ne seepage coefficient of water is a constant. Darcy's law of this form is very convenient on In groundwater seepage not convenient to use hydraulic slope and seepage coefficient to study the movement of oil, gas water and their mixture in formation, and using pressure gradient and the seepage coefficient which get rid of the impact of fluid property is better. It is apparently to see that K=K In the formula above: scale coefticient is K called permeability, it completely depends on the properties of porous media and has nothing to do with the properties of fluid, Then the Darcys to be (1-8) As the result that imagined. Many people repeated Darcys experiment after him, No matter the cylinder is put straight or inclined, the differential pressure should be the converted pressure that consider the position head 1.3.3 Seepage velocity and average true velocity According to the Darcys law, the seepage velocity t is 1 It shows the rate of flow through the unit cross-sectional area of rock. However, the seepage velocity is not the true velocity of the fluid mass point in pores. Because in the given cross section of rock, there are the section of pores that the fluid can pass through and the solid particles that the tluid cannot pass through. Taking any cross section of core and assume its pore area is A, then In the formula above n is called the transparence of the cross section, also called surface porosity.Apparently,the average true velocity on any cross section is: Becausc the hetcrogeneity of rock even though the arca a of cach section remains unchanged, the pore area of each section is different and the transparence is not the same theretore. the of transparence along the flow route is taken to determine the average true velocity of fluid 1」 And the pore volume of rock can be written CAnde =u(1I ndx)=AT/
According to the definition of porosity n =o. so the average true velocity of fluid along the flow routes 1 Apparently, the value of average true velocity of fluid is bigger than seepage velocity Darcy's law is obtained through the experiment of steady seepage llow using homogeneous iquid in uniform cross section sand layer, Under normal circumstances, the cross section of seepage is changing, the formation is not only heterogenous but also anisotropic, and the fluid properly sometimes changes with location. It is considered in this book that the Darcy s law right on the micro element volume, il is generally written as 人 grad(p+pgz)=-v(p+pgz) Dividing the velocily into three components along three directions of x, y and =, then D+pg);,=-a p+pg: ut dxp+pgz) The negative sign in the above formula means the direction of seepage velocity is always opposite with that of pressure gradient, which means is the direction of seepage velocity is always along the direction of pressure decline. The thing that needs to be noticed is: the infinitesimal body must include enough sand particles and enough fluid, if the infinitesimal body is really considered as infinitely small, maybe there is only solid particles or fluid material point in it. then it is not seepage flow, and this is the statistical property of the Darcys law 1.3. 4 The dimension of permeability Permeability is an important physical property of reservoir; it is directly related to oil and gas deliverability and is an important parameter in the reservoir evaluation and development According to the principle of dimension compatibility and use formula(1-7). the dimension of permeability can be obtained as Mass M. time T and length L are taken as three basic dimensions, then the right part of the formula above changes to be 」=1./T; Substitute the equations above into formula( 1-10). then ,/ It can be known from above, permeability K has the dimension of area, because it shows the ability of the lluid passing through the porous medium that permits, so it is reasonable for permeability to have the dimension of area. The unit of arca is m" in SI; it is too big for using herefore, the second power of micrometer um) is taken as the unit of permeability Kin reservoir engineering si system
