Patent Publication Number: US-8527205-B2

Title: Gravity interpretation workflow in injection wells

Description:
BACKGROUND OF THE DISCLOSURE 
     Grind and inject technology disposes of waste generated by oilfield operations. The goal is to decrease environmental impact and the overall footprint of new developments. Grind and inject technology entails crushing solid waste and adding water to make a slurry which is then injected into specially permitted disposal wells, where subsurface formations trap the injected slurries. However, the effect of such injection requires the reservoir to be monitored. That is, the injection is expected to change the reservoir density over time, both near the borehole and far from the borehole. In this context, there exists a need to monitor the displacement of mass in the subsurface and, in particular, indicate if the injected material is staying near the perforation zone or is taking advantage of a path induced by the injection, such that the slurry has moved vertically in the formation. 
     SUMMARY OF THE DISCLOSURE 
     One aspect is directed to a method that may include estimating a change in a characteristic of a subterranean formation into which a fluid has been injected via a well extending into the subterranean formation. The method may also include building a multi-dimensional model balancing mass of the injected fluid, wherein the model is based on the estimated characteristic change, and utilizing the model to determine the sensitivity of a borehole gravity tool in the well. The method may further include measuring gravity with the borehole gravity tool at a plurality of stations along the well, and utilizing the model and the gravity measurements to locate the injected fluid in the subterranean formation. 
     Another aspect is directed to a method that may include determining total porosity of a subterranean formation into which a fluid has been injected via a well extending into the subterranean formation, and estimating time-lapse variation of the porosity based on the total porosity, perforation data and injection data. The method may also include building a time-lapse density model based on the estimated time-lapse variation of the porosity, formation matrix density data and injected fluid density data, and projecting a borehole gravity tool response at a plurality of stations along the well based on the time-lapse density model. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present disclosure is best understood from the following detailed description when read with the accompanying figures. It is emphasized that, in accordance with the standard practice in the industry, various features are not drawn to scale. In fact, the dimensions of the various features may be arbitrarily increased or reduced for clarity of discussion. 
         FIG. 1  is a schematic view of apparatus according to one or more aspects of the present disclosure. 
         FIG. 2  is a schematic view of apparatus according to one or more aspects of the present disclosure. 
         FIGS. 3A and 3B  are schematic views of apparatus according to one or more aspects of the present disclosure. 
         FIG. 4  is a schematic view of apparatus according to one or more aspects of the present disclosure. 
         FIGS. 5A-5C  are schematic views of apparatus according to one or more aspects of the present disclosure. 
         FIG. 6  is a schematic view of apparatus according to one or more aspects of the present disclosure. 
         FIG. 7  is a schematic view of apparatus according to one or more aspects of the present disclosure. 
         FIG. 8  is a schematic view of apparatus according to one or more aspects of the present disclosure. 
         FIG. 9  is a schematic view of apparatus according to one or more aspects of the present disclosure. 
         FIGS. 10A-10D  are schematic views of apparatus according to one or more aspects of the present disclosure. 
         FIG. 11  is a schematic view of apparatus according to one or more aspects of the present disclosure. 
         FIG. 12  is a chart demonstrating one or more aspects of the present disclosure. 
         FIG. 13  is a schematic view demonstrating one or more aspects of the present disclosure. 
         FIG. 14  is a flow-chart diagram of at least a portion of a method according to one or more aspects of the present disclosure. 
         FIG. 15  is a flow-chart diagram of at least a portion of a method according to one or more aspects of the present disclosure. 
         FIG. 16  is a schematic view of apparatus according to one or more aspects of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     It is to be understood that the following disclosure provides many different embodiments, or examples, for implementing different features of various embodiments. Specific examples of components and arrangements are described below to simplify the present disclosure. These are, of course, merely examples and are not intended to be limiting. In addition, the present disclosure may repeat reference numerals and/or letters in the various examples. This repetition is for the purpose of simplicity and clarity and does not in itself dictate a relationship between the various embodiments and/or configurations discussed. Moreover, the formation of a first feature over or on a second feature in the description that follows may include embodiments in which the first and second features are formed in direct contact, and may also include embodiments in which additional features may be formed interposing the first and second features, such that the first and second features may not be in direct contact. 
     Oil exploration involves evaluating reservoirs to determine the movement or absence of oil, gas, or water as the reservoir fluids are produced. Gas, oil and/or water movement in a reservoir can be monitored by gravity methods which include determination of the acceleration due to gravity (henceforth referred to as gravity for the sake of brevity) within a borehole and at the surface of the reservoir. Borehole gravity data may be used to map out the vertical distribution of gas, oil and/or water at a well, and surface gravity may be used to understand the horizontal distribution of gas, oil and/or water. 
     Borehole gravity surveys comprise measuring local earth gravity at a series of stations in a borehole. The difference in gravity (Δg) and the vertical distance (Δz) between two successive stations yield sufficient information to determine the bulk density of the strata adjacent the borehole. Bulk density includes the density of the rock matrix and the density of the gas or fluid filling the pore space. The bulk density is then mapped out to determine the vertical distribution of water, oil and/or gas as the reservoir fluids are produced. 
     Bulk density, ρ, is given by the following expression:
 
ρ=( F−Δg/Δz )/(4 πG )
 
where Δg/Δz is the vertical gradient of gravity between two spaced apart stations, F is the free air gravity, and G is the universal gravitational constant. The free air gravity, F, may be determined during borehole gravity surveys, so that the only unknown is the bulk density, ρ.
 
