The application of the Laplace transformation to flow problems in reservoirs

T.P. 2732 THE APPLICATION OF THE LAPLACE TRANSFORMATION TO FLOW PROBLEMS IN RESERVOIRS A. F. VAN EVERDINGEN, SHELL OIL CO., HOUSTON, AND W. HURST, PETROLEUM CONSULTANT, HOUSTON, MEMBERS AIME ABSTRACT For several years the authors have felt the need for a source from which reservoir engineers could obtain fundamental theory and data on the flow of fluids through permeable media in the unsteady state. The data on the unsteady state flow are composed of solutions of the equation O'P + ~ oP = oP or' r Or at Two sets of solutions of this equation are developed, namely, for "the constant terminal pressure ca;;e" and "the constant terminal rate case." In the constant terminal pressure case the pressure at the terminal boundary is lowered by unity at zero time, kept constant thereafter, and the cumulative amount of fluid flowing across the boundary is computed, as a function of the time. In the constant terminal rate case a unit rate of production is made to flow across the terminal boundary (from time zero onward) and the ensuing pressure drop is computed as a function of the time. Considerable effort has been made to compile complete tables from which curves can be constructed for the constant terminal pressure and constant terminal rate cases, both for finite and infinite reservoirs. These curves can be employed to reproduce the effect of any pressure or rate history encountered in practice. Most of the information is obtained by the help of the Laplace transformations, which proved to be extremely helpful for analyzing the problems encountered in fluid flow. Tht' application of this method simplifies the mOTe tedious mathe- matical analyses employed in the past. With the help of La- place transformations some original developments were ob- tained (and presented) which could not have been easily foreseen by the earlier methods. INTRODUCTION This paper represents a compilation of the work done over the past few years on the flow of fluid in porous media. It concerns itself primarily with the transient conditions prevail- ing in oil reservoirs during the time they are produced. The study is limited to conditions where the flow of fluid obeys the Manuscript received at office of Petroleum Branch January 12, 1949. Paper presented at the AIME Annual Meeting in San Francisco, Febru- ary 13-17. 1949. 1 References are given at end of paper. diffusivity equation. Multiple-phase fluid flow has not been considered. A previous publication by Hurst' shows that when the pres- sure history of a reservoir is known, this information can be used to calculate the water influx, an essential term in the material balance equation. An example is offered in the lit- erature by Old' in the study of the Jones Sand, Schuler Field, Arkansas. The present paper contains extensive tabulated data (from which work curves can be constructed), which data are derived by a more rigorous treatment of the subject mat- ter than available in an earlier publication. ' The applicatIon of this information will enable those concerned with the analysis of the behavior of a reservoir to obtain quantitatively correct expressions for the amount of water that has flowed into the reservoirs, thereby satisfying all the terms that appear in the material balance equation. This work is likewise applicable to the flow of fluid to a well whenever the flow conditions are such that the diffusivity equation is obeyed. DIFFUSITY EQUATION The most commonly encountered flow system is radial flow toward the well bore or field. The volume of fluid which flows per unit of time through each unit area of sand is expressed by Darcy's equation as K oP v = fJ. Or where K is the permeability, fJ. the viscosity and oP lor the pressure gradient at the radial distance r. A material balance on a concentric element AB, expresses