Appendix I SPE Nomenclature and Units·

Standard letter symbols for reservoir engineering Gas volume and electric logging have been defined by the AI ME cubic foot) measured at 1 atmosphere (Society of Petroleum Engineers). Some non• cubic metre) and 60°F standard terms, subscripts and nomenclature are still MCF = thousands of cubic feet in use and may be encountered. MMCF = millions of cubic feet No effective standardization or metrication of (The billion is the American billion = 109 ; units has yet occurred, and the industry uses the trillion is the American trillion = 1012.) American mixed units to a large extent, although some metric units mixed with American still may be Pressure encountered. An application of the SI metric system pounds force per square in (psi) is found in the Journal of Petroleum Engineerinng atmosphere (1985) in the issues for August (p.1415) and October bar p.1801. Temperature degrees Fahrenheit OF UNITS degrees Rankine OR = 460 + OF degrees Kelvin K Volume acre-foot for large volumes Length barrel pipelines - miles, feet, kilometres cubic ft well depths - feet or metres cubic metre Diameters tubular diameters generally inches or centimetres Liquid volume feet/metres barrel = 5.615 cubic ft cubic metre = (35.31) fe Viscosity (Unless otherwise specified, an oil volume will be centipoise tank oil measured at 1 atmosphere and 60°F.) Density lb mass per cubic foot * Reprinted fromlournal of Petroleum Technology, 1984, kg mass per cubic metre pp. 2278-2323 by permission. © SPE-AlME, 1984. g per cubic centimetre

257 258 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE

Specific gravity Oil densities liquids relative to water (62.4lb/ft3) API gravity gases relative to air (0.0765Ib/ft3) API scale for tank oil 0API = 141.5 --,----.,.-- - 131. 5 (SG)oil Gas-oil ratio standard cubic feet of gas per stock tank barrel of oil SG = specific gravity of water = 1.0 cubic metres of gas (s.c.) per cubic metre tank oil Recommendation for metrication and appropriate Flow rate conversion factors for units are given: liquids - barrel per day (bid) cubic metres per day (m3/d) gases - standard cubic ft per day SCF/d, MCF/d and MSCFD/d, MMSCFD cubic metres per day (m3/d) MSCFD/d

Recommended units: conversions Quantity SI unit Industry SPE preferred Conversion unit unit factor (industry ~ preferred) Length m mile km 1.609344 metre m 1.0 foot m 0.3048 inch mm 25.4 Area m2 sq. mile km2 2.589988 acre km2 4.046873 x 103 sq.ft m2 0.0920304 sq. inch mm2 6.4516 x 102 Volume m3 m3 m3 1.0 acre foot m3 1.233482 x 103 barrel m3 1.589873 x 10.1 ft3 m3 2.831685 x 10.2 US gallon m3 3.785412 x 10.3 CapacityIlength m3/m barrels/ft m3/m 5.216119 x 10.1 ft3/ft m3/m 9.02903404 x 10.2 US gall.lft m3/m 1.241933 x 10.2 Mass kg Ibmass kg 4.535924 x 10.1 short ton Mg 0.9071847 Temperature gradient Kim °F/ft Kim 1.822689 Pressure Pa atmosphere kPa 1.013250x 102 bar kPa 1.0 x 102 kgf/sq. em kPa 9.806650 x 101 Ibf/sq. in. kPa 6.894757 dyne/sq. em Pa 1 x 10.1 Pressure gradient Palm Ibf/sq. in.lft kPaim 2.262059 x 101 Density kg/m3 Ibmlft3 kg/m3 1.601846 x 10-1 Ibm/USgal1. kg/m3 1 .198264 x 10 2 Volume rate m3/s bid m3/d 1.589873 x 10-1 US gall.!min m3/hr 0.2271247 Viscosity Pa.s cP Pa.s 1.0 x 10-3 Permeability m2 Darcy 11m2 9.869233 x 10-1 miliiDarcy 11m2 9.869233 x 10-4 SPE NOMENCLATURE AND UNITS 259 sPE SYMBOLS STANDARD original standards were published in 1956 following five years of intensive development. Additions Preface resulted from requests from members and from Objectives editorial reviews of the numerous papers submitted to SPE for publication. The primary objectives of the 1984 Symbols Stan• dards are to combine prior standards and supple• Principles of symbols selection ments into one publication so as to provide (1) consistency of usage and maximum ease of under• Once the original reservoir Symbols Standard was standing of mathematical equations for the readers established in 1956, the principles employed in the of technical papers, and (2) to codify symbols lists, selection of additional symbols have been as follows: rules and guides for the writers of technical papers. A. (1) Use single letters only for the main letter symbols. This is the universal practice of the Structure of lists American National Standards Institute (ANSI), The 1984 Symbol Standards are a consolidation of the International Organization for Standardiza• the 1956 Standard and all later supplements. Some tion (ISO) and the International Union of Pure of the cross-grouping and obsolete quantities have and Applied Physics (IUPAP) in more than 20 been eliminated. The complete symbols list is given formal Standards adopted by them for letter in four different forms as follows: symbols employed in mathematical equations. (2) Make available single and multiple sub• A. Symbols alphabetized by physical quantity, scripts to the main letter symbols to the extent B. Subscripts alphabetized by physical quantity, necessary for clarity. Multiple letters such as abbreviations are C. Symbols alphabetized by symbols, prohibited for use as the main symbol (kernel) D. Subscripts alphabetized by symbols. for a quantity. A few exceptions are some traditional mathematical symbols such as log, In The names or labels for the quantities are for and lim. Thus quantities that are sometimes identification only and are not intended as defini• represented by abbreviations in textual mate• tions. Defining equations are given in a few cases rial, tables or graphs are required in the SPE where further identifications may be needed. For the Symbols Standards to have single-letter kernels. present, the specification of units and conditions of Examples are: gas-oil ratio (GOR), bottom• measurement is left to the user. hole pressure (BHP), spontaneous potential For convenience in dimensional checking of equa• (SP), static SP (SSP), which, respectively, have tions, a column has been included giving the the following SPE Standard symbols: R,pbh, dimensions of each quantity in terms of mass, Esp, Essp. . length, time, temperature and electrical charge (m, B. Adopt the letter symbols of original or prior L, t, T, q). The term various also appears in this author usage, where not in conflict with princi• column for several symbols. This terminology per• ples C and D below. mits maximum flexibility for quantities that may C. Adopt letter symbols consistent or parallel with require different dimensions in different problems. the existing SPE Standard, minimizing conflicts Examples are symbols: (1) m for slope of a line (two with that Standard. variables of any dimensions can be related); (2) C D. Where pertinent, adopt the symbols already for concentration (dimensions might be m/L3 , standardized by such authorities as ANSI, ISO, dimensionless or other); (3) F (factor) when it or IUPAP (see A); minimize conflicts with represents ratio (dimensions might be L 3/m, m, these Standards. dimensionless or other). This flexibility in dimen• E. Limit the list principally to basic quantities, sions permits desirable shortening of the symbols avoiding symbols and subscripts for combina• list. tions, reciprocals, special conditions, etc. F. Use initial letters of materials, phase, processes, Additional standard symbols etc., for symbols and subscripts, as being The extraordinary growth in all phases of petroleum suggestive and easily remembered. and computer technology has necessitated the adop• G. Choose symbols that can be readily handwrit• tion of additional standard symbols, since the ten, typed, and printed. 260 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE

Principles of letter symbol standardization nent part of a complex mathematical formula A. Requirements for Published Quantity. - for example, as an exponent of a given Each published letter symbol should be: base. Instead, one may introduce locally, a 1. Standard, where possible. In the use of single non-conflicting letter to stand for such published symbols, authors of technical a complicated component. An explanatory works (including textbooks) are urged to definition should then appear in the immedi• adopt the symbols in this and other current ate context. standard lists and to conform to the princi• B. Secondary symbols. Subscripts and superscripts ples stated here. An author should give a are widely used and for a variety of convention• table of the symbols used and their respec• al purposes. For example, a subscript may tive interpretations, or else refer to a stan• indicate: (1) the place of a term in a sequence or dard list as a source for symbols used but not matrix; (2) a designated state, point, part, or explained. For work in a specialized or time, or system of units; (3) the constancy of developing field, an author may need sym• one independent physical quantity among bols in addition to those already contained in others on which a given quantity depends for its standard lists. In such a case the author value; (4) a variable with respect to which the should be careful to select simple suggestive given quantity is a derivative. Likewise, for symbols that avoid conflict in the given field example, a superscript may indicate: (1) the and in other closely related special fields. exponent for a power, (2) a distinguishing label, Except in this situation, the author should (3) a unit, or (4) a tensor index. The intended not introduce new symbols or depart from sense must be clear in each case. Several currently accepted notation. subscripts or superscripts sometimes separated 2. Clear in reference. One should not assign to a by commas may be attached to a single letter. A given symbol different meanings in such a symbol with a superscript such as prime (') or manner as to make its interpretation in a second ("), or a tensor index, should be en• given context ambiguous. Conflicts must be closed in parentheses, braces or brackets before avoided. Often a listed alternative symbol or an exponent is attached. So far as logical clarity a modifying subscript is available and should permits, one should avoid attaching subscripts be adopted. Except in brief reports, any and superscripts to subscripts and superscripts. symbol not familiar to the reading public Abbreviations, themselves standardized, may should have its meaning defined in the text. appear among subscripts. A conventional sign, The units should be indicated whenever or abbreviation, indicating the adopted unit necessary. may be attached to a letter symbol, or corres• 3. Easily identified. Because of the many ponding numeral. Reference marks, such as numerals, letters and signs that are similar in numbers in distinctive type, may be attached to appearance, a writer should be careful in words and abbreviations, but not to letter ca!ling for separate symbols that in published symbols. form might be confused by the reader. For C. Multiple subscript-position order. The wide example, many letters in the Greek alphabet variety and complexity of subject matter co• (lower case and capital) are practically indis• vered in the petroleum literature make it tinguishable from English letters; the zero is impossible to avoid use of multiple subscripts easily mistaken for a capital O. with many symbols. To make such usage less 4. Economical in publication. One should try to confusing, the following guides were employed keep at a minimum the cost of publishing for the order of appearance of the individual symbols. In particular: (1) Notations which letters in multiple subscripts in the symbols list. call for handsetting of movable type should Use of the same rules is recommended when it be rejected in favour of forms adapted to becomes necessary to establish a multiple sub• modern mechanical methods of composition. script notation that has not been included in this (2) No one work should use a great variety of list. types and special characters. (3) Handwrit• ing of inserted symbols, in copy largely 1. When the subscript r for 'relative' is used, it typewritten and to be reproduced in facsi• should appear first in subscript order. Ex• mile, should not be excessive. (4) Often a amples: Kr01· Krg. complicated expression appears as a compo- 2. When the subscript i for 'injection' or SPE NOMENCLATURE AND UNITS 261

'injected' or 'irreducible' is used, it should few distinct letters used from other alphabets, if appear first in subscript order (but after r for carefully made, should be self-explanatory. It is 'relative'). Examples: Big, formation important to select a type face that has italic volume factor of injected gas; Cig, compress• forms, and clearly distinguished upper case, ibility of injected gas. lower case and small capitals. Only type faces 3. Except for Cases 1 and 2 above (and with serifs are recommended. symbols Kh and Lv), phase, composition and system subscripts should generally appear E. Remarks. Quantity symbols may be used in mathematical expressions in any way consistent first in subscript order. Examples: Bgi, initial or original gas formation volume with good mathematical usage. The product of two quantities is indicated by writing abo The factor; B , initial or original oil formation oi quotient may be indicated by writing volume factor; CO,i' initial or original ox• ygen concentration; Bli, initial or original a -,alb or ab- 1 total system formation volume factor; PsE, b density of solid particles making up ex• perimental pack; also FaH G Lp' Gwgp, G Fi' If more than one solidus is used in any 4. Abbreviation subscripts (such as 'ext', 'lim', algebraic term, parentheses must be inserted to 'max', 'min'), when applied to a symbol remove any ambiguity. Thus, one may write already subscripted, should appear last in (a/b)/c, or a/bc, but not alb/c. subscript order and require that the basic symbol and its initial subscript(s) be first F. Special notes. Observe the following: enclosed in parentheses. Examples: (ia)max, 1. When the mobilities involved are on oppo• (Shr)min' site sides of an interface, the mobility ratio 5. Except for Case 4 above, numerical sub• will be defined as the ratio of the displacing scripts should appear last in subscript order. phase mobility to the displaced phase mobil• Examples: qoD3, dimensionless oil produc• ity, or the ratio of the upstream mobility to tion rate during time period 3; PR2, reservoir the downstream mobility. pressure at time 2; (ial)max, maximum air 2. Abbreviated chemical formulas are used as injection rate during time period 1. subscripts for paraffin hydrocarbons: C 1 for 6. Except for Cases 4 and 5 above, subscript D methane, C for ethane, C for propane ... for 'dimensionless' should usually appear 2 3 Cn for Cn H 2n+2 • last in subscript order. Examples: PID; qoD; 3. Complete chemical formulas are used as (qoD3)max' subscripts for other materials: CO2 for 7. Except for Cases 4, 5 and 6 above, the carbon dioxide, CO for carbon monoxide, following subscripts should usually appear O2 for oxygen, N2 for nitrogen, etc. last in subscript order: regions such as bank, 4. The letter R is retained for electrical resistiv• burned, depleted, front, swept, unburned ity in well logging usage. The symbol P is to (b, b, d, f, s, u); separation, differential and be used in all other cases and is that flash (d, f); individual component identifica• preferred by ASA. tion (i orQI other). Examples: EDb; Rsf, npJ- 5. The letter C is retained for conductivity in D. Typography. Letter symbols for physical quan• well logging usage. The symbol (J is to be tities, and other subscripts and superscripts, used in all other cases and is that preferred whether upper case, lower case, or in small by ASA. capitals, when appearing as light-face letters of 6. Dimensions: L = length, m = mass, q = the English alphabet, are printed in italic electrical charge, t = time, and T = temper• (sloping) type. Arabic numerals, and letters or ature. other alphabets used in mathematical express• 7. Dimensionless numbers are criteria for ions, are normally printed in vertical type. geometric, kinematic and dynamic similarity When a special alphabet is required, boldface between two systems. They are derived by type is to be preferred to German, Gothic, or one of three procedures used in methods of script type. In material to be reproduced in similarity: integral, differential, or dimen• facsimile, from copy largely typewritten, letters sional. Examples of dimensionless numbers that would be boldface in print may be indicated are Reynolds number (NRe) and Prandtl to be such by special underscoring, while the number (Npr). For a discussion of methods 262 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE of similarity and dimensionless numbers, see abbreviation. All three character positions "Methods of Similarity", by R.E. Schilson, must be employed. J. Pet. Tech. (August, 1964) 877. Fixed characters are utilized in this part of 8. The quantity x can be modified to indicate the notation when heat quantities, indexes an average or mean value by an overbar, X· and exponents are being assigned computer symbols. When a heat quantity is denoted, Principles of computer symbol H appears in the first character position, as standardization exemplified by thermal conductivity HCN. A. Symbol Structure. The computer symbols are Indexes such as resistivity index are denoted structured from four possible parts representing by X in the third character position. Expo• respectively arithmetic mode, mathematical nents are characterized by XP in the second operators, basic quantities and subscripts, ex• and third positions, such as porosity expo• clusive of time and space designations. Each of nentMXP. these parts has a defined number of characters 4. The fourth part of the notation (subscript and, when all are used in a single symbol, the field) is used to represent the subscripts of total length may be ten characters. Example the mathematical letter symbol and normal• ten-character notations are: ly consists of one of the three character positions. Computer symbol subscripts are XDELPRSTQQ,XDELCMPPRD normally designated by using the mathema• When any of the four parts are not used, the tical letter subscripts of the SPE Symbols remaining characters are to be right- or left• Standard. justified to form a string of characters without Though usually not required, more char• blank positions. acters may be used when necessary for In practice, the combined notations will not designation of multiple mathematical letter usually exceed six characters. In those cases subscripts. For example, dimensionless where the complete computer symbol does average reservoir pressure would be de• exceed six characters, and the computer lan• noted by PRSA VQ. guage being used will not allow more than six, a The computer subscript designation is shortened notation must be employed. The part placed immediately to the right of the of the notation representing the basic mathema• quantity symbol field with no intervening tical quantity (letter) symbol should be retained space. and the other parts of the notation shortened. Dimensionless numbers are denoted by Q Shortened symbols are no longer standard, and in the last required subscript position. A ver• therefore must be defined in the text or appen• age, maximum, minimum, extrapolated or dix as is appropriate. limiting values of a quantity are denoted respectively by A V, MX, MN, XT, of LM in 1. The first part of the notation consists of one the first two subscript positions; additional character position to define the arithmetic subscripting occurs immediately to the right mode of the complete computer symbol. It of these defined notations. Other than in is suggested that X be used for floating point these cases, the order of subscripting should variables and I for integers. This notation follow the rules given in the 'Multiple position should be used only if absolutely Subscripts - Position Order' . necessary, the preferred approach being the 5. No binding rule is made for the notation of use of a declaration within the program. space and time subscripts, since the method 2. The second part of the notation (operator of subscripting is often dictated by the field) consists of three characters and is used characteristics of a particular computer. for mathematical operators. The notation However, the vital importance of these should suggest the operation. subscripts makes it necessary to establish a 3. The third part of the notation (quantity standard and require an author to define symbol field) consisting of three characters, any deviations. The system outlined below is used to represent the basic mathematical should be used when the subscripts are not quantity (letter) symbol. The three letter implied by an array location or an index notation mnemonically denotes the quantity specified by the program logic. name as closely as possible. This part of the The following sketch indicates the coor• computer notation is thus of the nature of an dinate system used to denote special posi- SPE NOMENCLATURE AND UNITS 263 tion in multi-dimensional arrays. measure. Authors are urged to familiarize them• 1(I = 1, 2, 3, ... ,NX) selves with the SI System of units and use them as much as practical. The choice of units (Trans. A/ME 263 (1977) 1685) and their designation is, however, left to the author. C. Restriction to computer programs. Use of the computer symbols is restricted to the description of programming for computers. As a consequ• ence, the computer symbols must not be used in works of portions of papers where programming This convention was adopted so that the is not discussed or as abbreviations in text or page position of printed output obtained in graphical material. a normal I, J, K sequence would correspond D. Character set. The computer symbols must be to position as viewed on maps as normally constructed from the 26 English letters and 10 used in petroleum engineering. Similarly, I, Arabic numerical characters. Each complete K or J, K sequences would correspond to computer symbol must begin with a letter and cross-sections as normally used. not a numeral. The space and time subscripts are con• The computer symbols are always represented structed by placing a letter code (I, J, K, T) by vertical type in printed text. English capital before the following symbols: letters and Arabic numerals are used in hand or typewritten material. Machine E. Nonstandard symbols. The rules for establishing Symbol Definition the computer symbols contained in this standard P2 present location plus 2 are such that quantities not covered can, in most P3H . present location plus 3/2 instances, be given a notation that is compatible PI present location plus I with it. Such additional computer symbols are, PIH present location plus 112 by definition, nonstandard. MIH present location minus 112 Duplication of computer symbols for quanti• MI present location minus I ties that can occur simultaneously in an equation M3H present location minus 3/2 or computer program must be avoided. Elimina• M2 present location minus 2 tion of a duplication may lead to a computer symbol that is at variance with the standard; i.e., Hence, the subscript for the present time t a notation that is nonstandard. would be T, and that for subscript t-2 would be When nonstandard computer symbols occur in TM2. a technical work, they should be clearly defined an array contains information correspond• If in the text or appendix, as is appropriate, and in ing to points halfway between the normally indexed points, then the convention is to shift the program. F. Special notes. No computer symbols have been the plus-direction elements to the node being defined here for numerical quantities, functions, indexed. and arithmetic, relational, or logical operators. In the following example, the permeability When employed in programs, their usage should at the i_lh point would be referenced as be fully explained by comments in the program PRMIPIH(I - 1), and that for the Hl/2 point text. Some of these special cases are noted would be referenced as PRMIPIH(I). See sketch below. below: 1. No computer symbols to designate common i-I Ph H liz or natural logarithms have been estab• ---(0 I 0 ---I--~ lished. Rather, these functions should be 1-1 PRMIPIH(I-l) I PRMIPIH(I) designated by the notations compatible with the computer system being employed. The B. Units. Each complete computer symbol repre• notation used should be defined in the sents a mathematical letter symbol and its associ• paper. ated subscripts. The mathematical letter symbol 2. The computer symbol for dimensionless in turn designates a physical quantity. Neither numbers in general (unnamed dimension• the complete computer symbol nor the mathema• less numbers) is NUMQ. Named dimen- tical letter symbol implies any specific units of 264 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE sionless numbers have the mnemonic title designation in Computer Symbols Subscript designation in the field representing the List. (Only changes in the basic subscripts quantity and a Q in the last subscript are shown. Combination subscripts that position employed. Thus, Reynolds number contain these items are also changed accor• is designated as REYQ. Similarly, Prandtl dingly.) number could be designated as PRDQ, 2. Quantities represented by single symbol in Grashof number as GRSQ, Graetz number SPE Letter Symbols Standard but by as GRTQ. Any dimensionless number not symbol-subscript combination in Computer contained in this standard should be defined Symbols List. in the paper. 3. No computer subscript notations corres• ponding to these mathematical letter sub• scripts are established. See section G. SPEletter Computer Quantity 4. No mathematical letter subscripts corres• symbol symbol title pond to these computer subscripts. See G GASTI total inital gas in section G. place in reservoir L MOLL moles of liquid phase G. Permissible format changes. In preparing the N NUMO dimensionless number computer symbols it became necessary to modify in general the format of certain of the basic letter symbols, N 01 LTI initial oil in place subscripts or symbol-subscript combinations. in reservoir These changes are in accord with the General u VELV volumetric velocity (flow rate or flux, Principles of Computer Symbol Standardization. per unit area) They do not imply that changes in the form of the V M0 LV moles of vapour phase economics, well logging and formation evalua• W WTRTI initial water in place tion, reservoir engineering, or natural gas en• in reservoir gineering letter symbols as contained elsewhere x MFRL mole fraction of in this SPE Standard are authorized. Rather component in liquid these changes are shown as a matter of record to phase prevent confusion and to present examples of y MFRV mole fraction of permissible format changes in the computer component in vapour symbols that may be followed when it becomes phase z MFRM mole fraction of necessary to construct a computer notation not component in included in the list. mixture 1. Basic symbolic subscripts of SPE Letter Symbols Standard represented by different 3. Quantities represented by symbol-subscript SPEletter Computer combination in SPE Letter Symbols Stan- subscript symbol Subscript title dard but by a Computer Symbol Notation c CP capillary only. D 0 dimensionless quantity Dm OM dimensionless quantity at condition m E EX experiment SPEletter ext XT extrapolated symbol• F FU fuel subscript Computer lim LM limiting value combination symbol Quantity title m FU fuel (mass of) max MX maximum HC N thermal conductivity min minimum - MN p PAV mean or average pressure 4. Symbol-subscript combinations of SPE Let• pr PRO pseudo-reduced ter Symbols Standard represented by Com• r RO reduced puter Symbol-Subscript Notation wherein tD TQ dimensionless time subscript notations are not the same. SPE NOMENCLATURE AND UNITS 265 place of the full name of a quantity, unit, or other SPEletter symbol- entity. Abbrevi{ltions are not acceptable in mathe• subscript Computer Quantity matical equations. SPE provides a list of prefer• combination symbol title red abbreviations in its 'Style Guide' for authors.

