Heterogeneity, Geostatistics, Horizontal Wells, and Blackjack Poker
Authors: Dennis Beliveau 
Published: December 2007

This paper presents data on more than 1,000 horizontal wells specifically comparing their hydrocarbon production performance to offsetting vertical wells. The data are striking, revealing an approximate log-normal distribution of productivity improvement factors (PIF's). This distribution is ascribed primarily to geologic heterogeneities compounded by mechanical drilling and completion effects. Horizontal wells in conventional reservoirs show a mode or "most-likely" PIF=2; a median, or "50/50," PIF=3; and a mean, or "average," PIF=4. Somewhat higher PIF's are observed for heavy-oil horizontal wells and horizontal wells in heavily fractured fields. The data also show an operator's "expectation" should be based on the number of wells planned, with a larger number of wells yielding a higher "average" production per well.

In some cases, comparing actual production results with those predicted by the operator was possible. This also showed some rather startling results. Although our "predictive models" appear quite accurate when averaged over several wells, the error expected for any individual horizontal well is &gt;50%. Some simple gaming examples show that an error of this size should be expected for almost any reservoir calculation (perhaps a better phrase is reservoir estimate).

Introduction

Although reservoir engineers deal with technical uncertainty on a daily basis, experience shows that we don't properly appreciate its large impact on our business. In addition, we don't clearly communicate our "error bars" to other technical professionals or management. Underestimation of subsurface uncertainties often leads to underestimation of actual production results. We don't want to be wrong; but if we are, it's nice to have done better than predicted. However, if we don't recognize and communicate upside potential, it is often left on the table. By our very nature and training (not to mention corporate cultures), engineers tend to be conservative people.

Clearly, a certain amount of "gambling" or risk-taking is inherent in all aspects of our business. Two areas that jump to mind are forecasting hydrocarbon rates and reserves or prices and costs. As petroleum professionals, one part of our jobs is to identify prospect risks and potential rewards. After ranking prospects on the basis of their appropriate risks/rewards, we invest from the top of the list downward until we hit our minimum investment criteria or run out of money. Although we are still gambling, we have technically weighted the odds in our favor as much as possible by analysis of the potential risks and rewards. This is similar to several games of chance, including (1) blackjack poker that involves card-counting, a practice banned in casinos because it slants the odds toward the gambler; (2) backgammon, where the "doubling cube" allows the value of the game to increase at any time (i.e., you play at the higher risk/reward level if you like your odds or concede defeat and write off your current investment); and (3) certain craps games with dice, one of which will be discussed briefly in this article.

An example that neatly demonstrates the importance of subsurface uncertainties is horizontal well performance. This study is a technical "look back" at the performance of 1,306 horizontal wells from 230 fields around the world. The horizontal wells are in sandstone and carbonate reservoirs containing light and heavy-oil as well as gas. To be included in the survey, sufficient production data had to be available for estimation of the PIF of each horizontal well. The data show an approximate log-normal (positively skewed) distribution of PIF's and also that our current predictive methods are reasonably accurate when applied over several wells. However, the typical error associated with any individual horizontal well estimate is at least ±50%! Further, only half the time at best do actual individual well results turn out within ±50% of their forecasts! And, I suggest these just might be typical error bars for any reservoir engineering estimate.

Log-Normality and Uncertainty

A recent article by Clapp1 is a classic example of the large role played by uncertainty in drilling programs. The paper opens with a simple but powerful statement: "Uncertainty about the outcome of individual wells is a complicating factor in the development of performance measures." Think of your last major drilling or development program. Although the wells on average may have been successful, a (large) number were likely quite different from their expectations.

Clapp clearly explains why with a very simple example: assume an exploration program with 20 wells drilled, each with a probability of success (POS) of 20% and with each successful producer yielding 10 million bbl of reserves. Obviously, the "expected" reserves from this program would be 40 million bbl (20 wells×20%×10 million bbl). Clapp then asks, "How concerned should we be if our program results in a reserve of 24 million bbl, which is 40% less than our expectation?"

Fig. 1 shows a simple Monte Carlo analysis for this drilling program. Although our expectation was 40 million bbl, Fig. 1 shows that there is a 35% chance of achieving &lt;20 million bbl and a 51% chance of achieving &lt;30 million bbl. In other words, only a slightly better than even chance exists of achieving at least 24 million bbl; about the same as a coin toss. In addition, the results of this program will be within ±50% of the expectation less than half the time. This simple example bothers almost every engineer and manager who sees it; it even makes a lot of geologists squirm! Even when a wide distribution of results is expected, the large range from a simple example usually surprises people. In another fascinating article, Capen2 says, "Our desire for precision in such an unpredictable world may be leading us astray. Because we are paid to know, do we find it difficult to admit we don't?" Rather than letting our engineering egos (=) get in the way, perhaps we need to get more in touch with our geologic (&lt;, ?, and &gt;) sides.

Most reservoir parameters (e.g., porosity, gross/net thickness, permeability, and saturation) are log-normally distributed about their mean values. Because rates and reserves result from the multiplication of a number of these parameters, it should come as no surprise that they show similar distributions. Fig. 2 shows some properties of log-normal curves. Most importantly, the arithmetic mean (average) is larger than the median (50/50) value, which is larger than the mode (most-likely) value. The difference between the three values depends on how skewed the distribution is. And interestingly for our business, the distribution tends to be skewed toward the positive (dare I say upside?).

Human nature and, indeed, much of our engineering technical training and focus tend to have a "mean" perspective. However, the reality of log-normal distributions is that most individual outcomes will be closer to the mode/median values. It is important to realize these differences exist and to reconcile their effects on our reservoir engineering thinking processes and development planning. For example, the following observations will be supported by data in this paper.Even with a correct estimate of POS and outcome, expecting the result of any single well to approach the mean or "correct" value is unreasonable; i.e., the estimate you just gave your boss for the next well is wrong!Even with a correct estimate of POS and outcome, it is most likely the result of any single trial will be less than the mean or "correct" value; i.e., the estimate you just gave your boss is not only wrong, it's too high!!