarity solution. (7.41)is a second order equation requiring two boundary conditions. A typical situation ina river catchment is where there is drainage from a watershed to a river. A suitableproblem in two dimensions is

ht = (hhx)x + r, (7.42)

where the source term r represents recharge due to rainfall. It is given by

r = rD

ε2K, (7.43)

where rD is the rainfall rate and K = kρg/μ is the hydraulic conductivity. At thedivide (say, x = 0), we have hx = 0, whereas at the river (say, x = 1), the elevationis prescribed, h = 1 for example. The steady solution is

h = [1 + r − rx2]1/2

, (7.44)

and perturbations to this decay exponentially. If this value of the elevation of thewater table exceeds that of the land surface, then a seepage face occurs, where water


seeps from below and flows over the surface. This can sometimes be seen in steepmountainous terrain, or on beaches, when the tide is going out.

The Dupuit approximation is not uniformly valid at x = 1, where conditions ofsymmetry at the base of a valley would imply that u = 0, and thus px = 0. There istherefore a boundary layer near x = 1, where we rescale the variables by writing

x = 1 − εX, w = W

ε, h = 1 + εH, p = 1 − z + εP. (7.45)

Substituting these into the two-dimensional version of (7.35) and (7.36), we find

u = PX, W = −Pz, ∇2P = 0 in 0 < z < 1 + εH, 0 < X < ∞, (7.46)

with boundary conditions

P = H, εHt + PXHX = W

ε+ r on z = 1 + εH,

PX = 0 on X = 0,

Pz = 0 on z = 0,

P ∼ H ∼ rX as X → ∞.

(7.47)

At leading order in ε, this is simply

∇2P = 0 in 0 < z < 1, 0 < X < ∞,

Pz = 0 on z = 0,1,

PX = 0 on X = 0,

P ∼ rX as X → ∞.

(7.48)

Evidently, this has no solution unless we allow the incoming groundwater flux r

from infinity to drain to the river at X = 0, z = 1. We do this by having a singularityin the form of a sink at the river,

P ∼ r

πln

{X2 + (1 − z)2} near X = 0, z = 1. (7.49)

The solution to (7.48) can be obtained by using complex variables and the methodof images, by placing sinks at z = ±(2n + 1), for integral values of n. Making useof the infinite product formula (Jeffrey 2004, p. 72)

∞∏1

(1 + ζ 2

(2n + 1)2

)= cosh

πζ

2, (7.50)

where ζ = X + iz, we find the solution to be

P = r

πln

[cosh2 πX

2cos2 πz

2+ sinh2 πX

2sin2 πz

2

]. (7.51)


Fig. 7.2 Groundwater flowlines towards a river at X = 0,z = 1

The complex variable form of the solution is

φ = P + iψ = 2r

πln cosh

πζ

2, (7.52)

which is convenient for plotting. The streamlines of the flow are the lines ψ =constant, and these are shown in Fig. 7.2.

This figure illustrates an important point, which is that although the flow towardsa drainage point may be more or less horizontal, near the river the groundwaterseeps upwards from depth. Drainage is not simply a matter of near surface rechargeand drainage. This means that contaminants which enter the deep groundwater mayreside there for a very long time.

A related point concerns the recharge parameter r defined in (7.43). According toTable 7.1, a typical permeability for sand is 10−10 m2, corresponding to a hydraulicconductivity of K = 10−3 m s−1, or 3 × 104 m y−1. Even for phreatic slopes aslow as ε = 10−2, the recharge parameter r � O(1), and shallow aquifer drainage isfeasible.

However, finer-grained sediments are less permeable, and the calculation of r

for a silt with permeability of 10−14 m2 (K = 10−7 m s−1 = 3 m y−1 suggests thatr ∼ 1/ε2 1, so that if the Dupuit approximation applied, the groundwater surfacewould lie above the Earth’s surface everywhere. This simply points out the obvi-ous fact that if the groundmass is insufficiently permeable, drainage cannot occurthrough it but water will accumulate at the surface and drain by overland flow. Thefact that usually the water table is below but quite near the surface suggests that thelong term response of landscape to recharge is to form topographic gradients andsufficiently deep sedimentary basins so that this status quo can be maintained.

7.3 Unsaturated Soils

Let us now consider flow in the unsaturated zone. Above the water table, water andair occupy the pore space. If the porosity is φ and the water volume fraction perunit volume of soil is W , then the ratio S = W/φ is called the relative saturation. If

7.3 Unsaturated Soils 399

Fig. 7.3 Configuration of air and water in pore space. The contact angle θ measured through thewater is acute, so that water is the wetting phase. σws, σas and σaw are the surface energies of thethree interfaces

S = 1, the soil is saturated, and if S < 1 it is unsaturated. The pore space of an un-saturated soil is configured as shown in Fig. 7.3. In particular, the air/water interfaceis curved, and in an equilibrium configuration the curvature of this interface willbe constant throughout the pore space. The value of the curvature depends on theamount of liquid present. The less liquid there is (i.e., the smaller the value of S),then the smaller the pores where the liquid is found, and thus the higher the curva-ture. Associated with the curvature is a suction effect due to surface tension acrossthe air/water interface. The upshot of all this is that the air and water pressures arerelated by a capillary suction characteristic function which expresses the differencebetween the pressures as a function of mean curvature, and hence, directly, S:

pa − pw = f (S). (7.53)

The suction characteristic f (S) is equal to 2σκ , where κ is the mean interfacialcurvature: σ is the surface tension. For air and water in soil, f is positive as wateris the wetting phase, that is, the contact angle at the contact line between air, waterand soil grain is acute, measured through the water (see Fig. 7.3). The resulting formof f (S) displays hysteresis as indicated in Fig. 7.4, with different curves dependingon whether drying or wetting is taking place.

7.3.1 The Richards Equation

To model unsaturated flow, we have the conservation of mass equation in the form

∂(φS)

∂t+ ∇.u = 0, (7.54)

where we take φ as constant. Darcy’s law for an unsaturated flow has the form, nowwith gravitational acceleration included,

u = −k(S)

μ[∇p + ρgk], (7.55)


Fig. 7.4 Capillary suctioncharacteristic. It displayshysteresis in wetting anddrying

where k is a unit vector upwards, and the permeability k depends on S. If k(1) = k0(the saturated permeability), then one commonly writes k = k0krw(S), where krwis the relative permeability. The most obvious assumption would be krw = S, butthis is rarely appropriate, and a better representation is a convex function, such askrw = S3. An even better representation is a function such as krw = (

S−S01−S0

)3+, whereS0 is known as the residual saturation. It represents the fact that in fine-grainedsoils, there is usually some minimal water fraction which cannot be removed. It isnaturally associated with a capillary suction characteristic function pa − p = f (S)

which tends to infinity as S → S0+, also appropriate for fine-grained soils.In one dimension, and if we take the vertical coordinate z to point downwards,

we obtain the Richards equation

φ∂S

∂t= − ∂

∂z

[k0

μkrw(S)

{∂f

∂z+ ρg

}]. (7.56)

We are assuming pa = constant (and also that the soil matrix is incompressible).

7.3.2 Non-dimensionalisation

We choose scales for the variables as follows:

f = σ

dp

ψ, z ∼ σ

ρgdp

, t ∼ φμz

ρgk0, (7.57)

where dp is grain size and σ is the surface tension, assumed constant. The Richardsequation then becomes, in dimensionless variables,

∂S

∂t= − ∂

∂z

[krw

(∂ψ

∂z+ 1

)]. (7.58)

To be specific, we consider the case of soil wetting due to surface infiltration: ofrainfall, for example. Suitable boundary conditions for infiltration are

S = 1 at z = 0 (7.59)


if surface water is ponded, or

krw

(∂ψ

∂z+ 1

)= u∗ = μu0

k0ρwg= u0

K0, (7.60)

if there is a prescribed downward flux u0; K0 is the saturated hydraulic conduc-tivity. In a dry soil we would have S → 0 as z → ∞, or if there is a water tableat z = zp , S = 1 there.5 For silt with k0 = 10−14 m2, the hydraulic conductivityK0 ∼ 10−7 m s−1 or 3 m y−1, while average rainfall in England, for example, is≤1 m y−1. Thus on average u∗ ≤ 1, but during storms we can expect u∗ 1. Forlarge values of u∗, the desired solution may have S > 1 at z = 0; in this case pondingoccurs (as one observes), and (7.60) is replaced by (7.59), with the pond depth beingdetermined by the balance between accumulation, infiltration, and surface run-off.

7.3.3 Snow Melting

An application of the unsaturated flow model occurs in the study of melting snow.In particular, it is found that pollutants which may be uniformly distributed in snow(e.g. SO2 from sulphur emissions via acid rain) can be concentrated in melt wa-ter run-off, with a consequent enhanced detrimental effect on stream pollution. Thequestion then arises, why this should be so. We shall find that uniform surface melt-ing of a dry snowpack can lead to a meltwater spike at depth.

Suppose we have a snow pack of depth d . Snow is a porous aggregate of icecrystals, and meltwater formed at the surface can percolate through the snow packto the base, where run-off occurs. (We ignore effects of re-freezing of meltwater.)The model (7.58) is appropriate, but the relevant length scale is d . Therefore wedefine a parameter

κ = σ

ρgddp

, (7.61)

and we rescale the variables as z ∼ 1/κ , t ∼ 1/κ . To be specific, we will also take

krw = S3, (7.62)

and

ψ(S) = 1

S− S, (7.63)

based on typical experimental results.Suitable boundary conditions in a melting event might be to prescribe the melt

flux u0 at the surface, thus

krw

(∂ψ

∂z+ 1

)= u∗ = u0

K0at z = 0. (7.64)

5With constant air pressure, continuity of S follows from continuity of pore water pressure.


If the base is impermeable, then

krw

(∂ψ

∂z+ 1

)= 0 at z = h. (7.65)

This is certainly not realistic if S reaches 1 at the base, since then ponding mustoccur and presumably melt drainage will occur via a channelised flow, but we ex-amine the initial stages of the flow using (7.65). Finally, we suppose S = 0 at t = 0.Again, this is not realistic in the model (it implies infinite capillary suction) but it isa feasible approximation to make.

Simplification of this model now leads to the dimensionless Darcy–Richardsequation in the form

∂S

∂t+ 3S2 ∂S

∂z= κ

∂

∂z

[S(1 + S2)∂S

∂z

]. (7.66)

If we choose σ = 70 mN m−1, dp = 0.1 mm, ρ = 103 kg m−3, g = 10 m s−2,d = 1 m, then κ = 0.07. It follows that (7.66) has a propensity to form shocks,these being diffused by the term in κ over a distance O(κ) (by analogy with theshock structure for the Burgers equation, see Chap. 1).

We want to solve (7.66) with the initial condition

S = 0 at t = 0, (7.67)

and the boundary conditions

S3 − κS(1 + S2)∂S

∂z= u∗ on z = 0, (7.68)

and

S3 − κS(1 + S2)∂S

∂z= 0 at z = 1. (7.69)

Roughly, for κ � 1, these are

S = S0 at z = 0,

S = 0 at z = 1,(7.70)

where S0 = u∗1/3, which we initially take to be O(1) (and <1, so that surface pond-ing does not occur).

Neglecting κ , the solution is the step function

S = S0, z < zf ,

S = 0, z > zf ,(7.71)

and the shock front at zf advances at a rate zf given by the jump condition

zf = [S3]+−[S]+−

= S20 . (7.72)


Fig. 7.5 S(Z) given by(7.78); the shock frontterminates at the origin

In dimensional terms, the shock front moves at speed u0/φS0, which is in fact ob-vious (given that it has constant S behind it).

The shock structure is similar to that of Burgers’ equation. We put

z = zf + κZ, (7.73)

and S rapidly approaches the quasi-steady solution S(Z) of

−V S′ + 3S2S ′ = [S(1 + S2)S ′]′

, (7.74)

where V = zf ; hence

S(1 + S2)S′ = −S

(S2

0 − S2), (7.75)

in order that S → S0 as Z → −∞, and where we have chosen

V = S20 , (7.76)

(as S+ = 0), thus reproducing (7.72). The solution is a quadrature,

∫ S (1 + S2) dS

(S20 − S2)

= −Z, (7.77)

with an arbitrary added constant (amounting to an origin shift for Z). Hence

S − (1 + S20)

2S0ln

[S0 + S

S0 − S

]= Z. (7.78)

The shock structure is shown in Fig. 7.5; the profile terminates where S = 0 atZ = 0. In fact, (7.75) implies that S = 0 or (7.78) applies. Thus when S given by(7.78) reaches zero, the solution switches to S = 0. The fact that ∂S/∂Z is discon-tinuous is not a problem because the diffusivity S(1 + S2) goes to zero when S = 0.This degeneracy of the equation is a signpost for fronts with discontinuous deriva-tives: essentially, the profile can maintain discontinuous gradients at S = 0 becausethe diffusivity is zero there, and there is no mechanism to smooth the jump away.

Suppose now that k0 = 10−10 m2 and μ/ρ = 10−6 m2 s−1; then the satu-rated hydraulic conductivity K0 = k0ρg/μ = 10−3 m s−1. On the other hand, if


a metre thick snow pack melts in ten days, this implies u0 ∼ 10−6 m s−1. ThusS3

0 = u0/K0 ∼ 10−3, and the approximation S ≈ S0 looks less realistic. With

S3 − κS(1 + S2)∂S

∂z= S3

0 , (7.79)

and S0 ∼ 10−1 and κ ∼ 10−1, it seems that one should assume S � 1. We define

S =(

S30

κ

)1/2

s; (7.80)

(7.79) becomes

βs3 − s

[1 + S3

0

κs2

]∂s

∂z= 1 on z = 0, (7.81)

and we have S30/κ ∼ 10−2, β = (S0/κ)3/2 ∼ 1.

We neglect the term in S30/κ , so that

βs3 − s∂s

∂z≈ 1 on z = 0, (7.82)

and substituting (7.80) into (7.66) leads to

∂s

∂τ+ 3βs2 ∂s

∂z≈ ∂

∂z

[s∂s

∂z

], (7.83)

if we define t = τ/(κS30)1/2. A simple analytic solution is no longer possible, but

the development of the solution will be similar. The flux condition (7.82) at z = 0allows the surface saturation to build up gradually, and a shock will only form ifβ 1 (when the preceding solution becomes valid).

7.3.4 Similarity Solutions

If, on the other hand, β � 1, then the saturation profile approximately satisfies

∂s

∂τ= ∂

∂z

[s∂s

∂z

],

−s∂s

∂z=

{1 on z = 0,

0 on z = 1.

(7.84)

At least for small times, the model admits a similarity solution of the form

s = τaf (η), η = z/τb, (7.85)


Fig. 7.6 Schematicrepresentation of theevolution of S for both largeand small β

where satisfaction of the equations and boundary conditions requires 2a = b and2b = 1 = a, whence a = 1

3 , b = 23 , and f satisfies

(ff ′)′ − 1

3(f − 2ηf ′) = 0, (7.86)

with the condition at z = 0 becoming

−ff ′ = 1 at η = 0. (7.87)

The condition at z = 1 can be satisfied for small enough τ , as we shall see, be-cause Eq. (7.86) is degenerate, and f reaches zero in a finite distance, η0, say, andf = 0 for η > η0. As η = 1/τ 2/3 at z = 1, then this solution will satisfy the no fluxcondition at z = 1 as long as τ < η

−3/20 , when the advancing front will reach z = 1.

To see why f behaves in this way, integrate once to find

f

(f ′ + 2

3η

)= −1 +

∫ η

0f dη. (7.88)

For small η, the right hand side is negative, and f is positive (to make physicalsense), so f decreases (and in fact f ′ < − 2

3η). For sufficiently small f (0) = f0, f

will reach zero at a finite distance η = η0, and the solution must terminate. On theother hand, for sufficiently large f0,

∫ η

0 f dη reaches 1 at η = η1 while f is stillpositive (and f ′ = − 2

3η1 there). For η > η1, then f remains positive and f ′ > − 23η

(f cannot reach zero for η > η1 since∫ η

0 f dη > 1 for η > η1). Eventually f musthave a minimum and thereafter increase with η. This is also unphysical, so we re-quire f to reach zero at η = η0. This will occur for a range of f0, and we have toselect f0 in order that ∫ η0

0f dη = 1, (7.89)

which in fact represents global conservation of mass. Figure 7.6 shows the schematicform of solution both for β 1 and β � 1. Evidently the solution for β ∼ 1 willhave a profile with a travelling front between these two end cases.


