Autonomous Uncertainty-Aware Engine For Pressure Gradient Identification Using A Discrete Optimization Framework

Abstract

A method and system for identifying a fluid within a subterranean formation. The method may comprise obtaining one or more pressure measurements at one or more depths with a downhole fluid sampling tool, forming a depth-pressure measurement set form the one or more pressure measurements, creating a solution novelty threshold from at least the depth-pressure measurement set, constraining a solution space with the solution novelty threshold, and finding a solution-space-inscribed simplex within the solution novelty threshold. The method may further comprise generating a simplicial decomposition for a convex hull of the solution-space-inscribed simplex up to the solution novelty threshold, identifying at least one inscribed simplex within the convex hull of the solution-space-inscribed simplex, determining a novel simplex interior with the at least one inscribed simplex, and forming a plurality of solutions with the novel simplex interior.

Description

BACKGROUND

During oil and gas exploration, different types of information may be collected and analyzed. The information may be used to determine the quantity and quality of hydrocarbons in a reservoir and to develop or modify strategies for hydrocarbon production. For instance, collected information may be used for reservoir evaluation, flow assurance, reservoir stimulation, facility enhancement, production enhancement strategies, and reserve estimation.

Reservoir evaluation may utilize collected information from one or downhole fluid sampling tools that measure formation pressure. Collecting information involves measuring pressure or other fluid properties at correlated depths (i.e., depth-pressure measurements, or depth-composition measurements) in an exploration and or production operation and determining pressure gradients which represent the reservoir properties from the depth-pressure measurements. There are a variety of different tools that may be used for pressure measurements or other fluid property measurements during a production operation. Fundamentally, all the measurements are correlated through the physics of an equation of state, however, empirical correlations may also be appropriate. Downhole analysis of reservoir fluids may also be used to provide real-time fluid properties, thus augmenting laboratory measurements and also providing information mitigating delays associated with laboratory analysis. In addition, downhole fluid analysis may be acquired at more physical locations along the wellbore than may be sampled in the same amount of time.

Depth-pressure measurements may be indicative of reservoir fluids disposed in the formation. For example, a reservoir fluid may manifest as a pressure gradient and a pressure gradient may be determined from depth-pressure measurements. A pressure gradient may be directed to a first order linear equation. However, the pressure gradient may be affected by compositional changes, capillary pressure, and relative permeability yielding a tangential curve, or equations of state yielding to a second order an exponential behavior. Some pressure gradient models may be empirical, or semi-empirical, including those defined by machine learning models with historical information previously measured within the same measurement operation.

In examples, a slope or gradient obtained from a line that fits over a collection of depth-pressure measurement correlates directly to the density of the underlying fluid that is being measured (assuming said collection corresponds to a single fluid). Fluid density ranges may then be mapped to fluid labels. Thus, a given formation may be comprised of any number of reservoir fluid pressure gradients caused by the density of gas, oil, or water, compositional changes, or formation effects due to factors such as wettability, capillary pressure, or relative mobility.

However, multiple solutions for a single set of depth-pressure measurements are possible. A simple inversion for a single set of depth-pressure measurements may not be reliable. Therefore, interpreting depth-pressure measurements may require an expert opinion. The expert opinion is based on the given depth-pressure measurements as well as other downhole fluid measurements, and the experience of the expert. Other downhole fluid measurements may comprise pressure measurement or fluid density measurements or compositional measurements, or formation properties measurements, taken by a downhole fluid sampling tool, and which may be evaluated with other supporting measurements. Other supporting measurements may comprise compositional measurements of fluid or rock, nuclear measurements of the formation, electromagnetic measurements of the formation, acoustic measurements of the formation, or nuclear magnetic resonance measurements of the formation. An expert opinion is required to interpret depth-pressure measurements as a.

BRIEF DESCRIPTION OF THE DRAWINGS

These drawings illustrate certain aspects of some examples of the present disclosure and should not be used to limit or define the disclosure.

FIG. 1 illustrates a measurement operation utilizing a downhole fluid sampling tool disposed on a conveyance;

FIG. 2 illustrates a drilling operation utilizing a downhole fluid sampling tool disposed on a drill string;

FIG. 3 illustrates a schematic of downhole fluid sampling tool;

FIG. 4A illustrates a simplified example of a noise envelopes;

FIG. 4B illustrates a simplified example of larger noise envelopes;

FIG. 5 illustrates an approximating linear envelope example; and

FIG. 6 illustrates a workflow in which a solution-enumerating simplex-decomposition-generating algorithm is applied.

DETAILED DESCRIPTION

This disclosure is directed to methods and systems for subterranean operations and, more particularly, to providing a computational means to autonomously interpret a plurality of solutions, or pressure gradients from one or more depth-pressure measurements. Methods and systems discussed below compute the pressure gradients from a collection of depth-pressure measurements, taken during a measurement operation as that of solving a mixed-integer linear programming problem, although nonlinear programming methods may also be used. Methods and systems may determine a plurality of solutions all of which may represent plausible interpretations of the depth-pressure measurements while satisfying constraints. Herein, a plurality of solutions may be defined as at least one accurate pressure gradient interpretation of two or more depth-pressure measurements of reservoir fluids. Additionally, the plurality of solutions may satisfy all imposed constraints. The method may incorporate measurement noise models (of both pressure and depth) as a solution constraint in addition to physical domain constraints. Further, methods and systems disclosed below may process noise into characterized uncertainty of error expectation or prior expectation of uncertainty of error expectation. Additionally, methods and systems disclosed below may autonomize interpretation of depth-pressure measurements.

FIG. 1 illustrates a measurement operation utilizing a downhole fluid sampling tool 100 disposed on a conveyance 102. As illustrated, wellbore 104 may extend through subterranean formation 106. In examples, reservoir fluid may be contaminated with well fluid (e.g., drilling fluid) from wellbore 104. As described herein, the fluid sample may be analyzed to determine fluid contamination and other fluid properties of the reservoir fluid. As illustrated, a wellbore 104 may extend through subterranean formation 106. While the wellbore 104 is shown extending generally vertically into the subterranean formation 106, the principles described herein are also applicable to wellbores that extend at an angle through the subterranean formation 106, such as horizontal and slanted wellbores. For example, although FIG. 1 shows a vertical or low inclination angle well, high inclination angle or horizontal placement of the well and equipment is also possible. It should further be noted that while FIG. 1 generally depicts a land-based operation, those skilled in the art will readily recognize that the principles described herein are equally applicable to subsea operations that employ floating or sea-based platforms and rigs, without departing from the scope of the disclosure.

