Weber and Reinhardt [23] have presented a detailed analysis of water transport from lightweight aggregates to cement paste. The analysis presented below is a variant inspired by their approach, in which the water flow rate is equated to the value needed to maintain saturation in the hydrating cement paste. It is assumed that the cement paste and water reservoirs are both composed of a set of equi-size cylindrical pores as shown in Figure 10. In the case of the water reservoirs, the pore size is fixed while in the case of cement paste, the pore size decreases with continuing hydration.
We begin by considering the flow of water through a cylindrical pore (radius Rpaste and length L, both in units of m) in the hydrating cement paste in a unit volume (1 m3) of concrete. The volumetric flow in m3/s, Q, is given by:
 | (4) |
where ΔP = pressure drop (Pa), k = permeability (m2), and µ = fluid viscosity (0.001002 Pa·s for water at 20 ºC). If water is flowing from a pore of radius Rres (m) in the water reservoir (effectively creating a meniscus within this reservoir pore) into a pore of radius Rpaste in the hydrating cement paste, the pressure drop can be estimated by [23]:
 | (5) |
where γ is the surface tension of water (0.07275 Pa·m at 20 ºC) and a contact angle of 0 degrees has been assumed. Rpaste should logically be the largest pore size in the cement paste that desires to be water-filled at the expense of the pores in the water reservoirs.
It is desirable that this water flow exactly balance the water demand of the concrete due to the ongoing chemical shrinkage. We define a fill factor, ∂ε|/∂t, as the water demand per unit volume of concrete porosity (units of s−1) to maintain saturation. This will be given by the differential rate of water demand from equation (3) converted to a volumetric basis, divided by the capillary porosity of the concrete, Φconcrete or:
 | (6) |
where ∂α/∂t is the current hydration rate for the time period of interest (s−1) and ρw is the density of water (998.23 kg/m3 at 20 ºC). Then, extending equation (6) from a global porosity to a local individual pore level, we require that our volumetric flow rate within the individual pores, Q, also be:
 |
(7) |
Finally, equating equations (4) and (7), and solving for L yields:
 |
(8) |
With estimates of the permeability of the cement paste [24] and the pore sizes of the paste and water reservoirs, one can calculate first the pressure and then the pore length or water flow distance. Here, a pore radius of 10 µm was selected for the water reservoirs, but using a value of 100 µm, for example, resulted in only minor changes in the calculated flow distance values presented in Table I, which provides the calculated pore lengths (or water flow distances) for early, middle, late, and "worst case" hydration conditions for low w/b cement paste in concrete.
| Table I: Calculated water flow distances in cement pastes of various ages with the water reservoir pore radius set at 10 µm, chemical shrinkage (CS) of 0.07 kg water/kg cement and a cement factor (Cf) of 700 kg/m3. | |||||||
|---|---|---|---|---|---|---|---|
|
|
Paste |
Rpaste |
ΔP |
Hydration |
Concrete |
Fill |
Flow |
| early | 1.00E−17 | 1.0E−06 | −130950 | 6.94E−06 | 0.12 | 2.84E−06 | 21.4 |
| middle | 1.00E−20 | 1.0E−07 | −1440450 | 1.16E−06 |
0.06 |
9.47E−07 | 3.90 |
| late | 1.00E−22 | 2.00E−08 | −7260450 | 2.31E−07 | 0.015 | 7.58E−07 | 0.98 |
| worst | 1.00E−23 | 2.00E−08 | −7260450 | 2.31E−07 | 0.01 | 1.14E−06 | 0.25 |
One can observe that the "likely" water flow distances vary from tens of millimeters at early ages to millimeters at middle ages to hundreds of micrometers at later ages. While these estimates are quite approximate in nature due to high uncertainties in the proper values to use for paste permeability and pore radius, the early and middle age flow distances are quite similar to the observed penetration depths of drying fronts in a w/c=0.40 ordinary portland cement paste, based on X-ray absorption measurements [25]. In that study, a penetration depth on the order of 20 mm was observed for specimens immediately exposed to a drying environment while a penetration depth of about 4 mm was observed for specimens first cured under saturated conditions for 1 d or 3 d. Penetration depths of several millimeters during the first few days of sealed curing, as predicted for middle age (Table I), were reported in [26] based on a combined experimental and analytical evaluation.