GL N G L TI initial condensate B. Computer Symbols - (for use in computer prog• liquids in place rams) - a computer symbol is a letter or group of in reservoir letters and numerals used to represent a specific GLp NGLP cumulative condensate physical or mathematical quantity in the writing liquids produced and execution of computer programs. One com• NRe REya Reynolds number puter symbol may be employed to represent a (dimensionless number) group of quantities, properly defined. Computer Rsw GWRS gas solubility in water symbols are not acceptable as substitutes for letter symbols in the required mathematical (equation• al) developments leading up to computer prog• 5. Subscripts of SPE Letter Symbols Standard rams. At the present time, all SPE computer not assigned Computer Subscript Notations symbols employ capital letters and numerals. as a result of actions noted in 4. C. Dimensions - dimensions identify the physical nature of or the general components making up a specific physical quantity; SPE employs the five SPEletter basic dimensions of mass, length, time, tempera• subscript Subscript title ture, and electrical charge (m, L, t, T, q). * liquid produced, cumulative D. Letter symbols - (for use in mathematical equa• (usually with condensate, tions) - a letter symbol is a single letter, modified GLp) when appropriate by one or more subscripts or Re Reynolds (used with Reynolds superscripts, used to represent a specific physical number only, NRe) or mathematical quantity in a mathematical sw solution in water (usually with gas solubility in water, Rsw) equation. A single letter may be employed to represent a group of quantities, properly de• fined. The same letter symbol should be used 6. Letter operator-symbol combination of consistently for the same generic quantity, or SPE Letter Symbols Standard represented special values, being indicated by subscripts or by Computer Symbol Notation only. superscripts. E. Reserve symbols - a reserve symbol is a single SPEletter Computer letter, modified when appropriate by one or symbol symbol quantity Title more subscripts or superscripts, which can be used as an alternate when two quantities (occur• T A C interval transit time ring in some specialized works) have the same standard letter symbol. These conflicts may Distinctions between, and descriptions of, result from use of standard SPE symbols or abbreviations, computer symbols, dimensions, subscript designations that are the same for two letter symbols, reserve symbols,'unit different quantities, or use of SPE symbols that abbreviations and units conflict with firmly established, commonly used notations and signs from the fields of mathema• Confusion often arises as to the proper distinctions tics, physics, and chemistry. between abbreviations, computer symbols, dimen• To avoid conflicting designations in these sions, letter symbols, reserve symbols, unit abbre• cases, use of reserve symbols, reserve subscripts, viations and units used in science and engineering. The Society of Petroleum Engineers has adhered to and reserve symbol-reserve subscript combina• the following descriptions: tions is permitted, but only in cases of symbols conflict. Author preference for the reserve sym• A. Abbreviations - (for use in textual matter, tables, bols and subscripts does not justify their use. figures, and oral discussions) - an abbreviation is In making the choice as to which of two a letter or group of letters that may be used in quantities should be given a reserve designation, * Electrical charge is current times time, ISO uses: Mass (M), Length (L), Time (T), Temperature (8), Electric current (I), Amount of substance (N) and Luminous intensity (J). 266 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE an attempt should be made to retain the standard (ISO) and many other national and international SPE symbol for the quantity appearing more bodies concerned with standardization empha• frequently in the paper; otherwise, the standard size the special character of these designations SPE symbol should be retained for the more and rigidly prescribe the manner in which the basic item (temperature, pressure, porosity, per• unit abbreviations shall be developed and meability, etc.). treated. Once a reserve designation for a quantity is G. Units - units express the system of measurement employed, it must be used consistently through• used to quantify a specific physical quantity. In out a paper. Use of an unsubscripted reserve SPE usage, units have 'abbreviations' but do not symbol for a quantity requires use of the same have 'letter symbols'. Up to this time, SPE has reserve symbol designation when subscripting is not standardized a general system of units, nor required. Reversion to the standard SPE symbol units for individual quantities; it has signified or subscript is not permitted within a paper. For willingness, however, to join in a future national larger works, such as books, consistency within a effort to convert from the English to a metric chapter or section must be maintained. system of units. The symbol nomenclature, which is a required SPE's practices showing the above distinctions part of each work, must contain each reserve are illustrated in the table of example quantities. notation that is used together with its definition. Authors can materially aid themselves, editors, F. Unit Abbreviations - a unit abbreviation is a and readers by keeping the distinctions in mind letter or group of letters (for example, cm for when preparing papers for SPE review. Manu• centimeter), or in a few cases a special sign, that scripts submitted to SPE are subject to review on may be used in place of the name of a unit. The these aspects before being accepted for publica• International Organization for Standardization tion. Examples Letter Reserve Abbrev. symbol symbol for text, for used only in Computer tables, mathe- case of symbol Unit figures, matical symbols for Dimen- abbrev. Quantity oral use equations conflict programs sions and units' gas-oil ratio, producing GaR R none GaR none cu ftlBBL gas-oil ratiO, initial Rsi none GORSI none cu ftlBBL solution, initial solution GaR productivity index PI J j POX L4Vm bid/psi productivity index, SPI Js js POXS L3t1m b/d/psilft specific * Examples only; SPE has not standardized units. Contrasting symbol usage petroleum production. These ASA symbol stan• SPE and certain American Standards Association, dards are published by the American Society of American National Standards Institute and Interna• Mechanical Engineers, United Engineering Center, tional Organization for Standardization symbols lists 345 East 47th Street, New York, NY 10017. do not use the same letter symbols to represent The Society Board of Directors has approved the identical quantities. The variations in notations SPE 1984 Symbols Standards, and recommends result from the application of the SPE guides in them to the membership and to the industry. All choosing symbols as detailed herein, the lack of authors must include Nomenclatures in any manu• agreement between various ASA standards, the script submitted to SPE for publication. ASA's policy of allowing several symbols to repre• sent the same quantity in any list and the large number of quantities assigned symbols by the SPE. Acknowledgement It is to be emphasized that the symbols contained in The work done in sorting and combining the various the SPE list are standard for use in petroleum standard lists by Schlumberger Well Services engineering, but the symbols of other disciplines as Engineering personnel in Houston, Texas and sanctioned by the American Standards Association Schlumberger-Doll Research Center personnel in should be used when working outside the area of Ridgefield, Connecticut is gratefully acknowledged. SPE NOMENCLATURE AND UNITS 267 A. Symbols alphabetized by physical quantity

Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol w z ARR Arrhenius reaction rate velocity constant L3/m k K PRM absolute permeability (fluid flow) L2 g GRV acceleration of gravity Llt2 Za MPDA acoustic impedance m/L2t v V,U VAC acoustic velocity Lit a ACT activity FaF FACAFU air/fuel ratio various ia INJA air injection rate L3/t a Fa AIR air requirement various aE FaE AIREX air requirement, unit, in laboratory experimental L3/m run, volumes of air per unit mass of pack aR FaR AIRR air requirement, unit, in reservoir, volumes of air per unit bulk volume of reservoir rock IJ-a 11a VISA air viscosity miLt mk AMAK amortization (annual write-off of unamortized M investment at end of year k) A AMP amplitude various Ac AMPC amplitude, compressional wave various Ar AMPR amplitude, relative various As AMPS amplitude, shear wave various a [J,y ANG angle e [J,y ANG angle e ad ANGD angle of dip ec roYc ANGC angle, contact w angular frequency lit Kam Mam COEANI anisotropy coefficient IR INCK annual operating cash income, over year k M Gan fGan GMFAN annulus geometrical factor (muliplier or fraction) tascript t at TACA apparent interval transit time tiL Ca Oa ECNA apparent conductivity tq2 /mL3 Pa Da DENA apparent density m/L3 rwa Rwa RADWA apparent or effective wellbore radius (includes L effects of well damage or stimulation)

Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol ASYM asymptotically equal to Pa P a PRSA atmospheric pressure m/Lt2 Z ANM atomic number A AWT atomic weight (atomic mass, relative) m a Me< COEA attenuation coefficient IlL q Q RTEAV average flow rate or production rate L3/t AV average or mean (overbar) - P p PRSAV average pressure m/Lt2 PR p PRSAVR average reservoir pressure m/Lt2 * f3d DAZ azimuth of dip /.t M RAZ azimuth of reference on sonde n NGW backpressure curve exponent, gas well C CGW backpressure curve (gas well), coefficient of L3-2nt4n/m2n n NGW backpressure curve (gas well), exponent of loga base a, logarithm f3 y BRGR bearing, relative h d,e THK bed thickness, individual L Pwj P wj PRSWF bottom hole flowing pressure m/Lt2 Pbh P BH PRSBH bottomhole pressure m/Lt2 Pwj P wj PRSWF bottom hole pressure flowing m/Lt2 Piwj P iwj PRSIWF bottomhole flowing pressure, injection well miLe P iws P iws PRSIWS bottomhole static pressure, injection well m/Lt2 Pws P ws PRSWS bottomhole pressure at any time after shut-in m/Lt2 Pw P w PRSW bottomhole pressure, general m/Lt2 Pws P ws PRSWS bottomhole pressure, static miLt 2 Pww P ww PRSWW bottomhole (well) pressure in water phase m/Lt2 Tbh 8BH TEMBH bottomhole temperature T b w WTH breadth, width, or (primarily in fracturing) L thickness Pe P e PRSE boundary pressure, external m/Lt2 Te Re RADE boundary radius, external L Bgb Fgb FVFGB bubble-point formation volume factor, gas Bob Fob FVFOB bubble-point formation volume factor, oil Ph Ps,Ps,Pb PRSB bubble-point (saturation) pressure m/Lt2 bgb !gb,Fgb RVFGB bubble-point reciprocal gas formation volume factor at bubble-point conditions V bp Vbp VOLBP bubble-point pressure, volume at L3 Rsb Fgsb GORSB bubble-point solution gas-oil ratio I::J.tws I::J.t ws DELTIMWS buildup time; shut-in time (time after well is shut in) (pressure buildup, shut-in time) Pb Db DENB bulk density m/L3 K Kb BKM bulk modulus miLe Vb Vb VOLB bulk volume L3 V bE Vbt VOLBEX bulk volume of pack burned in experimental L3 tube run !v !Vb, V bt FRCVB bulk (total) volume, fraction of V Rb VRb VOLRB burned reservoir rock, volume of L3 Vb Vb,Ub VELB burning-zone advance rate (velocity of) Lit SPE NOMENCLATURE AND UNITS 269 *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol *

*C ECQ capacitance q2e/mL2 Qv Zv CEXV capacity, cation exchange, per unit pore volume QVt ZVt CEXUT capacity, cation exchange, per unit pore volume, total Pe PoPe PRSCP capillary pressure miLe Ci INVI capital investment, initial M Ck INVK capital investment, subsequent, in year k M C C INVT capital investments, summation of all M Ppv CFLPV cash flow, discounted M P CFL cash flow, un discounted M h INCK cash income, annual operating, over year k M I INC cash income, operating M Ia INCA cash income, operating, after taxes M h INCB cash income, operating, before taxes M Pe! Pet PRSCF casing pressure, flowing m/Lt2 Pes Pes RSCS casing pressure, static m/Lt2 Qv Zv CEXV cation exchange capacity per unit pore volume QVt ZVt CEXUT cation exchange capacity per unit pore volume, total m MXP cementation (porosity) exponent (in an empirical relation between FRand heat transfer m/eT D /L,a DFN coefficient, diffusion L2/t Ke MoKee COEC coefficient, electrochemical mL2/t2q KR MRa,C COER coefficient, formation resistivity factor (FRm) K KSP coefficient in the equation of the electro- me/t2q chemical component of the SP (spontaneous electromotive force) C CGW coefficient of gas-well backpressure curve L3-2nt4n/m2n V VT, Ve HTCU coefficient heat transfer, over-all m/eT I IT, Ie HTCI coefficient, heat transfer, radiation m/eT J3 b HEC coefficient, thermal cubic expansion liT K M COE coefficient or multiplier various SL PVSL SATL combined total liquid saturation log common logarithm, base 10 npj Npj MOLPJ component j, cumulative moles produced nj Nj MOLl component j, moles of x MFRL component, mole fraction of, in liquid phase z MFRM component, mole fraction of, in mixture y MFRV component, mole fraction of, in vapour phase C ne NMBC components, number of Ee *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol zp Zp ZEDPAV compressibility factor or deviation factor for gas, at mean pressure cf kfi Kf CMPF compressibility, formation or rock Lt2/m cg kg, Kg CMPG compressibility, gas Lt2/m Co ka> Ko CMPO compressibility, oil Lt2/m Cpr kpT> Kpr CMPPRD compressibility, pseudo-reduced C w kw. Kw CMPW compressibility, water Lt2/m Ac AMPC compressional wave amplitude various C c,n CNC concentration various Cel CCl CNCCI concentration, methane (concentration of various other paraffin hydrocarbons would be indicated similarly, Cel> CC3, etc.) C O2 CO2 CNC02 concentration, oxygen (concentration various of other elements or compounds would be indicated similarly, Ceo2, CN2 , etc.) C m Cm,nm CNCFU concentration, unit fuel various (see symbol m) GL gL NGLTI condensate liquids in place in reservoir, L3 initial GLp gLp NGLP condensate liquids produced, cumulative L3 C L CVnL CNTL condensate or natural gas liquids content various 0 Y SIG conductivity (other than logging) various C 0 ECN conductivity (electrical logging) tq2/mL3 C a Oa ECNA conductivity, apparent tq2/mL3 C fD CNDFQ conductivity, fracture, dimensionless kh A HCN conductivity, thermal (always with additional mLieT phase or system subscripts) w z ARR constant, Arrhenius reaction L 3/m rate velocity constant A C LAM constant, decay (I/ed) lit E DIC constant, dielectric q2elmL3 Y constant, Euler's = 0.5772 Dc DSCC constant-income discount factor h HPC constant, hyperbolic decline q = qJ[ 1 + -j;al r R RRR constant, universal gas (per mole) mL2/t2T C WDC constant, water-drive L 4elm C L WDCL constant, water-drive, linear aquifer L 4t2/m m FF FCM consumption, fuel various mE FFE FCMEX consumption of fuel in experimental tube run mlL3 m tg FFEg FCMEXG consumption of fuel in experimental tube run m (mass of fuel per mole of produced gas) mR FFR FCMR consumption of fuel in reservoir m/L3 e c r,Yc ANGC contact angle C L CL, nL CNTL content, condensate or natural gas liquids various C wg cwg,nwg CNTWG content, wet-gas various h hh,hT HTCC convective heat transfer coefficient m/eT SPE NOMENCLATURE AND UNITS 271 *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol gc GRVC conversion factor in Newton's second law of Motion B C COR correction term or correction factor (either additive or multiplicative) N n,C NMB count rate (general) lit NN NmCN NEUN count rate, neutron lit NCR Ny,Cc NGR count rate, gamma ray lit Sgc PgoSgc SATGC critical gas saturation Pc Pc PRSC critical pressure m/Lt2 Tc 8c TEMC critical temperature T Swc Pwoswc SATWC critical water saturation A S ARA Cross-section (area) L2 I S XSTMAC cross-section, macroscopic IlL a XSTMIC cross-section, microscopic IlL a S XNL cross-section of a nucleus, microscopic L2 j3 b HEC cubic expansion coefficient, thermal lIT GLp gLp NGLP cumulative condensate liquids produced C GFp gFp GASFP cumulative free gas produced L3 Ge ge GASE cumulative gas influx (encroachment) L3 Gi gi GASI cumulative gas injected C Rp Fgp,Fgop GORP cumulative gas-oil ratio Gp gp GASP cumulative gas produced L3 npj Npj MOLPJ cumulative moles of component j produced Ne ne OILE cumulative oil influx (encroachment) L3 Np np OILP cumulative oil produced L3 Qp FLUP cumulative produced fluids (where Np and Wp are not applicable) We We WTRE cumulative water influx (encroachment) C Wi Wi WTRI cumulative water injected L3 Fwop FACWOP cumulative water-oil ratio mLlt2 Wp wp WTRP cumulative water produced L3 Gwgp gwgp GASWGP cumulative wet gas produced L3 '\Ix curl I i script i,i CUR current, electric q/t rs Rs RADS damage or stimulation radius of well (skin) L Fs Fd DMRS damage ratio or condition ratio (conditions relative to formation conditions unaffected by well operations) Z D,h ZEL datum, elevation referred to L A C LAM decay constant (lhd) lit 'td td TIMD decay time (mean life) (111..) t tdN TIMDN decay time, neutron (neutron mean life) t h HPC decline constant, hyperbolic [from equation q = q;ll[ + -j;a·t r *

* d DECE decline factor, effective a DEC decline factor, nominal 8 DCR decrement various F DGF degrees of freedom 272 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol \l DEL del (gradient operator) td 'td TIMDY delay time t D DLV deliverability (gas well) L3/t P D DEN density mlL3 Pa Da DENA density, apparent m/L3 Pb Db DENB density, bulk mlL3 Pf Df DENF density, fluid m/L3 Pxo Dxo DENXO density, flushed zone mlL3 n N NMB density (indicating 'number per unit volume') lIL3 PF DF DENFU density, fuel mlL3 3 Pg Dg DENG density, gas mlL Pma Dma DENMA density, matrix (solids, grain) mlL3 nN NMBN density (number) of neutrons I/L3 PL Ih DENAVL density of produced liquid, weight -weighted avg. m/L3 PsE DsE DENSEX density of solid particles making up m/L3 experimental pack Po Do DENO density, oil m/L3 y s, Fs SPG density, relative (specific gravity) Pt Dt DENT density, true m/L3 Pw Dw DENW density, water mlL3 DE EDE depletion NR NF FUDR deposition rate of fuel m/L3t Dp EDP depreciation D y,H DPH depth L a rs SKD depth, skin (logging) L Z Z ZED deviation factor (compressibility factor) for gas (z = p VlnR1) zp Zp ZEDPAV deviation factor (compressibility factor) for gas, at mean pressure a ANGH deviation, hole (drift angle) Pd Pd PRSD dew-point pressure m/Lt2 d D DIA diameter L dh dH,Dh DIAH diameter, hole L di dbDi DIAl diameter, invaded zone (electrically equivalent) L ap Dp DIAAVP diameter, mean particle L (0 DIC dielectric constant q2t2/mL3 A DEL difference or difference operator, finite (ax = X2-XI or X-X2) D /-L,a DFN diffusion coefficient L2/t 'YJ DFS diffusivity, hydraulic (klcpc/-L or A/cpc) L2/t QLtD QLtD script I ENCLTQQ dimensionless fluid influx function, linear aquifer QtD ENCTQQ dimensionless fluid influx function at dimensionless time tD CfD CNDFQ dimensionless fracture conductivity qgD QgD RTEGQ dimensionless gas production rate N NUMQ dimensionless number, in general (always with identifying subscripts) (Example: Reynolds number, NRe) qoD QoD RTEOQ dimensionless oil production rate SPE NOMENCLATURE AND UNITS 273 *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol VpD VpD VOLPQ dimensionless pore volume PD PD PRSQ dimensionless pressure PtD PtD PRSTQQ dimensionless pressure function at dimensionless time tD qD QD RTEQ dimensionless production rate XD dimensionless quantity proportional to x rD RD RADQ dimensionless radius tD LD TIMQ dimensionless time tDm tDm TIMMQ dimensionless time at condition m qwD QwD RTEWQ dimensionless water production rate e ad ANGD dip, angle of e a ada ANGDA dip, apparent angle of a f3da DAZA dip, apparent azimuth of f3d DAZ dip, azimuth of Dc DSCC discount factor, constant-income D DSC discount factor, general Dsp DSCSP discount factor, sin~le-payment [1/(1 + i)k; or e-] ,j = In (1 + i)] Dspc DSCSPC discount factor, single-payment (constant annual rate) [e-jk( ei - 1)/j] i RTED discount rate Ppv CFLPV discounted cash flow K d DSP dispersion coefficient L2/t ~ DSM dispersion modulus (dispersion factor) s L DIS displacement L EDb l'JDb,eDb EFFDB displacement efficiency from burned portion of in situ combustion pattern EDu l'JDweDu EFFDU displacement efficiency from unburned portion of in situ combustion pattern ED l'JD,eD EFFD displacement efficiency: volume of hydrocarbons (oil or gas) displaced from individual pores or small groups of pores divided by the volume of hydrocarbons in the same pores just prior to displacement 8 Fd DPR displacement ratio 80b Fdob DPROB displacement ratio, oil from burned volume, volume per unit volume of burned reservoir rock 80u Fdou DPROU displacement ratio, oil from unburned volume, volume per unit volume of unburned reservoir rock 8wb Fdwb DPRWB displacement ratio, water from burned volume, volume per unit volume of burned reservoir rock d L d,L2 DUW distance between adjacent rows of injection and L production wells a LmLJ DLW distance between like wells (injection or L production) in a row L s,! script I LTH distance, length, or length of path L !l.r !l.R DELRAD distance, radial (increment along radius) L 274 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol V divergence rd Rd RADD drainage radius L atwl a'twl DELTIMWF drawdown time (time after well is opened to t production) (pressure drawdown) a ANGH drift angle, hole (deviation) ir RORI earning power or rate of return (internal, true, or discounted cash flow) s S,O SKN effect, skin d DECE effective decline factor rwa Rwa RADWA effective or apparent well bore radius (includes L effects of well damage or stimulation) kg Kg PRMG effective permeability to gas L2 ko Ko PRMO effective permeability to oil L2 kw Kw PRMW effective permeability to water L2 hydrocarbon in the same pores just prior to displacement E[ 'Y)b e[ EFFI efficiency, invasion (vertical): hydrocarbon pore space invaded (affected, contacted) by the injection fluid or heat front divided by the hydrocarbon pore space enclosed in all layers behind the injected- fluid or heat front ER 'Y)R,eR EFFR efficiency, over-all reservoir recovery: volume of hydrocarbons recovered divided by volume of hydrocarbons in place at start of project (ER = Ep E[ED = Ev ED) Ep 'Y)p,ep EFFP efficiency, pattern sweep (developed from areal efficiency by proper weighting for variations in net pay thickness, porosity and hydrocarbon saturation): hydrocarbon pore space enclosed behind the injected-fluid or heat front divided by total hydrocarbon pore space of the reservoir or project EVb 'Y)Vb,eVb EFFVB efficiency, volumetric, for burned portion only, in situ combustion pattern Ev 'Y)v,ev EFFV efficiency, volumetric: product of pattern sweep and invasion efficiencies E y ELMY elasticity, modulus of (Young's modulus) m/Lt2 SPE NOMENCLATURE AND UNITS 275 *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol I i script i,i CUR electric current q/t Ze ZE,'Y] MPDE electric impedance mL2/t~2 p R RHO electrical resistivity (other than logging) mL3tq R p,r RES electrical resistivity (electrical logging) mL3tq2 'te TORE electrical tortuosity di dbDi DIAl electrically equivalent diameter of the invaded L zone Kc MoKec COEC electrochemical coefficient mL2/t2q Ec energy mL2/t2 H I HEN enthalpy (always with phase or system subscripts) me/t2 Hs Is HENS enthalpy (net) of steam or enthalpy above mL2/t2 reservoir temperature h HENS enthalpy, specific L2/t2 s a HERS· entropy, specific L2/t2T S at HER entropy, total mL2/t2T ~ GE equal to or larger than !:S LE equal to or smaller than K k,Feq EQR equilibrium ratio (ylx) di dbDi DIAl equivalent diameter (electrical) of the L invaded zone tp 'tp TIMP equivalent time well was on production prior t to shut-in (pseudo-time) Rwe RWE equivalent water resistivity mL3tq2 erf ERF error function erfc ERFC error function, complementary En Euler number y Euler's constant = 0.5772 fJ b HEC expansion coefficient, thermal cubic IIT CPE fE,tE POREX experimental pack porosity n NGW exponent of back-pressure curve, gas well m MXP exponent, porosity (cementation) (in an empirical relation between FRand cP ) n SXP exponent, saturation eZ expz EXP exponential function 276 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol -Ei (-x) exponential integral 00 *

* dt, x positive xI ~ t Ei (x) exponential integral, modified *

*'"" [r.!...- dl + ~ dl Jx positive E~ 0 _00 t E t Pe Pe PRSE external boundary pressure mlLt2 'e Re RADE external boundary radius L Pext Pext PRSXT extrapolated pressure miLe z Z ZED factor, compressibility (gas deviation factor z = PVlnRT) D DSC factor, discount d DECE factor, effective decline a DEC factor, nominal decline ge GRVC factor, conversion, in Newton's second law of Motion FR FACHR factor, formation resistivity, equals RolRw (a numerical subscript to f indicates the value of Rw) f FACF factor, friction G fa GMF factor, geometrical (multiplier) ( electrical logging) Gan fGan GMFAN factor, geometrical (multiplier) annulus (electrical logging) Gi fGi GMFI factor, geometrical (multiplier) invaded zone (electrical logging) Gp fGp GMFP factor, geometrical (multiplier) pseudo (electrical logging) Gxo fGxo GMFXO factor, geometrical (multiplier) flushed zone (electrical logging) Gm fGm GMFM factor, geometrical (multiplier) mud (electrical logging) Gt fat GMFT factor, geometrical (multiplier) true (non-invaded zone) (electrical logging) F FAC factor in general, including ratios various (always with identifying subscripts) F8 FACB factor, turbulence w m MRT flow rate, mass mit Q q,fl> HRT flow rate, heat mL2/t3 u 'tp VELV flow rate or flux, per unit area Lit (volumetric velocity) q Q RTE flow rate or production rate L3/t qp Q--p RTEPAV flow rate or production rate at mean pressure L3/t q Q RTEAV flow rate or production rate, average L3/t Piw! Piw! PRSIWF flowing bottom-hole pressure, injection well miLe Pw! pw! PRSWF flowing pressure, bottom-hole m/Lt2 Pc! Pc! PRSCF flowing pressure, casing m/Lt2 Ptf pt! PRSTF flowing pressure, tubing miLe SPE NOMENCLATURE AND UNITS 277 *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol ll.twf ll.'twf DELTIMWFflowing time after well is opened t to production (pressure drawdown) F f FLU fluid (generalized) various vf Vf,uf VACF fluid interval velocity Lit Z D,h ZEL fluid head or height or elevation referred L to a datum tfscript t ll.tf TACF fluid interval transit time tiL Pf Df DENF fluid density m/L3 QtD ENCTQQ fluid influx function, dimensionless, at dimensionless time tD QLtD QltD script I ENCLTQQ fluid influx function, linear aquifer, dimensionless Qp QltD script I FLUP fluids, cumulative produced (where Np and Wp are not applicable) Pxo Dxo DENXO flushed-zone density mlL3 Rxo Pxmrxo RESXO flushed-zone resistivity (that part of the mL3tq2 invaded zone closest to the wall of the hole, where flushing has been maximum) Gw fGxo GMFXO flushed-zone geometrical factor (fraction or multiplier) u 'tV FLX flux various u 'tV VELV flux or flow rate, per unit area Lit (volumetric velocity) F Q FCE force, mechanical mLlt2 E V EMF force, electromotive (voltage) mL2/t2q CPR fER PORR formation or reservoir porosity cf kf,Kf CMPF formation or rock compressibility Lt2/m FR FACHR formation resistivity factor - equals RoIRw (a numerical subscript to Findicates the value Rw) KR MR,a,C COER formation resistivity factor coefficient (FR m) Rt Pt,rt REST formation resistivity, true mL3tq2 Ro po,ro RESZR formation resistivity when 100% saturated mL3tq2 with water of resistivity Rw Tf 8f TEMF formation temperature T Bgb Fgb FVFGB formation volume factor at bubble-point conditions, gas Bob Fob FVFOB formation volume factor at bubble-point conditions, oil Bg Fg FVFG formation volume factor, gas Bo Fo FVFO formation volume factor, oil Bt Ft FVFT formation volume factor, total (two-phase) B F FVF formation volume factor volume at reservoir conditions divided by volume at standard conditions Bw Fw FVFW formation volume factor, water f F FRC fraction (such as the fraction of a flow stream consisting of a particular phase) fg Fg FRCG fraction gas 278 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol h F L,ft script I FRCL fraction liquid Iv Ivb, Vbf FRCVB fraction of bulk (total) volume Iq,sh fugacity miLe NGR Ny,CG NGR gamma ray count rate lit y GRY gamma ray [usually with identifying subscript( s)] various G g GAS gas (any gas, including air) always with various identifying subscripts Sog Pog,Sog SATOG gas-cap interstitial-oil saturation Swg Pwg,Swg SATWG gas-cap interstitial-water saturation cg kg, Kg CMPG gas compressibility Lt2/m z Z ZED gas compressibility factor (deviation factor) (z = PVlnRT) *

*R RRR gas constant, universal (per mole) mL2/t2T pg Dg DENG gas density m/L3 zp Zp ZEDPAV gas deviation factor (compressibility factor) at mean pressure z Z ZED gas deviation factor (compressibility factor, z = PVlnRT) (deviation factor) kg Kg PRMG gas, effective permeability to L2 Bg Fg FVFG gas formation volume factor SPE NOMENCLATURE AND UNITS 279 *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol *

*Bgb Fgb FVFGB gas formation volume factor at bubble-point conditions fg Fg FRCG gas fraction G g GASTI gas in place in reservoir, total initial L3 G e ge GASE gas influx (encroachment), cumulative L3 flGe flge DELGASE gas influx (encroachment) during an interval L3 eg ig ENCG gas influx (encroachment) rate L3/t G i gi GASI gas injected, cumulative L3 flGi flg; DELGASI gas injected during an interval L3 ig INJG gas injection rate L3/t CL cL,nL CNTL gas liquids, natural, or condensate content various Ag MOBG gas mobility Ct/m fg Fg FRCG gas, fraction fg Fg MFRTV gas mole fraction [__ V_] L+V kglko KglKo PRMGO gas-oil permeability ratio *