     The further apart the station measurements are made, the further from the borehole in the horizontal plane is the zone of investigation. A five-foot interval may produce a radial zone of investigation of 0-5 feet from the borehole. The deeper zone of investigation may make it possible to determine the true gas-oil contact, free from borehole effects such as localized gas cone, mud, and casing. 
     Gravity measurements may be monitored in the microGal (10 −6  cm/s 2 ) range, which may ensure useable data that provide an indication of untapped pockets of oil or gas in the strata adjacent a borehole. This level of resolution in gravity measurements requires a highly precise gravity sensor and carefully implemented measuring techniques. For example, the gravity sensor may be oriented so that the sensitive axis of the sensor is vertically aligned. A deviation of the sensitive axis of the sensor by an angle α from the vertical corresponds to an error of g·(1−cos α), where g is the gravitational acceleration. Thus, a deviation by an angle α equal to 45 μrad (or 0.00258°) from the vertical would result in an error of about 1 microGal. In addition to keeping the sensitive axis of the gravity sensor aligned with the vertical during gravity measurements, the depth measurements of the stations may also be maintained accurate to within 1 mm, such that density may be obtained with accuracy of 0.01 g/cm 3 . 
       FIG. 1  is a schematic view of a sonde  10  according to one or more aspects of the present disclosure. The sonde  10  is suspended in a borehole  12  on the end of a wireline  14  that is supported at the surface  16 . The wireline  14  is used to lower and raise the sonde  10  within the borehole  12 . The positioning of the sonde  10  in the borehole  12  is controlled from the surface  16 . The borehole  12  may be cased, lined or open. 
     The sonde  10  includes an elongated, hollow, pressure vessel  18  configured to withstand the pressures, temperatures and fluids of the borehole environment. A gravity tool  20  disposed in the pressure vessel  18  comprises a rotatable portion  22  and a non-rotatable portion  24 . The rotatable portion  22  may rotate about the longitudinal axis  26  of the pressure vessel  18 . The rotatable portion  22  and the non-rotatable portion  24  are configured to travel simultaneously along the longitudinal axis  26  of the pressure vessel  18  to make gravity measurements. 
     The rotatable portion  22  comprises a gravity meter  28 , a gimbal drive assembly  30 , an electronic controller  32 , an accelerometer assembly  34 , a roll-axis drive  36  and a slip ring assembly  38 . The gravity meter  28  comprises a gravity sensor configured to measure gravity at stations along the borehole. The gimbal drive assembly  30  and the roll-axis drive  36  are collectively configured to align the sensitive axis of the gravity sensor with the vertical before gravity measurements are taken at a measuring station. 
     The electronic controller  32  is configured to monitor operation of the gimbal drive assembly  30  and the roll-axis drive  36  and respond to signals from the accelerometer assembly  34 . The signals from the accelerometer assembly  34  may be indicative of the inclination of the gravity tool  20  with respect to the vertical. The slip ring assembly  38  is configured to couple electrical signals between the rotatable portion  22  and the non-rotatable portion  24 . 
     The non-rotatable portion  24  comprises an elevator mechanism  40 , an optical encoder assembly  42  and a spring-loaded harness assembly  44 . The elevator mechanism  40  is configured to translate the entire gravity tool  20  from one station to the next inside the pressure vessel  18 . The optical encoder assembly  42  is configured to measure the displacement of the gravity meter  28  from one station to the next. The spring-loaded harness assembly  44  is configured to control the electrical wiring as the gravity tool  20  moves along the length of the pressure vessel  18 . 
     The gravity tool  20  is attached to a head assembly  46  which comprises an interface (not shown) through which electrical power may be supplied to the gravity tool  20  from the surface  16 . A nose assembly  48  serves as a shock absorber when the gravity tool  20  impacts the nose assembly  48 . 
     Although  FIG. 1  shows the sonde  10  and gravity tool  20  in a vertical borehole, it should be clear that one or more aspects of the present disclosure are not limited to a vertical borehole, and are applicable or readily adaptable to use in a deviated or horizontal borehole. 
       FIG. 2  is a sectional view of a portion of the gravity tool  20  shown in  FIG. 1 . As shown in  FIG. 2 , the gravity meter  28  comprises an outer Dewar  50 . A heater sleeve  52  disposed in the outer Dewar  50  is configured to slip over an inner Dewar  54  and an electronics board  56 . An outer stopper  58  anchors the heater sleeve  52 , the electronics board  56  and the inner Dewar  54  in the outer Dewar  50 . 
     A sensor housing  60  is anchored in the inner Dewar  50  by an inner stopper  62 . The sensor housing  60  may comprise a heater (not shown). A gimbal  64  supported on a gimbal shaft  66  is mounted for rotation inside the sensor housing  60 . A pivot axis  68  of the gimbal  64  is displaced at an angle relative to the longitudinal axis  26  of the pressure vessel  18 . The pivot axis  68  of the gimbal  64  may be orthogonal to the longitudinal axis  26  of the pressure vessel  18 . A gravity sensor  65  configured to measure gravity is supported inside the gimbal  64 . 
     The outer Dewar  50  and the inner Dewar  54  collectively define a temperature-stabilized chamber  70  for the gravity sensor  65 . The temperature of the chamber  70  may be maintained at 25° C. above the highest ambient temperature rating in the borehole. The temperature may be controlled to 0.001° C. and may be modeled to 10 −6 ° C. One or more heaters disposed in the sensor housing  60  and/or the heater sleeve  52  may be configured to compensate for small residual temperature changes in the chamber  70 . The stopper  58  in the outer Dewar  50  and/or the stopper  62  in the inner Dewar  54  may also be heated and/or otherwise serve to prevent heat flow through the ends of the Dewars. The electronics board  56  adjacent the inner Dewar  54  may be configured to control any heaters incorporated in the gravity meter  28 . 
     The heater sleeve  52  may comprise a magnetic shield configured to protect the gravity sensor  65  from magnetic fields in the borehole  12 . Magnetic fields in the borehole  12  can create torque on the gravity sensor which may result in errors in gravity measurements. However, embodiments within the scope of the present disclosure are not limited to those in which the heater sleeve  52  comprises the magnetic shield. For example, a magnetic shield may alternatively or additionally be located with the sensor housing  60 , the outer Dewar  50  and/or the inner Dewar  54 . 
     The gimbal shaft  66  supports a pulley  72 . A gimbal cable  74  is wound on the pulley  72  with the free ends of the gimbal cable  74  extending through the sensor housing  60  to the exterior of the gravity meter  28 . During operation of the gravity tool  20 , the free ends of the gimbal cable  74  are linked to the gimbal drive assembly  30 . The gimbal drive assembly  30  is configured to extend or retract the free ends of the gimbal cable  74  to cause the gimbal  64  to be rotated about its pivot axis  68  through a predetermined angle and in a predetermined direction. The diameter of gimbal cable  74  may be very small or otherwise configured to minimize heat loss and transfer between Dewars  50  and  54  and the environment. 
       FIGS. 3A and 3B  are schematic views of the gimbal drive assembly  30  shown in  FIG. 2 . Referring to  FIGS. 2 ,  3 A and  3 B, respectively, the gimbal drive assembly  30  comprises a gimbal drive frame  80 . The gimbal drive frame  80  comprises an upper portion  82  and a lower portion  84 . A reducing gear box  86 , stepper motor  87 , right-angle gear box  88  and bobbin  90  are coupled to the upper portion  82  of the gimbal drive frame  80 . The drive shaft of the stepper motor  87  is coupled to the gear box  88  which drives the bobbin  90 . The free end of a backlash cable  92  wound on the bobbin  90  is attached to a spring  94  by a turnbuckle  96 . The spring  94  is coupled to a bracket  98  that is coupled to the gimbal drive frame  80 . The spring  94  is configured to eliminate backlash when the stepper motor  87  is stopped or reversed. A connector  100  coupled to the upper portion  82  of the gimbal drive frame  80  is configured to allow power to be supplied to the stepper motor  87 . 
     A shaft  102  is rotatably coupled to the lower portion  84  of the gimbal drive frame  80  via a pair of ball bearings  103 . A sensor ring  104  which supports a sensor mounting ring  106  is coupled to the shaft  102 . A sensor adjuster ring stop  108  is supported on the shaft  102  and coupled to the top of the sensor mounting ring  106  so that the sensor adjuster ring stop  108  and the sensor mounting ring  106  can rotate together. 