7.4 Immiscible Two-Phase Flows: The Buckley–LeverettEquation

In some circumstances, the flow of more than one phase in a porous medium isimportant. The type example is the flow of oil and gas, or oil and water (or all three!)in a sedimentary basin, such as that beneath the North Sea. Suppose there are twophases; denote the phases by subscripts 1 and 2, with fluid 2 being the wetting fluid,and S is its saturation. Then the capillary suction characteristic is

p1 − p2 = pc(S), (7.90)

with the capillary suction pc being a positive, monotonically decreasing function ofsaturation S; mass conservation takes the form

−φ∂S

∂t+ ∇.u1 = 0,

φ∂S

∂t+ ∇.u2 = 0,

(7.91)

where φ is (constant) porosity, and Darcy’s law for each phase is

u1 = − k0

μ1kr1[∇p1 + ρ1gk],

u2 = − k0

μ2kr2[∇p2 + ρ2gk],

(7.92)

with kri being the relative permeability of fluid i.For example, if we consider a one-dimensional flow, with z pointing upwards,

then we can integrate (7.91) to yield the total flux

u1 + u2 = q(t). (7.93)

If we define the mobilities of each fluid as

Mi = k0

μi

kri , (7.94)

then it is straightforward to derive the equation for S,

φ∂S

∂t= − ∂

∂z

[Meff

{q

M1+ ∂pc

∂z+ (ρ1 − ρ2)g

}], (7.95)

where the effective mobility is determined by

Meff =(

1

M1+ 1

M2

)−1

. (7.96)

7.4 Immiscible Two-Phase Flows: The Buckley–Leverett Equation 407

Fig. 7.7 Graph ofdimensionless wave speedV (S) as a function of wettingfluid saturation, indicating thespeed and direction of wavemotion (V > 0 means wavesmove upwards) if the wettingfluid is more dense. Theviscosity ratio μr (see(7.100)) is taken to be 30

This is a convective-diffusion equation for S. If suction is very small, we obtainthe Buckley–Leverett equation

φ∂S

∂t+ ∂

∂z

[Meff

{q

M1+ (ρ1 − ρ2)g

}]= 0, (7.97)

which is a nonlinear hyperbolic wave equation. As a typical situation, supposeq = 0, and kr2 = S3, kr1 = (1 − S)3. Then

Meff = k0S3(1 − S)3

μ1S3 + μ2(1 − S)3, (7.98)

and the wave speed v(S) is given by

v = −(ρ2 − ρ1)gM ′eff(S) = v0V (S), (7.99)

where

v0 = (ρ2 − ρ1)gk0

μ2, V (S) = χ ′(S)

χ(S)2,

χ(S) = μr

(1 − S)3 + 1

S3 , μr = μ1

μ2.

(7.100)

The variation of V with S is shown in Fig. 7.7. For ρ2 > ρ1 (as for oil and water,where water is the wetting phase), waves move upwards at low water saturation anddownwards at high saturation.

Shocks will form, but these are smoothed by the diffusion term − ∂∂z

[Meffp′c

∂S∂z

],in which the diffusion coefficient is

D = −Meff p′c. (7.101)

As a typical example, take

pc = p0(1 − S)λ1

Sλ2(7.102)


with λi > 0. Then we find

D = k0p0S2−λ2(1 − S)2+λ1

[λ1S + λ2(1 − S)

μ1S3 + μ2(1 − S)3

], (7.103)

and we see that D is typically degenerate at S = 0. In particular, if λ2 < 2, theninfiltration of the wetting phase into the non-wetting phase proceeds at a finite rate,and this always occurs for infiltration of the non-wetting phase into the wettingphase.

A particular limiting case is when one phase is much less dense than the other,the usual situation being that of gas and liquid. This is exemplified by the problemof snow-melt run-off considered earlier. In that case, water is the wetting phase, thusρ2 − ρ1 = ρw − ρa is positive, and also μw ≈ 10−3 Pa s, μa ≈ 10−5 Pa s, whenceμa � μw (μr � 1), so that, from (7.98),

Meff ≈ k0S3

μw

, (7.104)

at least for saturations not close to unity. Shocks form and propagate downwards(since ρ2 > ρ1). The presence of non-zero flux q < 0 does not affect this statement.Interestingly, the approximation (7.104) will always break down at sufficiently highsaturation. Inspection of V (S) for μr = 0.01 (as for air and water) indicates that(7.104) is an excellent approximation for S � 0.5, but not for S � 0.6; for S � 0.76,V is positive and waves move upwards. As μr → 0, the right hand hump in Fig. 7.7moves towards S = 1, but does not disappear; indeed the value of the maximumincreases, and is V ∼ μ

−1/3r . Thus the single phase approximation for unsaturated

flow is a singular approximation when μr � 1 and 1 − S � 1.

7.5 Heterogeneous Porous Media

Perhaps the major concern in groundwater studies concerns the permeability.Whereas we tend to think of the permeability as a well-defined quantity which re-flects the local soil or rock properties, in reality it varies over many orders of mag-nitude on very small length scales. The consequence of this is that the value of thepermeability itself needs to be averaged in some way.

Permeability is so variable because of soil and rock heterogeneity. Because itscales with the square of the constituent grain size, clay and sand permeabilities arevastly different. And because sediments are lain down so slowly, over millions ofyears, sand and clay layers often lie in close proximity. The same is true for sedi-mentary rocks, which are simply the same sand and clay layers cemented togetherafter burial and consequent subjection to high pressure and temperature.

In seeking to quantify porous medium flow at the large scale, we need to av-erage the permeability in some way at the mesoscale: larger than the pore scale,but less than the macroscale. The simplest approach is to suppose that the per-

7.5 Heterogeneous Porous Media 409

meability in a mesoscale block has a random distribution, often assumed to bea lognormal distribution. An averaged permeability can be derived by supposing,for example, that fluctuations have small amplitude. One finds that the consequentaveraged permeability is a tensor, whose components depend on the direction offlow. In the following section, we consider a more specific model of the mesoscalestructure, where the heterogeneity is related to the occurrence of fractures in themedium. This leads to the idea of a secondary porosity associated with the frac-tures.

7.5.1 Dual Porosity Models

Take a walk on exposed basement rock: at the seaside, in the mountains. Rocks arenot uniform, but are inevitably fractured, or jointed. There are numerous reasons forthis. Sedimentary rocks are lain down over millions of years via the deposition ofoutwash clays, sands or calcareous microfossils in marine environments. Over thistime the deposition rate may average a millimetre or less per year. A metre of rockmay take a million, or ten million years, to accumulate. In this time, sea level mayrise or fall by tens or more of metres, and the land itself rises or falls because oftectonic processes: the crashing of continents, the uplift of mountains, the burial ofsedimentary basins.

It is no surprise that in an exposed sedimentary sequence, such as one sees incoastal cliffs, rocks form stratigraphic layers separated by unconformities markingdifferent sedimentary epochs. These unconformities are layers of weakness, andwhen the rocks are later subjected to tectonic compression and folding, fractureswill form.

It is not only sedimentary rocks which tear as they are stressed. Igneous rocksfracture as they solidify because of solidification shrinkage. They also form intru-sions such as dikes and sills, whose different erosional properties can cause subse-quent voidage.

The occurrence of faulting or jointing in rocks leads to a particular problem in thedescription of groundwater flow through them. The rock itself is porous, and admitsa Darcy flow through its pore space; but the fractures act as a second porosity, ad-mitting a secondary flow which would occur even if the rock itself was completelyimpermeable. The situation is illustrated in Fig. 7.8. It is because of this configura-tion that the system is called a double, or dual, porosity system, and the resultingmodel to describe the flow is called a dual porosity model.

In order to characterise porous flow through such a medium, we distinguish be-tween the blocks of the matrix and the cross-cutting fractures. We suppose the frac-tures are tabular, or planar, of width h, and the blocks are of dimension dB , andthat h � dB . We denote the blocks by the domain M , and the fractures as ∂M . Be-cause the fractures are narrow, ∂M essentially represents the external surfaces ofthe blocks. We also suppose that dB � l, where l is a relevant macroscopic length


Fig. 7.8 A doubly poroussystem. Porous matrix blocksare transected by (here) twosets of transverse fractures

scale. For these tabular cracks, we can define a fracture porosity

φf = h

dB + h. (7.105)

We define a matrix pore pressure pm, which is the locally averaged pore pressurein the matrix blocks, and a fracture pressure pf . There is then a matrix volume fluxper unit area um, and in the usual way we have Darcy’s law in the form

um = −km

μ∇pm, (7.106)

where km is the permeability of the fine-grained matrix. We have

km = d2p

τm

, (7.107)

where dp is grain size and τm is a tortuosity factor.We suppose that flow in the fractures is essentially Poiseuille flow, and this leads

to a prescription of fracture volume flux per unit transverse width of crack, qf ,(through a single fracture) as

qf = − h3

τf μ∇pf , (7.108)

where for a plane walled crack of width h, the fracture tortuosity τf = 12; for roughcracks, one can expect a higher value to be appropriate. The mean fracture velocity


(which is also the fracture volume flux per unit area of fracture) is thus

uf = qf

h= −kf

μ∇pf , (7.109)

where we define the fracture permeability parameter kf as

kf = h2

τf

. (7.110)

Now if we consider the total (averaged) Darcy flux u through such a doubly porousmedium, it is straightforward to show that

u = (1 − φf )〈um〉M + φf 〈uf 〉∂M, (7.111)

where the angle brackets denote averages: for um, a volume average over the matrixblocks; for uf , an average over the fractures. Since h � dB , we can effectivelyconsider the average of uf to be a surface average over the fracture surface denotedby ∂M , the external boundary of the matrix blocks M . Note that each fracture hastwo walls, and thus provides two external surfaces to M .

Our object is to characterise these averages in terms of macroscopic variables,if possible. Notice that we have already carried out a primary averaging in definingthe fluxes um and uf in the first place: um is averaged over the grain scale of thematrix, and uf is averaged over the width of the fractures. However, these fluxesstill represent values at a point within the larger block/fracture system. In particular,note that by its definition the fracture flux is parallel to the fracture, and this carriesthe implication that

∂pf

∂n= 0, (7.112)

where n is the normal to ∂M (and we take it to point from the matrix into thefracture).

We now want to average over the larger block scale dB . The ‘point’ fluxes um

and uf satisfy the conservation of mass equations

∇.um = 0 (7.113)

and

∇.(huf ) = um.n|∂M, (7.114)

where in (7.114) there is a flux um.n at the fracture surfaces from the matrix to thefractures. Some comment on this equation is necessary. It takes this form becauseof the fact that the fracture flux as defined in (7.108) is already averaged over thecross section of the fracture. Continuity of the fluxes at the block fracture interfaceproduces the source term in (7.114) through the integration of the fracture pointtransverse velocity across the fracture.


To use the ideas of homogenisation, we first define dimensionless variables. Wescale the variables as

uk ∼ U, pk ∼ P, x ∼ l, (7.115)

where l is the macroscopic length scale, and we define a second (now dimensionless)spatial variable X by putting

x = εX, (7.116)

where

ε = dB

l. (7.117)

The blocks thus have size X ∼ O(1). We write, with an obvious notation, the di-mensionless gradient operator in the form

∇ = ∇x + 1

ε∇X, (7.118)

where we are now using x and X as multiple spatial scales. We suppose that theblock structure is periodic in X, although this is inessential for the methodology.

Now the requirement of (7.112) that∂pf

∂n= 0 implies

n.

[∇xpf + 1

ε∇Xpf

]= 0, (7.119)

whence it follows that we can write, approximately,

pf = p(x) + εpf (X), (7.120)

and then

∂p

∂n≡ n.∇xp = −n.∇Xpf ≡ −∂pf

∂N. (7.121)

p is the macroscopic average pressure variable, and we may impose periodicity inX of pf with zero mean.

We have continuity of matrix and fracture pressure at ∂M , and therefore we canwrite

pm = p(x) + εpm(X), (7.122)

and the matrix pressure satisfies

∇2Xpm = 0 in M,

pm = pf on ∂M.(7.123)

To find the solution pm of (7.123), define a Green’s function G(X,Y) which satisfies

∇2YG = δ(X − Y) in M, G = 0 for Y ∈ ∂M. (7.124)


Then pm is given by

pm =∫

∂M

∂G(X,Y)

∂NY

pf (Y) dS(Y), (7.125)

where ∂∂NY

= n(Y).∇Y, and it follows from this that (for X ∈ ∂M)

∂pm

∂N=

∫∂M

K(X,Y) pf (Y) dS(Y), (7.126)

where

K(X,Y) = ∂2G(X,Y)

∂NX∂NY

. (7.127)

It remains to determine the fracture pressure perturbation pf . This involves solv-ing (7.109) and (7.114). Supposing h and kf are constant, these reduce, at leadingorder in ε, to

∇2Xpf =

(dBkm

hkf

)[∂pm

∂N+ ∂p

∂n

], (7.128)

subject to conditions of periodicity in X and zero mean. Note that the Laplacian in(7.128) is defined on the surface ∂M . Using (7.126), we can write (7.128) in theform

∇2Xpf = α

[∫∂M

K(X,Y) pf (Y) dS(Y) + ∂p

∂n

], (7.129)

where

α = dBkm

hkf

= τf dBd2p

τmh3 . (7.130)

The canonical microscale fracture problem to be solved on ∂M is thus

∇2∂Mq = α

[∫∂M

K(X,Y)q(Y) dS(Y) + n], (7.131)

with periodic boundary conditions and zero mean, and then

pf = q(X).∇xp. (7.132)

We now use these results to find the effective permeability of the medium. Aver-aging (7.106) over the matrix blocks yields (dimensionlessly)

U 〈um〉M = −kmP

μl∇xp, (7.133)

since pm is continuous across fractures and periodic in X. In a similar way,

U 〈uf 〉∂M = −kf P

μl

[∇xp + 〈∇Xpf 〉∂M

], (7.134)


but the surface average term in this expression does not obviously vanish. We have

∇Xpf = [∇Xpf − n(n.∇Xpf )] + n(n.∇Xpf ); (7.135)

the term in square brackets is a tangential derivative of pf along ∂M , and we sep-arate the terms in this way because pf is only defined on ∂M . In addition, becausepf = pm on ∂M , we could replace the subscript f by m in the square-bracketedexpression. Because of (7.121), we have

n(n.∇Xpf ) = −n(n.∇xp), (7.136)

and thus⟨n(n.∇Xpf )

⟩∂M

= −ei〈ninj 〉∂M∂p

∂xj

, (7.137)

where ei is the unit vector in the xi direction. We therefore have

U 〈uf 〉∂M = −kf P

μl

[(I − 〈nn〉∂M

).∇xp + ⟨∇Xpf − n(n.∇Xpf )

⟩∂M

], (7.138)

where I is the unit tensor, and nn is the tensor with elements ninj .We now substitute the expression for pf in (7.131) into (7.138); averaging over

∂M , we finally derive the expression for the mean fracture velocity in the dimen-sionless form

U〈uf 〉∂M = −kf P

μl

[(I − 〈nn〉∂M

) + ⟨{(I − nn).∇X

}q⟩].∇xp. (7.139)

Rewriting this in dimensional form, we have

〈uf 〉∂M = −kf

μk∗.∇p, (7.140)

where the fracture relative permeability tensor k∗ is defined by

k∗ij =

⟨(δik − nink)

(δjk + ∂qj

∂Xk

)⟩∂M

. (7.141)

Equally, the dimensional matrix flux is, from (7.133),

〈um〉M = −km

μ∇p. (7.142)

(7.140), (7.141) and (7.142) give the recipes for the averaged matrix and fracturefluxes in terms of the macroscopic pressure gradient and the solution of the blockscale fracture pressure problem (7.131).

If we take a representative volume consisting of many blocks, and integrate(7.113) over the matrix volume, and (7.114) over the fracture volume, we obtain

7.6 Contaminant Transport 415

the averaged (dimensional) equations for the averaged fluxes in the form

∇.[φf 〈uf 〉∂M

] = sf 〈um.n〉∂M,

∇.[(1 − φf )〈um〉M

] = −sf 〈um.n〉∂M,(7.143)

where sf is the specific fracture surface area (i.e., surface area per unit volume: heresf ∼ d−1

B ). Note that the source term in (7.143) is just

sf 〈um.n〉∂M = −kmsf P

μl

⟨∂pm

∂N

⟩∂M

, (7.144)

because 〈 ∂p∂n

〉∂M = 〈n〉∂M.∇xp = 0,6 and in the present case this is just zero becauseof (7.123).