*Rp Fgp, Fgop GORP gas-oil ratio, cumulative RF FgFlFgoF GORF gas-oil ratio, free producing (free-gas volume/ oil volume) R Fg,Fgo GOR gas-oil ratio, producing Rsb Fgsb GORSB gas-oil ratio, solution at bubble-point conditions Rs Fgs, Fgos GORS gas-oil ratio, solution (gas solubility in oil) Rs; Fgsi GORSI gas-oil ratio, solution, initial G p gp GASP gas produced, cumulative L3 flGp flgp DELGASP gas produced during an interval L3 G pE gpE GASPEX gas produced from experimental tube run C qg Qg RTEG gas production rate L3/t qgD QgD RTEGQ gas production rate, dimensionless bg fg,Fg RVFG gas reciprocal formation volume factor bgb fgb,Fgb RVFGB gas reciprocal formation volume factor at bubble-point conditions G pa gpa GASPUL gas recovery, ultimate L3 krg Krg PRMRG gas, relative permeability to Sg pg,Sg SATG gas saturation Sgc PgoSgc SATGC gas saturation, critical Sgr PgnSgr SATGR gas saturation, residual Rs Fgs, Fgos GORS gas solubility in oil (solution gas-oil ratio) Rsw GWRS gas solubility in water Yg sg,Fgs SPGG gas specific gravity /-tg 'YJ g VISG gas viscosity miLt /-tga 'YJga VISGA gas viscosity at 1 atm miLt C CGW gas-well back-pressure curve, coefficient of L3--2nt4n/m2n n NGW gas-well back-pressure curve, exponent of D DLV gas-well deliverability L3/t G wgp gwgp GASWGP gas, wet, produced, cumulative L3 h d,e THK general and individual bed thickness L N NUMQ general dimensionless number (always with identifying subscripts) 280 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol G fa GMF geometrical factor (multiplier) ( electrical logging) Gan fGan GMFAN geometrical factor (multiplier), annulus (electrical logging) Gxo fGxo GMFXO geometrical factor (multiplier), flushed zone (electrical logging) Gi fai GMFI geometrical factor (multiplier), invaded zoned (electrical logging) Gm fGm GMFM geometrical factor (multiplier), mud (electrical logging) Gt fat GMFf geometrical factor, (multiplier), true (non-invaded zone) (electrical logging) Gp fap GMFP geometrical factor (multiplier), pseudo (electrical logging) Gt fGt GMFf geometrical factor (multiplier), true (electrical logging) g Y GRD gradient various gG gg GRDGT gradient, geothermal T \1 gradient operator gT gh GRDT gradient, temperature T Pma Dma DENMA grain (matrix, solids) density m/L3 g GRV gravity, acceleration of Llt2 Y s,Fs SPG gravity, specific, relative density Yg sg,Fgs SPGG gravity, specific, gas Yo smFos SPGO gravity, specific, oil Yw sw,Fws SPGW gravity, specific, water ht dt,et THKT gross (total) pay thickness L Vu Ru GRRU gross revenue ('value') per unit produced M/e V R, Vt,Rt GRRT gross revenue ('value'), total M t1l2 TIMH half life t Q q,cfJ HRT heat flow rate mL2/e Lv Av HLTV heat of vaporization, latent L2/t2 a a, 'YJh HTD heat or thermal diffusivity L2/t C c HSP heat, specific (always with phase or system L2/t2T subscripts) h hh,hT HTCC heat transfer coefficient, convective m/t3T U UT,Ua HTCU heat transfer coefficient, over-all mleT I In/a HTCI heat transfer coefficient, radiation mleT Z D,h ZEL height, or fluid head or elevation L referred to a datum h d,e ZHT height (other than elevation) L A SH HWF Helmholtz function (work function) mL2/t2 y f HOL hold-up (fraction of the pipe volume filled by a given fluid: Yo is oil hold-up, Yw is water hold-up ~of all hold-ups at a given level is one) 8 ANGH hole deviation, drift angle dh dH,Dh DIAH hole diameter L 'YJ DFS hydraulic diffusivity (kl c {t or AC) L2/t SPE NOMENCLATURE AND UNITS 281 *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol rH RH RADHL hydraulic radius L TH TORHL hydraulic tortuosity hydrogen index h HPC hyperbolic decline constant (from equation) q = q) [1 + -j;a·t r g (z) script I imaginary part of complex number z Z MPD impedance various Za MPDA impedance, acoustic m/L2t Ze ZE,lj MPDE impedance, electric mL2/tq2 I I --X index (use subscripts as needed) If if,/F,iF FRX index, fracture IFf iFf FFX index, free fluid IH iH HYX index, hydrogen I IJX index, injectivity L4t/m n JL RFX index of refraction Icp icp PRX index, porosity ICPI iCPI PRXPR index, primary porosity I j PDX index, productivity L4t/m IR iR RXSH index, (hydrocarbon) resistivity R/Ro Icp2 iCP2 PRXSE index, secondary porosity IshGR ishGR SHXGR index, shaliness gamma-ray (Ylog - Yen)/(Ysh - Yen) Is is IJXS index, specific injectivity Ct/m Is js PDXS index, specific productivity L3t/m h d,e THK individual bed thickness L Ge ge GASE influx (encroachment), cumulative, gas L3 Ne ne OILE influx (encroachment), cumulative, oil L3 We We WTRE influx (encroachment), cumulative, water L3 AGe Age DELGASE influx (encroachment) during an interval, gas L3 ANe Ane DELOILE influx (encroachment) during an interval, oil L3 AWe AWe DELWTRE influx (encroachment) during an interval, water L3 QLtD Q'tD script I ENCLTQQ influx function, fluid, linear aquifer, dimensionless QtD Q'tD script I ENCTQQ influx function, fluid, dimensionless (at dimensionless time tD) e ENC influx (encroachment) rate L3/t eg ig ENCG influx (encroachment) rate, gas L3/t eo io ENCO influx (encroachment) rate, oil L3/t ew iw ENCW influx (encroachment) rate, water L3/t GL gL NGLTI initial condensate liquids in place in reservoir L3 Ci INVI initial capital investment M 282 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol N n OILTI initial oil in place in reservoir L3 Pi Pi PRSI initial pressure m/Lt2 GFi gFi GASFI initial reservoir free-gas volume L3 (=mNBoJ (=GBgJ Rsi Fgsi GORSI initial solution gas-oil ratio Llt2 W W WTRTI initial water in place in reservoir L3 Swi Pwi,swi SATWI initial water saturation Gi gi GASI injected gas, cumulative L3 AGi Agi DELGASI injected gas during an interval L3 Wi Wi WTRI injected water, cumulative L3 AWi AWi DELWTRI injected water during an interval L3 INJ injection rate L3/t ia INJA injection rate, air L3/t ig INJG injection rate, gas L3/t iw INJW injection rate, water L3/t Piwj Piwj PRSIWF injection well bottom-hole pressure, flowing m/Lt2 Piws Piws PRSIWS injection well bottom-hole pressure, static m/Lt2 I i IJX injectivity index L4t/m Is is IJXS injectivity index, specific L3t/m G L gL NGLTI in-place condensate liquids in reservoir, initial L3 G g GASTI in-place gas in reservoir, total initial L3 N n OILTI in-place oil in reservoir, initial L3 W W WTRTI in-place water in reservoir, initial L3 Fwo FACWO instantaneous producing water-oil ratio b y ICP intercept various IRCE interest rate, effective compound (usually annual) iM IRPE interest rate, effective, per period j r IRA interest rate, nominal annual Pj Pj PRSF interface or front pressure M/Lt2 0 Y,Y SFT interfacial, surface tension m/t2 *

* lim t t c~ 0 [~J ~ dt + ~· dt 1,x positive _00 t E t /q,sh internal energy mele Sog POWSog SATOG interstitial-oil saturation in gas cap SPE NOMENCLATURE AND UNITS 283 *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol Swg Pwg,Swg SATWG interstitial-water saturation in gas cap Swo Swb SATWO interstitial-water saturation in oil band tscript t at TAC interval transit time tIL tascript t ata TACA interval transit time, apparent tIL M maD SAD interval transit time-density slope (absolute tL2/m value) tfscript t atf TACF interval transit time, fluid tIL tmascript t atma TACMA interval transit time, matrix tIL tsh script t atsh TACSH interval transit time, shale tIL di dbDi DIAl invaded zone diameter, electrically equivalent L Gi lGi GMFI invaded zone geometrical factor (multiplier) ( electrical logging) Ri Pbri RESI invaded zone resistivity mL3tq2 E[ l'Jb e[ EFFI invasion (vertical) efficiency: hydrocarbon pore space invaded (affected, contacted) by the injected-fluid or heat front divided by the hydrocarbon pore space enclosed in all layers behind the injected-fluid or heat front Siw Piw,Siw SATIW irreducible water saturation v N VSK kinematic viscosity L2/t Ek ENGK kinetic energy mL2/t2 :z (y) script L Laplace transform of y 00 J y (t) e-stdt 0 s Laplace transform variable \l Laplacian operator > GT larger than Lv A.v HLTV latent heat of vaporization L2/t2 L s,fscript I LTH length, path length, or distance L T t TIMAV lifetime, average (mean life) t lim LM limit CL WDCL linear aquifer water-drive constant L4t2/m h h,f script I FRCL liquid fraction h F Ltf script I MFRTL liquid mole fraction L/(L + V) x MFRL liquid phase, mole fraction of component in L nL MOLL liquid phase, moles of SL PL,SL SATL liquid saturation, combined total GL gL NGLTI liquids, condensate, in place in reservoir, initial GLp gLp NGLP liquids, condensate, produced cumulative L3 loga logarithm, base a log logarithm, common, base to Ln logarithm, natural, base e I S XSTMAC macroscopic cross section tiL a s XNL macroscopic cross section of a nucleus L2 JL m PRMM magnetic permeability mLlq2 k I( SUSM magnetic susceptibility mLlq2 M I MAG magnetization mlqt Mf MAGF magnetization, fraction 284 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol m MAS mass m w m MRT mass flow rate mit tmascript !l.tma TACMA matrix interval transit time tiL Pma Dma DENMA matrix (solids, grain) density rnIL3 Vma Vma VOLMA matrix (framework) volume (volume L3 of all formation solids except dispersed clay or shale) 1:' t TIMAV mean life (average lifetime) t 1:' t TIMD mean life (decay time) (Ill') t - p P PRSAV mean or average pressure rnILt2 AV mean or average (overbar) IL MEN mean value of a random variable a Dp DIAAVP mean particle diameter L -x MENES mean value of a random variable, x, estimated F Q FCE mechanical force mLlt2 *

*CCI cej CNCCI methane concentration (concentration of various other paraffin hydrocarbons would be indicated similarly Cc , Cc , etc.) a XSTMIC mIcroscoPIc• • cross sectIOn• 2 3 e z MFRM mixture, mole fraction of component A MOB mobility (kilL) L 3t/m Ag MOBG mobility, gas L 3t/m Ao MOBO mobility, oil L3t/m M FA MBR mobility ratio, general (Adisplacin/AdisPlaced) Ms MDd,Msu MBRSAV mobility ratio, diffuse-front approximation [(AD + Ad)SWJ/(AdLnswept]; D signifies displacing; signifies displaced; mobilities are evaluated at average saturation conditions behind and ahead of front M FA MBR mobility ratio, sharp-front approximation (AD/Ad) Mt FAt MBRT mobility ratio, total, [(At)swep/(At)unswept]; 'swept' and 'unswept' refer to invaded and uninvaded regions behind and ahead of leading edge of displacement front At A MOBT mobility, total, of all fluids in a particular region L3t/m of the reservoir, e.g., (Ao + Ag + Aw) Aw MOBW mobility, water et/m K Kb BKM modulus, bulk rnILt2 'IjJ DSM modulus, dispersion, (dispersion factor) G Es ELMS modulus, shear m/Lt2 E y ELMY modulus of elasticity (Young's modulus) m/Lt2 VM Vm VOLM molal volume (volume per mole) e Fg MFRTV mole fraction gas V/(L + V) ~ FL,J; script I MFRTL mole fraction liquid LI(L + V) x MFRL mole fraction of a component in liquid phase z MFRM mole fraction of a component in mixture SPE NOMENCLATURE AND UNITS 285 *

*Letter feserve Computer Quantity Dimensions symbol 'PE letter letter symbol symbol y MFRV mole fraction of a component in vapor phase R N MRF molecular refraction L3 M MWT molecular weight (mass, relative) m ML MWTAVL molecular weight of produced liquids, m mole-weighted average n N NMBM moles, number of nj Nj MOLJ moles of component j npj Npj MOLPJ moles of component j produced, cumulative L nL MOLL moles of liquid phase V nv MOLV moles of vapor phase nt Nt NMBMT moles, number of, total ML MWTAVL mole-weighted average molecular weight m of produced liquids Rmc Pmormc RESMC mud-cake resistivity mL3tq2 hmc dmoemc THKMC mud-cake thickness L Rmf Pmf,rmf RESMF mud-filtrate resistivity mL3tq2 Gm fGm GMFM mud geometrical factor (multiplier) ( electrical logging) Rm pm,rm RESM mud resistivity mL3tq2 G fo GMF multiplier (factor), geometrical ( electrical logging) Gan fGan GMFAN multiplier (factor), geometrical, annulus (electrical logging) Gxo fGxo GMFXO multiplier (factor), geometrical, flushed zone (electrical logging) Gi foi GMFI multiplier (factor), geometrical, invaded zone (electrical logging) Gm fGm GMFM multiplier (factor), geometrical, mud (electrical logging) Gp fGp GMFP multiplier (factor), geometrical, pseudo (electrical logging) Gt fot GMFT multiplier (factor, geometrical, true (electrical logging) K M COE multiplier or coefficient various CL cL,nL CNTL natural gas liquids or condensate content various In natural logarithm, base e hn dmen THKN net pay thickness L NN NmCN NEUN neutron count rate lit nN NMBN neutrons, density (number) of tN tN,tn NFL neutron lifetime lit N mcj>ND SND neutron porosity-density slope (absolute value) elm N NEU neutron [usually with identifying subscript(s)] various gc GRVC Newton's Second Law of Motion, conversion factor in a DEC nominal decline factor (J s XNL nucleus cross section, microscopic L2 Z ANM number, atomic N NUMQ number, dimensionless, in general (always with identifying subscripts) 286 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter Letter symbol symbol n N NMB number (of variables, or components, or steps, or increments, etc.) n N NMB number (quantity) M m NMBCP number of compounding periods (usually per year) Cn C NMBC number of components nt Nt NMBM number of moles, total N Re REYQ number, Reynolds (dimensionless number) N n OIL oil (always with identifying subscripts) various Swo Swb SATWO oil band interstitial-water saturation Co kmKO CMPO oil compressibility Lf/m Po Do DENO oil density m/L3 80b Fdob DPROB oil displaced from burned volume, volume per unit volume of burned reservoir rock 80u Fdou DPROU oil displaced from unburned volume, volume per unit volume of unburned reservoir rock ko Ko PRMO oil, effective permeability to L2 Bo Fa FVFO oil formation volume factor Bob Fob FVFOB oil formation volume factor at bubble point conditions Rs Fgs, Fgos GORS oil, gas solubility in (solution gas-oil ratio) N n OILTI oil in place in reservoir, initial L3 Ne OILE oil influx (encroachment) cumulative L3 D.Ne D.ne DELOILE oil influx (encroachment) during an interval L3 eo io ENCO oil influx (encroachment) rate L3/t Ao MOBO oil mobility et/m Np np OILP oil produced, cumulative L3 D.Np D.np DELOILP oil produced during an interval L3 qo Qo RTEO oil production rate L3/t qoD QoD RTEOQ oil production rate, dimensionless bo fmFo RVFO oil reciprocal formation volume factor (shrinkage factor) Npa npa OILPUL oil recovery, ultimate L3 kro K ro PRMRO oil, relative permeability to So Pmso SATO oil saturation Sag Pog,Sog SATOG oil saturation in gas cap, interstitial Sor PonSor SATOR oil saturation, residual Yo smFos SPGO oil specific gravity iJ-o vA VISO oil viscosity miLt f INC operating cash income M fa INCA operating cash income, after taxes M f INCB operating cash income, before taxes M 0 XPO operating expense various Ou XPOU operating expense per unit produced MlL3 \1 2 operator, Laplacian U UT,Ue HTCU over-all heat transfer coefficient mlt3T ER lJR,eR EFFR over-all reservoir recovery efficiency: volume of hydrocarbons recovered divided by volume of hydrocarbons in place at start of project (ER = EpEs Eo = Ev ED) SPE NOMENCLATURE AND UNITS 287 *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol *

*CO2 CO2 CNC02 oxygen concentration (concentration of other various elements or compounds would be indicated as, CC02,CN , etc.) e02 Eo_ 2 UTL02 oxygen utilization dp Dp DIAAVP particle diameter, mean L L s,fscript I LTH path length, length, or distance L Ep 'Y/,ep EFFP pattern sweep efficiency (developed from areal efficiency by proper weighting for variations in net pay thickness, porosity and hydrocarbon saturation: hydrocarbon pore space enclosed behind the injected-fluid or heat front divided by total hydrocarbon pore space of the reservoir or project ht dt,et THKT pay thickness, gross (total) L hn dmen THKN pay thickness, net L T e PER period t k K PRM permeability, absolute (fluid flow) L2 kg Kg PRMG permeability, effective, to gas L2 ka Ka PRMO permeability, effective, to oil L2 kw Kw PRMW permeability, effective, to water L2 I-L m PRMM permeability, magnetic mLlq2 kika KiKa PRMGO permeability ratio, gas-oil k.Jka KwlKa PRMWO permeability ratio, water-oil krg Krg PRMRG permeability, relative, to gas kra Kra PRMRO permeability, relative, to oil krw Krw PRMRW permeability, relative, to water P NMBP phases, number of I-L v,a PSN Poisson's ratio Vp vp VOLP pore volume Vb - Vs L3 VpD VpD VOLPQ pore volume, dimensionless Qi qi FLUIQ pore volumes of injected fluid, cumulative, dimensionless *

* icj> PRX porosity index Icj>l icj>l PRXPR porosity index, primary Icj>2 icj>2 PRXSE porosity index, secondary *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol f POT potential or potential function various V U VLT potential difference (electric) mL2/~Jt Ep ENGP potential energy mL2/t Pbh Pbh PRSBH pressure, bottomhole m/Lt2 P P PRS pressure mlLt2 pa Pa PRSA pressure, atmospheric m/Lt2 P P PRSAV pressure, average or mean m/Lt2 P Pd PRSAVR pressure, average, reservoir m/Lt2 Pws Pws PRSWS pressure, bottomhole, at any time after shut-in m/Lt2 Pwt Pwt PRSWF pressure, bottom hole flowing m/Lt2 Piwt Piwt PRSIWF pressure, bottom hole flowing, injection well m/Lt2 Pw Pw PRSW pressure, bottom hole general m/Lt~ Pws Pws PRSWS pressure, bottomhole static miLt Pww Pww PRSWW pressure, bottomhole (well), in water phase miLe Piws Piws PRSIWS pressure, bottomhole static, injection well m/Lt2 Pb Ps,Ps,Pb PRSB pressure, bubble-point (saturation) m/Lt2 Pe PoPe PRSCP pressure, capillary m/Lt2 Pet Pet PRSCF pressure, casing flowing m/Lt2 Pes Pes PRSCS pressure, casing static miLe Pe Pe PRSC pressure, critical miLe Pd Pd PRSD pressure, dew-point m/Lt2 PD PD PRSQ pressure, dimensionless Pe Pe PRSE pressure, external boundary m/Lt2 Pexe Pext PRSXT pressure, extrapolated m/Lt2 Pwt Pwt PRSWF pressure, flowing bottomhole mlLt2 Pet Pet PRSCF pressure, flowing casing m/Lt2 Pet Ptt PRSTF pressure, flowing tubing m/Lt2 Pt Pt PRSF pressure, front or interface miLe PeD PtD PRSTQQ pressure function, dimensionless, at dimensionless time tD Pi Pi PRSI pressure, initial m/Lt2 Ppe Ppe PRSPC pressure, pseudo-critical m/Lt2 Ppr Ppr PRSPRD pressure, pseudo-reduced miLe f!..r Pr PRSRD pressure, reduced PR P PRSAVR pressure, reservoir average m/Lt2 Psp Psp PRSSP pressure, separator m/Lt2 Pse Pse PRSSC pressure, standard conditions miLe Pws Pws PRSWS pressure, static bottom-hole m/Lt2 Pes Pes PRSCS pressure, static casing m/Lt2 Pes Pes PRSTS pressure, static tubing miLe Pt! Pet PRSTF pressure, tubing flowing m/Lt2 Pts Pes PRSTS pressure, tubing static m/Lt2 1<1>1 i*