the net fluid traversing the surfaces A and B, which must equal the fluid lost from within the element. Thus, if the density of the fluid is ex- pressed by p, then the weight of fluid per unit time and per unit sand thickness, flowing past Surface A, the surface near- est the well bore, is given as 2~rp ~ ~~ = 2~fJ.K ( pr ~~) The weight of fluid flowing past Surface B, an infinitesimal distance or, removed from Surface A, is expressed as oP o( pr g; ) [pr - + or or 2~K or] December, 1949 PETROLEUM TRANSACTIONS, AIME 305 T.P. 2732 THE APPLICATION OF THE LAPLACE TRANSFORMATION TO FLOW PROBLEMS IN RESERVOIRS The difference between these two terms, namely, o( pr 'O_~) 27rK or - -- -------- or, p or is equal to the weight of fluid lo:t by the element AB, ()j' OP - 27rfr -- or aT where f is the porosity of the formation. This relation gives tf:e equat:on of continuity for the radial system, namely, a (pr .Q~-) K Or OP - ---- fr --- (II-I) p or aT From the physical characteristics of fluids. it is known that density is a function of pressure and that the density 01 a fluid decreases with decreasing pressure due to the fact that the fluid expands. This trend expres~ed in exponential form is p = p"e-"(I',,-I') (II-2) where P is less than P,,, and c the compressibility of the fluid. If we substitute Eq. II-2 in Eq_ II-I, the diffusivity equation can be expressed using density as a function of radius and time. or ( 02p + 2:.. 2!_) ~_ = ~_ (I1-3) or' r Or fllc aT For liquids which are only slightly compressible, Eq. II-2 simplifies to p ~ Po [1- c (Po - P)] which further modifies Eq. 1I-3 to give ( o~_ -+ _1 __ OP ) ~ = 1l.!'... Furthermore, if the or- r or fpc aT radius of the well or field. R h, is referred to as a unit radius, then the relation simplifies to o'P 1 oP oP - - + -- -- == ------ or' r Or at (II-4) where t = KT /fJlcR,,' and r now expresses the distance as a multiple of R h , the unit radius. The units appearing in this paper are always med in connection with Darcy's equation, so that the permeability K must be expressed in darcys; the time T in seconds. the porosity f as a fraction, the viscosity f' in centipoises. the compressibility c as volume per volume per atmosphere, and the radius Rb in centimeters. LAPLACE TRANSFORMATION In all publications, the treatment of the diffusivity equation has been essentially the orthodox application of the Fourier- Bessel series. This paper presents a new approach to the solution of problems encountered in the study of flowing fluids, namely, the Laplace transformation, since it was recognized that Laplace transformations offer a useful tool for solving difficult problems in less time than by the use of Fourier- Bessel series. Also, original developments have been obtained which are not easily foreseen by the orthodox methods. If p(t) is a pressure at a point in the sand and a function of time, then its Laplace transformation is expressed by the infinite integral (III-l) where the constant p in this relationship is referred to as the operator. If we treat the diffusivity equation by the process implied by Eq. Ill-I, the partial differential can be trans- formed to a total differential equation. This is performed by multiplying each term in Eq. II-4 by e-'" and integrating with respect to time between zero and infinity, as follows; 'L _ . ., (o'P 1 oP ) ,ie' -,-+--- o Or- r or x oP dt = f e-;'t --dt o· at (III-2) Since P is a function of radius and time, the integration with respect to time will automatically remove the time function and leave P a function of radius only. This reduces the left side to a total differential with respect to r, namely, x O'l' J e-:" oar' Jo a')' 1 e-JO ' P dt f d'P,JO) dt = -----._- = _.- or' dr' and Eq. HI-2 hecomes dr' P, PRESSURE q(t), RATE I dP"" r dr dP dt t, t2 t3 t, TIME dt etc. FIG. lA - SEQUENCE CONSTANT TERMINAL PRESSURES. 1 B - SEQUENCE CONSTANT TERMINAL RATES. 