The usefulness of all this methodology is that it carries across to other, morecomplicated averaging problems (as we see below), but in the present case of in-compressible double porosity flow, it may be somewhat unnecessary. The reasonfor this is that fracture relative permeability depends on the solution of (7.131), andthus on the fracture geometry and the single dimensionless parameter α given by(7.130). Assuming small fracture porosity, the ratio of matrix flow to fracture flowis, from (7.111), (7.109), (7.105) and (7.106), of the order of

um

φuf

∼ kmdB

kf h= α. (7.145)

If α is large, then very little flow occurs through the fractures anyway, and the sec-ondary porosity is of little concern. If α is small, the blocks are essentially imper-meable, and the fracture network is crucial; but then the solution of (7.131) is justq = O(α), and the relative permeability is simply

k∗ = I − 〈nn〉∂M, (7.146)

which only differs from the unit tensor if the medium is anisotropic. It is only in thecase α = O(1) that the competition between the two systems becomes important. Ifwe use values τf = 102, dB = 1 m, h = 10−3 m and km = 10−12 m2 (cf. Table 7.1),we get α ≈ 0.01. This might be appropriate for a fractured sandstone on a regionalscale. Generally, the primary and secondary (fracture) permeability will only becomparable if the host rock is itself quite permeable.

7.6 Contaminant Transport

Much of the interest in modelling groundwater flow lies in the prediction of solutetransport, in particular in understanding how pollutants will disperse: for example,

6The average of n over ∂M is zero because the normals on opposite sides of a fracture are inopposite directions. More specifically,

∫∂M

φndS = ∫M

∇φ dV , thus∫∂M

ndS = 0.


how do nitrates used for agricultural purposes disperse via the local groundwatersystem? Mostly simply, one would simply add a diffusion term to the advection ofthe solute concentration c:

φct + u.∇c = ∇.[φD∇c]. (7.147)

The diffusive width �l of a sharp front travelling at speed u after it has trav-elled a distance l is of the order of �l ∼ (Dl/u)1/2; if we take D ∼ 10−9 m2 s−1,u ∼ 10−6 m s−1 (30 m y−1), l = 103 m, then �l ∼ 1 m, and the diffusion zone isrelatively narrow. For a more porous sand, the diffusion width is even smaller.

In fact, as velocity increases, the effect of diffusion increases. That this is so isdue to a remarkable phenomenon called Taylor dispersion, described by G.I. Taylorin 1953. Consider the diffusion of a solute in a tube of circular cross section throughwhich a Poiseuille flow passes. If the mean velocity is U and the tube is of radius a,then the velocity is 2U(1 − r2/a2), and the concentration satisfies the equation

ct + 2U(1 − r2/a2)cx = D

(crr + 1

rcr + cxx

), (7.148)

where x is measured along the tube, and r is the radial coordinate. Taylor showed,rather ingenuously, that when the Péclet number Pe = aU/D is large, then the effectof the diffusion term in (7.148) is to disperse the mean solute concentration diffu-sively about the position of its centre of mass, x = Ut , with a dispersion coefficientof a2U2/48D. Aris later improved this to

DT = a2U2

48D+ D, (7.149)

which is asymptotically valid for x a. The dispersive mechanism is due to theradial variation of the velocity profile, which can disperse the solute even if thediffusion coefficient is very small.

Typically, this is generalised for porous media (where we think of the pores asbeing like Taylor’s tube) by writing the dispersion coefficient as

DT = D∗ + D‖, (7.150)

where D∗ represents molecular diffusion and D‖ dispersion in the direction of flow.The tortuosity of the flow paths and the possibility of adsorption on to the solidcauses D∗ to be less than D, and ratios D∗/D between 0.01 and 0.5 are commonlyobserved. In porous media, remixing at pore junctions causes the dependence of D‖on the flow velocity to be less than quadratic, and a relation of the form

D‖ = αum, (7.151)

where u is the Darcy flux, fits experimental data reasonably well for values 1 <

m < 1.2. A common assumption is to take m = 1. Mixing at junctions also causestransverse dispersion to occur, with a coefficient D⊥ which is measured to be lessthan D‖ by a factor of order 102 when Pe 1. Dispersion is thus a tensor property.


If we write

D‖ = α‖|u| (7.152)

for the longitudinal dispersion coefficient, and

D⊥ = α⊥|u| (7.153)

for the lateral dispersion coefficient, then a suitable tensor generalisation is

DTij = α⊥|u|δij + (α‖ − α⊥)

uiuj

|u| , (7.154)

where δij is the Kronecker delta.The conservation of solute equation is then

φ∂c

∂t+ u.∇c = ∇.

(φDT.∇c

] = ∂

∂xi

(φDT

ij

∂c

∂xj

). (7.155)

For a one-dimensional flow in the x direction, c satisfies

∂c

∂t+ v

∂c

∂x= ∂

∂x

[D‖

∂c

∂x

]+ ∂

∂y

[D⊥

∂c

∂y

]+ ∂

∂z

[D⊥

∂c

∂z

](7.156)

(v = u/φ is the linear velocity) and if the dispersivities are constant, then the solu-tion for release of a mass M at the origin at t = 0 is

c = M

8φ(πD‖)1/2D⊥t3/2exp

[− (x − vt)2

4D‖t− r2

4D⊥t

], (7.157)

where r2 = y2 + z2.

7.6.1 Reactive Dual Porosity Models

Let us now consider the reactive transport of a contaminant of concentration c withina fractured soil or rock which has dual porosity. We follow the ideas of averagingand homogenisation in Sect. 7.5.1. The point forms of the equations are taken in theform

∂c

∂t+ ∇.(cu) = ∇.[D∇c] + Sf ,

∂cm

∂t+ ∇.(cmu) = ∇.[Dm.∇cm] + Sm,

(7.158)

where Sf , Sm represent source or sink terms due to chemical reaction, D is themolecular diffusion coefficient of the contaminant within the fractures, and Dm is


the dispersivity within the matrix blocks. The concentrations within the fracturesand matrix are denoted by c and cm, respectively.

The first thing to do is to average the fracture concentration equation across thewidth of the fracture. When we do this, we effectively regain the Taylor dispersionequation, with the addition of the reaction terms, and also a solute flux deliveredfrom the matrix:

∂cf

∂t+ ∇.(cf uf ) = ∇.[Df .∇cf ] + Sf − 1

h

[(n.Dm.∇cm) − cmum.n

]∣∣∂M

. (7.159)

This equation is analogous to (7.114). It differs from (7.158) in that cf is the cross-sectional average concentration (actually, in the derivation of the Taylor dispersionequation, one finds the concentration is cross-sectionally uniform, so that cf = c);uf is the cross-sectionally averaged fracture velocity, just as before; and Df is thelocal dispersion coefficient: it will be modified again at the larger macroscale. Thereaction term Sf depends on cross-sectionally averaged concentrations, which equaltheir point forms, so that Sf is unchanged.

Now we write down the equivalents of (7.143). We define the block averagedconcentrations

cm = 〈cm〉|M, cf = 〈cf 〉|∂M, (7.160)

where as before 〈c〉|M denotes an average of c over the matrix blocks M , and 〈c〉|∂M

denotes the average of c over the fracture surface ∂M . Then we have the equationfor cf :

∂

∂t(φf cf ) + ∇.[φf cf uf ] = ∇.[φf Df .∇cf ] + φf Sf

+ sf⟨n.{cmum − Dm.∇cm}⟩∣∣

∂M. (7.161)

We can make use of (7.143)1, and the fact that cf = cm on ∂M , to simplify this to

φf

[∂cf

∂t+ uf .∇cf

]= ∇.[φf DF .∇cf ] + φf Sf − sf 〈n.Dm.∇cm〉|∂M. (7.162)

The specific fracture surface area is defined as sf , as before. The macroscopic frac-ture dispersivity DF here is distinct from Df , in the same way that Taylor dispersionin a tube is distinct from that in a porous medium; in this case it is because of remix-ing of fracture fluid at the junctions between fractures at the block boundaries. Theformal averaging assumption which is made is

〈Df .∇cf − cf uf 〉|∂M = DF .∇cf − cf 〈uf 〉|∂M, (7.163)

and this has some justification insofar as it is just this result which emerges in thestudy of Taylor dispersion.


In a similar way to the derivation of the average fracture concentration equation,the matrix averaged concentration satisfies

(1 − φf )

[∂cm

∂t+ um.∇cm

]

= ∇.[(1 − φf )Dm.∇cm

] + (1 − φf )Sm + sf 〈n.Dm.∇cm〉|∂M. (7.164)

The result of averaging is the two Eqs. (7.162) and (7.164) for the averagefracture and matrix concentrations. In principle, the block average fracture dis-persivity DF should be calculable by solving the local block problem, althoughin practice one would assume a value by analogy with assumptions about porousmedium dispersion coefficients. However, unlike the incompressible dual porositymass flow equations (7.143), the source term sf 〈n.Dm.∇cm〉|∂M is non-zero, andthis must be constituted, ideally by solving the block scale problem, which is givenby Eqs. (7.158)2 and (7.159); these can be slightly simplified to the forms

∂cm

∂t+ um.∇cm = ∇.[Dm.∇cm] + Sm,

∂cf

∂t+ uf .∇cf = ∇.[Df .∇cf ] + Sf − 1

h(n.Dm.∇cm)|∂M.

(7.165)

The boundary condition for cm is that

cm = cf on ∂M, (7.166)

and cf is periodic over ∂M .The basis for the method of homogenisation is the expansion of the local block

problem in terms of the parameter ε = dB/l, where dB is the block scale and l isthe macroscopic length scale. However, because of the complexity of the equationsto be solved, the application of this method must be done judiciously.

To illustrate this point, suppose that the dispersion coefficient tensors are allisotropic, and equal to DT I, where DT is constant, and that suitable (macroscopic)scales for the variables are c ∼ c0, x ∼ l, u ∼ U , t ∼ l/U , and suppose to be precisethat the reaction kinetics are first order, i.e., S = −rc, and that the specific frac-ture surface area is constant, sf = 1/dB . The dimensionless equation for the matrixaverage cm is then

(1 − φf )Pe

[∂cm

∂t+ um.∇xcm

]

= (1 − φf )∇2x cm − PeΛ(1 − φf )cm + 1

ε2

⟨∂cm

∂N

⟩∣∣∣∣∂M

, (7.167)

where ∂cm

∂Ndenotes n.∇Xcm, and

Pe = Ul

DT

, Λ = rl

U. (7.168)


The local block equation (7.165)1 rewritten in the block variables

X = xε, T = t

ε, (7.169)

is then

εPe

[∂cm

∂T+ um.∇Xcm

]= ∇2

Xcm − ε2PeΛcm. (7.170)

What is obvious is that all the terms cannot balance in both local and global prob-lems, and this leads to simplifications. The simplest case is where the macroscopicPéclet number and reaction number Λ are both taken to be O(1); then (7.167) sug-gests that in (7.170) we put

cm = c(x) + ε2cm(X) (7.171)

(and thus also cf = c(x)+ ε2cf (X)); both inner and outer problems are well scaled,and the advective term can be neglected in solving the inner problem, which in factreduces approximately to Poisson’s equation. Specifically, using multiple scales inboth x and t , we have at leading order in ε,

∇2Xcm = Pe

[∂c

∂t+ um.∇xc

]+ PeΛc − ∇2

x c (7.172)

subject to cm = cf on ∂M and conditions of periodicity on M . (If we integrate(7.172) over M , we regain the averaged equation (7.164) as an integrability condi-tion for (7.172), as expected via homogenisation.)

In a similar way, the leading order local equation for the fracture perturbed con-centration cf is

∇2Xcf − (1 − φf )

φf

∂cm

∂N

∣∣∣∣∂M

= Pe

[∂c

∂t+ uf .∇xc

]+ PeΛc − ∇2

x c (7.173)

(note that a term(1−φf )

εφf

∂c∂n

|∂M ∝ ∇xc.n|∂M is identically zero, because the normals

n on the two faces of a fracture cancel each other). cf satisfying (7.173) is subject toperiodicity in ∂M and zero mean. As for the fluid flow, we can solve (7.172) usinga Green’s function for the Laplacian in M , so that the boundary derivative termin (7.173) becomes an integral convolution in terms of cf . We then solve (7.173)using a Green’s function for the Laplacian on ∂M . This allows us to determine thehomogenised equation for c (see Question 7.13).

In fact, it is rarely the case that Pe and Λ are O(1): more commonly they are bothlarge. Suppose for example that ε = 10−4, Pe = 104, Λ = 104: not unreasonablevalues (as we shall see below). Putting

p = εPe, λ = εΛ, (7.174)

7.7 Environmental Remediation 421

it seems natural to take λ ∼ p ∼ O(1). The block problem becomes (in terms of T

and X only)

p

[∂cm

∂T+ um.∇Xcm

]= ∇2

Xcm − λpcm, (7.175)

which implies that reaction occurs on the block scale. This is a linear equation for cm

which can in principle be solved to give the boundary flux term in the cf equationas a convolution integral in terms of cf . In a strict sense, this distinguished limitdescribes the structure of a reaction front in which both reaction and dispersion areimportant. Outside this front, reaction and dispersion are negligible, and the reactantsimply advects with the flow.7 Importantly, it implies that the speed of the front isdetermined by the local diffusion and reaction within the block.

If Λ is even larger than this, so that λ 1, then the reaction is fast at the blockscale, and occurs in a thin rind within the block; for a single species, as here, thisrind must be on the boundary, as the interior reactant concentration reaches zerorapidly. Bearing in mind that the normal coordinate n at ∂M points into the fracture,the boundary layer solution for cm is just

cm ≈ cf exp[(λp)1/2n

], (7.176)

and thus the flux term in the local fracture equation derived from (7.165)2 is

− 1

φf

∂cm

∂n

∣∣∣∣∂M

= − 1

φf

(λp)1/2cf . (7.177)

Hence cf satisfies the local equation

∇2Xcf − 1

φf

(λp)1/2cf = p

[∂cf

∂T+ uf .∇Xcf

]+ λpcf . (7.178)

As is perhaps obvious, the reaction is fast in the fractures also and cf rapidly ap-proaches zero.

7.7 Environmental Remediation

Environmental pollution is now an endemic phenomenon. All over every industri-alised country, spills of hydrocarbons, effluents, and industrial waste have causedpollution of underlying groundwater. The resulting ‘plumes’ move slowly with theprevailing background groundwater flow, and in course of time will enter streams,rivers and lakes, with consequent health risk via the contamination of drinking wa-ter. Short of more drastic measures such as pumping out polluted groundwater and

7Actually, this only makes sense for multi-species reactions of the form A + B → P , for then thereaction rate ∝AB , and can be zero by virtue of A = 0 on one side of the front, and B = 0 on theother.


treating it, natural bioremediation anticipates that microbial action will eventuallybreak down most pollutants, rendering them harmless. The issue for the environ-mental scientist is to predict the future movement of the plume, and the likelihoodof microbial breakdown before it reaches drinking water sources.

In so doing, groundwater flow modelling is essential, because of its ability topredict into the future, and also because accurate monitoring of subsurface contam-ination is expensive and not straightforward. Against this, subsurface soil and rockis usually an extremely heterogeneous medium, both physically and chemically, andthe validation of computational results is difficult.

7.7.1 Reactive Groundwater Flow

The general context of many subsurface pollution problems of concern is similar,and we therefore begin with some generalities. Contaminants may be aqueous, inwhich case they mix with the groundwater, or non-aqueous, in which case theydo not. Hydrocarbons, for example, are non-aqueous. Amongst the non-aqueousphase liquids (NAPLs), one distinguishes dense liquids (DNAPLs) from light ones(LNAPLs). DNAPLs will sink into the saturated zone, whereas LNAPLs, such ashydrocarbons, will sink to the base of the unsaturated zone, and there sit on thewater table, from where their constituents may diffuse downwards.

The contaminant plume will typically consist of a cocktail of different chemicals,which flow with the local groundwater flow, disperse within it, and react with oxy-gen and other substances in the soil via the agency of microbial action. The typicalsort of model of concern is thus the reaction-advection-dispersion equation

R∂c

∂t+ ∇.(cu) = ∇.(D .∇c) + S, (7.179)

where c is one of a sequence of reactants, u is the local groundwater flux (given byDarcy’s law), D represents dispersion, and is typically anisotropic, in the sense thatdispersion in the longitudinal direction is larger than lateral dispersion. Dispersionitself is partly due to molecular diffusion, but more importantly (at high grain scalePéclet number) is due to grain scale shear-induced distortion of the fluid associatedwith Taylor dispersion, together with remixing at pore junctions. Typically, the lon-gitudinal dispersion coefficient D‖ ∼ dp|u|, where dp is grain size, while lateraldispersion D⊥ is a factor of 10–100 smaller.

