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Hydraulic fracturing , also called fracking , hydrofracking , and hydrofracturing , is a well stimulation technique involving the fracturing of bedrock formations by a pressurized liquid. The process involves the high-pressure injection of "fracking fluid" primarily water, containing sand or other proppants suspended with the aid of thickening agents into a wellbore to create cracks in the deep-rock formations through which natural gas , petroleum , and brine will flow more freely.
When the hydraulic pressure is removed from the well, small grains of hydraulic fracturing proppants either sand or aluminium oxide hold the fractures open. Hydraulic fracturing began as an experiment in , [2] and the first commercially successful application followed in As of , 2. However, hydraulic fracturing is highly controversial.
There is considerable uncertainty about the scale of methane leakage associated with hydraulic fracturing, and even some evidence that leakage may cancel out the greenhouse gas emissions benefits of natural gas relative to other fossil fuels. EDF has recently announced a satellite mission to further locate and measure methane emissions. Increases in seismic activity following hydraulic fracturing along dormant or previously unknown faults are sometimes caused by the deep-injection disposal of hydraulic fracturing flowback a byproduct of hydraulically fractured wells , [25] and produced formation brine a byproduct of both fractured and nonfractured oil and gas wells.
Fracturing rocks at great depth frequently becomes suppressed by pressure due to the weight of the overlying rock strata and the cementation of the formation. This suppression process is particularly significant in "tensile" Mode 1 fractures which require the walls of the fracture to move against this pressure.
Fracturing occurs when effective stress is overcome by the pressure of fluids within the rock. The minimum principal stress becomes tensile and exceeds the tensile strength of the material.
Most mineral vein systems are a result of repeated natural fracturing during periods of relatively high pore fluid pressure. The impact of high pore fluid pressure on the formation process of mineral vein systems is particularly evident in "crack-seal" veins, where the vein material is part of a series of discrete fracturing events, and extra vein material is deposited on each occasion.
Stress levels rise and fall episodically, and earthquakes can cause large volumes of connate water to be expelled from fluid-filled fractures. This process is referred to as "seismic pumping". Minor intrusions in the upper part of the crust , such as dikes, propagate in the form of fluid-filled cracks.
In such cases, the fluid is magma. In sedimentary rocks with a significant water content, fluid at fracture tip will be steam. Fracturing as a method to stimulate shallow, hard rock oil wells dates back to the s. Dynamite or nitroglycerin detonations were used to increase oil and natural gas production from petroleum bearing formations.
Edward A. Roberts received a patent for an " exploding torpedo ". Later still the same method was applied to water and gas wells. Stimulation of wells with acid, instead of explosive fluids, was introduced in the s. Due to acid etching , fractures would not close completely resulting in further productivity increase. Mitchell are each considered to have pioneered hydraulic fracturing innovations toward practical applications.
The relationship between well performance and treatment pressures was studied by Floyd Farris of Stanolind Oil and Gas Corporation. This study was the basis of the first hydraulic fracturing experiment, conducted in at the Hugoton gas field in Grant County of southwestern Kansas by Stanolind.
The experiment was not very successful as deliverability of the well did not change appreciably. The process was further described by J. Clark of Stanolind in his paper published in A patent on this process was issued in and exclusive license was granted to the Halliburton Oil Well Cementing Company. On 17 March , Halliburton performed the first two commercial hydraulic fracturing treatments in Stephens County, Oklahoma , and Archer County, Texas. In contrast with large-scale hydraulic fracturing used in low-permeability formations, small hydraulic fracturing treatments are commonly used in high-permeability formations to remedy "skin damage", a low-permeability zone that sometimes forms at the rock-borehole interface.
In such cases the fracturing may extend only a few feet from the borehole. In the Soviet Union , the first hydraulic proppant fracturing was carried out in Massive hydraulic fracturing also known as high-volume hydraulic fracturing is a technique first applied by Pan American Petroleum in Stephens County, Oklahoma , US in The definition of massive hydraulic fracturing varies, but generally refers to treatments injecting over short tons, or approximately , pounds metric tonnes , of proppant.
