bottom hole is Ap(r)=p(r) While the total pressure loss from the supply boundary to the bottom hole i Assuming that the ratio of the pressure loss from the distance r lo bolo hole to tolal pressure loss is /. the following expression can be obtained: ×100 Substitute r, / and r into the expression above, the results are shown as Table 2-1 Table 2-1 The calculation results 81.8 447 (2)Assuming that the increasing percent of production if the well diameter is extended one time is q, the following expression can be obtained -×100% ×100%=10.0.3% In From the example above we can see clearly that the pressure loss of radial fluid flow is mainly xhausted around the area of bottom hole. If the radius of bottom hole is extended one time the increase of drilling cost is far more one time, while the production is only enhanced 10.0%e. It the stimulation treatments are adopted such as fracturing, the enhancing times will increace more than 10. 03%. if we do this the cost is far more less than that of extending the well diameter So extending the well diameter is not treated as a stimulating mcasure at home and abroad It can be known from the formula(2-26) that the average true flow velocity of the fluid from the well where the distance is r in formation is dr K(p.-P, spuN Assuming that the location is ro of a given fluid mass point when time is 1=0, while the ocation changes to be r after time 1, substitute the data into the formula(2-28) and integrate the following expression can be obtained Kip, -Pv) cuIn Combine the formula(2-16) and formula(2-21), the following expression can be
h(ro-r) ? (2-29b) The formula(2-29a )or the formula(2-29b) shows the movement laws of fluid mass int of radial fluid seepage tlow, The numerator of the formula(2-29) shows the lluid volume between the cross section of original location whose radius is ro of lluid mass point and the cross section of present location whose radius is r, when the numerator divides by the flow rate @, the needed time that the flow rate exhausts the fluid in the given interval is obtained, and the time is exactly equal to the time that the fluid mass point moves form /n to 1. There is always irreducible water in actual reservoir; the ratio of the pore volumes that contain oil to pore volume is called oil turation. so in order to know the moving time of oil in formation, the right numerator of the formula(2-29) need to multiply the oil saturation If the original location of the tluid material point is r, =r, it moves to the bottom hole(=I) after time T. and the following expression can be obtained It is not easy to ascertain the boundary pressure pe of actual reservoir, even if the boundary pressure is a constant, when there are numbers of wells producing together, it is impossible to measure the boundary pressure when shutting down one of the wells. Provided there is only a well producing, the boundary pressure can be measured when shutting down the well. So, the avcrage formation pressure is used to substitute the boundary pressure it is also called supply boundary pressure ) when there are numbers of wells producing simultaneously. The so called average formation pressure is volume weighted avcrage of pressure, that is In the expression above: P is the pressure of a given point in formation: dv is the infinitesimal of volume around the point, V is the volume of the whole formation. As to the circular homogencous, isopachous and isotropic formation, because the constant pressure surface is a ylindrical surface, so the following expression can be obtained I1=2Torhdr And the average formation pressure is Substitute the formula (2-24)in to the formula abov (rlr,-rInrw/(:-Tw)
B r-sr into the tollowing expression can be obtained P≈P (2-30a) The formula can be also wrilten as PeW 2n 2-30c) Usually r//>10. So 2In ->13. 8, hence, the following expression can be obtained ! 0.93 It can be know from the expression above: the average formation pressure is close to the boundary pressure. that is P=Pe. In actual tield condition, the average formation pressure is usually used to replace the supply boundary pressure Combine the formula (2-22)and the formula (2-30b)and climinating(P-P).the the following expression can be obtained The formula above can be also written as lakh(P-pw) (2-31b) The formula(2-3Ib) is the production formula of planar radial fluid flow shown with average formation pressure. In practice. the following two points should be paid more attention 1) All the derivation are set up under the condition that there is only a producing well in the center of circular formation but the actual reservoir is abnormal and there are numbers of well producing simultaneously, at this time each well is considered as the center, and the connection between the center of neighboring well is considered as the drainage region of the well, and alter the area of drainage region to a circular, so the radius of the circular drainage region is supply radiu In actual field condition, sometimes average of half of the well spacin pply radius, because the log of supply radius usually appears in formula, although there is crror between the two. there is no much influence on production. (2)The production Q in the formula (2-22) is the production in formation. In actual work it is easy to measure the surface production Qc, if replace the in the formula(2-22)with
