Principles of Well Spacing By MORRIS MUSKAT*

(New York Meeting. February 1939)

ALTHOUGH the problem of well spacing is one of the most important involved in the production of oil, it must be considered at the present time as still subject to further development. The published literature on this question is so voluminous that we cannot enter here into a review of it, except to refer to the recent papers by L. L. Foleyl and E. A. Stephen­son,2 where other references are cited. However, a decisive and conclu­sive answer to the problem of well spacing in any general form does not seem to have been developed. And while we shall be unable to present the desired solution in the present paper, it is nevertheless felt that the material to be given here not only provides the physical ground work for the ultimate solution, but shows qualitatively the essential factors that enter the well-spacing problem.

Field studies of the well-spacing problem, in which the ultimate recoveries from different fields with different well spacings have been compared, have generally suffered from the lack of knowledge as to the similarity of the inherent characteristics of the producing reservoirs being compared, such as the sand volumes, the sand porosities, the sand permeabilities, the original reservoir pressures, the presence or absence of gas caps, the presence or absence of effective water drives, and the economic limits of production rates at which the various fields were considered to have yielded their ultimate recoveries. Certainly no one who is evaluating originally the economic significance of any oil reservoir would deliberately ignore these phases of the problem. And it is equally certain that no one would reasonably expect the ultimate recoveries from two reservoirs, even with the same well spacing, to be identical regardless of these other factors.

More reliable results might be expected from comparisons of the recoveries obtained from different leases with different well spacings, but producing from the same reservoir sand. Unfortunately, however, even conclusions for such studies may be subject to serious errors. The reason simply is that however much one may insist that an operator is entitled only to the oil immediately underneath his surface acreage he will nevertheless drain the surrounding properties as long as his reservoir

Manuscript received at the office of the Institute Jan. 26, 1939. Issued as T.P. 1086 in PETRoLEml TECH:-;OLOGY, August 1939.

* Gulf Research and Development Co., Pittsburgh, Pa. I References are at the end 'If the paper.

37

38 PRINCIPLES OF WELL SPACING

pressures are maintained below that of his neighbors. * In other words, oil will migrate from the surrounding acreage and be brought up through his wells unless the surrounding leases maintain average reservoir pres­sures that are no higher than his. The intent of the operator with regard to this migration is entirely irrelevant. If he does not know the reservoir pressures in his and the neighboring leases it will go on without his knowl­edge. If he does know them he may still be forced to induce such migra­tion even against his will if his neighbors are unable to develop and produce their properties as fast as he finds it expedient and desirable to produce his own. Such a situation will always obtain when different parts of a single reservoir are drained at rates that are not all proportional to the oil present under the surface acreage and the wells are produced in such a manner as to create pressure gradients across the reservoir.

Thus let us suppose that one-half of a uniform reservoir that is not subjected to effective edge water or gas-cap drives is drilled twice as densely as the other half. If all the wells are produced at the same rate, it follows that while these rates are maintained the densely drilled acreage will remove twice as much oil as its neighbor. If the sands are otherwise uniform the reservoir pressure in the densely drilled half will fall at a rate that is at least twice as great as that in the other half, provided these two halves are separated by an impermeable membrane. If such a membrane were to really exist, the total recovery from either half, when production had become unprofitable, would directly reflect the effect of the different well spacings used in the two halves. However, in reality no such mem­brane exists. On the contrary, it is just as easy for oil to flow across the boundary between the two halves in consequence of pressure gradients as it is for the oil to be driven into the producing wells by the pressure gradients directed toward these wells. In other words, as soon as the pressures in the densely drilled half begin to fall below those in the other half oil will begin to migrate from the latter into the former.

In performing such an experiment one would therefore obviously anticipate that the closely drilled half would produce more oil in a limited time than the sparsely drilled half, even if inherently the well density had no effect upon the recovery of the oil that was originally in place in either side of the reservoir. And, indeed, this is the situation that has been frequently observed in field practice and has been universally quoted as supporting the claim that close well spacing will result in higher recovery than will wide spacing. Until the actual recoveries have been corrected for the migration into or out of any acreage that is being used in the study of well spacing, it is clear that the conclusions drawn will not be

* The existence of this factor of migration in the interpretation of field data with respect to the problem of well spacing seems to have been first pointed out by A. C. Rube13 and subsequently critically examined by R. D. Wyckoff. 4 In fact, the above discussion is essentially nothing more than a restatement of Wyckoff's conclusions.

MORRIS MUSKAT 39

valid in indicating the value of either close or wide well spacing. As this has not been done in studies made heretofore, and since it is not clear how such corrections can be made accurately, one is forced to question seriously the ultimate significance of tests or studies of this kind.

In view of these difficulties of obtaining significant data on the problem of well spacing from records of actual field performance, * it appears neces­sary to resort to deductions made from laboratory studies of the problem. It is the purpose of this paper to develop this point of view. Since the laboratory approach to the problem of well spacing can of itself indicate only the nature of the physical solution to the problem, one must obviously supplement this with economic considerations in order to make the results of practical interest. However, it will be well to consider first the nature of the physical ultimate recoveries of oil reservoirs in relation to the well spacing of the drainage system in order to have a proper basis for the discussion of the economic phase of the problem. For this purpose the sands will be considered as uniform throughout the extent of the reservoir. When the producing horizon is broken up into lenses, each of which would certainly contain enough oil to pay for the cost of drilling and operating at least a single well, it is obvious that the spacing must be made sufficiently small to insure a high probability that each individual lens is penetrated by at least one well. It is further to be understood that neither effective water nor gas-cap drives are present as aids in the production. Only the gas originally dissolved in the oil will be considered as a source of energy for draining the sand of its liquid.

