Tuesday, October 15, 2013

Gas-Fired Electricity: A Problem with Demand and Supply

California's power market is not a typical example, it's an extreme example, but useful to analyze because it is large and it represents something most states are gradually moving toward.

Enthusiasm for gas-fired power growth is high.  The projections for load growth and coal retirements make it seem like gas has a golden, inevitable future.  There are risks to this vision however.

  • Higher gas prices destroying demand
  • Higher electricity prices (not cause by gas prices) destroying demand
  • Other things destroying demand (DR incentives, economic malaise, conservation, technology)
But a threat being overlooked, or underestimated, is the threat from renewables.  I previously posted a snapshot of California power stats in early October.  Here's another glimpse:

This is a picture of a nightmare for natural gas.  Load not growing, and renewables increasing rapidly.
It's not like this everywhere, California is special.  They have prime solar geography, and powerful carrots and sticks to ensure renewables growth.  They also have very high power prices.

Power generation isn't growing on trend right now.  And the private sector is efficiently capitalizing on all the government incentives for renewables.  

Solar and wind are the two horsemen of the gas apolocalypse.  (bad/worse for coal too.).  

Solar and wind are not much alike however.  Solar power is much more valuable than wind (more predictable and nearer to peak load hours), and takes longer to construct.  This gives us much greater visibility into the solar pipeline (as it were).  There is A LOT of solar coming on line in the next 3 years.  Most of it in California, but increasingly elsewhere.

And the off-grid issue is becoming material.  The solar panels being installed by the thousands on California rooftops are not going to ever go away.  It is costing the system a fortune in rebates, and the supply + peak shaving potential of that power is very dangerous to the existing structure of the power generation sector.

So gas will gain from coal retirements (how much how soon is debatable), but stands to lose from both demand loss, peak loss, and renewables displacement.  

Notice in the graphs above, that on similar month-to-date weather, California is behind -689 GWH in total generation demand, and solar is up 172%, or +155 GWH.  Since wind capacity is also up YOY, that leaves the thermal power piece of the pie shrinking faster than coal retirements, thus natural gas is still a net loser.

Monday, October 14, 2013

3rd Quarter Baker Hughes Well Counts released

On Friday Baker Hughes released the preliminary Q3 well spud counts by basin, along with average rig counts in each.  No distinction is made between oil and gas rigs, so the data is less useful in basins with separate oily and gassy windows, like the Eagle Ford. Nevertheless, here are the implied drill times in several gassy basins, with very little improvement shown over Q2:

Friday, October 4, 2013

Oil and Gas Rig Counts

Today's rig report from Baker Hughes contained few changes.  Detailed reports at:

California's Changing Power Generation Mix

California's power market is notable for its renewable initiatives, high retail prices, and flat demand trends.  It is an often cited example as both a success and a failure, and will be the focus of attention as various mandates are implemented through the rest of the decade.  Here is a quick look at the start of October, and how it begins to compare to October 2012.  Some changes are obvious, but a full month of data and some weather comps will be necessary to draw more conclusions, so the graphs will be revisited in four weeks.

Tuesday, October 1, 2013

Retail Electric Power Prices in Texas and California

As we look for a return to trend growth in power generation nationally, it is interesting to note widening retail price divergence in retail electricity prices for two major power markets (both of them natural gas intensive).

Average prices in California are now over 14 cents/mwh, while they have fallen below 9 cents in Texas.

Monday, September 30, 2013

EIA July Natural Gas Demand

The EIA released the monthly data for July 2013 today, with few surprises.  The electrical and industrial demand numbers were as follows:

Electrical generation demand was well below 2012, but only 1 BCFD below July 2011, despite far fewer cooling degree days.

NOAA's climate prediction center records these CDD counts for the last 3 years:

July 2011:  +411
July 2012:  +408
July 2013:  +351

Friday, September 27, 2013

Canadian Natural Gas Drilling Activity Up

Canadian natural gas drilling activity trends have been diverging noticeably from the US.  With today's US rig count down a steep 10 rigs to 374 gas rigs, near the 3 year low, the Canadian NG rig count rose 13 to 158, near a 3 year seasonal high:

Should we look for an increase in output from the north?  Perhaps, although the rig fleet has only recently begun the ramp up, so we may see nothing for some time.  Precision Drilling (PDS) is dominant in the canadian drilling market, and their most recent comments were positive, including an expectation that more rigs will soon be working to delineate the northern BC/AB shale gas plays in preparation for LNG export.