As illustrated, a hoist 108 may be used to run downhole fluid sampling tool 100 into wellbore 104. Hoist 108 may be disposed on a vehicle 110. Hoist 108 may be used, for example, to raise and lower conveyance 102 in wellbore 104. While hoist 108 is shown on vehicle 110, it should be understood that conveyance 102 may alternatively be disposed from a hoist 108 that is installed at surface 112 instead of being located on vehicle 110. Downhole fluid sampling tool 100 may be suspended in wellbore 104 on conveyance 102. Other conveyance types may be used for conveying downhole fluid sampling tool 100 into wellbore 104, including coiled tubing and wired drill pipe, for example. downhole fluid sampling tool 100 may include a tool body 114, which may be elongated as shown on FIG. 1. Tool body 114 may be any suitable material, including without limitation titanium, stainless steel, alloys, plastic, combinations thereof, and the like. Downhole fluid sampling tool 100 may further include one or more sensors 116 for measuring properties of the fluid sample, reservoir fluid, wellbore 104, subterranean formation 106, or the like. In examples, downhole fluid sampling tool 100 may also include a fluid analysis module 118, which may be operable to process information regarding fluid sample, as described below. Downhole fluid sampling tool 100 may be used to collect fluid samples from subterranean formation 106 and may obtain and separately store different fluid samples from subterranean formation 106.

In examples, fluid analysis module 118 may include at least one sensor that may continuously monitor a reservoir fluid. Such sensors include optical sensors, acoustic sensors, electromagnetic sensors, conductivity sensors, resistivity sensors, selective electrodes, density sensors, mass sensors, thermal sensors, chromatography sensors, viscosity sensors, bubble point sensors, fluid compressibility sensors, flow rate sensors. Sensors may measure a contrast between drilling fluid filtrate properties and formation fluid properties.

In examples, fluid analysis module 118 may be a gas chromatography analyzer (GC). A gas chromatography analyzer may separate and analyze compounds that may be vaporized without decomposition. Fluid samples from wellbore 104 may be injected into a GC column and vaporized. Different compounds may be separated due to their retention time difference in the vapor state. Analyses of the compounds may be displayed in GC chromatographs. In examples, a mixture of formation fluid and drilling fluid filtrate may be separated and analyzed to determine the properties within the formation fluid and drilling fluid filtrate.

Fluid analysis module 118 may be operable to derive properties and characterize the fluid sample. By way of example, fluid analysis module 118 may measure absorption, transmittance, or reflectance spectra and translate such measurements into component concentrations of the fluid sample, which may be lumped component concentrations, as described above. The fluid analysis module 118 may also measure gas-to-oil ratio, fluid composition, water cut, live fluid density, live fluid viscosity, formation pressure, and formation temperature. Fluid analysis module 118 may also be operable to determine fluid contamination of the fluid sample and may include any instrumentality or aggregate of instrumentalities operable to compute, classify, process, transmit, receive, retrieve, originate, switch, store, display, manifest, detect, record, reproduce, handle, or utilize any form of information, intelligence, or data for business, scientific, control, or other purposes. For example, fluid analysis module 118 may include random access memory (RAM), one or more processing units, such as a central processing unit (CPU), or hardware or software control logic, ROM, and/or other types of nonvolatile memory.

Any suitable technique may be used for transmitting signals from the downhole fluid sampling tool 100 to surface 112. As illustrated, a communication link 120 (which may be wired or wireless, for example) may be provided that may transmit data from downhole fluid sampling tool 100 to an information handling system 122 at surface 112. Information handling system 122 may include a processing unit 124, a monitor 126, an input device 128 (e.g., keyboard, mouse, etc.), and/or computer media 130 (e.g., optical disks, magnetic disks) that can store code representative of the methods described herein. Information handling system 122 may act as a data acquisition system and possibly a data processing system that analyzes information from downhole fluid sampling tool 100. For example, information handling system 122 may process the information from downhole fluid sampling tool 100 for determination of fluid contamination. Information handling system 122 may also determine additional properties of the fluid sample (or reservoir fluid), such as component concentrations, pressure-volume-temperature properties (e.g., bubble point, phase envelop prediction, etc.) based on the fluid characterization. This processing may occur at surface 112 in real-time. Alternatively, the processing may occur downhole hole or at surface 112 or another location after recovery of downhole fluid sampling tool 100 from wellbore 104. Alternatively, the processing may be performed by an information handling system in wellbore 104, such as fluid analysis module 118. The resultant fluid contamination and fluid properties may then be transmitted to surface 112, for example, in real-time.

It should be noted that in examples a gas chromatograph 132 may be disposed on surface 112 and analyze samples captures by downhole fluid sampling tool 100. For example, fluid analysis module 118 may capture fluid samples and bring them to the surface 112 for analysis at the wellsite. As illustrated, gas chromatograph 132 may be disposed in vehicle 110. However, gas chromatograph 132 may be a standalone assembly that may be available at the wellsite. Additionally, information handling system 122 may be connected to gas chromatograph 132 through communication link 120. In examples, gas chromatograph 132 may operate and function as described above. Other surface testing is also possible by devices such as a liquid chromatograph, a mass spectrometer, or a microfluidic system. Composition, and physical properties including PVT or phase behavior properties may be measured. Properties include chemical properties such as individual component information, CO₂, H₂S, N₂, C₁, C₂, C₃, C₄, C₅, C₆ etc., plus fractions, such as C₆+ or C₇+, organic acids, chemical classes such as Saturates, Aromatics, Resins and Asphaltenes (SARA), acidity or total acid number, eH, pH, or may include physical properties such as bubble point, phase envelop, density, viscosity, compressibility, speed of sound, asphaltene precipitation pressure, wax precipitation temperature, GOR, API gravity, molecular weight of the oil or a fraction of the oil etc.