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol GpE gpE GASPEX produced gas from experimental tube run L3 Gwgp 8Jygp GASWGP produced gas, wet, cumulative e 'ih D DENAVL produced-liquid density, weight-weighted mlL3 average npj Npj MOLPJ produced moles of component j, cumulative Np np OILP produced oil, cumulative L3 D.Np tmp DELOILP produced oil during an interval e Wp wp WTRP produced water, cumulative L3 Awp Awp DELWTRP produced water during an interval L3 Gwgp gwgp GASWGP produced wet gas, cumulative L3 R Fg,Fgo GOR producing gas-oil ratio RF FgHFgoF GORF producing gas-oil ratio, free (free-gas volume/oil volume) Fwo FACWO producing water-oil ratio, instantaneous qi Qi RTEI production rate at beginning of period L3/t qa Qa RTEA production rate at economic abandonment L3/t qD QD RTEQ production rate, dimensionless qg Qg RTEG production rate, gas L3/t qgD QgD RTEGQ production rate, gas, dimensionless qo Qo RTEO production rate, oil L3/t qoD QoD RTEOQ production rate, oil, dimensionless q Q RTE production rate or flow rate L3/t qjJ Qp RTEPAV production rate or flow rate at mean pressure L3/t q Q RTEAV production rate or flow rate, average L3/t qw Qw RTEW production rate, water L3/t qwD QwD RTEWQ production rate, water, dimensionless AtwJ A. DELTIMWFproduction time after well is opened to production (presure drawdown) tp .p TIMP production time of well, equivalent, prior to shut-in (pseudo-time) J j PDX productivity index L4t/m Pk PRAK profit, annual net, over year k M fpk PRAPK profit, annual, over year k, fraction of unamortized investment P PI PRFT profit, total M ex: proportional to Js js PDXS productivity index, specific L3t/m Tpe 8pe TEMPC pseudo-critical temperature T Ppe Ppe PRSPC pseudo-critical pressure m/U2 Gp fop GMFP pseudo-geometrical factor (multiplier) (electrical logging) cpr Kpr>Kpr CMPPRD pseudo-reduced compressibility Ppr Ppr PRSPRD pseudo-reduced pressure Epsp sp EMFP pseudo-SP mL2/qt2 Tpr 8pr TEMPRD pseudo-reduced temperature T tp .p TIMP pseudo-time (equivalent time well was on t production prior to shut-in) fs Q,x QLTS quality (usually of steam) 290 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol t:.r t:.R DELRAD radial distance (increment along radius) L I ly,Ie HCTI radiation heat transfer coefficient mleT r R RAD radius L rwa Rwa RADWA radius, apparent or effective, of well bore L (includes effects of well damage or stimulation) rD RD RADQ radius, dimensionless re Re RADE radius, external boundary L rH RH RADHL radius, hydraulic L rd Rd RADD radius of drainage L rwa Rwa RADWA radius of wellbore, apparent or effective L (includes effects of well damage or stimulation) rs Rs RADS radius of well damage or stimulation (skin) L rw Rw RADW radius, well L ia INJA rate, air injection Cit 1 k; RTE rate: discount, effective profit, of return, reinvestment, etc; use symbol i with suitable subscripts q Q RTE rate, flow or production L3/t NGR Ny,CG NGR rate, gamma ray count lit eg ig ENCG rate, gas influx (encroachment) L3/t ig INJG rate, gas injection L3/t qg Qg RTEG rate, gas production L3/t qgD QgD RTEGQ rate, gas production, dimensionless e i ENC rate, influx (encroachment) L3/t x MENES random variable, mean value of x, estimated INJ rate, injection L3/t IRCE rate, interest, effective compound (usually annual) iM IRPE rate, interest, effective, per period j r IRA rate, interest, nominal annual w m MRT rate, mass flow mit u 'lj! VELV rate of flow or flux, per unit area Lit (volumetric velocity) Q q,CI> HRT rate of heat flow mL2/t3 ir RORI rate of return (internal, true, or discounted cash flow) or earning power eo io ENCO rate, oil influx (encroachment) L3/t qo Qo RTEO rate, oil production L3/t u 'lj! VELV rate per unit area, flow (volumetric velocity) Lit qoD QoD RTEOQ rate, oil production, dimensionless q Q RTE rate, production or flow L3/t qjJ Qp RTEPAV rate, production, at mean pressure L3/t q Q RTEAV rate, production, average Cit qD QD RTEQ rate, production, dimensionless qs Qs RTES rate, segregation (in gravity drainage) L3/t y e SRT rate, shear lit Vb Vb,Ub VELB rate (velocity) of burning-zone advance Lit ew iw ENCW rate, water influx (encroachment) L3/t iw INJW rate, water injection L3/t SPE NOMENCLATURE AND UNITS 291 *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol qw Qw RTEW rate, water production L3/t qwD QwD RTEWQ rate, water production, dimensionless FaF FACAFU ratio, air-fuel various Fs Fd DMRS ratio, damage ('skin' conditions relative to formation conditions unaffected by well operations) S Fd DPR ratio, displacement Sob Fdob DPROB ratio, displacement, oil from burned volume, volume per unit volume of burned reservoir rock Sou Fdou DPROU ratio, displacement, oil from unburned volume, volume per unit volume of unburned reservoir rock Swb Fdwb DPRWB ratio, displacement, water from burned volume, volume per unit volume of burned reservoir rock K k,Feq EQR ratio, equilibrium (y/x) RF FgHFgoF GORF ratio, free producing gas-oil (free-gas volume/oil volume) Rp Fgp, Fgop GORP ratio, gas-oil, cumulative Rsi Fgsi GORSI ratio, gas-oil, initial solution kglKo Kglko PRMGO ratio, gas-oil permeability R FWFgo GOR ratio, gas-oil producing Rsb Fgsb GORSB ratio, gas-oil, solution, at bubble-point conditions Rs Fgs,Fos GORS ratio, gas-oil, solution (gas solubility in oil) M FA MBR ratio, mobility, general (AdisplacingfAdisplaced) Ms MDd,Msu MBRSAV ratio, mobility, diffuse-front approximation [(AD + Ad)swep/(Ad)unswept]; D signifies dIsplacing; d signifies displaced; mobilities are evaluated at average saturation conditions behind and ahead of front M FA MBR ratio, mobility, sharp-front approximation (AD/Ad) Mt FAt MBRT ratio, mobility, total [(At)swep/(AtLnswept]; 'swept' and 'unswept' refer to invaded and uninvaded regions behind and ahead of leading edge of a displacement front m Fpl,Fgo MGO ratio of initial reservoir free-gas volume to initial reservoir oil volume F FAC ratio or factor in general (always with identifying various subscripts) kglko KglKo PRMGO ratio, permeability, gas-oil R Fg,Fgo GOR ratio, producing gas-oil kw/ko Kw/Ko PRMWO ratio, permeability, water-oil Rsb Fgsb GORSB ratio, solution gas-oil, at bubble-point conditions Rs Fgs, Fgos GORS ratio, solution gas-oil (gas solubility in oil) Rsi Fgsi GORSI ratio, solution gas-oil, initial FwF FACWFU ratio, water-fuel Fwop FACWOP ratio, water-oil, cumulative 292 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol kw/ko Kw/Ko PRMWO ratio, water-oil permeability Fwo FACWO ratio, water-oil, producing, instantaneous X XEL reactance ML2/tq2 k r,j RRC reaction rate constant LIt !Yl (z) script R real part of complex number z b f,F RVF reciprocal formation volume factor, volume at standard conditions divided by volume at reservoir conditions (shrinkage factor) bg lFg RVFG reciprocal gas formation volume factor bgb gb,Fgb RVFGB reciprocal gas formation volume factor at bubble-point conditions j w reciprocal permeability l/L2 bo fmFo RVFO reciprocal oil formation volume fator (shrinkage factor) ER 'l']R,eR EFFR recovery efficiency, reservoir over-all; volume of hydrocarbons recovered divided by volume of hydrocarbons in place at start of project. (ER = EpE/ED = EvED) Gpa gpa GASPUL recovery, ultimate gas Pr Pr PRSRD reduced pressure Tr ar TEMRD reduced temperature a RED reduction ratio or reduction term asp REDSP reduction, SP (general) due to shaliness R N MRF refraction, molecular n JL RFX refraction index aSPsh REDSH reduction ratio, SP, due to shaliness Ar AMPR relative amplitude A AWT relative atomic mass (atomic weight) M MWT relative molecular weight (molecular weight) f3 y BRGR relative bearing y s,Fs SPG relative density (specific gravity) krg Krg PRMRG relative permeability to gas kro Kro PRMRO relative permeability to oil krw Krw PRMRW relative permeability to water t2 V2 TIMAV relaxation time, free-precession decay t tt '1:t TIMRP relaxation time, proton thermal t a Fa AIR requirement, air aE FaE AI REX requirement, unit air, in laboratory experimental L3/m run, volumes or air per unit mass of pack aR FaR AIRR requirement, unit air, in reservoir, volumes of air per unit bulk volume of reservoir rock GFi gFi GASFI reservoir initial free-gas volume (=mNBoi) L3 *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol VRu VRu VOLRU reservoir rock unburned, volume of L3 TR 9R TEMR reservoir temperature T Sgr PgnSgr SATGR residual gas saturation Shr Phnshr SATHR residual hydrocarbon saturation Sor PonSor SATOR residual oil saturation Swr PwnSwr SATWR residual water saturation r R RST resistance ML2/t~2 R p,r RES resistivity (electrical) mL3tq Ran Pam ran RESAN resistivity, annulus mL3tq2 Ra Pmra RESA resistivity, apparent mL3tq2 Rz pz,rz RESZ resistivity, apparent, of the conductive mL3tq2 fluids in an invaded zone (due to fingering) KR MR,a,C COER resistivity factor coefficient, formation mL3tq2 (FRcj>m) FR FACHR resistivity factor, formation, equals RoIRw a numerical subscript to F indicates the Rw Rxo Pxmrxo RESXO resistivity flushed zone (that part of the mL3tq2 invaded zone closest to the wall of the borehole, where flushing has been the maximum) Ro po,ro RESZR resistivity, formation 100% saturated mL3tq2 with water of resistivity Rw Rt Pt,rt REST resistivity, formation, true mL3tq2 IR iR RSXH resistivity index (hydrocarbon) equals R/Ro Ri pi,ri RESI resistivity, invaded zone mL3tq2 Rm Pm,rm RESM resistivity, mud mL3tq2 Rmc Pmormc RESMC resistivity, mud-cake mL3tq2 Rmf Pmf,rmf RESMF resistivity, mud-filtrate mL3tq2 Rsh psh,rsh RESSH resistivity, shale mL3tq2 Rs ps,rs RESS resistivity, surrounding formation mL3tq2 Rw pw,rw RESW resistivity, water mL3tq2 Vu Ru GRRU revenue, gross ('value'), per unit produced MlL3 V R, VI1 Rt GRRT revenue, gross ('value'), total M NRe REYQ Reynolds number (dimensionless number) cf kfiKf CMPF rock or formation compressibility Lt2/m C c,n CNC salinity various S p,s SAT saturation n SXP saturation exponent Sg Pg,Sg SATG saturation, gas Sgc PgoSgc SATGC saturation, gas, critical Sgr PgnSgr SATGR saturation, gas, residual Sog PogtSog SATOG saturation, interstitial-oil, in gas cap Swg Pwg,Swg SATWG saturation, interstitial-water, in gas cap Sh Ph,Sh SATH saturation, hydrocarbon Shr Phnshr SATHR saturation, residual hydrocarbon So PmSo SATO saturation, oil Sor PonSor SATOR saturation, oil, residual 294 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol Pb Ps,Ps,Pb PRSB saturation or bubble-point pressure mlLt2 SL PL,SL SATL saturation, total (combined) liquid Sw pw>sw SATW saturation, water Swc Pwoswc SATWC saturation, water, critical Swi Pwbswi SATWI saturation, water, initial Siw Piw,Siw SATIW saturation, water irreducible Swr Pw"swr SATWR saturation, water, residual I 2 i2 PRXSE secondary porosity index qs Qs RTES segregation rate (in gravity drainage) L3/t Psp Psp PRSSP separator pressure mlLt2 Ish script t Atsh TACSH shale interval transit time tiL Rsh psh,rsh RESSH shale resistivity mL3tq2 IShGR ishGR SHXGR shaliness gamma-ray index (Ylog - Ycn)/(Ysh - Yen) G Es ELMS shear modulus miLe y e SRT shear rate lit As AMPS shear wave amplitude various bo fmFo RVFO shrinkage factor (reciprocal oil formation volume factor) Pws Pws PRSWS shut-in bottomhole pressure, at any time mlLt2 Atws A'tws DELTIMWS shut-in time (time after well is shut in) t (pressure buildup) Dsp DSCSP single payment discount factor Dspc DSCSPC single payment discount factor (constant annual rate) 8 rs SKD skin depth (logging) L S S,a SKN skin effect various rs Rs RADS skin radius (radius of well damage or stimulation) L m A SLP slope various M meD SAD slope, interval transit time vs density (absolute tL2/m value) N meND SND slope, neutron porosity vs density (absolute L3/m value) < LT smaller than PsE DsE DENSEX solid particles density of experimental rock m/L3 Vs VS VOLS solid(s) volume (volume of all formation solids) L3 Pma Dma DENMA solids (matrix, grain) density m/L3 Rs Fgs, Fgos GORS solubility, gas in oil (solution gas-oil ratio) Rsw GWRS solubility, gas in water Rsb Fgsb GORSB solution gas-oil ratio at bubble-point conditions Rs Fgs, Fgos GORS solution gas-oil ratio (gas solubility in oil) Rsi Fgsi GORSI solution gas-oil ratio, initial Ec *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol Yo sOJFos SPGO specific gravity, oil Yw sw,Fws SPGW specific gravity, water C c HSP specific heat capacity (always with phase or L2/eT system subscripts) Y k HSPR specific heat capacity ratio Is is IJXS specific injectivity index L3t/m Is js PDXS specific productivity index Ct/m v VS SPY specific volume L3/m Fwv Y WGTS specific weight mL2/t2 Essp stress, normal and general m/Lt2 1: SS STSS stress, shear m/Lt2 k SUM summation (operator) u 1jJ VELV superficial phase velocity (flux rate of a Lit particular fluid phase flowing in pipe; use appropriate phase subscripts) qse qmQse RTESC surface production rate L3/t (J Y,Y SFT surface tension, interfacial m/t2 Rs ps,rs RESS surrounding formation resistivity mL3t~2 k K SUSM susceptibility, magnetic mLlq T 8 TEM temperature T Tbh 8BH TEMBH temperature, bottomhole T Te 8e TEMC temperature, critical T Tf 8f TEMF temperature, formation T gT gh GRDT temperature gradient TIL Tpe 8pe TEMPC temperature, pseudo-critical T Tpr 8pr TEMPRD temperature, pseudo-reduced T Tr 8r TEMRD temperature, reduced T TR 8R TEMR temperature, reservoir T Tse 8se TEMSC temperature, standard conditions T (J Y,Y SFT tension, surface (interfacial) m/e X tensorofx kh A HCN thermal conductivity (always with additional mLlt3T phase or system subscripts) f3 b HEC thermal cubic expansion coefficient liT a a,'Yjh HTD thermal or heat diffusivity L2/t h d,e THK thickness (general and individual bed) L ht doet THKT thickness, gross pay (total) L 296 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol hmc dmoemc THKMC thickness, mud-cake L hI dt,et THKT thickness, pay, gross (total) L hn dmen THKN thickness, net pay L t 't TIM time t Atwf i>'twf DELTIMWFtime after well is opened to production t (pressure drawdown) Atws A'tws DELTIMWS time after well is shut in (pressure build-up) t 't 'tc TIMC time constant t 'td td TIMD time, decay (mean life) (111..) t td 'td TIMD time, delay t At A't DELTIM time difference t (time period or interval, fixed length) tD 'tD TIMQ time, dimensionless tDm 'tDm TIMMQ time, dimensionless at condition m ts 'ts TIMS time for stabilization of a well t rscript t At TAC time, interval transit tiL ta script t Ata TACA time, interval transit, apparent tiL ryscript t Atf TACF time, interval transit, fluid tIL tina script t Atma TACMA time, interval transit, matrix tIL Tsh script t Atsh TACSH time, interval transit, shale tIL tdN TIMDN time, neutron decay (neutron mean life) t 'tp,tpo TIMPO time, pay-out (pay-off, pay-back) t !t A't DELTIM time period or interval, fixed length t tp 'tp TIMP time well was on production prior to shut-in, t equivalent (pseudo-time) 't TOR tortuosity 'te TORE tortuosity, electric 'tH TORHL tortuosity, hydraulic SL PL,SL SATL total (combined) liquid saturation S HER total entropy L2/t2T At A MOBT total mobility of all fluids in a particular region L3t/m ofthe reservoir, e.g., (1"0 + I..g + I..w) Mt Ft..t MBRT total mobility ratio [(I..t)swep/(I..t)unsweptl; 'swept' and 'unswept' refer to invaded and uninvaded regions behind and ahead of leading edge of a displacement front ht dt,et THKT total (gross) pay thickness L V R, Vt,Rt GRRT total gross revenue ('value') M G g GASTI total initial gas in place in reservoir L3 n nt, Nt NMBM total moles CPt ft,Et PORT total porosity Bt Ft FVFT total (two-phase) formation volume factor h hh,hT HTCC transfer coefficient, convective heat rnIeT U UT,Ue HTCU transfer coefficient, heat, over-all rnIt3T I Ir,ls HTCI transfer coefficient, heat, radiation rnIt3T tscript t At TAC transit time, interval tIL ta script t Ata TACA transit time, apparent, interval tIL ryscript t Atf TACF transit time, fluid interval tIL tina script t Atma TACMA transit time, matrix interval tIL Tsh script t Atsh TACSH transit time, shale interval tIL 00 ::z (y) script L transform, Laplace of y Jy (t)e-stdt 0 SPE NOMENCLATURE AND UNITS 297 *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol s transform, Laplace, variable T T TRM transmissivity, transmissibility various Pt Dt DENT true density mlL3 Rt Pt,rt REST true formation resistivity mL3tq2 Gt fat GMFf true geometrical factor (multiplier) (non-invaded zone) (electrical logging) Ptt Ptt PRSTF tubing pressure, flowing mlLt2 Pts Pts PRSTS tubing pressure, static mlLt2 FB FACB turbulence factor Bt Ft FVFf two-phase or total formation volume factor Gpa gpa GASPUL ultimate gas recovery L3 Cuk INVUK unamortized investment over year k P un discounted cash flow M VRu VRu VOLRU unburned reservoir rock, volume of L3 aE FaE AIREX unit air requirement in laboratory experimental L3/m run, volumes of air per unit mass of pack aR FaR AIRR unit air requirement in reservoir, volumes of air per bulk volume of reservoir rock Cm cm,nm CNCFU unit fuel concentration (see symbol m) various R RRR universal gas constant (per mole) mL2/t2T e02 E02 UTL02 utilization, oxygen z VAL valence y MFRV vapour phase, mole fraction of component V MOLV vapour phase, moles of L{ Av HLTV vaporization, latent heat of L2/t2 0 VAR variance of a random variable S2 VARES variance of a random variable, estimated x vectorofx v V,u VEL velocity Lit v V,u VAC velocity, acoustic Lit Va VmUa VACA velocity, acoustic apparent (measured) Lit Vt Vt,Ut VACF velocity, acoustic fluid Lit Vma Vma,uma VACMA velocity, matrix acoustic Lit Vsh Vsh,Ush VACSH velocity, shale acoustic Lit Vb Vb,Ub VELB velocity (rate) of burning-zone advance Lit El 'YJbel EFFI vertical (invasion) efficiency: hydrocarbon pore space invaded (affected, contacted) by the injected-fluid or heat front divided by the hydrocarbon pore space enclosed in all layers behind the injected-fluid or heat front /La 'YJa VISA viscosity, air miLt /LjJ 'YJjJ VISPAV viscosity at mean pressure miLt /L 'YJ VIS viscosity, dynamic miLt /Lg 'YJg VISG viscosity, gas miLt /Lga 'YJga VISGA viscosity, gas, at 1 atm miLt v N VSK viscosity, kinematic L2/t /La 'YJa VISO viscosity, oil miLt /Lw 'YJw VISW viscosity, water miLt V v VOL volume L3 Vbp Vbp VOLBP volume at bubble-point pressure L3 Vb Vb VOLB volume, bulk L3 298 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol VbE VbE VOLBEX volume, bulk, of pack burned in L3 experimental run Ve Vpe, Ve VOLG volume, effective pore L3 V iv,Fv VLF volume fraction or ratio (as needed, use same various subscripted symbols as for 'volumes'; note that bulk volume fraction is unity and pore volume fractions are <1>]) GFi gFi GASFI volume, free-gas, initial reservoir L3 (=mNho) Vgr Vgr VOLGR volume, grain (volume of all formation solids L3 except shales) Vig Vig VOLIG volume, intergranular (volume between grains; L3 consists of fluids and all shales) (Vb-Vgr) Vim Vim VOLIM volume, intermatrix (consists of fluids and L3 dispersed shale) (Vb - V rna) Vma Vma VOLMA volume, matrix (framework) (volume of all formation solids except dispersed shale) Vne Vpne' Vne VOLNE volume, noneffective pore (Vp - Ve) L3 VRb VOLRB volume of reservoir rock burned L3 VRu VOLRU volume of reservoir rock unburned C VM VOLM volume per mole (molal volume) L3 Vp vp VOLP volume, pore (Vb - Vs) L3 VpD VpD VOLPQ volume, pore, dimensionless Vshd Vshd VOLSHD volume, shale, dispersed L3 Vshl'script I Vshi script I VSHLAM volume, shale, laminated L3 Vshs Vshs VOLSHS volume, shale, structural L3 Vsh Vsh VOLSH volume, shale(s) (volume of all shales: C structural and dispersed) Vs VS VOLS volume, solid(s) (volume of all formation L3 solids) v VS SPY volume, specific Clm EVb 'YJVb,eVb EFFVB volumetric efficiency for burned portion only, in situ combustion pattern Ev 'YJv,ev EFFV volumetric efficiency: product of pattern sweep and invasion efficiencies q Q RTE volumetric flow rate Cit qdh qw/,qDH,Qdh RTEDH volumetric flow rate downhole L3/t qsc qmQsc RTESC volumetric flow rate, surface conditions Cit M HSPV volumetric heat capacity mlLeT u 'P VELV volumetric velocity (flow rate or flux, Lit per unit area) W w WTR water (always with identifying subscripts) various Cw kw,Kw CMPW water compressibility Lt2/m Pw Dw DENW water density mlL3 Owb FWb DPRWB water displaced from burned volume, volume per unit volume of burned reservoir rock C WDC water-drive constant L4elm 4 CL WDCL water-drive constant, linear aquifer L elm kw Kw PRMW water, effective permeability to L2 SPE NOMENCLATURE AND UNITS 299 *

*Letter Reserve Computer Quantity Dimensions symbol SPE letter letter symbol symbol Bw Fw FVFW water formation volume factor FwF FACWFU water-fuel ratio various Rsw GWRS water, gas solubility in W W WTRTI water in place in reservoir, initial L3 We We WTRE water influx (encroachment), cumulative L3 J1We J1we DELWTRE water influx (encroachment) during an interval L3 ew iw ENCW water influx (encroachment) rate L3/t Wi Wi WTRI water injected, cumulative L3 J1Wi <1 Wi DELWTRI water injected during an interval L3 iw INJW water injection rate L3/t Aw MOBW water mobility L3t/m kwlko KwlKo PRMWO water-oil permeability ratio Fwop FACWOP water-oil ratio, cumulative Fwo FACWO water-oil ratio, producing, instantaneous Wp wp WTRP water produced, cumulative L3 J1Wp J1wp DELWTRP water produced during an interval L3 qw Qw RTEW water production rate L3/t qwD QwD RTEWQ water production rate, dimensionless krw Krw PRMRW water, relative permeability to Rw pw,rw RESW water resistivity mL3tq2 Sw Pw,sw SATW water saturation Swc Pwoswc SATWC water saturation, critical Swi Pwi,Swi SATWI water saturation, initial Swo Swb SATWO water saturation (interstitial) in oil band Swg Pwg,Swg SATWG water saturation in gas cap, interstitial Siw Piw,Siw SATIW water saturation, irreducible Swr PwnSwr SATWR water saturation, residual Yw Sw.Fws SPGW water specific gravity ILw 1']w VISW water viscosity mILt A WVL wave length (I/o) L 0 v WVN wave number (III...) I/L W w,G WGT weight (gravitational) m/Lt2 3 fh 15L DENAVL weight-weighted average density mlL of produced liquid A AWT weight, atomic m M MWT weight, molecular m rw Rw RADW well radius L rs Rs RADS well radius of damage or stimulation (skin) L Is 1: TIMS well stabilization time t rwa Rwa RADWA wellbore radius, effective or apparent (includes L effects of well damage or stimulation Cwg cwg,nwg CNTWG wet-gas content various Gwgp gwgp GASWGP wet gas produced, cumulative L3 b W WTH width, breadth, or (primarily in fracturing) L thickness W W WRK work mL2/t2 E y ELMY Young's modulus (modulus of elasticity) mlLt2 di dbDi DIAl zone diameter, invaded, electrically equivalent L Ri Pbri RESI zone resistivity, invaded mL3tq2 300 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*B. Subscripts alphabetized by physical quantity *

*Subscript de£mition Letter ReserveSPE Computer subscript subscript letter subscript abandonment a A A acoustic a A, «alpha A activation log, neutron NA na NA active, activity, or acting a A after taxes a A air a A A air-fuel aF AFU altered a A amplitude log A a A angle, angular, or angular coordinate () theta THE anhydrite anh AH anisotropic ani ANI annulus apparent (from log readings; an AN AN use tool description subscripts) apparent (general) a ap A apparent wellbore (usually with wellbore radius) wa WA areal A A atmosphere, atmospheric a A A average or mean pressure 2- PAY average or mean saturation S s,p rho SAY band or oil band b B B bank or bank region b B base b r, f3 beta B before taxes b B B bond log, cement CB cb CB borehole televiewer log TV tv TV bottom hole bh w,BH BH bottom-hole, flowing (usually with pressure or time) wi WF bottom-hole, static (usually with pressure or time) ws WS boundary conditions, external e 0 E breakthrough BT bt BT bubble b B bubble-point conditions, oil at (usually with ob OB formation volume factor, Bob) bubble-point conditions, solution at (usually sb SB with gas-oil ratio, Rsb ) bubble point (saturation) b s,bp B bubble-point or saturation (usually with bp B volume, Vbp ) bulk (usually with volume Vb) b B,t B burned in experimental tube run (usually bE BEX with volume, VbE) burned or burning b B B burned portion of in situ combustion pattern, displacement Db DB from (usually with efficiency, E Db) burned portion of in situ combustion pattern, volumetric Vb VB of (usually with efficiency, Evb ) burned reservoir rock Rb RB SPE NOMENCLATURE AND UNITS 301 *

*Subscript definition Letter ReserveSPE Computer subscript subscript letter subscript burned volume, oil from (usually ob OB with displacement ratio, Oob) burned volume water from (usually wb WB with displacement ratio, Owb) calculated C calc CA caliper log C c C capillary (usually with capillary pressure, Pc) c C CP capture cap C carbon dioxide CO2 CO2 carbon monoxide CO CO casing or casinghead c cg CS casing, tlowing (usually with pressure) cf CF casing, static (usually with pressure) cs CS cement bond log CB cb CB chemical c C chlorine log CL cl CL clay cl cla CL clean en cln CN coil C c C compaction cp CP compensated density log CD cd CD compensated neutron log CN en CN component(s) C C componentj j J component j produced pj PJ (usually with moles, npj) compressional wave c C C conditions for infinite dimensions 00 INF INF conductive liquids in invaded zone z Z constant c C C contact c C C (usually with contact angle, 8c) contact log, microlog, minilog ML mlscript I ML convective C conversion (usually with conversion factor in c C Newton's law of motion, gc) core c C C corrected cor COR critical c cr CR cumulative intlux (encroachment) e i E cumulative injected i I cumulative produced p P cumulative produced free value Fp FP (usually with gas, GFp ) cumulative produced liquid Lp (usually with condensate, GLp) damage or damaged (includes 'skin' conditions) s d S decay d D deep induction log ID id ID deep laterolog LLD Il'd script II LLD delay d odelta D 302 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Subscriptdeflnition Letter ReserveSPE Computer subscript . subscript letter subscript density prho RHO density log, compensated CD cd CD density log D d D depleted region, depletion d 6 delta D dew-point d D differential separation d D differential temperature log DT dt DT diffusivity 'YJ eta ETA dimensionless pore value pD PQ (usually with volume VpD) dimensionless quantity D Q dimensionless quantity at condition m Dm QM dimensionless time tD TQ dimensionless water wD WQ dip (usually with angle, ad) d D diplog, dipmeter DM dm DM directional survey DR dr DR dirty (clayey, shaly) dy dty DY discounted value, present worth, or present value PV pv PV dispersed d D D dispersion K d K displaced d s,D DD displacement from burned portion of in situ Db DB combustion pattern (usually with efficiency, EDb) displacement from unburned portion of in situ Du DU combustion pattern (usually with efficiency, EDu) displacing or displacement (efficiency) D s, (J sigma DN dolomite dol DL down-hole dh DH DH drainage (usually with drainage radius, rd) d D dual induction log DI di DI duallaterolog DLL dll'script II DLL earth e E E effective (or equivalent) e E electric, electrical e E E electrochemical c ec C electrode E e E electrokinetic k ek K electrolog, electrical log, electrical EL el, ES EL survey electromagnetic pipe inspection log EP ep EP electron el e/script el E empirical E EM EM encroachment (influx), cumulative e i E entry e E E epithermal neutron log NE ne NE eqivalent eq BV EV estimated E est ES ethane C2 C2 experimental E EX EX SPE NOMENCLATURE AND UNITS 303 *

*Subscript definition Letter ReserveSPE Computer subscript subscript letter subscript experimental value per mole of produced gas Eg EXG (usually with fuel consumption, mEg) external, outer boundary conditions e 0 E extrapolated ext XT fast neutron log NF nf NF fill-up F f F finger or fingering f F F flash separation f F F flowing bottom-hole (usually with pressure or time) wf WF flowing casing (usually with pressure) cf CF flowing conditions, injection well (usually with iwf IWF pressure, Piwf) flowing conditions, well (usually with time) wf f WF flowing tubing (usually with pressure) if TF fluid f fl F fluids in an invaded zone, conductive z Z flushed zone xo XO formation 100% saturated with ozero 7ZR water (used in Ro only) formation (rock) f fm F formation, surrounding s S fraction or fractional f r F fracture, fractured or fracturing f F FR free (usually with gas or gas-oil ratio quantities) F f F free fluid Ff f FF free value, cumulative produced, Fp FP (usually with gas, GFp ) free value, initial (usually with gas, G n) Fi FI front, front region, or interface f F F fuel, mass of (usually with fuel concentration, em) m FU fuel (usually with fuel properties, such as PF) F FU gamma-gamma ray log GG gg GG gamma ray log GR gr GR gas g G G gas at atmospheric conditions ga GA gas at bubble-point conditions gb GB gas cap, oil in (usually with saturation, Sag) og OG gas cap, water in (usually with saturation, Swg) wg WG gas, dimensionless gD GQ gas-oil, solution (usually with gas-oil ratios) s S gas-water, solution sw (usually with gas solubility in water, Rsw) geometrical G G geothermal G T GT grain gr GR grain (matrix, solids) ma MA gravity meter log GM gm GM gross (total) t T T guard log G g G gypsum gyp GY half 112 H 304 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Subscriptdeflnition Letter ReserveSPE Computer subscript subscript letter subscript heat or thermal h T, e theta HT heavy phase HP hp HP hole h H H horizontal H h H hydraulic H HL hydrocarbon h H H hydrogen nuclei or atoms H HY hydrocarbon, residual hr HR hydrogen sulphide H2S H2S imbibition I i script i I induction log, deep investigation 1D id ID induction log I i I induction log, dual D1 di DI induction log, medium investigation 1M im 1M infinite dimensions, conditions for 00 INF influx (encroachment), cumulative e E initial conditions or value i I initial free value (usually with gas, GFi) Fi PI initial solution (usually with gas-oil ratio, Rsi) si SI initial value or conditions I injected, cumulative I I injection, injected or injecting i inj I injection well, flowing conditions (usually with pressure, iwf IWF Piw/) injection well, static conditions (usually with pressure, iws IWS Piws) inner or interior i l iota, t script i I interface, front region, or front f F F interference I i, t script i I intergranular ig IG intermatrix 1m 1M internal i l iota, { script i I intrinsic int I invaded I I invaded zone i I I invaded zone, conductive liquids in an z Z invasion (usually with invasion efficiency, E /) I I irreducible ir, l iota, t script i IR jth component j J jth component, produced pj PJ junction j J laminar ('script 1 L LAM laminated, lamination ('script L L LAM lateral (resistivity) log L ('script 1 L laterolog (add further tool configuration LL If script II LL subscripts as needed) laterolog, dual DLL d Il"script II DLL lifetime log, neutron, TDT PNL n i'script 1 PNL light phase LP i'p script 1 LP limestone Is 1st LS limiting value lim LM SPE NOMENCLATURE AND UNITS 305 *