     An angular tilt sensor  110  is coupled to the sensor mounting ring  106 . The angular tilt sensor  110  may be or comprise a uniaxial accelerometer configured to provide an error signal indicating departure of the sensitive axis of the gravity sensor  65  from the vertical. When the sensitive axis of the gravity sensor  65  is aligned with the vertical, the angular tilt sensor  110  is level and the output voltage of the tilt sensor is equal to V offset . As the pressure vessel  18  traverses a deviated borehole, the output voltage of the tilt sensor becomes V offset +V tilt , where V tilt  is proportional to the tilt angle of the sensitive axis of the gravity sensor  65  with respect to a fixed reference. The angular tilt sensor  110  uses the Earth&#39;s gravitational field as a reference. 
     In operation, a drive cable  112  is wound on the bobbin  90 . The free ends of the drive cable  112  are attached to turnbuckles  114 . The free ends of the gimbal cable  74  from the gravity meter  28  pass through a first set of slots  116  in the lower portion  84  of the gimbal drive frame  80  and a second set of slots  118  in the upper portion  82  of the gimbal drive frame  80  to attach to turnbuckles  120 . The turnbuckles  120  are linked to turnbuckles  114  by connectors  122 . A portion of one of the free ends of the gimbal cable  74  is wound once around the sensor adjuster ring stop  108  to allow the angular tilt sensor  110  and the gimbal  64  in the gravity meter  28  to rotate concurrently. The drive cable  112  and the gimbal cable  74  are appropriately tensioned to eliminate backlash in the cable system when the stepper motor  87  is stopped or reversed. A bushing idler  124  ensures that the cables  74  and  112  follow a straight course as they extend and retract. In an alternative embodiment, cable  92 , spring  94 , turnbuckle  96 , bracket  98  and idler  124  may be omitted. 
     Signals from the angular tilt sensor  110  are sent to the electronic controller  32 . The electronic controller  32  uses these signals to determine if the gimbal drive assembly  30  should be operated to drive the gimbal  64  to maintain the vertical orientation of the gravity sensor  65 . The electronic controller  32  may send an electrical pulse to the stepper motor  87  to cause the drive shaft of the stepper motor  87  to rotate through a predetermined fixed angle. As the drive shaft of the stepper motor  87  rotates, the drive cable  112  winds on or unwinds from the bobbin  90 . The movement of the drive cable  112  is transmitted to the gimbal cable  74 , causing the gimbal  64  and the angular tilt sensor  110  to rotate about their respective pivot axes. The angular tilt sensor  110  and the gimbal  64  can rotate a full 360° about their pivot axes, if necessary. As the gimbal  64  and the angular tilt sensor  110  rotate, feedback signals are sent to the electronic controller  32  by the angular tilt sensor  110 . When the angular tilt sensor  110  sends a signal that indicates that the angular tilt sensor  110  is level, the electronic controller  32  stops the stepper motor  87 . The electronic controller  32  and the gimbal drive assembly  30  are collectively configured to maintain the gravity sensor vertical to within 48.5 μrad (or 0.00278°). 
     As described above, the gimbal drive assembly  30  is configured to rotate the gimbal  64  about the pivot axis  68 . However, other mechanisms, such as push rods, rack and pinion, and gear sets, may also be used to rotate the gimbal  64  within the scope of the present disclosure. The gravity sensor  65  may also be provided with a built-in tilt meter which may be controlled to align the sensitive axis of the gravity sensor with vertical; however, the typical range of a built-in tilt meter is of the order of 4.85 mrad (or 0.278°). The range of the angular tilt sensor  110  which tracks the position of the gravity sensor  65  with respect to vertical may be 360°, which may enable the gimbal  64  to effectively align the sensitive axis of the gravity sensor  65  in any deviated or horizontal borehole. Also, the present disclosure is equally applicable or readily adaptable to applications where a sensor may need to be at any predetermined angle to vertical since the angular tilt sensor may be arranged to give continuous feedback signals indicative of the departure of the sensor from vertical. 
       FIG. 4  is a schematic view of the accelerometer assembly  34  shown in  FIG. 1 . The accelerometer assembly  34  comprises a sensor frame  126  to which a first sensor ring  128 , a second sensor ring  130  and a third sensor ring  132  are coupled. Single-axis accelerometers  134 ,  136  and  138  are coupled to the sensor rings  128 ,  130  and  132 , respectively. The sensitive axes of the three accelerometers  134 ,  136  and  138  may be orthogonal to each other, and the sensitive axis of the accelerometer  134  may be coincident with the longitudinal axis  26  of the pressure vessel  18 . The accelerometers  134 ,  136  and  138  are configured to measure instantaneous acceleration along their corresponding sensitive axes. This information is sent to the electronic controller  32  to, for example, determine the pitch and roll inclinations of the gravity tool  20 . A power distribution board (not shown) may be mounted inside the sensor frame  126  to distribute power to the accelerometer assembly  34  and the electronic controller  32 . 
       FIGS. 5A-5C  are schematic views of the slip ring assembly  38  shown in  FIG. 1 . Referring to  FIGS. 5A-5C , collectively, the slip ring assembly  38  may be or comprise a multi-conductor slip ring/brush block assembly, which may comprise a slip ring housing  160  comprising an upper end  162  and a lower end  164 . A tube  166  inside the slip ring housing  160  is arranged to receive a shaft. The upper end of the tube  166  is provided with a plurality of apertures  167 . The lower end  164  of the slip ring housing  160  is provided with a plurality of apertures  168 . Electrical wires extending through the slip ring housing exit through the apertures  167  and  168  at the upper and lower ends  162  and  164  of the slip ring housing  160 , respectively. Ball bearings (not shown), such as may be disposed between the slip ring housing  160  and the tube  166 , may support the slip ring housing  160  for rotation about the tube  166 . Rotors, stators, brushes and slip rings (all of which are not shown) that conduct electrical signals in the slip ring assembly  38  may be located between the walls of the tube  166  and the slip ring housing  160 . A sleeve  170  is bolted to the upper end  171  of the tube  166 . Wave springs  172  and a Teflon washer  174  positioned between the sleeve  170  and the tube  166  may prevent backlash when the roll-axis drive  36  driving the rotatable portion  22  of the gravity tool  20  is stopped. 
       FIG. 6  is a schematic view of a portion of the gravity tool shown in  FIG. 1 , including the slip ring assembly  38  and the elevator mechanism  40 . A coupling assembly  180  which couples the slip ring assembly  38  to the elevator mechanism  40  is at the upper end  162  of the slip ring housing  160 . The coupling assembly  190  comprises an upper portion  182  and a lower portion  184 . The lower portion  184  comprises a shaft  186  which mates with the tube  166  in the slip ring housing  160 . The shaft  186  comprises an internal bore  190  configured to receive electrical wires from the slip ring assembly  38 . The bore  190  communicates with a channel  192  in the upper portion  182  of the coupling assembly  180 . Electrical wires extending out of apertures  167  in the upper end of the tube  166  (see  FIG. 5B ) enter the channel  192  through a slot  194  which communicates with the bore  190  and slots  196  which are circumferentially arranged about the portion  198  of the coupling assembly  180 . The wires in the channel  192  extend out of slots  200  in the upper portion of the coupling assembly  180  and are received in channels  202  in the elevator mechanism  40 . The coupling assembly  180  is coupled to the sleeve  170  that is bolted to the tube  166  by a pair of circular plates  204 . 
     A motor mount  206  which houses the roll-axis drive  36  is coupled to the lower end  164  of the slip ring housing  160 . The roll-axis drive  36  comprises a stepper motor  208  which drives a transmission system  210 . The transmission shaft  212  of the transmission system  210  is coupled to the shaft  186  of the coupling assembly  180  by a shaft coupling  214 . 
     In operation, electrical pulses are sent to the stepper motor  208  of the roll-axis drive  36 , causing the drive shaft of the stepper motor  208  to rotate through a predetermined angle. The drive shaft of the stepper motor  208  in turn drives the transmission system  210 . The transmission shaft  212  attempts to rotate the shaft  186  of the coupling assembly  180 . However, the coupling assembly  180  is coupled to the non-rotatable portion  24  of the gravity tool  20  so that the shaft  186  of the coupling assembly  180  does not rotate. Instead, the resultant torque generated between the driven transmission shaft  212  and the shaft  186  of the coupling assembly  180  causes the motor mount  206  which supports the transmission shaft  212  to rotate. As the motor mount  206  rotates, the slip ring housing  160  coupled to the motor mount  206  also rotates, as does the accelerometer assembly  34 , the electronic controller  32 , the gimbal drive assembly  30  and the gravity meter  28 . The tube  166  does not rotate with the slip ring housing  160 . 
