Approach will be presented in four steps:

1. Determine whether equilibrium, kinetics, or mass transport is controlling, 2. Relate precipitation to the physical variables in the field or laboratory core, 3. Relate the physical variables that affect scale formation or dissolution to space and time in the reservoir or core material, and 4. Combine the two into a single description explicitly and implicitly.

Two physical geometries will be examined algebraically, 1st(left). linear flow as might occur in a laboratory column, and 2nd(right). radial flow as might ideally represent flow into or out from a well in a reservoir. Pictorially, these can be represented as follows:

height(m)

re,Pe,equilibium

rw,Pw(well)

r,P(pressure)

radius, r(m)

flow, Q(m3/s)

height(m)

re,Pe,equilibium

rw,Pw(well)

r,P(pressure)

radius, r(m)

flow, Q(m3/s)

Two flow regimes will be considered for each geometry: A. no gas phase present, as is common with laboratory cores or in deep reservoirs at pressure, and B. with a gas phase is present. The difference is important to CaCO3, FeCO3, FeS, etc., but will no direct effect upon BaSO4, CaSO4’s, NaCl, etc.

Objective: Express the fractional change in porosity, n/n, in terms of flow conditions, brine composition, and time. Specifically, the calculation of n/n can be considered to be composed of four parts, illustrated here for calcite deposition or dissolution:

Summary of approach: the precipitation of, e.g., calcite, d(Ca2+,M), will be expressed as a function of pressure change, dP(psi), which will be expressed in terms of flow rate, Q(m3/s, or B/d) and reservoir properties, such as permeability, k(m2, or md) and porosity.

1. Relate kinetics and mass transport. First, estimate the average distance, or time, that brine will flow for half of the amount that can precipitate to precipitate, i.e., t1/2, and from this show that equilibrium is generally fast compared with reservoir flow.

The heterogeneous mass transport equation for precipitation in the presence of solid core material can generally be modeled with a mass transport-type equation. Calcite, CaCO3, will be used as an example precipitate material, wherein the number of moles of calcite precipitated per liter of brine, d(CaCO3), is equal to the change in the molarity of calcium ions, d(Ca2+,M) per second, dt(s):

Where km(cm/s) is a mass transfer coefficient for the ions in solution and can be estimated by any many dimensionless groups, e.g., the Sieder and Tate heat/mass trasfer analogy, Asolid(cm2) is the surface area of the solid reservoir material with a corresponding volume of brine, Vbrine(cm3), SSA(cm2/g) is the specific surface area of the reservoir material, rs/w(g/cm3) is the ratio of the solid mass to the corresponding volume of water, solid is the density of the solid, and n is the porosity of the reservoir. k1st order(s-1) is a first order rate constant for precipitation. For illustration purposes, assume solid 2.5 g/cm3, n = 0.25, and km = 0.001 cm/s (a conservative estimate):

k1st order(s-1) 0.0075SSA(cm2/g)

-Typically, when we measure SSA of core material by BET isotherm the value is on the magnitude of ½ to 1.0 m2/g = 104 cm2/g, but a range of values will be examined for radial flow (a similar table would apply to linear flow):

SSA (cm2/g)| K1st order (s-1)| t1/2 (sec)| Dist. Traveled (cm) at v = 90 ft/d* (r2 ft)| Dist. Traveled (cm) at v = 360 ft/d* (r0.5 ft)| 10,000=1 m2/g| 75.| 0.009| 0.00029| 0.00115|

1,000| 7.5| 0.09| 0.0029| 0.0115|

100| 0.75| 0.9| 0.029| 0.115|

10| 0.075| 9.0| 0.29| 1.15|

*v(ft/d) = Q(ft3/d)/{nA(ft2)} = Q(ft3/d)/{2nr(ft)b(ft)}, with b(ft) representing the reservoir thickness. In this table, b = 20 ft; Q = 1,000 B/d have been assumed.

Flows at less than r = 0.5 ft...