The coefficient R is the retardation factor, and it is a slowing rate due to the ad-sorption of aqueous phase concentration on solid particles. Specifically, we actuallyhave two separate conservation laws for solid and aqueous concentrations cs and cl ,respectively, thus

∂cl

∂t+ ∇.(clu) = ∇.(D.∇cl) + S + kdcs − kacl,

∂cs

∂t= −kdcs + kacl,

(7.180)

7.7 Environmental Remediation 423

where kd and ka are desorption and adsorption rates, respectively. (7.180) specifi-cally assumes that the reaction occurs only in the aqueous phase, by way of example.

Now the point is that if the sorption rates are very large (and constant), then thesecond equation in (7.180) tells us that

cs ≈ kacl

kd

, (7.181)

and the sum of the two equations then gives us (7.179), with the retardation factor

R = 1 + ka

kd

. (7.182)

The source term S represents reaction driven sources and sinks. It is typicallythe case that there are many, many reactions and reactants. Equally typically, manyof the reaction rates are not well known, and the rates may be widely disparate. Ingeneral this will imply that many reactions can be taken to be in equilibrium, withonly the slowest (rate-controlling) reaction being of dynamical importance.

7.7.2 Biomass Modelling

The reaction terms S in (7.180) are mediated by bacteria within soil, which consumethe various nutrients provided through metabolic reactions. Like all living things,microbes survive by generating energy from nutrients through a variety of such re-actions. This process involves a network of redox (oxidation–reduction) reactions,and involves the overall exchange of electrons between two distinct chemical fuelswhich are consumed in the reactions; the metabolic process is in this case calledrespiration. While there may be a number of such fuels, there is a hierarchy in theiruse. Dissolved oxygen is commonly the terminal electron acceptor (as the exter-nally sourced oxidant is typically referred to), while an organic carbon compoundis the electron donor. When these preferred substrates are absent or depleted, othercompounds can be used instead. When the same organic compound is used as bothdonor and acceptor, the metabolic process is called fermentation.

Many bacteria are able to use several reaction pathways independently, givingthem a degree of flexibility to differing conditions. This capability is very species-dependent, and competition ensures that the species which are best adapted tolocal conditions become dominant. Microbial growth depends heavily on energymetabolism but also requires the uptake of other substrates needed to generate newbiomass. Growth rate is generally limited by the supply of one or more substrates,but saturates to a maximum growth rate in conditions of ample supply. The depen-dence of bacterial growth rate is commonly taken, by analogy with simple enzymekinetic uptake rates, as proportional to c

K+c, where c is the relevant nutrient con-

centration, and K is a constant; such kinetics are called Monod kinetics. When two


nutrients control growth, as in respiration, it is usual to take the growth rate as pro-portional to the product of two Monod factors, thus (for example)

S = −r0Xc1

K1 + c1

c2

K2 + c2, (7.183)

in which r0 is a reaction rate constant, and X is the biomass density.More complex models also consider the growth of the microbial population X

in terms of the nutrient supply, in particular in the form of a microbial mat whichbecomes attached to the soil grains, and usually called a biofilm. In particular, theconcept of biodegradation, for example of oil spills in sea water, or of factory efflu-ent in groundwater, commonly involves the development of bacterial colonies whichare able to efficiently use the offending contaminant to promote their own growth.

7.7.3 Non-dimensionalisation

It is commonly the case that the water table is at a depth of 10–20 metres, whereas aplume may have spread (or we are concerned with whether it will spread) a distanceof order kilometres. Therefore these plumes generally have high aspect ratio, a causeboth of computational stress and analytical simplification. The latter may be offsetby the decreased lateral dispersion coefficient, as we now show.

Let us consider the scalar contaminant equation (7.179) in two dimensions, withhorizontal and vertical coordinates x and z, corresponding Darcy fluxes u and w,and longitudinal (horizontal) and transverse (vertical) dispersivities D‖ and D⊥,which we take to be constant. We suppose l and d are suitable horizontal and verticallength scales, U is a horizontal Darcy flux scale (and therefore mass conservationimplies that hU/l is a suitable vertical flux scale), Rl/U is then the convective timescale, and we take S0 to be a measure of the reaction rate term, and c0 to be a typicalcontaminant concentration. The units of concentration are mol l−1 (moles per litre),and the units of S0 are mol l−1 s−1.

We define dimensionless variables by writing

u = Uu∗, w = hU

dw∗, c = c0c

∗, t = Rl

Ut∗,

x = lx∗, z = dz∗, S = S0S;(7.184)

substituting these into (7.179) (and forthwith dropping the asterisks), we obtain thedimensionless equation

∂c

∂t+ u

∂c

∂x+ w

∂c

∂z= 1

Pe‖∂2c

∂x2 + 1

Pe⊥∂2c

∂z2 + ΛS, (7.185)

where

Pe‖ = Ul

D‖, Pe⊥ = Ud2

D⊥l, Λ = S0l

Uc0. (7.186)

7.8 Three Specific Remediation Problems 425

The aspect ratio d/l and transverse dispersivity ratio D⊥/D‖ compete againsteach other, and generally we might suppose Pe⊥ ∼ Pe‖; but also if D‖ ∼ dpU , thenwe can expect that usually Pe‖ 1, heralding the existence of thin boundary layersin which dispersion is effective. The parameter Λ is the ratio of advection timeto reaction time, and will often be very large for microbially mediated reactionsof interest. These two observations cause numerical difficulties in solving (7.186),but can aid analytical insight. In the following section, we discuss three specificgroundwater contamination problems of recent concern.

7.8 Three Specific Remediation Problems

Four Ashes Four Ashes is a site in the Midlands of England where a fifty yearold plume of phenol (C6H60) and other contaminants, about 500 metres long, lieswithin the saturated zone. The plume is thought to lie between depths of 10 m and30 m at 130 m from the source, but is sinking as it moves west (perhaps becauseof surface recharge), reaching depths 21–44 m at 350 m distance (this informationcomes from two boreholes drilled at the site). Degradation is very slow: perhapsonly 5% of the total contaminant load has so far been degraded. This is thoughtto be due to toxicity of the phenol within the plume, and to supply limitation ofelectron acceptors (oxygen and nitrate) at the plume fringe.

The overall reaction scheme which is considered to apply is that the phenol isoxidised at the plume boundary by oxygenated groundwater, producing TIC (to-tal inorganic carbon). Within the plume, phenol is fermented to produce hydrogenamongst other things, which then reduces various oxides within the soil via mi-crobial agency. A suitable set of reactions to describe the situation consists of thefollowing:

C6H6O + 7O2 + 3H2Or1→ 6HCO−

3 + 6H+,

C6H6O + 5.6NO−3 + 0.2H2O

r2→ 6HCO−3 + 0.4H+ + 2.8N2,

C6H6O + 5H2Or3→ 3CH3COOH + 2H2,

C6H6O + 17H2Or4→ 6HCO−

3 + 6H+ + 14H2,

H2 + MnO2(s) + 2H+ r5→ 2H2O + Mn2+,

H2 + 2FeOOH(s) + 4H+ r6→ 4H2O + 2Fe2+,

H2 + SO2−4 + 0.25H+ r7→ H2O + 0.25HS−,

H2 + CO2−3 + 0.5H+ r8→ 0.75H2O + 0.25CH4,

(7.187)

and r1–r8 are the reaction rates, defined using Monod kinetics. In terms of the reac-tion rates, the source term in each reactant equation is then

Sj =∑

l

sj lrl, (7.188)


where sjl is the (sign dependent) stoichiometric coefficient for reactant j in reac-tion l.

The maximum (saturated) rates kj of the Monod rates rj are thought to rangefrom about 10−13 mol l−1 s−1 to about 10−10 mol l−1 s−1. The background phenolconcentration p0 is of the order of 10−10 mol l−1, and the aqueous dissolved oxygenlevel c0 is of order 10−4 mol l−1. The horizontal Darcy flux is estimated to be U =10 m y−1. The dispersivities are taken in the form D = αU , and values of α‖ = 1 mand α⊥ = 4×10−4 m are assumed.8 Also, we take l = 500 m, d = 20 m, and R = 1.With all of these values, we have

Pe‖ = l

α‖= 500, Pe⊥ = d2

α⊥l= 2000, Λ ∼ 106–109, (7.189)

where the definition of Λ is based on p0, i.e.,

Λ = S0l

Up0. (7.190)

As suggested, Pe‖ ∼ Pe⊥ 1, and the reaction rate parameter Λ is extremely large.To get an idea of the solution behaviour when Pe 1 and Λ 1, we consider thesimpler reaction

C6H6O + σO2r→ products, (7.191)

where σ is a suitable stoichiometric coefficient. The corresponding version of(7.185) is simply9

∂p

∂t+ ∇.(pu) = 1

Pe∇2p + ΛS,

∂c

∂t+ ∇.(cu) = 1

Pe∇2c + νΛS,

(7.192)

where p and c are the concentrations of the two species (phenol and oxygen), andthe reaction term is just

S = −(

p

kp + p

)(c

kc + c

); (7.193)

the parameter ν is given by

ν = σp0

c0∼ 10−6. (7.194)

8For both theoretical and experimental reasons, we expect α‖ ∼ dp , the pore or grain size (Bear1972, p. 609), with α⊥/α‖ ∼ 0.01–0.05 (Sahimi 1995, p. 225). The large value of α‖ here suggestsflow in sub-parallel fractures with spacing on the order of a metre.9We can take Pe‖ = Pe⊥ by choosing d appropriately.


The asymptotic structure of the solution of (7.192) is easily given. For Λ 1, thereaction region is very thin, and will occur only at the fringe of the plume. Outsidethe plume, p = 0, while inside the plume c = 0, so that S = 0 everywhere apartfrom the reaction front, which is located at the plume boundary. Denoting the plumeby P and its boundary by ∂P , we thus have to solve

∂p

∂t+ ∇.(pu) = 1

Pe∇2p, x ∈ P,

∂c

∂t+ ∇.(cu) = 1

Pe∇2c, x /∈ P,

(7.195)

and p and c both satisfy p = c = 0 on ∂P . The location of P has to be determined(it is a free boundary), but this is simply done because of the precise conservationlaw for c − νp, which follows from (7.192):

∂(c − νp)

∂t+ ∇.

[(c − νp)u

] = 1

Pe∇2(c − νp). (7.196)

Integrating this across the reaction front yields the extra condition10 to determinethe front ∂P :

∂c

∂n= ν

∂p

∂n. (7.197)

Note that the normal derivative here has to be treated with some attention whennon-dimensionalised. If the plume fringe is at dimensionless position z = ζ(x, t),then the dimensional normal component of the dispersive flux of oxygen (in termsof dimensionless variables) is just

(D.∇c).n = c0Ud

l(1 + δ2ζ 2x )1/2

[1

Pe⊥∂c

∂z− 1

Pe‖∂c

∂x

∂ζ

∂x

], (7.198)

where δ is the aspect ratio. There is a similar expression for ∂p/∂n. It follows fromthis that more generally, the conservation boundary condition (7.197) can be ex-pressed as

1

Pe⊥∂c

∂z− 1

Pe‖∂c

∂x

∂ζ

∂x= ν

[1

Pe⊥∂p

∂z− 1

Pe‖∂p

∂x

∂ζ

∂x

], (7.199)

and to interpret this as (7.197) requires the unit normal n to be suitably defined.The problem is even easier when (as here) Pe 1. Then diffusive boundary lay-

ers of (dimensionless) thickness O(1/Pe1/2) adjoin the plume boundary, and awayfrom these, the reactants are simply advected with the flow.

In the present case, the parameter ν � 1. This modifies the discussion as follows.Reaction must still occur in thin regions, outside which the reactants obey (7.195),

10In this particular case, it is even simpler, since we just integrate the convective-diffusion equa-tion (7.196) with appropriate external boundary conditions, and ∂P is located by the curve wherec − νp = 0.


but there cannot be a front (in the sense of a moving boundary) at which c = ∂c∂n

= 0.If, for example, we have the one-dimensional problem

ct + ucx = 1

Pecxx, (7.200)

with c = 1 at t = 0 and as x → ∞, and c = 0 at x = 0, then if Pe ∼ O(1), reactionoccurs near x = 0, oxygen supply being sufficient to remove the phenol. If Pe 1,then a front moves away from x = 0 at speed u. Behind this front, c ≈ 1, ahead ofit c ≈ 0, and the front itself is purely diffusional, of width O{(t/Pe)1/2}, having anerror function profile for solution.

Rexco The site was a former coal carbonisation plant (between 1935 and 1970)where ammonium liquor had been allowed to drain on site. The liquid drainedthrough the unsaturated zone of some 18 metres depth to the saturated zone, whereit is spreading as a plume of some 25 metres depth in a sandstone aquifer. At theplume fringe, ammonium ions (NH+

4 ) ions are subject to oxidation, releasing nitro-gen. Phenol is also present in front of the ammonium, which is highly retarded (witha retardation factor of R = 5). Well extraction by a new factory on site is causingthe groundwater flow field to be time dependent, and this is an issue in modellingstudies.

The modelling scheme is similar to the Four Ashes site, but less is known of thereaction rates. The reactions which are considered to be important for the ammo-nium are

NH+4 + 2O2

r1→ NO−3 + H2O + 2H+,

NO−3 + 1.25CH2O + H+ r2→ 0.5N2 + 1.25CO2 + 1.75H2O, (7.201)

NH+4 + 0.6NO−

3

r3→ 0.8N2 + 1.8H2O + 0.4H+.

The kinetics (only) of these reactions is described by the five ordinary differ-ential equations for the five reactants, ammonium NH+

4 , oxygen O2, nitrate NO−3 ,

hydrogen ion H+ and organic carbon CH2O:

dcNH+4

dt= −r1 − r3,

dcO2

dt= −2r1,

dcNO−3

dt= r1 − r2 − 0.6r3,

dcH+

dt= 2r1 − r2 + 0.4r3,

dcCH2O

dt= −1.25r2.

(7.202)


To estimate Péclet numbers, we take d = 25 m, l = 1000 m, R = 5, U =100 m y−1. Values of dispersion coefficients are uncertain, so we choose for il-lustration D‖ = 100 m2 y−1, D⊥ = 10 m2 y−1, i.e., α‖ = 1 m (like Four Ashes)and α⊥ = 0.1 m. The (unretarded) time scale l/U is then 10 years, and Pe‖ ∼ 103,Pe⊥ ∼ 6. These values are not too reliable, and perhaps are consistent with a highlyfractured aquifer. For a homogeneous medium, we would expect much smaller α,and thus much larger Pe.

It is possible to extend the Four Ashes discussion of the simple phenol/oxygenreaction to the more complicated scheme above. Reaction rates are not well known.Most of the reactants are present at the level of mmol l−1 (millimole per litre). Spe-cific estimates are

cNH+4

∼ 12 mmol l−1, cO2 ∼ 0.3 mmol l−1, cNO−3

∼ 0.15 mmol l−1,

cH+ ∼ 10−4 mmol l−1, cCH2O ∼ 1 mmol l−1.

(7.203)

If we use cNH+4

= 12 mmol l−1 as a concentration scale in the definition of Λ in

(7.186) and arbitrarily pick a reaction rate scale of S0 = 10−10 mol l−1 s−1 (com-parable to phenol degradation rate at Four Ashes), then we have Λ ∼ O(1), and thesimple description for Four Ashes is inappropriate.

Despite this, it is thought that reactions are fast, and we will suppose that in factΛ 1. In particular, there are data from a borehole measurement which appearto be consistent with the idea that there is a thin reaction zone at the upper plumeboundary. Oxygen (presumably) diffuses to this front from above, and ammoniumfrom below; in the reaction zone there is a huge spike of nitrate (see Fig. 7.9). Wewish to see whether the existence of this nitrate spike is consistent with the model,assuming the measurement was realistic. (Figure 7.9 actually suggests the existenceof two fronts, with nitrate produced at the upper front diffusing down to the secondfront, but we will focus only on the upper front.)

