A local conservation law is usually expressed mathematically as a continuity equation, a partial differential equation
which gives a relation between the amount of the quantity and the
"transport" of that quantity. It states that the amount of the conserved
quantity at a point or within a volume can only change by the amount of
the quantity which flows in or out of the volume.
From Noether's theorem, each conservation law is associated with a symmetry in the underlying physics.
Conservation laws as fundamental laws of nature
Conservation laws are fundamental to our understanding of the
physical world, in that they describe which processes can or cannot
occur in nature. For example, the conservation law of energy states
that the total quantity of energy in an isolated system does not change,
though it may change form. In general, the total quantity of the
property governed by that law remains unchanged during physical
processes. With respect to classical physics, conservation laws include
conservation of energy, mass (or matter), linear momentum, angular
momentum, and electric charge. With respect to particle physics,
particles cannot be created or destroyed except in pairs, where one is
ordinary and the other is an antiparticle. With respect to symmetries
and invariance principles, three special conservation laws have been
described, associated with inversion or reversal of space, time, and
charge.
Conservation laws are considered to be fundamental laws of
nature, with broad application in physics, as well as in other fields
such as chemistry, biology, geology, and engineering.
Most conservation laws are exact, or absolute, in the sense that
they apply to all possible processes. Some conservation laws are
partial, in that they hold for some processes but not for others.
One particularly important result concerning conservation laws is Noether theorem, which states that there is a one-to-one correspondence between each one of them and a differentiable symmetry
of nature. For example, the conservation of energy follows from the
time-invariance of physical systems, and the conservation of angular
momentum arises from the fact that physical systems behave the same
regardless of how they are oriented in space.
Exact laws
A partial listing of physical conservation equations due to symmetry that are said to be exact laws, or more precisely have never been proven to be violated:
translation of weak potential field along axis in phase space
Conservation of CPT parity
CPT invariance
1
simultaneous inversion of space, time, charge coordinates
Conservation of the difference between baryon and lepton numbers B - L
U(1)B-L Gauge invariance
1
Approximate laws
There are also approximate conservation laws. These are
approximately true in particular situations, such as low speeds, short
time scales, or certain interactions.
The
total amount of some conserved quantity in the universe could remain
unchanged if an equal amount were to appear at one point A and simultaneously disappear from another separate point B.
For example, an amount of energy could appear on Earth without
changing the total amount in the Universe if the same amount of energy
were to disappear from some other region of the Universe. This weak
form of "global" conservation is really not a conservation law because
it is not Lorentz invariant, so phenomena like the above do not occur in nature. Due to special relativity, if the appearance of the energy at A and disappearance of the energy at B are simultaneous in one inertial reference frame, they will not be simultaneous
in other inertial reference frames moving with respect to the first.
In a moving frame one will occur before the other; either the energy at A will appear before or after the energy at B disappears. In both cases, during the interval energy will not be conserved.
A stronger form of conservation law requires that, for the amount
of a conserved quantity at a point to change, there must be a flow, or flux of the quantity into or out of the point. For example, the amount of electric charge at a point is never found to change without an electric current into or out of the point that carries the difference in charge. Since it only involves continuous local changes, this stronger type of conservation law is Lorentz invariant; a quantity conserved in one reference frame is conserved in all moving reference frames. This is called a local conservation law.
Local conservation also implies global conservation; that the total
amount of the conserved quantity in the Universe remains constant. All
of the conservation laws listed above are local conservation laws. A
local conservation law is expressed mathematically by a continuity equation,
which states that the change in the quantity in a volume is equal to
the total net "flux" of the quantity through the surface of the volume.
The following sections discuss continuity equations in general.
In continuum mechanics, the most general form of an exact conservation law is given by a continuity equation. For example, conservation of electric charge q is
where ∇⋅ is the divergence operator, ρ is the density of q (amount per unit volume), j is the flux of q (amount crossing a unit area in unit time), and t is time.
