This article was written 4 years ago in Chinese based on some research and calculations I did, way before I got my Google Data Analytics certificate in 2021. It is by no means a serious thesis although my tone may suggest otherwise. This is a translated version with all identifying information removed.
1. Background#
Geological Survey for project A found out that the ground was karst-rich and recommended a per-pile survey be carried out (per-column survey for podium building, per-pile when a karst is found). Project A has a total of 482 bored pile foundations and 433 holes were drilled for the survey, 148 of which found karst underneath, with 49 holes detected karsts over 3 metres deep, 19 holes detected multiple karsts. Most karsts were found on the east side of the site and the east side had a very well-developed karst zone where many enormous karsts and stringed karsts were found. Around 70% of holes in the east side detected karsts. Most karsts were half-filled or empty.
The engineers proposed that these karsts be filled with chip stones and C15 concrete (1:1 in volume) to 1 metre above the karsts. Upon said concrete reaches strength equivalent to C10, pile foundations shall be cast. An estimation of about 4,537.8 cubic metres was also given for the karsts. During procurement, an estimation of around 5,000 cubic metres was listed in the tender document.
Project A started construction on March 22nd, 2019. All parties involved decided in a meeting prior to construction that the engineers' proposal was too time-consuming and pouring the same concrete designed for pile foundations (that is, pouring concrete as if there were no karsts until the concrete reached the designated pile top).
At the beginning of the construction phase, too much concrete was poured into drilled holes and many previously unknown karsts were discovered at multiple locations. As of May 20nd, 2019, a total of 216 pile foundations were constructed and karsts were found under 33 of them. Over 2,000 cubic metres of extra concrete was poured into pile foundations and the total pile concrete volume might grow out of control. I decided to make an estimation of how much concrete may be poured into the pile foundations, how much it might cost, and how long it might take. If my estimation went above 10,000 cubic metres, it might be worth considering switching to a different method.
2. Model Building#
Assumptions and Model#
Thus, I considered building a model of all the underground karsts and attempted to calculate the total volume of karsts and extra concrete.
Per-pile survey holes were merely 150mm in diametre and could be considered 'points' in the grand scheme of things. I built a model using the 'Surface Fitting' tool in MatLab and using linear interpolation in between all survey holes.
Considering the complexity of underground conditions, a simplified approach was proposed where we delete all data regarding karst depth and keep only survey hole coordinates and the total height of karst (that is, the total length of empty segments in each hole). Such a decision was made based on simple calculations:
Let's say the straight lines on the left and right are adjacent survey holes and karsts were discovered under both of them. The distance between them measured l. The total height of karsts is d1 on the left and d2 on the right. Linear interpolation would tell us the shaded area in between were karsts. Since they took thousands, if not millions of years to form they should be relatively smooth with no sharp cliffs.
If we remove all data regarding karst depth, the total volume of karsts would turn into the shaded area on the right, which has the exact same area since they were both trapezoids and their areas were both calculated as (d1 + d2) * l/2.
Such simplifications also stand in 3-dimensional space. We will not go into details of calculations but feel free to contact me should you have any questions.
The simplified model is shown below. The total volume of all craters equals that of all underground karsts. It predicted several undetected karsts on the east side but was not so accurate on the south.
How Concrete Flows#
Concrete should spread in a circle under gravity, above ground, or several metres underground. Thus, an estimation of how far the concrete had spread could be made with existing data from 216 piles that were already finished. I tried to form a relation between karst height and distance spread (the radius of the circle) with linear fitting:
3. Making Estimations#
A rough estimation should be made first to determine whether switching to a faster or cheaper method was in order. This estimation didn't have to be accurate but should give a number under the worst-case scenario, which, all things considered, should be when all karsts were filled with concrete.
There was hardly any karst discovered west so I divided the site into 5 sections. There was quite some distance between section III and V so section III's south boundary was 2 metres south of the southmost piles in that section. Section V's north boundary was 2 metres north of the piles in that section.
The total volume of karsts in section I ~ V was 9,000 cubic metres, less than the threshold where we must consider an alternate method.
As the construction progressed, the spread radius displayed a different tendency in relation to karst height. After 300 piles, another graph was made using polynomial fitting to discover that relation:
I would say there was no relation between spread radius and karst height as the radius was always very close to 1.5 metres but for the sake of accuracy, let's pretend that there was a correlation between them.
The minimum distance between pile foundations was 4 metres so there shouldn't be any interference between different piles. The total volume of extra concrete poured into a single pile should be calculated as the volume of a cylinder - 1.5 metres in radius and the same height as the karst underneath - minus the volume of the pile foundation.
The result, as per MatLab, was 5,730 cubic metres.
4. The Result#
On July 14th, 2019, all pile foundations were finished. A total of 5866.87 cubic metres of extra concrete was poured into the ground. My estimation was a mere 2.33% offset from the actual number.
5. Other Things#
This is by no means an entirely scientific approach to estimating concrete volume when casting pile foundations in karst-rich ground. I may be close out of pure genius or dumb luck. Please take everything with a grain of salt and try to form your own opinions and theories.