American geologists gradually became aware that there were huge volumes of gas-saturated sandstones with permeability too low generally less than 0. Massive hydraulic fracturing quickly spread in the late s to western Canada, Rotliegend and Carboniferous gas-bearing sandstones in Germany, Netherlands onshore and offshore gas fields , and the United Kingdom in the North Sea. Horizontal oil or gas wells were unusual until the late s. Then, operators in Texas began completing thousands of oil wells by drilling horizontally in the Austin Chalk , and giving massive slickwater hydraulic fracturing treatments to the wellbores.
Horizontal wells proved much more effective than vertical wells in producing oil from tight chalk; [49] sedimentary beds are usually nearly horizontal, so horizontal wells have much larger contact areas with the target formation.
Hydraulic fracturing operations have grown exponentially since the mids, when technologic advances and increases in the price of natural gas made this technique economically viable.
Hydraulic fracturing of shales goes back at least to , when some operators in the Big Sandy gas field of eastern Kentucky and southern West Virginia started hydraulically fracturing the Ohio Shale and Cleveland Shale , using relatively small fracs.
The frac jobs generally increased production, especially from lower-yielding wells. In , the United States government started the Eastern Gas Shales Project , which included numerous public-private hydraulic fracturing demonstration projects. In , Nick Steinsberger, an engineer of Mitchell Energy now part of Devon Energy , applied the slickwater fracturing technique, using more water and higher pump pressure than previous fracturing techniques, which was used in East Texas in the Barnett Shale of north Texas.
Griffin No. Mitchell has been called the "father of fracking" because of his role in applying it in shales. As of , massive hydraulic fracturing is being applied on a commercial scale to shales in the United States, Canada, and China. Several additional countries are planning to use hydraulic fracturing.
According to the United States Environmental Protection Agency EPA , hydraulic fracturing is a process to stimulate a natural gas, oil, or geothermal well to maximize extraction. The EPA defines the broader process to include acquisition of source water, well construction, well stimulation, and waste disposal. A hydraulic fracture is formed by pumping fracturing fluid into a wellbore at a rate sufficient to increase pressure at the target depth determined by the location of the well casing perforations , to exceed that of the fracture gradient pressure gradient of the rock.
The rock cracks, and the fracture fluid permeates the rock extending the crack further, and further, and so on. Fractures are localized as pressure drops off with the rate of frictional loss, which is relative to the distance from the well. Operators typically try to maintain "fracture width", or slow its decline following treatment, by introducing a proppant into the injected fluid — a material such as grains of sand, ceramic, or other particulate, thus preventing the fractures from closing when injection is stopped and pressure removed.
Consideration of proppant strength and prevention of proppant failure becomes more important at greater depths where pressure and stresses on fractures are higher.
The propped fracture is permeable enough to allow the flow of gas, oil, salt water and hydraulic fracturing fluids to the well. During the process, fracturing fluid leakoff loss of fracturing fluid from the fracture channel into the surrounding permeable rock occurs. This may result in formation matrix damage, adverse formation fluid interaction, and altered fracture geometry, thereby decreasing efficiency.
The location of one or more fractures along the length of the borehole is strictly controlled by various methods that create or seal holes in the side of the wellbore. Hydraulic fracturing is performed in cased wellbores, and the zones to be fractured are accessed by perforating the casing at those locations. Hydraulic-fracturing equipment used in oil and natural gas fields usually consists of a slurry blender, one or more high-pressure, high-volume fracturing pumps typically powerful triplex or quintuplex pumps and a monitoring unit.
Associated equipment includes fracturing tanks, one or more units for storage and handling of proppant, high-pressure treating iron [ clarification needed ] , a chemical additive unit used to accurately monitor chemical addition , low-pressure flexible hoses, and many gauges and meters for flow rate, fluid density, and treating pressure.
