Methods and systems for characterizing multiple properties of cement using p-waves of multiple frequencies

A method comprising:

• providing a cement composition in a sample cell;
• before the cement composition reaches a percolation threshold,
• transmitting at least a first p-wave having a first frequency and a second p-wave having a second frequency through the cement composition while allowing the cement composition to at least partially hydrate, wherein the first frequency is higher than the second frequency;
• determining a velocity of the first p-wave through the sample and a velocity of the second p-wave through the cement composition;
• after the cement composition reaches the percolation threshold,
• transmitting at least a third p-wave having a third frequency through the cement composition while allowing the cement composition to at least partially hydrate, wherein the third frequency is higher than the second frequency;
• determining at least a velocity of the third p-wave through the cement composition;
• calculating a first compressibility of the cement composition (1/Kb) before its percolation threshold according to a formula:
• V
• s
• =
• (
• K
• b
• +
• 4
• 3
• ⁢
• G
• [
• (
• 1
• -
• ϕ
• )
• ⁢
• ρ
• s
• +
• (
• ϕ
• -
• 2
• +
• α
• ϕ
• )
• ⁢
• ρ
• f
• ]
• )
• 0
• .
• 5
• wherein Vs is the velocity of the second p-wave through the cement composition, ϕ is a porosity of the cement composition, α is a geometric factor and is related to ϕ as α=½(1+1/ϕ), ρs is a weighted average density of solid constituents of the cement composition, ρf a density of one or more fluid components of the cement composition, and G approaches zero;
• calculating a Poisson's ratio of the cement composition (ν) using the formula:
• v
• =
• 1
• 2
• ⁡
• [
• (
• V
• f
• V
• s
• ⁢
• h
• ⁢
• e
• ⁢
• a
• ⁢
• r
• )
• 2
• -
• 2
• (
• V
• f
• V
• shear
• )
• 2
• -
• 1
• ]
• wherein Vf is the velocity of the third p-wave and Vshear is a velocity of a shear wave through the cement composition;
• calculating a Young's modulus of the cement composition (E) using the formula:
• E=2Vshear2ρ(1+ν)
• wherein ρ is a density of the cement composition;
• calculating a shear modulus of the cement composition (G) using the formula:
• G=Vshear2ρ
• calculating a second compressibility (1/Kb) of the cement composition after its percolation threshold using the formula:
• K
• b
• =
• V
• f
• 2
• ⁢
• ρ
• +
• 4
• 3
• ⁢
• G
• determining an attenuation of one or more of the first p-wave and the third p-wave that has been transmitted through the cement composition;
• based at least in part on the attenuation of the first p-wave or third p-wave, determining a sonic gel strength of the cement composition;
• providing a fixed amount of an inert gas in a cavity in communication with the sample cell;
• measuring an initial pressure of the inert gas within the cavity;
• measuring a second pressure of the inert gas within the cavity after the cement composition has been allowed to at least partially hydrate;
• and
• based at least in part on the initial pressure and the second pressure of the inert gas, calculating a change in volume of the cement composition that has been allowed to at least partially hydrate.