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BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The invention relates generally to subterranean well tools such as inflatable packers, bridge plugs or the like, which are set through the introduction of an actuating fluid into an expandable elastomeric bladder and, more particularly, to an apparatus and method that utilize a multi-stage piston with multiple operating surfaces in contact with hydrostatic well pressure for maintaining a relatively uniform fluid pressure in the bladder when the tool is subjected to thermal variants after setting. 
     2. Description of Problems 
     It is known among those skilled in the use of these types of inflatable devices that they are subject to changes in inflation pressure when the temperature of the inflation fluid varies from its initial inflation temperature. Typically, an increase in fluid temperature results in increased inflation pressures, and a decrease results in decreased inflation pressures. An increase in inflation pressure can make the tool susceptible to burst failure. A decrease in inflation pressure can diminish anchoring between the tool and the well bore to a point where the tool is not able to provide its intended anchoring function. In both instances, significant changes in temperature in the inflation fluid can result in compromised tool performance and possible tool failure. These failures can result in significant monetary loss and possible catastrophe. 
     The magnitude of temperature change needed to adversely affect the performance of an inflatable tool depends upon a number of parameters, such as, for example (1) the expansion ratio of the inflation element, (2) the relative stiffness of the steel structure of the inflation element compared with the compressibility and thermal expansion coefficient of the inflation fluid, (3) the relative stiffness of the casing and/or formation compared with the compressibility and thermal expansion coefficient of the inflation fluid, and (4) the inelastic properties of the elastomeric components in the inflation element. There are other factors of lesser significance known to those skilled in the relevant art. 
     Regardless of the specific values of the aforementioned parameters, conventional inflatable tools cannot tolerate positive or negative temperature changes greater than about 10-15 F.° (5.6-8.3 C.°) from the initial temperature at the end of their inflation cycle. If the temperature of the inflation fluid varies by more than this amount, the tool is subjected to excessive inflation pressures or insufficient inflation pressures, which could result in tool performance problems of the nature described above. 
     In addition, cycling the inflation fluid temperature within a ±15 F.° of the initial temperature upon expansion can cause stress cycling in the steel structure of the inflation element and in the bladder. There is the potential for a serious problem when the inflation element survives routine thermal cycling for a finite period oftime, during which cyclic damage in the tool accumulates. In such a case, failure can occur at some time after the rig has departed from the well site. Thus, an inflatable tool can provide short term functional performance during low magnitudes of thermal cycling. However, cumulative damage phenomena can occur in steel structures and/or elastomeric components and eventually cause device failure. 
     A time delayed failure can be more costly and possibly more catastrophic than one which occurs within a short time after the initial setting of the tool. Replacement of the failed device would entail performing a second project about equal in size and expense to the first service operation, instead of the case of a short-lived tool which would fail before the rig is broken down and moved off the site. Operations of this type can cost in excess of one hundred thousand dollars, and as high as several millions of dollars. 
     There are many operations in the oil and gas industry that successfully use pressure isolation devices which routinely encounter substantial thermal excursions and substantial magnitudes of combined positive and negative thermal cycling. Typically, inflatable devices are excluded as candidates for such projects. Typical projects are listed below. 
     large volume stimulation projects, n 
     selective zone treatment projects, n 
     large volume cement squeeze projects, n 
     production packer service in oil and/or gas wells experiencing cooling from Joules-Thompson expansion and cooling of gases, n,c 
     production packer service in oil and/or gas wells experiencing heating from deeper produced fluids, p,c 
     conversion of a producing well to an injection well and temporary isolation between perforation intervals, n,c 
     huff/puff steam injection methods for producing viscous oil formations, p,c 
     [n=these operations typically result in a large negative thermal excursion (cooling) in the pressure isolation device.] 
     [p=these operations typically result in a large positive thermal excursion (heating) in the pressure isolation device.] 
     [c=these projects typically repeated multiple thermal cycling in the pressure isolation device over long periods of time.] 