PHYSICAL ULTIMATE RECOVERIES OF EXTENDED UNIFORM SANDS

By the detailed calculation of the production history of a column of sand of uniform properties it has been found that the physical ultimate recovery of liquid from the sand is practically constant over its whole extent. 5,U By the "physical ultimate recovery" we mean that which obtains after an infinite time of production. This in turn implies that the pressure has fallen everywhere to the value maintained at the bottom of the producing wells. The economic phase of the question obviously enters into the problem raised in the attainment of such an ultimate depletion, since it involves the production from the reservoir during its later history at extremely low rates. However, for the present we shall ignore this question and shall suppose this ultimate depletion to be a

* A type of field data pertinent to the problem of well spacing that would auto­matically take care of differences in sand volume, sand porosity and initial reservoir pressure would consist of the ratio of the initial gas-oil ratio of the field to its average value during its production history. For these ratios would be direct measures of the fractional oil recovery. However, such data, too, would suffer from the complicating ~ffects of free gas zones and water drives in comparisons of different fields, and from mterlease migrations in the study of producing tracts in a single field.

40 PRINCIPLES OF WELL SPACING

practicable achievement. There is indeed some variation of the ultimate liquid saturation at the state of complete depletion and, in particular, the recovery of liquid appears to be somewhat higher near the drainage sur~ face than at distant points. However, this increased recovery is con~ centl'ated about the outflow surface, and one may reasonably approximate the final result by saying that the ultimate recovery is uniform over the extent of the producing sand.

While t,his conclusion has thus far been expressed essentially as a consequence 'of a particular theoretical calculation and similar direct experimental laboratory tests, it may be shown that they are not to be associated with the special problem for which the calculation was carried through in detail. On the contrary, they have considerable general valid­ity. The proof of this may be deduced directly from the character of the general differential equations governing the flow of gas-liquid mixtures through sands, which have been derived as direct consequences of labora­tory experiments. These are the equations governing the space and time distribution of the fluid pressure and liquid saturation in any sand carry­ing a gas-liquid mixture which, under the assumption of uniform sand permeability, become:

V· {pFg(p)Vp) + aV· {pFI(p)Vp) = a{:l :tp{p + ~(1 - p)} [1]

V. {FI(P)Vp) = IMI ap [2] ko at

where p is the fluid pressure, p the liquid saturation expressed as the frac­tion of the local pore volume occupied by liquid, t the time, a = 8Mg/C}J.I,

}J.g, IJ.I being the viscosities of the gas and liquid phases, 8 the solubility of the former in the latter, C the density of the gas at atmospheric pressure, 1 the porosity of the sand, ko its homogeneous fluid permeability, and the differential operator V refers to the space coordinates. It is also assumed for simplicity in constructing equations 1 and 2 that the gas is ideal and obeys Henry's law. The functions koFl(p) and koFg(p) are the empirically detennined values of the permeability of the sand to the liquid and free gas phases expressed as functions of the liquid saturation. These func­tions represent the hydrodynaInic definition of the sand as the carrier of a heterogeneous fluid, just as the permeability ko is that for a homo­geneous fluid (liquid or gas) flow. The technique required to determine these functions has been described by Wyckoff and Botset (ref. 5, p. 325).

For the linear system these equations reduce to:

MORRIS MUSKAT 41

where x, t are the dimensionless variables defined by: x =xIL; t = kotlfjJ.zL2. It is also assumed, of course, that the pressure P is measured relative to some unit pressure, which is to be considered as multiplied into the right side of the expression for 't.

It follows that for a given set of physical constants, included in ex, and a certain mode of production-defined, for example, by the character of the variation in pressure or flux at the outflow surface-the saturation distribution at any time and position will be expressed by the universal function p = p(ex, Pi, x, t), where Pi is the initial reservoir pressure. The pressure distribution will be given by a similar function, it being explicitly noted that the absolute length of the system L does not enter into these functions, the parameters of significance being the dimensionless quanti­ties x and t. The total ultimate recovery of a unit cross-section column of sand of porosity 1 and length L, drained by a single" well" at its center, will therefore be given by:

P = 21 1L1l1 - p(ex, pi, ~, (0) JdX l = IL 11[1 - p(a, pi, x, 00 )]dx = F(f, ex, Pi)L

[4]

the 00 indicating that p refers to the state of ultimate depletion. Thus we see that the total ultimate recovery will be directly propor­

tional to the length of the sand; that is, the average recovery per unit length of sand will be independent of the total length of the system. Furthermore, if there are n identical" wells" draining the column of sand, the total ultimate recovery will still be:

fLl2n[ ( 2nx Pro = 2nl Jo 1 - p ex, pi, T'

= IL 101 [1 - p(ex, pi, x, 00 )]dx [5]

which is the recovery obtainable by a single well. With regard to the absolute ultimate recovery of the linear system, there is, therefore, no gain whatever in using more than one well to deplete the column of sand.