*SubscriptderlOition Letter ReserveSPE Computer subscript su!'script Htter subscript linear, lineal L tscript I L liquid or liquid phase L tscript I L liquids, conductive, invaded zone Z Z liquid produced, cumulative (usually with Lp condensate G Lp) location subscripts, usage is secondary to that for 1,2,3, etc. representing times or time periods log LOG log L lower tscript I L L magnetism log, nuclear NM nm NM mass of fuel (usually with fuel concentration, m FU Cm) matrix (solids, grain) ma MA matrix [solids, except (nonstructural) ma MA clay or shale] maximum max MX mean or average pressure PAY mean or average saturation ~ S, prho SAY medium investigation induction log 1M im 1M methane C1 Cl microlaterolog MLL md'script II MLL microlog, minilog, contact log ML mtscriptl ML micro-seismogram log, signature log, variable VD vd VD density log minimum min MN mixture M z,m M mobility Alambda M LAM molal (usually with volume, V M) M M Mth period or interval M m M mud m M mud cake me MC mud filtrate mf MF net n N neutron N n N neutron activation log NA na NA neutron lifetime log, TDT PNL ntscript I PNL neutron log, compensated CN en CN neutron log N n N neutron log, epithermal NE ne NE neutron log, fast NF nf NF neutron log, sidewall SN sn SN neutron log, thermal NT nt NT nitrogen N2 N2 noneffective ne NE nonwetting nw NW NW normal n N normal (resistivity) log N n N (add numerical spacing to subscript to N; e.g., N16) normalized (fractional or relative) n r,R N nth year, period, income, payment, or unit n N N nuclear magnetism log NM nm NM 306 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Subscript definition Letter ReserveSPE Computer subscript subscript letter subscript numerical subscripts (intended primarily 1,2,3, etc. to represent times or time periods; available secondarily as location subscripts or for other purposes) observed DB OB oil at bubble-point conditions (usually with formation ob OB volume factor, Bob) oil, dimensionless oD 00 oil 0 N,n 0 oil from burned volume (usually with displacement ob OB ratio, Sob) oil from unburned volume (usually with displacement ou OU ratio, Sou) oil in gas cap (usually with saturation, Sog) og OG outer (external) e 0 E oxygen O2 02 particle (usually with diameter, dp) p P particular period, element, or interval k K K pattern (usually with pattern efficiency, Ep) P P pay-out, pay-off, or pay-back p po PO permeability k K K phase or phases P P pipe inspection log, electromagnetic EP ep EP pore (usually with volume, Vp) p P P pore value, dimensionless (usually with volume, pD PO VpD ) porosity phi f, E epsilon PHI porosity data phi j, E epsilon P pressure, mean or average p PAY primary 10ne p,pri PR produced p P P produced component j (usually with moles, npj) pj Pl produced, cumulative p P produced free value, cumulative Fp FP (usually with gas, GFp ) produced in experiment pE PEX produced liquid, cumulative Lp (usually with condensate, G Lp) produced water-oil (cumulative) wop WOP (usually with cumulative water-oil ratio, Fwop) production period (usually with time, tp) p P P profit - unamortized investment Pk PK proximity log P p P pseudo p P pseudo-critical pc PC pseudo-dimensionless pD PO pseudo-reduced pr PRD pseudo-SP pSP PSP radial r R R radius, radial, or radial distance r R R rate of return r R R SPE NOMENCLATURE AND UNITS 307 *

*Subscriptdeflnition Letter ReserveSPE Computer subscript subscript letter subscript recovery (usually with recovery efficiency, R R ER ) reduced r RD reference r b, prho R relative r R R reservoir R r R reservoir rock, burned Rb RB reservoir rock, unburned Ru RU residual r R R residual hydrocarbon hr HR resistivity R R resistivity log R r, p rho R Reynolds (used with Reynolds number Re only, NRe) rock (formation) f fm F sand sd sa SD sandstone ss sst SS saturation, mean or average S 5, prho SAY saturation or bubble point b s B saturation or bubble point (usually with bp BP volume, Vbp ) scattered, scattering sc SC secondary 2 two s,sec SE segregation (usually with segregation s S, a sigma S rate, qs) separator conditions sp SP shale sh sha SH shallow laterolog LLS It s script II LLS shear s 1: tau shear wave s 1: tau S sidewall S SW SW sidewall neutron log SN sn SN signature log, micro-seismogram log, VD vd VD variable density log silt sl sit SL single payment sp SP skin (stimulation or damage) s S S slip or slippage s a sigma S slurry (,mixture') M z,m M solid( s) (all formation solids) s a sigma S solids in experiment sE SEX solids (matrix, grain) ma MA solution at bubble-point conditions (usually with sb SB gas-oil ratio, Rsb ) solution in water (usually with gas solubility sw in water, Rsw) solution, initial (usually with gas-oil si SI ratio, Rsi ) solution (usually with gas-oil ratios) s S sonde, tool T t T sonic velocity log SV sv SV 308 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Subscriptdeflnition Letter ReserveSPE Computer subscript subscript letter subscript SP SP sp SP spacing s L specific (usually with J and l) s S SSP SSP SSP stabilization (usually with time) s S S standard conditions sc (J sigma SC static bottom-hole (usually with pressure or time) ws WS static casing (usually with pressure) cs CS static conditions, injection well (usually with pressure) iws IWS static or shut -in conditions (usually with time) ws s WS static tubing (usually with pressure) ts TS static well conditions (usually with time) ws s WS steam or steam zone s S S stimulation (includes 'skin' conditions) s S S stock-tank conditions st ST storage or storage capacity S s, (J sigma S strain £ epsilon e EPS structural st s ST surface s (J sigma S surrounding formation s S swept or swept region s S, (J sigma S system s (J sigma S TDT log, neutron lifetime log PNL pnfscript I PNL televiewer log, borehole TV tv TV temperature T h, 8 theta T temperature log T t,h T temperature log, differential DT dt DT thermal (heat) h T, 8 theta HT thermal decay time (TDT) log PNL pnfscript I PNL thermal neutron log NT nt NT time, dimensionless tD TQ times or time periods 1,2,3, etc. tool-description subscripts: see individual entries such as 'amplitude log', 'neutron log,' , etc. tool, sonde T t T total initial in place in reservoir ti TI total (gross) t T T total, total system t T T transmissibility T t T treatment or treating 1: tau T true (opposed to apparent) t tr T tubing flowing (usually with pressure) if TF tubing or tubinghead t tg T tubing, static (usually with pressure) ts TS turbulence (used with Fonly, FB ) B B ultimate a ul UL unamortized u U U unburned u U unburned portion of in situ combustion pattern Du DU displacement from (usually with efficiency, E Du) unburned reservoir rock Ru RU SPE NOMENCLATURE AND UNITS 309 *

*Subscript deDnition Letter ReserveSPE Computer subscript sub~cript letter subscript unburned volume, oil from (usually with ou OU *

* displacement ratio, 60u) unit u U U unswept or unswept region u U U upper u U U vaporization, vapour, or vapour phase v V V variable density log, micro-seismogram log, VD vd VD signature log velocity v V V velocity, sonic or acoustic log SV sv SV vertical V v V volumetric of burned portion of in situ combustion Vb VB pattern (usually with efficiency, Evb) volume or volumetric V v V water w W W water, dimensionless wD WQ water from burned volume (usually with displacement wb WB ratio,6wb) water-fuel wF WFU water in gas cap (usually with saturation, Swg) wg WG water-oil (usually with instantaneous producing wo WO water-oil ratio, Fwo) water-oil produced (cumulative) wop WOP (usually with cumulative water-oil ratio, Fwop) water, solution in (usually with gas solubility sw SW in water, Rsw) water-saturated formation, 100% ozero zr ZR weight W w W well conditions w W well, flowing conditions (usually with time) wI I WF well, static conditions (usually with time) ws s WS well, injection, flowing conditions iwl IWF (usually with pressure Piw/) well, injection, static conditions iws IWS (usually with pressure Piws) well, static conditions (usually with time) ws WS wellbore, apparent (usually with wellbore wa WA radius, rwa) wellhead wh th WH wet gas (usually with composition or content, wg WG Cwg) wet gas produced wgp WGP wetting w W W Young's modulus, refers to Y Y zero hydrocarbon saturation ozero zr ZR zone, conductive fluids in an invaded z Z zone, flushed xo XO zone, invaded I I Appendix 2 Solutions to Examples *

*Chapter 2 *

*Solution 2.1 Although this problem should place probabilistic ranges on the given data and assumptions, it will be calculated deterministically. We will assume that the combination of oil expelled from source rocks and trapped in potential structures represents some 8% of the converted source rocks, i.e.: Oil converted for source rock = 5 x 4500 x 12 x 106 m3 Trapped oil ( = OIP) = 0.085 x 4500 x 12 x 106 m3 = 2.16 x 1010 m3 *

*Assuming an average formation volume factor of 1.4 rm3/sm 3 this yields a stock tank oil in place of 1.54 x 1010 sm3. For an assumed overall technical recovery factor of 0.35 this yields a recoverable reserve of 1.54 x 1010 x 0.35 = 5.4 x 109 sm3 (This is equivalent to 34 x 109 STB.) (N.B. The UK Government's 1983 'Brown Book' indicates a probable range of technically recoverable reserves between 11 and 23 x 109 STB, assuming an oil formation volume factor of 1.4 rm3/sm 3 .) Chapter 3 *

*Solution 3.1 Casing Design Example (a) The buoyancy factor (BF) is given by SGsteel- SGfluid BF=----- SGsteel For the external system: 7.84 - 1.92 BF = = 0.755 7.84 and for the internal fluid system: 7.84 - 1.15 BF = 7.84 = 0.853 SOLUTIONS TO EXAMPLES 311 *

*The neutral point (NP) is thus the depth at which the string above is in tension and below in compression. NP = 13000 x BF = 13000 x 0.755 = 9820 ft This is rounded off to 9800 ft. *

*(b) For the design weight of casing (CWT) we have CWT = weight in air x BR where the buoyancy ratio BR is given by BF for outside mud system 0.755 BR = = --= 0 885 BF for internal fluid system 0.853 . *

*(c) In the lower section we can check criteria: (i) Collapse The external mud gradient is SG x 0.433 psi/ft = 1.92 x 0.433 = 0.831 psi/ft The collapse limit of the P-110 casing of the various weights is given from Table A3.1 as 9570 0.831 = 11520 ft for 20 ppf casing, and *

*11630 0.831 = 14000 ft for 23 ppf casing *

*:. Use 23 ppf casing from bottom to 11520 ft, that is (13000 - 11520) = 1480 ft (NB no tension problem since neutral point is at 9800 ft.) (ii) Burst check Since a more dense mud is used outside the casing then the greatest internal:external pressure difference is at the top of each section. At 11 520 ft, internal differential is: (max surface pressure) + (internal fluid head) - (external fluid head) Internal pressure gradient = (SG x 0.433) = 1.15 x 0.433 = 0.498 psi/ft :.8000 + 11520 [0.498 - 0.831] = 4164 psi As burst pressure of 23 ppf casing is given as 11780 psi no problem arises. (iii) Joint strength calculation check Since the entire section is below the neutral point, tension is not a problem so an API joint with long threads is sufficient. (iv) Design weight for the section (CWT) CWT = Design length x wt per foot x BR = 1480 x 23 x 0.885 = 301251bs. (d) For the next section N-80, 23 ppf has the next highest collapse pressure to P-110, 20 ppf and can be set below the neutral point (see Table A3.1). 8370 (i) Collapse limit = 0.831 = 10072 ft *

*Rounding off, we can propose a section length of 11 520 - 10 070 = 1450 ft (ii) Burst check 8000 + 10 070 [0.498 - 0.831] = 4647 psi no problem arises. 312 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*(iii) As we are below the neutral point no joint strength problem. (iv) Design weight for this section 1450 x 20 x 0.855 =24 795lb Total weight calculated so far = (30 125 + 24 795) = 54 920 lb. (e) In the next section we might consider the use of P-ll 017 ppf but only a relatively short section could be used. It is considered more economical to design for N-80, 20 ppf. 6930 (i) Collapse limit = 0.831 = 8339 ft, round to 8340 ft *

*This is above the neutral point and therefore subject to the weight of casing above. We calculate the ratio (R) for unit tensile stress to minimum yield strength using the ellipse of biaxial yield stress curve (Fig. A3.1) to obtain the percent offull collapse pressure that is appropriate. From Table A3.1 the plain end area (A) of 20 ppfN-80 is 5.828 in2 • For the minimum yield strength (Ym ) of 80000 psi we have: weight in air of casing above neutral point R= Y .A m Assume casing above neutral point is 20 ppf 20 (9800 - D) R = 80000 (5.828) *

*We have to choose D such that the reduction factor (FR ) correlated with R to obtain the effective collapse depth is consistent: . 6930 J 20 (9800 - D)} I.e. 0.831 X FR = f(R) = f \80000 (5.828) This is solved by trial and we might choose D to be 7900 ft 20 (9800 - 7900) R = 80 000 (5.828) = 0.0815 From Fig. A3.1 the value of FR corresponding to 0.0815 is 0.956% 6930 Collapse limit is 0.956 x 0.831 = 7972 ft *

*We could converge a little better but might accept 7900 ft as a suitable depth, giving 2170 feet of casing required between 7900 and 10 070 ft. (ii) Burst check for internal differential at 7900 ft = 8000 + 7900 [0.498 - 0.831] = 5369 psi This is within the tolerance of both 20 and 23 ppf N-80 (iii) Joint strength check Section design weight = (2170 x 20 x 885) = 38 409lb Total design weight = 38 409 + 54 920 = 93 329lb We can see that the joint strengths of 20 and 23 ppfN-80 casing are both greater than the design weights (Table A3.1): 23 ppf : 251 000 lb 20 ppf: 214000 lb (f) In abnormal pressure wells, a depth can be reached where either collapse or burst may control. A design trial for the next section is made using 17 ppf N -80. 5240 (i) Collapse check 0.831 = 6305 ft SOLUTIONS TO EXAMPLES 313 *

*05 *

* g ~go ~ -0 0; .;;' 0.1 E ~ ·c ·E ~ 0.05 o o ~ ;; ·w'" i .'§ 0,02 'l; o a:c *

*0.85 0.90 1.00 Of full collapse pressure *

*We can converge on a reduced setting depth of 5430 feet. 20(9800 - 7900) + 17(7900 - 5430) .. R = 4.962 (80 000) = 0.202, gIvmg FR = 0.884 and a collapse limit of 5573 ft which is in tolerance. The possible length of this section is thus (7900 - 5430) = 2470 ft (ii) Burst check Internal differential at 5430 ft = 8000 + (5430 [0.498 - 0.831]) = 6192 psi The burst strength of 17 ppf N-80 is quoted in Table A3.1 as 6180 psi. We must check the depth at which burst governs, i.e. the depth equivalent to a burst strength of 6180 psi. 8000 - 6180 Depth = 0.831 _ 0.498 = 5466 ft *

*The depth that 17 ppf N-80 will withstand the internal pressure differential is below its allowable collapse depth and this grade cannot be used in this part of the design. We must therefore consider using 20 ppf N-80 as we know that this is collapse designed down to 7900 ft. The burst strength for this is 7400 psi. 8000 -7400 Depth = 0.831 _ 0.498 = 1802 ft, round up to 1820 ft *

*This means that we could design a section of length (7900 - 1800) = 6080 ft (iii) Joint strength check Design weight for section is (6080 x 20 x 0.885) = 107 616lb Total weight is 107 616 + 93329 = 200 945lb *

*The joint strength for 20 ppf N-80 is given in Table A3.1 as 214000 lb. We have so far designed 11180 ft of the total well depth of 13000 ft. The remaining 1820 ft are considered using P-110, 17 ppf grade casing. 314 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE (g) (i) Collapse check 20(9800 - 1820) + 17(1820) R = 11 000 (4.962) = 0.35 *

*FR = 0.78 7000 (0.78) Setting depth = 0.831 = 6570 ft *

* a proposed setting at 1820 ft is acceptable. *

*(ii) Burst check Internal difference at top of string is 8000 psi (max) and Table A3.1 gives burst rating as 8500 psi, therefore design is acceptable. (iii) Joint strength check Design weight of section added = 1820 x 17 x 0.885 = 27382lb Total string weight = 27 382 + 200 945 ------Section Length (ft) Casing Grade Joint strength of P-ll0 L = 247 OOOlb design is acceptable. (h) We can summarize the design as follows: Surface -1820 1820 17 ppfP-ll0L 1820-10 070 8250 20 ppfN-80L 10 070 - 11 520 1450 23 ppfN-80L 11 520 - 13 000 1480 23 ppf P-11O L *

*It should be emphasized that this design is one of many combinations which may be acceptable and optimization in terms of economics is possible. Solution 3.2 The average gradients give a pore pressure at 13 000 ft of 13 000 x 0.455 = 5915 psi and a fracture pressure at 13 000 ft of 13 000 x 0.80 = 10 400 psi The minimum setting depth is given by equating, above 13 000 feet, the gas and fracture gradients to a common pressure. If the distance above 13 000 ft is D' then *

*Pg = 5915 - (0.1 x D') Plr = 10 400 - (0.8 X D') *

*Setting Pg = Plr we have 10400 - 5915 D' = = 6407ft 0.8 - 0.1 Minimum setting depth is 13 000 - 6407 = 6593 ft. *

*TABLE A3.1 Casing data for example (Grade NSO-L/PllO-L 5.5 in. OD.) Weight Wall thickness Collapse incl. Burst strength Joint strength Section (lblft) (in) safety factor into wk. press (incl. S.F.) lOOOlb area (psi) (incl. S.F.) psi (in2) 17.0 PlIO 0.304 7000 8500 247 4.962 17.0 N80 0.304 5240 6180 174 4.962 20.0 PlIO 0.361 9570 10 180 274 5.828 20.0N80 0.361 6930 7400 214 5.828 23.0 PlIO 0.415 11 630 11 780 322 6.630 23.0N80 0.415 8370 8570 251 6.630 *

*Minimum yield strength (Ym) = 80 000 psi for N-80 = 110 000 psifor P-ll0 SOLUTIONS TO EXAMPLES 315 Chapter 4 Solution 4.1 *

*141.5 API = SG - 131.5 *

*SG API SG API 0.70 70.6 0.80 45.4 0.72 65.0 0.82 41.0 0.74 59.7 0.84 36.9 0.76 54.7 0.86 33.0 0.78 49.9 0.88 29.2 0.90 25.72 *

*NB API gravity is non-linear, inverse scale. Water SG = 1.0; API = 10. Solution 4.2 *

*Yj MW YjMW Pc YjPci Tc YjTcj *

*C1 0.90 16 14.4 673 605.7 343 308.7 ~ 0.05 30 1.5 708 35.4 550 27.5 C3 0.03 44 1.32 617 18.5 666 19.9 C4 0.02 58 1.16 551 11.0 765 15.3 (a) L = 18.38 (c) L = 670.6 (c) L = 371.5 MW 18.38 (b) Specific gravity = 28.97 =28.97 =0.634 *

*. _ m _ MP _ 18.38 x 14.7 _ -2 3 Gas denSIty - V - RT - 10.732 x 520 - 4.8 x 10 Ibft (d) At 2000 psia and 595°R 595 Tpr = 371.5 = 1.60 2000 P pr = 670.6 = 2.98 *

*(e)Fromgraphsz=0.825 (fig 4.7) . _ MP _ 18.38 x 2000 _ 3 (t) DenSIty - zRT - 0.825 x 10.732 x 595 - 6.9771bft 6.977 (~sBg ~a( ;,r( ~r~ ~ro~::~;'5~:it®6 ~ 6.9 xI~' vowvoL = 6.9 x 10-3 5.615 = 1.235 BBLIMSCF *

*(h) From graphs, Itl = 0.0116 (Fig. 4.8) and Ratio ~ = 1.3 (Fig. 4.9) 1-11 *

*Therefore 1-1 = 0.015 cp 316 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*1 1 dz (i) Compressibility cg = PI - (~ ~P 1 dz ) *

*= Ppc Ppr - -; dPpr *

*1 (Ppc 1 dz ) = Ppc p--; dPpr *

* from graph (Fig. 4.7) of z vs. reduced pro1erties, by graphical differentiation 1 (670.6 1 cg = 670.6 2000 - 0.825 X ( - 0.01) *

*= 5.2 X 10-4 psi-! *

*0) At 4100 ft SS aquifer pressure would be 0.44 x 4100 = 1804 psi *

*Since gas has a smaller density than water, it will lie above water. At gas-water contact, pressures are equal. From the given data clearly this gas-water contact will be below 4100 ft. Let this extra distance be x ft. Assume too that density of gas is a constant over the distances concerned, and that the reservoir temperature is 135°F, thus the density takes the value calculated in (f), 6.9771b ft3 (or gradient 0.0485 psi ft-!). *

*Pressure balance at gas-water contact: *

*(4100 + x) 0.44 = 2000 + 0.0485 x *

*:. x = 196/0.3915 = 500 ft *

*Therefore gas-water contact depth = 4600 ft SS *

*(k) From 0), gas-water contact is at 4600 ft SS, and pressure is 4600 x 0.44 = 2024 psi *

*Assuming the gas density remains constant for 1000 ft, pressure due to gas = 0.0485 x 1000 = 48.5 psi *

*Therefore pressure at crest of structure = 2024 - 48.5 = 1975.5 psi *

*Therefore pressure of mud at this point will be = 1975.5 + 500 = 2475.5 psi *

*Assuming the mud to be incompressible, let density of mud = p Ibslcu ft *

*Pressure exerted by mud at 3600 ft = 1~ x 3600 = 2475.5 psi *

*Therefore p = 99.0 Ibslcu ft *

* i.e. specific gravity of mud = 1.58 *

*Solution 4.3 *

*6 Cg = IIp = 1/1923 = 520 x 10- psi-! *

*(a) Total compressibility (b) Effective hydrocarbon compressibility CT 171.5 x 10-6 Coe = = 225 X 10-6 psi-! 1 - Swi 0.76 *

*= 10-6 [5 + 0.45(10) + 0.24(3) + 0.31(520)] *

*= 171.5 X 10-6 pS(1 SOLUTIONS TO EXAMPLES 317 *

*Solution 4.4 13"'+,'-+_f--+7--f:";C",~--t==j"'" ".. (a) From graphs (Fig. 4.21, 4.22) or correlation equations for '''+--+-~''=f-=-b-!''''f'''=+--l API = 38°; GOR = 750; T = 175°F; and Yg = 0.7: ""'¥7""b-=t---t-I---1--t--l ,OOO"----'-----'----'-_L-.-'-----'----' Pp,:{ps;.) &00 bubble point pressure = 2800 psia I~ ..." formation volume factor = 1.4 RB/STB 1-"':::: ~ ~ "- '" 141.5 ~ ::::::-- specific gravity oftank oil = 131.5 + 38 = 0.834 300 12 -.::::: ::-:::: f::: weight of oil and gas in SOlution} 100 120 140 1SO t80 200 220 240 (b) Density of reservoir oil = ( . MOlECULAA WEIGHT volume of oIl reservoir conditions Fig. A4.1 Pseudo critical properties of hydrocarbon liquids Weight of one barrel of water = 5.615 x 62.4 = 350.4 pounds (density of fresh water is 62.4lb/fe and 5.615 cu ft = 1 barrel). *

*From specific gravity of tank oil, weight of one barrel of oil is 350.4 x 0.834 = 292.2 lb. *

*Avogadro's law states that lIb-mole of any ideal gas occupies 379.4 cu ft at 60°F and 14.7 psia. *

*:. weight of gas which will dissolve in 1 STB of tank oil is given by the number of moles of gas times its molecular weight. The molecular weight of gas is the gas gravity x molecular weight of air :. weight of gas/STB = (R,I379.4) x 0.7 x 28.971bs = 0.05345 Rslbs. Volume of 1 STB oil at reservoir conditions = Bo BBL [292.2] + [750 x 0.053445] :. Density of reservoir condition oil = 1.400 lbs/BBL density at reservoir conditions :. SG = 350.4 = 0.677 *

*The reservoir oil gradient is therefore 0.677 x 0.433 psi/ft where 0.433 is the fresh water gradient :. oil gradient = 0.293 psi/ft. *

*For an oil-water contact of 7000 ft SS the hydrostatic pressure is 7000 x 0.465 = 3255 psi. *

*The bubble point pressure is the pressure of oil saturated with gas in equilibrium at the gas-oil contact :. pressure at top of oil column = 2800 psi. 8255 - 2800 For constant oil gradient, height of oil zone = 0.293 = 1550 ft *

*:. GOe = 7000 - 1550 = 5450 ft SS *

*For a molecular weight of 180 and 38° API oil the liquid critical temperature is 12200R and the liquid critical pressure is 310 psia (460 + 175) 4000 Tpr = 1220 = 0.52 and P pr = 310 = 12.9 *

*The reduced compressibility from charts (Fig. A4.1) is given at this Tpn P pr condition as CR = 0.002. 0.002 Since CR = Co· Pc then Co = 310 = 6 x 10-6 psia-I *

*From a constant oil compressibility between 2800 and 4000 psia 318 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Bo = Bob (1 - Co!).P) *

*= 1.40 ( 1.0 - 6 X 10-6 (4000 - 2800)) *

*= 1.389 RB/STB *

*From graphs, viscosity of dead oil at reservoir conditions = 1.4 cP *

*:. viscosity ofreservoir crude = 0.6 cPo *

*Solution 4.5 *

*From graph of system pressure vs. system volume the bubble point is estimated by inflexion at 2500 psi. *