FIG. 2 illustrates a drilling operation utilizing a downhole fluid sampling tool 100 disposed on a drill string 200. Downhole fluid sampling tool 100 may be used to obtain a fluid sample, for example, a fluid sample of a reservoir fluid from subterranean formation 106. The reservoir fluid may be contaminated with well fluid (i.e., drilling fluid) from wellbore 104. As described herein, the fluid sample may be analyzed to determine fluid contamination and other fluid properties of the reservoir fluid. As illustrated, a wellbore 104 may extend through subterranean formation 106. While the wellbore 104 is shown extending generally vertically into the subterranean formation 106, the principles described herein are also applicable to wellbores that extend at an angle through the subterranean formation 106, such as horizontal and slanted wellbores. For example, although FIG. 2 shows a vertical or low inclination angle well, high inclination angle or horizontal placement of the well and equipment is also possible. It should further be noted that while FIG. 2 generally depicts a land-based operation, those skilled in the art will readily recognize that the principles described herein are equally applicable to subsea operations that employ floating or sea-based platforms and rigs, without departing from the scope of the disclosure.

As illustrated, a drilling platform 202 may support a derrick 204 having a traveling block 206 for raising and lowering drill string 200. Drill string 200 may include, but is not limited to, drill pipe and coiled tubing, as generally known to those skilled in the art. A kelly 208 may support drill string 200 as it may be lowered through a rotary table 210. A drill bit 212 may be attached to the distal end of drill string 200 and may be driven either by a downhole motor and/or via rotation of drill string 200 from the surface 112. Without limitation, drill bit 212 may include, roller cone bits, PDC bits, natural diamond bits, any hole openers, reamers, coring bits, and the like. As drill bit 212 rotates, it may create and extend wellbore 104 that penetrates various subterranean formations 106. A pump 214 may circulate drilling fluid through a feed pipe 216 to kelly 208, downhole through interior of drill string 200, through orifices in drill bit 212, back to surface 112 via annulus 218 surrounding drill string 200, and into a retention pit 220.

Drill bit 212 may be just one piece of a downhole assembly that may include one or more drill collars 222 and downhole fluid sampling tool 100. Downhole fluid sampling tool 100, which may be built into the drill collars 22) may gather measurements and fluid samples as described herein. One or more of the drill collars 222 may form a tool body 114, which may be elongated as shown on FIG. 2. Tool body 114 may be any suitable material, including without limitation titanium, stainless steel, alloys, plastic, combinations thereof, and the like. Downhole fluid sampling tool 100 may be similar in configuration and operation to downhole fluid sampling tool 100 shown on FIG. 1 except that FIG. 2 shows downhole fluid sampling tool 100 disposed on drill string 200. Alternatively, downhole fluid sampling tool 100 may be lowered into the wellbore after drilling operations on a wireline. In examples, logging while drilling (LWD)may also be implemented.

As previously described, downhole fluid sampling tool may be deployed on a wireline conveyance system or logging while drilling conveyance system, may further comprise one or more sensors 116 for measuring properties of the fluid sample reservoir fluid, wellbore 104, subterranean formation 106, or the like. The properties of the fluid are measured as the fluid passes from the formation through downhole fluid sampling tool 100 and into either the wellbore or a sample container. As fluid is flushed in the near wellbore region by the mechanical pump, the fluid that passes through downhole fluid sampling tool 100 generally reduces in drilling fluid filtrate content, and generally increases in formation fluid content. Downhole fluid sampling tool 100 may be used to collect a fluid sample from subterranean formation 106 when the filtrate content has been determined to be sufficiently low. Sufficiently low depends on the purpose of sampling. For some laboratory testing below 10% drilling fluid contamination is sufficiently low, and for other testing below 1% drilling fluid filtrate contamination is sufficiently low. Sufficiently low also depends on the nature of the formation fluid such that lower thresholds may be generally needed, the lighter the oil as designated with either a higher GOR or a higher API gravity. Sufficiently low also depends on the rate of cleanup in a cost benefit analysis since longer pumpout times to incrementally reduce the contamination levels may have prohibitively large costs. As previously described, the fluid sample may include a reservoir fluid, which may be contaminated with a drilling fluid or drilling fluid filtrate. Downhole fluid sampling tool 100 may obtain and separately store different fluid samples from subterranean formation 106 with fluid analysis module 118. Fluid analysis module 118 may operate and function in the same manner as described above. However, storing of the fluid samples in downhole fluid sampling tool 100 may be based on the determination of the fluid contamination. For example, if the fluid contamination exceeds a tolerance, then the fluid sample may not be stored. If the fluid contamination is within a dynamic tolerance, then the fluid sample may be stored in downhole fluid sampling tool 100. Dynamic tolerance may range from 0.001% contamination to 1% contamination, 1% contamination to 5% contamination, 5% contamination to 15% contamination, 15% contamination to 50% contamination, 50% contamination to 99% contamination, or any range between. In examples, depth-pressure measurements may be made while sampling after sampling or before sampling operations. Pressure measurements may be conducted as part of the sampling process or made independent of the sampling process. Pressure measurements may be made on a transient system, a steady state system, intermediate system, or any combination of system thereof.

As previously described, information from downhole fluid sampling tool 100 may be transmitted to an information handling system 122, which may be located at surface 112. As illustrated, communication link 120 (which may be wired or wireless, for example) may be provided that may transmit data from downhole fluid sampling tool 100 to an information handling system 111 at surface 112. Information handling system 140 may include a processing unit 124, a monitor 126, an input device 128 (e.g., keyboard, mouse, etc.), and/or computer media 130 (e.g., optical disks, magnetic disks) that may store code representative of the methods described herein. In addition to, or in place of processing at surface 112, processing may occur downhole (e.g., fluid analysis module 118). In examples, information handling system 122 may perform computations to estimate clean fluid composition. In the presence of drilling fluid filtrate contamination, the effects on fluid or rock properties measured may need to be deconvoluted by means such as but not limited to trend fitting or equation of state deconvolution.

As previously described above, a gas chromatograph 132 (e.g., referring to FIG. 1) may be disposed on surface 112 and analyze samples captures by downhole fluid sampling tool 100. For example, fluid analysis module 118 may capture fluid samples and bring them to the surface 112 for analysis at the wellsite. As illustrated, gas chromatograph 132 may be a standalone assembly that may be available at the wellsite. Additionally, information handling system 122 may be connected to gas chromatograph 132 through communication link 120. In examples, gas chromatograph 132 may operate and function as described above. As previously described other instrumentation includes liquid chromatograph, mass spectrometer, microfluidic system, or any combination therein.