       FIG. 7  is a schematic view of the elevator mechanism  40  shown in  FIG. 1 . The elevator mechanism  40  comprises a motor  232  coupled within an elevator housing  230 . The motor  232  may be or comprise a brushless DC motor and/or a stepper motor. The drive shaft of the motor  232  is coupled to a reduction gear box  214  that drives a pair of worm gears  236 . Each worm gear  236  drives a spur gear  238 . A wheel  240  on each spur gear  238  is configured to contact the inside surface of the pressure vessel  18 . When the spur gears  238  are driven, the wheels  240  ride up and down along the length of the pressure vessel  18 . 
     The wheels  240  are preloaded against the wall of the pressure vessel  18  using Belleville springs  242 . The Belleville springs  242  are supported on a rod  243 . Levers  244  on the ends of the rod  243  are connected to the shafts  245  of the spur gears  238  and to the shafts  246  of the worm gears  236 . This arrangement allows the springs  242  to exert force on the levers  244  to push the wheels  240  against the inside diameter of the pressure vessel  18 . The force applied to the wall of the pressure vessel  18  by the springs  242  is sufficient to provide traction to lift the weight of the gravity tool  20  when the gravity tool  20  is in the vertical position. 
     The motor  232  may be provided with a brake  247  configured to prevent the motor  232  from turning when it is on station. The worm gears  236  may also function as a brake if a gear pitch is selected that does not back-drive when the gravity tool  20  is vertical. The channels  202  on the sides of the elevator housing  230  receive electrical wires from the coupling assembly  180  (shown in  FIG. 6 ). 
     While the illustrated embodiment shows the elevator mechanism  40  as being linked to the gravity meter  28  so as to move the gravity meter  28  inside the pressure vessel  18 , it should be clear that the scope of the present disclosure is not limited to embodiments using the elevator mechanism  40  to move the gravity meter  28  inside the vessel  18 . For example, in another embodiment within the scope of the present disclosure, the elevator mechanism  40  may be sealed within an oil-filled enclosure and mounted external to the pressure vessel  18  and the gravity meter  28  can be held at a fixed position inside the vessel  18 . The externally mounted elevator mechanism would then support and translate the pressure vessel along the length of the borehole to make gravity measurements. This may provide a greater depth of investigation, since the gravity sensor could be moved to stations beyond that achievable inside the pressure vessel. 
     A conveyance mechanism, such as a cable supported on pulleys or a rotatable winch at the surface, may also or alternatively be used to move the pressure vessel along the length of the borehole instead of or in addition to the elevator mechanism. The pressure vessel may be quickly lowered into the borehole by the aid of a casing collar locator  47  (shown in  FIG. 1 ) which may be mounted on the pressure vessel. The casing collar locator, which may be or comprise an electromagnetic pickup or acoustic transducer or mechanical feeler gauge, may be configured to find casing collars that are located at known depths inside the borehole. Once the casing collars are located, the measuring stations can be accurately located to within 1 mm or 2 mm. Also, the elevator mechanism may be used inside the pressure vessel to move the gravity sensor along the length of the pressure vessel while a conveyance mechanism is used to move the pressure vessel along the length of the borehole. 
       FIG. 8  is a schematic view of the optical encoder assembly  42  shown in  FIG. 1 . The optical encoder assembly comprises a mounting frame  250 . A lever  252  is spring mounted on the mounting frame  250 . The lever  252  supports an optical encoder  254 . An encoder wheel  256  is connected to the optical encoder  254  by a shaft  258 . The encoder wheel  256  may be made of a material that does not change dimensions with temperature, which may eliminate the need for temperature correction on the measured displacement. For example, the encoder wheel  256  may comprise invar. As the wheel  256  rotates, the shaft  258  also rotates. The optical encoder  254  delivers electrical pulses which are proportional to the speed of the shaft  258  at its output terminal. A connector  260  is mounted inside the mounting frame  250  for connecting electrical wires from the elevator mechanism  40  to the optical encoder assembly  42 . An electronics board (not shown) may also be provided to record readings from the optical encoder  254 . 
     Several other means exist for measuring the displacement of the gravity meter inside the pressure vessel within the scope of the present disclosure. For example, if a stepper motor is used in the elevator mechanism  40 , the steps required to move from one station to the next may be counted and translated to displacement. Alternatively, or additionally, a magnetic or optical pickup may measure the rotation of the worm gear or spur gear of the elevator mechanism  40  as the elevator mechanism moves the gravity tool  20 . An electrical encoder can also be used in place of or in addition to an optical encoder. 
       FIG. 9  is a schematic view of the harness assembly  44  shown in  FIG. 1 . The harness assembly  44  comprises an upper portion  270  and a lower portion  272  which are linked by a flexible helical spring (not shown). Supports  276  may be inserted in the helical spring at spaced intervals along the length of the helical spring to keep the helical spring from vibrating during gravity measurement. The spring rate of the helical spring may be configured such that the elevator mechanism  40  does not have to support the weight of the harness assembly  44  when the helical spring is fully extended. 
     A rod  278  attached to the lower portion  272  moves with the elevator mechanism  40  inside the channel created by the coils of the helical spring as the elevator mechanism  40  translates the gravity meter  28  from one station to the next inside the pressure vessel  18 . The rod  278  is arranged to contact a limit switch  280  in the upper portion  270  of the harness assembly  44  when the gravity meter  28  has reached the maximum upper limit or home position. 
     A cable containing insulated electrical wires runs from the upper portion  270  to the lower portion  272 . The cable may be pre-coiled such that it fits inside the channel created by the coils of the helical spring and over the rod  278 . The cable is configured to stretch or recoil as the gravity meter  28  is translated inside the pressure vessel  18 . 
     The overall design of the gravity tool may allow the diameter of the tool to be fairly small, perhaps about 3⅜″, and possibly scalable to 1 11/16″, although other sizes are also within the scope of the present disclosure. An embodiment of the assembled gravity tool  20  is shown in sequential segments in  FIGS. 10A-10D , although other embodiments are also within the scope of the present disclosure. 
     As shown in  FIG. 10A , the gravity meter  28  is at the downhole end of the gravity tool  20 . Teflon pads  303  are provided on the gravity meter  28  to space the surface of the gravity meter  28  from the inner surface of the pressure vessel  18 . Coupled to one end of the gravity meter  28  is the gimbal drive assembly  30  which aligns the gravity sensor in the gravity meter  28  with the vertical. The gimbal drive assembly  30  is coupled to the electronic controller  32  by a coupling assembly  304 . The coupling assembly  304  has a flange portion  306  and a shaft portion  308 . The flange portion  306  is coupled to the gimbal drive assembly  30  and the shaft portion  308  is coupled to the mounting bracket  309  of the electronic controller  32 . 
     As shown in  FIG. 10B , the mounting bracket  309  of the electronic controller  32  is coupled to the sensor frame  126  of the accelerometer assembly  34 . The sensor frame  126  is coupled to the motor mount  206  which houses the roll-axis drive  36 . As shown in  FIG. 10C , the motor mount  206  is coupled to the slip ring housing  160  of the slip ring assembly  38 . The stationary tube  166  (shown in  FIG. 5A ) in the slip ring housing is coupled to the elevator mechanism  40  by the coupling assembly  180 . The optical encoder assembly  42  is coupled to the elevator mechanism  40 . As shown in  FIG. 10D , the optical encoder assembly  42  is coupled to the lower portion  272  of the spring-loaded harness assembly  44 . 
     Roller assemblies  310 ,  312 ,  314  and  316  are configured to center the gravity tool  20  inside the pressure vessel  18 . Roller assemblies  310  and  312  support the gravity assembly  28 , gimbal drive assembly  30  and electronic controller  32  and permit axial and rotational movement of the rotatable portion  22  of the gravity tool  20  in the pressure vessel  18 . Roller assemblies  314  and  316  support the non-rotatable portion  24  of the gravity tool  20  and permit axial movement in the pressure vessel  18 . 
       FIG. 11  is a schematic view of the roller assembly  310  (also roller assembly  312 ) that supports the gimbal drive assembly  30  and the gravity meter  28  for rotation about the longitudinal axis  26  of the pressure vessel  18 . The roller assembly  310  comprises a body  330  which is provided with an internal bore  332 . Ball bearings  334  inside the bore  332  support the shaft portion  308  of the coupling assembly  304  (shown in  FIG. 10A ). 