306 PETROlEUM TRANSACTIONS, AIME December, 1949 A. F. VAN EVERDINGEN AND W. HURST T.P. 2732 Furthermore, if we consider that P (l) is a cumulative pressure drop, and that initially the pressure in the reservoir is every· where constant so that the cumulative pressure drop p(t~O)=O, the integration of the right hand side of the equation becomes dP 00 As this term is also a Laplace transform, Eq. III·2 can be writ· ten as a total differential equation, or d'P(p) + 1 dP,p) dr' r dr (III.3) y 8 i! PLANE c~ ________ ~----~ --------------________ hM __ ~~(T~O~)--x Dr-----~~------~ A FIG. 2 - CONTOUR INTEGRATION IN ESTABLISHING THE CONSTANT TERMINAL RATE CASE FOR INFINITE EXTENT. y i! PLANE -1~~rt-+-1~-+-+~~~4-~~--+---x (cr ,0) FIG. 3 - CONTOUR INTEGRATION IN ESTABLISHING THE CONSTANT TERMINAL RATE CASE FOR LIMITED RESERVOIR. The next step in the development i, to reproduce the boun· dary condition at the wdl bore or field radius, r = 1, as a Laplace transformation and introduce this in the general solu· tion for Eq. III·3 to give an explicit relation By inverting the term on the right by the Mellin's inversion formula, or other methods, we obtain the solution for the cumulative pressure drop as an explicit function of radius and time. ENGINEERING CONCEPTS Before applying the Laplace transformation to develop the necessary work·curves, there are some fundamental engineer· ing concepts to be considered that will allow the interpreta· tion of these curves. Two cases are of paramount importance in making reservoir studies, namely, the constant terminal pressure case and the constant terminal rate case. If we know the explicit solution for the first case, we can reproduce any variable pressure history at the terminal boundary to deter· mine the cumulative influx of fluid. Likewise, if the rate of fluid influx varies, the constant terminal rate case can be used to calculate the total pressure drop. The constant terminal pressure and the constant terminal rate calOe are not inde· pendent of one another, as knowing the operational form of one, the other can be determined, as will be shown later. Constant Terminal Pressure Case The constant terminal pre3sure case is defined as follows: At time zero the pressure at all points in the formation is con· stant and equal to unity, and when the well or reservoir is opened, the pressure at the well or reservoir boundary, r = 1, immediately drops to zero and remains zero for the duration of the production history. If we treat the constant terminal pressure case symbolically, the solution of the problem at any radius and time is given by P = p(,.,t). The rate of fluid influx per unit sand thickness under these conditions is given by Darcy's equation q(T) = 21TK (r OP) " (IV.I) /L or r = 1 If we wish to determine the cumulative influx of fluid in absolute time T, and having expressed time in the diffusivity equation as t = KT/f/LcRb" then T 21TK f,acRo' t Q('I') = f q(T) dT = --x-~ J o· /L K 0 = 21TfcR h 2 Q(t) where ( OP) -- dt or r = 1 (IV·2) Q«) = / (OP ) dt (IV.3) o or r = 1 In brief, knowing the general solution implied by Eq. IV·3, which expresses the integration in dimensionless time, t, of the pressure gradient at radius unity for a pressure drop of one atmosphere, the cumulative influx into the well bore or into the oil.bearing portion of the field can be determined by Eq. IV·2. Furthermore, for any pressure drop, f,P, Eq. IV·2 expresses the cumulative influx as Q('I') = 21TfcR,,' f,P Q", (IV·4) per unit sand thickness.