Tuesday, September 24, 2013

Texas Drilling Permits and Completions

As previously discussed, Texas publishes a helpful monthly summary of drilling permits and well completions.  The catch is that there are three types of permits and only two types of completions.  Permits can be for oil, gas, or oil+gas.  Completions are classified as either gas or oil.

Here is a statewide look at the permit history and the completion history, with monthly data and a 3 month moving average line:

Tuesday, September 17, 2013

Texas Production and Drilling Trends

Texas remains the epicenter of oil and gas activity in the US.  About half the 1,700+ active rigs in the nation are working there. But the rig mix is changing significantly.  From about even three years ago, oil rigs now outnumber gas about 6 to 1:

The Texas Railroad Commission collects and reports oil and gas statistics, but their data is not close to finalized for many months, so recent production is typically understated.  This has let some to wrongly conclude that production is in decline.  The EIA uses a methodology for estimating unreported production, and their data is less subject to revision and more accurate for recent periods.  Here is the monthly history since 2010 of the EIA data vs. the TX RRC data for natural gas production.  Over time, the RRC number has averaged about 0.2 BCFD below the EIA estimate.  But look at the current period:

So it seems that unless a new source of error has crept in, the RRC data is not reliable for close to a year.

It can be difficult to accurately assess oil vs gas activity levels in Texas, part because drilling permits can be one of three types:
  • Oil
  • Gas
  • Oil & Gas
But well completions are classified as either oil or gas, not both.  When a well completion is considered an oil completion, the production therefrom is classed as "Oil" and "Casinghead Gas".  A gas well completion has production classified as "Gas Well Gas" and "Condensate".  

To look strictly at oil wells, we see that the amount of casinghead gas produced with the oil has maintained a steady percentage as oil production has grown.  Here is the %, on a BTU basis, of casinghead gas produced from oil wells 2010 to present:

This is a high associated gas content.  It implies that even with a low gas rig count, a significant volume of natural gas will be produced as Texas oil production continues to rise.

Now total liquids is another point worth making.  Here is how the RRC reports it, separating oil and condensate.  (There is some discussion about the classification of the liquids coming out of the Eagle Ford.)

Monday, September 16, 2013

Rig Count Classifications

US rig counts have been under scrutiny as reliable predictors of future production for two reasons:
  • Classification of the rig, like the drilling permit itself, is at the driller's discretion in many cases.  And drillers are motivated to find more liquids and less gas, and convince investors that they are doing so.
  • Drilling targets contain meaningful amounts of both oil and gas, more so than in the past.
  • Production results don't seem to be well correlated to rig counts.
This issue has been widely reported, but not quantified.  Here is an example that is more than anecdotal, and it gives some insight into what is really going on.

I looked for an example that had these characteristics:
  • Both oil and gas targets
  • Both oil and gas rig counts
  • Good data
  • A basin where gas and oil were found together in the zones being targeted.
I picked the epicenter of the Granite Wash, Wheeler County Texas.  Here are the basics.  (Oil includes condensate.)

Some initial observations:
  • Rig counts are very current (through Sep 13), but production goes only through April.  Some production is reported through June, but it is likely not a complete account, so there is the lag is reporting, plus the obvious lag between drilling and production.
  • Drill times are averaging around 25-30 days overall in the Granite Wash.
  • Oil rig counts were zero or thereabouts for nearly all of 2011, yet oil production steadily grew from about 21 KBOPD to 34.  
  • In January 2013, an abrupt reclassification of 8-10 rigs took place, from gas to oil.
From a BTU standpoint, Wheeler still produces about 3x as much gas as oil, but gas has been in apparent decline since Jan '12, and oil is trailing off but less so.  There may be gas takeaway constraints in this area, but that would not explain a decline in gas production, unless dry gas was in decline and wet gas production was increasing.  

Chesapeake produces almost half the oil in the county, so their own rig classification and production history will explain this in further detail.  