If we assume that the motion u of the charge is a continuous function of position and time, then
In the one-dimensional space a conservation equation is a first-order quasilinearhyperbolic equation that can be put into the advection form:
where the dependent variable y(x,t) is called the density of the conserved (scalar) quantity, and a(y) is called the current coefficient, usually corresponding to the partial derivative in the conserved quantity of a current density of the conserved quantity j(y):
the conservation equation can be put into the current density form:
In a space with more than one dimension the former definition can be extended to an equation that can be put into the form:
where the conserved quantity is y(r,t), ⋅ denotes the scalar product, ∇ is the nabla operator, here indicating a gradient, and a(y) is a vector of current coefficients, analogously corresponding to the divergence of a vector current density associated to the conserved quantity j(y):
In the general case a conservation equation can be also a system of this kind of equations (a vector equation) in the form:
where y is called the conserved (vector) quantity, ∇y is its gradient, 0 is the zero vector, and A(y) is called the Jacobian of the current density. In fact as in the former scalar case, also in the vector case A(y) usually corresponding to the Jacobian of a current density matrixJ(y):
and the conservation equation can be put into the form:
For example, this the case for Euler equations (fluid dynamics). In the simple incompressible case they are:
where:
u is the flow velocityvector, with components in a N-dimensional space u1, u2, ..., uN,
Conservation
equations can be also expressed in integral form: the advantage of the
latter is substantially that it requires less smoothness of the
solution, which paves the way to weak form, extending the class of admissible solutions to include discontinuous solutions. By integrating in any space-time domain the current density form in 1-D space:
In a similar fashion, for the scalar multidimensional space, the integral form is:
where the line integration is performed along the boundary of the domain, in an anticlockwise manner.
Moreover, by defining a test functionφ(r,t) continuously differentiable both in time and space with compact support, the weak form can be obtained pivoting on the initial condition. In 1-D space it is:
In the weak form all the partial derivatives of the density
and current density have been passed on to the test function, which with
the former hypothesis is sufficiently smooth to admit these
derivatives.
A blowout is the uncontrolled release of crude oil and/or natural gas from an oil well or gas well after pressure control systems have failed. Modern wells have blowout preventers intended to prevent such an occurrence. An accidental spark during a blowout can lead to a catastrophic oil or gas fire.
Prior to the advent of pressure control equipment in the 1920s,
the uncontrolled release of oil and gas from a well while drilling was
common and was known as an oil gusher, gusher or wild well.
History
Gushers were an icon of oil exploration during the late 19th and early 20th centuries. During that era, the simple drilling techniques, such as cable-tool drilling, and the lack of blowout preventers
meant that drillers could not control high-pressure reservoirs. When
these high-pressure zones were breached, the oil or natural gas would
travel up the well at a high rate, forcing out the drill string and
creating a gusher. A well which began as a gusher was said to have
"blown in": for instance, the Lakeview Gusherblew in in 1910. These uncapped wells could produce large amounts of oil, often shooting 200 feet (61 m) or higher into the air. A blowout primarily composed of natural gas was known as a gas gusher.
Despite being symbols of new-found wealth, gushers were dangerous
and wasteful. They killed workmen involved in drilling, destroyed
equipment, and coated the landscape with thousands of barrels
of oil; additionally, the explosive concussion released by the well
when it pierces an oil/gas reservoir has been responsible for a number
of oilmen losing their hearing entirely; standing too near to the
drilling rig at the moment it drills into the oil reservoir is extremely
hazardous. The impact on wildlife is very hard to quantify, but can
only be estimated to be mild in the most optimistic
models—realistically, the ecological impact is estimated by scientists
across the ideological spectrum to be severe, profound, and lasting.
