1. Archimedes porosity determination
In the Archimedes porosity determination, one measures or is given the following quantities:
(1), (3), (5), and (7).
(1) is the weight of the air-dried core:
𝑊𝑑 = 𝑑𝑟𝑦 𝑔𝑟𝑎𝑖𝑛 𝑤𝑒𝑖𝑔ℎ𝑡 𝑁𝑒𝑢𝑡𝑜𝑛𝑠 (𝑁, 𝑘𝑔 ∙
𝑚
𝑠2
) (1)
𝑊𝑝𝑓 = 𝑝𝑜𝑟𝑒 𝑓𝑙𝑢𝑖𝑑 𝑤𝑒𝑖𝑔ℎ𝑡 (𝑁, 𝑘𝑔 ∙
𝑚
𝑠2
) (2)
𝑊𝑠𝑎𝑡 = 𝑠𝑎𝑡𝑢𝑟𝑎𝑡𝑒𝑑 𝑐𝑜𝑟𝑒 𝑤𝑒𝑖𝑔ℎ𝑡 (𝑁, 𝑘𝑔 ∙
𝑚
𝑠2
) (3)
𝑊𝑝𝑓 = 𝑊𝑠𝑎𝑡 − 𝑊𝑑 (𝑁, 𝑘𝑔 ∙
𝑚
𝑠2
) (4)
In (5), 𝑔 (
𝑚
𝑠2 ) is the local gravitational acceleration. 𝜌𝑓 (
𝑘𝑔
𝑚3 ) is the fluid density.
𝛾𝑓 = 𝑔𝜌𝑓 = 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑓𝑙𝑢𝑖𝑑 𝑤𝑒𝑖𝑔ℎ𝑡 (
𝑁
𝑚3
) (5)
𝑉𝑝 =
𝑊𝑝𝑓
𝛾𝑓
=
𝑊𝑠𝑎𝑡 − 𝑊𝑑
𝛾𝑓
= 𝑝𝑜𝑟𝑒 𝑣𝑜𝑙𝑢𝑚𝑒 ( 𝑚3) (6)
𝑊𝑠𝑢𝑏 = 𝑠𝑢𝑏𝑚𝑒𝑟𝑔𝑒𝑑 𝑐𝑜𝑟𝑒 𝑤𝑒𝑖𝑔ℎ𝑡 (𝑁, 𝑘𝑔 ∙
𝑚
𝑠2
) (7)
𝑉𝐵 = 𝑏𝑢𝑙𝑘 𝑣𝑜𝑙𝑢𝑚𝑒 ( 𝑚3) (8)

2. 𝐵 𝐹 = 𝑏𝑢𝑜𝑦𝑎𝑛𝑡 𝑓𝑜𝑟𝑐𝑒 (𝑁) (9)
Archimedes principle for submerged core:
𝐵 𝐹 = 𝛾𝑓 𝑉𝐵 (𝑁) (10)
Force balance on submerged core (Figure 1):
𝑊𝑠𝑢𝑏 = 𝑊𝑠𝑎𝑡 − 𝐵 𝐹 (𝑁, 𝑘𝑔 ∙
𝑚
𝑠2
) (11)
Figure 1: Force Balance on Submerged Core
Subbing (10) into (11):
𝑊𝑠𝑢𝑏 = 𝑊𝑠𝑎𝑡 − 𝛾𝑓 𝑉𝐵 (𝑁, 𝑘𝑔 ∙
𝑚
𝑠2
) (12)
𝐵 𝐹
𝑊𝑠𝑎𝑡

3. Rearranging:
𝑊𝑠𝑎𝑡 − 𝑊𝑠𝑢𝑏
𝛾𝑓
= 𝑉𝐵 (𝑚3
) (13)
(13) may be used as a check on volume measured by calipers or by fluid displacement.
Archimedes porosity is determined by:
𝜙 𝑎 = 𝑝𝑜𝑟𝑜𝑠𝑖𝑡𝑦 =
𝑉𝑝
𝑉𝐵
(14)
Subbing (6) and (13) into (14):
𝜙 𝑎 =
(
𝑊𝑠𝑎𝑡 − 𝑊𝑑
𝛾𝑓
)
(
𝑊𝑠𝑎𝑡 − 𝑊𝑠𝑢𝑏
𝛾𝑓
)
(14)
Simplifying:
𝜙 𝑎 =
𝑊𝑠𝑎𝑡 − 𝑊𝑑
𝑊𝑠𝑎𝑡 − 𝑊𝑠𝑢𝑏
(15)
You can use anything proportional to weight ( 𝑊, 𝑁𝑒𝑢𝑡𝑜𝑛𝑠) in (15), observing consistent units.
For example:
𝜙 𝑎 =
𝑀𝑠𝑎𝑡 − 𝑀𝑔
𝑀𝑠𝑎𝑡 − 𝑀𝑠𝑢𝑏
(16)
Here, 𝑀 indicates mass. 𝑀𝑠𝑢𝑏 indicates the apparent submerged mass.

4. Once 𝜙 is known, we may state:
𝑉𝑔 = 𝑔𝑟𝑎𝑖𝑛 𝑣𝑜𝑙𝑢𝑚𝑒 = (1 − 𝜙) 𝑉𝐵 (𝑚3
) (17)
Further:
𝛾𝑔 = 𝑔𝜌 𝑔 = 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑔𝑟𝑎𝑖𝑛 𝑤𝑒𝑖𝑔ℎ𝑡 (
𝑁
𝑚3
) (18)
Therefore:
𝑊𝑑 = 𝛾𝑔 𝑉𝑔 = 𝑔𝜌 𝑔 𝑉𝑔 (𝑁, 𝑘𝑔 ∙
𝑚
𝑠2
) (19)
Solving for grain density:
𝑊𝑑
𝑔𝑉𝑔
= 𝜌 𝑔 (
𝑘𝑔
𝑚3
) (20)