     Three slots  336  in the wall of the body  330  are spaced 120° apart along the circumference of the body  330 . Mounting blocks  338  on either side of the slots  336  project outwardly from the wall of the body  330 . The mounting blocks  338  may be integrally formed with the body  330 . The mounting blocks  338  comprise bores for receiving the ends of axles  340 . Each axle  340  is supported on preloaded ball bearings  342  that are fixed to a side of the mounting blocks  338 . The axles  340  may be stiff bow springs to help eliminate radial play. 
     A roller  344  is mounted on each axle  340 . The rollers  344  fit into the slots  336  in the wall of the body  330 , and are configured to ride along the wall of the pressure vessel in a direction parallel to the longitudinal axis of the pressure vessel. 
     One of the axles  340   a  is eccentrically mounted on its supporting bearing to allow for tight fitting of the roller assembly  310  with the inside diameter of the pressure vessel  18 . The position of the eccentrically mounted axle  340   a  may be adjusted by loosening the screw  342  which locks a sprocket  344  in place on the side of one of the mounting blocks. When the screw  342  is loosened, the axle  340   a  can be adjusted so that the rollers  344  fit tightly with the inside diameter of the pressure vessel  18 . 
     The roller assemblies  314  and  316  are similar to the roller assemblies  310  and  312 , except that their bores are not lined with bearings and the roller assembly bodies are fixedly attached to the coupling assembly  180  and the optical encoder assembly  42 , respectively. As such, the coupling assembly  180  and optical encoder assembly  42  do not rotate when the roll-axis drive  36  turns the rotatable portion of the gravity tool  20 . 
     In operation, the sonde  10  is lowered into the borehole  12  on the end of a wireline  14 . As the sonde  10  is lowered, the electronic controller  32  is continually receiving signals from the angular tilt sensor and the sensor assembly and using the gimbal drive assembly  30  and the roll axis drive  36  to align the gravity sensor with the vertical. 
     In conducting gravimetric surveys, the sonde  10  is lowered to a certain desired depth in the borehole on a wireline. The sonde  10  is then clamped to the borehole by a suitable clamping mechanism. The clamping mechanism ensures that the gravity sensor  65  is stable when gravity readings are taken. After the sonde  10  is secured to the borehole, the elevator mechanism  40  translates the gravity tool  20  inside the pressure vessel  18  until the gravity sensor  65  is aligned with a station. At the same time, the optical encoder  42  records the distance moved by the gravity tool  20 . 
     At the measuring station, the accelerometers in the accelerometer assembly  34  measure instantaneous acceleration in three orthogonal directions. The electronic controller  32  uses the instantaneous accelerations from the accelerometers to determine the pitch and roll angles of the gravity tool  20  from a fixed reference. Based on the roll angle, the electronic controller  32  energizes the stepper motor of the roll-axis drive  36  to incrementally rotate the gravity tool  20  about an axis coincident with the longitudinal axis  26  of the pressure vessel  18 . Also, based on the pitch angle, the electronic controller  32  energizes the stepper motor of the gimbal drive assembly  30  to incrementally rotate the bobbin  90  which in turn rotates the angular tilt sensor  110  and the gimbal  64 . As the electronic controller  32  controls the roll-axis drive  36  and the gimbal drive assembly  30  to align the sensitive axis of the gravity sensor  65  with the vertical, the angular tilt sensor  110  sends signals indicative of the magnitude of departure of the sensitive axis of the gravity sensor  65  with respect to the vertical. 
     When the angular tilt sensor  110  indicates that the sensitive axis of the gravity sensor  65  is aligned with the vertical, the electronic controller  32  stops the gimbal drive assembly  30  and the roll-axis drive  36 . The electronic controller  32  may send a signal to the surface to indicate that the gravity sensor  65  is aligned with the vertical. The gravity sensor  65  may then be activated from the surface or otherwise to measure gravity. After measuring gravity, the elevator mechanism  40  moves the gravity tool  20  inside the pressure vessel  18  again until the gravity sensor  65  is aligned with the next measuring station. The optical encoder assembly  42  monitors the position of the gravity sensor  65  as the gravity sensor  65  moves inside the pressure vessel  18 . Again, the electronic controller  32  ensures that the sensitive axis of the gravity sensor  65  is aligned with the vertical before gravity readings are taken. The process of translating the gravity tool  20  inside the pressure vessel  18 , aligning the sensitive axis of the gravity sensor  65  with the vertical, and activating the gravity sensor to measure gravity may continue until the gravity tool  20  touches the nose assembly  48 . The distance between successive measuring stations in the pressure vessel may be about 1 m or more. 
     Other embodiments of the gravity tool  20  are also within the scope of the present disclosure. In one such example, the rotatable portion  22  of the gravity tool  20  may be extended to include the spring loaded harness assembly  44  so that the slip ring assembly  38  is not necessary to couple signals between the rotatable portion  22  and the non-rotatable portion  24  of the gravity tool  20 . 
     The interpretation workflow described below presents a methodology to create, under specific conditions, three-dimensional (“3D”) time-lapse density models used to forward model the response of the gravity tool  20  in a grind and inject well. These models may be built based on initial porosity logs, injection history and on perforation zone location. Each model represents the density change during a time interval due to a given mass of slurry injected into the subsurface at a certain depth interval. This workflow is applicable or readily adaptable to applications where another fluid is injected in the subsurface, such as with water injection. For example, such models may be built based on initial porosity logs, injection history, perforation zone location and an estimation of the initial water saturation along the well. 
     As used herein, “3D” may include models varying in each of three dimensions, such as along each of the x, y and z axes of a Cartesian system. For example, such models may utilize cells or other elements which have or represent height, width and depth. Other models, however, are also within the scope of the present disclosure. For example, other 3D models may be based on a cylindrical coordinate system. 3D models may also include those which vary in two dimensions but be considered to have a constant value in a third dimension. For example, in one such model based on a cylindrical coordinate system, cells or other elements of these models may be defined by axial position along the wellbore, angular orientation within the wellbore (e.g., azimuth), and radial distance from the wellbore, wherein the data may be assumed to be constant at all angular orientations (e.g., the data does not vary dependently upon azimuth). Such an example may be referred to in the industry as “2½D”. For the sake of brevity, however, all of these systems may be referred to herein as “multi-dimensional.” 
     The inputs may comprise one or more well deviation surveys, one or more neutron and/or density porosity logs at an initial time T 0  before slurry injection, injection history (e.g., mass injected per year), and the zone of injection in the well. The workflow generally comprises two steps: estimation of the change in porosity along the well over time, due to slurry injection; and building of a multi-dimensional time-lapse density model, respecting the total mass injected each year and using the estimation of the time-lapse porosity variation to compute an approximate time-lapse density variation. 
     In the following description of the estimate of time-lapse variation of porosity near a grind and inject well, it is supposed that the formation crossed by the well is shaly sand composed of a succession of sand and shale zones. A petrophysical interpretation based on a dual water model allows computation along the well of the volume of sand, the volume of dry clay and the total porosity. The sum of these three quantities is equal to 1. Based on these quantities, the time-lapse change of porosity due to an injection of slurry can be estimated. 
       FIG. 12  is a typical petrophysical interpretation positioning three points on a crossplot of the neutron porosity φ N  and the density porosity φ D : a clean matrix point, a dry clay point and a fluid point. The fluid point is a fresh water point with φ N =1 and φ D =1, the dry clay point is taken to be φ Ndcl =0.40 and φ Ddcl =−0.10, and the clean matrix point is φ N =0 and φ D =0. 
     Using this crossplot, the total porosity φ, and the volume of dry clay V dcl  are computed with the following formulas: 
     