     The first five project categories are very common in the industry. Thousands of them are performed per year. The bottom two categories are relatively infrequent with respect to world wide activities. 
     If conventional packers and bridge plugs are not able to provide service for a given well configuration, because they are not able to pass through restrictions and subsequently set in casing, it is common to use a rig to pull tubing and perform a costly work-over project. The use of thru-tubing inflatable devices provides well known benefits and versatility to the oil and gas industry. Their lack of service worthiness for operations that include thermal cycling and thermal excursions exclude them from a substantial portion of the remedial service sector. An invention that would eliminate the deleterious effects of routine thermal excursions and thermal cycling, would eliminate the aforementioned problems, augment the benefits and versatility of inflatable devices and provide substantial cost savings to operators in the industry. 
     3. Description of the Prior Art 
     Subterranean well tools, such as conventional packers, bridge plugs, tubing hangers, and the like, are well known to those skilled in the art and may be set or activated a number of ways, such as mechanical, hydraulic, pneumatic, or the like. Many of such devices contain sealing mechanisms which expand radially outwardly upon the introduction of a substantially incompressible actuating fluid for setting the device in the well to provide a seal in the annular area of the well between the exterior of the device and the internal diameter of well casing, if the well is cased, other tubular conduit, or along the wall of open borehole, as the case may be. 
     Frequently, the seal is established subsequent to the setting of such device in the well and will be adversely effected by temperature variances of the device or in the vicinity of the device. Such temperature variances can cause expansion or contraction of the sealing mechanism, thus jeopardizing the sealing and even anchoring integrity of the device over time. For example, such devices are typically utilized in well stimulation jobs in which an acidic composition is injected into the formation or zone adjacent a well packer or bridge plug. As the stimulation fluid is injected into the zone, the temperature of the device and the well bore in the vicinity of the formation will be reduced. 
     If, for example, the well tool utilizes a sealing mechanism that includes an inflatable elastomeric bladder, the temperature of the actuating fluid utilized to inflate the bladder and retain same in set position in the well is affected by the temperature reduction during the stimulation job, causing a reduction of pressure within the interior of the bladder, fluid chambers and communicating passageways within the tool. This reduction in pressure, in turn, causes the bladder to contract from the initial setting position. In more dramatic situations, anchoring of the device in the well bore can be lost and the differential pressures across the device can cause “corkscrewing” of the coiled tubing or work string, resulting in project failure, expensive solution of the corkscrew problem and substantial operational risks. 
     On the other hand, the same inflatable tool is also adversely affected by an increase in device temperature during certain types of secondary and tertiary injection techniques utilizing, for example, the injection of steam. As the steam is injected into the zone of the well immediate the set packer or well plug, the zone and accompanying devices, including tubing, quickly become exposed to the increased temperature. Some prior art devices containing inflatable packer components have been known to have the inflatable bladder element actually rupture, due to exposure to increased pressure within the bladder and interconnected chambers and passageways as steam flows through the device and is injected into the well zone. 
     In U.S. Pat. No. 4,655,292, entitled “Steam Injection Packer Actuator and Method,” a device is shown and disclosed, which addresses the problems associated with the prior art by providing a mechanism incorporating a compressible fluid, such as nitrogen gas. The fluid is used to accommodate an increase in temperature during steam injection and other operations for preventing the packer mechanism from rupturing as a result of exposure to enhance pressures resulting from the increase of temperature of inflation fluid and device components as stream flows through the device. 