The dimensionless equations equivalent to (7.192) for the reaction scheme(7.201) are

R∂cNH+

4

∂t+ ∇.(cNH+

4u) = 1

Pe⊥

∂2cNH+4

∂z2 + 1

Pe‖

∂2cNH+4

∂x2 − Λ(r1 + r3),

∂cO2

∂t+ ∇.(cO2 u) = 1

Pe⊥∂2cO2

∂z2 + 1

Pe‖∂2cO2

∂x2 − 2ΛνO2r1,

∂cNO−3

∂t+ ∇.(cNO−

3u) = 1

Pe⊥

∂2cNO−3

∂z2 + 1

Pe‖

∂2cNO−3

∂x2

+ ΛνNO−3(r1 − r2 − 0.6r3),

∂cH+

∂t+ ∇.(cH+u) = 1

Pe⊥∂2cH+

∂z2+ 1

Pe‖∂2cH+

∂x2+ ΛνH+(2r1 − r2 + 0.4r3),

∂cCH2O

∂t+ ∇.(cCH2Ou) = 1

Pe⊥∂2cCH2O

∂z2+ 1

Pe‖∂2cCH2O

∂x2− 1.25ΛνCH2Or2,

(7.204)


Fig. 7.9 Data from borehole 102 at Rexco, May 2003, courtesy of David Lerner and ArnëHuttmann, GPRG, University of Sheffield. Units are metres for depth, mg l−1 for concentrations.Since the molecular weights of ammonium and nitrate are 18 and 62 (g mole−1), respectively, then1 mg NH+

4 l−1 = 118 mmol NH+

4 l−1, 1 mg NO−3 l−1 = 1

62 mmol NO−3 l−1. Nitrate shows a sharp

spike (of about 10 mmol l−1) at a depth of 19 m. There appears to be a second front at 23 m: thenitrate produced at 19 m diffuses there and takes out (according to (7.201)) either the inorganiccarbon CH2O or the acid H+

where the parameters νi are defined analogously to (7.194), i.e.,

νi = ci

cNH+4

, (7.205)

ci being the scale for ci . From the values given above, we can suppose that all theνi � 1.

We now mirror the Four Ashes discussion, supposing that Λ 1. The reactionfront at the plume fringe should then be thin as before, with oxygen diffusing tothe front from outside the plume, and ammonium diffusing to it from inside, andboth concentrations being zero at the front. The fringe position is unknown, and weseek a conservation law to provide an extra condition for it analogous to (7.197).Denote the right hand sides of (7.202) by e1–e5. There are five reactants but onlythree reactions, therefore there are two (linear) relationships between the ei : theseprovide suitable jump conditions across the fringe. The equations

∑i βiei = 0 are

satisfied for any β1 and β2 provided β3 = −β1 + 12β2, β4 = β1 + 3

4β2, and β5 = −β2.Selecting (β1, β2) = (1,0) and (0,1), we thus have

e1 − e3 + e4 = 0,

e2 + 1

2e3 + 3

4e4 − e5 = 0,

(7.206)

and it follows from this that in consideration of (7.204) there are two conservedquantities across the fringe (i.e., flux in equals flux out); explicitly (with the same


caveat concerning the normal n as expressed in (7.199)), we have

[∂cNH+4

∂n− νNO−

3

∂cNO−3

∂n+ νH+

∂cH+

∂n

]+

−= 0,

[νO2

∂cO2

∂n+ 1

2νNO−

3

∂cNO−3

∂n+ 3

4νH+

∂cH+

∂n− νCH2O

∂cCH2O

∂n

]+

−= 0,

(7.207)

where [j ]+− denotes the jump in j across the fringe.Consulting (7.201), we can write the dimensionless reaction rates ri in terms of

Monod rates

Mi = ci

Ki + ci

(7.208)

in the following way:

r1 = MNH+4MO2 , r2 = k2MNO−

3MH+MCH2O, r3 = k3MNH+

4MNO−

3,

(7.209)

where k2 and k3 are dimensionless constants (the ratios of the saturated maximakmax of reactions 2 and 3 in (7.201) to that of reaction 1). The requirement that re-action rates vanish on either side of the fringe is satisfied if cNH+

4= 0 outside the

plume (then r1 = r3 = 0), cO2 = 0 within the plume (so r1 = 0) and if cNO−3

= 0 onboth sides (then r2 = r3 = 0). This simplifies the situation to one where pre-existingsoil nitrate concentrations are low, and assumes all the nitrate produced in the re-action front by reaction 1 is consumed there by reaction 2. An alternative would bethat the produced nitrate diffuse away on either side, but this is not consistent witha fast reaction 2 unless there is a natural source for nitrate production (e.g., from

surface composting).11 Since from (7.203), νH+ ∼ 10−5, this suggests that∂c

NH+4

∂n

is approximately continuous across the fringe, which is thus diffusive in character(providing Pe 1). In that case the fringe is located simply by advection of ammo-nium.

The second jump condition then provides a flux boundary condition for cCH2O,which is not constrained to be zero at the front. In addition, existence of a spike of ni-trate within the front must be determined by solution of the reaction equations withinthe front. The reaction front is of thickness ∼(1/ΛPe)1/2 (taking Pe‖ = Pe⊥ for sim-plicity), and within this front, the nitrate concentration is given approximately by

∂2cNO−3

∂N2 + νNO−3(r1 − r2 − 0.6r3) = 0, (7.210)

11Alternatively, there may be more than one reaction front. This is suggested by Fig. 7.9.


where N is a suitably rescaled normal variable. We can expect this to be solv-able for cNO−

3subject to cNO−

3→ 0 as N → ±∞, since for example if we lin-

earise the nitrate Monod coefficient, MNO−3

= cNO−3

, then the solution can be writtenusing a Green’s function which depends also on all the other reactant concentra-tions.

There remains the issue of determining a boundary condition for H+ at the front.Counting conditions, the first flux condition in (7.207) determines the location of thefringe. We already have boundary conditions for cNH+

4and cO2 (both equal zero),

and the nitrate spike is determined from (7.210). Thus, in principle we know thejump in flux of ammonium, oxygen and nitrate, and by integrating (7.210) and itsequivalents for ammonium and oxygen through the front, we know that

∫ ∞

−∞(r1 + r3) dN =

[∂cNO−3

∂N

]+

−,

νO2

∫ ∞

−∞2r1 dN =

[∂cO2

∂N

]+

−, (7.211)

νNO−3

∫ ∞

−∞−(r1 − r2 − 0.6r3) dN = 0.

Thus we know the values of∫ ∞−∞ ri dN for i = 1,2,3 and this tells us (by integrating

through the reaction front) the values of∫ ∞−∞ ri dN for i = 4,5, and thus the jump in

flux of ∂cH+∂N

; this provides the extra boundary condition we seek. The jump in fluxof cCH2O is also given by

∫ ∞−∞ 1.25νCH2Or2 dN , but this is equivalent to (7.207)2. In

this way, the approximate model provides all the conditions necessary to determinethe solution.

St. Alban’s At St. Alban’s, there has been a petroleum spillage (at a filling station)into an underlying chalk aquifer. The fluids are LNAPLs: hydrocarbons, BTEX (acancer-forming aromatic hydrocarbon12 and MTBE.13 BTEX is retarded comparedto MTBE and thus forms a secondary plume within the MTBE plume. The LNAPLshave seeped through the unsaturated zone and sit on top of the chalk aquifer, act-ing as a source (via dissolution) of contaminant to the underlying groundwaterflow.

The sequence of reactions appears to be similar to those of the other examples,with oxidation by oxygen and nitrate at the plume fringe, and by Mn (manganese),Fe (iron) and SO2−

4 (sulphate) in the plume core. The sequence of reactions which

12More specifically, BTEX refers to a suite of volatile hydrocarbons, the acronym referring to ben-zene, toluene, ethylbenzene and xylene, with chemical formulae C6H6 (benzene), C7H8 (toluene),C8H10 (ethylbenzene and xylene); we use toluene in the chemical reaction model.13Methy tert-butyl ether, C5H12O.


are modelled is the following:

C7H8 + 9O2 + 3H2Or1→ 7CO2−

3 ,

C7H8 + 7.5H2Or2→ 2.5CO2−

3 + 4.5CH4 + 5H+,

C7H8 + 36Fe3+ r3→ 7CO2−3 + 36Fe2+ + 50H+,

CH4 + 2O2r4→ CO2−

3 + 2H+ + H2O,

CH4 + 8Fe3+ + 3H2Or5→ CO2−

3 + 8Fe2+ + 10H+,

Fe2+ + O2 + H+ r6→ Fe3+ + 0.5H2O,

C7H8 + 36FeOOH(s) + 58H+ r7→ 36Fe2+ + 7CO2−3 + 51H2O,

CH4 + 8FeOOH(s) + 7H+ r8→ 8Fe2+ + CO2−3 + 5H2O.

(7.212)

Maximum reaction rates vary over about four orders of magnitude, from about10−12 mol l−1 s−1 (reactions 2, 3, 7) to 10−8 mol l−1 s−1 (reaction 8). The princi-pal oxidising reaction rate r1 ∼ 10−10 mol l−1 s−1. Contaminant levels are of order10−7 mol l−1, with oxygen level of order 10−6 mol l−1. Plume depth and length ared ∼ 20 m, l ∼ 300 m, dispersivity parameters are taken as α‖ ∼ 1 m, α⊥ ∼ 0.2 m.The hydraulic conductivity of the upper chalk aquifer is 25 m d−1 (that of the lowerpart is only about 0.1 m d−1), and the hydraulic gradient is 1.75 × 10−3, thus thelongitudinal flux scale is U ∼ 16 m y−1. With these values, the parameters definedby (7.186) have values

Pe‖ ∼ 300, Pe⊥ ∼ 10, Λ ∼ 106. (7.213)

As now seems monotonously to be the case, reactions are fast, longitudinal disper-sion is small, but transverse dispersion may be effective because of the high aspectratio l/d .

The distinguishing feature of this particular site is that, although the chalk is veryporous (φ � 0.3), it has a very low effective permeability, presumably due to chem-ical adsorption by the chalk. On the other hand, the chalk matrix is dissected intoblocks by numerous fractures, and thus acts as a dual porosity system. Contaminantcan diffuse into the pore space of the matrix, and the issue of concern is whether andhow fast this happens, since storage in matrix blocks will act as a residual source ofcontamination after the fracture system has been flushed.

We have considered this problem before, in Sect. 7.6.1. Let us suppose for sim-plicity that dispersivities of matrix and fractures are constant and isotropic but notnecessarily equal. (Additionally, we suppose the local fracture dispersivity Df isequal to the macroscopic fracture dispersivity DF .) The point forms of the equationsdescribing a single reactant are given by (7.165), and the block scale and averaged


equations for the matrix concentration are given by (7.167) and (7.170):

Pem

[∂cm

∂t+ um.∇xcm

]= ∇2

x cm + Pem ΛSm + 1

ε2〈n.∇Xcm〉|∂M,

εPem

[∂cm

∂T+ um.∇Xcm

]= ∇2

Xcm + ε2Pem ΛSm,

(7.214)

where

Pem = Ul

Dm

, Λ = S0l

Uc0. (7.215)

Analogous dimensionless forms for the fracture average and block scale equationsare, from (7.162) and (7.165)2,

Pef

[∂cf

∂t+ uf .∇xcf

]= ∇2

x cf + Pef ΛSf − Dm

ε2Df φf

〈n.∇Xcm〉|∂M,

εPef

[∂cf

∂T+ uf .∇Xcf

]= ∇2

Xcf + ε2Pef ΛSf − Dm

Df φf

(n.∇Xcm)|∂M,

(7.216)

and Pef = Ul/Df .The question is how the interfacial transport term should be modelled. In princi-

ple we solve the block equation for cm with cm = cf on ∂M , yielding the interfacialterm as an integral convolution of cf . Putting this into the block equation for cf andsolving this then determines cf and thus gives the interfacial term, which closes thedescription of the averaged equations.

The upshot of our earlier discussion was that the details of the homogenisationprocess depend on the relation between the parameters ε = dB/l, Pe and Λ. Clas-sical homogenisation theory as in Sect. 7.6.1 assumes all the parameters are O(1)

apart from ε, but this is unlikely ever to be appropriate. Estimates for St. Alban’smay be Pef ∼ 102, Λ ∼ 106, ε ∼ 10−4; in addition, um � uf and Dm � Df . Thusthe reaction terms in the macroscopic equations are always large, and this suggeststhat, as before, reactions will be restricted to thin fronts. A complication in (7.214)is that the interfacial transport term may be large also, so this now needs to be de-termined.

To be specific, let us consider a simple two component reaction similar to (7.191),i.e.,

C7H8 + σO2r→ products, (7.217)

and let p denote dimensionless BTEX concentration and c denote dimensionlessoxygen concentration. Just as in (7.192), we can put Sm = νS in (7.221), and thereis a corresponding reaction term S in the equivalent equation for p: S can be takenas the product of Monod rates in (7.193), and for simplicity we take the linear rates,thus S = −pc.


The critical parameter in (7.214)2 is ε2Pem Λ; we write

Bm = ε2PemΛ, Bf = ε2Pef Λ; (7.218)

if Pef ∼ 102 then Bf ∼ O(1), and Bm 1: reaction in the block is fast.Just as in (7.196), cm − νpm satisfies an advective-diffusion conservation equa-

tion in the block, diffusion acting on a time scale t ∼ ε2Pem, where t is the macro-scopic time variable. The parameter ε2Pem is crucial. If we suppose that molec-ular diffusion applies in the blocks, then Dm ∼ 10−9 m2 s−1, Pem ∼ 105, andε2Pem ∼ 10−3 is small. Therefore on the long macroscopic time scale, diffusionsmoothes cm − νpm, and we can take it to be locally constant in the blocks (and thusalso the fractures). Since reaction is fast in the blocks, this implies

cm ≈ νpm, (7.219)

and therefore the block reaction rate for cm can be written as

ε2Pem ΛSm ≈ −Bmc2m. (7.220)

When Bm is large there is a reaction boundary layer like a rind at the blocksurface, within which cm satisfies

∂2cm

∂N2 − Bmc2m ≈ 0. (7.221)

In the interior of a block inside the plume the oxygen is depleted. and we havecm → 0 as N → −∞. The first integral of (7.221) determines the flux to the blocksas

∂cm

∂N=

√2Bm

3c

3/2f , (7.222)

using the fact that cm = cf on ∂M . The matrix concentration is negligible, and theaverage fracture concentration satisfies the Eq. (7.216)1

Pef

[∂cf

∂t+ uf .∇xcf

]= ∇2

x cf − Bf

ε2 c2f − 1

ε2φf

√2Bf Pef

3Pem

c3/2f , (7.223)

where we take Sf = −c2f as in the matrix, and we have supposed that

c2f = c2

f , c3/2f = c

3/2f . (7.224)

With Bf ∼ O(1), Pef ∼ 102, Pem ∼ 105, the flux term to the blocks,Bf

ε2 c2f 1 and

the fracture reaction term 1ε2φf

√2Bf Pef

3Pemc

3/2f 1; the implication is that the fracture

concentration of oxygen within the plume rapidly decreases within both blocks and


fractures to very small levels. The dual porosity appears to have little effect on thecharacteristics of the reactant distributions.

7.9 Precipitation and Dissolution

A sedimentary basin, as illustrated in Fig. 7.10, refers to an accumulation of sedi-ments (of typical depth 10 km) derived from river outwash sands and silts, or marinedeposited muds and microfossils. Sedimentary basins are everywhere in continen-tal rocks: we have the Paris basin, the London basin, the North Sea, and so on.As sediments accumulate and are buried, they are subjected to increasing heat andpressure, and these two factors enable the formation of intergranular cements, whichthus convert the sediments to rock (sand to sandstone; clay to shale, marine organ-isms to limestone). As they are buried, the sediments also compact, expelling porewater. Depending on the permeability, this can lead to pore pressures above hydro-static, a situation which is of concern in oil-drilling operations (and is discussedfurther in Sect. 7.11).

Diagenesis refers generally to the process of chemical alteration to rock, and inthis section we study the effects of diagenesis on the groundwater flow of accumu-lating sediments within a sedimentary basin. The particular type of diagenesis whichwe discuss is the conversion of smectite (a form of hydrated clay) to illite (a dehy-drated clay) via a dewatering reaction. The resultant release of water is also a poten-tial cause of excess pore pressures, but the main purpose of the present discussionis to show how the use of an approximation which we may call the weak solubilitylimit allows enormous simplification of quite complicated reaction schemes.