*Liquid volume at standard conditions = 29l;~) *

*At 3000 psi a liquid compressibility Co = - V \dP T 4000 *

*3500 (404-410) _1_ -6 '-1 ~ ::l Co = (4000 _ 2500) . 408 = 9.8 X 10 pSI til 3000 ~ 0. 408 ~ 2500 B03000 psi a = 295 = 1.383 RB/STB If) (f)'"' 2000 410 = = Bo2500 psia 295 1.390 RB/STB 1500!-:::-;;---'------::*o::------'-~---"" 400 26.275 System volume Rs = 295 (10-3) = 89.06 v/v = 89.06 (5.615) = 500 SCF/STB *

*At 2000 psia 388 430 Bo = 295 = 1.315 RB/STB ; Bt = 295 = 1.457 RB/STB 21 Rs = 295 X 10-3 X 5.615 = 400 SCF/STB *

*:. Bt = Bo + (Rsi - Rs) Bg . Bt - Bo (1.457 - 1.315)(295) -3 .. Bg = (Rsi _ Rs) = (26.275 _ 21.0)103 = 7.94 x 10 v/v *

*(2000) (520) -3_ Z = (Pl)Tl . (T2)P2 . (Vl)V2 . -_ 660 . 14.7 ·7.94 x 10 - 0.85 *

*ChapterS *

*Solution 5.1 *

*F: 30 19.3 12.5 8.4 6.0 *

*Plot either on log: log scales, or log F: log *

*100 \ 80 \,\ Slope' 60 ,, length Faxis -17.95 m = length axis = ~ 40 \\ = -1.53 o 30 ._,, Intercept at '" = 1 ,, a = 0.774 20 0., ,, '0 \ 10 ,, t 8 0,\ LL 6 '0 '\, , 4 ,, 3 \ ,, ,, 2 ,, \ \ \ , 1 ,, 0.8 --a 0.6 0.5 '------'--'---'--L.L--'-l0=-'.OO:-:1-----'----'---'---'----'---L--L..1 1,J.0 *

*Fig.A5.1 Fvs. *

*From plot m = - 1.53 a = 0.774 Substitute back into laboratory data to calculate check values of F. *

*Check 0.092 0.120 0.165 0.205 0.268 Calculate F 29.8 19.8 12.2 8.7 5.80 *

*If the true resistivity is 1.29 Qm and water resistivity is 0.056 Qm then Ro 1.29 F = Rw = 0.056 = 23.04 = 0.109 1 If I = ~ where exp n = 2 w R/ = 11.84 Qm Ro = 1.29 Qm 11.84 then I = 1.29 = 9.18 1]0.5 Sw = [I = 0.330 If exp = 1.8 Sw = 0.292 *

*Ifexp = 2.2 Sw = 0.365 320 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE Solution 5.2 *

*(a) *

*Data values Calculated values *

*Log values GR FDC SNP C1LD R1Ld V SHGB VSHDas: CPDIN Shale 102 2.52 29.0 1100 0.91 1.00 1.00 Zone A 52 2.22 22.5 150 6.67 0.39 0.00 0.26 B 72 2.37 20.5 350 2.86 0.63 0.31 0.14 C 20 2.20 21.0 4650 0.215 0.00 0.00 0.25 *

*Bul k density grams/cc Porosity % r------T------Correction -0.5 0 +0.5 *

*2.0 2.5 3.0 Sidewall *

*Fig.A5.2.1 SOLUTIONS TO EXAMPLES 321 *

*Gamma ray Resistivity Conductivity 20 API units 120 Depth Ohms mlm Millimhos 1m 10 divisions lS"normol Induction conductivity o 40" spacing Radiation intensity o increases ~ 4000 o o Oil bose mud Induction resistivity 8000 4000 Temp =226 0____ 1C;i:.s.E!!.~'!L __19 0______1<2.0 *

*I I I I I, I I :I I I I I I ,I \ " ... _--- ..... A :--.,,,,--~ B ,,/ I I I I I I I I I I I I " C *

*,, ,. Fig. AS.2.2 '--__---"'"--' I *

*2.0 Pr=1.0g/cc *

*2.2 *

*Shale *

*Matrix point *

*2.8 *

*.,.~., ~~ ~~"i Fig. AS.2.3 3q';;10=---L--;0~~L-~:;----L----;:!;=----1.-~:-----'--~_...J() '" Density/SNP crossplot. Sidewall neutron apparent limestone porosity (%) 322 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*(b) For zone C, point plots close to clean sandstone line with cJ> = 0.25. Assuming C to be water bearing Ro = FRw = 1/cJ>2 . {Rw} Rw = cJ>2Ro = 0.262 X 0.215 = 0.0145 (taking R1Ld as Ro) *

*(c) Shale values are listed above. *

*(d) GRclean = 20, GRshale = 102 GR - GRclean GR-20 VshGR = ------GRshale - GRclean 82 *

*VshGR values calculated are tabulated above. *

*(e) See Fig. A5.2.3 for shale point. Only level B shows a significant displacement from clean line. Graphically Vsh for zone B = XB/XS = 1.25/4 = 0.31. *

*(f) Taking the minimum shale indication (from DIN) gives only B as shaly. Presumably there are radioactive minerals in the sands (such as feldspar) so the GR overestimates shale content. *

*As above graphically for level B, Vsh = 0.31. The porosity is given by point Yon the clean sandstone line where BY is parallel to the matrix shale line, i.e. cJ> = 0.14. The graphical construction is complicated by the curve on the sandstone line. More rigorously convert density and neutron values to sandstone matrix cJ>D = 16.5, cJ>N = 24.2. cJ>NSH = 32, cJ>DSH = 7.5, cJ> = cJ>N - VSH cJ>NSH, cJ> = cJ>D - VSH cJ>DSH Solving the equations for unknown VSH cJ>N - cJ>D 24.2 - 16.5 VSH = = = 0.31 cJ>NSH - cJ>DSH 32 - 7.5 *

* cJ> = cJ>N - VSHcJ>NSH = 24.2 - 0.31 x 7.5 = 0.14 *

*(g) Saturation calculations *

*LevIe! Ala~ eq)uatiO:S reduce t~ I(r~~: ()VSH ~ ~) / Rw 1 '\ /0.0145 *

*:·Rr = FRw ·Sw :.Sw= VR; =~VR;= 0.26 V6.67 =0.18 *

*Level B with n =2, Rw =0.0145, Rt =2.86, RSH =0.91, VSH =0.31, =0.14. Archie ~, ~ (F~w) .Sw' :. 0.35 ~ 1.352 S.' :. Sw = 0.51 *

* r~"O:~)· S.' + (~::) :. 035 ~ I.352S.' + 0341 :. Sw = 0.082 *

*~o~ifi(e~ Si)mandzoux(VSH ) . _ z R - FR . Sw + R . Sw .. 0.35 - 1.352 Sw + 0. 341Sw t w SH . . Solvmg quadratIc + ve root only :. Sw = 0.376 SOLUTIONS TO EXAMPLES 323 Poupon and Leveaux (Indonesia) 1 1 VSH(1-VsH/2) 'I!Rr = YFRw Sw + VRsH . Sw :. 0.592 = 1.163 Sw + 390 Sw :. Sw =0.38 *

* where 1 1 1 -=- =0.350;~=0.592 Rt 2.86 VRt V (1-VsHI2) V SH = 0.341 ; VSH(l - VSH/2) = 0.372; s~ = 0.390 RSH SH 1 ~2 1 FRw = Rw = 1.352 ; YFRw= 1.163 Thus the modified Simandoux and Indonesia equations give similar Sw's which are less than the Archie Sw. The shale conductance in the basic Simandoux is already near to the measured conductance so the solution gives an unlikely optimistic value for a shaly sand. *

*Solution 5.3 *

*Waxman and Thomas equation with a = 1, m = 2, n = 2 ~ =~ 2+BQv Sw R t FR w Sw F *

*= -1 { - 1 Sw 2 + BQvSw ) F Rw *

*BQv = 0.046 x 0.3 mho.cm2.meq-!.meq/cc = 0.0138 mho cm-! or ohm-! cm-! = 100 x 0.0138 = 1.38 ohm-) m- l 1 1 2 :. Rt = F (10 Sw + 1.38 Sw) *

*F = 1I~2 = 1/0.262 = 14.79 Rt = 5 :. 0.2 = 0.0676 (10 S} + 1.38 Sw) = 0.676 S} + 0.0933 Sw :.0.676 Sw 2 + 0.0933 Sw - 0.2 = 0 Solving the quadratic _ (-0.0933 ± \1[0.09332 - 4.676 . ( -0.2) 1) = 0.4 Sw - 2(0.676) *

*'\ 1FRw (see Archie solution, Sw = VIi; = 0.544) *

*Modified Simandoux model *

*~_~ 2 VSH R t FRw Sw + RSH . Sw Comparing with the Waxman Thomas equation 324 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*BQv VSH BQv p= RSH :. VSH=RsHp 1.5 X 1.38 VSH = 14.79 = 0.140 i.e. it would take 14% shale with resistivity 1.5 ohm-m to get the same result as the Waxman Thomas equation .. ! _~ 2 VSH.. _ 0.0676 2 0.14 BaslcSlmandoux Rt - FRw Sw + RSH , .. 0.2 - 0.1 Sw + 1.5 0.2 = 0.676 Sw2 + 0.0933 *

*1 1[0.2 - 0.0933] Sw = V 0.676 = 0.397 *

*Solution 5.4 (a) Prove From Darcy's law: -kA JP q=--- !.t Jx Assuming Boyle's law: 3000ff QscPo = qP and Po = 1 atm. Hence: -kA JP 1000 ft:-I ______----' Q =-P- sc !.t Jx 1501~ kA p/- P12 orQsc =-; 2L ~-5750 Qsc!.t2 L (b) k = A(P12 - p/) 6.2 x 2 x 0.018 x 2.54 *

*XX 127' «(~:r -1) =0.2D Solution 5.5 The problem requires correction of pressure so that the linear Darcy law can be used. In field units: kAAP q = 1.127 X lO-3 -; L BBLId *

*Assuming average water gradient of 0.45 psi/ft (0.433 x 1.038) and referring to a HWC datum of 5250 ft SS, static pressure at the outcrop is: PS2so = 0.45 x 5250 = 2362.5 psi But pressure = 1450 psi at 5250 3 750 x 3000 x 65 (2362.5 - 1450) Hence, q = 1.127 x lO- x 1 x 52 800 --\ _------j Poutcrop q = 2848.5 BBLId ------P HWC .::::/// -/ f P outcrop at HWC datum ....."";------10 miles ----" SOLUTIONS TO EXAMPLES 325 Solution 5.6 Using the equation: Qsc 2 ilL (6.4/60) x 2 x 0.018 x 2.54 k = A(PI2 - pl) 2 (861)2 2 3t 1.27 (760 - 1 ) *

*Sc for rate 1 = 0.0068 D = 6.8 mD Scfor rate 2 = 6.02mD Scforrate 3 =5.0mD This is because of the Klinkenberg effect. *

*Plotting k against 11P mean gives k L as 11P mean ~ 0 as 3 mD. Solution 5.7 Assume cross-sectional area A. dh q = -A dt where q is flow rate and h is current height measured from bottom of core plug. Flow across core is: -kA I1P q=-- Il L But I1P = datum correction pressure difference, so: -kA pgh dh q=--=-A- Il L dt L dh kPg'Jt so- -=- dt J h ilL ho 0 ho kpg' or log., It = L t ho ho Il (lOg -- log so k = IlL ~[log (holh)] = ilL e h2 e -JhI pg' ~t pg' I1 t Note: pg' has to be in units such that pg' h = atm. 1 x 2 X 106 loge84 -loge 15.5 Hence k = 1.02 x 981 x 4500 =0.8D Note: a plot of log.,h against t would be best. Solution 5.8 50 Poil = 50 lb/fe = .144 psi/ft = 0.3472 psi/ft (a) Correct well pressures to 5750 ft = 1750 + 0.3472 x 750 = 2010.4 psi (b) Flowing gradient kA I1P q = -;-L 1.127 X 10-3 *

*I1P = qllL 1000 x 1.135 x 7 x 3000 1.127 X 10-3 x k x A = 1.127 X 10-3 x 150 x 150 x 1000 *

*= 94 psi 326 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*So, Powc = 2010.4 + 94 = 2104.4 psi PV res = 3000 X 1000 X 150 X cp Equating production Vp = VTc· llP, then 3000 X 1000 X 150 x PVaquifer = (2104.4 _ 500) 3 X 10-6 = 93.5 X 109 x ft3 *

*Solution 5.9 Q k dP Darcy's equation A = - -; dx for non-compressible flow (a) Linear beds - parallel flow P, Q = qi + q2 + q3 Assume infinitely thin barriers between layers q, ---fIto-\ Q llP llP --... Q q2--~-\ r------~------~k, Q=qI+q2+" .=kIAIf..tL +k2A 2 f..tL + ... *

* llP =k'A- f..tL where k' is the apparent permeability and A the total area. Hencek'A = kiAI + k2A2 + ... n Lk;A; Therefore k' = ~ LA; I Lkh· or if beds all same width = f *

*(b) Series flow *

*Assume equal areas P, P2 P2 P3 P3 p. Al =A2 =. - . o D B B q, = qi = q2 = q3 - .. -\LJ~U-U Now PI - P4 = (PI - P2) + (P2 - P3) + (P3 - P4) ..• L, L2 Using Darcy's law L f..t LI f..t L2 f..t qtAk' = qi Aki + q2 A ki + ... Since flow rates, cross-sections and viscosities are equal in all beds *

*(c) Radial flow parallel From the figure, it is noted that the same terms appear in the radial flow network as in the linear system. 2Jtkh (Pe - Pw) Q= f..tln(re/rwJ e - external boundary w - internal boundary SOLUTIONS TO EXAMPLES 327 *

* hi -qi k, 1 h2 -q2 k2 ht *

* h3 -q3 k3 j *

*The only difference in the two systems is the manner of expressing the length over which the pressure drop occurs. All these terms are the same in each case. "[k·h· Therefore k' = --'-' hI *

*(d) Radial flow series By same reasoning as in the linear case k' = _In_(.o...r::....,/r...::w~)- 't In (r/rj_l) j=1 kj *

*Bed Depth/ Horizontal permeability; mD Length of bed 1 250 25 2 250 50 3 500 100 4 1000 200 *

*For radial systems, wellbore = 6", and radius of effective drainage 2000' and bed 1 is adjacent to wellbore. Linear flow - parallel, and radial flow - parallel, take data lengths as bed depths and bed lengths and radii to be equal. *

*Linear flow in parallel k' = 250 x 25 + 250 x 50 + 500 x 100 + 1000 x 200 = 134.4 mD 2000 Radial flow in parallel "[kh k'=T 250 x 25 + 250 x 50 + 500 x 100 + 1000 x 200 k' = 2000 268750 k' = 2000 = 134.4 mD 328 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Linear flow in series 2000 2000 -=80mD k' = 250 250 500 1000 25 2s +50 +100+ 200 Radial flow in series In (200010.5) k'=------~----~------In 25010.5 + In 500/250 + In 10001500 + In 200011000 25 50 100 200 = 30.4mD i.e. permeability near wellbore most important. *

*Chapter 6 *

*Solution 6.1 *

*Pc 0 4.4 5.3 5.6 7.6 10.5 15.7 35.0 Sw 100 100 90.1 82.4 60.0 43.7 32.2 29.8 h 0 33.3 40.2 42.4 57.5 79.6 119.0 265.3 *

*Pc (h = ) (Pw - Po)/144 *

*- Crest 200 - - - - - 150 r- - l- .e I- \\-+-\-----samPle location Sw =0.31 a; > I- o ~... I- ~c ~ 100 I- CI) ~ - CI) > - 0 .Q \0 c - 1: - c> ~o 'CD :I: 50- ~ "-----0 -0 ~o OWC at 33 ft relative *

*I J I I I I I I J 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Sw (fraction)---*

*Fig. A6.1 Saturation distribution. SOLUTIONS TO EXAMPLES 329 *

*Note that the oil-water contact is at Sw = 1.0, not at Pc =0. At 100 ft above owe, Sw = 0.31 (135 ft relative) - ISw dh S =-- w h From area under Sw against h curve: Sw = 0.37 Solution 6.2 *

*Sw 100 100 90.1 82.4 60.0 43.7 32.2 29.8 (Pc)O-w 0 4.4 5.3 5.6 7.6 10.5 15.7 35.0 0 65.1 78.4 82.9 112.5 155.4 232.4 518.0 (PdH g f(J)=PcYf 0 1534.4 1847.9 1954.0 2651.7 3662.8 5477.7 12209.0 (PC)Hg 0 110.7 133.3 140.9 191.2 264.1 395.0 880.4 *

* for25mD and 0 = 0.13 Solution 6.3 For the laboratory data YkTcj>c = (150/0.22) 0.5 = 26.11 and using ](Sw) = PC(Sw~ Vi with CJ cos e = 72 dyne/cm the ](sw) vs Sw relationship is calculated. CJ cos cj> ] (Sw) = 0.363 Pc (Sw)Jab *

*1.0 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.2 o 0.363 1.451 2.176 2.901 3.445 4.862 4.968 5.984 8.341 36.27 *

*](sw> CJ cos e At reservoir conditions PC(Sw)", = v'kicj> for CJ cos e = 26 and v'kTcj> = 44.72 Pc (Sw)", = 0.581] (Sw) and the reservoir condition Pc curve is therefore calculated as *

*1.0 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.2 o 0.211 0.843 1.264 1.685 2.00 2.823 2.886 3.451 4.846 21.07 *

*For the reservoir specific gravity of oil and water given Ap = (1.026-0.785) = 0.241 The relationship between capillary pressure and height H above FWL is, in the units required, pc(sw) = 0.433 HAp :. H= Pc(sw) 0.104 *

*Using the threshold value of pc(sw) (= Pct) as the observed oil water contact, then *

*0.211 Howe = -- = 2 ft above the FWL 0.104 4.85 HTIZ = 0-- = 46.5 ft above the FWL .104 330 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE Chapter 7 Solution 7.1 *

*From Darcy's law modified for effective permeability in horizontal linear flow qo!-loL qw!-lw L Ko (s) = A I1P and Kw (s) = A I1P o w *

*Assuming zero capillary pressure (Pc = 0 = Po - P w) so I1Po = I1P w = I1P, and using Darcy units of eels for rate and atmospheres for I1P, then: q!-l [(4) (9) (1000)] Ke (md) = I1P n: (3.2)2 3600 *

*For oil Ko = ~P (9.14) qw For water Kw = I1P (5.0) Ko Kw For Kro= ~and Krw = ~ O(cw) o (cw) 90 Ko(cw) = 49.25 (9.14) = 16.7 md *

*15.0 1.0 o 19.8 0.452 0.017 25.1 0.30 0.025 32.1 0.20 0.049 41.0 0.12 0.075 54.9 0.05 0.156 68.1 o 0.249 *

*These data are plotted in Fig. A 7.1 *

*1.0 0 *

*0.9 0.8 *

*0.7 t 0.6 ~cpl.x:g 0.5 II OA ~ 0.3 0 0.2 *

*0.1 *

*:::::---'--=,-:--'--::,-~'o I I 0.8 10 sw-*

*Fig. A7.1 Steady-state relative permeability. SOLUTIONS TO EXAMPLES 331 Solution 7.2 For pressure maintenance, the oil rate in RB/D is 10 000 x 1.2765 = 12765 RB/D The end points of the relative permeability curve are *

*K ro' = 0.9 at Swi = 0.28 Krw' = 0.7 at Sor = 0.35 *

*The ratio Ilw is then calculated from the given end point mobility ratio of 2.778. flo krw' flo flw krw' 0.7 Since M' = flw • kro' , then flo = M' kro ' = 2.778 (0.9) = 0.28 *

*The fractional flow curve can now be calculated for the horizontal reservoir: 1 fw = f k } 1 + 0.28 l k:: *

*0.28 0.30 0.35 0.45 0.55 0.60 0.65 o 0.082 0.295 0.708 0.931 0.984 1.00 *

*A line tangential to the fractional flow curve from Sw = 0.28 gives the tangent at Swf = 0.4 (fw = 0.535) and the intercept withfw = 1 at Sw = 0.505. The gradient of this tangent[dfwfdSwlswis 4.44. From Buckley-Leverett theory the constant rate frontal advance of the 40% saturation front is: q(t) (5.615) [df ] Xflday = (A)(*

*(12765) (5.615) (4.44) Xflday = (5280) (50) (0.25) = 4.82 ftlday *

*For a system le~gth of 5280 ft, breakthrough therefore occurs in 1095 days (= 3 years) At year 4 the pore volume injected is _ _4-,-(3_65...:.)-,-(1_2_7_65.:....) ..:.-(5_.6_15..;...)_ =0.3PV (5280) (50) (5280) (0.25) *

* and dflds] = _1_ = 3.33 Swe 0.3 The tangent of gradient 3.33 to the fractional flow curve at saturations greater than frontal occurs at Swe = 0.45 dfw (from a plot of dS vs Sw)· w At this saturation (Swe),fwe = 0.71, the reservoir condition water cut. The average saturation remaining in the reservoir is given by the Welge equation as: - foe Sw = Swe + [dfwfdSwlswe _ (1-0.71)_ Sw = 0.45 + 3.33 :. Sw = 0.537 *

*The reco~ry factor is thus: S -S . RF = W WI = 0.36 1- Swi 332 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE Solution 7.3 The critical injection rate for gas is given in field units of SCF/D as: 4 4.9 x 10- k k r/ A (Yg - Yo) sin a: q SCFID = Ilg Bg (M - 1) where Bg is in units of RBISCF and a: is negative for updip injection. *

*The density difference in terms of specific gravity is: 17 -48 Ay = 62A = -0.4968 0.5 (1.8) M' = 0.028 (0.9) = 35.71 *

* sin (-10°) = -0.1736 4.9 x 10-4 (800) (0.5) (8000) (100) (-0.4968) (-0.1736) :. qcrit = (0.028) (35.71 - 1) (7.5 x 10-4) = 18.589 MMSCF/D *

*The rate of injection proposed (15 MMSCFID) is less than the critical rate and might almost lead to a stable displacement. *

*The oil rate expected prior to breakthrough is therefore: 15 x 106 x 7.5 X 10-4 Qo = 1.125 = 10 MSTB/D *

*Solution 7.4 *

*1.0 *

*I \ Distribution after 0.5 yrs O. 8 :--1:." I .. ~ t 0.6 \ I i ./ Calculated frontal iY position i 0.4 i i i Initio I j I distribution 0.2 ---.l....-_!.-*

*Fig. A7.2 Saturation distributions *

*From the given data the saturation is plotted as shown in Fig. A 7.2 ql = 9434 rbld Dip = 6° 110 = 1.51 cp h = 100' k = 276 mD Ilw = 0.83 cp w = 8000' = 0.215 Ay = 0.04 A= 800 000 ft2 The fractional ~ow curv(e is calculated as fO~I~:~ [ ]1 3 fw = Ilw kro 1 + 1.127 X 10- qt 110 - 0.4335 Ay sina: 1+-·- krw 110 SOLUTIONS TO EXAMPLES 333 *

*The results are shown in Fig. A 7.3 1.0 / ..... 0.9 *

*0.8 0.7 •I 0.6 t 0.5 ~ - 0.4 •/ 0.3 0.2 / 0.1 /. 0 • ...,.. I I I I I 0.2 0.4 0.6 0.8 1.0 Sw *

*Therefore: Fig. A7.3 Fractional flow curve. *

*Sw 0.16 0.25 0.35 0.45 0.55 0.65 0.75 0.79 fw 0 0.036 0.127 0.344 0.64 0.88 0.98 1.0 *

*Since there is no uniform saturation distribution initially a material balance solution is used: 5.615 q, At [Llfw (LlX)s"'j = A [AfwLlS 1 = 0.308 At LlS 1 for At in days *

*2.5 2.0 t 1.5 ;-',~ -(/) 1.0 -0 ~ *

*0.5 *

*1.0 Sw Fig. A7.4 Slopes of fractional flow curve. *

*The slope of the fractional flow curve as a function of saturation is plotted in Fig. A 7.4. Selecting saturations *

*ForSw = 0.79 ForSw = 0.75 ForSw = 0.7 t(yrs) X t (yrs) X t(yrs) X o 10ft o 12 ft o 15 ft 0.5 10 + 23.5 = 33.5 0.5 12 + 36.5 = 48.5 0.5 15 + 56.2 = 71.2 1.0 10 + 47 = 57 1.0 12 + 73.0 = 85 1.0 15 + 112.4 = 127.4 2.0 10 + 94 = 104 2.0 12 + 146 = 158 2.0 15 + 225 = 240 334 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*At 0.5 years the saturation distribution is shown on Fig. A7.2 and is represented in 10' increments. 5.615 qt Llt ) ( Note: A = 56.22 *

*Llx LLlx Swi Sw (0.5 yr) Llx (Sw - SWi) LLlX (Sw - SWi) 10 10 0.79 0.79 0 0 10 20 0.70 0.79 0.9 0.9 10 30 0.56 0.79 2.3 3.2 10 40 0.45 0.78 3.3 6.5 10 50 0.375 0.755 3.8 10.3 10 60 0.33 0.730 4.0 14.3 10 70 0.30 0.710 4.10 18.4 10 80 0.278 0.690 4.12 22.52 10 90 0.254 0.675 4.21 26.72 10 100 0.24 0.650 4.10 30.83 10 110 0.23 0.640 4.10 34.93 10 120 0.215 0.630 4.15 39.08 10 130 0.205 0.620 4.15 43.23 10 140 0.20 0.613 4.13 47.36 10 150 0.195 0.605 4.10 51.46 10 160 0.190 0.600 4.10 55.56 10 170 0.183 0.595 4.12 59.68 *

*{ 56.22 - 55.56) Interpolation :. Xf = 160 + 10 59.68 _ 55.56 = 161.6 ft from owe *

*From Fig. A 7.2, at Xf = 161.6 ft, Swf = 0.60 Solution 7.5 For the particular example the problem reduces to the following tabulation, numbering layers n, from n = 0 to n = N = 5, bottom to top. n 5 - _ 0.7n + 0.15 (5 - n) . - _ 0.5 f. kj . - _ 'hf'j Swn - 5 ,Krwn - ___ , K ron - 0.9_ 5 5 ~ kj ~ kj *