     PCT application, Ser. No. WO/98/36152, the description and drawings of which are incorporated herein as though fully set forth, describes a thermal compensating apparatus that utilizes hydrostatic well pressure for maintaining a relatively constant pressure in the bladder of an inflatable tool. The apparatus has a piston with a pair of opposed surfaces, which are respectively in contact with the fluid used to actuate the tool and the surrounding well fluid below the tool. The surface in contact with the well bore fluid is proportionately larger in surface area than the surface in contact with the actuating fluid, at a ratio of about 1.4:1 to 1.8:1. Relatively constant hydrostatic well pressure bears on the larger of the surfaces. Referencing off of the hydrostatic well pressure, the piston moves in response to any change in volume and concomitant pressure in the actuating fluid due to temperature changes in the vicinity of the tool, for maintaining a substantially constant pressure in the actuating fluid. 
     However, the apparatus in the PCT application is not suitable for smaller-diameter thru-tubing tools such as, for example, tools 2⅛″ in diameter which are commonly run through 2⅞″ tubes that have internal diameter restrictions of 2{fraction (5/16)}″ and set in a 7″ casing. These thru-tubing tools are inflated to high expansion ratios and therefore are filled with a substantial volume of actuation fluid. The volume of actuation fluid is exceptionally high when compared to the area and volume sweeping capacity of the pressure maintaining piston in a single state device having an intensification ratio of 1.4:1 to 1.8:1. These types of tools do not have a large enough diameter to provide a differential surface area on the respective fluid contact surfaces that is great enough to compensate for temperature variances greater than 10-15 F.°. Because temperature variances in excess of 20 F.° are not uncommon, there is a need for an apparatus that utilizes hydrostatic well pressure for maintaining a relatively constant pressure in small diameter thru-tubing tools in service operations that experience substantial variances in tool temperature while in service. 
     The present invention addresses these problems associated with the prior art devices, and maintains a relatively constant inflation pressure even when the device experiences single and/or multiple thermal excursions of substantial magnitude. The invention operates to abate the adverse effects of any combination of heating and cooling, both quasi-static and dynamic cycling. 
     SUMMARY OF THE INVENTION 
     The present invention provides an improved thermal compensating apparatus over one described in PCT patent application, Ser. No. WO/98/36152. As in the apparatus in the PCT application, the present invention utilizes opposing surfaces with a differential surface area ratio, also referred to as intensification ratio, that is set at the differential between the pressure in the actuating fluid used to set the tool and the relatively constant hydrostatic well pressure. However, a multi-stage piston is utilized so that the surrounding well fluid bears on more than one piston surface so that a relatively constant actuation pressure can be maintained in tools that encounter the most extreme combinations of tool diameter, expansion ratio, and substantial temperature variations, and even at unusually high intensification ratios. 
     When pressure in the actuating fluid changes due to temperature variations in the vicinity of the tool, the hydrostatic well pressure is in contact with more than one surface of the piston so that the same differential ratio can be utilized as in the apparatus of the PCT application, but in a tool having a much smaller diameter. Instead of utilizing only a pair of opposing surfaces for providing the differential surface area, the improved apparatus utilizes multiple surfaces, arranged tandem, in contact with the hydrostatic well pressure. In this way, a tool having a smaller diameter with contact surfaces having surface areas can be utilized for accommodating temperature variances as great as 200 F.°, even at high intensification ratios. 
