auger drill mess in basement

A few months ago I had to call a plumber for a stopped up problem in my kitchen sink. Before i called, I used Liquid Plumber, Drano Max Gel without success. I took him to the basement and he removed a plug in a tapered pipe hooked up to the kitchen drain. That plug was put there by another plumber years ago. He did his thing with a electric drill. Once done, he plugged the hole with plumbers black tape. I remembered this as I watched. Now, months later its blocked again and this time I bought a snake that gets attached to a drill, and also the same tape. I made a hole in that tape where the hole was, entered the snake and almost all 25' was used. With hole opened, I tested the sink with water and of course i knew some water would come out of that hole. I wanted to be sure that the blockage wasn't closer to the sink. I covered the hole with some new tape. I've been hesitating to use the sink still, because I see drippings around that tape. First, if that pipe is 'cleaned' wouldn't the water keep moving in a downward trend, despite its small pitch? So, why does water still drip down even after not useing the sink drain for hours.
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If the cleanout is still dripping, three may be a downstream clog. Perhaps other sinks or toilets are connected to the same pipe, adding to the water level. Or the tape leaks at 0 atm, due to its leaky nature... Are you sure it is only the sink draining to it? Also, cleanouts should be plugged with something more durable and pressure resistant than tape, or you risk major flooding in the basement. Any hardware sells screw-in expandable rubber plugs for this purpose, and the plumber should have installed or recommended one.. Once in place, they resist leaking, can stand more hydraulic pressure, and may even allow the water to build up until the obstruction is blown out, downstream of the plug.
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Thanks, you be suprised how many people don't know what an expandable plug is, when asked in home depot, loews. I did get one in a local hardware store. Will see, if this does the trick. BTW, yes, the pipe I speak of is the only pipe hooked up to my kitchen sink. Of course, so is my dishwasher. They all eventually merge into the main drain. But I am not having problem with the others..just the kitchen sink was slow to drain. Oh yes, I seemed to recall now, the last time I used the dishwasher, it did create a mess under the kitchen sink by backing up. I hope that cleaning out the pipe that I've been talking about would stabilize the situation..thanks again:)
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