The Effect of Heat Transfer To A Nearby Layer on Heat Efficiency

Transcription

1 he Effect of eat ransfer o A Nearby Layer on eat Efficiency Michael Prats SPE Michael Prats Assocs Inc Summary his paper provides an analytic solution in Laplace space for the heat distribution in two nearby layers undergoing heat injection at variable rates he heat content in the layers undergoing thermal injection divided by the cumulative net-injected heat is known as the heat efficiency In this treatment the thermal properties of the layers the thickness of the layers undergoing heat injection the intervening layer can have any values he overburden underburden are semi-infinite Examples are given for two sets of thermal properties three values of the intervening layer s thickness Only a constant rate of heat injection is considered in the examples Results indicate that significant thermal interaction occurs at an intervening layer thickness as great as 45 ft with the effect being faster more pronounced the thinner the intervening layer Introduction Marx Langenheim developed equations showing how the heat content of a single reservoir layer undergoing steam injection increases with time he results took into account the heat losses to the adjacent formations considered the entire heated volume to be at the injection temperature he fraction of the injected heat within the injection layer was called the heat efficiency Prats later showed that although the parameters appearing in the equation depend on the process the equation for the heat efficiency is independent of the process used to introduce heat into the reservoir hese results make use of vanishing vertical temperature gradients within the heated zone an approximation first made by Lauwerier 3 for hot fluid injection eat efficiencies provide a quick indication of the attractiveness of potential steam-injection projects are useful in estimating steamflooding performance 45 In practice especially in steamflooding operations steam is injected into nearby ss either simultaneously or sequentially 67 his paper examines how the heat losses from one layer affect the heat content of a second layer losmann 8 developed an analytical model that describes the volumes of the heated zones for simultaneous equal injection into an infinite number of equal equally spaced layers Because of the symmetry of his model losmann only considers two different sets of thermal properties one in the reservoir layers another in the intervening layers ere all five formations (two injection layers an overburden an underburden a center layer may have arbitrary properties Assumptions Steam is injected into a well open to two ss Layers which are separated by an impermeable center layer of nonzero thickness More generally heat is injected or generated in the two layers by any means but this discussion is in terms of steam injection he thickness thermal properties (volumetric heat capacity thermal conductivity of the five formations may differ but are constant uniform within a formation Fig is a schematic of the system considered eat transfer within the injection layers is by both convection conduction he temperature within Layer resulting from steam injection is independent of the vertical position within the layer (the Lauwerier assumption his temperature varies with time opyright Society of Petroleum Engineers Original SPE manuscript received for review 4 January Revised manuscript received 7 June Paper SPE 6968 peer approved 9 July within the lateral extent of the layer he same is true for temperatures in Layer that result from steam injection eat losses to the overburden underburden are considered to behave as though they were semi-infinite No assumptions are made about the direction of the heat flow within the overburden underburden center layer Some of the heat lost from Layer is conducted through the center layer into Layer beyond vice versa Before the breakthrough of heat at the producing wells the fluid flow within the layers does not affect the overall heat balance After heat breakthrough the heat balances are still correct if the heat-injection rate is considered to be that due to the difference between the heat injected that produced Of course the heat production rate is difficult to predict for arbitrary well configurations so the analyses results have more significance before heat breakthrough Because fluid flow does not affect the overall heat balance before breakthrough gravity plays no role during this period Gravitational effects would however affect the time at which heat breakthrough occurs the subsequent heat-production rate With this proviso the layers may be tilted at any angle to the horizontal plane Analyses of the results are based on horizontal layers but the term vertical can be generalized to mean normal to the layers with corresponding comments for horizontal Solution Outline he solution has two steps In the first steam injection is into Layer only he total heat content of Layer is determined as a function of time taking into consideration the differences in the thermal properties of Layer the center layer the overburden the underburden his heat content is denoted by he heat transferred from Layer to Layer is also determined as a function of time is denoted by In the second step steam injection is only into Layer In a similar manner the heat content in Layer owing to injection in Layer that in Layer owing to injection in Layer are determined Because the systems are linear the heat contents during simultaneous steam injection are additive so the total heat content in Layer j is j(t j j with j or A full discussion of the solution is provided in the Appendix Laplace transforms are used extensively employing time solutions obtained with the Stehfest inversion algorithm 9 Results Examples illustrate the thermal interaction between nearby layers undergoing steam injection wo sets of thermal properties are used (able In set P volumetric heat capacities thermal conductivities are the same every In set P the properties of Layers are the same but those of the other formations reflect possible values of shales ss lower porosities Note that the properties are not symmetric in set P hree sets of geometries are used (able Layers are 5 ft thick with the thickness of the center layer at ft he overburden underburden are semi-infinite onsider a constant net rate of heat injection into Layer no steam injection into Layer Fig shows the fraction of the cumulative net-injected heat present in the various zones as a function of time for property set P geometry set G (case PG he descending solid line is the fractional heat content of the injection layer (Layer referred to in the literature as the heat efficiency It is noted that the heat content of Layer after injecting steam for year is less than half the heat injected his result 6 September SPE Journal