or example. assuming a kind of liquid whose viscosity is ImPa. s passes though a rock sample whose length is Icm, cross-scction area is Icm, and its pressure diffcrence of both ends is 10 Pa, if the rate of tlow of the liquid is Icm/s, then the permeability of rock sample is K=4=101mx10的1×10=10-m2=1m 1. 4 The limitations of Darey's law and non Darcy seepage flow The Darcy s law shows that the pressure loss is entirely determined by the viscous resistance which is consistent with the characteristic that the porous medium has big specilic area. The contact area of liquid and solid is big, so is the viscous loss. Therefore in most cases the Darcys Law is obeyed. I lowever, the second characteristic of porous medium is that its channel diameter loss)is big. The rclationship between viscous loss and velocity is direct ratio. while the relationship between inertia loss and the second power of velocity is direct ratio. When the seepage ine loss can be ignored. Wh ty is bigge the inertia loss cannot be omitted. Therefore the Darcy's Law will lose effectiveness with the increase of seepage velocity The failure of d porous media In fact, the failure of Darcy's law is mainly due to the large inertia loss under laminar flow condition. and experiment also confirmed this In lluid mechanics, the criterion judging the flow pattern of tlow is the Reynolds number Re When studying whether the fluids flow in porous media obey Darcys law, the Reynolds number ( ratio of viscosity force and inertia force) Re is also adopted. When the Reynolds number reaches a certain value, the Darcy's law loses effectiveness. It can be known from the know ledge of mechanIcs Re- fidr (1-11) Where: p density of liquid, kg/m 从 hydraulic diameter of porous media channel, m: L ,True velocity of liquid, m/s Although the above formula is got, however, because the seepage channel is very irregular even it is very ditficult to give the pore diameter, and also to measure the pore diameter directly so an ideal model is adopted usually to compute the average channel diameter. The Ideal model is he simplest simulation of actual core, and it is far from reality as shown in Figure 1-7,(a)is the actual core;( b) is the ideal model, and assuming: (I) The geometry of the actual core and the ideal model is the same: ( 2)The ideal model consists of a cluster of isodiametric capillary and
pore volume of actual core and the ideal model is cqual; (3) :.. Under the same pressure difference, the llow rate of liquid r:-以" L-.I1. passing through actual core and the ideal model is the same. The 1101 45 -- diameter of capillary d meeting the above conditions of the ideal model is the equivalent average diameter of formation pore actual core Under the condiuion of laminar llow the rate of llow passing though a single capillary tube is In the formula above: r is the radius of capillary. Assumi that there are N capillary tubes in the unit cross section of porou nedium, and the total amount of the capillary is NA. then the (b)ideal model total rate of flow passing the ideal model Fi ideal model (1-12) olume of the actual corc and the ideal model. the following equation can be obtained MARl=(AL After simplifying the equation above. the following equation can be obtained Bring the equation above to the formula(1-12). then bAr-A Bul While according to the Darcys law, the rate of tlow passing though the actual core is 人1▲ (1-14) According to the assuming condition (3), make the formula( 1-13) equals to formula (1-14),then Bring the formula(1-15)and =>into formula(1-11).then Re 4人 (1-16) If kg/cm is used as the unit of tluid density, m/s is used as the unit of seepage velocily um" is used as the unit of permeability. mPa. s is used as the unit of viscosity. then the formula -l6) changes to 03a人
La The experiment proves that, the critical value of Reynolds Number is 0.2-0.3. and this alue has big difference with the value 2320 in ordinary hydromechanics and the transitional range of the failure of Darcys law is very big Figure 1-8), it also shows that the main reason of the failure of Darcy's law is not the occurrence of turbulent Hlow, but the increase of inertia loss 3×10 7N| 00135b7 Figure 1-8 The relational diagram of the Reynolds number and resistance factor Many scholars are trying to find a law that takes both viscous loss and inertia loss into onsideration, and the Darcy's law is treated as its special case. The binomial seepage law is considered not only to think about viscous loss but also inertia loss, and it is accepted generally as the cquation that can reflect the actual (1-18) In the formula above: gradp is the pressure gradient, h is a constant depending on the properties of porous medium and is confirmed from experiment; p is the density of fluid. The first lem of the right part of formula(1-18) means viscous loss, and the second item means inertia loss. When the seepage velocity is very small and 1 =>1. the binomial law will change to be the Darcys law. If a third power column is added after the binomial law, the result will be accordant with the experimental outcome, but the physical meaning become vague. Sometimes, exponent seepage law is also used for the convenience of mathematical handling