To complicate matters further, the free flowing oil was—and is—in danger of igniting. One dramatic account of a blowout and fire reads,
With a roar like a hundred express trains racing across
the countryside, the well blew out, spewing oil in all directions. The
derrick simply evaporated. Casings wilted like lettuce out of water, as
heavy machinery writhed and twisted into grotesque shapes in the blazing
inferno.
The development of rotary drilling techniques where the density of the drilling fluid
is sufficient to overcome the downhole pressure of a newly penetrated
zone meant that gushers became avoidable. If however the fluid density
was not adequate or fluids were lost to the formation, then there was
still a significant risk of a well blowout.
In 1924 the first successful blowout preventer was brought to market. The BOP valve affixed to the wellhead could be closed in the event of drilling into a high pressure zone, and the well fluids contained. Well control
techniques could be used to regain control of the well. As the
technology developed, blowout preventers became standard equipment, and
gushers became a thing of the past.
In the modern petroleum industry, uncontrollable wells became
known as blowouts and are comparatively rare. There has been significant
improvement in technology, well control techniques, and personnel
training which has helped to prevent their occurring. From 1976 to 1981, 21 blowout reports are available.
Notable gushers
A
blowout in 1815 resulted from an attempt to drill for salt rather than
for oil. Joseph Eichar and his team were digging west of the town of Wooster, Ohio,
US along Killbuck Creek, when they struck oil. In a written retelling
by Eichar's daughter, Eleanor, the strike produced "a spontaneous
outburst, which shot up high as the tops of the highest trees!"
Oil drillers struck a number of gushers near Oil City, Pennsylvania, US in 1861. The most famous was the Little & Merrick well, which began gushing oil on 17 April 1861. The spectacle of the fountain of oil flowing out at about 3,000 barrels (480 m3)
per day had drawn about 150 spectators by the time an hour later when
the oil gusher burst into flames, raining fire down on the oil-soaked
onlookers. Thirty people died. Other early gushers in northwest
Pennsylvania were the Phillips #2 (4,000 barrels (640 m3) per day) in September 1861, and the Woodford well (3,000 barrels (480 m3) per day) in December 1861.
The Shaw Gusher in Oil Springs, Ontario,
was Canada's first oil gusher. On January 16, 1862, it shot oil from
over 60 metres (200 ft) below ground to above the treetops at a rate of
3,000 barrels (480 m3) per day, triggering the oil boom in Lambton County.
Lucas Gusher at Spindletop in Beaumont, Texas, US in 1901 flowed at 100,000 barrels (16,000 m3)
per day at its peak, but soon slowed and was capped within nine days.
The well tripled U.S. oil production overnight and marked the start of
the Texas oil industry.
Dos Bocas
in the State of Veracruz, Mexico, was a famous 1908 Mexican blowout
that formed a large crater. It leaked oil from the main reservoir for
many years, continuing even after 1938 (when Pemex nationalized the Mexican oil industry).
Lakeview Gusher on the Midway-Sunset Oil Field in Kern County, California, US of 1910 is believed to be the largest-ever U.S. gusher. At its peak, more than 100,000 barrels (16,000 m3)
of oil per day flowed out, reaching as high as 200 feet (61 m) in the
air. It remained uncapped for 18 months, spilling over 9 million barrels
(1,400,000 m3) of oil, less than half of which was recovered.
A short-lived gusher at Alamitos #1 in Signal Hill, California, US in 1921 marked the discovery of the Long Beach Oil Field, one of the most productive oil fields in the world.
The Barroso 2 well in Cabimas, Venezuela, in December 1922 flowed at around 100,000 barrels (16,000 m3) per day for nine days, plus a large amount of natural gas.
Baba Gurgur near Kirkuk, Iraq, an oilfield known since antiquity, erupted at a rate of 95,000 barrels (15,100 m3) a day in 1927.
The Yates #30-A in Pecos County, Texas, US gushing 80 feet through
the fifteen-inch casing, produced a world record 204,682 barrels of oil a
day from a depth of 1,070 feet on 23 September 1929.