Fracturing equipment operates over a range of pressures and injection rates, and can reach up to megapascals 15, psi and litres per second 9. A distinction can be made between conventional, low-volume hydraulic fracturing, used to stimulate high-permeability reservoirs for a single well, and unconventional, high-volume hydraulic fracturing, used in the completion of tight gas and shale gas wells.
High-volume hydraulic fracturing usually requires higher pressures than low-volume fracturing; the higher pressures are needed to push out larger volumes of fluid and proppant that extend farther from the borehole. Horizontal drilling involves wellbores with a terminal drillhole completed as a "lateral" that extends parallel with the rock layer containing the substance to be extracted.
For example, laterals extend 1, to 5, feet to 1, m in the Barnett Shale basin in Texas, and up to 10, feet 3, m in the Bakken formation in North Dakota. In contrast, a vertical well only accesses the thickness of the rock layer, typically 50— feet 15—91 m.
Horizontal drilling reduces surface disruptions as fewer wells are required to access the same volume of rock. Drilling often plugs up the pore spaces at the wellbore wall, reducing permeability at and near the wellbore.
This reduces flow into the borehole from the surrounding rock formation, and partially seals off the borehole from the surrounding rock. Low-volume hydraulic fracturing can be used to restore permeability. The main purposes of fracturing fluid are to extend fractures, add lubrication, change gel strength, and to carry proppant into the formation.
There are two methods of transporting proppant in the fluid — high-rate and high- viscosity. High-viscosity fracturing tends to cause large dominant fractures, while high-rate slickwater fracturing causes small spread-out micro-fractures.
Water-soluble gelling agents such as guar gum increase viscosity and efficiently deliver proppant into the formation. Fluid is typically a slurry of water, proppant, and chemical additives. This process is called waterless fracturing. When propane is used it is turned into vapor by the high pressure and high temperature.
None of the chemicals used will return to the surface. Only the propane used will return from what was used in the process. The proppant is a granular material that prevents the created fractures from closing after the fracturing treatment.
Types of proppant include silica sand , resin-coated sand, bauxite , and man-made ceramics. The choice of proppant depends on the type of permeability or grain strength needed. In some formations, where the pressure is great enough to crush grains of natural silica sand, higher-strength proppants such as bauxite or ceramics may be used. The most commonly used proppant is silica sand, though proppants of uniform size and shape, such as a ceramic proppant, are believed to be more effective.
The fracturing fluid varies depending on fracturing type desired, and the conditions of specific wells being fractured, and water characteristics. The fluid can be gel, foam, or slickwater-based. Fluid choices are tradeoffs: more viscous fluids, such as gels, are better at keeping proppant in suspension; while less-viscous and lower-friction fluids, such as slickwater, allow fluid to be pumped at higher rates, to create fractures farther out from the wellbore.
Important material properties of the fluid include viscosity , pH , various rheological factors , and others. Water is mixed with sand and chemicals to create hydraulic fracturing fluid. Approximately 40, gallons of chemicals are used per fracturing. The most common chemical used for hydraulic fracturing in the United States in — was methanol , while some other most widely used chemicals were isopropyl alcohol , 2-butoxyethanol , and ethylene glycol.
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In computer science , a k -d tree short for k-dimensional tree is a space-partitioning data structure for organizing points in a k -dimensional space. The k -d tree is a binary tree in which every node is a k -dimensional point.
Every non-leaf node can be thought of as implicitly generating a splitting hyperplane that divides the space into two parts, known as half-spaces. Points to the left of this hyperplane are represented by the left subtree of that node and points to the right of the hyperplane are represented by the right subtree.
The hyperplane direction is chosen in the following way: every node in the tree is associated with one of the k dimensions, with the hyperplane perpendicular to that dimension's axis.
So, for example, if for a particular split the "x" axis is chosen, all points in the subtree with a smaller "x" value than the node will appear in the left subtree and all points with a larger "x" value will be in the right subtree. In such a case, the hyperplane would be set by the x value of the point, and its normal would be the unit x-axis. Since there are many possible ways to choose axis-aligned splitting planes, there are many different ways to construct k -d trees.