2 h 3 Layer h h enter ABLE ERMAL PROPERIES Property Set P Set P M Btu/ft 3 ºF λ Btu/ft ºF day 35 3 M Btu/ft 3 ºF λ Btu/ft ºF day 35 5 M Btu/ft 3 ºF λ Btu/ft ºF day M Btu/ft 3 ºF λ Btu/ft ºF day 35 3 M O Btu/ft 3 ºF λ O Btu/ft ºF day Layer ABLE LAYER IKNESS F Set G Set G Set G3 Layer enter Layer Layer Fig Schematic diagram of the -layer heat injection system all other results shown in Fig are independent of the rate of net heat injection provided the latter is constant independent of the injection temperature he rest of the net heat injected is conducted to the zones above below Layer he fractional heat content of the underburden exceeds that of Layer in just under 3 years Although not specifically shown the combined fractional heat content of the three zones above Layer also exceeds it just before 3 years Of interest is the fractional heat content of Layer which increases to approximately % of the heat in Layer at approximately 5 years his means that the equal simultaneous injection of steam into both Layers would result in a significant increase in the heat content from that time forward Somewhat different results are obtained when other thermal reservoir properties are used Fig 3 shows similar curves for case PG Figs 4 through 7 show similar curves for cases PG3 PG PG PG3 he range of properties considered affect the heat content of Layer the underburden only to a small extent As the thickness of the center layer increases its fractional heat content increases primarily because of its increased volume At a given time the heat content of Layer is reduced as the thickness of the center layer increases with a corresponding effect on the total heat content of Layers For property set P the improvement in the heat efficiency of Layer owing to equal net heat-injection rates into both layers is given by the curve labeled Layers At years the increase in the combined heat efficiency relative to that of only one layer is % for a center layer thickness of ft respectively For property set P the increases are % he improvements are smaller at earlier times hus the interaction between layers is significant for an intervening layer thickness of as much as 45 ft One of the effects of increasing the thickness of the center layer is to delay from approximately to 5 years the time at which the combined heat efficiency increases by % Discussion onclusions he results are valid for any manner of heat injection are presented in terms of steam injection he range of properties considered in the examples is both reasonable limited Equations are developed for calculating the heat content of two layers undergoing variable steam injection with the layers separated by an impermeable formation Examples are presented for only the same constant net rate of heat injection into each of the two layers he fractional heat content of the layers is independent of the actual heat-injection rate temperature when the net heat-injection rate is constant he numerical results presented are valid for the total heat content in the injection layers do not describe in any way the areal temperature distributions within the layers he effect of steam override may be approximated by considering an effective thickness of the steam zone (for example from van Lookeren rather than the actual layer thickness A user could select different injection rates in the two layers to meet some objective For example suppose the user wants the equivalent heat fronts to be at approximately the same position in both layers near a time of t days For the two constant differ- Fractional eat ontent Injection Into Only Layer Layer enter Layer Layer Layers Fractional eat ontent Injection Into Only Layer Layer enter Layer Layer Layers ime yr Fig Fractional distribution of heat for ase PG ime yr Fig 3 Fractional distribution of heat for ase PG September SPE Journal 63

3 Injection Into Only Layer Injection Into Only Layer Fractional eat ontent Layer enter Layer Layer Layers Fractional eat ontent Layer enter Layer Layer Layers ime yr Fig 4 Fractional distribution of heat for ase PG3 Injection Into Only Layer ime yr Fig 5 Fractional distribution of heat for ase PG Injection Into Only Layer Fractional eat ontent Layer enter Layer Layer Layers Fractional eat ontent Layer enter Layer Layer Layers ime yr Fig 6 Fractional distribution of heat for ase PG ent rates of heat injection that give equal front positions at t (obtained by trial error the ratio of the areas of the two layers near that time would then be given by: ( D ( t D h h E h h ( D ( D A E t E t A E t ( he user may also consider variable rates of heat injection in optimizing objectives by using well-known superposition methods For a constant net rate of heat injection into two 5-ft layers separated by a center layer of or 45-ft thickness for reasonable thermal layer properties the following is found hermal interaction between layers as far apart as 45 ft can be significant within years after the onset of steam injection Increasing the distance between injection layers from 35 to 45 ft increases the time at which the total heat efficiency first increases by % relative to that of a single layer from approximately to 5 years 3 At a separation of 5 ft the improvements are significant after year of injection 4 Even faster more pronounced improvements would result from shorter separations between the layers than the 5-ft minimum used in the examples 5 It is expected that the heat efficiency could be optimized by varying the injection rate into the layers Nomenclature A function defined by Eq A-49 B function defined by Eq A-65 function defined by Eq A-54 D function defined by Eq A-57 h i position of upper boundary of layer i ft h is the base of layer h gross thickness of s ft ime yr Fig 7 Fractional distribution of heat for ase PG3 heat content of layer Btu M volumetric heat capacity of layer Btu/ft 3 F Q j rate of heat injection with respect to dimensionless time into layer j Btu Q l rate of heat loss from injection layer Btu/day R ij ~ j (h i/ ~ j (h j s Laplace variable dimensionless t time days t D G t dimensionless time temperature rise F u heat flux Btu/ft day x y z artesian coordinates with z increasing upward /M thermal diffusivity ft /day > ij i i/( j j dimensionless G s/ /ft G defined by Eq A- days / thermal conductivity Btu/ft F day Subscripts Layers underburden O overburden center layer r reference xyz directional components Superscripts function in Laplace space u vertical average value of function u u ~ areal average of u Q rate of change of Q with respect to real time Q rate of change of Q with respect to dimensionless time t D 64 September SPE Journal