FIG. 3 illustrates a schematic of downhole fluid sampling tool 100, which may be used for fluid sampling operations and/or pressure test operations. In examples, downhole fluid sampling tool 100 includes a power telemetry section 302 through which downhole fluid sampling tool 100 communicates with other actuators and sensors 116 in drill string 200 or conveyance 102 (e.g., referring to FIGS. 1 and 2), the drill string's telemetry section 302, and/or directly with a surface telemetry system (not illustrated). In examples, power telemetry section 302 may also be a port through which the various actuators (e.g., valves) and sensors (e.g., temperature and pressure sensors) in the downhole fluid sampling tool 100 may be controlled and monitored. In examples, power telemetry section 302 includes a computer that exercises the control and monitoring function. In one embodiment, the control and monitoring function is performed by a computer in another part of the drill string or wireline tool (not shown) or by information handling system 122 on surface 112 (e.g., referring to FIGS. 1 and 2).

In examples, downhole fluid sampling tool 100 may comprise probe 304. Probe 304 may be defined as dual probes, a single probe, a focused probe, a packer, or multi probe section with shapes defined to seal against the wellbore comprising but not limited to circular probes, oval probes, elliptical probes, elongated probes or radial probes. Probe 304 may extract fluid from the reservoir and deliver it to a channel 306 that extends from one end of downhole fluid sampling tool 100 to the other. Without limitation, probe 304 includes two probes 318, 320 which may extend from downhole fluid sampling tool 100 and press against the inner wall of wellbore 104 (e.g., referring to FIG. 1). Probe channels 322, 324 may connect probes 318, 320 to channel 306. The high-volume bidirectional pump 312 may be used to pump fluids from the reservoir, through probe channels 322, 324 and to channel 306. Alternatively, a low volume pump 326 may be used for this purpose. Two standoffs or stabilizers 328, 330 hold downhole fluid sampling tool 100 in place as probes 318, 320 press against the wall of wellbore 104. In examples, probes 318, 320 and stabilizers 328, 330 may be retracted when downhole fluid sampling tool 100 may be in motion and probes 318, 320 and stabilizers 328, 330 may be extended to sample the reservoir fluids at any suitable location in wellbore 104. Other probe sections include focused sampling probes, oval probes, or packers.

In examples, channel 306 may be connected to other tools disposed on drill string 200 or conveyance 102 (e.g., referring to FIGS. 1 and 2). In examples, downhole fluid sampling tool 100 may also include a quartz gauge section 308, which may include sensors to allow measurement of properties, such as temperature and pressure, of fluid in channel 306. Additionally, downhole fluid sampling tool 100 may include a flow-control pump-out section 310, which may include a high-volume bidirectional pump 312 for pumping fluid through channel 306. In examples, downhole fluid sampling tool 100 may include two multi-chamber sections 314, 316, referred to collectively as multi-chamber sections 314, 316 or individually as first multi-chamber section 314 and second multi-chamber section 316, respectively.

In examples, multi-chamber sections 314, 316 may be separated from flow-control pump-out section 310 by sensor section 332, which may house one or more sensors 334. Sensor 334 may be displaced within sensor section 332 in-line with channel 306 to be a “flow through” sensor. In alternate examples, sensor 334 may be connected to channel 306 via an offshoot of channel 306. Without limitation, sensor 334 may include optical sensors, acoustic sensors, electromagnetic sensors, conductivity sensors, resistivity sensors, selective electrodes, density sensors, mass sensors, thermal sensors, chromatography sensors, viscosity sensors, bubble point sensors, fluid compressibility sensors, flow rate sensors, microfluidic sensors, selective electrodes such as ion selective electrodes, and/or combinations thereof. In examples, sensor 334 may operate and/or function to measure drilling fluid filtrate, discussed further below.

Additionally, multi-chamber section 314, 316 may comprise access channel 336 and chamber access channel 338. Without limitation, access channel 336 and chamber access channel 338 may operate and function to either allow a solids-containing fluid (e.g., mud) disposed in wellbore 104 in or provide a path for removing fluid from downhole fluid sampling tool 100 into wellbore 104. As illustrated, multi-chamber section 314, 316 may comprise a plurality of chambers 340. Chambers 340 may be sampling chamber that may be used to sample wellbore fluids, reservoir fluids, and/or the like during measurement operations. It should be noted that downhole fluid sampling tool 100 may also be used in pressure testing operations.

For example, during pressure testing operations, probes 318, 320 may be pressed against the inner wall of wellbore 104 (e.g., referring to FIG. 1). Pressure may increase at probes 318, 320 due to formation 106 (e.g., referring to FIG. 1 or 2) exerting pressure on probes 318, 320. As pressure rises and reaches a predetermined pressure, valves 342 opens so as to close equalizer valve 344, thereby isolating fluid passageway 346 from the annulus 218. In this manner, valve 342 ensures that equalizer valve 344 closes only after probes 318, 320 has entered contact with mudcake (not illustrated) that is disposed against the inner wall of wellbore 104. In examples, as probes 318, 320 are pressed against the inner wall of wellbore 104, the pressure rises and closes the equalizer valve in fluid passageway 346, thereby isolating the fluid passageway 346 from the annulus 218. In this manner, the equalizer valve in fluid passageway 346 may close before probes 318, 320 may have entered contact with the mudcake that lines the inner wall of wellbore 104. Fluid passageway 346, now closed to annulus 218, is in fluid communication with low volume pump 326.

As low volume pump 326 is actuated, formation fluid may thus be drawn through probe channels 322, 324 and probes 318, 320. The movement of low volume pump 326 lowers the pressure in fluid passageway 346 to a pressure below the formation pressure, such that formation fluid is drawn through probe channels 322, 324 and probes 318, 320 and into fluid passageway 346. The pressure of the formation fluid may be measured in fluid passageway 346 while probes 318, 320 serves as a seal to prevent annular fluids from entering fluid passageway 346 and invalidating the formation pressure measurement.

With low volume pump 326 in its fully retracted position and formation fluid drawn into fluid passageway 346, the pressure will stabilize and enable pressure transducers 348 to sense and measure formation fluid pressure. The measured pressure is transmitted to information handling system 122 disposed on downhole fluid sampling tool 100 and/or it may be transmitted to the surface via mud pulse telemetry or by any other conventional telemetry means to an information handling system 122 disposed on surface 112.