The canonical method of k -d tree construction has the following constraints: [2]. This method leads to a balanced k -d tree, in which each leaf node is approximately the same distance from the root.
However, balanced trees are not necessarily optimal for all applications. Note that it is not required to select the median point. In the case where median points are not selected, there is no guarantee that the tree will be balanced. To avoid coding a complex O n median-finding algorithm [3] [4] or using an O n log n sort such as heapsort or mergesort to sort all n points, a popular practice is to sort a fixed number of randomly selected points, and use the median of those points to serve as the splitting plane.
In practice, this technique often results in nicely balanced trees. Given a list of n points, the following algorithm uses a median-finding sort to construct a balanced k -d tree containing those points. It is common that points "after" the median include only the ones that are strictly greater than the median in the current dimension.
For points that lie on the median in the current dimension, it is possible to define a function that compares them in all dimensions.
In some cases, it is acceptable to let points equal to the median lie on one side of the median, for example, by splitting the points into a "lesser than" subset and a "greater than or equal to" subset. This algorithm creates the invariant that for any node, all the nodes in the left subtree are on one side of a splitting plane , and all the nodes in the right subtree are on the other side.
Points that lie on the splitting plane may appear on either side. The splitting plane of a node goes through the point associated with that node referred to in the code as node. Alternative algorithms for building a balanced k -d tree presort the data prior to building the tree. Then, they maintain the order of the presort during tree construction and hence eliminate the costly step of finding the median at each level of subdivision. Two such algorithms build a balanced k -d tree to sort triangles in order to improve the execution time of ray tracing for three-dimensional computer graphics.
These algorithms presort n triangles prior to building the k -d tree , then build the tree in O n log n time in the best case. It then maintains the order of these k presorts during tree construction and thereby avoids finding the median at each level of subdivision. The above algorithm implemented in the Python programming language is as follows:. One adds a new point to a k -d tree in the same way as one adds an element to any other search tree.
First, traverse the tree, starting from the root and moving to either the left or the right child depending on whether the point to be inserted is on the "left" or "right" side of the splitting plane.
Once you get to the node under which the child should be located, add the new point as either the left or right child of the leaf node, again depending on which side of the node's splitting plane contains the new node. Adding points in this manner can cause the tree to become unbalanced, leading to decreased tree performance.
The rate of tree performance degradation is dependent upon the spatial distribution of tree points being added, and the number of points added in relation to the tree size.
If a tree becomes too unbalanced, it may need to be re-balanced to restore the performance of queries that rely on the tree balancing, such as nearest neighbour searching. To remove a point from an existing k -d tree, without breaking the invariant, the easiest way is to form the set of all nodes and leaves from the children of the target node, and recreate that part of the tree.
Another approach is to find a replacement for the point removed. For the base case where R is a leaf node, no replacement is required. For the general case, find a replacement point, say p, from the subtree rooted at R. Replace the point stored at R with p. Then, recursively remove p. For finding a replacement point, if R discriminates on x say and R has a right child, find the point with the minimum x value from the subtree rooted at the right child.
Otherwise, find the point with the maximum x value from the subtree rooted at the left child. Balancing a k -d tree requires care because k -d trees are sorted in multiple dimensions so the tree-rotation technique cannot be used to balance them as this may break the invariant.
Several variants of balanced k -d trees exist. Many of these variants are adaptive k-d trees. The nearest neighbour search NN algorithm aims to find the point in the tree that is nearest to a given input point. This search can be done efficiently by using the tree properties to quickly eliminate large portions of the search space.
Generally the algorithm uses squared distances for comparison to avoid computing square roots. Additionally, it can save computation by holding the squared current best distance in a variable for comparison. The algorithm can be extended in several ways by simple modifications. It can provide the k nearest neighbours to a point by maintaining k current bests instead of just one. A branch is only eliminated when k points have been found and the branch cannot have points closer than any of the k current bests.