Wednesday, September 11, 2013

Shale Gas: The Future is Low Gas Prices and Overproduction

Shale gas math has proven hard for producers, consumers, and investors to accept.  Resource scarcity was the reigning paradigm, and the rules were:
  • Hydrocarbons getting harder to find
  • Each well costs more (inflation, deeper/remote drilling)
  • Each well finds less (infill drilling, downspacing, weaker prospects, depleted fields)
The shale revolution came, but it didn't seem sufficient to offset production declines because:
  • Decline rates were high
  • Prospects were few (for a long time just the Barnett and Fayetteville)
  • Wells were expensive
But production grew, and grew.  New reserves were found onshore.  Technology advanced, costs came down.  So gas prices came down to the point that demand was elastic by competing with coal.

But whether it is the human tendency to anchor on the status quo, or to doubt revolutionary claims, or something else, the situation is setting up again for natural gas.  Old metrics die hard, and the legacy cost structures and productivity seem to be blinding us to the (increasingly) obvious.

Exhibit A:  Shale basins and rig productivity, from Baker Hughes Q2 2013:

Consider the best known basins now, the Marcellus and Utica.  608 wells were drilled in these two plays in Q2, and rigs were averaging a little less than 2 wells per month, so an average of 113 rigs were working, about 30% of the gas rig fleet.  

What is the average well going to produce?  This is an important question.  The Utica is perhaps a little early to set a type curve on, with about 500 wells drilled and only about 125 producing.  But the early wells are also typically improved upon over time, and the best wells to date were drilled by Antero Resources, delivering what are frankly mind-boggling flow rates, around 8-10,000 BOE per day, much better than the best Marcellus wells.

On the Marcellus side, there are 3 or 4 well types at this point:
  • Dry Northeast
  • Dry Southwest
  • Rich Southwest
  • Super-Rich Southwest
These wells are generating 4 -15 BCF reserve estimates per well (dry gas only).  The drilling is moving to the wetter regions, but still plenty of dry gas wells being drilled, and obviously the ability to migrate rigs quickly should gas prices perk up.  

Here then is the basic truth:

200 wells per month, at 5 BCF dry gas reserve estimates = 1 Trillion cubic feet developed per month.  At that constant development rate, production would peak at over 30 BCFD, multiples of what is being produced now.  And wells are going down faster, cheaper, and finding more gas as these plays develop and operators improve completion techniques.

There is also a lot of undeveloped acreage, so prime drilling locations will not deplete quickly.  

We are using the best play as the example, granted.  But the data is lagging, and operators are projecting better results going forward.  There are so many other upside risks to this, including stacked pay:  The Upper Devonian is above the Marcellus, and early results are looking great.  This can multiply reserves very quickly, and take advantage of existing infrastructure.

We could go through a similar exercise for the Eagle Ford, where drill times are much faster and the rig fleet is double that of the Marcellus/Utica (including oil rigs).  The dry gas area of the Eagle Ford is being neglected now in favor of liquids to the north, but the reserves have been proved, and the locations are there when prices justify it and bottlenecks are cleared.

What is going to happen?  Gas demand will steadily and noticeably rise due to electricity generation, industrial use, and finally LNG export.  But the many BCFs these sources will eventually consume will not overwhelm production capacity, even at very low prices.  The dreams of $5 gas are very very far off.  Another expected outcome is a reduction in price volatility based on low gas inventory.  If anything, the volatility will come when demand evaporates (unseasonable weather for example), and there is nowhere to put the produced gas.  A few of these events seem inevitable.  

Monday, September 2, 2013

Eagle Ford Natural Gas Production

Texas currently produces almost 30% of the nation's marketed natural gas, about 20 BCFD.  The state's overall output has been relatively steady for the last three years, though activity has fluctuated substantially from basin to basin.  The Barnett and Haynesville have been in noticeable decline.

The Eagle Ford is one area where production has been growing quickly.  But rigs have been steadily moving from the dry to the wet side of the play.  Transportation constraints are a factor as well, and gas has a harder time getting out of the basin.  Here is what the Texas RRC reports for gross production by month from the major counties in the play.  Note that Liquids = Crude and Condensate, not NGLs.  NGLs are still in the gas stream at the well head:

90% of the natural gas is produced in 7 counties, with Webb producing the lion's share.  At May 2013, here is the relative contribution:

Gas production from the EF will look as though it is growing year-over-year at the end of 2013, but the slope is currently negative, and gas rig counts are down to a meager 37 (up from a 3 year low of 23):

The drilling results from the wetter regions have wide ranging GORs, but the coveted acreage can produce over 90% liquids, so it is unclear whether associated gas will make a major contribution going forward.