     The apparatus and method provide a multi-stage piston arrangement with multiple surfaces in contact with the surrounding well fluid. This is accomplished by a multi-stage piston with a first surface in contact with the actuating fluid and a multi-stage second piston that has two or more surfaces that remain in contact with the surrounding well fluid. This arrangement allows the use of a relatively large surface area on the first piston in contact with the actuation fluid when compared with the surface area of the same piston area of the apparatus described in the PCT application. This multi-stage piston has two or more surfaces that are exposed to the surrounding well pressure, so that the intensification ratio, which is ratio of the surface areas exposed to the surrounding well fluid to the surface area of the piston exposed to the actuation fluid can be much larger even when the diameter of the invention is small compared with the set diameter of the inflatable tool and when the invention must provide substantial swept volume to maintain a relatively constant actuation pressure when the temperature of the tool varies by as much as ±200 F.°. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     A better understanding of the invention can be obtained when a detailed description of preferred embodiments described below is considered in conjunction with the appended drawings, in which: 
     FIG. 1 is a plan view, partially in section, of an expanded tool, such as an inflatable packer, to which a prior art thermal compensating apparatus is connected, such as the one in FIG. 1 in PCT application WO98/36152; 
     FIG. 2 is a sectional view of the relative positions of the components in the prior art thermal compensating apparatus shown in FIG. 2 of the PCT application, after actuating fluid has expanded the inflatable packer into contact with the well casing; 
     FIG. 3 is a sectional view of the relative positions of the components in the prior art thermal compensating apparatus shown in FIG. 3, of the PCT application, when the actuating fluid is subjected to a decrease in temperature; 
     FIG. 4 is a sectional view of a second embodiment of the prior art, single-stage thermal compensating apparatus shown in FIG. 4 of the PCT application; 
     FIG. 5 is a sectional view of the second embodiment of the prior art thermal compensating apparatus shown in FIG. 5, of the PCT application, after the piston is moved upwardly when the actuating fluid is subjected to a decrease in temperature; 
     FIG. 6 is a sectional view of the improved, multi-stage thermal compensating apparatus of the present invention; and 
     FIG. 7 is a sectional view of the improved thermal compensating apparatus shown in FIG. 6, after the actuating fluid in the tool has been subjected to a decrease in temperature. 
     FIG. 8 is a sectional view of the improved thermal compensating apparatus shown in FIG. 6, after the actuating fluid in the tool is subjected to thermal expansion. 
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
     The multi-stage thermal compensating apparatus of the present invention is an improvement over the single-stage apparatus described in PCT application WO98/36152, the drawings and description of which are incorporated herein by reference as though fully set forth. The improved apparatus has particular applicability for service conditions where the diameter of the inflatable tool is less than about 50% of the diameter of the set inflatable tool and the intensification ratio is greater then 1.4:1 and the temperature of the inflatable tool is expected to cycle or significantly depart from its initial temperature at the end of the setting operation, for example the invention is ideally suited for a 1{fraction (11/16)}″ diameter tool which is run through 2⅜″ tubing or the like, and set in 4″ or larger casing. The prior art single stage invention cannot maintain constant actuation fluid pressure when the tool temperature varies by a significant amount. However, the present invention is not limited to tools of that size, and can be used in tools of any size in which a multi-stage piston arrangement can be used for obtaining a suitable intensification ratio for the areas of the surfaces that are in contact with the actuating fluid and surrounding well fluid. 
     Before describing the apparatus of the present invention, the prior art apparatus described in the PCT application will be described for background information. First, referring to FIGS. 1-3, one embodiment of the prior art thermal compensating apparatus is shown as being connected to an inflatable downhole tool  10 , such as a packer, bridge plug or the like. The tool  10  has been inflated in a known manner with a suitable incompressible actuating fluid for setting the tool  10  inside a casing  12 . When the tool  10  is inflated as shown schematically in FIG. 1, it establishes and maintains a seal across the internal cross-section of the casing  12 . The tool  10  may be set, for example, above a formation zone that produces water or other undesired fluid. As shown in FIG. 1, the tool  10  is connected at its upper most end to a length of coiled tubing  14  or the like, through which a well known type of actuating fluid is transmitted for expanding the tool  10  as shown. 
     FIG. 2 shows the internal components of a thermal compensating apparatus  16 . An upper housing section  20  is connected to the lower most end of the tool  10  in a known way. A first upper piston  22  is positioned for up and down movement in a portion  20 ′ of the upper housing section  20 . A pair of channels  24 ,  24 ′ extend in the upper housing section  20  between a cavity  10 ′ formed in the tool  10  and a chamber  26 , which has an internal surface  20 ′″ and is defined at one end by a downward-facing end surface  20 ″ within the housing  20 , and on the opposite end by an upward-facing end surface  22 ′ of the first piston  22 . As described below, the piston end surface  22 ′ is influenced by the fluid pressure inside the tool  10  and the chamber  26 . 
