How do I calculate OG when adding fermentables?
I have a question about calculating OG when making late additions of fermentables to your beer. I was reading in Extreme Brewing about this being one way to raise the ABV. For instance, if my OG was 1.07ish in three gallons of wort and I add three pounds of DME every five days (or after the krausen falls) for 20 days (boiled in a half gallon of water) and after five more days add three pounds of brown sugar or candy sugar boiled again in a half gallon of water and repitch champagne yeast. How do I calculate the OG accurately? Do I aerate the new fermentables that I add? Thanks in advance for the advise.

Sorry I'm not good with the numbers but for OG purposes I don't know why you couldn't add the ingredients into beersmith/beer calculus/your beer math app of choice and get an accurate overall gravity of the fermentables. Calculating FG might be a little trickier with the extra yeast being pitched.
However, dang thatll be pretty high gravity. 1.070 + 12 lbs dme + 3 lbs brown/candy sugar in 5.5 gallons? Am I reading that right? 
If you add too much sugar, you will end up killing the yeast and have a very sweet drink that will not carbonate.
First: Only aerate at first pitch. Aerating any time later will result in oxidation and give sherry or wet cardboard tastes. How to calculate additional gravity ........  Find out how many gravity points the additive has. That is; if dissolved in 1 gallon of water, what would the gravity be. For DME its 1.045, or 45 gravity points. For brown or table sugar its 1.046 or 46 gravity points. LME = 36, Corn sugar = 36, Honey = 33.  Say you add 3 lbs of DME (dissolved in o.5 gallons of water) to 3 gallons of 1.070 wort. In 3 gallons of wort you have 70x3 = 210 gravity points. 3 lbs of DME = 3x45 = 135 gravity points. Total gravity points = 210+135 = 345. Since your new volume is 3.5 gallons, divide that by 3.5 = 345/3.5 = 99, or will give you an effective OG of 1.099. 
You can figure that 1 lb. of dme equals approx 1.045 in 1 gal. of water, and 1 lb. cane sugar is around 1.046 in 1 gal. water, same is true of brown sugar. So, in the case of dme (3lbs in 1 gal. is 3x45=1.135)/(.5)=1.270 That number would then be devided by the total volume of liquid in the beer and added to the 1.07ish amount. So if the amount that is fermenting is 6 gals., divide 1.270 by 6 to give you an additional gravity of 1.045 which you would add to 70 for a grand total of 1.115 in 6.5gals. Do the same for each addition of sugar. Secondly, I'm not sure about the addition of oxygen to the fermented beer. That could lead to negative flavor components.
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So, if I'm following the formula out correctly for all four additions and a total wort volume of five gallons I should have a SG of 1.1509? My intent of the champagne yeast and candy sugar is to help dry out the beer so that it isn't like drinking hopped syrup.

Here is how BeerSmith works it out.
7lbs 12oz 2 Row in 3 gal = 1.070 or a 6.84 ABV with FG = 1.018 + 3lbs DME = 1.099 OG for 3.5 gallons, 9.69 ABV, FG = 1.025 + 3lbs DME = 1.120 OG for 4 gallons, 11.87 ABV, FG = 1.030 + 3lbs DME = 1.136 OG for 4.5 gallons, 13.59 ABV, FG = 1.033 + 3lbs DME = 1.149 OG for 5 gallons, 14.98 ABV , FG = 1.036 + 3lbs Sugar = 1.155 OG for 5.5 gallons, 15.63 ABV , FG = 1.038 Note, Beersmith does not do sugars correctly. Sugar will ferment out 100% and add alcohol which will lower FG. BeerSmith said my Blonde Ale with a 21% load of inverted sugar would have a FG of 1.011 and it was 1.002 when I bottled it. The BeerSmith estimated ABV was 3.88% and actual was 4.94%. BTW; use brewers’ candy or invert your sugars first. Unless you like cider flavors in your brew. I have one brew going now with a 42% invert sugar load and it tastes great. Down from 1.108 to 1.020 so far and it should go much lower. Previous brews with as little as 13% noninverted cane sugar had a very off cider like flavor 
I disagree with the other posts saying not to oxygenate multiple times. Adding 15 pounds of fermentables over three weeks is asking a lot of the yeast. Instead of adding o2 at the time of additions, I'd add it over the first few days of fermentation while the yeast are replicating. This is when the yeast will use it to make strong cell walls allowing them to go the distance. O2 additions during yeast activity won't lead to oxidation as it will be used up by the yeast.
Also, champagne yeast may not be the best choice for your second strain. While it is alcohol tolerant, it is a wine yeast that doesn't do well with maltose. I believe Shea Comfort spoke of this on one of the sessions. To aid in attenuation you'd probably be better off with one of the high gravity beer strains such as wlp099. Champagne yeast in beer is best left to bottle conditioning where it will consume the simple priming sugars in the presence of higher alcohol without adding additional flavor. 
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Adding fermentables during the fermentation instead of all at once can sometimes help the yeast along, but this is a LOT. Beers like this can be insanely difficult, and many fail to attenuate fully. Champagne yeast should be the absolute last thing tried though, specifically BECAUSE it doesn't like maltose. If fermentation stalls, your best bet is just pitching a lot of fresh yeast already fermenting at high krausen. 
At an estimated 16% abv, I'm not sure what he can use except for a wine yeast to finish it off. I know WLP099 is supposed to be fine for high alcohol, but adding it late will not really help.
I'd like to know what the OP is trying to make. Why make a 16% abv beer (or whatever it might be), it is very difficult to get the bittering, hop and malt profile correct; just throwing in DME tells me it probably is not balanced. I'm sure he would get far more enjoyment from 10 gallons of 8% abv beer, and it would certainly be easier to ensure it fermented out. 
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I was going to say, "the most enjoyable beer he possibly can," but I'm guessing that this beer (if he can even pull it off) is probably perfectly suited to his idea of enjoyment. 
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