For the radial flow system the analytical equivalent of the above result cannot be derived as a rigorous consequence of the differential equations. However, it can be shown to remain valid within the accuracy of practical interest. Thus, returning to the fundamental eqs. 1 and 2, it is readily seen that the significant dimensionless variables corresponding to x and t in eq. 3 are: f = rlr., and t = kotllJJ.lr.2 where r. is the radius of the external boundary delimiting the sand system. Furthermore, as there will be in general two boundaries to the system, defined by the absolute dimensions r = rw , r = re, their ratio (3 = r.lrw may be expected

42 PRINCIPLES OF WELL SPACING

to enter as a parameter in the saturation distribution. The final equi­librium saturation after depletion will therefore be a function as: p = pea, pi, (J, r, 00). Actually, however, the ratio {J may be shown to drop out from the function p. For from the earlier study5 of eqs. 1 and 2 it was found that the equilibrium saturation at a closed boundary, which is also an equipressure surface, depends only on the ratio of the initial to final pressure in the system and not on its geometrical properties. Now, the only way in which these geometrical properties enter at the external boundary (r = 1) is through {3. Hence it follows that (3 must drop out of p, leaving for the ultimate depletion saturation p = pen, pi, r, 00). The total ultimate recovery may therefore be written for unit sand thick­ness as:

P = 21rf II 1 - p( a, pi, ;.' 00 ) }dr! = 2'IIJr.2~1[1 - pea, pi, r, 00 )]rdr

fJ

[6]

To see now the manner in which P varies with the area of the system being drained, we note first from eq. 6 that:

oP --;- = 21rfrw[1 - pea, Pi, 1/{3, 00)] ~ 21rfrw

ur", [7]

The maximum variation of P with rw therefore corresponds to the change in volume of liquid in the sand due to changes in well radius, which, of course, is entirely negligible for all practical purposes. The variation of P with r., on the other hand, is given by:

oP = 2P + 21rfrw2[1 _ pea Pi 1 00 )] ,...., 2P or. r. r. ' , , r. [8]

It immediately follows that Par.2, so that again the average ulti­mate recovery, per unit area, is independent of the total absolute area which the well drains. While it is not possible to fill out a plane area completely by means of strictly circular units, it is clear that for practical purposes the fact that the average recovery in a circular region is inde­pendent of the total area of the region also implies that the total ultimate recovery from a large tract with a number of wells in it will be no greater than if that tract were drained by a single well at its center. Here, too, therefore, a close well spacing would not lead to a larger physical ultimate recovery than would wide spacing.

It thus appears that under similar conditions of production the total amount of oil that can be displaced from a sand filled with a liquid that

MORRIS MUSKA T 43

is saturated with gas to a given pressure is entirely independent of the number of wells that are used to withdraw that oil. In other words, the question of well spacing does not really exist from a strictly physical point of view.

ECONOMIC ULTIMATE RECOVERIES AS AFFECTED BY WELL SPACING

While the conclusions just drawn appear to follow from eqs. 1 and 2, it must nevertheless be admitted that in themselves they are essentially only of academic interest. For it has been explicitly assumed in their derivation that the comparisons between the ultimate recoveries under various conditions of well spacing are made only after an infinite time of production. * As previously mentioned, this assumption implies that one has waited until the pressures throughout the sand have become equalized and fallen to those maintained at the bottoms of the wells. A considerable portion of the total recoveries thus obtained would corre­spond to extremely low rates of withdrawal from the wells, and such as would be not only unprofitable but would moreover definitely incur economic losses in the operations. Indeed, it has been not an uncommon experience to find, especially in tight sands, the pressures between wells to be of considerable magnitude even though it has become necessary to abandon the original wells draining the reservoir because they could no longer yield oil at profitable rates. To have continued to operate these original wells until the pressures between them had fallen to the low values immediately surrounding the wells, so that the depletion through­out the whole sand would have attained its ultimate value predicted by the above theory, would, of course, have been gross economic folly.

It is obviously pertinent, therefore, to inquire whether such theo­retical considerations as developed above can also be subjected to the economic limitations of the problem of oil production. For only under such conditions will the implications of the laboratory studies have practical significance. Unfortunately, the mathematical analysis involved in the solutions of the fundamental eqs. 1 and 2 has not reached the stage of development wherein any arbitrary geometrical system can be treated. In fact, only the simplest case, in which the producing reservoir is in the form of a long column of sand, has thus far been analyzed in detail. For this case, however, as we shall see presently, it is possible to apply economic considerations to the problem of well spacing.

The economic restriction that we shall now impose is that the recovery of the oil will not be profitable unless the rate of production per well is

* While the additional assumptions of strictly constant liquid viscosity and ideal kinetic theory behavior of the gas and liquid reservoir fluids also limit the generality of the conclusions even from the physical point of view, it seems very unlikely that the mere removal of these assumptions would lead to physical ultimate recoveries which vary appreciably with the well spacing.

44 PRINCIPLES OF WELL SPACING

equal to or exceeds a preassigned minimum value. * The question to be answered, then, is: How will the total recovery from a tract that is to be obtained with rates of production from the individual wells above this minimum value depend upon the number of wells in the tract? For simplicity, it will be assumed that all the wells in any particular tract are produced in exactly the same manner. Moreover, it will be supposed that this mode of production is that in which all the wells are drilled simultaneously and the bottom-hole pressures are reduced immediately

4-+--+-+ I ~

a 2 3 456 loot

7 8 9 10

FIG. I.-DECLINE IN "FLUX FROM LINEAR CHANNEL ORIGINALLY SATURATED TO A PRESSURE OF 10 UNITS AND EXPOSED AT ONE END TO A UNIT PRESSURE AND CLOSED AT THE OTHER.