     The housing  20  is threadedly connected at its lower end to the upper end of a lower housing section  27 . As shown, the lower housing section  27  has a greater internal area than the internal surface  20 ′″ of the upper housing section. A second lower piston  30  is positioned for up-and-down movement in the lower housing section  27 . 
     The lower housing section  27  has a tapered end  27 ′, which is formed with a central opening  32 , so that the lower most end surface  30 ′ of the second piston  30  is continuously influenced by the hydrostatic pressure within the well. The lower piston  30  isolates the well fluids with the actuating fluid that is located in the cylinder  26 . 
     The pistons  22 ,  30 , are connected to each other by means of a central piston rod  34 , so that the pistons  22 ,  30 , move up and down in tandem. A space  31  is formed between the pistons  22 ,  30 , which contains air at atmospheric pressure. 
     The end surface  30 ′ of the lower piston  30  has a substantially larger surface area than the end surface  22 ′ of the upper piston  22 . For example, the piston surface  30 ′ may have a surface area 1.1 to 2.0 times larger than the piston surface  22 ′. This differential maintains the pistons  22 ,  30 , in equilibrium in the position shown in FIG. 2, within the well at the predetermined hydrostatic well pressure and actuating fluid pressure. 
     When there is a temperature change in the vicinity of the tool  10 , which causes the pressure in the actuating fluid to change, the pistons  22 , 30 , automatically move and maintain a substantially constant pressure in the actuating fluid. This pressure compensation is provided by the pistons  22 ,  30 , and the tubular piston rod  34 . This piston-based pressure compensator, working with the hydrostatic well pressure as the reference pressure, absorbs or reduces the effective cooling or heating of the actuating fluid used for setting the downhole tool  10 . In this way, a relatively constant pressure is maintained within the tool so that its functions are not adversely affected. 
     One application found for this device is when, for example, water is injected into the formation at a point plugged by the tool  10 , so as to displace oil or gas in a secondary recovery project. In such case, the injection water cools the actuating fluid within the downhole tool  10 . This in turn causes the actuation fluid to contract. In a conventional tool not having a pressure compensating device this contraction will cause the actuation pressure to decrease. When such a reduction in pressure occurs, there is a risk that the seal provided by the tool  10  may be lost. If the temperature decrease is 15 F.° or greater, the seal and anchoring functions will most certainly be lost. On the other hand, there are conditions when the temperature in the vicinity of the tool  10  is increased relative to the ambient temperature in the well, this would cause an over pressure situation within the tool  10 . If the temperature increase is 10 F.° or greater, the tool will most certainly fail in burst. 
     By way of example, FIG. 3 shows the positions of the components of the thermal compensating apparatus  16  when there is a decrease of the temperature of the actuating fluid of the tool  10 . As shown, the tool  10  has been set by the introduction of pressurized actuating fluid, so that the fluid also flows through channels  24 ,  24 ′, and into the chamber  26 . The hydrostatic well pressure below the set tool  10  remains relatively constant. When the volume of the actuating fluid is decreased due to a decrease in temperature in the vicinity of the tool  10 , the pressure of the hydrostatic well pressure fluid bearing on the underside  30 ′ of the piston  30  causes the piston  22  to move upward as shown in FIG.  3  and force actuating fluid from the chamber  26  into the internal cavity  10 ′ for maintaining a substantially constant pressure within the tool  10 . In this way, the pressure of the actuating fluid is automatically maintained at a substantially constant level through the action of the hydrostatic well pressure. 
     The opposite occurs when there is an increase in the temperature in the vicinity of the tool  10 . The volume of the actuating fluid increases, causing the fluid to expand into the chamber  26  and move the pistons  22 ,  30 , downwardly. Thus, by using the hydrostatic well pressure as the reference fluid, a substantially constant pressure can be maintained within the tool  10  through the use of the movable pistons  22 ,  30 , as described. 