One view of the smectite–illite reaction is to treat it using first order kinetics, thus

SS → IS + nH2O, (7.225)

where each mole conversion yields n moles of water: S denotes smectite, I denotesillite, and the superscript S denotes the solid phase (likewise, L will denote an aque-ous phase). Such a scheme is not inconsistent with at least some experimental data,and the rate factor involved depends on temperature, with an activation energy inthe range 60–80 kJ mol−1.

However, it is likely that the transformation of smectite to illite occurs througha compound sequence of precipitation and dissolution reactions; one possible de-

Fig. 7.10 Schematic of asedimentary basin. Sedimentaccumulates from outflowfrom rivers, and also throughthe settlement of marineorganisms

7.9 Precipitation and Dissolution 437

scription is the following:

SS R1−→ XL + nH2O,

KFsR2−→ K+L + AlO−L

2 + s SiOL2 ,

K+L + AlO−L2 + f XL R3−→ f IS + SiOL

2 ,

SiOL2

R+4�

R−4

Qz.

(7.226)

The smectite SS dissolves to form a hydrous silica combination XL, such asSi4O10(OH)2. Additionally, illite precipitation requires potassium ions, and thesemay be obtained from the dissolution of potassium feldspar (in the second reaction);the aluminium hydroxyl ions AlO−L

2 act in the same way. The hydrous silica com-bination now combines with the potassium and aluminium to form illite precipitate,together with aqueous silica SiOL

2 , which itself precipitates as quartz Qz. Takingsuitable multiples of the reactions and adding to eliminate the aqueous phases, theoverall reaction is found to be

S + f −1KFsR−→ I + nH2O + f −1(s + 1)Qz. (7.227)

Ideally we would like to be able to write kinetics for (7.227) analogously to (7.225),with a recipe for the effective reaction rate R. Note that all the reactions in (7.226)are precipitation or dissolution reactions, and therefore the reaction rates are pro-portional to grain surface area.

It turns out, at least for this reaction scheme (but we might suspect more gener-ally), that the weak solubility limit allows such a recipe to be found. To illustrate themethod, note that conservation equations for the concentrations S, X, F , K , A, L,Q, I of the substances smectite, hydrous silica, feldspar, potassium ions, aluminiumhydroxyl ions, aqueous silica, quartz and illite satisfy equations of the type

∂

∂t

[(1 − φ)S

] + ∇.[(1 − φ)Sus

] = −r1,

∂

∂t(φX) + ∇.

(φXul

) − ∇.(φD.∇X) = r1 − f r3,

(7.228)

and so on, where the reaction rates ri are scaled with respect to the surface ratesRi by the specific interfacial surface areas Σi (thus ri = ΣiRi ). ul and us are theliquid and solid Darcy fluxes (i.e., the volume fluxes per unit area). We allow anon-zero solid velocity in order to cater for the effects of compaction of the porousmatrix. There are six other equations of the type in (7.228), together with a waterconservation equation (this is the equation for φ). Of the total of nine equations,four are for aqueous concentrations. That for A is identical to that for K , and weignore it henceforth.


The weak solubility limit is associated with the observations that the aqueousdissolved species X (hydrous silica), K (potassium) and L (silica) are typicallypresent in trace quantities of the order of 10–100 ppm (1 ppm = 10−3 kg m−3),14

and thus the concentrations of the aqueous phases are much less than those of thesolid phases. When the model is suitably non-dimensionalised, the result is that thetransport terms for the aqueous phases are very small, so that the correspondingreaction terms can be taken to be in equilibrium. The reaction terms in the equationsfor X, K and L are, respectively, r1 −f r3 (as above), r2 −r3 and sr2 +r3 −r+

4 +r−4 .

From this, we obtain the three relationships

r1 ≈ f r3,

r2 ≈ r3, (7.229)

r+4 − r−

4 ≈ (s + 1)r3.

Since the rate of dissolution of S is r1 and the rate of precipitation of I is f r3,this immediately shows that first order kinetics of the form (7.225) does apply, withthe reaction rate being f r3. Since r+

4 is a precipitation rate, and r−4 the dissolution

rate of the same mineral pair SiOL2 ↔ Qz, and since either r+

4 = 0 or r−4 = 0, it is

apparent that (7.229)3 determines r+4 and r−

4 together. Thus all the reaction ratescan be written in terms of r3, and this then determines the aqueous phase pseudo-equilibrium concentrations of X, L and K in terms of r3 and various temperaturedependent rate factors, since the kinetic rates ri are prescribed in terms of these.

If the smectite equation (7.228)1 is written in terms of smectite volume fractionφS , then it becomes

∂φS

∂t+ ∇.

[φSus

] = −MS

ρS

f r3, (7.230)

where MS is the molecular weight of smectite, and ρS its density. An equivalentequation for illite is

∂φI

∂t+ ∇.

[φI us

] = fMI

ρI

r3, (7.231)

and there are two similar equations for φF and φQ (with the right hand sides beingproportional to r3). In addition, the porosity φ satisfies

∂φ

∂t+ ∇.

(φul

) = nMw

ρw

f r3. (7.232)

It remains to determine the rate constant r3 in terms of the reactant concentrations.

14Actually, 1 ppm = 1 mg kg−1, but 1 m3 of water weighs 103 kg, so for aqueous solutions, this isequivalent to 10−3 kg m−3.

7.9 Precipitation and Dissolution 439

We assume that the reaction rates take the general form (in which D denotesdissolution, and P precipitation)

rDi = ΣiRi

[1 − ci

cis

]+,

rPi = ΣiRi

[ci

cis

− 1

]+,

(7.233)

where ci is the relevant aqueous concentration of phase i and cis is the associatedsolubility limit, i.e., the saturation concentration of aqueous phase i in the presenceof solid phase s. The rate factor Ri generally depends on temperature. (7.233) statesthat precipitation occurs when a solution is oversaturated, and dissolution occurs ifit is undersaturated.

For the specific case of the precipitation/dissolution scheme in (7.226), we sup-pose that

rD1 = φSRSD[1 − ψX]+,

rD2 = φf RFD[1 − ψK ]+[θL − ψL]+,

rP3 = φIRIP[ψK − θK ]+[ψX − θX]+,

rP4 = φQRQP[ψL − 1]+,

rD4 = φQRQD[1 − ψL]+.

(7.234)

In these equations we have assumed that specific surface area is proportional tovolume fraction, thus ΣS = φS/dp , and we have absorbed the grain size dp intothe reaction rates. The reaction rate subscripts describe what they represent: SD,smectite dissolution; FD, feldspar dissolution; IP, illite precipitation; QP, quartzprecipitation; QD, quartz dissolution.

The quantities ψD represent dimensionless aqueous concentrations (of phase D),scaled with a suitable solubility limit: cX is scaled with cXS , the solubility limit forhydrous silica in the presence of smectite; cL is scaled with cLQ, the solubilitylimit for silica in the presence of quartz. Potassium is more complicated, becausein (7.226), we see that dissolution of feldspar produces two aqueous phases, andprecipitation of illite requires two. We can in fact define four further solubility limits:cLF , that of silica in the presence of feldspar; cKF , that of potassium in the presenceof feldspar; cKI , that of potassium in the presence of illite; and cXI , that of hydroussilica in the presence of illite. The assumption is then that, for example, feldspar willdissolve only if both potassium and silica are undersaturated, i.e., cK < cKF andcL < cLF . In writing (7.234), we have scaled cK with cKF , and the three solubilityratios in (7.234) are therefore defined by

θL = cLF

cLQ

, θK = cKI

cKF

, θX = cXI

cXS

. (7.235)


Various possibilities can now occur depending on the values of the different solu-bility limits. We suppose that the solubility of hydrous silica X with respect to illiteis much less than that with respect to smectite (thus θX � 1), and we suppose thisis also true for potassium, that is to say, θK � 1. On the other hand, we supposethat the solubilities of silica with respect to feldspar or quartz are comparable, sothat θL ∼ 1, and in fact we will take θL > 1 as appears to be appropriate (indeedotherwise the smectite–illite transition will not occur in this model).

The equilibrium equations (7.229) are

φSRSD[1 − ψX]+ = f r3,

φF RFD[1 − ψK ]+[θL − ψL]+ = r3,

φQRQP[ψL − 1]+ − φQRQD[1 − ψL]+ = (s + 1)r3,

r3 = φIRIP[ψK − θK ]+[ψX − θX]+,

(7.236)

and these are four equations for the reaction rate r3 and the three aqueous concen-trations ψX , ψL and ψK . Assuming non-zero reaction rates, and taking θK, θX � 1,and θL > 1, we deduce from (7.236) that ψL > 1 (thus quartz precipitates fromsolution), and

r3 = φIRIPψKψX,

ψX = 1 − f r3

φSRSD,

ψL = 1 + (s + 1)r3

φQRQP,

ψK = 1 − r3

φF RFD{θL − 1 − (s+1)r3

φQRQP

} ,

(7.237)

whence the basic reaction rate r3 is determined by

r3 = φIRIP

[1 − f r3

φSRSD

][1 − r3

φF RFD{θL − 1 − (s+1)r3

φQRQP

}]. (7.238)

The right hand side of this expression is a decreasing function of r3 if it is positive,while the left hand side is increasing; therefore this expression defines the rate r3uniquely in terms of the solid phase concentrations of feldspar, quartz, smectite andillite.

A general complication in solving for φS and the other reactant porosities is thatr3 depends on φS,φI ,φQ and φF . Given us , (7.230) is a hyperbolic equation for φS

with boundary condition φS = φ0S on the upper surface z = h of a sedimentary basin

b < z < h. The equations for the solid fractions φY , Y = S, I,Q,F are all of theform

∂φY

∂t+ ∇.

[φY us

] = αY r3, (7.239)

7.10 Consolidation 441

for certain constants αY , and if φY = φ0Y on z = h, then φY can be written as a linear

combination of φS and φI ,

φY = (αY φ0S + φ0

Y )φI + (αIφ0Y − αY φ0

I )φS

αIφ0S + φ0

I

. (7.240)

Therefore, the reaction rate r3 can generally be written explicitly as a function of φS

and φI , and the diagenesis model collapses to equations for φS,φI and φ, togetherwith Darcy’s law.

Diagenesis (at least in this theory) turns out to have a minor quantitative effecton groundwater flow, essentially because the source term in (7.232) is relativelysmall. What is perhaps of more interest is that a fairly complicated sequence ofprecipitation/dissolution steps can be reduced, in the limit of weak solubility, to amodel with first order kinetics, albeit with a complicated (but explicitly defined)reaction rate. In fact, this observation is likely to be true in general. Suppose wehave a sequence of precipitation and dissolution steps for solids Si and liquids Lj :

L1 + · · · R1−→ S1 + · · · ,

S2 + · · · R2−→ L2 + · · · .

(7.241)

Each reaction step necessarily involves at least one aqueous phase component, andthus all the reaction rates R1, . . . ,Rn occur in the conservation equations for theaqueous phase components. Since these can all be taken to be in equilibrium, thenif there are k different aqueous phase components, we obtain k relations for the n

reactions. If k = n − 1, then all the reaction rates can be written in terms of theoverall production rate, and first order kinetics will apply.

In the present example (7.226), there are five reaction steps, and three aqueouscomponents (lumping K+L and Al(OH)−L

4 together), but the precipitation/dissolu-tion of quartz is effectively one reaction (either but not both at once can occur),and so the condition n = k + 1 is effectively met. More generally, we see that theproduction of solid precipitate P from solid substrates S through a sequence ofintermediate dissolution/precipitation steps may often lead to this situation.

7.10 Consolidation

Consolidation refers to the ability of a granular porous medium such as a soil tocompact under its own weight, or by the imposition of an overburden pressure. Thegrains of the medium rearrange themselves under the pressure, thus reducing theporosity and in the process pore fluid is expelled. Since the porosity is no longerconstant, we have to postulate a relation between the porosity φ and the pore pres-sure p. In practice, it is found that soils, when compressed, obey a (non-reversible)relation between φ and the effective pressure

pe = P − p, (7.242)

where P is the overburden pressure.


Fig. 7.11 Form of therelationship between porosityand effective pressure. Ahysteretic decompression-re-consolidation loop isindicated. In soil mechanicsthis relationship is oftenwritten in terms of the voidratio e = φ/(1 − φ), andspecificallye = e0 − Cc logpe , where Cc

is the compression index

The concept of effective pressure, or more generally effective stress, is an ex-tremely important one. The idea is that the total imposed pressure (e.g., the overbur-den pressure due to the weight of the rock or soil) is borne by both the pore fluid andthe porous medium. The pore fluid is typically at a lower pressure than the overbur-den, and the extra stress (the effective stress) is that which is applied through grainto grain contacts. Thus the effective pressure is that which is transmitted throughthe porous medium, and it is in consequence of this that the medium responds to theeffective stress; in particular, the characteristic relation between φ and pe representsthe nonlinear pseudo-elastic effect of compression.

As pe increases, so φ decreases, thus we can write (ignoring irreversibility)

pe = pe(φ), p′e(φ) < 0. (7.243)

Taking the fluid density ρ to be constant, we obtain from the conservation of massequation the nonlinear diffusion equation

φt = ∇.

[k(φ)

μ

∣∣p′e(φ)

∣∣∇φ

], (7.244)

assuming Darcy’s law with a permeability k, ignoring gravity, and taking P as con-stant. This is essentially the same as the Richards equation for unsaturated soils.

The dependence of the effective pressure on porosity is non-trivial and involveshysteresis, as indicated in Fig. 7.11. Specifically, a soil follows the normal consol-idation line providing consolidation is occurring, i.e. pe > 0. However, if at somepoint the effective pressure is reduced, only a partial recovery of φ takes place.When pe is increased again, φ more or less retraces its (overconsolidated) path tothe normal consolidation line, and then resumes its normal consolidation path. Herewe will ignore effects of hysteresis, as in (7.243).

When modelling groundwater flow in a consolidating medium, we must takeaccount also of deformation of the medium itself. In turn, this requires prescriptionof a constitutive rheology for the deformable matrix. This is often a complex matter,but luckily in one dimension, the issue does not arise, and a one-dimensional modelis often what is of practical interest. We take z to point vertically upwards, and let vl

7.10 Consolidation 443

and vs be the linear (or phase-averaged) velocities of liquid and solid, respectively.Then ul = φvl and us = (1 − φ)vs are the respective fluxes, and conservation ofmass of each phase requires

∂φ

∂t+ ∂(φvl)

∂z= 0,

−∂φ

∂t+ ∂{(1 − φ)vs}

∂z= 0;

(7.245)

Darcy’s law is then

φ(vl − vs

) = − k

μ

[∂p

∂z+ ρlg

], (7.246)

while the overburden pressure is

P = P0 + [ρs(1 − φ) + ρlφ

]g(h − z); (7.247)

here z = h represents the ground surface and P0 is the applied load. (7.247) assumesvariations of φ are small. More generally, we would have ∂P/∂z = −[ρs(1 − φ) +ρlφ]g. The effective pressure is then just pe = P − p.

We suppose these equations apply in a vertical column 0 < z < h, for whichsuitable boundary conditions are

vl = vs = 0 at z = 0,

p = 0, h = vs at z = h,(7.248)

and with an initial condition for p (or φ).The two mass conservation equations imply

vs = − φvl

1 − φ. (7.249)

Substituting this into (7.246), we derive, using (7.245),

∂φ

∂t= ∂

∂z

[k

μ(1 − φ)

{∂p

∂z+ ρlg

}]. (7.250)

If we assume the normal consolidation line takes the commonly assumed form (seeFig. 7.11)

φ

1 − φ= e0 − Cc ln

(pe/p

0e

), (7.251)

then we derive the consolidation equation

∂pe

∂t= pe

Cc(1 − φ)2

∂

∂z

[k

μ(1 − φ)

{∂pe

∂z+ �ρ(1 − φ)g

}], (7.252)

where �ρ = ρs − ρl .


If Cc is small (and typical values are in the range Cc ≤ 0.1) then φ varies little,and the consolidation equation takes the simpler form

∂pe

∂t= cv

∂2pe

∂z2 , (7.253)

where

cv = k

μ

pe

Cc(1 − φ)(7.254)

is the coefficient of consolidation.Suitable boundary conditions are

∂pe

∂z+ �ρ(1 − φ)g = 0 at z = 0,

pe = P0 at z = h,

(7.255)

and if the load is applied at t = 0, the initial condition is

pe = �ρ(1 − φ)g(h − z) at t = 0. (7.256)

The equation is trivially solved. The consolidation time is

tc ∼ h2

cv

= μCc(1 − φ)h2

kpe

, (7.257)

and depends primarily on the permeability k. If we take k ∼ 10−14 m2 (silt),Cc = 0.1, φ = 0.3, μ = 10−3 Pa s, P0 = 105 Pa (a small house), then cv ∼10−5 m2 s−1, and te ∼ 1 year for h ∼ 10 m.