It can also be converted to an approximation algorithm to run faster. For example, approximate nearest neighbour searching can be achieved by simply setting an upper bound on the number points to examine in the tree, or by interrupting the search process based upon a real time clock which may be more appropriate in hardware implementations.
Nearest neighbour for points that are in the tree already can be achieved by not updating the refinement for nodes that give zero distance as the result, this has the downside of discarding points that are not unique, but are co-located with the original search point.
Approximate nearest neighbour is useful in real-time applications such as robotics due to the significant speed increase gained by not searching for the best point exhaustively. One of its implementations is best-bin-first search. A range search searches for ranges of parameters. For example, if a tree is storing values corresponding to income and age, then a range search might be something like looking for all members of the tree which have an age between 20 and 50 years and an income between 50, and 80, Since k-d trees divide the range of a domain in half at each level of the tree, they are useful for performing range searches.
Analyses of binary search trees has found that the worst case time for range search in a k -dimensional k -d tree containing n nodes is given by the following equation. Finding the nearest point is an O log n operation on average, in the case of randomly distributed points, although analysis in general is tricky. In high-dimensional spaces, the curse of dimensionality causes the algorithm to need to visit many more branches than in lower-dimensional spaces.
In particular, when the number of points is only slightly higher than the number of dimensions, the algorithm is only slightly better than a linear search of all of the points.
Otherwise, when k -d trees are used with high-dimensional data, most of the points in the tree will be evaluated and the efficiency is no better than exhaustive search, [12] and, if a good-enough fast answer is required, approximate nearest-neighbour methods should be used instead. Additionally, even in low-dimensional space, if the average pairwise distance between the k nearest neighbors of the query point is significantly less than the average distance between the query point and each of the k nearest neighbors, the performance of nearest neighbor search degrades towards linear, since the distances from the query point to each nearest neighbor are of similar magnitude.
In the worst case, consider a cloud of points distributed on the surface of a sphere centered at the origin. Every point is equidistant from the origin, so a search for the nearest neighbor from the origin would have to iterate through all points on the surface of the sphere to identify the nearest neighbor — which in this case is not even unique.
To mitigate the potentially significant performance degradation of a k -d tree search in the worst case, a maximum distance parameter can be provided to the tree search algorithm, and the recursive search can be pruned whenever the closest point in a given branch of the tree cannot be closer than this maximum distance.
This may result in a nearest neighbor search failing to return a nearest neighbor, which means no points are within this maximum distance from the query point. Instead of points, a k -d tree can also contain rectangles or hyperrectangles.
The tree is constructed the usual way with all the rectangles at the leaves. In an orthogonal range search , the opposite coordinate is used when comparing against the median. For example, if the current level is split along x high , we check the x low coordinate of the search rectangle.
If the median is less than the x low coordinate of the search rectangle, then no rectangle in the left branch can ever intersect with the search rectangle and so can be pruned.
Otherwise both branches should be traversed. See also interval tree , which is a 1-dimensional special case. It is also possible to define a k -d tree with points stored solely in leaves. The midpoint splitting rule [15] selects on the middle of the longest axis of the space being searched, regardless of the distribution of points. This guarantees that the aspect ratio will be at most , but the depth is dependent on the distribution of points.
A variation, called sliding-midpoint, only splits on the middle if there are points on both sides of the split. Otherwise, it splits on point nearest to the middle. Maneewongvatana and Mount show that this offers "good enough" performance on common data sets. From Wikipedia, the free encyclopedia.
Multidimensional search tree for points in k dimensional space. A 3-dimensional k -d tree. The first split the red vertical plane cuts the root cell white into two subcells, each of which is then split by the green horizontal planes into two subcells. Finally, four cells are split by the four blue vertical planes into two subcells.
Since there is no more splitting, the final eight are called leaf cells. This section needs expansion. You can help by adding to it. November February Main article: Range searching. Communications of the ACM. S2CID Computational Geometry. ISBN August
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