     A second embodiment of the prior art device describe in the PCT application is shown in FIGS. 4 and 5 which are reproductions of FIGS. 4 and 5 in the PCT application. Briefly, this embodiment is different from the one described above in conjunction with FIGS. 1-3 in the configuration of the pistons, the provision of a central through passage for transmittal of surrounding well fluids, and the use of two axially-spaced seals. 
     As shown in FIG. 4, a central, tubular piston rod  34   a  is formed with a piston  36  that includes a first piston surface  36 ′ which is in contact with the actuating fluid for the tool  10 . The piston surface  36 ′ has a considerably smaller surface area than a second piston  36 ″ which is in contact with the surrounding well fluid. As in the embodiment shown in FIGS. 1-3, the surface area proportion is preferably 1:6. 
     The upper end of the piston rod  34   a  is movable within a lower section  38 ′ of a concentric inner tube  38  located in the upper housing section  20 . The inner tube  38  is connected end-to-end to a co-axial tube  40 , which has a bore  40 ′ that extends through the inflated tool  10 . The tube section  38 ′ has a relatively large diameter so that the piston  34   a  can move up and down within the tube section  38 ′. The tube section  38 ′ also is surrounded by longitudinal channels  24 ,  24 ′ (or alternatively by a concentric annulus, not shown), which as shown in FIG. 4 are connected through a cylinder bore  42  and into contact with the piston surface  36 ′. 
     In FIG. 5, the piston rod  34   a  and piston  36  are shown in their uppermost position in the upper housing section  20 . This embodiment is particularly suited when two spaced-apart tools  10  are connected to each other. FIG. 5 shows the upper tool with the lower one (not shown) being connected through a lower conical-downward tapering end portion  27 ′ in a tight-fitting manner so that the opening  32  is not exposed to the surrounding well fluid. Instead, the surrounding well fluid is in contact with a cylinder bore  44  through radially-extending ports  46 ,  46 ′. A seal  48  is located between the piston  36  and the inner surface of the lower piston housing section  27  for preventing the surrounding well fluid from flowing downwardly into the lower piston housing section  27 . Actuating fluid is thus able to flow downwardly through bore  40 ′ and bore  32  in order to set the lower-most tool  10  (not shown), without leaking into a space formed between the tools. 
     This embodiment operates essentially the same way as the ones shown in FIGS. 1-3. When temperature in the vicinity of the tool  10  decreases, the pressure of the actuating fluid bearing on the piston surface  36 ′ is decreased. The relatively constant surrounding hydrostatic well pressure forces the piston  36  to move upwardly by exerting force against the lower piston surface  36 ″ in order to move the piston upwardly to the position shown in FIG.  5 . The opposite occurs when there is an increase in temperature in the vicinity of the tool  10 , forcing the piston  36  to move downwardly within the cylinder bore  42 . 
     Because of the typical differential pressures between the actuating fluid used to set the tool  10  and the hydrostatic well pressure, the differential surface areas on the opposing surfaces of the pistons described must be relatively large (for example at a proportion of about 1.4 to 2.0) in order to provide a relatively constant pressure within the tool  10  throughout temperature fluctuations up to ±200 F.°. These design constraints require the diameter of the tool and of the pistons that move up and down within the tool to be relatively large, which prevents them from being used in thru-tubing tools. Single stage apparatuses like those shown in FIGS. 1-5 are limited in serviceability. They are not able to provide pressure maintenance in most thru-tubing inflatable service applications like those described earlier in this text, for reasons also described earlier in this text. 
     In accordance with the invention, a multi-stage pressure maintenance device is provided which has a wide range of serviceability including but not limited to thru-tubing applications where the relative size of the tool is small, the intensification ratio can be as high as and exceed 2:1, the swept volume of the first piston can be substantial, and actuation fluid pressure can be maintained constant even when the temperature varies by as much as ±200 F.°. 