* * The set of symbols now introduced and the symbo~s reoorted in Hurst's1 earlier paper on water-drive are related as follows: t G(o;' O/R') = Q(l) and G(o;' B/R') r Q(t) dt where o· 0;' e/R' = t December, 1949 PETROlEUM TRANSACTIONS, AIME 307 T.P. 2732 THE APPLICATION OF THE lAPLACE TRANSFORMATION TO FLOW PROBLEMS IN RESERVOIRS When an oil reservoir and the adjoining water-bearing for- mations are contained between two parallel and sealing fault- ing planes, the flow of fluid is essentially parallel to these planes and is "linear." The constant terminal pressure case can also be applied to this case. The basic equation for linear flow is given by O'P Ox' oP at (IV-S) where now t = KT / fl'c and x is the absolute distance meas· ured from the plane of influx extending out into the water- bearing sand. If we assume the same boundary conditions as in radial flow, with P = P(x, t) as the solution, then by Darcy's law, the rate of fluid influx across the original water- oil contact per unit of cross-sectional area is expressed by qUi = ~ ( ~:-) x=o (IV-6) The total fluid influx is given by ! K fl'c .t ( oP ) Q(T) = j q('l') dT = --. --- j -- dt o I' K 0 Ox x=o = f C Q(l) (IV-7) where Q(" lS the generalized ~olution for linear flow and is equal to ~ ( OF ) Q(l) = J .- o OX dt (IV-8) x==o Therefore, for any over-all pressure drop L.F, Eq. IV-7 gives Q{'j') = fcL.P Q,,) (IV-9) per unit of cross-sectional area. Constant Terminal Rate Case In the constant terminal rate ca:-;e it is likewise assumed that initially the pressure everywhere in the formation is constant but that from the time zero onward the fluid is withdrawn from the well bore or reservoir boundary at a unit rate. The pressure drop is given by P = p(,.,t), and at the boundary of the field, where r = 1, (OP/Or)..=l = -1. The minus sign is introduced because the gradient for the pressure drop rela- tive to the radius of the well or reoervoir is negative. If the cumulative pressure drop is expressed as L.P, then .' (IV-IO) where q(t) is a constant relating the cumulative pressure drop with the pressure change for a unit rate of production. By applying Darcy's equation for the rate of fluid flowing into the well or reservoir per unit sand thickness where q(T) is the rate of water encroachment per unit area of cross-ECction, and P tt ) is the cumulative pressure drop at the sand face per unit rate of production. Superposition Theorem With these fundamental relationships available. it remams to be shown how the constant pressure case can be interpreted for variable terminal pressures, or in the constant rate case, for variable rates. The linearity of the diffusivity equation al- lows the application of the superposition theorem as a se- quence of constant terminal pre~sures or constant rates in such a fashion that it reproduces the pressure or production hiHory at the boundary, r = 1. This is essentially Duhamel's principle, for which reference can be made to transient electric circuit theory in texts by Karman and Biot,S and Bush." It has been applied t olhe flow of fluids by Muskat,' Schilthuis and Hurst,' in employing the variable rate case in calculating the pressure drop in the East Texas Field: The physical significance can best be realized by an appli- cation. Fig. I-A shows the pressure decline in the well bore or a field that has been flowing and for which we wish to ob- tain the amount of fluid produced. As shown, the pressure history is reproduced as a series of pressure plateaus which repre~ent a sequence of constant terminal pressures. Therefore, hy the application of Eq. IV-4, the cumulative fluid produced in time t by· the pressure drop L.P", operative since zero time, is expre,'ed hy Q(T) = 27rfcR b' ,0,1'" Q't). If we next consider r-Q(t) 30~--------~------------~r--------' q(T! = -21rK ( QL.P) =-21rK q(,) (oP(r,t)) I' Or" = 1 I' or r = 1 101---/ h · h ' l·fi q('nl' Th f w IC sImp I es to q(t) = --. ere ore, for any constant 21rK rate of production the cumulative pressure drop at the field radius is given by P _ qcnl' P ,0, - 27rK (t) (IV-ll) Similarly, for the constant rate of production m linear flow, the cumulative pressure drop is expressed by L.P = qcnl' p K (ti (IV-I2) 0~1----------~5-------------J10~------~ FIG, 