Real flux in cubic centimeters per second per unit area of sand column is k o/ I'lL

times ordinates. t = dimensionless time.

after completing the wells to a tenth of their initial reservoir pressure and thereafter maintained indefinitely until the field is abandoned. Con­sidering this final pressure to be that of the atmosphere, the assumption corresponds to supposing that the initial reservoir pressure in the sand is 10 atmospheres, or approximately 150 lb. These specific assumptions will be used here simply because the only case for which a detailed numeri­cal solution of the fundamental differential eqs. 1 and 2 is available was also developed under these assumptions.

For this particular case it has been found that the velocity of outflow in dimensionless units from such a column of sand as a function of the dimensionless time t is given by the curve of Fig. 1. The real flux into the well in cubic centimeters per second per unit area of sand column

* A condition of this kind and the general type of economic interpretation of the well-spacing problem presented here has recently been applied to gas fields by D. T. MacRoberts. 7

MORRIS MUSKAT 45

is ko/p.l£ times the ordinates in this figure, where ko is the homogeneous fluid permeability of the sand, /J.! the viscosity of the oil, and L the length of sand column on either side of the well, it being supposed that the well bisects the sand and that the latter is closed at its distant terminals.

If there are n wells uniformly spaced in the total sand column of length L, the rate of production from each well will be given by:

Q = 4nko(F ap) [9] " p.l£ lax 0

where again the quantity in parenthesis is given by the ordinates of Fig.!. The total rate of recovery from the sand will obviously be nQ". The total recovery of oil at any dimensionless time t will be given by:

fi( a) - fl -P = fL Jo F!a~ odt = fL Jo [1 - p(a, Pi, x, O]dX. [10]

Its variation with t is shown in Fig. 2, for the case corresponding to Fig.!.

0·28

0·24

0·20

0·16 P

fL 0·12

0·08

0·04

/ I

i i

, !

I ,/

V

v /

II

2

I..-./

3 4

I I--VC-

5_ 6 loot

r---

r : I

I

I

7 8 9 10

FIG. 2.-VARIATION OF FRACTIONAL LIQUID RECOVERY FROM LINEAR SYSTEM WITH TIME.

P = total liquid recovery per unit area of sand column; f = sand porosity; L = length of sand column; t = dimensionless time.

We are now ready to impose the economic limitations mentioned above; namely, that the ultimate recovery is to be considered as having been obtained when the production rate Q" has fallen to a limiting minimal value, which we may denote by Q"o. This means, by eq. 9, that the minimal value of the ordinates of Fig. 1 of economic significance will be that corresponding to:

(Flap) = p.ILQ"o == Qo ax 0 4kon n

[11]

46 PRINCIPLES OF WELL SPACING

where Qo is the minimum production rate expressed in dimensionless terms, and it is supposed that tqe pressures p have been taken throughout as being measured relative to a unit pressure. Thus choosing QnO, and therefore Qo, one can easily find from Fig. 1 the dimensionless time t at which the recovery is to be considered as complete from an economic point of vie,,, for any number of wells n. From Fig. 2 can be obtained immediately the total fractional recovery up to that time, or the total economic ultimate recovery. * Moreover, this procedure also gives the

28 I [ ; I ,I i

1 -LH-- '

I ! i 1.9P+-I

I I Y I 'Qo·~ I--r--I--I--~ I -24

! I V: ; V1\",~~ V l.--l-I--~----c-7~'-/ I-r V iii ! , /' , -'

( I I V .'l-3.-V ~ 1 :/ ill ~i i VI

1 I I II 1/ I--'" I ./

II )' 1/ ...... 1--'" i !I Vi !/1 '3rr

il V /' I--'"

4 ( Vi I j...--( -r-1 r I I i 1

00 2 3 4 5 6 7 8 9 10 n = Number Of Wells

FIG. 3.-VARIATIO~ OF ECONOMIC ULTIMATE RECOVERY OF LIQUID FROM LINEAR SYSTEM WITH NUMBER OF WELLS n.

Qo = p./LQno/4ko; ko = sand permeability; P.I = liquid viscosity; L = total length of system; Q .. o = minimal production rate (per unit sand cross section) per well at which the well can be produced profitably.

absolute value of the time by which this economic ultimate recovery will be obtained, since:

[12]

The ultimate recoveries obtained as just described are plotted as a function of the number of wells draining the linear system in Fig. 3.

* These two steps could be combined analytically into the single equation:

p = - fL ('" vOddt dvo where (F 1 aa~) is denoted by Vo. However, this would require J'Qo/n Vo x °

calculating the slopes ddt from a curve (Fig. 1) which is not known very accurately, Vo

whereas the integrated form: P = fL[Qol + r co idVo] would still involve using Fig. 1, n JQo/n

together with that for ftdvo.

MORRIS MUSKAT 47

The practical significance of the dimensionless minimal production rate Qo is given by the observation that a minimal rate of 0.1 bbl. per day per square foot of sand of an oil of 5 centipoise viscosity produced from a sand with a permeability of one darcy and 1000 ft. long corresponds to a value of Qo = 7.55.