The largest known 'wildcat' oil gusher blew near Qom, Iran, on 26 August 1956. The uncontrolled oil gushed to a height of 52 m (171 ft), at a rate of 120,000 barrels (19,000 m3) per day. The gusher was closed after 90 days' work by Bagher Mostofi and Myron Kinley (USA).
On October 17, 1982, a sour gas well AMOCO DOME BRAZEAU RIVER
13-12-48-12, being drilled 20 km west of Lodgepole, Alberta blew out.
The burning well was finally capped 67 days later by the Texas
well-control company, Boots & Coots.
One of the most troublesome gushers happened on 23 June 1985, at well #37 at the Tengiz field in Atyrau, Kazakh SSR, Soviet Union, where the 4,209-metre deep well blew out and the 200-metre high gusher self-ignited two days later. Oil pressure up to 800 atm and high hydrogen sulfide
content had led to the gusher being capped only on 27 July 1986. The
total volume of erupted material measured at 4.3 million metric tons of
oil and 1.7 billion m³ of natural gas, and the burning gusher resulted in 890 tons of various mercaptans and more than 900,000 tons of soot released into the atmosphere.
Petroleum or crude oil is a naturally occurring, flammable liquid consisting of a complex mixture of hydrocarbons of various molecular weights, and other organic compounds, found in geologic formations
beneath the Earth's surface. Because most hydrocarbons are lighter than
rock or water, they often migrate upward and occasionally laterally
through adjacent rock layers until either reaching the surface or
becoming trapped within porous rocks (known as reservoirs) by
impermeable rocks above. When hydrocarbons are concentrated in a trap,
an oil field forms, from which the liquid can be extracted by drilling
and pumping. The downhole pressure in the rock structures changes
depending upon the depth and the characteristics of the source rock. Natural gas (mostly methane)
may be present also, usually above the oil within the reservoir, but
sometimes dissolved in the oil at reservoir pressure and temperature.
Dissolved gas typically comes out of solution as free gas as the
pressure is reduced either under controlled production operations or in a
kick, or in an uncontrolled blowout. The hydrocarbon in some reservoirs
may be essentially all natural gas.
Formation kick
The downhole fluid pressures are controlled in modern wells through the balancing of the hydrostatic pressure provided by the mud
column. Should the balance of the drilling mud pressure be incorrect
(i.e., the mud pressure gradient is less than the formation pore
pressure gradient), then formation fluids (oil, natural gas, and/or
water) can begin to flow into the wellbore and up the annulus (the space
between the outside of the drill string and the wall of the open hole or the inside of the casing), and/or inside the drill pipe. This is commonly called a kick. Ideally, mechanical barriers such as blowout preventers
(BOPs) can be closed to isolate the well while the hydrostatic balance
is regained through circulation of fluids in the well. But if the well
is not shut in (common term for the closing of the blow-out preventer), a
kick can quickly escalate into a blowout when the formation fluids
reach the surface, especially when the influx contains gas that expands
rapidly with the reduced pressure as it flows up the wellbore, further
decreasing the effective weight of the fluid.
Early warning signs of an impending well kick while drilling are:
Sudden change in drilling rate;
Reduction in drillpipe weight;
Change in pump pressure;
Change in drilling fluid return rate.
Other warning signs during the drilling operation are:
Returning mud "cut" by (i.e., contaminated by) gas, oil or water;
Connection gases, high background gas units, and high bottoms-up gas units detected in the mudlogging unit.
The primary means of detecting a kick while drilling is a relative
change in the circulation rate back up to the surface into the mud pits.
The drilling crew or mud engineer keeps track of the level in the mud
pits and closely monitors the rate of mud returns versus the rate that
is being pumped down the drill pipe. Upon encountering a zone of higher
pressure than is being exerted by the hydrostatic head of the drilling
mud (including the small additional frictional head while circulating)
at the bit, an increase in mud return rate would be noticed as the
formation fluid influx blends in with the circulating drilling mud.
