2)The reduced radius / ne: because the incomplete well usually happen the spatial llow near the well and planar radial fluid tlow a little far away the oil well, so the incomplete wells are always treated as the completely well. The radius of the completely penetrating well is expressed with r. and is called reduced radius or eflective radius ) Hence the production formula of incomplete well can also be written as 2Tkh( If make the formula(2-33) and the formula (2-35) cqual, then the skin factor S and the reduced radius /r have a relationship as The change of the seepage flow resistance caused by incomplete well equals the well radius changes from r. to r:, And because the change of well diameter has little intluence on seepage tlow resistance. in order to make the effect of reducing well radius in line with the resistance change caused by incomplete, the reduced radius is always a very little value (3)The flow efficicncy incomplete factor ) n: it is the proportionality of the production of incomplete well and that of completely penetrating well under the same pressure difference, that is The incomplete factor connect n not only with S, but also with re/r. Although it cannot reflect the intluence of incompletion exactly, it is very illustrative, so it is still used in tield 2.5 The systematic well testing( Step rate testing) A very important clement that determines the commcrcial valuc of reservoir is the well roduction and the water injection rate ol injection well. With the development of reservoir, the two capacities change all the time. To understand precisely the two capacities is very important in the phases of reservoir evaluation, development and prod Provided the properties of formation and the lluid in it are known, the oil production and the water injction rate can be obtained using the theoretical formula that have learned and going to be learned( the Dupuit Formula for example ) Actually it is very difficult to understand the operties of formation. For example, the permeability measured in laboratory using core is absolute permeability, while there is various degrees of irreducible water in actual oil layer, it is relative permeability that af fects deliverability: Secondly, the heterogenetic of oil layer is usually very serious, the difference in permeability is very big between well points, Thirdly, the number cored well is usually quite few which is less than one tenth, So far the permeability mcasured with geophysical means is all indirect whose error is more than 50%c, Even the division of net pay thickness also has many human factors. For example, in the early stage of development, the oil
layer with high permeability yields oil while the oil layer with low permeability does not. Aller plugging the high water cut layer, the non-effective pay layer without oil changes to be effective pay layers. It is very difficult to ascertain the skin factor S theoretically; it is also not easy to obtain the parameters when using theoretical formula The best way to obtain the formation parameters is to directly use the measured production and pressure data and then to predict the production and pressure under new working condition with the obtained formation parameters, This kind of method is called well testing( it is also called antiproblem of percolation mechanics in porous medium theoretically ) Well testing is not confined in the development phase; it can be run on the early, middle and late phase, Secondly production performance and it is of high accuracy. Fourthly, the formation parameters obtained stand for the average value of the area that well affects which has great representativeness. So well lesting has become an important means that reservoir engineer and production engineer understand the reservoir and determine stimulation etfect The oil producing capacity should be in line with the lifting ability in well bore, so an optimal working condition is nceded, such as an optimal choke size, the optimal gas in ection for gas lift and the optimal pumping parameters and so on. All these need to be ascertained by well testing but it is belong to the research area of production engincerit As the anti-problem of percolation mechanics in porous medium, well testing mainly determines the formation parameters and skin factor and so on according to the measured production and bottom hole pressure. The initial formation pressure and average reservoir pressure can be obtained according to the shut-in data of short time so as to ascertain the reservoir boundary and determine the effect of reservoir stimulation That every measured pressure and production do not change with time in the process of well testing is called systematic well testing: while that the production or pressure changes with time is called unsteady well test. The testing technique of unsteady well test is casier than the systematic well testing, and its operation is more convenient and also can solve more problems Change the working conditions of production and injcction wells. Such as changing the choke of flowing well, the pumping parameters and the injection pressure or injection and so on, after the production attaining steady. measure the production under each working condition and its corresponding bottom hole flowing pressure. Provided the production condition of the surrounding ing the time of well testing and the pr working condition really attain steady, then the production and pressure have a linear relationship the valid limit of the Darcy s law, If the production pressure difference points under each steady system are drawn on the chart which its X-coordinate is production and the Y-coordinate is pressure difference, a straight line can be obtained, and this line is called index curve. With the increase of pressure difference and production the Darcys law will be invalid which makes the index curve break away Irom the straight line and toward the pressure ditference coordinate because the bottom hole pressure is lower than saturation pressure, will also make the index curve