     As shown in FIGS. 6-8, such an apparatus is provided, which utilizes a multi-stage piston that can be formed with a smaller outside diameter that heretofore possible. 
     In FIG. 6, a thermal compensating apparatus  52  is shown which is adapted to be connected at its upper end  54  to a tool (not shown) of the type described above. The apparatus  52  includes an upper piston housing  56  that is threadedly connected to the upper end  54 , an intermediate piston housing  58  that is connected to the upper piston housing  56  through a guide  60 , and a lower piston housing  62  connected to the intermediate piston housing  58  through a guide  64 . These sections are all threadedly connected to each other in a known manner. The apparatus  52  also includes a bottom plug  66 , is connected to the lower piston housing  62 , which includes a rod  68  that is held in place in the plug  66  through a pin  70 . 
     The upper end  54  of the apparatus  52  includes an elbow-shaped bore  72 , which is in fluid communication with the actuating fluid used to the set the tool. A rupture disk  74  is located within the bore  72 , in a known way, which ruptures when actuating fluid at pre-determined pressure is transmitted to the tool. A check valve mechanism and control head sub-assembly (not shown but of known generic construction to those skilled in the art) will facilitate inflation of the tool with actuation fluid that is in the conduit bore immediately above bore  72 . When rupture disk  74  breaches the check valve mechanism will automatically and simultaneously close and trap a finite volume of actuation fluid in the tool and cavity  78 . 
     A second bore  76  extends through the upper portion  54  for providing fluid communication for the actuating fluid between the tool and a chamber  78  formed in the upper piston housing  56 . Actuating fluid in the chamber  78  bears against an upper surface  80 ′ of upper piston  80 . A rod  82  rigidly connects the upper piston  80  with an intermediate piston  84  located in the intermediate piston housing  58 , and a lower piston  86  located in the lower piston housing  62 . 
     All three pistons  80 ,  84  and  86 , all move in tandem through their connections to rigid rod  82 . The rod  82  passes through the guides  60 ,  64 , for maintaining alignment as the pistons  80 ,  84  and  86  move up and down within their respective piston housings. 
     The piston  84  moves within a chamber  88  formed in the intermediate piston housing  58 , and the piston  86  moves within a chamber  90  formed within the lower piston housing  62 . The underside of each of the pistons  80 ,  84  and  86 , remain in contact with the surrounding well fluid through passageway  92  in the upper piston housing  56 , passageway  94  in the intermediate piston housing  58 , and passageway  96  in the lower piston housing  62 . In this way, the underside  80 ″ of the piston  80 , the underside  84 ″ of the piston  84 , and the underside  86 ″ of the piston  86  are exposed to hydrostatic well pressure. The space above the pistons  84 ,  86  within their respective chambers, is void, i.e., a vacuum exists in the space above pistons  84  and  86 . Each of the pistons and guides includes appropriate O-ring seals for isolating each of the chambers and the portions on opposite sides of the pistons from each other. 
     The apparatus  52 , as shown in FIG. 6, is in the “run-in” position before actuating fluid is used to set the tool and before the tool is exposed to hydrostatic well pressure. 
     The apparatus  52 , as shown in FIG. 7, is shown in an intermediate position. The inflatable tool has been expanded. Device  52  is essentially force balanced at the desired inflation pressure after piston face  80 ′ has separated from the bottom of sub  54  and prior to piston face  86 ″ touching item  68 . The multi-stage piston rod assembly is force balanced when it resides between these two described end points. The force balance is described by the following equation. 