       
         
           
             
               
                 
                   
                     ϕ 
                     t 
                   
                   = 
                   
                     
                       
                         
                           ϕ 
                           D 
                         
                         × 
                         
                           ϕ 
                           Ndcl 
                         
                       
                       - 
                       
                         
                           ϕ 
                           N 
                         
                         × 
                         
                           ϕ 
                           Ddcl 
                         
                       
                     
                     
                       
                         ϕ 
                         Ndcl 
                       
                       - 
                       
                         ϕ 
                         Ddcl 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
             
               
                 
                   
                     V 
                     dcl 
                   
                   = 
                   
                     
                       
                         ϕ 
                         N 
                       
                       - 
                       
                         ϕ 
                         D 
                       
                     
                     
                       
                         ϕ 
                         Ndcl 
                       
                       - 
                       
                         ϕ 
                         Ddcl 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     The volume of sand is deduced with:
 
 V   sand =1− V   dcl −φ t   (3)
 
     To create a scenario of time-lapse porosity variation, it is supposed that this variation is proportional to the porosity with a proportional parameter varying along the well, depending on the dry clay volume and bounded by a maximum value B with 0≦B≦1. 
     The porosity at time T 0  is deduced directly from the total porosity log. Then, from year to year the porosity at times T 1 , . . . , T n  is deduced from the porosity at times T 0 , . . . , T n-1  with the following formulas:
 
If  V   dcl ≦0.1 then, φ T     n   =(1 −B ) φ T     n-1    
 
If 0.1 ≦V   dcl ≦0.5 then, φ T     n   =(1−2.5· B (0.5 −V   dcl ))φ T     n-1    
 
If  V   dcl ≧0.5 then, φ T     n   =φ T     n-1     (4)
 
The parameter B can be adjusted and has been taken equal to 0.1 in the present discussion.
 
     These equations state that the porosity is not likely to change significantly in the zones where the volume of dry clay is important and greater than 0.5. The porosity may change as much as 10% year to year in the clean sand zones containing very little volume of dry clay (e.g., less than 0.1). The porosity changes linearly from 0% to 10% in zones where the volume of dry clay is between 0.5 and 0.1. 
     If the petrophysical interpretation presented before is not valid in the formation surrounding the grind and inject well, or if one of the porosity logs is not available, the variation of porosity with time may be estimated by supposing that this variation is proportional to the porosity with a proportional constant parameter along the well. In other words, Equation (4) can be simplified to φ T     n   =(1−B)φ T     n-1    for all points along the well. 
     After the change in porosity along the well over time is estimated, a one-year time-lapse multi-dimensional model may be built. 
     The following equation relates density to porosity:
 
ρ=(1−φ)ρ m +φ( S   w ρ w +(1− S   w )ρ f )  (5)
 
with ρ the bulk density, φ the porosity, ρ m , the rock matrix density, S w  the water saturation, ρ w  the water density, and ρ f  the density of the fluid which is not water (e.g., oil or gas).
 
     To build a time-lapse density model, the following approximations are made:
         The density of the solids injected with the slurry is close to the density of the rock matrix ρ m .   The density of the water injected with the slurry is close to the density of the water in the formation, ρ w .   The density of the fluid ρ f  in the formation remains the same throughout the injection process.       

     Under these assumptions, the time-lapse density change can be written:
 
ρ T     1   −ρ T     0   =(φ T     1   −φ T     o   )(ρ f −ρ m )+(φ T     1     S   wT     1   −φ T     0     S   wT     0   )(ρ w −ρ f )  (6)
 
     For the grind and inject application, this equation simplifies. That is, the wells have been produced and we assume that the only fluid present near the wells is water. Therefore, water saturation equals 1 at all times. Consequently, the time-lapse density change is proportional to the time-lapse porosity change:
 
ρ T     1   −ρ T     0   =(φ T     1   −φ T     0   )(ρ w −ρ m )  (7)
 
The estimation of the time-lapse porosity change, (φ T     n   −φ T     n-1   ), due to slurry injection is done using some petrophysical interpretation, as described above and in Equation (4).
 
     For a water injection application, Equation (6) would also simplify, because there is no porosity change over time. The time-lapse density change would be proportional to the time-lapse change of water saturation with:
 
ρ T     1   −ρ T     0   =φ( S   wT     1     −S   wT     0   )(ρ w −ρ f )  (8)
 
     Thus, the time-lapse increase of water saturation due to water injection could be estimated roughly with for example a proportional change:
 
If  CS   wT     0   ≦1 then,  S   wT     1     =CS   wT     0    
 
If  CS   wT     0   ≧1 then,  S   wT     1   =1  (9)
 
C is a constant greater than 1. Its value would have to be adjusted depending on the local conditions and should reflect the expected increase in water saturation with time. Also, a more advanced petrophysical interpretation could be done to adjust its value depending on the characteristics of the formation along the well.
 
     The multi-dimensional models may now be created. In the following paragraphs, the grind and inject application is taken as the primary application. However, the steps followed to create a multi-dimensional time-lapse model could be applied as-is to different applications, with either a slurry injection or a fluid injection, such as: injection of proppants during hydraulic fracturing of a subterranean formation and injection of liquid surfactants to improve hydrocarbon mobility, among others. 
     For building the input models for borehole gravity modeling, it is assumed that the injected slurry primarily expands horizontally in the perforation zone, and then in a second step goes up and down along the well. It is supposed that the slurry is expanding circularly from the well. 
     These assumptions can be translated into equations relating the radial extent r of the injected mass to the depth z and to the following parameters: z p , the depth of the perforation middle point; h p , the height of the perforation zone; and φ(z), the porosity. The radial extent of the mass injected is maximal in the perforation zone and varies according to the following function: 
                     r   ⁡     (   z   )       =         R   0     ⁢     ϕ   ⁡     (   z   )         A             (   10   )               
This radial extent is decreasing in
 
             1          z   -     z   p                  
outside the perforation zone. Combining the depth and porosity dependence results in the following:
 
                     r   ⁡     (   z   )       =         R   0     ⁢     ϕ   ⁡     (   z   )           (     A   +          z   -     z   p              )               (   11   )               
It is linearly depending on the porosity everywhere.
 
     The parameter R 0  is adjusted for mass balance such that: 
                       ∑     i   ,   j   ,   k       ⁢       Δρ     i   ,   j   ,   k       ⁢     V     i   ,   j   ,   k           =     M   injected             (   12   )               
where M injected  is the total mass injected in one year, V i,j,K  is the volume of cell (i,j,k) and Δρ i,j,k  is the change of density in cell (i,j,k) for that year. The indices (i,j,k) represent a single grid cell of the multi-dimensional gridded model.
 