4 - RADIAL FLOW, CONSTANT TERMINAL PRESSURE CASE, INFIN- ITE RESERVOIR, CUMULATIVE PRODUCTION VS. TIME. 308 PETROlEUM TRANSACTIONS, AIME December, 1949 A. F. VAN EVERDINGEN AND W. HURST T.P. 2732 the pressure drop ,6P" which occurs in time t" and treat this as a separate entity, but take cognizance of its time of incep- tion t " then the cumulative fluid produced by this increment of pressure drop is Q(t) = 2trfcRb ' ,6P, Q(t-tl)' By super- imposing all the.'e effects of pressure changes, the total influx in time t is expressed as Q(T) = 27rfcR h' [,6Po Q(t) + ,6P,Q(t-t,) + ,6P,Q(tt,) + ,6P,Q(t-t 3 ) + ] (IV-I3) when t > I,. To reproduce the smooth curve relationship of Fig. I-A, these pressure plateaus can be taken as infinitesim- ally small, which give the summation of Eq. IV-13 by the integral , ~ o,6P QfT) = 27rfcR,,- j ---- Q(t-t') dt' . o· at' (IV-I4) By considering variable rates of fluid production, such as shown in Fig. I-B, and reproducing these rates as a series of constant rate plateaus, then by Eq. IV -11 the pressure drop in the well bore in time t, for the initial rate q" is ,6Po = qoP(t). At time t" the comparable increment for constant rate is ex- pressed as .q, - qo, and the effect of this increment rate on the corresponding increment of pressure drop is ,6P, = (q, - qJ p(t-tl)' Again by superimposing all of these effects, the determination for the cumulative pressure drop is ex- pressed by ,6P = q(o) P tt ) + [q, (t, ) - q(O)] p(t-t,) + [q(t,) - q(t ,)] p(t-t .. ) + [q(t3) -q(f,)] p(t-t,) + (IV-I5) rr===-Q-(t)-,------r------,----~~----,_--__. 35~---+---~---~ 3.01-----+-----+----,~1__7"-------+_--__+---_____l 2.5f--------+---V;:L--+---------::~---====+===1 2.01----+---I'---T"---t------ir-------f-----__+--------j 1.5r----__ -----!lr----__ --=l=~--;I~t_---A~SYrM-T-.:0-T~IC~VA-.:L;..:U-=E-I:.:..5::00".::J.\~ "R =2.0 I. OJ----f--+-----+-----f-------+-----__+--------l ASYMTOTIC VALUE 0.625 R = 1.5 o 0 0;;--------;I-';;.0;-------:2t.0;;------;f3.0;;-----~40;;------;05L,;0:------d6.0 FIG. 5 - RADIAL FLOW, CONSTANT TERMINAL PRESSURE CASE, CUMULATIVE PRODUCTION VS. TIME FOR LIMITED RESERVOIRS. If the increments are infinitesimal, or the smooth curve rela- tionship applies, Eq. IV-I5 becomes t dq(t') ,6P = q(o) P(t) + J -- p(t-t') dt' o dt' If q(o) = 0, Eq. IV-I6 can also be expressed as t (IV-I6) ,6P = J q(t') p'(t-t') dt' (IV-I7) o where p'(t) is the derivative of Pit) with respect to t. Since Eqs. IV-I3 and IV-I5 are of such simple algebraic forms, they are most practical to use with production history in making reservoir studies. In applying the pressure or rate plateaus as shown in Fig. 1, it must be realized that the time interval for each plateau should be taken as small as possible, so as to reproduce within engineering accuracy the trend of the curves. Naturally, if an exact interpretation is desired, Eqs. IV-I4 and IV-I6 apply. FUNDAMENTAL CONSIDERATIONS In applying the Laplace transformation, there are certain fundamental operations that must be clarified. It has been stated that if P (t) is a pressure drop, the transformation for Pit) is given by Eq. III-I, as To visualize more concretely the meaning of this equation, if the unit pressure drop at the boundary in the constant termi- nal pressure case is employed in Eq. III-I, its transform is given by 00 -pt -e 1 (V-I) PiP) = J e- pt 1 dt = --- 1 o p p o The Laplace transformations of many transcendental functions have been developed and are available in tables, the most com- plete of which is thc tract by Campbell and Foster.' It is there- fore often possible after solving a total differential such as Eq. 1I1-3 to refer to a ~et of tables and transforms and deter- mine the invcrse of PCP) or Pit). It is frequently necessary to simplify PiP) before an inversion can be made. However, Mel- lin's inversion formula is always applicable, which requires analytical