4 References Marx JW Langenheim R: Reservoir eating by ot Fluid Injection rans AIME ( Prats M: he eat Efficiency of hermal Recovery Processes JP (March ; rans AIME 46 3 Lauwerier A: he ransport of eat into an Oil Layer aused by Injection of ot Fluid Applied Science Research (955 A Myhill NA Stegemeier GL: Steam-Drive orrelation Prediction JP (February Jones J: Steam Drive Model for -eld Programmable alculators JP (September ook DL: Influence of Silt Zones on Steam Drive Performance pper onglomerate Zone Yorba Linda Field alifornia JP (November Spivak A Muscatello JA: Steamdrive Performance in a Layered Reservoir A Simulation Sensitivity Study SPERE (August losmann PJ: Steam Zone Growth During Multiple-Layer Steam Injection SPEJ (March 967 ; rans AIME 4 9 Stehfest : Algorithm 368 Numerical Inversion of Laplace ransforms ommunications of the AM ( van Lookeren J: alculation Methods for Linear Radial Steam Flow in Oil Reservoirs SPEJ (June Appendix eat Interaction Between Layers eat Balance in Layer he differential equation (DE within injection Layer is: u u u + + M x y z t the u ijthe components of the heat flux vector θ As in Prats we define the following three quantities: the Laplace transform of (t D as ( s ( t D the vertical average over the thickness of Layer as an areal average defined by h dx dy (A-5 h ran undefined reference thickness ake the Laplace transform of Eq A- integrate over the layer thickness to obtain u u x y h + + uz( h uz( x y Now integrate each term of Eq A-6 over the areal extent he first term leads to the Laplace transform of the net heat injection rate into Layer x y z Mθ smθ h M h u h + dxdy Q x y θ Q td h dz r λ + M λ M h ux y (A- (A- (A-3 (A-4 (A-6 (A-7 ere Q represents the Laplace transform of the time derivative of the cumulative heat injected as Q represents the Laplace transform of the derivative of the cumulative heat injected with respect to the dimensionless time t D he second third terms lead to the Laplace transform of the rate of heat loss to the overlying underlying formations respectively : uz ( h dxdy θ Q ( h he fourth term leads to the Laplace transform of the rate of change of the heat content of Layer owing to injection into Layer he sum of the four terms leads to the global heat balance: eat ransfer to Semi-Infinite Layers In the underburden the DE is the same as Eq A- but subscripted instead of Instead of integrating vertically over the thickness the vertical heat flux is represented by Fick s Law of conduction (z: u sθ M h dxdy sθ z which after integration over the areal extent noting that the only heat entering the underburden laterally is from wellbore heat transfer (considered negligible gives s M (A-3 θ / λ γ z with solution because z is never positive in the underburden so that the heat flux into the underburden is From Eq A-9 it follows that the Laplace transform of the heattransfer rate at the boundary between the underburden Layer is Q θ Q u h u ( h h z r z θ u ( dxdy Q ( u ( h z r h r sθ Mh dz λ z z e γ ( γ ( u λ h γ λ r In a similar fashion the Laplace transform of the heat-transfer rate at the boundary between the overburden L is Q h u h h z θ 3 Oz 3 r z h γ λ h r O O O( 3 s Q Q h Q (A-8 (A-9 (A- (A- (A-4 (A-5 r (A- (A-6 (A-7 eat ransfer to enter Layer he transformed DE is similar to Eq A-3 but subscripted for the center layer In terms of the temperatures at the boundaries of the center layer the temperature distribution is given by September SPE Journal 65