The fact that the curves with smaller Qo lie above those with larger values of Qo simply means that, as would have been anticipated, the ultimate economic recovery for the linear system under consideration will increase as the minimal production rate for abandonment decreases. Likewise, for a given minimal production rate the ultimate fractional recovery will decrease as the permeability decreases, since byeq. 11 Qo is inversely proportional to the permeability. And finally, the recovery will decrease with increasing viscosity of the oil, for a given number of wells and for a fixed minimum production rate at which the operations would become unprofitable. On the other hand, it is to be noted that the ultimate recoveries will not vary in a simple manner with the value of Qo, the actual dependence on Qo being determined by the number of wells. Thus, whereas for low well densities the recovery is roughly inversely proportional to Qo, the variation is much less for the higher well densities. Hence one cannot simply conclude that by doubling the minimal produc­tion rate the economic ultimate recovery will be necessarily halved, or that. by doubling the sand permeability the ultimate recovery will be doubled.

Of more importance than these features of Fig. 3 is the variation of the ultimate recovery from the column of sand with the number of wells n. Here we find a marked dependence upon the absolute value of Qo. For small values of Qo the recovery increases rapidly to approximate satura­tion as n increases, whereas for large values of Qo the increase in recovery as the number of wells increases is much more gradual. While the details of these curves must be accepted simply as consequences of the analysis and the calculations of the decline history of the particular system as expressed in Figs. 1 and 2, their general features can be given a reasonable physical interpretation. Thus the sharp rise and marked flattening of the curves for small values of Qo implies that for a highly permeable sand column little additional recovery would be gained by increasing the number of wells if the minimal production rate is kept fixed. Or, if we consider sand columns of the same permeability, it means that when the minimal production rate has been set at a low value one approaches, even with a small number of wells, so closely to the physical ultimate recovery that further drilling will add but little. On the other hand, the flatness of the curves for large values of Qo when n is small indicates that if the sand column is very tight one must make the spacing rather close to obtain appreciable recoveries before the operations become unprofitable. Likewise, if the limiting production rate is for some reason

50 PRINCIPLES OF WELl, SPACING

trend will uudoubtedly be the same. On the other hand, we do not yet know how to translate a specific rate of production for the radial system such as, for example, 1 bbl. per day per foot of sand, into an equivalent rate for the linear system; i.e., in terms of barrels per day per sq uare foot of sand. It is impossible, therefore, to apply the present theory numerically to radial systems and hence to practical well-spac­ing programs.

With regard to the effect of the manner of production upon the well­spacing problem, it is again impossible to state definite conclusions. From the point of view of the actual analysis, the method of production will play an important role. Thus if we should suppose that instead of flowing the wells wide open their production is prorated to a fixed rate throughout their whole life, the production-decline curve obviously will lose its meaning. Rather, one will have to construct pressure-decline curves, and determine the total recoveries at the time the bottom-hole pressures have fallen to their minimal practical values with various daily prorated production rates. Such generalizations must also be left to future analytical and experimental work. However, it appears unlikely that the well-spacing problem will be seriously affected by the details of the mode of production.

Finally, it should be emphasized that the numerical results given here, or those obtained by any equivalent theory, will depend exclusively upon the details of the decline curve such as shown in Fig. 1. In fact, any type of quantitative prediction with regard to the well-spacing problem can be obtained by suitably varying the character of the dimensionless decline curve with which one begins. Thus, if one begins with the decline curve corresponding to a linear compressible homogeneous liquid system, one obtains total economic ultimate recovery curves similar in type to Fig. 4, but sufficiently different in detail to lead to per well recoveries that uniformly decrease with increasing well density rather than to maxima in the curves such as indicated in Fig. 4. And the curves for the time of recovery corresponding to Fig. 5 also show appreciable changes. As previously mentioned, the maxima in the curves of Figs. 4 and 5 prob­ably have no physical significance but arise from the approximations inherent in the numerical solutions of eqs. 3, which underlie all the quantitative results of this paper. Moreover, in any development of an actual well-spacing program it would be necessary to take into account further economic factors such as interest on investment and price of oil, which have not been explicitly included in the above theory.

No claim is made, therefore, that the well-spacing problem has been given a numerical or quantitative solution here. The purpose of this paper has not been to derive such an ultimate solution, but rather to present the physical bases upon which the well-spacing problem for any particular system should be treated and to show by an idealized example

MORRIS MUSKA T 51

how the analytical program could be carried through. On the other hand, it may not be an entirely fortuitous circumstance that the numerical results that have been derived for the linear system parallel so closely the general opinions heretofore declared with regard to the well-spacing problem, even though they have been founded upon questionable intpr­pretations of field experience.

SUMMARY

As a result of the study reported here, it may be concluded that from a strictly phYRical point of view there is no basis for believing that the absolute ultimate recovery of oil that can be produced from a uniform and nonlenticular sand through the agency of the dissolved gaseR will materially depend upon the number of wells used to drain the reservoir. There is, however, a definite variation of the economic ultimate recoveries that can be derived from a given reservoir with the number of wells, if we consider the economic ultimate recovery to refer to that which can be obtained with the individual well-production rates exceeding a preassigned mmlmum. For a linear system, whirh has been treated in detail, it has been found that this economic ultimate recovery will increase as the number of wells t hat are used increases. When the minimum limiting production rate is small or the sand columns are highly permeable, the economic ultimate recovery rapidly rises as the number of wells is first increased, but quickly attains values that are thereafter no longer appreciably increased by further drilling. For tight sand columns, or if the lowest production rate at which the operation of a well would still be profitable is high, one must make the well spacing fairly small in order to approach ultimate recoveries that would be obtainable in highly perme­able sands or with low minimal production rates.