7.11 Compaction

Compaction is the same process as consolidation, but on a larger scale. Other mech-anisms can cause compaction apart from the rearrangement of sediments: pressuresolution in sedimentary basins, grain creep in partially molten mantle (see Chap. 9).The compaction of sedimentary basins is a problem which has practical conse-quences in oil-drilling operations, since the occurrence of abnormal pore pressurescan lead to blow-out and collapse of the borehole wall. Such abnormal pore pres-sures (i.e., above hydrostatic) can occur for a variety of reasons, and part of thepurpose of modelling the system is to determine which of these are likely to berealistic causes. A further distinction from smaller scale consolidation is that thevariation in porosity (and, particularly, permeability) is large.

The situation we consider was shown in Fig. 7.10. Sediments, both organic andinorganic, are deposited at the ocean bottom and accumulate. As they do so, theycompact under their weight, thus expelling pore water. If the compaction is fast (i.e.,

7.11 Compaction 445

the rate of sedimentation is greater than the hydraulic conductivity of the sediments)then excess pore pressure will occur.

Sedimentary basins, such as the North Sea or the Gulf of Mexico, are typicallyhundreds of kilometres in extent and several kilometres deep. It is thus appropriateto model the compacting system as one-dimensional. A typical sedimentation rateis 10−11 m s−1, or 300 m My−1, so that a 10 kilometre deep basin may accumulatein 30 My (30 million years). On such long time scales, tectonic processes are im-portant, and in general accumulation is not a monotonic process. If tectonic upliftoccurs so that the surface of the basin rises above sea level, then erosion leads to de-nudation and a negative sedimentation rate. Indeed, one purpose of studying basinporosity and pore pressure profiles is to try and infer what the previous subsidencehistory was—an inverse problem.

The basic mathematical model is that of slow two-phase flow, where the phasesare solid and liquid, and is the same as that of consolidation theory. The effec-tive pressure pe is related, in an elastic medium, to the porosity by a functionpe = pe(φ). In a soil, or for sediments near the surface up to depths of perhaps500 m, the relation is elastic and hysteretic. At greater depths, more than a kilo-metre, pressure solution becomes important, and an effective viscous relationshipbecomes appropriate, as described below. At greater depths still, cementation oc-curs and a stiffer elastic rheology should apply.15 In addition, the permeability is afunction k = k(φ) of porosity, with k decreasing to zero fairly rapidly as φ decreasesto zero.

Let us suppose the basin overlies an impermeable basement at z = 0, and that itssurface is at z = h; then suitable boundary conditions are

vs = vl = 0 at z = 0,

pe = 0, h = ms + vs at z = h,(7.258)

where vs and vl are solid and liquid average velocities, and ms is the prescribedsedimentation rate, which we take for simplicity to be constant.

If we assume a specific elastic compactive rheology of the form

pe = p0{ln(φ0/φ) − (φ0 − φ)

}, (7.259)

then non-dimensionalisation (using a depth scale d = p0(ρs−ρl)g

and a time scale dms

)and simplification of the model leads to the nonlinear diffusion equation, analogousto (7.250),

∂φ

∂t= λ

∂

∂z

{k(1 − φ)2

[1

φ

∂φ

∂z− 1

]}, (7.260)

where the permeability is defined to be

k = k0k(φ), (7.261)

k0 being a suitable scale for k.

15Except that at elevated temperatures, creep deformation will start to occur.


The dimensionless parameter λ is given by

λ = K0

ms

, (7.262)

where K0 = k0(ρs − ρl)g/μ is essentially the surface hydraulic conductivity, andwe can then distinguish between slow compaction (λ � 1) and fast compaction(λ 1). Typical values of λ depend primarily on the sediment type. For ms =10−11 m s−1, we have λ ≈ 0.1 for the finest clay, λ ≈ 109 for coarse sands. In gen-eral, therefore, we can expect large values of λ. The associated boundary conditionsfor the model become

φz − φ = 0 at z = 0,

φ = φ0, h = 1 + λk(1 − φ)

[1

φ

∂φ

∂z− 1

]at z = h.

(7.263)

Slow Compaction, λ � 1 When λ is small, overpressuring occurs. A boundarylayer analysis is easy to do, and shows that φ ≈ φ0 in the bulk of the (uncompacted)sediment, while a compacting boundary layer of thickness

√λt exists at the base.

Fast Compaction, λ � 1 The more realistic case of fast compaction is also themore mathematically interesting. Most simply, the solution when λ 1 is the equi-librium profile

φ = φ0 exp[h − z]; (7.264)

the exponential decline of porosity with depth is sometimes called an Athy profile,but it only applies while λk 1. If we assume a power law for the dimensionlesspermeability of the form

k = (φ/φ0)m, (7.265)

then we find that λk reaches one when φ decreases to a value

φ∗ = φ0 exp

[− 1

mlnλ

], (7.266)

and this occurs at a dimensionless depth

Π = 1

mlnλ (7.267)

and time

t∗ = Π − φ0(1 − e−Π)

1 − φ0. (7.268)

Typical values m = 8, λ = 100, φ0 = 0.5, give values φ∗ = 0.28, Π = 0.58, t∗ =0.71. In particular, for a reasonable depth scale of 1 km (corresponding to p0 =2 × 107 Pa = 200 bars), this would correspond to a depth of 580 m. Below this, the

7.11 Compaction 447

Fig. 7.12 Solution of (7.260)for λ = 100 at timest = t∗ ≈ 0.71 and at t = 2.The porosity (horizontal axis)is plotted as a function of thescaled vertical height z/h(t).The solid lines are numericalsolutions, whereas the dottedlines are the large λ

equilibrium profiles. There isa clear divergence at depth fort > t∗

profile is not equilibrated, and the pore pressure is elevated. Figure 7.12 shows theresulting difference in the porosity profiles at t = t∗ and t > t∗, and Fig. 7.13 showsthe effect on the pore pressure, whose gradient changes abruptly from hydrostatic tolithostatic at the critical depth.

Fig. 7.13 Hydrostatic,overburden (lithostatic) andpore pressures at t = 5 andλ = 100, as functions of thescaled height z/h(t). Thetransition from equilibrium tonon-equilibrium compactionat the critical depth isassociated with a transitionfrom normal to abnormalpore pressures. The dashedlines represent two distinctapproximations to the porepressure profile, respectively,valid above and below thetransition region


If we take φ∗ = O(1) and λ 1, then formally m 1, and it is possible toanalyse the profile below the critical depth. One finds that

φ = φ∗ exp

[− 1

m

{lnm + O(1)

}], (7.269)

which can explain the flattening of the porosity profile evident in Fig. 7.12, andwhich is also seen in field data.

Viscous Compaction Below a depth of perhaps a kilometre, pressure solutionat intergranular contacts becomes important, and the resulting dissolution and lo-cal reprecipitation leads to an effective creep of the grains (and hence of the bulkmedium) in a manner analogous to regelation in ice. For such viscous compaction,the constitutive relation for the effective pressure becomes

pe = −ξ∇.us . (7.270)

In one dimension, the resulting dimensionless model is

−∂φ

∂t+ ∂

∂z

[(1 − φ)u

] = 0,

u = −λk

[∂p

∂z+ 1 − φ

],

p = −Ξ∂u

∂z,

(7.271)

where p is the scaled effective pressure. The compaction parameter is the same asbefore, and the extra parameter Ξ can be taken to be of O(1) for typical basin depthsof kilometres. Boundary conditions for (7.271) are

u = 0 on z = 0,

p = 0, φ = φ0, h = 1 + u at z = h.(7.272)

This system can also be studied asymptotically. When λ � 1, compaction is slowand a basal compaction layer again forms. When λ 1, explicit solutions can againbe obtained. There is an upper layer at equilibrium, but now the porosity decreasesconcavely with depth.16 As before, there is a transition when φ = φ∗, and below this

φ = φ∗ exp

[− 2

m

{lnm + O(1)

}], (7.273)

similar to (7.269).

16In view of Chap. 6, we need to be careful here. The function is mathematically concave, i.e., therate of decrease of porosity with depth increases as depth increases.

7.12 Notes and References 449

Fig. 7.14 Evolution of theporosity as a function ofdepth h − z, with a viscousrheology, at λ = 100. Theupper concave part is inequilibrium, whileoverpressuring occurs wherethe profile is flatter below this

The main distinction between viscous and elastic compaction is thus in the formof the rapidly compacted equilibrium profile near the surface (Fig. 7.14). The con-cave profile is not consistent with observations, but we need not expect it to be, asthe viscous behaviour of pressure solution only becomes appropriate at reasonabledepths. A more general relation which allows for this is a viscoelastic compactionlaw of the form

∇.us = − 1

Ke

dpe

dts− pe

ξ. (7.274)

7.12 Notes and References

Flow in porous media is described in the books by Bear (1972) and Dullien (1979).More recent versions, for example by Bear and Bachmat (1990) have developed ataste for more theoretical, deductive treatments based on homogenisation (see be-low) or averaging, with a concomitant loss of readability. The classic geologists’book on groundwater is by Freeze and Cherry (1979) and the classic engineeringtext is by Polubarinova-Kochina (1962). A short introduction, of geographical style,is by Price (1985). A more mathematical survey, with a variety of applications, is byBear and Verruijt (1987). The book edited by Cushman (1990) contains a wealth ofarticles on topics of varied and current interest, including dispersion, homogenisa-tion, averaging, dual porosity models, multigrid methods and heterogeneous porousmedia. Further information on the concepts of soil mechanics can be found in Lambeand Whitman (1979).


Homogenisation The technique of homogenisation is no more than the tech-nique of averaging in the spatial domain, most often formulated as a multiple scalesmethod. Whole books have been written about it, for example those by Bensoussanet al. (1978) and Sanchez-Palencia (1983). For application to porous media, see, forexample, Ene’s article in the book edited by Cushman (1990).

Piping Many dams are built of concrete, and in this case the problems associatedwith seepage do not arise, owing to the virtual impermeability of concrete. Earthand rockfill dams do exist, however, and are liable to failure by a mechanism calledpiping. The Darcy flow through the porous dam causes channels to form by erodingaway fine particles. The resultant channelisation concentrates the flow, increasingthe force exerted by the flow on the medium and thus increasing the erosion/collapserate of the channel wall. We can write Darcy’s law as a force balance on the liquidphase,

0 = −φ∇p − φμ

kvl − φρlgk (7.275)

(k being vertically upwards) and φμvl/k is an interactive drag term; then the cor-responding force balance for the solid phase is

0 = −(1 − φ)∇ps + φμ

kvl − (1 − φ)ρsgk, (7.276)

where ps is the pressure in the solid. For a granular solid, we can expect grainmotion to occur if the interactive force is large enough to overcome friction andcohesion; the typical kind of criterion is that the shear stress τ satisfies

τ ≥ c + pe tanφ, (7.277)

but in view of the large confining pressure and the necessity of dilatancy for soildeformation, the piping criterion will in practice be satisfied at the toe of the dam(i.e. the front), and piping channels will eat their way back into the dam, in muchthe same way that river drainage channels eat their way into a hillslope. A simplercriterion at the toe then follows from the necessity that the effective pressure on thegrains be positive. A lucid discussion by Bear and Bachmat (1990, p. 153) indicatesthat the solid pressure is related to the effective pressure pe which controls graindeformation by

pe = (1 − φ)(ps − p), (7.278)

and in this case the piping criterion at the toe is that pe < 0 in the soil there, or∂pe/∂z > 0. From (7.275), (7.276) and (7.278), this implies piping if

μv

k> (ρs − ρl)(1 − φ)g, (7.279)

where v is the vertical component of vl . This criterion is given by Bear (1972).More generally, piping can be expected to occur if pe reaches 0 in the soil interior(ignoring cohesion). Sellmeijer and Koenders (1991) develop a model for piping.

7.12 Notes and References 451

Taylor Dispersion Taylor dispersion is named after its investigation by Taylor(1953), who carried out experiments on the dispersal of solute in flow down a tube.The dispersion is enabled by the combination of differential axial advection by thedown tube velocity, typically a Poiseuille flow, and the rapid cross stream diffusionwhich renders the cross-sectional concentration profile radially uniform. The theoryof Taylor is somewhat heuristic; it was later elaborated by Aris (1956). For a formalderivation using asymptotic methods, see Fowler (1997, p. 222, Exercise 2).

Its application to porous media stems from the conceptual idea that the pore spaceconsists of a network of narrow tubules connected at pore junctions. If the tubes areof radius a and length dp , the latter corresponding to grain size, then the Darcyflux |u| ∼ πφU , while the pore radius a ∼ dp

√φ. This would suggest a Taylor

dispersion coefficient of

DT ≈ a2U2

48D∼ d2

p|u|248π2Dφ

, (7.280)

as opposed to the measured values which more nearly have DT ∼ |u|. Taylor dis-persion in porous media has been studied by Saffman (1959), Brenner (1980) andRubinstein and Mauri (1986), the latter using the method of homogenisation.

Biofilm Growth Monod kinetics was described by Monod (1949), by way ofanalogy with enzyme kinetics, where one considers the uptake of nutrients as oc-curring through a series of fast intermediary reactions; when two nutrients controlgrowth, as in respiration, it is usual to take the growth rate as proportional to theproduct of two Monod factors (Bader 1978). A variety of enhancements to thissimple model have also been proposed to account for nutrient consumption due tomaintenance, inactivation of cells in adverse conditions, and other observed effects(Beeftink et al. 1990; Wanner et al. 2006).

Bacteria in soils commonly grow as attached biofilms on soil grains, with a thick-ness of the order of 100 µ. A variety of models to describe biofilm growth have beenpresented, with an ultimate view of being able to parameterise the uptake rate ofcontaminant species in soils and other environments (Rittmann and McCarty 1980;Picioreanu et al. 1998; Eberl et al. 2001; Dockery and Klapper 2001; Cogan andKeener 2004).

Remediation Sites The three sites described in Sect. 7.8 are under study by theGroundwater Restoration and Protection Group at the University of Sheffield, led byProfessor David Lerner. The site at Four Ashes is described by Mayer et al. (2001),that at Rexco by Hüttmann et al. (2003), and that at St. Alban’s by Wealthall et al.(2001).

The description of the two species reaction front given by (7.192) is similar tothat for a diffusion flame (Buckmaster and Ludford 1982) in combustion, and alsocorrosion in alloys (Hagan et al. 1986). It is not conceptually difficult to extendthis approach to an arbitrary number of reactions, although it may become awkwardwhen multiple reaction fronts are present (see, for example, Dewynne et al. 1993).


Diagenesis The first order reaction kinetics (7.225) for the smectite–illite transi-tion was proposed by Eberl and Hower (1976). Information on solubility limits isgiven by Aagaard and Helgeson (1983) and Sass et al. (1987). The asymptotic ap-proximation called here the weak solubility limit is called solid density asymptoticsby Ortoleva (1994). Details of the use of the weak solubility approximation can befound in Fowler and Yang (2003).

Compaction Interest in compaction is motivated by its occurrence in sedimentarybasins, and also by issues of subsidence due to groundwater or natural gas extraction(see, for example, Baú et al. 2000). The constitutive law used here for effectivepressure is that of Smith (1971); it mimics the normal consolidation behaviour ofcompacting sediments (such as soils), and is further discussed by Audet and Fowler(1992) and Jones (1994).

Athy’s law comes from the paper by Athy (1930). Smith (1971) advocates the useof the high exponent m = 8 in (7.265). Further details of the asymptotic solution ofthe compaction profiles are given by Fowler and Yang (1998). Freed and Peacor(1989) show examples of the flattened porosity profiles at depth.

Early work on pressure solution in sedimentary basins was by Angevine and Tur-cotte (1983) and Birchwood and Turcotte (1994). More recently, Fowler and Yang(1999) derived the viscous compaction law. An extension to viscoelastic compactionhas been studied by Yang (2000).