5 he areal integral of the vertical heat flux is then hus the areal integral of the heat flux at zh is given by that at zh by In these equations eat ransfer to Layer he transformed DE in layer before steam injection in that layer is similar to Eq A-3 but subscripted he areal integral of the temperature vertical heat-flux distributions are similar to Eqs A-8 A-9: ( z h h γ h [ λ ( z h ] + h z h h λ γ ( u z λγ h3 h h γ hus the areal integral of the heat flux at zh 3 is given by that at zh by + λγ h h In these equations γ θ s / α (A-7 γ θ sm / λ θ s/ α ( γ (A- + λγ h coth h ( z h h γ h 3 λ( z h [ λ ] + h z h ( γ ( z h λ ( γ h z h u ( h λγ ( h coth γ h z λγ ( h csch γ h u ( h λγ h csch γ h ( γ u z h h h [ λ ( z h ] λγ γ h ( u h coth λγ h γ h z ( u h csch λγ h γ h z z 3 λγ h [ λ z h ] (A-9 + λγ ( h csch coth ( γ (A- (A- (A-8 (A-3 (A-4 (A-5 (A-6 Boundary onditions here is continuity of both temperature heat fluxes across common boundaries between layers he Laplace transform of the areal integral of these quantities is also continuous Accordingly At zh 3 from Eqs A-5 A-5 A-8 A-3 we obtain from which the boundary temperatures are related by At zh using Eqs A- A-6 A-9 A-33 we obtain using Eq A-37 one obtains u h u h Q h θ h (A-3 u h u h Q h θ h (A-33 R / r u h u h Q h θ h / r u u Q θ / h r (A-8 h h 3 O 3 (A-9 h h (A-3 h h (A-3 / 3 O 3 3 r λγ λγ coth ( γ h h h 3 O O 3 h R h 3 3 coth ( γ csch ( γ h h h h λγ h cot h γ / λγ (A-4 h R h β csch β coth W β λ γ / λ γ O O O β λ γ /( λ γ (A-44 ( h ( γ h + 3 R / β γ h + γ h 3 O (A-39 R λγ ( h3csch γ h / λγ csch ( γ (A-36 + λγ h h (A-37 W (A-34 (A-35 (A-38 (A-4 (A-4 (A-43 eat ontent of Layer Owing to Injection Substitution of Eqs A-7 A- A-34 A-35 in Eq A- yields 66 September SPE Journal

6 s Now substitute EqA-4 in A-45 use the Lauwerier assumption to obtain hus he parameter A can be represented in a manner more easily recognizable as approaching s for large s (small times by rearranging the last equation as A eat ontent of Layer Owing to Injection in Layer he heat balance in Layer before injecting steam into it may be deduced from Eq A-3 to be Substitution of the definition of Eq A-48 of MR R + M h in Eq A-5 yields + Q h λ γ r θ Q A W Q s + A λ γ W + λ γ A θ M h s h3 r M h dz Mh ( h R + 3 Q s + A M γ W + M γ θ M h r 3 Mh h R R + r 3 γ h (A-5 γ γ h ( Mh h h Mh h r r ( θ h h γ h λγ hr ( γ h R γ h (A-48 Mh h + h r h s Q λ γ W + λγ (A-46 (A-49 r 3 (A-47 (A-5 (A-5 (A-53 (A-54 (A-55 eat ontent of Layers In a similar manner the heat content of Layer owing to injection into Layer is / θ (A-45 / h he heat content of Layer owing to injection into Layer is BQ (A-64 s + D MR RO+ B (A-65 M h λ γ Y + λ γ D Y Y csch / h h ( γ ( γ h β λ γ / ( λ γ (A-6 β λ γ / ( λ γ (A-6 / β ( γ + ( γ (A-63 R h h O h R O γ h (A-66 γ γ h he total heat content of Layer is adding Eqs A-46 A-6 Q B Q + + (A-67 s + A s+ D that of Layer is adding Eqs A-55 A-56 Q Q + + (A-68 s + D s+ A SI Metric onversion Factor Btu/ft 3 F E kj/m 3 K Btu/ft F day E W/m K ft 348 m year 3655 E day onversion factor is exact ( γ h + βcoth Y R Q sq s + D s+ D O O θ M h β ( γ h R γ h (A-56 (A-57 (A-58 (A-59 (A-6 SPEJ Michael Prats is president of Michael Prats Assocs Inc of ouston worked for Shell o from 948 to mikep@mpratscom e has contributed a number of publications on reservoir engineering topics especially thermal recovery e is on the faculty of both Stanford the of exas at Arlington Prats holds a BS degree MA degree in physics from the of exas at Austin e is an SPE onorary Member is the recipient of the Anthony F Lucas Gold Medal September SPE Journal 67