*5 where: L kj = 50 + 500 + 1500 + 2000 + 500 = 4550 mD. 1 *

* n N j Lk· f. k n+l 1 5 5 n N n L kj Lkj k k L kj Lk·1 1 1 rwn ron SWn 1 n+l*

*0 0 4550 0 1.000 0 0.900 0.15 1 50 4550 0.0110 0.989 0.0055 0.8901 0.26 2 550 4000 0.1209 0.879 0.0605 0.7911 0.37 3 2050 2500 0.4505 0.5494 0.2253 0.4940 0.48 4 4050 500 0.8901 0.1099 0.4451 0.0989 0.59 5 4550 0 1.00 0 0.50 0 0.70 *

*Sw vs j(rw and j(ro The resultant pseudo-relative permeability is plotted as n n n SOLUTIONS TO EXAMPLES 335 ChapterS *

*Solution 8.1 *

*Using the relationship h + 139 = 164/sinh x the saturation vs height relation is calculated as follows: *

*X (frac) 0.33 0.40 0.50 0.60 0.70 0.80 0.90 1.0 sinh x 0.3360 0.4108 0.5211 0.6367 0.7586 0.8881 1.0265 1.1752 h (ft) 349 260.2 175 118 77 45 20.8 0.55 *

*Fig. A 8.1 shows the plot of water saturation and porosity as a function of depth. Fig. A. 8.2 shows the plot of isopach value vs area contained within the contour. In the absence of a phinimeter to measure area use metric graph paper in a simplified approach. Take 50 ft intervals from base to crest. Count squares to determine volume for each interval. Assign appropriate value of *

*Porosity (cp) ~ 0.20 *

*320 *

*280 t 240 -.... ti 200 c c: -0 u ... 160 Q) -c ~ Q) > 120 0 .J:l C .s:: -CI 80 ·CP ::I: 40 *

*0 *

*Water saturation (Sw) ~ *

*Fig. AS.1 Sw and vs. depth. 336 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Area within contour (acres) *

*Interval No. ofsquares Gross rock volume Saturation Porosity Hydrocarbon (lffacreft) (Sw) (~) (volume x UP HHLs) 0- 50 46 115.00 0.875 0.160 16.73 50-100 35 87.50 0.70 0.178 36.25 100-150 26.7 66.75 0.57 0.197 43.87 150-200 21.5 53.75 0.49 0.215 45.72 200-250 17.0 42.50 0.43 0.234 43.98 250-300 11.3 28.25 0.39 0.252 33.69 300-350 3.0 7.50 0.36 0.271 10.09 L =401.250 L =230.326 *

*Hydrocarbon in place = 230326250 BBLs reservoir oil = 170 x 106 BBLs stock tank/oil Solution 8.2 The oil in place at stock tank conditions is evaluated using the relationship 7758Ahcjl So N=--~--=- Hoi where N is in STB A is in acres h is in feet cjlSo is a fraction Hoi is in RBISTB The recoverable reserve is N.(RF) where RFis the recovery factor (fraction). Deterministically, the minimum, 'most likely', and maximum values are calculated as: minimum 43 x 106 STB 'most likely' 116 x 106 STB maximum 274 x 106 STB SOLUTIONS TO EXAMPLES 337 *

*The distribution functions of the reservoir parameters are shown in Fig. A 8.3. These data are interrogated randomly using a Monte Carlo approach in the recoverable reserve calculation. The resulting cumulative frequency greater than a given value plot is shown in Fig. A 8.4. The values associated with the 90%, 50% and 10% levels are as follows: at 90% the recoverable reserve is at least 72 x 106 STB at 50% the recoverable reserve is at least 120 x 106 STB at 10% the recoverable reserve is at least 185 x 106 STB *

*100i 100 , .- hne! Area t o~ ~ ~ a. I-'l • *

*• ~., 0 *

*100 100 \ • cf>So RF t t 0\ ~ ~ ~ a. 50 0: 50 I-'l \ • 0\ I-'l *

*•\ ~o 0 *

*100 100 cf> *

* t ~BO ~ ~ ~ a. 50 50 I-'l \0 o~~ *

*• \ • 0 0 *

*Fig. A8.3 Distribution functions. 338 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*100 *

*-0 .l!! c 0 90 'ii .E c: c 80 ~ .,..... C ~ 70 c> .!!!., => 60 ~ C ~ 50 E: £i c ..c 40 0 .,a. c> c 30 C., e ., 20 .,a. .~ C :; 10 E => u 20 260 106 STS---- Fig. A8.4 Recoverable reserves distribution. Chapter 9 Solution 9.1 Kt to = *

* with (a) to = 1481 (b) to = 14815 (c) to = 7.4 X 10-3 *

*Solution 9.2 *

* p;-p= 4:~h [-+ ~:':) 1 *

*Hence AP = 22.72 atmospheres For (b) x = 0.4375 From graph - E; (-x) = 0.62 *

*Hence AP = 2.875 atmospheres For (c) x = 0.49 From graph - E; (-x) = 0.55 *

*Hence AP = 64 atmospheres SOLUTIONS TO EXAMPLES 339 Solution 9.3 *

*From the plot shown in Fig. A 9.1, of P wi vs 10glOt m = 18 psi/cycle 162.6 (500) (0.5) (1.7535) ThenKh = 18 *

*= 3960 mD ft 3960 KO=60=66mD *

*4940 *

*4930 m = 18 psi /cycle t • ~ 4920 .".~ • *

*Fig. A9.1 PwtVS 1091Ot. *

*Solution 9.4 f HAt) From a graph of P vs llog ---;;;:r with the points in the table calculated, the slope is determined as 21. 7 psi/cycle ( = m). For a reservoir rate q of 500 (1.454) rb/d (= 727 rb/d) 162.6q(..t Then, kh = = 3800 mD.ft m *

*For h = 120 ft then Ko = 32 mD. The value of P{h' corresponding to a Homer time function of 3.16 is 4981 psi 4981 - 4728 32 ) S = 1.151 21. 7 10glO (0.135)(0.7)(17 X 10-6)(0.5)2 + 3.23 = +7 340 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*!!.Ps = 0.87 m S = 132 psi 4981 - 4728 - 132 Efficiency = 4981 - 4728 = 0.5 (approx.) t+!!.t [t+!!.t] Data points for Horner plot on semi-log paper, P vs At or a plot of p vs log At on linear scales are as follows: *

* t = 60 x 24 = 1440 hours *

*Time (h) t+!!.t logJO P (psi) !!.t {(t+ !!.t)/ !!.t} 0.25 5761 3.76 4967 0.5 2881 3.46 4974 1.0 1441 3.16 4981 1.5 961 2.98 4984 2.0 721 2.88 4987 3.0 481 2.68 4991 6.0 241 2.38 4998 9.0 161 2.21 5002 18.0 81 1.91 5008 36.0 41 1.61 5014 48.0 31 1.49 5017 *

*Solution 9.5 Examination of the data shows that: !!.P/day = 3 psi Assuming 1 - Sw = 0.7 We have NBoi = NpBo/(co).!!.P and (co)e = 15 x 10-6/0.7 = 21.4 x 10-6 and Np = 500 bId. For Bo = Boi then 500 6 N = 21.4 X 10-6 x 3 = 7.8 x 10 BBL Solution 9.6 Slope = 7 psi/cycle from Homer plot (MSCFID) (!!.p2) total Rate Q 162.6 (q Bg) IA. 1 7290 42181 HenceKh = m 2 16737 126120 3 25724 237 162 0.00504zT 4 35522 391616 Bg = P BBLlscf = 0.00103· *

*Kh = 14 500 K= 72mD Assume tflow prior to build up is 4.5 hours: 2 2_14241A.zTQ { ) NowPe -Pw - Kh InO.606re/rw +Sl *

*Time since P = 0.855 Q {8.93 + Sl} shut in Pe2 - Pw2 1 0.7404 2509.7 Or Sl = 0.855 Q - 8.93 1.5 0.6201 2510.7 2 0.5119 2511.3 2.5 0.4472 2511.7 Rate Q 3 0.3979 2512.1 1 -2.16 7290 4 0.3274 2512.5 2 -0.12 5 0.2788 2513.0 3 +1.85 6 0.2430 2513.2 4 +3.96 35522 SOLUTIONS TO EXAMPLES 341 *

*D = AS/AQ = 2.16 x 10.4 S = -3.7 *

* f3 = DhWw = 2.865 x 109 2.22510 15 KYg 48211 f3theoretical = <1>5.5 v'K = 1.80 X 109 This is order of magnitude agreement. *

*The inertial pressure term Ap2inertial is calculated from B as follows: 3.16 x1O-1ZygTzf3 B= 2 h rw *

*= 0.000185 Hence (Ap2)inertial is as follows: *

*Rate Q(MSCFID) 1 7290 9851 42181 2 16737 51928 126120 3 25724 122665 237162 4 35522 233906 391616 *

*Comparison between the numbers shows that at high rates the inertial drop is over half the total drop, and that in this case only the inertial drop is close to the total drop of the previous rate. The AOF plot is shown in Fig. A 9.2 and when Ap2 is equal to Pe2 (6.32X106psi2) then QAOF = 220 X 106 SCFld *

*AOF= 220mm------e------. SCF/D C B i I I I I I I I I I I o'" I n =0.65 [= distance AB] ! distance Be I I I I I I I I I I I I tAI 107~--~L--L~~~~~~---L--~-L-L~~~----~~--L-~~LU ~ 1~ ~ ~ *

*Fig. A9.2 AOF determination. 342 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE Chapter 10 *

*Solution 10.1 *

*Volume ofreservoir = V = 100 x (5280? x 500 cu.ft Volume ofreservoir available for fluid = (1 - Sw) cj>V = Vr = 0.65 x 0.12 x V *

*_ 2000 l520 _ 12 Vsc - 14.7 0.825595 - 15.6 x 10 SCF (1) Assume no water influx, *

*Initial moles in place ni = Pi . RTVr = (PVi)RT Z, r SId Vi - gas in place measured at standard conditions. a Abandonment moles left in place na =--n=P Vr (PVa)RT Za r SId Gas recovered !1n = -Vr (Pi - - -Pj = -Ps !1 V RTr Zi Za RTs Recoverable gas measured at stan dar conditions *

*=--Vr Ts (Pi--- Pa) TrPs Zi Za 500 At 500 psi, reduced pressure Ppr = 671.6 = 0.75 Z = 0.94 *

*Vr TsPi Z, Pa) Therefore recoverable gas = -TP 1 - - -P r s Zl Za I *

*_ X 12( 0.825 500) - 15.6 10 1 - 0.94 ·2000 *

*= 15.6 X 1012 (1 - 0.219) *

*= 12.2 X 1012 SCF 12.2 Recovery factor = 15.6 = 78% *

*Solution 10.2 *

*2nkoh 2nk h !1Pg Radial flow of oil q 0 = --B• Radial flow of gas qg = =..:::£.::B -~'-- flo 0 flg g re log - e rw and if the capillary pressure gradient is negligible, and the pressure drop over the same radii are considered, *

*~ _ kgfloBo qo - ko flgBg SOLUTIONS TO EXAMPLES 343 *

*To this must be added the gas evolved from solution in the oil. The total measured gas-oil ratio will then be: kg flo Bo + Rs ko flg Bg For the figures given: (96)(0.8)(1.363) (1000)(0.018)(0.001162) + 500 = 5005 + 500 = 5505 SCF/STB *

*Solution 10.3 *

*We = Np B, + Bg(Rp - Rsi) - N(B, - Bo;) *

*(i) At cumulative 1.715 x 106 BBL (P = 1600) Wei = (1.715 X 106) [1.437 + 0.0015(878 - 690)] - 14.5 X 106[1.437 - l.363] = 1.875 X 106 *

*(ii) At cumulative 3.43 X 106 BBL (P = 1300) *

*We2 = (3.43 X 106) [1.594 + 0.0019(996 -690)] - 14.5 [1.594 - l.363] = 4.112 x 106 *

*At P = 1000 estimated water influx = 6.375 x 106 (from trend) N(B, - Boi ) + We Np= B, + Bg(Rp - Rsi ) 14.5(1. 748 - l.363) x 106 + 6 375 000 1.748 + 0.0025(1100- 690) *

*= 4.312 X 106 BBL Solution 10.4 *

*Total hydrocarbon in place = i:n: ,-2h (1 - Sw) 9 750 x 0.17 x 0.76 = :n: 3(528W 5.615 = 4.54 x 109 BBL *

*Since bubble-point is 1850 psi, this must be pressure at any gas-oil contact. Elevation of gas-oil contact above oil-water contact is: *

*(1919 - 1850) 144 43.4 = 229 ft *

*This is less than hydrocarbon column so gas-oil contact exists at 4031 ft SS Height of gas zone = 750 - 229 = 521 ft r2h2 h3 (520)3 Ratio gas/total = ~h: = h~ = 750 = 0.34 *

*0.66 x 4.54 x 109 Therefore, oil in place = 1.363 = 2.198 x 109 STB 344 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

* m = 0.5 (= 113 +- 2/3) Material balance Np[B/ + BiRp - Rs;)] - We + Wp Bw N= mBoi (B/ - Bo;) + B. (Bg - Bgi) gl At 1600 psi: B/ + BiRp - Rs;) = 1.437 + 0.00150(1100 - 690) = 2.0520 m Boi 0.5 (1.363) B/ - Boi + Bgi (Bg - Bg;) = 1.437 - 1.363 + 0.00124 (0.0015 - 0.00124) *

*= 0.0740 + 0.1429 = 0.2169 *

*We = (2.052 X 3.1 X 108 + 31 X 106) - 2.198 X 109 X 0.2169 = 1.904 X 108 BBL *

*At 1300 psi: B/ + Bg(Rp - Rsi) = 1.594 + 0.0019(1350 - 690) = 2.8480 m Boi 0.5 (1.363) B/ - Boi + Bgi (Bg - Bgi) = 1.594 - 1.363 + 0.00124 (0.0019 - 0.00124) *

*= 0.5937 *

*We = 5.5 X 108 X 2.8480 + 55 X 106 - 2.198 X 109 X 0.5937 = 3.164 X 109 *

*This is not simply linear with pressure but extrapolation is reasonably straightforward and water influx at 1000 psi is estimated at 3.75 X 109 BBL B/ + BiRp - Rs;) = 1.748 + 0.0025(1800 - 690) = 4.523 m Boi (0.5)1.363 B/ - Boi + Bgi (Bg - Bgi) = 1.748 - 1.363 + 0.00124 (0.00250 - 0.00124) = 1.0775 (denominator term) N X denom.) + We - Wp Bw N=------'-----'-- P B/ + Bg (Rp - Rsi) 2.198 X 109 X 1.0775 + 3.75 X 108 - 63 X 106 4.5230 = 5.926 X 108 = 590 X 106 STB Solution 10.5 *

*. . . _ GBgi _ 120.7 X 109 X 6.486 X 10-4 _ Gas cap. OIl zone ratio m - NBoi - 300 X 106 X 1.3050 - 0.2 *

*From PVT data the values of Bo, Rs and Bg at 4300 psi can be estimated by linear interpolation as: Bo = 1.228 RBISTB; Rs = 338 SCF/STB; Bg = 7.545 X 10-4 RB/SCF SPE NOMENCLATURE AND UNITS 345 *

*From production data the value of Rp is calculated as GpfNp to give the following table. *

*Time pepsi) RpCSCFISTB) R.(SCFISTB) 1.1.80 5000 o o o 500 1.1.81 4300 o 21.9 550 338 1.1.82 4250 25.55 43.8 600 325 *

*Using the relationship F = N(ET) + We + WinjBwinj the following is calculated where: *

*ET = mEg + Eo + Efw *

*Units 1.1.B1 1.1.B2 (a) F = Np Bo + (Rp - Rs)Bg 106 RB 30.28 62.41 (b) Eo = (Bo - Boi) + (Rsi - Rs) Bg RB/STB 0.0420 0.0435 RB/STB 0.2131 0.2302 (c) E,~ Bo; [~-ll *

*RB/STB 0.0061 0.0065 (d) [c.$W + cf Efw = (1 + m) Boi t!..P 1 - Sw 1 *

*(e) ET = mEg + Eo + Efw RB/STB 0.0887 0.0960 *

*6 (f) We = F - N (ET) - Winj Bwinj 10 BBL 3.67 8.06 *

*Solution 10.6 The dimensionless radius ratio is: r aquifer 81000 re = =--=9 D r oil zone 9000 The dimensionless time tD is related to real time by: 2.309 k t (years) 2.309 (707t) tD = ql, ~ = (0.18)(7x 10-6) (0.4) (900W = 40t The instantaneous pressure drops which at the start of each year are equivalent to the continuous pressure declines are: *

*Pi - PI 5870 - 5020 . t!..Po = --2- = 2 = 425 pSI *

*Pi - P2 5870 - 4310 t!..P = ---= = 780 psi I 2 2 PI - P3 5020 - 3850 t!..Pz = --2-= 2 585 psi *

*The aquifer constant is: U = 1.119 f h c rb U = 1.119 x 1 x 0.18 x 200 x (7 x 10-6) x (900W U = 22841 BBLIpsi *

*From tables or charts for dimensionless influx at reo = 9 we have: *

*40 21 80 29 120 34 346 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE j=n-I From We = U L APWD (TD - (Dj) j=O *

*Wei = 22841 [425 (21)] = 203.9 X 106 BBL *

*We2 = 22841 [425 (29) + 780 (21)] = 655.7 X 106 BBL We3 = 22841 [425 (34) + 780 (29) + 585 (21)] = 1127.3 X 106 BBL *

*Chapter 11 *

*Solution 11.1 *

*PI=-1 [0.00708 k kro h 1 " re ,,"0 In -- 0.75 + S rw For re = 1500 ft rw=0.5ft S= +4 Kro = 0.6 h = 100ft k= 1325mD 50 PI=• !.to *

*0.5 5 50 500 5000 .. ------PI 100 10 1 0.1 0.01 *

*Solution 11.2 *

*The injectivity index is given in field units by: 0.00708 k krw h II=----~[~----~----- !'w In ~ - 0.75+ s] *

*Assuming all other factors equal then *

*Solution 11.3 *

*Use is made of the plot in Fig. 11.4 which correlates areal sweep efficiency E A as a function of end point mobility ration (M') for different fractional injection volumes, V D. Kw' !.to 0.4 3.4 M'=-·- =--·-=4 !.tw Ko' 0.4 0.85 The volume of injected fluid, in reservoir barrels, after 10 years is: 10 X 365.25 X 53 000 x 1.005 = 1.945 x 108 RB SOLUTIONS TO EXAMPLES 347 *

*The displaceable pore volume (= PV (I-SoT - Swi» is given in reservoir barrels as follows: *

*(4 x 5280) (1 x 5280) (98) (0.25) 5.615 [1 - 0.3 - 0.3] *

*= 1.946 x lOS RB 1.945 X 108 VD = 1.946 X 108 - 1 *

*From Fig. 11.4 the value of EA corresponding to M' = 4 and V D = 1 is 0.7 Solution 11.4 For stable cone formation ~ ' = g' X (Pw - Po) For ~' (in psi), and cone height X (in feet) and density difference as specific gravities then 62.4 ~' = 144 (1.01 - 0.81) 50 *

*= 4.33 psi *

*Chapter 12 Solution 12.1 *

*(a) In field units 1.25 (4000) U= 70(1500) 0.0476 BID - ft3 *

*The viscous-gravity force ratio is calculated from 2050 UItaL Rv_g = ( _ ) Po Ps kh 2050 (0.0476) (0.5) (1500) = (0.8 - 0.4) (130) (70) = 20 *

*(a) (b) Solvent *

*Oil Oil *

*Regions I and n Region m *

*Region I : Single gravity override tongue (c) Region n : Single tongue but sweepout independent of RV- G for given M Solvent Regionill: Transition region with secondary fingers below main tongue RegionN: Multiple fingers with sweepout independent of RV- Gfor given M Region N Fig. A12.1 Displacement regimes. 348 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*For a mobility ratio, M represented by !1ot'1ls (= 25), Figure A 12.2 shows a breakthrough sweep efficiency of about 15% and a flow dominated by gravity tonguing. (Fig A 12.1) *

*(b) In field units 1.25 (1000) 2 u = 30 (2000) = 0.0208 BID - ft *

*The viscous gravity force ratio requires an approximation of permeability as: *

* k=VKv·Kh :. k = «1) (3»°·5 = 1. 73md *

*Then 2050 (0.0208) (0.36) (2000) Rv_g = (0.75 - 0.64) (1.73) (30) = 5378 *

*For a mobility ratio of M (IlJIls = 0.36/0.055 = 6.55), Figures A 12.1 and A 12.2 show a breakthrough sweepout efficiency of around 50% and a flow dominated by viscous fingering. *

*100 *

*~ ~ >- u c: Q) :::·u Q) -; 60 t-Regionill-i------Region N 0 0- Q) Q) ~ til M=6.5 .£: CI> e:::J .£: ~ M =27 Region N 0 E co I-----Regionll---·I.... • -----Region ill------<-t-''-i *

*10 100 1000 10000 . . . .. 2050UJ-L L (B/O-FT2)(CP)(FT) Viscous-gravity force ratiO (RV- G)' field units, _ 0 , '----,3:;--'-'----'--'----' At kh (G/cm )(md) (FT) *

*Fig. A12.2 Breakthrough sweep efficiency. *

*Solution 12.2 The tie lines for the system join the equilibrium compositions of systems A and B in the two phase region. The compositions are plotted in Figure A12.3 (a) The critical point (CP) is estimated where the limiting tie line becomes tangential to the phase envelope and has the composition, wt%, 21 % surfactant, 67% oil; 12% brine. (b) The point with the composition 4% surfactant and 77% oil is given on Figure A12.3 as point A. From the slope of tie lines in this region the equilibrium phase compositions are AI and A2 with weight percents estimated as: AI 10% oil; 10% surfactant; 80% brine A2 97% oil; 2% surfactant; 1 % brine For an original 200 g mixture containing 8g surfactant, 154 g oil, 38 g brine SOLUTIONS TO EXAMPLES 349 *

*100% Surfactant *

*\ Wt% Wt% Brine Surfactant *

*100% 100t-~~~::;Z::::::::::::TW~O=p:~:se:,~eg~io:n:~~~~~~~~~~ 100% Brine 0 30 40 50 60 70 80 Oil Wt% Oil~ *

*Fig. A12.3 Ternary diagram. *

*The tie line ratios give: wt of AI phase 3/13 x 200 = 46 g wt of A2 phase 10/13 x 200 = 154 g :. Composition of AI = 4.6 g oil 4.6 g surfactant 36.8 g brine :. Composition of A2 = 149.5 g oil 3.0 g surfactant 1.5 g brine *

*(c) On Figure A 12.3 the composition 20% oil and 80% brine is shown at location B. A line from B to the 100% surfactant point leaves the two phase region at location B', having a composition oil 16.5%, surfactant 17.5%, brine 66%. The oil + brine weight is 100 g and would constitute 82.5% of the mixture, so surfactant needed is 0.175 (100/0.825) = 21.2 g. *

*(d) On Figure A 12.3, location 1 is 10% oil, 40% surfactant and location 2 is 50% oil, 40% surfactant. They are in a single phase region and the resulting mixture contains 30% oil, 40% surfactant and 30% brine, as denoted by position 3. *

*(e) On Figure A 12.3, location 4 is 12% surfactant, 5% oil and location 5 is 20% surfactant, 77% oil. The mixture weight is 200 g and contains 41 % oil, 16% surfactant, and 43% brine. It is shown as location 6. The mixture is in the two phase region and equilibriates to compositions C and D on the equilibrium tie line through location 6. The compositions are: C: 58% brine; 21.5% oil, 20.5% surfactant (146 g total) D: 94% oil; 5% surfactant; 1% brine (54 g total) 350 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE Solution 12.3 For conventional production *

*208.71AO.S tJ.o In I - 0.964 *

*(0.003541 (1000)(60» 700 = 161 rbld For 9 acre spacing and a 200 psi differential. 150 ... H2O':': (3») - 0964] *

*For thermal stimulation and steam injection a 5 fold improvement in flow resistance between producers and injectors would lead to rates around 800 bid. To determine the steady state production/injection time at which such rates will lead to 50% of the pattern volume being occupied by steam we can conduct the following analysis: The cumulative heat injected into the reservoir, Q;, can be calculated from heat injection rate, using the mass rate of injection Wi Qi [ ] t = Wi Cw ~T+ fsdhLYdh *

*= qinj (5.615) (62.4) [C w (380 - 100) + 0.75 (845)] *

*The average specific heat, Cw, over the temperature range 380 - 100°F is given by: C = hw(Ts) - hw(Tres) 355 - 69 w 1.02 Btullb m - degF Ts - T res 380 - 100 *

*:. Qi = t· qinj . 322118 Btu *

*The ratio of latent heat to total energy injected, fhv is calculated from: _ { Cw ~T )"1 _ { (1.02 (380 - 100) )"1 fhv - 1 + fsdb LVdb - 1 + 0.75 (845) = 0.689 *

*Figure A 12.4 can now be used to estimate the thermal efficiency of the steam zone, Ehs, at different values of dimenSionle[:sti]~e;sto. The values of to are given from: *

* to = 4t MR h2 *

*45]2 [0.75] = 4t [35 (60? = 0.00138t days or 0.504 t years *

*The following table may now be constructed usingfhv = 0.689 on Fig A 12.4. *

* t (yr) t(days) to 1.0 365.25 0.5 0.64 233.8 1.5 547.9 0.75 0.59 323.3 2.0 730.5 1.0 0.56 409.1 2.5 913.1 1.25 0.52 474.8 *

*The volume of a steam zone, V., is in general given by: QiEhs Vs = 43560MR AT SOLUTIONS TO EXAMPLES 351 *

*1.0 *

*.c:... W oJ c:: 0 N E c 2 If) 0.6 fhv (ratio latent heat to 15 total energy injected) = >- ()c:: 1.O .<3Q) 0.50 ;;:: 0.4 0.33 Q) A 0.23 -c 0.167 E 0.091 Q) .c:: 0.2 I-*

*0 0.01 0.1 100 Dimensionless time, tD *

*Fig. A12.4 Thermal efficiency *

*For the case of 50% steam volume in the pattern of area A acres then _ (43560 MR Qi - 0.5Ah E AT) hs *

*Equating values of Qi we obtain the relationship 0.5 (9) (60) (43560) (35) (280) 322118 qinj . t = E hs where t is in days 357817.5 That is *

*The injection rates needed to provide 50% pattern volume of steam at the following times are therefore as shown in the following table. *

* t (yr) qinj (rblD) 1.0 1531 1.5 1107 2.0 875 2.5 753 *

*These data may be further evaluated in terms of steam injection equipment capacity and project economics. *

*Solution 12.4 The wet condensate gas volume is obtained from the volumetric calculation: *