Dry gas reserves on the other hand are ample, so a price signal and pipeline should bring the rigs back quickly.  Drill times in the EF are much faster than Haynesville or Marcellus, so a small rig count can be deceiving.  The best in class wells are drilling around 10 days to TD now, and that will undoubtedly improve.  A crude model of potential production looks a little like this:

37 Rigs @ 2 wells per month x 5 BCF per well = 370 BCF developed per month.  At that constant drilling rate, production would asymptotically approach 12 BCFD  (370 BCF per month / 30 days).  So that's still impressive.  The Haynesville has about the same rig count, but wells take about twice as long to drill, and have a similar EUR of 5-6 BCF.

Thursday, August 29, 2013

Legacy Shale

US natural gas production is conveniently divided into Shale Gas and Conventional Gas.  Usually the point is to show the inexorable growth of shale, and twilight of conventional production.  The significant exception to that is the Haynesville Shale.  It was rapidly developed in a $4+ gas environment, 2009-2011.  When gas prices fell, oil prices rose, and better shales were discovered, the rigs left and production quickly began to decline.

The rig counts fell from over 150 to 32, now at 42:

The play is split between Louisiana and Texas.  Louisiana has the larger piece, and better public data.  So looking at production in the major haynesville producing parishes (counties) over the last few years we see this:

It must be noted that the above includes perhaps 1 BCFD of legacy production, so the net production from the shale itself in May was probably only around 4 BCFD, a decline of 2.2 BCFD in 18 months.

The play was so young that most wells were in the steepest segment of the decline curve.  The Barnett Shale experienced a similar decline in rigs, but production didn't decline there (meaningfully) because the average well age was much much older.  

I find several aspects of the Haynesville situation important:

  • The production declines must be nearly over.  Not only are wells aging, and production much lower, but the current rig count and average drill time suggests that it will only take about 25 rigs to maintain production around 4 BCFD from the shale.  
  • The steep production decline has masked growth in other shales thusfar, but will not going forward.
  • This is the cleanest example of drilling behavior in dry gas plays, with so many conflicting opinions about what 'break even' gas prices are in various basins.  I think it shows that there are some economic opportunities in the 'core of the core' even below $4 gas.  Now that gas seems to be hitting the skids again, rigs are unlikely to be added.
  • The HVS is probably best positioned for the coming LNG boom, and we might expect to see more JV deals and production lockups for the gulf coast LNG terminals very soon, with Cheniere projecting a late 2015 start date for exports.  We might see more drilling activity sooner, if the forward price curve begins to rise in '15 and beyond.  

Monday, August 26, 2013

Drilling Rig Efficiency

Further to a previous post, here is another look at some rig efficiency stats, calculated from the Baker Hughes rig and spud data, recently released.  When compared to the reported drill times of the most prominent operators in each basin, and the known EUR data points, it is reasonable to suppose that basin-wide productivity and yields will continue to improve significantly, with the possible exception of the older shale basins like the Barnett and Fayetteville.

Friday, August 23, 2013

Natural Gas Rig Counts

Natural gas rig counts released by Baker Hughes today showed US Gas Rigs down just 1 to 387, and Canadian Gas Rigs up 10 to 137.

Canadian rig counts are rising quickly and well above last year.  That should begin to show up in Western Canadian production soon.  We are not seeing that at this time.  Field receipts on the TC Nova system in Alberta, which makes up over 80% of production, show this:

Gas prices are very low in WC now, just above $2, due to the transport tariff issue, so maybe we are seeing some curtailments in the production data.  Inventories are swelling as well, so a day of reckoning may lie ahead this season.

Natural Gas Storage Projection

Yesterday's 57 BCF injection reported by the EIA was well below consensus.  Upcoming weather is also warmer than normal.  This lowers the season end storage range, and the midpoint is now 3.904 TCF on Nov 15th:

Reasonable weather variations, price-driven changes, and other changes in supply and demand result in a range of 3.73 to 4.07 as a confidence interval.  That's about a 350 BCF range.

Injections for the next three weeks will remain low due to warm weather.  September should bring weekly injections up into the 80 - 100 BCF range.