As to the straight part of the index curve, the tollowing expression can be obtained =/p In the formula above: J is called prodt to an injection well it is called injectivity index, It means tion of oil well or the injection of injection well under unit producing pressure difference, its unit is m/(d. MPa). Comparing the formula (2-38 )and the Figure 2-l| the index curve of oil well Dupuit formula it can be seen that Gencrally. half of the average value of well spacing between wells and its adjacent well considered as /, the skin factor can only be obtained from unsteady well test or looking up some theoretical curves according to the perforation parameters. Thus, the formation flow coefticient near the well Kh/u can be obtained from measured producing index. If the net pay thickness h is obtained according to the log data, then the formation mobility K/u can also be obtained. If the tluid viscosity is known first, then the effective permeability K can be obtained. If the index curve is bent, the producing index is no longer a constant. It can be seen from igure 2-ll that productivity index is tangent of angle betwcen the straight line section and ordinate. If the index curve bends, the productivity index is a variable and is the slope of each I the index curve. With the increase of pressure difference, the producing index decrease The key point of systematic well testing is: the production under cvcry working condition must attain stcady, while to attain the steady needs rather long time, the less permeability is , the longer time to steady is, which is the drawback of systematic well testing However, it is nonreplaccable in the respect of choosing advisable oil-well working system and needed to be run As to the supply boundary pressure, it is best to use the average reservoir pressure measured when shutting in. If the supply boundary pressure pe is replaced with average reservoir pressure the value 1/2 should be deducted in the parentheses of denominator in formula(2-39) 2.6 The differential equation of single- phase fluid seepage flow In section 2. I and section 2.2 of this chapter, the seepage [low theory ol planar one dimensional and radial lluid steady seepage flow of single-phase incompressible lluid are introduced However, actual reservoir is very complicated, the boundary of reservoir is very irregular, the well pattern is varied, the effective pay thickness, permeability and epth everywhere of reservoir are diffcrent, and there are numbers of well producing simultancously in a reservoir and now the
tlow law and flow net are definitely very complicated. In order to develop reservoir economically and effectively, it is very important to understand the seepage law under all possible conditions 2. 6. 1 The continuing equation of seepage flow of single-phase fluid The change and movement of all things in nature is regular. All the physical and chemical changes within macroscopy obey the law of energy conservation and law of mass conservation. In he course of seepage flow of the underground fluid also obeys the law of mass conservation.Fluid annot vanish and also cannot produce in the course of seepage flow. This law is also the foundation of researching the movement law of fluid If an arbitrary confining surface is laken Irom seepage llow field, then in a definite time. the more( less )fluid mass flowing out must be equal to the decreasing( increasing )amount of fluid mass in the contining surface. Apparently, the velocity and density of each point on the surface is generally not the same and also within the surface. In the view of microscope, within a very tiny domain, around a point the law of mass conservation also stands expressed with mathematical which means the fluid mass difference flowing in and out around a point should be cqual around a point, the point must include enough solid particles and pore volume. It cannot be infinitely small. or the point will be taken from solid or from lluid which will break away from the limit of fluid seepage mechanics. However, its size also cannot be too large. This is the method that considers the porous medium as continuous mcdium. Provided the relationship betwcen rs around a point is understood, it then can be expanded into the whole reservoir by adopting advisable method. It is an universal researching method in classical physics Understanding the relationship of multiple parameters of a physical phenomenon around a point is setting up differential equation. and resolving concrete problems is solving the differential quation under given conditions or called the integration of differential equation. This is the eneral mcthod of from point to surface then to volume In order to understand the relationship of velocity, pressure and the formation and tluid propertics of a given point, a small enough contining surface should be taken surrounding this oint. Because the surface is small enough, a parallelepiped composed by the surfaces parallel lapu dr random confining surface. the error is high pus ax suming that there is only puf 2 ay fluid in oil laver, as to an arbitrary fluid material point M(x,, 2) in oil layer, consider the point I dpv. pu,? ax M as center taking an infinitesimal hexahedron rallel with respectively dx, dy and dz as shown in Figure Figure 2-12 The chart of formation infinitesimal hexahedron, then as to an arbitrary time 1, the
hanging amount( increasing amount)of fluid mass in the pore volume of hexahedron alter time dr should be equal to the fluid mass difference flowing in and out the hexahedron within time d/ Assuming the fluid density is p on the location M, the tlow velocity is t, then the mass velocity is pr, the components are respectively pr,, pr, and pr, on the coordinate x, y and z, and consider the positive direction of e, u, and r, as the positive direction of coordinate t, r and 2. Because the dr. dy and dz of hexahedron is very small. so after the high order infinitely small milled, the variability of mass velocity with dislance along the direction x in the infinitesimal back of hexahedron along the direction r is While the flowing out mass velocity from the front of hexahedron is The flowing in fluid mass from the back of the time dt is -1=)da While the flowing out fluid mass from the front of the time d is ax]dy didn the ditference of tluid mass flow ing in and out in time dt along the direction r is dxdvdzdt Il T he same principle, difference of fluid mass flowing in and out in time dr along the direction d z respectively is d/ tIxdrdd:. opt rxdrdeda The total difference of fluid mass flowing in and out of in time dz is opedxdydzdr Because porosity c and the fluid density in pores are relevant with formation the formation pressure in hexahedron changes, the fluid mass in it will also change. The fluid mass in hexahedron droids Assuming the variability 2p: of fluid mass changing in time dr is a constant, the hanging amount: increasing amount ) of fluid mass in time dr is dxdidzd 35一