Seals One process which we have not described is the formation of high pressureseals. In certain circumstances, pore pressures undergo fairly rapid jumps acrossa ‘seal’, typically at depths of 3000 m. Such jumps cannot be predicted within theconfines of a simple compaction theory, and require a mechanism for pore-blocking.Mineralisation is one such mechanism, as some seals are found to be mineralisedwith calcite and silica (Hunt 1990). In fact, a generalisation of the clay diagenesismodel to allow for calcite precipitation could be used for this purpose. As it stands,(7.232) predicts a source for φ, but mineralisation would cause a corresponding sinkterm. Reduction of φ leads to reduction of diffusive transport, and the feedback isself-promoting. Problems of this type have been studied by Ortoleva (1994), forexample.

7.13 Exercises

7.1 Show that for a porous medium idealised as a cubical network of tubes, the per-meability is given (approximately) by k = d2

pφ2/72π , where dp is the grainsize. How is the result modified if the pore space is taken to consist of pla-nar sheets between identical cubical blocks? (The volume flux per unit widthbetween two parallel plates a distance h apart is −h3p′/12μ, where p′ is thepressure gradient.)

7.13 Exercises 453

7.2 A sedimentary rock sequence consists of two types of rock with permeabilitiesk1 and k2. Show that in a unit with two horizontal layers of thickness d1 andd2, the effective horizontal permeability (parallel to the bedding plane) is

k‖ = k1f1 + k2f2,

where fi = di/(d1 + d2), whereas the effective vertical permeability is givenby

k−1⊥ = f1k

−11 + f2k

−12 .

Show how to generalise this result to a sequence of n layers of thicknessd1, . . . , dn.

Hence show that the effective permeabilities of a thick stratigraphic se-quence containing a distribution of (thin) layers, with the proportion of layershaving permeabilities in (k, k + dk) being f (k) dk, are given by

k‖ =∫ ∞

0kf (k) dk, k−1

⊥ =∫ ∞

0

f (k) dk

k.

7.3 Groundwater flows between an impermeable basement at z = hb(x, y, t) anda phreatic surface at z = zp(x, y, t). Write down the equations governing theflow, and by using the Dupuit approximation, show that the saturated depth h

satisfies

φht = kρg

μ∇.[h∇zp],

where ∇ = (∂/∂x, ∂/∂y). Deduce that a suitable time scale for flows in anaquifer of typical depth h0 and extent l is tgw = φμl2/kρgh0.

I live a kilometer from the river, on top of a layer of sediments 100 m thick(below which is impermeable basement). What sort of sediments would thoseneed to be if the river responds to rainfall at my house within a day; within ayear?

7.4 A two-dimensional earth dam with vertical sides at x = 0 and x = l has areservoir on one side (x < 0) where the water depth is h0, and horizontal dryland on the other side, in x > l. The dam is underlain by an impermeablebasement at z = 0.

Write down the equations describing the saturated groundwater flow, andshow that they can be written in the dimensionless form

u = −px, ε2w = −(pz + 1),

pzz + ε2pxx = 0,

and define the parameter ε. Write down suitable boundary conditions on theimpermeable basement, and on the phreatic surface z = h(x, t).

Assuming ε � 1, derive the Dupuit–Forchheimer approximation for h,

ht = (hhx)x in 0 < x < 1.


Show that a suitable boundary condition for h at x = 0 (the dam end) is

h = 1 at x = 0.

Now define the quantity

U =∫ h

0p dz,

and show that the horizontal flux

q =∫ h

0udz = −∂U

∂x.

Hence show that the conditions of hydrostatic pressure at x = 0 and con-stant (atmospheric) pressure at x = 1 (the seepage face) imply that

∫ 1

0q dx = 1

2.

Deduce that, if the Dupuit approximation for the flux is valid all the way tothe toe of the dam at x = 1, then h = 0 at x = 1, and show that in the steadystate, the (dimensional) discharge at the seepage face is

qD = kρgh20

2μl.

Supposing the above description of the solution away from the toe to bevalid, show that a possible boundary layer structure near x = 1 can be de-scribed by writing

x = 1 − ε2X, h = εH, z = εZ, p = εP,

and write down the resulting leading order boundary value problem for P .7.5 I get my water supply from a well in my garden. The well is of depth h0

(relative to the height of the water table a large distance away) and radius r0.Show that the Dupuit approximation for the water table height h is

φ∂h

∂t= kρg

μ

1

r

∂

∂r

(rh

∂h

∂r

).

If my well is supplied from a reservoir at r = l, where h = h0, and I withdrawa constant water flux q0, find a steady solution for h, and deduce that my wellwill run dry if

q0 >πkρgh2

0

μ ln[l/r0] .Use plausible values to estimate the maximum yield (litres per day) I can useif my well is drilled through sand, silt or clay, respectively.

7.13 Exercises 455

7.6 A volume V of effluent is released into the ground at a point (r = 0) at time t .Use the Dupuit approximation to motivate the model

φ∂h

∂t= kρg

μ

1

r

∂

∂r

(rh

∂h

∂r

),

h = h0 at t = 0, r > 0,

∫ ∞

0r(h − h0) dr = V/2π, t > 0,

where h0 is the initial height of the water table above an impermeable base-ment. Find suitable similarity solutions in the two cases (i) h0 = 0, (ii) h0 > 0,h − h0 � h0, and comment on the differences you find.

7.7 Fluid flows through a porous medium in the x direction at a linear velocity U .At t = 0, a contaminant of concentration c0 is introduced at x = 0. If the longi-tudinal dispersivity of the medium is D, write down the equation which deter-mines the concentration c in x > 0, together with suitable initial and boundaryconditions. Hence show that c is given by

c

c0= 1

2

[erfc

{x − Ut

2√

Dt

}+ exp

(Ux

D

)erfc

{x + Ut

2√

Dt

}],

where

erfc ξ = 2√π

∫ ∞

ξ

e−s2ds.

[Hint: you might try Laplace transforms, or else simply verify the result.]Show that for large ξ , erfc ξ = e−ξ2 [ 1√

πξ+ · · · ], and deduce that if x =

Ut + 2√

Dt η, with η = O(1), then

c

c0≈ 1

2erfcη + O

(1√t

).

Hence show that at a fixed station x = X far downstream, the measured profileis approximately given by

c ≈ c0

[1 − 1

2erfc

{1

2

(U 3

DX

)1/2(t − X

U

)}].

This is called the breakthrough curve, and indicates that dispersion causesbreakthrough to occur over a time interval (at large distance) of order �tb =(DX/U3)1/2. If D ≈ aU , show that the ratio of �tb to tb = X/U is �tb/tb ∼(a/X)1/2.

7.8 Rain falls steadily at a rate q (volume per unit area per unit time) on a soilof saturated hydraulic conductivity K0 (= k0ρwg/μ, where k0 is the saturated


permeability). By plotting the relative permeability krw and suction character-istic σψ/d as functions of S (assuming a residual liquid saturation S0), showthat a reasonable form to choose for krw(ψ) is krw = e−cψ . If the water tableis at depth h, show that, in a steady state, ψ is given as a function of the di-mensionless depth z∗ = z/zc, where zc = σ/ρwgd (σ is the surface tension, d

the grain size), by

h∗ − z∗ = 1

2ψ − 1

cln

[ sinh{ 1

2 (ln 1q∗ − cψ)

}sinh

{ 12 ln 1

q∗}

],

where h∗ = h/zc, providing q∗ = q/K0 < 1. Deduce that if h zc, then ψ ≈1c

ln 1q∗ near the surface. What happens if q > K0?

7.9 Derive the Richards equation

ρwφ∂S

∂t= − ∂

∂z

[k0

μkrw(S)

{∂pc

∂z+ ρwg

}]

for one-dimensional infiltration of water into a dry soil, explaining the mean-ing of the terms, and giving suitable boundary conditions when the surface fluxq is prescribed. Show that if the surface flux is large compared with k0ρwg/μ,where k0 is the saturated permeability, then the Richards equation can be ap-proximated, in suitable non-dimensional form, by a nonlinear diffusion equa-tion of the form

∂S

∂t= ∂

∂z

[D

∂S

∂z

].

Show that, if D = Sm, a similarity solution exists in the form

S = tαF (η), η = z/tβ,

where α = 1m+2 , β = m+1

m+2 , and F satisfies

(FmF ′)′ = αF −βηF ′, FmF ′ = −1 at η = 0, F → 0 as η → ∞.

Deduce that

FmF ′ = −(α + β)

∫ η0

η

F dη − βηF,

where η0 (which may be ∞) is where F first reaches zero. Deduce that F ′ < 0,and hence that η0 must be finite, and is determined by

∫ η0

0F dη = 1

α + β.

What happens for t > F(0)−1/α?

7.13 Exercises 457

7.10 Write down the equations describing one-dimensional consolidation of wetsediments in terms of the variables φ,vs, vl,p,pe, these being the porosity,solid and liquid (linear) velocities, and the pore and effective pressures. Ne-glect the effect of gravity.

Saturated sediments of depth h lie on a rigid but permeable (to water) base-ment, through which a water flux W is removed. Show that

vs = k

μ

∂p

∂z− W,

and deduce that φ satisfies the equation

∂φ

∂t= ∂

∂z

[(1 − φ)

{k

μ

∂p

∂z− W

}].

If the sediments are overlain by water, so that p = constant (take p = 0) atz = h, and if φ = φ0 +p/K , where the compressibility K is large (so φ ≈ φ0),show that a suitable reduction of the model is

∂p

∂t− W

∂p

∂z= c

∂2p

∂z2,

where c = K(1−φ0)k/μ, and p = 0 on z = h, pz = μW/k. Non-dimensionalisethe model using the length scale h, time scale h2/c, and pressure scaleμWh/k. Hence describe the solution if the parameter ε = μWh/k is small,and find the rate of surface subsidence. What has this to do with Venice?

7.11 Write down a model for vertical flow of two immiscible fluids in a porousmedium. Deduce that the saturation S of the wetting phase satisfies the equa-tion

φ∂S

∂t+ ∂

∂z

[Meff

{q

Mnw

+ g�ρ

}]= − ∂

∂z

[Meff

∂pc

∂z

],

where z is a coordinate pointing downwards,

pc = pnw − pw, �ρ = ρw − ρnw, M−1eff = (

M−1w + M−1

nw

),

q is the total downward flux, and the suffixes w and nw refer to the wettingand non-wetting fluid, respectively. Define the phase mobilities Mi . Give acriterion on the capillary suction pc which allows the Buckley–Leverett ap-proximation to be made, and show that for q = 0 and μw μnw , waves typ-ically propagate downwards and form shocks. What happens if q �= 0? Is theBuckley–Leverett approximation realistic—e.g. for air and water in soil? (As-sume pc ∼ 2γ /rp , where γ = 70 mN m−1, and rp is the pore radius: for clay,silt and sand, take rp = 1 µ, 10 µ, 100 µ, respectively.)

7.12 A model for snow-melt run-off is given by the following equations:

u = k

μ

[∂pc

∂z+ ρlg

],


k = k0S3,

φ∂S

∂t+ ∂u

∂z= 0,

pc = p0

(1

S− S

).

Explain the meaning of the terms in these equations, and describe the assump-tions of the model.

The intrinsic permeability k0 is given by

k0 = 0.077 d2 exp[−7.8ρs/ρl],where ρs and ρl are snow and water densities, and d is grain size. Take d =1 mm, ρs = 300 kg m−3, ρl = 103 kg m−3, p0 = 1 kPa, φ = 0.4, μ = 1.8 ×10−3 Pa s, g = 10 m s−2, and derive a non-dimensional model for meltingof a one metre thick snow pack at a rate (i.e. u at the top surface z = 0) of10−6 m s−1. Determine whether capillary effects are small; describe the natureof the model equation, and find an approximate solution for the melting of aninitially dry snowpack. What is the (meltwater flux) run-off curve?

7.13 Consider the following model, which represents the release of a unit quan-tity of groundwater at t = 0 in an aquifer −∞ < x < ∞, when the Dupuitapproximation is used:

ht = (hhx)x,

h = 0 at t = 0, x �= 0,∫ ∞

−∞hdx = 1

(i.e., h = δ(x) at t = 0). Show that a similarity solution to this problem existsin the form

h = t−1/3g(ξ), ξ = x/t1/3,

and find the equation and boundary conditions satisfied by g. Show that thewater body spreads at a finite rate, and calculate what this is.

Formulate the equivalent problem in three dimensions, and write down theequation satisfied by the similarity form of the solution, assuming cylindricalsymmetry. Does this solution have the same properties as the one-dimensionalsolution?

7.14 The tensor Dij (i, j = 1,2,3) has three invariants

DI = Dii, DII = DijDij , DIII = DijDjkDki .

(Summation over repeated indices is implied.) Show that the invariants of thetensor

Dij = α⊥uδij + (α‖ − α⊥)uiuj

u,

7.13 Exercises 459

where u = |u| and δij is the Kronecker delta (= 1 if i = j , = 0 if i �= j ), arethe same as those of the tensor

D =⎛⎝α‖u 0 0

0 α⊥u 00 0 α⊥u

⎞⎠ .

7.15 Suppose that a doubly porous medium consists of a periodic sequence ofblocks M with boundaries (fractures) ∂M . The concentration of a chemicalreactant c is taken to be a function of the fast space variable X and the slowspace variable x = εX, and we assume that c = c(x) + ε2ck(X), where thesuffix k refers to fractures (f ) or matrix block (m).

Let G(X,Y) be a Green’s function satisfying

∇2YG = δ(X − Y) in M, G = 0 for Y ∈ ∂M,

and suppose that

∇2Xψ = 0 in M,

ψ = χ on ∂M.

Show that

ψ =∫

∂M

∂G(X,Y)

∂NY

χ(Y) dS(Y),

where ∂G∂NY

= n.∇YG(X,Y), and hence show that

∂ψ

∂N

∣∣∣∣∂M

=∫

∂M

K(X,Y)χ(Y) dS(Y),

where

K(X,Y) = ∂2G(X,Y)

∂NX∂NY

.

Now suppose that the fluctuating matrix and fracture concentrations of thechemical reactant are given by

∇2Xcm = Pe

[∂c

∂t+ um.∇xc

]+ PeΛc − ∇2

x c ≡ Rm in M,

subject to cm = cf on ∂M , and

∇2Xcf − (1 − φf )

φf

∂cm

∂N

∣∣∣∣∂M

= Pe

[∂c

∂t+ uf .∇xc

]+ PeΛc − ∇2

x c ≡ Rf on ∂M,

subject to conditions of periodicity with zero mean.


Show that, if we define c∗ to be the solution of

∇2Xc∗ = 1 in M,

with

c∗ = 0 on ∂M,

then

cm =∫

∂M

∂G(X,Y)

∂NY

cf (Y) dS(Y) + Rmc∗,

and deduce that, for X ∈ M ,

φf ∇2Xcf − (1 − φf )

∫∂M

K(X,Y)cf (Y) dS(Y)

= φf Rf + (1 − φf )Rm

∂c∗

∂N

∣∣∣∣∂M

.

By integrating this equation over ∂M , show that the condition of periodicityof cf implies that the equation to determine c is

Pe

[∂c

∂t+ u.∇xc

]= ∇2

x c − PeΛc,

where u = φf uf + (1 − φf )um.7.16 The reaction rates in the reactions

SS r1−→ XL + nH2O,

KFsr2−→ K+L + AlO−L

2 + s SiOL2 ,

K+L + AlO−L2 + f XL r3−→ f IS + SiOL

2 ,

SiOL2

r+4�

r−4

Qz,

are related by

r1 ≈ f r3,

r2 ≈ r3,

r+4 − r−

4 ≈ (s + 1)r3.

The reaction rate r3 is given by

r3 = φIR3

[1 − f r3

φSR1

][1 − r3

φF R2{θL − 1 − (s+1)r3

φQR+4

}],

7.13 Exercises 461

where φi are porosities, Rk are rate factors (such that rk ∝ Rk), and the stoi-chiometric constants f and s, and the constant θL, may be taken as O(1) (andθL > 1). Show that r3 can be written explicitly in the form

2

r3=

{1

(θL − 1)γF

+ s + 1

(θL − 1)γQ

+ f

γS

+ 1

γI

}

+[{

1

(θL − 1)γF

+ s + 1

(θL − 1)γQ

+ f

γS

+ 1

γI

}2

+ 4(s + 1)

(θL − 1)γQ

(f

γS

+ 1

γI

)]1/2

,

where the coefficients γY represent the porosity weighted rate factors, i.e.,

γI = φIR3, γS = φSR1, γQ = φQR+4 , γF = φF R2.

Deduce that the slowest reaction of the four (as measured by γY ) controls theoverall rate, and give explicit approximations for r3 for each of the consequentfour possibilities.

mathematical geoscience.pdf

Documents