As the Atlantic remains quiet nearly 50% into hurricane season, the significant fundamental influences on gas  are going to be any heat waves in the distant forecast, and the critical influence of price on natural gas use in the electric power sector during the shoulder months of Sep-Oct-Nov.

Thursday, August 22, 2013

Renewable Energy Growth

Today the EIA released Electric Power Monthly for June 2013.  As expected, total generation was down from June 2012 and 2011, primarily on weather.  

June wind power was up about 15% Year on Year, and solar continued its predictable growth, up 56% over June 2012.  It still contributes only a quarter of 1% of total generation, so its easy to ignore.  But quite a bit of additional solar will be coming online in the next two years, along with the small but growing 'problem' of off-grid capacity.  Installed PV (photo-voltaic)  capacity has been doubling annually for a while.  That can't continue, but significant growth will.  

Solar and wind make up all the growth in US renewables, and small as they may seem, they are taking important market share away from fossil fuels.  Here is the 12 month trailing average of renewables (ex-hydro) market share.  It is a threat to traditional utilities and pricing models in the industry, as substantial capacity is scheduled to be added in the next two years.

Wednesday, August 21, 2013

Natural Gas Production Estimates from the EIA

Each month the U.S. Energy Information Administration reports historical monthly natural gas production data as part of the Natural Gas Monthly Report.  They also issue forecasts in the monthly Short Term Energy Outlook.Like most private analysts, they underestimated the growth in shale gas production. 

Because so much capital is being committed to natural gas drilling in north america and prices are so depressed, accurate forecasting of future production is critical.  Several pieces of the old formula appear to be broken, as rig counts have fallen, prices have fallen, and still production grows month after month.  Many one-time factors have been blamed for the disconnect:
  • Materials were in short supply so completions were delayed.  From pipe and casing to frac sand, the industry was often short.
  • Completion crews and pressure pumping capacity.  This definitely delayed completions and created a large inventory of non-producing wells.  This varied by basin.
  • Gas treating capacity.  The construction of cryogenic processors to strip liquids is lagging production in new basins.
  • Pipeline takeaway capacity.  Well known, self explanatory.
As gas storage surpluses were worked off in the spring of 2013, many were eager to bet the turn, and producers were certainly ready to receive their beautiful reward after suffering the pain for so long.  Analysts and CEOs were comfortable declaring that the fix is in, and low prices cured low prices etc...  Consensus forecasts for Henry Hub were into the $4's for the latter half of 2013 and beyond.

At $3.45 today and facing the possibility of record storage breaching 4 TCF in Nov, consensus is falling and the 2014 Strip is at $3.89.  More forecasts are acknowledging further production growth in 2014.

What is the EIA saying?  Two graphics.  The first is historical marketed production, and EIA's most current projection through Dec 2014.  It calls for further increases, with some monthly noise for hurricanes and maintenance.  The second graph shows the evolution of the EIA forecast for Dec '13 production.  They first issued estimates in January 2012, so we have the original estimate and 19 monthly revisions.  The trend will be evident.

What is the point?  The point is that acceptance of the shale gas revolution has been so reluctant, and underestimated time and again.  And the point of that point is to ask, "Are we still doing so?".  Drillers continue to find more gas, faster, cheaper, and more quickly to market.  The slope of the learning curve doesn't seem to be flattening out quite yet.  And something smells fishy about the broad consensus that production has plateaued.  That is a distinct possibility, but perhaps not so certain.

Be careful on this next graph.  Every data point is the same month, December 2013.  Only the production forecast has changed, and by a lot, over 20 months:

Tuesday, August 20, 2013

2013 Natural Gas Storage Peak Projection

With about 90 days remaining in injection season, the peak natural gas inventory level is still uncertain.  Weather and prices both will play an important part.  Price, because that determines whether utilities will burn coal or gas at a given power load, and weather because both heating and cooling demand will soon come into play, along with the season risks in the tropics.  In the current era, so much of the offshore infrastructure is storm-resistant that shut-ins are usually brief, and cyclones can suppress air conditioning demand after landfall.

The most recent storage report shows US inventories at 3.006 TCF on August 9th.  Peak storage is usually reached between October 25th and November 22nd.  With increasing uncertainty in distant weeks, here is the current range forecast storage range based on a variety of factors.  Clearly still a wide window.