Although but little practical significance can be attributed to the numerical results derived, because of the various assumptions underlying the details of the analysis, their broad features show a close correlation with general opinion regarding the well-spacing problem that has been previously expressed in the literature.

ACKNOWLEDGMENT

The author is indebted to Dr. Pa.ul D. Foote, Executive Vice Presi­dent, Gulf Research & Development Co., for permission to publish this paper.

REFERENCES

1. L. L. Foley: OillVeekly (Oct. 11, 1937) 100. 2. E. A. Stephenson: Oil lV eekly (June 27, 1938) 25. 3. A. C. Rubel: Amer. Petro Inst. Prod. Bull. 209 (1932) 19. 4. R. D. Wyckoff: Amer. Pet.r. Inst. Prod. Bull. 216 (1935) 109.

52 PRINCIPLES OF WELL SPACING

5. M. Muskat and M. W. Meres: Physics (1936) 7,346. 6. M. Muskat, R. D. Wyckoff, H. G. Bot:;et and M. W. Meres: Trans. A.I.M.E.

(1937) 123, 69. 7. D. T. Mac Roberts: Trans. A.I.M.E. (1938) 127, 146.

DISCUSSION

(B. C. Craft presiding)

R. J. SCHILTHUIS, * Houston, Texas.-Dr. Muskat's paper presents an excellent analysis and statement of the physical principles governing the behavior and the localized and regional movements of gas and oil within reservoirs. It throws the proper light on the significance of this behavior relative to the problem of well spacing anrl its influence on ultimate recovery.

Well-spacing practices in this country, particularly in Texas, have been motivated C'lItirely by competitive economic conditions with no proper consideration to either physical efficiency or the over-all economic efficiency of the operations. The result of this situation has been a great deal of over-all economic waste, as well as often to lead to physical inefficiency.

The paper is very timely in laying the groundwork for, and indicating the true answer to, the physical aspects of the well-spacing problem and its influence on recov­ery. The fact that well-spacing practice in the past has given way almost entirely to purely competitive economic conditions, ignoring sound physical as well as economic considerations is rapidly bringing on a critical situation in many areas today relative to per well allowables, payouts, etc. For this reason, Dr. Muskat's paper and con­clusions are deserving of very careful attention and consideration by the oil operators, as well as the various state regulatory bodies

In connection with the purely physical considerations in Dr. Muskat's paper, there is one question I would like to raise; the possibility of the importance of the effects of gravitational forces in bringing about zonal segregation between the oil and evolved gases during the course of production and the possible rC'lation of this effect to the well­spacing problem.

T. A. POLLARD, San Francisco, Calif.-Dr. Muskat has shown a certain "ultimate recovery" for a well or wells, predicated on a number of conrlitions, one of which was the reduction of the bottom-hole well pressure in the beginning to a fraction of thC' original reservoir pressure, and maintenance of the well pressure at the same vahlC' throughout the life of the well. 'Vhat woul(l be the effect on the "ultimate recovery" if the well pressure were first reduced to, say, 90 per cent of the original reservoir pres­sure, then 80 per cent, and so on to rlepietion, thus simulating a restricted or pro­rated condition?

N. D. DRAKE,t New York, N. Y.--This paper must be regarded as one more of the series of extremely valuable contributions to our knowledge of reservoir fluid behavior that have been made by Dr. Muskat and his co-workers. It would seem that it is now possible to show beyond a reasonable doubt the factors involved in reservoir drainage in an ideal reservoir, and the principal remaining obstacle to wide­spread practical application of the fundamental principles already disclosed would seem to be the question of physical changes in the reservoir system during productive life. Among these changes already appreciated are the changes in permeability to oil caused by variation in saturation, and the separation of oil and gas within the sand The latter is perhaps least susceptible at the present time to analysis, or perhaps it

* Humble Oil and Refining Co. t Standard Oil Development Co.

DISCUSSION 53

may be better said that it is most difficult to estimate the effect of t~is phenomenon in actual operations.

The discussions that have been carried on in recent years regarding the possible drainage of energy through stratification or slippage of gas without the drainage of oil are well known, and a case often cited as an illustration is the Mansion area at Oklahoma City. If in certain types of reservoirs there is going to be a substantial separation of oil and gas within the oil sand, it is to be inferred that spacing considera­bly closer than that indicated by mathematical analysis would be required to effi­ciently drain the oil, but the burden of proof on this point would seem to lie on those favoring close spacing for this or some other reason. It can be shown that any degree of stratification of oil and gas within the sand with subsequent flow of gas unaccom­panied by oil to tJw well bore is a function of the gravity gradient, the horizontal or flowing gradient and the physical characteristics of the reservoir. Looking at these factors, it seems that for any given set of conditions the relation between these two pressure gradients at, say, 1000 and 2000 ft. from the well bore would be very slight, hence the inference would be that the wider spacing would cause very little, if any, loss in oil recovery because of stratification. Putting it another way, it could be said that given a certain reservoir condition-that is, a certain degree of dip and vertical­horizontal permeability relationship-any well spacing, except one ridiculously close, will allow a certain degree of separation of oil and gas in the reservoir at some distance from the well bore, but as between, say, 20 and 40 or 80 acres per well, the difference is slight.