     The parameter A is an adjustable parameter which controls the shape of the volume of mass injected.  FIG. 13  illustrates the radial extent of the mass injected around the well  12  near the perforation zone  502 . The true vertical axis is depicted as “z”. For a value of A=5, the shape of the volume in the subsurface where some mass has been injected is shown by lines  504 . Lines  506  depict a value of A=7, and lines  508  depict a value of A=10. In multiple dimensions (e.g., 3D), this creates a gridded “circular shape” along the borehole based on this radial extent. Note that for all values of the parameter A, the mass contained in the multi-dimensional volume delineated by the various curves r(z) is identical. 
     The multi-dimensional time-lapse density model can be used to calculate the time-lapse response of a borehole gravity tool in the grind and inject well using the following equations: 
                     g   Z     =     G   ×       ∑     i   ,   j   ,   k       ⁢         (       Z     i   ,   j   ,   k       -     Z   tool       )     ×     Δρ     i   ,   j   ,   k       ×     V     i   ,   j   ,   k             (         (       X     i   ,   j   ,   k       -     X   tool       )     2     +       (       Y     i   ,   j   ,   k       -     Y   tool       )     2     +       (       Z     i   ,   j   ,   k       -     Z   tool       )     2       )       3   2                     (   13   )                       ⁢       Δ   ⁢           ⁢       g   Z     ⁡     (   z   )         =         g   Z     ⁡     (     z   +     Δ   ⁢           ⁢     z   /   2         )       -       g   Z     ⁡     (     z   -     Δ   ⁢           ⁢     z   /   2         )                   (   14   )               
where G is the Universal Gravitation Constant, g z  is the component in the z direction of the acceleration due to gravity, Δg z (z) is the differential gravity at the depth of z, and Δz is the distance between two station depths where the measurements are conducted.
 