During this interval, pressure transducers 348 may continuously monitor the pressure in fluid passageway 346 until the pressure stabilizes, or after a predetermined time interval. When the measured pressure stabilizes, or after a predetermined time interval, for example at 1800 psi, and is sensed by pressure transducer 348 the drawdown operation may be complete. Depth of downhole fluid sampling tool 100 is correlated with the sensed pressure and may be transmitted to information handling system 122 to form a log of one or more depth-pressure measurements.

Depth-pressure measurements recorded by information handling system 122 during a measurement operation may be used in a depth-pressure measurement set by information handling system 122. Once complete, fluid for the pressure test in fluid passageway 346 may be dispelled from of downhole fluid sampling tool 100 through the opening and/or closing of valves 342 and/or equalizer valve 344 as low volume pump 326 returns to a starting position. Downhole fluid sampling tool 100 may then move to a plurality of new depths and the above process is repeated to take a plurality of depth-pressure measurements which are used to form the measurement set. In examples, information handling system 122 may be communicatively connected, as discussed above, to an information handling system 122 disposed at the surface. The depth-pressure measurements may be processed into a measurement set in real time or in previously formed measurement sets to identify pressure gradients at any location in subterranean formation 106.

In examples, information handling system 122 may form multiple pressure gradients within the depth-pressure measurement set to evaluate solution uncertainty. Herein solution uncertainty may be the ambiguity in defining a particular pressure gradient or the ambiguity in partitioning the complete depth-pressure measurement set into depth-contiguous subsets, then mapping said subsets to distinct fluids. A pressure gradient may then be fitted to each depth subset. A pressure gradient over a given depth subset refers to the linear regime governing the relationship between depth and pressure values over the underlying depth subset range. Pressure gradients may be evaluated to determine whether or not a sampling operation may be performed with downhole fluid sampling tool 100, and if so, at which locations to sample in wellbore 104 (i.e., referring to FIG. 1 or FIG. 2). Partitioning to form one or more pressure gradients need not span the entire depth-pressure measurement set, rather some of the data points may not be assigned to any pressure gradient. These unassigned data points may be defined as outliers

Methods and systems disclosed below may allow for an automation of identifying a fluid that may be disposed within subterranean formation 106 (e.g., referring to FIGS. 1 and 2). The automation may be performed on information handling system 122 (e.g., referring to FIGS. 1 and 2). The automation may accept depth-pressure measurements for the and form one or more pressure gradients without any assumption about the data including, no sparsity requirement, no requirement on outlier removal, and/or allowance of potential faults. Additionally, the automation may utilize first principles as inputs as constraints. In examples, first principles may be physical slope ranges, inter-gradient constraints, general noise definitions, and/or depth-based gradient partitioning. The data may be utilized to create a mathematical framework.

Building a mathematical framework may be accomplished with an input, a discrete optimization framework implemented as a mixed integer linear programming. The input may comprise a set of real time or previous depth-pressure measurements coupled with a measured noise envelope or a historical noise envelope. Solving for the mathematical framework determines a single linear pressure gradient or a set of linear pressure gradients. Each linear pressure gradient may be utilized to determine what fluids may be present in subterranean formation 106. A potential problem may be uncovering multiple sets of pressure gradients over a collection of depth-pressures measurements, wherein each pressure gradient fits a distinct subset of points from the input measurement collection. As noted above, a noise envelope or a historical noise envelope may be utilized in conjunction with the depth-pressure measurements in determining what fluid may be disposed in subterranean formation 106.

Measurement noise characterization is one of the primary constraints for the disclosed method. Herein, minimal-level information may be defined as a conservative approach to noise as opposed to having a precise noise model. Therefore, instead of a noise model, noise envelopes may be utilized. Noise envelopes may contain measurement errors. Measurement errors may help define the mathematical framework, which may be utilized to determine all inherent interpretation ambiguities. In examples, minimal levels of information may comprise the extent of the expected error to be utilized as opposed to relying on super-refined noise definitions that introduce undue assumption risk. Herein, a pressure gradient may be solved from the mathematical framework of the depth-pressure measurement set. As previously described, a pressure gradient are manifestations of reservoir fluids. Therefore, a single pressure gradient may solve for reservoir fluids.

Within the depth-pressure measurement set a pressure measurement error may be determined from previous empirically observed knowledge. For example, pressure measurement error may obey the Boltzmann-Maxx family of distributions with error support interval ranging from a minimum of −10 psi to a maximum of +50 psi. Thus, noise constraints may be applied with fixed Botlzmann-Maxx distribution chosen from an adjustable parameter value. When the adjustable parameter value cannot be prefixed, several instances of distribution model may be used in the generation of noise data. Further, assuming that no gradient is expected to contain more than a fixed maximum number of inputs from the depth-pressure measurement set, the pressure measurements may be sampled from the fixed distribution anywhere from one sample up to the fixed maximum of pressure measurement samples. Then, the adjustable parameter value is applied iteratively across each sample from the depth-pressure measurement set. The preceding process may be re-iterated for a pre-designated number of iterations, generating an error sample collection with plausible variable-size pressure reconstruction error samples by any gradient.

In an example, using simulated data, a simulated error sample collection may be processed before a noise constraint is constructed. Processing an error sample collection may comprise, counting the number of points that fall in each of pre-designated error bins within the full distribution error support range (e.g., −10 psi to +50 psi with uniform bin width of 5 psi). Regardless of the initial continuous error distribution assumption, the effect of discretizing it into bins results in bin count sequences that may be multinomially distributed with parameters being a function of the original distribution shape. Further, because the original distribution parameter is varied, the bin count sequences are non-identically multinomially distributed. One unifying view of the bin count sequences is to interpret them as points forming vectors emanating from a vector space with dimension equal to the number of bins. The noise constraint may then be obtained by constructing a boundary (i.e., a noise envelope) around these points in their intrinsic vector space. As discussed below, the boundary may be both convex and linear. The boundary need not include all points but may be probabilistically defined so as to exclude a tolerable fraction of least probable points while achieving a tighter boundary. High-dimensional boundaries (i.e., a noise envelope) may be difficult to visualize in higher dimensions.