One phase of well-spacing studies frequently misunderstood is the effect of the time factor. It has been shown by Dr. Muskat and others that ultimate recovery in the ideal reservoir is to a large extent independent of the well spacing, time being the major consideration in the depletion of the sand. Under ideal conditions, the time required for recovery of a given amount of oil varies approximately as the square of the well spacing; that is, with twice the distance between wells the time required for drainage will be four times as great, and this fact has often been used to argue against 40-acre as compared to, say, 20-acre, spacing. The main point over­looked in such an argument when considering practical time limits is the fact that the time required to drain the oil with the closer spacing must be ascertained before any assumptions as to the time element on the wider spacing are justified. For example, a 20-acre spacing, purely from the standpoint of hydrodynamics, may be able to drain a given sand in fiye years even though under proration or some other restriction 20 or 30 years may be assigned for depletion. In such an area, doubling the spacing distance would result in the drilling of only one-fourth the number of wells, yet under the four­fold rule, the wider spacing would be entirely adequate to yield the recoverable oil within the 20 or 30-year period.

R. A. CATTELL, * Washington, D. C.-Perhaps what I am going to say should be prefaced with the suggestion that we may be placing too much emphasis on maximum ultimatc recovery in our consideration of well spacing. The objective, from a national standpoint, should be to obt.ain the grcatest benefit from our petroleum reserves rather than thc great cst number of barrels of oil. The nation and its people need a dependa­ble supply of oil at reasonable prices over a long period-not a flood of oil at low prices in one period and a shortage, with resultant high prices, in another. The operator's financial welfare dcpl'nds upon the return on his investment rather than upon the quantity of oil hl' produces.

I do not wish to say that it is necessary to reduce ultimate recovery to accomplish the objective of a relativel~· uniform and dependable supply over a long period

* l-. S. Bureau of Mines.

.,)4 PRINCIPI,ES OF WET,I, SPACING

However, if it were necessary to sacrifice something in ultimate recovery by current methods of operation to accomplish that objective, perhaps as a nation we could afford to make the sacrifice. A considerable part of the oil left underground when fields are depleted by usual methods may be a reserve that can be drawn upon when conditions demand or justify expenditures for mining and other methods of recovery that now are too costly for general adoption.

Dr. Muskat has mentioned the difficulty of obtaining conclusive evidence from analysis of field data. H. B. Hill and R. K. Guthrie, of the Bureau of Mines office at Dallas, have plotted recoveries in barrels per acre-foot of sand as ordinates, against spacing in acres per well as abscissas, for groups of fields that have been operated substantially to economic depletion under open-flow methods and are similar in sand characteristics and other attributes. When average values for the individual fields in such a group of similar fields are plotted, the curve trends downward with decreasing slope in the direction of wider spacing. That is, such curves are concave upward and tend to flatten as the well density is decreased. The hypothetical recoveries in barrels per acre-foot with an infinite number of wells, indicated by extrapolating such curves to the ordinate, seem reasonable.

If, using the same ordinates, recoveries per acre-foot of sand from tracts with different spacings in the same field are plotted against acres per well, the curves are steeper than those based on average data for entire fields, showing more pronounced increase in ultimate recovery with the closer spacing. This clearly reflects regional drainage.

Dr. Muskat indicated that he would expect higher ultimate recoveries in fields with closer spacing if the fields considered were comparable as to sand conditions and other features, because operation of a well ceases when the economic limit of rate of production is reached, instead of continuing to infinite time. However, there is one element that should not be overlooked in plotting field data in the manner I mentioned. The tendency of the operator is to select closer spacing in the areas where sand condi­tions are better, so the upward trend of the curve in the direction of closer spacing may be due to two influences: First, a higher ultimate recovery as a result of closer spacing, and second, a higher ultimate recovery in the more closely spaced areas due to better sand conditions in those areas. However, critical examination of some of the data giving curves of the type mentioned fails to disclose any superiority of sand conditions in the more closely spaced fields.

If curves of this type can be accepted as showing a relation of spacing to recovery (we are not ready to accept them without further study) it is possible with correspond­ing decline curves to make an economic analysis to determine the spacing that would give the greatest profit under the older methods of production. Also, if we can deter­mine what departures from such curves result from controlled and improved methods of operation, the economic analysis can be extended to apply to fields operated in accordance with the later practice.

Such an economic analysis must be based on an assumed price for oil. If an analysis based on the current price for oil should disclose that the greatest profit comes with closer spacing than is now gencrally practiced, and operators were to change their spacing accordingly, one might then expect a decrease in price of oil, which would invalidate the economic analysis on which the closer spacing was based.

The Bureau is endeavoring to obtain data concerning fields operated under con­trolled-production methods for comparison with data from the older fields. The difficulty is that the history of fields operated by the new methods is short, aljd such fields have not been produced to exhaustion. Comparisons must be based upon estimated rather than proved ultimate recoveries in the newer fields, and a long "fore­sight" must be projected from a short "backsight." However, our engineers have osme data that indicate that fields operated with pressure maintenance are

DISCUSSION 55

higher recoveries than fields with the same spacing operated under the older methods. With data of that kind it may be practicable to make economic analyses in which the cost of pressure maintenance can be compared with the cost of additional wells to gain the same recovery.