Conversely, if the rate of returns is slower than expected, it means
that a certain amount of the mud is being lost to a thief zone somewhere
below the last casing shoe.
This does not necessarily result in a kick (and may never become one);
however, a drop in the mud level might allow influx of formation fluids
from other zones if the hydrostatic head is reduced to less than that of
a full column of mud.
Well control
The
first response to detecting a kick would be to isolate the wellbore
from the surface by activating the blow-out preventers and closing in
the well. Then the drilling crew would attempt to circulate in a heavier
kill fluid to increase the hydrostatic pressure (sometimes with the assistance of a well control
company). In the process, the influx fluids will be slowly circulated
out in a controlled manner, taking care not to allow any gas to
accelerate up the wellbore too quickly by controlling casing pressure
with chokes on a predetermined schedule.
This effect will be minor if the influx fluid is mainly salt
water. And with an oil-based drilling fluid it can be masked in the
early stages of controlling a kick because gas influx may dissolve into
the oil under pressure at depth, only to come out of solution and expand
rather rapidly as the influx nears the surface. Once all the
contaminant has been circulated out, the shut-in casing pressure should
have reached zero.
Capping stacks are used for controlling blowouts. The cap is an open valve that is closed after bolted on.
Types of blowouts
Well blowouts can occur during the drilling phase, during well testing, during well completion, during production, or during workover activities.
Surface blowouts
Blowouts can eject the drill string out of the well, and the force of the escaping fluid can be strong enough to damage the drilling rig.
In addition to oil, the output of a well blowout might include natural
gas, water, drilling fluid, mud, sand, rocks, and other substances.
Blowouts will often be ignited from sparks from rocks being
ejected, or simply from heat generated by friction. A well control
company then will need to extinguish the well fire or cap the well, and
replace the casing head and other surface equipment. If the flowing gas
contains poisonous hydrogen sulfide, the oil operator might decide to ignite the stream to convert this to less hazardous substances.
Sometimes blowouts can be so forceful that they cannot be
directly brought under control from the surface, particularly if there
is so much energy in the flowing zone that it does not deplete
significantly over time. In such cases, other wells (called relief wells)
may be drilled to intersect the well or pocket, in order to allow
kill-weight fluids to be introduced at depth. When first drilled in the
1930s relief wells were drilled to inject water into the main drill well
hole.
Contrary to what might be inferred from the term, such wells generally
are not used to help relieve pressure using multiple outlets from the
blowout zone.
Subsea blowouts
The two main causes of a subsea blowout are equipment failures and imbalances with encountered subsurface reservoir pressure. Subsea wells have pressure control equipment located on the seabed or between the riser pipe and drilling platform. Blowout preventers
(BOPs) are the primary safety devices designed to maintain control of
geologically driven well pressures. They contain hydraulic-powered
cut-off mechanisms to stop the flow of hydrocarbons in the event of a
loss of well control.
Even with blowout prevention equipment and processes in place,
operators must be prepared to respond to a blowout should one occur.
Before drilling a well, a detailed well construction design plan, an Oil
Spill Response Plan as well as a Well Containment Plan must be
submitted, reviewed and approved by BSEE and is contingent upon access
to adequate well containment resources in accordance to NTL 2010-N10.
The Deepwater Horizon well blowout in the Gulf of Mexico in April 2010 occurred at a 5,000 feet (1,500 m) water depth.
Current blowout response capabilities in the U.S. Gulf of Mexico meet
capture and process rates of 130,000 barrels of fluid per day and a gas
handling capacity of 220 million cubic feet per day at depths through
10,000 feet.
Underground blowouts
An
underground blowout is a special situation where fluids from high
pressure zones flow uncontrolled to lower pressure zones within the
wellbore. Usually this is from deeper higher pressure zones to shallower
lower pressure formations. There may be no escaping fluid flow at the
wellhead. However, the formation(s) receiving the influx can become
overpressured, a possibility that future drilling plans in the vicinity
must consider.
