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be made to remove the undesirable low- gravity solids by screening before their particle size is reduced within the size range of the API barite particles. Hydrocyclones cannot be used alone on weighted systems because their cut point falls in the particle-size range of the API barite as shown in Fig. 2.26. However, they sometimes are used in con- junction with a shaker screen to increase the flow rate capacity of the solids removal equipment. A series arrangement of a hydrocyclone and a shaker screen is called a mud cleaner. It is suited best for muds of moderate density (below 15 Ibm/gal). The fine solids that pass through the screen can be handled by dilution and deflocculation. At higher densities, the mud cleaners are much less efficient. Much of the coarse solids in the mud remain in the liquid stream exiting the top of the unit and, thus, bypass the screen. Also, dilution requires • 68 ~ ~ z "' u a: "' Q. (f) 0 ::; 0 (f) I MICRON lmm I em 01 0.1 10 100 1000 •10,000 2 4 68 2 4 68 2 4 68 2 4 68 2 4 68 2 4 68 ~ r--~ p'-'if-> r 1'-'-~Q Ql< y ) ---- SILT FINE , COURSE GRAVEL SAND SAND ~~~~~ER 200 DISCARD 100 MESH 60 MESH 20 MESH f;ENTRIFUGE DESI TER UNDER fLOW OVERFLOW DESANDER UIURFLOW TOBACCO SMOKE MILLED FLOUR BEACH SAND SETTLING RATE OF DRILLED SOLIDS IN 68°F WATER, FEET PER MINUTE .01 0.1 I 10 30 5090 Fig. 2.26-Particle size range for common solids found in weighted water-base muds. discarding a large volume of API barite with a portion of the old mud and the cost of dilution can become quite high. In this situation, centrifuges often are employed to separate the particles having sizes that fall in the API barite range from the liquid and extremely fine solids. In this manner the mud stream is divided into (1) a low-density overflow slurry (approximately 9.5 Ibm/gal) and (2) a high- density slurry (approximately 23.0 Ibm/gal). The high-density slurry is returned to the active mud system, and the low-density slurry usually is discarded. An example solids removal system for weighted clay /water muds is shown in Fig. 2.27. About three-fourths of the bentonite and chemical content of the mud is discarded with the fine solids when the centrifuge is used. New bentonite and chemicals must be added to prevent depleting the mud. Also, since some of the API barite and drilled solids are discarded in the overflow, the volume of mud reclaimed from the underflow will be less than the volume of mud processed. A small additional volume of new mud must be built in order to maintain the total mud volume constant. A material balance calculation can be made to determine the proper amounts of API barite, clay, chemicals, and water required to reconstruct a barrel of mud that has been processed with a centrifuge. 2.3.13.1 Centrifuge Analysis. Consider the flow diagram for a centrifuge shown in Fig. 2.28. Dilution water and mud enter the centrifuge and a high- density slurry (pu) containing the API barite exits the underflow while a low-density slurry (p0 ) containing the low-specific-gravity solids and most of the water APPLIED DRILLING ENGINEERING Shaker OIS<:Ord Fig. 2.27-Schematic arrangement of solids-control equip- ment for weighted mud systems (after Ref. 5). Po qm qo MUD IN Pm BARITE WATER IN q., P., qu fa WATER q •2 API BARITE w8 Pm MUD CLAY we qm OUT ADDITIVE WI Fig. 2.28-Fiow diagram of a centrifuge. and chemicals exits the overflow. Thus, the flow rate of the overflow is the sum of the mud flow rate and water flow rate into the centrifuge less the underflow rate: qo =qm +qwl -qu. · · · · · · · · ·- · · · · · · · · (2.24) The total mass rate into the centrifuge is given by mass rate in= qwl Pw +QmPm · Similarly, the total mass rate out of the centrifuge is given by mass rate out= q uPu + Q 0 P0 • For continuous operation, the mass rate into the centrifuge must equal the mass rate out of the centrifuge. Equating the expressions for mass rate in and mass rate out gives QwJPw +QmPm =QuPu +QoPo· · · · · · · · · · (2.25) Substitution of Eq. 2.24 for the overflow rate in Eq. 2.25 and solving for the underflow rate, qu, gives the following equation. qm (Pm -po) -qwl (Po -pw) Qu = ..... (2.26) <Pu- Po) This equation allows the calculation of the underflow rate from a knowledge of ( 1) water flow rate and mud flow rate into the centrifuge, (2) the densities of the water and mud entering the centrifuge and (3) the densities of the underflow and overflow slurries exiting the centrifuge. The reconstruction of the mud from the centrifuge underflow occurs in a pit downstream of the cen- I DRILLING FLUIDS trifuge. It is desired to obtain a final mud flow rate from the mixing pit equal to the mud feed rate into the centrifuge. In addition, the final mud density should be equal to the density of the feed mud. A knowledge of volume fraction of feed mud, dilution water, and API barite in the underflow stream simplifies the calculations required for reconstruction of a mud with the desired properties. Consider the underflow stream to be composed of (1) old mud, (2) dilution water, and (3) additional barite stripped from the discarded mud. The density of the underflow can be expressed by Pu =pmfum +Pwfuw +PBfuB• · · · · · · · · · · (2.27) wherefum•fuw• andfuB are the volume fraction of mud, dilution water, and API barite in the underflow stream. Furthermore, if we assume perfect mixing of the feed mud and dilution water in the centrifuge, then the diluted feed mud in the underflow should consist of feed mud and dilution water in the same ratio as they existed going into the centrifuge. From this assumption we obtain qwl fuw =fum-· · · · · · · · · · · · · · · · · · · · .. (2.28) qm Also, since all the volume fractions must sum to one, thenfuB can be expressed by fuB = 1-fum- fuw = 1- fum- fum qwl · .. (2.29) qm Substituting these expressions for fuw and fuB into Eq. 2.27 and solving for fum gives (pB-Pu) fum=-------------------- qwl PB-Pm+- (pB-Pw) qm ...... (2.30) The flow rate of old mud, dilution water, and API barite to the pit from the centrifuge underflow are given by qufm, qufw, and quf8 , respectively. The fraction of the underflow stream composed of old mud already has Wyoming bentonite, defloc- culants, filtration control additives, etc., at the desired concentration. Thus, only sufficient additives to treat the remaining portion of the mud stream from the mixing pit are required. The fraction of the mud stream from the mixing pit composed of old mud is given by fm = qufum . qm If ci is the desired concentration in pounds per barrel of a given additive in the mud stream, then this additive must be added to the mixing pit at the following mass rate. Wi =ciqm ( 1- fm) =ci (qm -qufum) • · · · · (2.31) where q m and q u are expressed in barrels per unit time. This expression also can be used to obtain the mass rate at which commercial clay should be added. The volume of water and API barite needed to main- tain the density of the mud leaving the mixing pit at the same density as the mud entering the centrifuge can be determined through a balance of the material entering 69 and exiting the mixing pit. The total flow rate exiting the mixing pit can be expressed by WB We n Wi q m = q u + q w2 + -- + -- + .E PB Pe i=l Pi Similarly, the mass rate of material exiting the mixing pit is given by n QmPm =QuPu +qw2Pw + WB +We+ E wi. i=l Solving these simultaneous equations for unknowns q w2 and w 8 yields +(pB-Pw), .................. (2.32) and n WB=(qm-qu-qw2-~- .E ~ )PB· Pe i=l Pi ............................