The formula(2-10a)or the formula (2-10b) is called the moving law of fluid mass point of planar onc dimensional steady seepage flow, and the formula(2-10) can also be written as The numerator of formula(2-11) is the fluid volume between the sections of original location and the present section of a arbitrary mass point. When dividing it by llow rate q, the time that evacuating the total fluid between two scctions of formation is obtained while the time is just equal to the time that the fluid mass point moving from xo tox. Especially, when the fluid material point studied moves from the original location xn =0 to x=L after the time T, after substituting the value into equation (2-11), the following expression can be obtained 7=业1 Example 2-1 Assuming that there is a rigid water drive oil laver as shown in Figure 2-5. The pressure of formation supply boundary is P =20MPa P.=16MPa, the permeability suddenly changes at the o length Ly: when rs Li, K=K1=0.5um: when x> Figure 2-5 The distribution of permeabili Li, K=K,=2um. Assuming that L,=500m, L=1000m, the height of formation is h=5m width is B 100m, the viscosity of crude oil is u=4mPa. s. Please solve the following problems: (1) the production formula of fluid seepage flow and calculate its daily production (2) the pressure distribution formula and draw the pressure distribution diagram Solution:(1) Because the permeability K changes, so this formation can not meet the assumption of homogeneous formation, but it can meet the assumption of homogeneous seepage tlow in the area of Osx<L and L>t>L, and according to the continuity principle, the flow rate is cqual in the two areas. Assuming that the pressure is p, when x=L, then the following xpression can be obtained ( k1P-P1)k21{p1-P-) (L-1) Defor the formula (2-13) (2-14) K,1 Then the following production formula can be obtained using proportion by addition law to formula 2-14)
Substituting the values given into formula(2-15) an be obtained from the formula (2-13) K,(L-Lu)p+K,Lp O.5×10-2(1000-500)×20+2×10-12×500×16 0.5×10-2(1000-500)+2×10-2×500 16.8(HPa) When0≤x≤L1, the pressure distribution is (x)=/x=20-20-=20-6.4×10-x 500 When I1≤x≤L (x)=P =16.0+-500(100-x) 16.0+1.6×10(1000-x) Example 2-2 Assuming that there is a homogencous, and isotropic core angled c with al direction whose length is L(Figure 2-7). A pic ctively the inlet end and delivery end. Fluid in the core moves as incompressible one-dimensional steady that the density is p, and the height ditference of the two piezometer is AH, please derive the flow rate formula of the tluid pas Hy Figure 2-6 The pressure distribution with unequal permeability Figure 2-7 The figure of Example 2-2
Solution: In order not lo lose the generality and lor convenience, consider the axial direction of the core as the x axes, and y axes is vertical with x and get the origin at the inlet oo is the base level. because the fluid only ilow along the x direction and direction, so the reduced pressure of a arbitrary point on a arbitrary cross section that is vertical with x axes is equal. According to the Darcys law, the flow velocity of a arbitrary point in the core Is kdD (2-16) Assuming that the reduced pressure of the inlet end and delivery end is respectively pr and p, substituting ?=0/A into the formula(2-16) and separating the variables and integrating then the equation below can be obtained k1( 2-17) According to the detinition of reduced pressure Pr =p+pg pg(Z1+h1-22-h2) In the formula above: h, and hi, is respectively the height from the fluid level of the piezometer at inlet end and delivery end to the axial line of the core; Z and Z, is respectively the height f'rom the base level of the inlet end and delivery end to oO:(Z+h,)and(Z, +hi,) respectively the height Irom the fluid level of the piezometer of inlet end and delivery end to base level oo, the distance between the two tluid levels. marked as AH. So from the formul (2-17)and the formula(2-18). the following expression can be obtained From the expression, we can see that flow rate has nothing to do with the angle of inclination and is only in connection with the height difference of the lluid level of the two piczomctcr 2.3 The planar radial fluid flow of single-phase incompressible fluid As introduced before, the area around cach well in actual reservoir can be approximately considered as radial fluid flow. so the study of radial fluid flow has more significance. In order to set up the model of radial fluid flow, the following assumptions are needed The formation is homogeneous, isopachous and isotropic, (2) There is only one kind of homogeneous and incompressible fluid flowing in formation, and the compressibility of formation is not considered (3)There is no physical and chemical action between fluid and rock. (4 The production well barefoot well and the shape of formation is shown as Figure 2-8, The radius of the supply boundary is r. the well lies in the center of formation and its radius is r, the height of formation
is h, permeability is K, porosity is the pressure of supply boundary is p and the botom hole pressure is p The way study the radial fluid llow is exactly the same as the way study the planar one-dimensional seepage flow Because the formation is homogeneous, isopachous and tropic,so the constant pressure surface should be the cvlinder surface which is concentric with well. As to the arbitrary point on the cylinder surface which the distance is from the center of the formation the flow velocity can be expressed by the Darcys law k dp d (2-19) Dillerent with the one-dimensional seepage llow, the cross section is not constant now But continuity principle, the tlow rate passing through arbitrary n should not change bitrary cross scction whose radius is r should be (2-20) The minus in the formula above is because the direction f the tlow velocity is opposite with that of coordinate Combine the formula(2-19)and formula(2-20) and separate the variables Integrate the both sides of the formula(2-21). the integrating interval of r is r./, the integrating interval of p is PP, so the production formula of radial fluid flow is 2丌Kh(P The formula (2-22) is also called the Dupuit formula. If the integrating interval of formula (2-21)is changed from I, I to LI, r or[r, r, the following expression can be P Substitute the formula (2-22)into formula(2-23 ). the following expression can be
(2-24) pOr)is the pressure where the radius is r, the formula(2-24)is the pressure distribution formula and the corresponding pressure distribution curve is shown as Figure 2-8. Actually, the pressure distribution is a surface of revolution of log curve surrounding hole axis, and it is also called depression hopper. According to the formula(2-24), the pressure p and the distance logarithm r have a linear relationship, which the distance changes by geometric series and pressure difference changes by arithmetic series. Differentiate the both sides of formula (2-24) with respect to distance r In the formula (2-25): the pressure gradient is in inverse proportion to radius that is the closer to the bottom hole, the bigger is the pressure gradient. Because the closer to the bottom hole, the smaller is the cross section, the bigger is flow rate and the denser are the isobars in the seepage flow tield. From the Figure 2-9 we can know. mos of the pressure loses around the area of bottom hole. It also illustrates theoretically the importance of protecting the formation from damage in the process of drilling and roducing Figure 2-y The seepage tlow field of Substitute the formula(2-25) into formula (2-19) radial fluid flow the following expression can be obtaine (2-26) Because the isobars are concentric circles and the streamlines are naturally the rays that fluxes on original point( Figure 2-9), so the streamlines become denser toward the botton Example 2-3 Assuming that there is a circular reservoir, and the radius of supply boundary is /=200m, the radius of bottom hole is /w=0. Im, the pressure of supply boundary is pe, and the pressure of bottom hole is P,. Please solve the following problems:(1)when the distance Irom the center of well r=100m, 50m, 25m, 12. 5m, 6. 25m, 3m and Im. what is the ratio of the pressure loss from the distance to bottom hole lo total pressure loss?(2) If the well diameter is extended one time. what will the production change correspondingly? Solution:( 1)According to the formula (2-24), the pressure loss of a arbitrary point to the