FIGS. 4A and 4B illustrate a simplified example of noise envelopes. Both FIGS. 4A and 4B comprise examples of planar projections of noise envelopes 400 and 402 created by pressure and obtained from the depth-pressure measurement set. In FIG. 4A, noise envelope 400 may be generated from Bins 4 & 5. Likewise, in FIG. 4B, noise envelope 402 may be generated from Bins 5 & 6. Bin 4 ranges over 5-10 psi, bin 5 over 10-15 psi, and bin 6 over 15-20 psi. Herein, a vector v may be at least part of a depth-pressure measurement. Every vector v represents a set of error counts (one count per dimension). Each dimension represents an error range. For example, along the bin 4 dimension an error may range from 5-10 psi, as described below. Each point mapped between bins 4 & 5 within pressure noise envelope 400 represents a possible direction and amplitude for vector v to be mapped to. For example, a vector v=(3,10) (not illustrated) over bins 4 & 5 may be able to sample thirteen points. Of those thirteen points, three may have errors in the range of bin 4 and 10 may have errors in the range of bin 5. Pressure noise envelope 400 represents high-dimensional boundaries which provides a constraint for vector v. Similarly, FIG. 4B represents possible amplitude and direction a vector v may be mapped to over bins 5 & 6, but within pressure noise envelope 402.

Pressure noise envelopes 400 and 402 are geometrical constraints derived from the depth-pressure measurement set. In examples, pressure noise envelopes 400 and 402 may be derived probabilistically and/or constructed from point cloud data. From the point cloud statistical analysis may be run to exclude obvious (spatial) outliers. Forming pressure noise envelopes 400 and 402 yields pressure reconstruction constraints. In the description below, x_j denotes the j^th depth measurement where j spans all n depth samples. Similarly, y_j denotes the j^th of all n pressure measurements. Initially, seeking p gradients for an arbitrary gradient indexed by i, the reconstruction error may be defined via the difference: y_j−(m_ix_j+b_i), where mi denotes the gradient slope and b_i the gradient offset, or the difference between actual and predicted values. Further, assuming that there is a depth measurement error ε_j in each raw measurement x_j and therefore the reconstruction error becomes y_j−(m_i(x_j+ε_j)+b_i). Requiring that the last difference be particularly in the error bin spanning the range of −10 psi to −5 psi translates to following double inequality for Equation (1).

(−10)≤y_i−(m_i(x_j+ε_j)+b_i)≤(−05)   (1)

The above double inequality may be enforced or relaxed by resorting in part to a sufficiently large constant B, as in Equation (2) below. Herein r_i,j is a binary value which identifies whether or not the associated constraint may utilize relaxation.

(−10)−r_i,j,−5B≤y_j−(m_i(x_j+ε_j)+b_i)≤(−05)+r_i,j,−05B   (2)

When r_i,j,−05 is 0, the double inequality is enforced, and when r_i,j,−05 is 1, the double inequality is relaxed. Thus, the reconstruction difference of pressure point y_j at depth measurement x_j by the i^th gradient may satisfy the system of Equation (3) for some suitable values of the binary variables.

$\begin{array}{cc}\left\{\begin{array}{c}\begin{array}{c}\begin{array}{c}\left(-10\right)-r_{i,j,-05}B\le y_j-\left(m_i\left(x_j+{\epsilon }_j\right)+b_i\right)\le \left(-05\right)+r_{i,j,-05}B\\ \left(-05\right)-r_{i,j,-00}B\le y_j-\left(m_i\left(x_j+{\epsilon }_j\right)+b_i\right)\le \left(-00\right)+r_{i,j,-00}B\end{array}\\ \dots \end{array}\\ \left(+45\right)-r_{i,j,+50}B\le y_j-\left(m_i\left(x_j+{\epsilon }_j\right)+b_i\right)\le \left(+50\right)+r_{i,j,+50}B\end{array}\right\}& \left(3\right)\end{array}$

let k denote the bin index associated with variable r. Herein, the inequalities from Equation (3) may be referred to as measurement constraints. The above collection of double inequalities in (m, b, r, ε) is to be utilized over all indices i, j, and k. The quantity (1−r_i,j,k) is indicative of whether or not the measurement point pair (x_j, y_j) is reconstructed to within the k-th bin by gradient i. An additional constraint may be that a generic gradient i may reconstruct a minimum number of points within any of the pre-designated bins. Equation (4) accomplishes that for some minimum threshold A of minimum points per gradient for all gradient indices i.

Σ_j,k(1−r_i,j,k)≥Δ  (4)

Similarly, a certain minimum fraction δ of all n data points may be reconstructed by some gradient within one of its error bins (i.e., only an upper-bounded fraction of points is to be discounted as outliers) to form Equation (5). Herein, δn may be manually adjusted to any fraction. In examples, if δn is ⅕ not more than more than 20% of the data may be labeled as outliers.

Σ_i,j,k(1−r_i,j,k)≥┌δn┐  (5)

The total reconstruction error counts at a bin k by gradient i is given by Σ_j(1−r_i,j,k). Therefore, the bin count vector for gradient i may be expressed in Equation (6).

rΣ_j(1−r_i,j,1),Σ_j(1−r_i,j,2), . . . ,Σ_j(1−r_i,j,d)   (6)

Any r may satisfy convex linear envelope constraints. Generically, this may be specified in Equation (7) for some envelope boundary specifications H_error and h_error.

H_error(r)^T≤h_error   (7)

In examples, another constraint is to assume that ε_j is normally distributed. Convex linear envelope may easily be obtained for such a constraint expressed in Equation (8).

H_normal(ε)^T≤h_normal   (8)

Moving further with the constraints, the pressure reconstruction in Equation (9) contains a bilinear term: m_iε_j. In the interest of problem linearity, a linear envelope may approximate such a term. Let θ_i,jm_iε_j. Envelopes may then be constructed as Equation (9).

$\begin{array}{cc}\{\begin{array}{c}{m}^{\min }{\epsilon }_j+m_j{\epsilon }^{\min }-m^{\min }{\epsilon }^{\min }\\ m^{\max }{\epsilon }_j+m_j{\epsilon }^{\max }-m^{\max }{\epsilon }^{\max }\end{array}\le {\theta }_{i,j}\le \{\begin{array}{c}{m}^{\max }{\epsilon }_j+m_j{\epsilon }^{\min }-m^{\max }{\epsilon }^{\min }\\ m^{\min }{\epsilon }_j+m_j{\epsilon }^{\max }-m^{\min }{\epsilon }^{\max }\end{array}& \left(9\right)\end{array}$

where, m^min≤m_i≤m^max, ε^min≤ε_j≤ε^max for all i and j.