     
       
           AP×A   1   =BHP  ( A   1   +A   2   +A   3 ) 
       
     
     where: 
     AP=the pressure of the actuation fluid 
     BHP=(bottom hole temperature)=well bore pressure outside the tool  52   
     A 1 =projected area determined by the bore diameter of housing  56   
     A 2 =the projected area determined by the bore diameter of housing  58  less the projected area of piston connecting rod  82   
     A 3 =the projected area determined by the bore diameter of housing  62   
     The intensification ratio is determined by:        IR   =           A   1     +     A   2     +     A   3         A   1       =     AP   BHP                              
     where: 
     IR=intensification ratio which is the ratio of the actuation pressure divided by the bottom hole pressure immediately below the tool 
     A force balance always exists in device  52 . It is because of the force balance that constant force, i.e., pressure is always exerted on the fluid in chamber  78  and in the tool above. When the volume of the actuation fluid expands or contracts piston  80  travels so as to sweep through a volume equal to the magnitude of volume expansion or contraction while maintaining a constant force on the fluid in chamber  78  and therein maintaining a constant pressure in the actuation fluid. 
     At the end of the setting cycle piston face  86 ″ presses atop rod  68  which is shear pinned in place by pin  70 . The actuation pressure will be caused to increase by continued pumping into the tool. The rupture disk breaches once the actuation fluid pressure reaches the breaching pressure of the rupture disk. As described earlier, the check valve in the control head simultaneously closes with the breach of the rupture disk and the actuation fluid resides in the tool and in bore  76  and cavity  97 . Remembering that piston face  86 ″ is atop rod  68 , it is evident that contraction of the volume of actuation fluid will cause the piston rod assembly (composed of  80 ,  82 ,  84  and  86 ) to stroke upward away from rod  68  while the device  52  maintains constant pressure of the actuation fluid. While expansion of the volume of actuation fluid will cause the piston rod assembly to press down upon rod  68  and to shear pin  70 . Once pin  70  is sheared, rod  68  is unsecured and offers no resistance to downward motion of the piston rod assembly. This is shown in FIG.  8 . 
     The combination of rupture disk  74 , rod  68  and shear pin  70  allows the positioning of the piston rod assembly so that the desired initial actuation pressure (the initial setting pressure) can be achieved while positioning the piston rod assembly so that contraction and expansion of the actuation fluid can be accommodated after the tool is set. 
     If a decrease in temperature occurs within the vicinity of the tool, causing a contraction of the actuating fluid, surrounding well fluids move in the direction of arrows F (FIG. 7) and bear against the undersides  80 ″,  84 ″ and  86 ″, of the pistons  80 ,  84  and  86 , respectively, and cause the piston  80  to move upwardly for maintaining a substantially constant pressure within the actuating fluid. The converse occurs if there is a temperature increase within the vicinity of the tool, causing an expansion of the actuating fluid, which in turn causes the pistons  80 ,  84  and  86 , to move downwardly for maintaining a substantially constant pressure within the actuating fluid. 
     In this way, by utilizing a multi-stage piston arrangement, a much smaller diameter thermal compensating apparatus can be used in conjunction with thru-tubing tools, which has heretofore not been possible. In this way, the integrity of the seal of the downhole tool is maintained, without danger of rupture, due to pressure variance within the vicinity of the tool. 
     Although the invention has been described in terms as specified embodiments which are set forth in detail, it should be understood that this is by illustration only and that the invention is not necessarily limited thereto, since alternative embodiments and operating techniques will be come apparent to those skilled in the art in view of the disclosure. Accordingly, modifications are contemplated which can be made without departing from the spirit of the described invention.

Summary:
A thermal compensating apparatus and method for maintaining a substantially constant fluid pressure within a subterranean well tool of the type that includes a bladder that is selectively expandable upon the introduction of pressurized actuation fluid for actuating said tool at a location in a well. A multi-stage piston is movable in a housing. The piston includes a first surface in contact with the actuating fluid and a plurality of second surfaces in contact with well fluid surrounding the apparatus. The combined surface areas of the second surfaces are greater than the surface area of the first surface, so that expansion or contraction changes in the volume of the actuating fluid caused by temperature changes in the vicinity of the tool will result in movement of the piston for maintaining the actuating fluid at a relatively constant pressure.