*Ahn (,5) V = g sc B . g. *

*In terms of standard cubic feet this is: 1 Vsc = s. [It(3 x 5280? 300 (0.18) (0.75)] gl 352 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*3.1937 X 1010 Vsc = B SCF g; *

*In order to find Bg; we need the super compressibility factor z which can be obtained from Fig 4.7 using the reservoir condition molecular weight or gas gravity. *

*The oil molecular weight is given by 44.3 PL Mo = (1.03 - PL) 141.5 NOWPL= API + 131.5 0.75 :.Mo = 119 *

*The weight associated with a stock tank barrel of liquid is given by: W = (5.615 x 62.4 = 0.75) 5000 (0.58) (28.97) + 379.4 = 262.78 + 221.44 = 484.22 *

*The number of moles associated with this weight is 5000 (62.4) (0.75) (5.615) n = 379.4 + 119 n = 13.18 + 2.21 n = 15.39 W 484.22 :. MW(res) = -;;= 15.39 = 31.46 MW(res) 31.46 and Yg(res) = 28.97 = 28.97 = 1.086 *

* i.e. Yg(res) = 1.09 *

*From Fig 4.7, P pc = 620 and Tpc = 465 *

*From reservoir datum conditions 4500 670 P pr = 620 = 726 and Tpr = 465 = 1.44 The dry gas volume So, from Fig 4.7 z = 0.925 G - [ 8.197 x 10 12] [5000/379.4]15.39 Then: (0.02829) (0.925) (670) G = (8.197 x J(p) (0.8563) g B ;= 4500 G = 7.019 X 1012 SCF = 3.8962 x 10-3 RCF/SCF Similarly the oil volume 3.1937 x 1010 Vsc 8.197 x 1012 Vsc = 3.8962 X 10-3 NX R = 5000 = 8.197 X 1012 SCF N = 1.639 X 109 STB SOLUTIONS TO EXAMPLES 353 Chapter 13 *

*Solution 13.1 *

*Using the relationship that the depth equivalent of the total head is equal to the sum of the depth equivalents of the well head pressure and the well depth, then: *

*DT = D whp + Dwell (a) From Fig. A 13.1 at a well head pressure of 400 psi then DwhQ = 3700 ft. Since Dwell = 6000ft then DT = 9700 ft. At the GOR of 200 scf/stb the pressure at a depth equivalent of Y700 ft is read as 2400 psi. (b) From Fig. A 13.2 at the bottom hole pressure of 1200 psi and GOR of 500 scflstb the depth equivalent Dr. is read as 8900 ft. Since Dwell is 5000 ft then Dwhp is 3900 ft. The well head pressure is read from the graph at 3900 ft as 360 psi. *

*Vertical flawing_pressure gradients (all oil) Vertical flowing_pressure gradients (all oill Tubing Size 4 in.I.D. Producing Rate 2000 Bbls/day Tubing Size 4 in.I.D. Oil API Gravity 35° API Producing Rate 3000 Bbls/day Gas Specific Gravity 0.65 Oil API Gravity 35° API 3 Average Flowing Temp. 140° F Gas Specific Gravity 0.65 3 Average Flowing Temp. 140°F Q; Q; ~ 4 ~ 4 0 0 0 0 $2 $2 5 .!: .!: 5 .c C. .c c: C. Q) c: 6 Q) ..J ..J 6 *

*7 7 *

*8 8 *

*9 9 10 10 *

*Fig. A13.1 Fig.A13.2 *

*Solution 13.2 The maximum production rate qrnax can be evaluated using the Vogel relationship, withp, the static pressure, i.e. *

* q/qm~~ H.2 [~]- 08 [~]' *

*~ 1 - 0.2 [~: ]-08 [= r = 0.619 3315 therefore, qrnax = 0.619 = 5355 bid 354 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

*Pressure in 100 PSIG *

*Verlical flowing_p.ressure gradienls (all oil) Verlical flowing_pressure gradienls (all oil) Tubing Size 4 in.I.O. Tubing Size 4in.1.0. Producing Rale 4000 8bls/day Producing Rale 1000 8bls/day Oil API Gravily 35° API Oil API Gravily 35° API Gas Specific Gravily 0.65 Gas Specific Gravily 0.65 Average Flowing Temp. 140°F 3 Average Flowing Temp. 140°F 3 Q; ~ 4 0 Q; 4 0 ~ $2 0 .S; 5 0 .t:: $2 5 "6> c .S; Q) ...J 6 .t:: "6>c 6 Q) ...J 7 7 8 8 9 9 10 *

*10 *

*Fig. A13.3 *

*Fig. A13.4 *

*Verlical flowing_pressure gradienls (all oil) Tubi~g Size 4 in. 1.0. Producing Rale 5000 8bls/day Oil API Gravily 35° API Gas Specific Gravily 0.65 Average Flowing Temp. 140°F 3 *

*Q; ~ 4 0 0 $2 .S; 5 .t:: "6> c Q) 6 ...J *

*7 *

*8 *

*9 *

*10 *

*Fig.A13.5 SOLUTIONS TO EXAMPLES 355 *

*From Fig. A 13.1 to A 13.5 the different vertical flowing pressure gradient curves at different rates are found for 4 in. tubing and a GOR of 200 SCF/STB. The total head depth is obtained as the sum of the well depth and the depth equivalent to a tubing head pressure of 400 psig. The flowing bottom hole pressure equivalent to the total head depth is recorded as a function of flow rate. It can be seen that the bottom hole pressure is essentially independent of rate at this condition and is 2200 PSi[. (2200 ) (2200 )2] Hence q = qmax 1 - 0.2 2600 - 0.8 2600 *

*= 1400 bid Solution 13.3 For a residence time of 3 min. the volume of oil in the separator will be: (1000) (3) 3 Vo = (24) (60) = 2.083 m At 40°C and 20 bar the volumetric rate of associated gas will be V (1000) (95) (313.15) (1) 3 --II. = (24) (60) (60) (273.15) (20) = 0.06303 m Is *

*At separator conditions the gas density Pg is given by (273.15) Pg = 1.272 (0.75) (20) (313.15) *

*= 16.682 kg/m3 The maximum velocity equation is then used: 796 - 16.682]0.5 Umax = 0.125 [ 16.682 mls *

*= 0.8544m/s Since cross-sectional area = volume ratelvelocity then for an interface half way up the separator we have: n D2 0.06303 (2) (4) 0.8544 :.D = 0.4334 m Total volume of the separator is thus twice the oil volume for an interface half way up the separator :. Vsep = 2Vo = 4.166 m3 Design length for LID = 3 gives (4.166) (4) 3D = L = nD2 . 3_(4.166)(4) .. D - 3n *

*:. D = 1.209m and L = 3.627 m Design length for LID = 4 gives D3 = (4.166)(4) 4n :. D = 1.099 and L = 4.396 m In practice the separator design would be based on a standard size selected to be nearest the size calculated. Index *

*Abandonment pressure 159 Capillary number 191,193 absolute permeability 102 capillary pressure 93 AFE (authorisation for expenditure) document 23, 24-S and residual fluids 111-12 Amerada gauge 147, 148 defined 92 API (American Petroleum Institute) gravity and oil density 14 capillary pressure data (given rock type, correlation 99 aquifer characteristics capillary pressure hysteresis 97-8 correlation with model 167 capillary suction pressure see imbibition wetting phase threshold determination of 165-6 pressure aquifers and pressure change 165 carbon dioxide in miscible displacement 195, 196 areal sweep efficiency 176, 182-3 casing a well, reasons for 2S casing eccentricity 35-6 Back pressure equation 143-4, 221 casing selection 27 barrel 14 main design criteria 28 bedforms, grain size and stream power 242 casings 23, 25, 26, 28 biocides and injection water 229 caustic solutions 196 biopolymers 197 cementation problems 35-6 black oil reservoir modelling, uncertainties in 24&-7 chemical flood processes 196-200 black oil systems 42 choke assembly 146 blow-out 35 Christmas tree 36 blow-out preventers 34-5 coalescer 227 blowdown 210 Coates and Dumanoir equation 86 Boltzmann transformation 134 combination drive material balance equation 166 bond number 191 compaction drive 161 BOPs see blow-out preventers complete voidage replacement 173 bottlenecks 219 completion 28,29 bottom-hole sampling 52 completion for production (permanent, normal) 36 Boyle's law method and grain volume 73 composite cores 111 Brent Sand reservoirs 10, 11 compressibility 42-3,55 brine disposal 186 Compton scattering 76 bubble-point 41, 51, 53, 54, 55,159,220,221 conceptual models 233,245 bubble-point pressure 52, 54-5, 56-7,160,163,221 condensate analysis 208 in volatile oil reservoirs 211 condensate reservoirs and liquid drop-out 208 Buckley-Leverett theory 105 condensate systems 42 Buckley-LeverettlWelge technique 107, 109 condensing gas drive 194-5 cone height, critical 182 coning 181-2 core analysis and permeability distribution 83-4 Footnote: Numbers in italic indicate figures; Numbers in bold routine 69-71, 81 indicate tables presentation of results 70,71 *

*357 358 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE core data and palaeogeographical reconstruction 237-8 displacement principles 173-5 and recognition of sand body type 238 drawdown testing 138 core-derived data 68 drill bits 22, 32-3 core floods and surfactant testing 200 drill collars 23 core for special core analysis 67, 68 drill stem testing 145 core length and imbibition processes 110-11 testing tools and assemblies 145-7 core log 64,68 drilling, turbine versus rotary 33 core plug experiments, concern over drilling costs 23, 24, 25 laboratory-derived data 113-14 drilling fluid see drilling mud core plugs 68 drilling logs 30 analysis on 65 drilling mud pressure, excessive 29 and effective permeability 109 drilling muds 22-3 and fluid saturation 93-4 control of 28-9 and oil saturation 193 main constituents 67 and permeability 81 drilling muds and cements, rheology of 29-30 and porosity 72 drilling optimization 32-3 and residual saturation 174 drilling, special problems in cementation problems 35--{i core porosity, compaction corrected 131 pressure control and well kicks 34-5 core preservation 67-8 stuck pipe and fishing 33-4 core recovery, fluids for 31 drillstring 23 Coregamma surface logger 68 drive mechanisms 159 cores 62 dry gas reservoirs 41-2 composite 111 dual porosity systems 71,73 correlation with wireline logs 63,65,75 and gravity drainage 164-5 data obtainable from 63 diversity of information available 64 Early (transient) time solution 138 and geological studies 68-9 economic factors and oil production rates 180 and heavy oil reservoirs 202 effective permeability 102 residual fluid saturation and wettability 108 determination 69 enhanced oii recovery schemes and uncertainty 247 coring equity, distribution of, petroleum reservoirs 130-1 the case for 65 exploration well drilling 7, 8 conventional and oriented 66 of development wells 65--{i Faults, identification of 238 of exploration wells 65 faults (in-reservoir), effect on injection/production well locations coring decisions 64--{i 180 coring mud systems 66-7 field processing 224 corresponding states, law of 44-5,47 filtration, injection water treatment 229 Cricondenbar 41 , 42 flash liberation at reservoir temperature 52-3 Cricondentherm 41 flash separation tests 53-4 critical displacement rate 177 flooding efficiency ratio 110 critical displacement ratio 112 flow equations, linear and radial 80-1 critical gas (equilibrium) saturation 159 flow string 145 critical production rate (coning) 182 fluid contacts 12-13 crude oil multiple 12 flow of in wellbore 221, 223 fluid flow in porous media 78-9 metering of 229 fluid pairs 93 processing 226-8 fluid pressure and overburden load 11-12 cushion 147 fluid pressures, hydrocarbon zone 12-13 cuttings logs 31 fluid saturation, laboratory measurements and relationship with cyclic steam stimulation 205 reservoir systems 93--{i fluids, recovery of by depletion 211 Darcy (def. )79 Forcheimer equation 143 Darcy's equation 79 formation breakdown pressure 30 data acquisition during drilling 30-1 formation density logs and interpretation of porosity 202-3 datum correction 79-80 formation density tool response 75--{i deltaic environments, division of 238,240, 241,242 formation factor see formation resistivity factor deltaic models, use of 238-43 formation interval tester (FIT) 148 deltaic system model 242, 244 formation resistivity factor 74 demulsifiers and heavy oil processing 228 formation tester (FT) 148 depositional processes and reservoir rocks 7 formation volume factor 14, 55 dew-point 41 two-phase 55--{i dew-point locus 42 formation volume factors B 49-51 diamond coring 33 formation waters 14 differential liberation at reservoir temperature 53 fractional flow 104--{i displacement calculations, validation of relative permeability data analysis methods 105--{i for 113-14 effect of dip angle and wettability 175, 177 free water level (FWL) 12,95 INDEX 359 *

*Gas cap expansion drive 163-4 hydrocarbon volume in place calculations 127-8 gas compressibilities 48-9 hydrocarbons, migration of (modelled) 93-4 gas condensate, critical properties of 210 hydrocarbons (commercial reservoirs), gas condensate and volatile oil reservoirs, geological characteristics 62 uncertainties in 247 hydrostatic gradient, regional 10-11 gas condensate reservoirs 207-11 production methods for 209-11 Ideal gas law (and modification) 43 gas deviation factor Z 46, 47 imbibition processes gas expansion during production 157 and core length 110-11 gas flow and gradient 159 liquid 104 gas flow and permeability 81 imbibition wetting phase threshold pressure 97 gas flow rate, measurement of 150, 229 in-place volume 122 gas formation volume factor 157 inflow performance relationship, 220 gas formation volume factor Bg 49-50 dimensionless, for oil wells 220-1 gas properties 45 for gas wells 221 gas recycling, gas condensate reservoirs 210 injection fluids, compatibility with reservoir fluids 183-4 gas reinjection 186 injection fluids, quality of 183-6 gas reservoirs, recovery from 157-9 injection water, viscosity of 184 gas viscosities 47-8 injection water treatment 229 gas-kicks 12 injectivity index 174, gas-oil ratio 14,51-2,54,159 insert bits 33 gas-oil systems and relative permeability 103-4 isobaric thermal expansion coefficient 43 gas well testing 143-5 isocapacity maps 126 gases, behaviour of 43-4 isochores 124 gases, flow of in wellbore 221 isochronal testing 144 geological model, development of 237-8 isoliths 124 geothermal gradient and hydrocarbon generation 7, 9 isopachs 124 geothermal gradient and reservoir temperature 13 isoporosity maps 125 GOR see gas-oil ratio isosaturation lines 99 grain density 71 isosaturation maps 126 grain volume and Boyle's law method 73 isothermal compressibility 43 gravity drainage and dual porosity systems 164-5 isothermal retrograde condensation 42 gravity segregation and recovery efficiencies 164-5 gravity stabilization and reservoir dip 175 Kay's rule 45 Head loss in wellbores 221 kelly 23 heavy crude oil kick 34-5 Kimmeridge Clay 7, 9 characteristics of UKCS heavy crude oils 201 general classification 200 Klinkenberg correction 81, 82 Yen classification 200, 201 Kolmogorov-Smirnoff test 84 heavy oil processing 228 heavy oil recovery 200-2 Lasater correlation (bubble-point pressure) 55 heavy oil reservoirs leak off tests 30 examples of 201 Leverett J-function correlation 99 permeability increase and production improvement 204 light oil processing 226 production characteristics of 203-4 foaming problems 227-8 properties of 202-3 separator design considerations 227 and thermal energy 204-7 wax problems 228 and uncertainty 247 line source solution (fluid flowing in a porous medium) 134-5 heavy oil systems and thermal energy addition 204 development of 135-6 HKW (highest known water) 12,13 liquid drop out 208 homogeneous reservoirs and coning 181-2 liquids systems, generalized correlations 54-8 Horner analysis 13 lithofacies representation 125 hydrates 224 LKO (lowest known oil) 12,13 hydrocarbon accumulation and sedimentary basins 7 low interfacial tension (Iff) systems 193 hydrocarbon accumulations and formation waters 14 hydrocarbon exploitation, types of interactions 16 Material balance, reservoirs with water encroachment or water hydrocarbon field 7 injection 165-8 hydrocarbon generation and geothermal gradient 7, 9 material balance calculations hydrocarbon pore thickness (HPT) 126--7 generation of data 52 hydrocarbon pore volume maps 126--7 sources of error 168-9 hydrocarbon properties 47 material balance equation 158 hydrocarbon recovery, improved 191-211 combination drive 166 hydrocarbon reservoir fluids 15 gas cap expansion drive 163-4 hydrocarbon systems solution gas drive 161-3 volumetric and phase behaviour 40-1 material balance residual oil saturation 174 applications to field systems 41-2 mathematical models 233-4 360 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE mercury injection and porosimetry 73,96,97 oil saturation, local, influences on 191 meters 229 oil viscosity 56 microemulsion 198 oil-water contact (OWC) 96, 98-9 middle (late transient) time solution 139 oil-water systems and relative permeability 102-3 miscible displacement mechanisms 194-5 open-hole tests 145 miscible displacement processes 193 optimal salinity 198 miscible floods 194 orifice meters 229 applications 195-6 overpressure 11, 12 examples 196 miscible fluids, properties of 195 Packer 146 mobility ratio 104-5, 107, 175,176 Peng and Robinson equation 44 and polymers 197 permeabilities, averaging of 83 modelling of reservoirs 130-1 permeability 7, 78-86 models 233--4 and critical displacement ratio 112 mole (def.) 44 anistropy 82-3 Monte Carlo distributions 83--4 approach, probabilistic estimation 127 improvement 193--4 technique and recoverable reserves estimate 130 laboratory determination of 81-2 movable hydrocarbon formula (MHV) 130 ratios 104-5 mud cake 36 variation, effects of 106-8 mud circulation system 22, 23 permeameter 81 mud composition, general limitations on 67 petroleum mud logging 30-1 migration of 9-10 mud systems, bland (unreactive) and core recovery 31-2,67 origin and formation of 7 multicomponent systems, phase behaviour 41 recovery 5 multimodal porosity 78 petroleum engineering multirate data, analysis of 144-5 function of 1 multiphase flow, equations of 234-5 problem solving in 3 phase (def.) 14 Natural gas phase inversion temperature (PIT) 198 calorific value 226 physical models 233 dehydration 224-5 piston displacement, stratified reservoirs 107-8 onshore processing 225-6 planimeter 124, 127 sales specification 224 polyacrylamides 197 sweetening 225 polymer fluids 193 natural gas processing 224-6 polymer systems and adsorption 197 nitrogen in miscible displacement 195, 196 pool see reservoir non-wetting phase fluid 94 pore fluid pressures 11 non-wetting phase saturation 102 pore pressure, significance in drilling and well completion 26, 28 North Sea, heavy oil reservoirs 202 pore size distribution 96-7 North Sea, hydrocarbon fields pore space characteristics and equilibrium saturation distribution Beryl field 196 92-3 Brent field 196 pore volume compressibility 160 Buchan field 37 of reservoir rocks 203 Dunlin field 131,178 poro-perm data, validity of 242 Forties field 249 porosity 7, 71-8 Fulmar field 249-51 and permeability, relationship between 84-6 Magnus field 184 cut-off 124 Maureen field 187 distributions 77-8 Montrose reservoir (RFf data) 151 logs 75-7 Murchison field 125 main logging tools for 75 Rough gas field 123, 124, 126, 127 measurement of 72-3 Statfjord field 196, 245, 246 potential gradient 174 Thistle oil reservoir 122, 123, 125 pressure (abnormal) and d-exponent 25-6 North Sea, oil correlations, recent 56-8 pressure build-up analysis 139-40 North Sea, reservoirs, fluid choice for miscible displacement pressure build-up (testing) 149 196 pressure control and well kicks 34-5 North Sea, reservoirs and surfactants 198, 199 pressure decline, rates of 137 pressure depletion 210 ODT (oil down to) 13 pressure drawdown and reservoir limit testing 142-3 offshore production/injection system, pressure equilibrium, static system 12 principle components of 184,185,186 pressure gauges 137, 147 offshore system 21 (downhole), characteristics of 136 oil bank formation 195 pressure gradients and heterogeneity of reservoir pore space 129 oil density 14 pressure maintenance 173 oil flow rate, measurement of 150 pressure regimes, abnormal 11-12 oil formation factor Bn 51 primary recovery, oil reservoirs 159-64 INDEX 361 probabilistic estimation 127-8, 129, 130 reservoir rocks, characteristics of 62-86 produced fluids and offshore processing 184-{5 pore volume compressibility 203 produced water treatment 228 reservoir simulation modelling 233-7 producing rates (well inflow equations/pressure loss calculations) reservoir simulation and vertical communication 243, 245 174-5 reservoir temperatures 13 production costs, significance of 1, 3 reservoirs 7-18 production engineering, and well performance 220-1 areal extent of 122-4 production engineering described 218 residual oil 53, 191 production operations, influencing factors 218-29 influence of recovery mechanism 191, 193 production rate effects 180-2 residual oil saturation 192 production rates, technical and economic factors 219 average 174 production system 218-19 and material balance 174 production testing 150-1 measurement of 191, 192 productivity index (PI) 245 residual saturations 111-112 and inflow performance 220 resistivity factor see formation resistivity factor pseudo-critical temperatures and pressures 45-7 resistivity index 74 pseudo-relative permeability in dynamic systems 115 retrograde condensation 208 pseudo-relative permeability functions 177,178, 243,245 reverse circulating sub 146 static 115-16 rotary table 23 pseudo-relative permeability relationships and thicker sands 107 PVT analysis 52-4 Safety joints and jars 147 PVTrelationships, single and multicomponent systems 40-1 salinity and water viscosity 56 samplers 147 Radial equations in practical units 136 sand body continuity 180 radial flow in a simple system 134-5, 137 importance of 238,239-40 recombination sampling 52 sand body type recovery efficiency, water reservoirs 168 effect on injected water and oil displacement 178-80 recovery factors and reserves 128-30 recognition of 238 recovery string 34 saturation distributions in reservoir intervals 98-9 recovery targets 191 saturation gradients 164 Redlich-Kwong equation 44 saturation pressure see bubble-point pressure relative permeability 102-4,106-7 scribe shoe 66 effect of temperature 204 sea water as injection water 184 relative permeability seawater floods (continuous) and low surfactant concentration data, laboratory determination of 109-11 199-200 from correlations 112-13 secondary recovery and pressure maintenance 173-86 improvement, heavy oil reservoirs 204 secondary recovery techniques 173 relative spreading concept 93 sedimentary basins repeat formation tester (RFf) 148-50 and hydrocarbon accumulation 7 reservoir behaviour in production engineering 220-1 origin of7 reservoir condition worldwide 2 material balance techniques 160 segregated displacement 177 volumetric balance techniques 160-1 sensitivity studies 246-7 reservoir data, sources 14-15, 17 shaliness, effect of 13 reservoir (def.) 7 Shinoda diagrams 198 reservoir description in modelling 237-45 simulators uncertainty in 245-7 applications 235 reservoir development, costs of3, 4 classification of 235,236 reservoir dip angle 175,177 single component systems, phase behaviour 40-1 reservoir flow rate, effect of 181 skin effect 140-2 reservoir fluid properties, negative factors 142 measurement and prediction of 43-9 skin zone 194 reservoir fluids slabbing 68 and compressibility 42-3 solution gas drive, analysis by material balance 159-63 nature of 14 solution gas-oil ratio 53, 54, 55 properties of 40-58 Standing-Katz correlations 46, 47 reservoir geometry and continuity 180, 238-45 Standing'S data (bubble-point correlation) 55 reservoir heterogeneity 177-80 STB (stock tank barrel) 14 reservoir mapping and cross-section interpretation 245-6, 247 steady state permeability tests 110 reservoir modelling steam flooding 205 analysis and data requirements 237 steam properties 206, 207 application in field development 248-51 steamdrive analysis, example data requirements 207 concepts in 233-48 Stiles technique 107-8 reservoir performance analysis 157-68 stock tank oil 54 reservoir pore volume and change in fluid pressure 42-3 and retrograde condensation 208 reservoir pressures 10-12 stock tank oil in place and equity 362 PETROLEUM ENGINEERING: PRINCIPLES AND PRACTICE *

* determination 130 Walther's law offacies 238 stock tank units 14 water drive and gas condensate reservoirs 209,210 stock tank volume 53 water drive reservoirs 167 Stratapax bits 33 recovery efficiency of 168 stratified reservoir analysis 106 water formation factor Bw 50-1 stripping 191 water influx 165, 166 structure contour maps 122 water influx, gas reservoir 158-9 stuck pipe and fishing 33-4 water injection 166, 178 summation of fluids and porosity 72-3, 74 water saturation distribution, homogeneous reservoir 96 superposition technique 140 water viscosity 56 surfactant concentration (low) and waterflooding 178, 179,180 continuous seawater floods 199-200 Welge analysis 106 surfactant flooding 198-200 Welge's equations 174 surfactant phase systems 197-8 well arrangements, dipping reservoirs 181 surfactant processes 197-200 well classification 20 surfactants 193 well description log 31, 32 synthetic 199 well drilling operations 20-3 sweetening, natural gas 225 well locations and patterns 182-3 well performance, radial flow analysis of 134-51 Tester valve 146 well productivity improvement 193-4 thermal energy 204-7 well test methods, applications of analytical solutions 136-9 thermal injection processes 204-6 well test procedures 145-50 thickness maps 124 data analysis 147-8 threshold capillary pressure (reservoir rocks) 95 well testing and pressure analysis 150-1 threshold pressure 94 well/reservoir responses, different reservoir systems 139 traps (structural and stratigraphic) 10 wellbore, altered zone 141 tricone bits 32, 33 wellbore flow 221-3 trip gas 34 wellbore inflow equations 174 turbine meters 229 wellsite controls and core recovery 68 wettability 175 Ultimate recovery formula see movable hydrocarbon formula change in 67, 196 uncertainty in reservoir model description 245-8 degree of 93 unitization 130-1 wettability control, in situ 112 universal gas constant, values of 43 wettabilityeffects 108 unsteady state relative permeability tests 109-10 wettability preference 93 USA, heavy oil resource distribution 202 wetting phase fluid 93 wetting phase saturation 94 Van der Laan method (volume in place) 128 wetting preference 175 vaporizing gas drive 194, 195 wireline logs, correlation with cores 63, 65, 75 vapour phase 42 wireline testing 148-50 vertical bed resolution 76 WUT (water up to) 13 vertical permeability variation and fractional flow curve 177 vertical pressure logging 148-50 Xanthan gums 197 Viking Graben area (N North Sea) 10 Vogel dimensionless IPR 220-1 Zonation 99, 131,242,243,245 volatile oil reservoirs 211 Forties reservoir 249 volatile oil systems 42 and geological core study 68-9 volumetric balance techniques 160 and histogram analysis 84 vugular carbonates and whole core analysis 69 and permeability distributions 84 *