One dimensional dissolution diffusion transport for quartz

vein formation

The porosity of the host rock is , the silica concentration is m in units mol m-3

, the

supersaturation of the host rock prior to fracturing is m2, and D is the diffusion coefficient

D in units m2 s

-1 . m2 is related to pressure solution

The left hand side of Eq. (1) is a diffusion equation and the right hand side is a source

term when m<m2. The dissolution process is represented by a source term that is

proportional to both a reaction rate kd (in units mol m-3

s-1

) and the concentration

difference m - m2. Source term may represent other dissolution process as well

One boundary condition could be

m (x=) = m2 (concentration in the host rock far away from fracture)

The second boundary condition is at x=0 (on the fracture) is essentially a mass

conservation equation i.e., mass of silica input to the fracture by diffusion equals the

mass deposited on the fracture

(2)

Precipitation of quartz in the fracture is controlled by the reaction rate kp (units mol m

-2 s

-

1), the effective surface area for precipitation per unit area fracture wall a1 (units m

2/m

2)

and the difference m - m1 between the concentration of silica in the fracture fluid, m, and

the equilibrium concentration of silica in the racture, m1.

Precipitation process is fully reversible i.e. net of the forward (dissolution) and backward

(precipitation) - dissolution has a different rate than precipitation since precipitation

depends on surface area.

Conversion of the equation to dimensionless form (characteristic length l0 and

characteristic time

Concentration m can be represented by unit concentration

Concentration c=0 corresponds to m=m1 and c=1 to m=m2

With the imposed boundary condition the equation can be written as

(dimensionless equation) (3)

C=1 at x (cap) =

Where dimensionless distance x(cap) and are given by x(cap) = x/l0 and

= t/t0

td and tp are the characteristic time for the dissolution and precipitation and

are given by

The ratios t0/td and t0/tp are the only parameters in the model they are the Damkohler

numbers

…. (4)

These numbers measure the characteristic time for the diffusion process relative to the

characteristic time for the dissolution process and the precipitation process respectively.

The dimensionless equation has time dependent general solution but taking a stationary

state (time much longer than the characteristic time of any of the process) where the c/t

= 0 the equation reduces to

(the last two are the boundary condition where ld and lp are the characteristic lengths of

dissolution-transport and precipitation respectively and given by)

ld is the length scale into the host rock where there is a concentration difference because

of the cementation process.

The inverse length scale 1/lp can be interpreted as a characteristic concentration gradient

into the fracture.

By letting l0 = ld be the characteristic length of the system, we see that Nd = 1 and Np =

ld/lp. The number Np is therefore the relevant parameter for the cementation process over

long time spans where the silica concentration is almost stationary. This parameter (Np)

is now denoted Da and it can be written

…(5)

The solution for the stationary equation becomes

…(6)

The concentration of silica on the fracture cf and mf are given by

Da-number defines two different regimes in terms of the silica concentration in the

fractures, depending on whether Da is much less than 1 or much larger than 1. cf = 0 if

Da>> 1 and cf= 1 if Da <<1 – the latter is precipitation dominated regime and the former

is dissolution transport dominated regime

The stationary solution to the concentration difference between the host rock and fracture

has the characteristic length l0.

The concentration difference between the fracture (mf) and deep into the host rock (m2) is

reduced to the half at the distance x = ln2l0. It is seen from Nd=1 that the characteristic

time for diffusion and dissolution are equal for the characteristic length l0.

The Damkohler number

The rate kd is now replaced by the Da-number and the quartz cementation process is

studied in terms of the Da-number. The Da-number allows for a study of the vein

cementation process without going into details of the dissolution process. The quartz

cementation process can be studied using the following Da-numbers 0.01, 0.1, 1 and 10,

100 and 1000 where the precipitation limited regime is represented by Da = 0.01 and 0.1,

and

the diffusion limited regime is represented by Da = 10, 100 and 1000, and where Da = 1

is an intermediate regime. The characteristic time and length can be represented as

follows using the Da-number

…(7)

Numbers for the parameters D, m1, m2, kp and a1 are needed before the characteristic

time, length and cementation rates can be calculated. A simplified system where quartz is

the dominant mineral is assumed,

H2O + SiO2 = H4SiO4

for which you can calculate the K

very simplistically, you can assume activities of water and silica to be unity and also the

solution to be sufficiently diluted for taking activity coeff. of SiO2 = 1, then the

equilibrium conc. of vein fluid (m1) can be taken as equal to K for which the temperature

dependence is known

kp can be expanded by the Arrhenius law as

problem with the laboratory data for quartz kinetics applied to sandstones could be that

simple geometrical models for the pore space over-estimates the effective surface area for

quartz precipitation

D- is also a difficult parameter to estimate

Rate of Fracture Cementation

The rate at which the fracture is filled is given by

…..(8)

Where vq is the molar volume of quartz (multiplication by 2 is because

fracture is filled bothways from the fracture centre)

Using the previous solution (stationary condition)( eq 6)

For Da<<1

Characteristic time for dissolution and transport process in the host rock is much longer

than the characteristic time for precipitation in the fracture when Da>>1. The

cementation rate in this regime taking Da+1=Da

Using the definition of Da (lp/ld) as previously given,

We can also find out the dissolution rate for values of Da with D and kp known by using

the equation previously derived (eq 5).