FIG. 5 illustrate an approximating linear envelope example based on the disclosure above. The linearizing envelope for the bilinear terms is comprised of a total of four terms. Two of these four terms may be two under-estimators. One of the two under-estimators is the upper bound defined as m^minε_j+m_jε^min−m^minε^min and the lower bound defined as m^maxε_j+m_jε^max−m^maxε^max. The other two bounds may be two over-estimators. One of the two over-estimators is the upper bound defined as m^maxε_j+m_jε^min−m^maxε^min and the lower bound defined as m^minε_j+m_jε^max−m^minε^max.Therefore, the bilinear term may be any function which satisfies the two under-estimators and the two over-estimators.

The list of constraints may thus be comprised of reconstruction error constraints from a noise envelope, maximum number of outliers, minimum number of points per gradient, and the linearization of the bilinear term. An additional constraint may be that of contiguous depth range partitioning. Therefore, the partitioning constraint may utilize the collection of measurement point pairs by partitioning the collection of measurement point pairs based at least in part on depth and where each gradient index i effectively indexes a subset in the partition. Without loss of generality, assume the set of depth measurements {x_j|j} is readily sorted in terms of index j. Further, defining j₀≤j₁≤j₂≤ . . . ≤j_p to be an integer-indexing of a p-subset partitioning of the complete depth range with j₀=0 and j_p=n. The i^th subset in the partitioning is indexed by the integer set J_i{j_i−1+1, . . . , j_i}. In examples, the partitioning constraint may be utilized for any index set J_i, therefore the number of points reconstructed by all gradients other than gradient i over J_i is zero. Thus, Equation (10) may be formed below, herein i′ denotes any gradient index other than fixed gradient index i.

Σ_j∈Ji,k(1−r_i′,j,k)=0∀i≠i′  (10)

Further, two sets of inter-gradient constraints may be defined. Both sets of inter-gradient constraints may have an absence of discontinuities in the pressure profile. The first set amounts to requiring increasing slope constraints by depth (i.e., m₁≤m₂≤ . . . ≤m_p). The second set of inter-gradient constraints implements the fact that for any two consecutively indexed gradients, depth-pressure points belonging to each gradient should lie in the upper halfspace with respect to the other gradient. Such constraints are produced in Equations (11) and (12).

m_ix_j+b_i≥m_i+1x_j+b_i+1∀i,∀j∈J_i   (11)

m_i+1x_j+b_i+1≥m_ix_j+b_i∀i,∀j∈J_i+1   (12)

Gathering all constraints thus far, the complete set of inequalities in terms of all problem variables which we collectively denote by S may be formed and defined in Equation (13):

$\begin{array}{cc}S\stackrel{△}{=}\left\{\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}\left(-10\right)-r_{i,j,-05}B\le y_j-\left(m_i\left(x_j+{\epsilon }_j\right)+b_i\right)\le \left(-05\right)+r_{i,j,-05}B\\ \left(-05\right)-r_{i,j,-00}B\le y_j-\left(m_i\left(x_j+{\epsilon }_j\right)+b_i\right)\le \left(-00\right)+r_{i,j,-00}B\end{array}\\ \dots \end{array}\\ \left(+45\right)-r_{i,j,+50}B\le y_j-\left(m_i\left(x_j+{\epsilon }_j\right)+b_i\right)\le \left(+50\right)+r_{i,j,+50}B\end{array}\\ r_{i,j,k}:\mathrm{binary}\forall i,j,k\end{array}\\ {\sum }_{j,k}\left(1-r_{i,j,k}\right)\ge \Delta \forall i\end{array}\\ {\sum }_{i,j,k}\left(1-r_{i,j,k}\right)\ge \lceil \delta n\rceil \end{array}\\ {H_{\mathrm{error}}\left(\stackrel{\_}{r_l}\right)}^T\le h_{\mathrm{error}}\end{array}\\ {H_{\mathrm{normal}}\left(\epsilon \right)}^T\le h_{\mathrm{normal}}\end{array}\\ \begin{array}{c}\{\begin{array}{c}{m}^{\min }{\epsilon }_j+m_j{\epsilon }^{\min }-m^{\min }{\epsilon }^{\min }\\ m^{\max }{\epsilon }_j+m_j{\epsilon }^{\max }-m^{\max }{\epsilon }^{\max }\end{array}\le {\theta }_{i,j}\le \\ \{\begin{array}{c}{m}^{\max }{\epsilon }_j+m_j{\epsilon }^{\min }-m^{\max }{\epsilon }^{\min }\\ m^{\min }{\epsilon }_j+m_j{\epsilon }^{\max }-m^{\min }{\epsilon }^{\max }\end{array}\end{array}\end{array}\\ {\sum }_{j\in J_i,k}\left(1-r_{i^{\prime },j,k}\right)=0\forall i\ne i^{\prime }\end{array}\\ j_0\le j_1\le j_2\le \dots \le j_p,j_0=0,j_p=n,j_i:\mathrm{integer}\forall i\\ m_1\le m_2\le \dots \le m_p\end{array}\\ \{\begin{array}{c}{m}_i{x}_j+b_i\ge m_{i+1}{x}_j+b_{i+1}\forall i,\forall j\in J_i\\ m_{i+1}{x}_j+b_{i+1}\ge m_i{x}_j+b_i\forall i,\forall j\in J_{i+1}\end{array}\end{array}\right\}& \left(13\right)\end{array}$

where r_i and J_i are defined in Equations (14) and (15) below.

r_iΣ_j(1−r_i,j,1),Σ_j(1−r_i,j,2),. . . ,Σ_j(1−r_i,j,d)   (14)

J_i{j_i−1+1, . . . ,j_i}  (15)

All constraints used in Equation (13) define a complete for the plurality of solutions, any vector (m, b, θ, ε, r, J), that is feasible over S represents a possible solution. Therefore, choosing a single solution reduces to computing a vector contained within S. Herein, the inequalities from Equation (13) may be referred to as physical fluid gradient constraints. In addition to requiring the error be within the allowable envelope, minimizing the error may further improve the solution. Minimizing the error may be performed in Equation (16) below:

$\begin{array}{cc}\underset{\left(m,b,\theta ,\epsilon ,r,J\right)\in S}{\arg \min }{\sum }_{i,j,k}\left(1-r_{i,j,k}\right){\mathrm{Bin}}_k& \left(16\right)\end{array}$

where, Bin_k is defined as the error representation at the k^th bin. The newly created linear proxy variable may simultaneously satisfy all bounding terms. The linear proxy variable may be solved for in Equation (16) in the case of the single solution or by the process of FIG. 6 in case of plurality of solutions, to be discussed below.