surface production @, the following expression can be obtained In the formula above b is the volume factor of oil under the condition of formation. 2. 4 The incomplete well It is assumed that the oil well is barefoot finishing in the former section when if comes to planar radial lluid seepage flow. this type of oil well is called seepage flow complete well. In actual work. due to the reasons of technology, all oil( gas layers can not be drilled out by well or even all the oil layers are drilled out, and the bottom hole structure is changed by perforation completion after well cementation. Otherwise, because the mud plug during the process of drilling or the stimulation treatments acidize and fracturing technology for example)adopted in the process of producing, the reservoir properties are changed in partial area aroune he well with changed bottom hole structure or changed reservoir properties around the bottom hole is called seepage tlow incomplete well, and also called incomplete well for short. Most of oil wells are incomplete well actually .4. 1 The classification of incomplete well many types of incomplete well, as to the oil/ gas wells with changed bottom hole structure, according to the ditference of bottom hole structure, the incomplete well can be classified as three types as below 1)The incomplete penetrating well. The whole layer is not drilled out, and the drilled part of the oil layer is totally exposed as shown in Figure 2-10( a). The incompletion of well is called incomplete penetrating; such type of wells is mostly seen in reservoir with active bottom water =二 (a)Incomplete penetrating well (b)Incomplete characteristic well (c)Double incomplete well Figure 2-10 The incomplete well (2)The incomplete characteristic well. The whole layer is drilled out, but the perforation method used to complete. so the fluid in formation can only llow into bore hole through perforation, such incomplete of well is called incomplete characteristic as shown in Figure 2-10(b). Such type of oil wells is mostly scen in sandstone reservoir
3)The double incomplete well. The well does not drill out the whole oil layer, and the drilled part is completed by perforation. such incompletion of well is called double incomplete well as shown in Figure 2-10(c) II can be found by theoretical analysis and simulated experiment, a common unique feature of the three types of incomplete well is that the streamlines bend near the bottom hole. generally to the partial penetration well, the streamlines are basically parallel with each other the beyond I 2 limes formation height. As to the incomplete characteristic well. the seepage llow of lluid around the bottom hole is three -dimensional llow the ilow direction changes constantly. so the tance will increase caused by this, Under same conditions that the formation properties, fluid properties, producing pressure difference and the bottom hole radius the production of these incomplete wells must be lower than that of completely well Generally speaking. the mud contamination in the process of drilling will cause the increase of seepage resistance near the area of oil well. Recent years, because of the widely application of he fracturing technologics and acidizing and the improvement of perforation, the production of incomplete well is more than that of complete well. So, the concept of incomplete cannot be interpreted as that its production is lower than that of incomplete well 4.2 The research method of incomplete well As to the research of incomplete wells, lots of scholars have done considerable works. For the incomplete wells, the following methods are mainly used at prescr hole. when talked about the incomplete wells, the formula(2-22) is usually written as -2mk(P=/) The s in the formula (2-33) is called skin factor ( skin cffect) which is an important index hen measuring the degree of incomplete well. As to the partial penetration well, because it has a connection with the drilling depth b, bottom- hole radius and the formation height h, that is S(r b, h). As to the incomplete characteristic well. the skin factor is the function of perforating density n. the perforated depth i in formation, the diameter of perforation d and bottom hole radius r, that is S(n, 1, d, /). Although the curve of S(l, / d, I') can be obtained through electrolysis analogy, but because /, /(especially /) is very difticult to ascertain, so it is not casy to obtain S using this method. The skin factor is usually oblained through well testing. Generally speaking when the well is contaminated s>0. after fracturing and acidizing s<O It is also can be seen from the formula(2-33). when the incomplete well producing with e production Q. the llow resistance needed to overcome is diflerent from that of the complete well with the same production needed to overcome, and the difference of the two is 2丌Kh The Ap, in the formula above is called the additional resistance caused by skin factor S