Sunday, August 18, 2013

Natural Gas: Rig Counts and Production

Rig counts were once closely watched for their strong correlation to natural gas and oil production levels.  That disconnected with the shale and hydrofracturing revolutions in the past five years, and more recently the advent of liquids-rich natural gas.  Nevertheless, drilling remains the only known method of producing oil and gas, so working rigs are still the key.  Helpfully, Baker Hughes has recently added well count data to their weekly rig count data release.  That enables rig productivity calculations to be attempted.

Most basically, we have two of the three variables needed to estimate developed reserves, and from that, production:

  • Working Rig Count
  • Rig Productivity (wells per rig)
  • Reserves per Well
What did the Baker Hughes data show, and not show?  
It shed no light on one large problem, that of well classification.  The data is provided by basin, but without and oil/gas breakout.  We are a few years into a collapsing gas rig count and a spiking oil rig count, so basins that have an oil, dry gas, and wet gas window cannot be easily analyzed from this data.
It does show average rig counts, and total well spuds, by basin.  The data is by calendar quarter, and begins in Q1 2012, so there are 6 quarters of complete data.

Even where the rig productivity data isn't clean or doesn't pass the common sense test, the well spud count itself is useful.  Here are the well spuds for the most recent quarter, Q2 2013:

Rig productivity is clearly improving, but the data show rates of efficiency and implied drill times that are much lower than the best-in-class operators are reporting publicly.  Here are the well spuds per rig in a few well known basins:

The lesson from history is that the best operators continue to improve, and the worst operators improve more (if they live), so it appears there is substantial room for improvement in these plays.  The younger plays also improve more, due to the steeper slope of the learning curve, and the decreasing need to devote rigs to R&D science projects.

In particular consider the Haynesville.  The most recent quarter indicates 3 well spuds per rig, thus an implied 30 day spud-to-spud time.  Q2 2012 spuds were 1.8, and implied 50 days per well.  The rigs left this play quickly in 2011, and then stabilized, currently at 42.  Per Baker Hughes Rig Count......

If we guess that the wells being drilled have a 6 BCF ultimate recovery potential, we could say that:

42 rigs  x  30 days per well  x 6BCF per well  = 250 BCF per month of developed resource.  Over time, the developed resource would approximate the production level, and 250 BCF per month is 8.3 BCF per day.  Of course there are nuances and important other factors that make this a theoretical maximum only, but it demonstrates that a few dozen rigs can accomplish a lot in an environment where all 3 variables are constantly improving.  

Sunday, August 11, 2013

US Electricity Generation: A Zero Sum Game

The electricity industry has an uncertain future that belies the historically stable, low beta performance of utilities.  The past was stable in part because growth was strong and reliable....from 334 BKwH  in 1950 to 4,157 BKwH in 2007, with only 2 years of negative growth.

Beginning with the financial crisis in '08, things changed.  As in many sectors, growth turned negative.  Unlike many sectors, it has not recovered.

And through July of 2013 there doesn't appear to be any change in the red slope for this year.  This has important implications for the industry, and for the feed stock components of power generation where smaller revolutions are taking place already.

So five significant variables for the industry:

  • Uncertain growth in aggregate power demand
  • Regulatory opposition to fossil fuels (chiefly coal)
  • Subsidy and protection for renewables
  • Shale gas revolution 
  • Everything else (technology risks, ageing grid, conservation, macro, nuclear, ham sandwiches)
Virtually all of these issues are urgent, and generating immediate impacts on the energy industry.  And when the pie is not growing, as is the case for now, any growth in market share for one component is theft from another.  Up to now, the threat posed by renewables to fossil fuels was negligible because power demand was growing and renewable market share was meaninglessly small.  And the regulatory/environmental opposition to coal fired power was strong but primarily preventing expansions.  

Now, coal is being crushed by the combination of cheap shale gas, relentless regulation, a decline in exports, and the presence, at last, of renewables on the radar.  Coal's worst nightmare is wind.  And wind is where the growth is in renewables.  

Fossil fuels are losing market share:

And coal is quickly ceding its share of the fossil fuel component:

In 2014, a significant number of coal fired generating stations will be retired, and the price of natural gas is low enough to encourage coal to gas switching in some areas of the country.  This phenomenon is very price sensitive, and very self-correcting, so it cannot be reliably predicted very far into the future when prices are near the boundary.