These remarks are more or less premature, and I have no desire that any definitp conclusions be drawn from them. My main purpose in this diseussion is to give a suggestion of some of the studies the Bureau's engineers are making, and some of th(' factors that are puzzling us, primarily with the hope that as a result we may obtain further data and ideas that will bear upon the work. If any of you can refer H. B. Hill, H. C. Miller, or other members of our staff who are engaged in Bureau work that bears upon weil spacing, to information that discloses differences in recovery between field~ of similar types produced by open-flow methods and by controlled methods, we shall be appreciative.

H. H. POWER, * Austin, Texas.-It is evident that Dr. Muskat's paper applie~ principally to a pool where the principal source of energy to drive the oil toward the weil is the gas dissolved in and produced with the fluid. As I understand it, his analysis is rather definite, in so far as it applies to radial two-dimensional flow under such con­ditions of reservoir control. To what extent have reservoirs under varying degrees of hydraulic control been analyzed, and what fundamental issues are involved in so far as the spacing pattern is concerned, and, more particularly, what would the author consider to be the radius of drainage of such weils for proper and efficient drainagp of the oil content?

M. MusKAT (author's reply).-Mr. Schilthuis' reaction that this paper provides a sound attack upon the well-spacing problem is indeed gratifying, in view of the exten­sive fundamental researches by himself and his co-workers on the principles of oil production. His question regarding the effects of gravitational forces in bringing about zonal segregation between the oil and evolved gases during the course of produc­tion is well taken. Indeed, such effects must be present. Although we cannot yPi­

estimate quantitatively their magnitude, we may be certain that in general gas segre­gation will be more pronounced under wide well-spacing conditions than for close well spacings. As such segregation, moreover, will be conducive to gas by-passing and hence inefficient oil recovery, the close weil-spacing program may for this reason possess an advantage over wide-spacing plans. In practice, however, this difference Illa~' Iw entirely insignificant, as its magnitude may be so smail as to be entirely countcrbal­aneed by the economic factors related to the cost of drilling and the time of pa~'out, etc. Moreover, the physical effect in itself will be largely eliminated in producing formations separated by shale breaks or where the vertical permeability as a whole is appreciably less than the horizontal permeability.

With regard to Mr. Poilard's question concerning the effect on the ultimat" recovery of stepwise reductions of the bottom-hole flowing pressures, I can only repeut from the text of the paper the feeling that it appears unlikely that the well-sparing problem will be seriously affected by the details of the mode of production. We an' attempting to generalize the analysis so as to obtain more quantitatiYe predictions for such effects, but we are not yet able to draw any general conclusions.

Mr. Drake's discussion of the effect of gas segregation is in agreement with our point of view. Quantitatively, however, we would hesitate to predict with ccrtainty the exact range of practical conditions where this phenomenon would or would not be serious. Mr. Drake's comments regarding the time element in well-spacing considcra­tions are also well taken. At the present time the extended periods of payouts result-

* Professor of Petroleum Engineering, University of Texas.

56 PRINCIPLES OF WELL SPACING

ing from proration cannot be ignored. Qualitatively, proration control would certainly tend to decrease the gap between the life periods of fields producing under wide and close spacing. On the other hand, it may still be true, under certain conditions, that the restrictions imposed by proration may automatically disappear in the early history of a field, owing to the failure of the wells to make their allowables even under open­flow conditions. In such cases, which, of course, exclude effective water-drive fields like East Texas, the depletion time for the closely spaced wells may still remain litPpreciably smaller than for widely spaced wells. These are admittedly only possi­bilities and it is not proposed that they represent any general situation or rule.

Mr. Cattrell's brief summary of the recent work of the Bureau of Mines on well spacing is interesting. The distinct difference between the curves for wells obtained from different fields and those from tracts in the same field is certainly a gratifying confirmation of our interpretation. of the significance of field data. We also agree with the desirability of ultimately applying the physical data of recovery vs. well spacing to practical economic situations to see what the curves of profit vs. well spacing might look like. We are planning to carry out this type of calculation on the basis of our theoretical analysis of the well-spacing problem, and it will be interesting to see how they will compare with those obtained by the Bureau of Mines in using the results they are gathering from field experience. Of course, we agree that the oil recovery observed with any particular field developed over its actual production history does include the effect of the well spacing characterizing the development. Our only fear with regard to the use of such field data lies simply in the uncertainty as to the degree to which the recovery reflects the well spacing and that to which it reflects the sand and fluid characteristics, the structural features of the field, and such other items as edge­water drives or gas-cap drives. If the work of the Bureau of Mines satisfactorily eliminates these other factors, without question the field data should be entirely trust­worthy and significant.

Answering Professor Power, I should like to stress that our treatment of the well­spacing problem as given in this paper is explicitly restricted to linear systems. For this reason we feel that it can correctly give only the principles and trends involved in the problem rather than any quantitative magnitudes. We are attempting to extend the analysis to radial flow systems and if that should be successful we shall have results of more immediate practical applicability. As to the matter of reservoir control, it is true that we have considered only the phase of the production involving the evolution and flow of the gas originally dissolved in the oil. The inclusion of the effects of edge-water encroachment appears at present to be extremely difficult to carry out quantitatively. Qualitatively, however, our studies indicate that whatever changes water drives may make upon the general picture, it will be such that the differences between the economic recoveries under close and wide spacing will be decreased by the effects of water drives as compared to those obtained when water drives are entirely absent.