     The time-lapse density model and its associated borehole gravity tool response can be used in job planning for a borehole gravity survey. For example, it can be used to determine whether a survey is warranted based on the amplitude of the expected gravity response and the sensitivity of the gravity measurement instrument. Alternatively, the methods described above can be applied to a surface gravity survey. 
     Once a multi-dimensional model has been constructed, an initial gravity response is computed based on estimates for the values of parameters A, B and C. If actual measured gravity data is obtained, then it is possible to iteratively adjust the model parameters A, B and C to better match the computed gravity response to the measured data. The resulting model with optimized parameters A, B and C will be a representation of the actual location of injected materials to the extent that other assumptions in the model are correct. 
       FIG. 14  is a flow-chart diagram of at least a portion of a method  600  according to one or more aspects of the present disclosure. The method  600  may be or comprise an implementation of one or more of the aspects described above, and may be performed by or in conjunction with the apparatus shown in  FIG. 1  and/or otherwise within the scope of the present disclosure. 
     The method  600  comprises a step  610  in which formation porosity and/or water saturation change is estimated. Such change may be estimated on an annual basis, such as by estimating the formation porosity and/or water saturation at year-long intervals. Step  610  may comprise obtaining well log data, core data, perforation data, injection data, well survey data, formation matrix density data, injected fluid density data, and/or combinations of these. 
     One or more time-lapse density models are then built in a step  620 . The change in formation porosity and/or water saturation that was estimated during step  610  is utilized as at least one of the bases for the model(s), as well as balancing the mass of the fluid injected within the time period of investigation (e.g., during each of the intervals utilized in step  610 ). 
     The method  600  then proceeds to a step  630  during which the time-lapse density model(s) is (are) utilized to evaluate the sensitivity of a borehole gravity tool (e.g., the sonde  10  and/or gravity tool  20  shown in  FIG. 1  and elsewhere in the present disclosure). In a subsequent step  640 , the borehole gravity tool is utilized to perform gravity measurements at a plurality of stations along the well. 
     Thereafter, in a step  650 , the time-lapse density model(s) is (are) calibrated using the borehole gravity tool measurements. For example, differences between projected and actual gravity measurements may be utilized to adjust one or more parameters of the model(s) (e.g., parameters A, B and/or C in the description above). The calibration of the time-lapse density model(s) performed in step  650  may include iterative adjustments to the one or more parameters of the model(s), such as where the incremental adjustments may iteratively repeat until the differences between the projected and actual gravity measurements fall below a predetermined threshold. Such threshold may be, for example, about 1%, although others are also within the scope of the present disclosure. 
     The method  600  may also include a step  660  in which the calibrated or otherwise adjusted model(s) is (are) utilized to determine the location of the injected materials. For example, the method  600  may be executed in the above-described context of grind and inject wells, such that the location and/or movement of the injected slurry of oilfield operations waste materials may be determined. 
     Other embodiments of the method  600  are also within the scope of the present disclosure. An exemplary embodiment of the method  600  may comprise steps  610 ,  620  and  630 , and possibly other steps, but may omit steps  640 ,  650  and  660 . 
       FIG. 15  is a flow-chart diagram of at least a portion of a method  700  according to one or more aspects of the present disclosure. The method  700  may be or comprise an implementation of one or more of the aspects described above, and may be performed by or in conjunction with the apparatus shown in  FIG. 1  and/or otherwise within the scope of the present disclosure. The method  700  may comprise or be performed in conjunction with one or more steps of the method  600  shown in  FIG. 14 . 
     The method  700  comprises a step  705  during which well log and core data is obtained. Such data may comprise natural gamma-ray data, neutron porosity data, gamma-gamma density data, resistivity data and/or core data. From this data, the initial total porosity of the formation may then be determined in subsequent step  710 . Step  710  may further comprise correcting the total porosity of the formation based on an estimated and/or measured clay content of the formation. 
     Perforation and injection data is then obtained in subsequent step  715 , although this may also occur prior to step  710 , at least in part. The perforation data may comprise location of the various perforations that have been previously formed in the well, and may further comprise the time elapsed since each perforation was formed. The injection data may comprise the amount of fluid injected, the location of each injection (e.g., in relation to the perforations), the density of the fluid utilized for each injection, and/or the time elapsed since each injection. The total porosity, the perforation data and the injection data is then utilized in a subsequent step  720  to estimate a time-lapse variation of porosity of the formation resulting from the series of injections. 
     Well survey data, formation matrix density data and/or injected fluid density data is then obtained in a subsequent step  725 , although this may also occur prior to steps  720  and/or  710 , at least in part. This data may comprise a well survey obtained during or after drilling, matrix density obtained through core analysis and injected fluid density measured from fluid samples taken at the surface. From this data, one or more time-lapse density models are built during subsequent step  730 , as described above. 
     In a subsequent step  735 , a borehole gravity tool response at various stations along the well trajectory is projected using the one or more time-lapse density models built during step  730 . Thereafter, in step  740 , actual gravity measurements are obtained with the borehole gravity tool at the same or similar stations. The projected and actual gravity measurements are then compared in a step  745  to assess the accuracy of the time-lapse density model(s) developed during step  730 . 
     The method  700  may also comprise a step  750  during which the one or more time-lapse density model(s) developed during step  730  are adjusted to account for differences between the projected and actual gravity measurements of steps  735  and  740 . For example, one or more of the above-described parameters A, B and C may be adjusted to bring the projected gravity measurements of step  735  into accord with the actual gravity measurements of step  740 . 
     Such adjustment may comprise an iterative procedure. For example, one or more of the parameters A, B and C may be incrementally adjusted to obtain new projected gravity measurements based on the adjusted time-lapse model(s). These new projected gravity measurements may then be compared to the actual gravity measurements to reassess the accuracy of the adjusted time-lapse model(s). If differences still exist between the projected and actual gravity measurements, or if the differences do not fall within a predetermined threshold, the model parameters may again be adjusted and the process repeated as necessary to sufficiently bring the model(s) into accord with the actual gravity measurements. In this manner, the time-lapse model(s) may be calibrated. Thereafter, the one or more time-lapse models may be utilized to assess the location and/or movement of the injected fluid within the formation. 
     Other embodiments of the method  700  are also within the scope of the present disclosure. An exemplary embodiment of the method  700  may comprise steps  705 ,  710 ,  715 ,  720 ,  725 ,  730  and  735 , and possibly other steps, but may omit steps  740 ,  745  and  750 . 
       FIG. 16  is a schematic view of at least a portion of an example computing system P 100  that may be programmed to carry out all or a portion of the methods of the present disclosure. For example, the computing system P 100  shown in  FIG. 16  may be used to implement surface components (e.g., components located at the Earth&#39;s surface) and/or downhole components (e.g., components located in a downhole measuring tool) of a distributed computing system. The computing system P 100  may be used to implement all or a portion of the electronics and processing components or system shown in  FIG. 1  and/or otherwise within the scope of the present disclosure. 
     The computing system P 100  may include at least one general-purpose programmable processor P 105 . The processor P 105  may be any type of processing unit, such as a processor core, a processor, a microcontroller, etc. The processor P 105  may execute coded instructions P 110  and/or P 112  present in main memory of the processor P 105  (e.g., within a RAM P 115  and/or a ROM P 120 ). When executed, the coded instructions P 110  and/or P 112  may cause one or more components of the apparatus  10  of  FIG. 1  and/or otherwise within the scope of the present disclosure to perform at least a portion of the method  600  of  FIG. 14  and/or at least a portion of the method  700  of  FIG. 15 , among other methods within the scope of the present disclosure. 
     The computing system P 100  may also include an interface circuit P 130 . The interface circuit P 130  may be implemented by any type of interface standard, such as an external memory interface, serial port, general-purpose input/output, etc. One or more input devices P 135  and one or more output devices P 140  are connected to the interface circuit P 130 . The example output device P 140  may be used to, for example, display, print and/or store on a removable storage media one or more of gravity, porosity and/or density data as described above. 
     The processor P 105  may be in communication with the main memory (including a ROM P 120  and/or the RAM P 115 ) via a bus P 125 . The RAM P 115  may be implemented by dynamic random-access memory (DRAM), synchronous dynamic random-access memory (SDRAM), and/or any other type of RAM device, and ROM may be implemented by flash memory and/or any other desired type of memory device. Access to the memory P 115  and the memory P 120  may be controlled by a memory controller (not shown). The memory P 115 , P 120  may be used to store one or more of the operational parameters of the gravity sonde  20  of  FIG. 1  and readings of the acceleration due to gravity within a subterranean formation, among other things. 
     Further, the interface circuit P 130  may be connected to a telemetry system P 150 , including, for example, the multi-conductor cable  14  of  FIG. 1 . The telemetry system P 150  may be used to transmit measurement data, processed data and/or instructions, among other things, between the surface and downhole components of the distributed computing system. 
     In view of all of the above and the figures, it should be clear that the present disclosure introduces a method comprising: estimating a change in a characteristic of a subterranean formation into which a fluid has been injected via a well extending into the subterranean formation; building a multi-dimensional model balancing mass of the injected fluid, wherein the model is based on the estimated characteristic change; utilizing the model to determine the sensitivity of a borehole gravity tool in the well; measuring gravity with the borehole gravity tool at a plurality of stations along the well; and utilizing the model and the gravity measurements to locate the injected fluid in the subterranean formation. The subterranean formation characteristic may be porosity and/or density. Building the model may be based on an estimated change in density of the subterranean formation which may be based on the estimated change in water saturation. The change in the characteristic of the subterranean formation may be estimated in intervals no shorter than about three months. The method may further comprise determining physical progression of the injected fluid on a year-by-year basis based on the model and the gravity measurements. The injected fluid may comprise a slurry, ground solid waste, or a slurry resulting from grinding solid waste generated by oilfield operations. The multi-dimensional model may be a 3D model or a 2½D model. The present disclosure also introduces apparatus comprising means for performing such a method. 
     The present disclosure also introduces a method comprising: determining total porosity of a subterranean formation into which a fluid has been injected via a well extending into the subterranean formation; estimating time-lapse variation of the porosity based on the total porosity, perforation data and injection data; building a time-lapse density model based on the estimated time-lapse variation of the porosity, formation matrix density data and injected fluid density data; and projecting a borehole gravity tool response at a plurality of stations along the well based on the time-lapse density model. Determining the total porosity of the subterranean formation may be based on well log data. The total porosity may be corrected to account for clay within the subterranean formation. The method may further comprise measuring gravity with the borehole gravity tool at each of the stations along the well. The method may further comprise adjusting a parameter of the time-lapse density model based on differences between the projected borehole gravity tool response and the measured gravity. Adjusting the parameter of the time-lapse density model may comprise iteratively adjusting the parameter of the time-lapse density model until the differences between the projected borehole gravity tool response and the measured gravity fall below a predetermined threshold. Adjusting the parameter of the time-lapse density model may comprise adjusting a plurality of parameters of the time-lapse density model based on the differences between the projected borehole gravity tool response and the measured gravity. Adjusting the plurality of parameters of the time-lapse density model may comprise iteratively adjusting the plurality of parameters of the time-lapse density model until the differences between the projected borehole gravity tool response and the measured gravity fall below a predetermined threshold. The multi-dimensional model is a 3D model or a 2½D model. The present disclosure also introduces apparatus comprising means for performing such a method. 
     The present disclosure also introduces an apparatus comprising means for performing at least one of a first method and a second method, wherein the first method comprises: estimating a change in a characteristic of a subterranean formation into which a fluid has been injected via a well extending into the subterranean formation; building a multi-dimensional model balancing mass of the injected fluid, wherein the model is based on the estimated characteristic change; utilizing the model to determine the sensitivity of a borehole gravity tool in the well; measuring gravity with the borehole gravity tool at a plurality of stations along the well; and utilizing the model and the gravity measurements to locate the injected fluid in the subterranean formation; and wherein the second method comprises: determining total porosity of the subterranean formation into which the fluid has been injected via the well; estimating time-lapse variation of the porosity based on the total porosity, perforation data and injection data; building a time-lapse density model based on the estimated time-lapse variation of the porosity, formation matrix density data and injected fluid density data; and projecting a borehole gravity tool response at a plurality of stations along the well based on the time-lapse density model. Such apparatus or means may be further configured to performed one or more aspects of other methods within the scope of the present disclosure. 
     The foregoing outlines features of several embodiments so that those skilled in the art may better understand the aspects of the present disclosure. Those skilled in the art should appreciate that they may readily use the present disclosure as a basis for designing or modifying other processes and structures for carrying out the same purposes and/or achieving the same advantages of the embodiments introduced herein. Those skilled in the art should also realize that such equivalent constructions do not depart from the spirit and scope of the present disclosure, and that they may make various changes, substitutions and alterations herein without departing from the spirit and scope of the present disclosure. 
     The Abstract at the end of this disclosure is provided to comply with 37 C.F.R. §1.72(b) to allow the reader to quickly ascertain the nature of the technical disclosure. It is submitted with the understanding that it will not be used to interpret or limit the scope or meaning of the claims.