Blowout control companies
Myron M. Kinley
was a pioneer in fighting oil well fires and blowouts. He developed
many patents and designs for the tools and techniques of oil
firefighting. His father, Karl T. Kinley, attempted to extinguish an oil
well fire with the help of a massive explosion—a method still in common
use for fighting oil fires. Myron and Karl Kinley first successfully
used explosives to extinguish an oil well fire in 1913. Kinley would later form the M. M. Kinley Company in 1923. Asger "Boots" Hansen and Edward Owen "Coots" Matthews also begin their careers under Kinley.
Paul N. "Red" Adair
joined the M. M. Kinley Company in 1946, and worked 14 years with Myron
Kinley before starting his own company, Red Adair Co., Inc., in 1959.
Red Adair Co. has helped in controlling offshore blowouts, including:
The 1968 American film, Hellfighters,
which starred John Wayne, is about a group of oil well firefighters,
based loosely on Adair's life; Adair, Hansen, and Matthews served as
technical advisors on the film.
After the Macondo-1 blowout on the Deepwater Horizon,
the offshore industry collaborated with government regulators to
develop a framework to respond to future subsea incidents. As a result,
all energy companies operating in the deep-water U.S. Gulf of Mexico
must submit an OPA 90 required Oil Spill Response Plan with the addition
of a Regional Containment Demonstration Plan prior to any drilling
activity.
In the event of a subsea blowout, these plans are immediately
activated, drawing on some of the equipment and processes effectively
used to contain the Deepwater Horizon well as others that have been
developed in its aftermath.
In order to regain control of a subsea well, the Responsible
Party would first secure the safety of all personnel on board the rig
and then begin a detailed evaluation of the incident site. Remotely operated underwater vehicles (ROVs) would be dispatched to inspect the condition of the wellhead, Blowout Preventer
(BOP) and other subsea well equipment. The debris removal process would
begin immediately to provide clear access for a capping stack.
Once lowered and latched on the wellhead, a capping stack uses
stored hydraulic pressure to close a hydraulic ram and stop the flow of
hydrocarbons.
If shutting in the well could introduce unstable geological conditions
in the wellbore, a cap and flow procedure would be used to contain
hydrocarbons and safely transport them to a surface vessel.
The Responsible Party works in collaboration with BSEE and the United States Coast Guard to oversee response efforts, including source control, recovering discharged oil and mitigating environmental impact.
Several not-for-profit organizations provide a solution to effectively contain a subsea blowout. HWCG LLC and Marine Well Containment Company operate within the U.S. Gulf of Mexico waters, while cooperatives like Oil Spill Response Limited offer support for international operations.
Use of nuclear explosions
On Sep. 30, 1966, the Soviet Union experienced blowouts on five natural gas wells in Urta-Bulak, an area about 80 kilometers from Bukhara, Uzbekistan. It was claimed in Komsomoloskaya Pravda that after years of burning uncontrollably they were able to stop them entirely. The Soviets lowered a specially made 30 kiloton nuclear physics package
into a 6-kilometre (20,000 ft) borehole drilled 25 to 50 metres (82 to
164 ft) away from the original (rapidly leaking) well. A nuclear
explosive was deemed necessary because conventional explosives both
lacked the necessary power and would also require a great deal more
space underground. When the device was detonated, it crushed the
original pipe that was carrying the gas from the deep reservoir to the
surface and vitrified
the surrounding rock. This caused the leak and fire at the surface to
cease within approximately one minute of the explosion, and proved to be
a permanent solution. An attempt on a similar well was not as
successful. Other tests were for such experiments as oil extraction
enhancement (Stavropol, 1969) and the creation of gas storage reservoirs
(Orenburg, 1970).