FIG. 6 illustrates, workflow 600 in which a solution-enumerating simplex-decomposition-generating algorithm is applied to the depth-pressure measurement set to determine a plurality of solutions within a solution space. As previously described, a plurality of solutions may be at least one accurate pressure gradient interpretation of two or more depth-pressure measurements. The inequalities from Equations (1)-(3) and (13) may be used within workflow 600. Workflow 600 may be performed on information handling system 122. Workflow 600 may begin with block 602. In block 602 a solution novelty threshold may be obtained from the measurement set. For example, enumerating a plurality of solutions within a solution space may be performed instead of singling out a solution that is optimum according to a certain measure. All enumerated solutions may be pairwise novel (i.e., dissimilar) for some solution novelty threshold. Herein, solution novelty threshold may be defined as sufficient dissimilarity according to said chosen measure. Such a solution novelty threshold may be obtained in block 602. Obtaining a solution novelty threshold may comprise determining constraints for a gradient slope differentiation threshold and for a gradient offset differentiation threshold, as discussed in the inequalities from Equation (1). Further, obtaining a solution novelty threshold may be found by determining a constraint for when two index sets are partitioned, as discussed in the inequalities from Equation (10). The solution novelty threshold may be a synthesis of a single threshold or a combination of thresholds comprising a gradient slope differentiation threshold, a gradient offset differentiation threshold, or a partitioning differentiation threshold. The solution novelty threshold is a metric applied to the plurality of solutions over solution space as a constraint. Once obtained, the solution novelty threshold may be used for enumerating a plurality of solutions to provides a direct characterization of the uncertainty such that each solution from the plurality of solutions is plausible and non-redundant.

In block 604, a solution space inscribed simplex is found. Constructing a simplex-decomposition-generating algorithm to perform the enumeration such that the linearly constrained solution space may have been a polytope if it were not for the integer constraints. As a result, finding a solution-space-inscribed simplex in block 604 may be a union of polyhedral slices. However, precise geometry of the solution-space-inscribed simplex may not be utilized. Regardless, the solution-space-inscribed simplex is a union of slices forming a geometrical shape and may satisfy the solution novelty threshold. The slices may follow any of the inequalities from Equation (3) and/or (13). The simplex-decomposition-generating algorithm may compute a point lattice over the underlying solution space to determine the solution-space-inscribed simplex. If a solution from the plurality of solutions is completely composed of vectors m, b, and J. Then the solution space to be enumerated is the projection of S over the vector subspace spanning all dimensions in (m, b, J). Each dimension m, b, and J may be a slice from of the geometrical shape, and their origin may be the solution-space-inscribed simplex. The solution space may further be adjusted in block 606.

In block 606, iteratively generating a simplicial decomposition of the convex hull of the solution-space-inscribed simplex up to a threshold may be performed in the solution space. Thus, block 606 may comprise computing a new inscribed simplex within the solution space's convex hull. This may be performed by iteratively adding a new simplex point that maximizes a hyperplane of normal in the null space of the lower dimensional simplex constructed up until the preceding iteration. For each full-face of the initial simplex, a new simplex may be found by finding a new simplex point on the opposite halfspace as the original point. The process is repeated until the entire convex hull of the solution-space-inscribed simplex is be generated to within the solution novelty threshold obtained in block 602. As a result, a simplicial decomposition to the novelty-based approximation for the convex hull of the solution-space-inscribed simplex up to the solution novelty threshold is calculated, providing simplices for the solution space. Further, processing for all simplices within the simplicial decomposition determined in block 606 may be performed. In other examples, simplicial decomposition may be performed without the initial inscribed simplex. For example, the simplex-decomposition-generating algorithm may identify a simplicial decomposition for the convex hull of the solution-space-inscribed simplex solution space to within the novelty threshold.

For example, in block 608, each simplex may undergo delimitation to determine a novel simplex interior by forming the constraints that ensure the determine a novel simplex interior is sufficiently within the solution novel threshold with respect to the associated simplex's vertices. After constraining the novel simplex interior, the simplex-decomposition-generating algorithm is recursively applied over the novel simplex interior for all simplices. Further, finding one or more interior points within each simplex may satisfy the solution novel threshold for each simplex. The process terminates when none of the simplices in a decomposition may be sufficiently large to contain interior novel solutions.

The resulting novel simplex interior comprises the plurality of solutions and represents all plausible interpretations of the acquired depth-pressure measurements. Forming the plurality of solutions may be performed by adding every individual component of the novel simplex interior to form each solution. Each solution may be an interpretation of the acquired depth-pressure measurements and may yield density of reservoir fluids and fluid labels to identify all plausible reservoir fluids. In examples, constraint formulation may be readily extended to the discrete pressure profile setting. Additionally, a recommendation engine for new depth sample recommendation may be devised based on a characterization of prediction uncertainty from the enumerated solution collection.

In further examples, additional physical constraints may be defined such that gas may lie on top of oil and oil may lie on top of water having fluid contact points. Other examples may include fluids that are not in contact (i.e., discontinuous gradients separated by non-fluid bearing sections). In other examples, the lower bound of gas density may be defined by an equation of state for methane which is typically defined by temperature and pressure. Additionally, Bayesian constraints may be applied wherein water has a density range of 0.95 g/cc to 1.2 g/cc typically, gas typically is from 0.05 g/cc to 0.35 g/cc or outliers of pressure measurements usually are at higher pressure more than they are at a lower pressure, or shales are usually not mobile-fluid-bearing, that low resistivity zones are typically water and high resistivity zones are typically oil or gas. In further examples, probabilistic or statistical constraints may be applied wherein two gradients in contact may satisfy a minimum confidence that the slope and intercept may not be distinct such as 85% or 90% or 95%, or that discontinuous fluids may satisfy a defined gradient intercept such as 85%, or 95% that the intercepts are distinct, or that the outliers be a confident distance from the gradient.

Constraints may be applied as hard or soft constraints. Hard constraints do not allow a solution to move over a single bound. Soft constraints weight the solutions toward an expectation value reducing the probability of a solution deviating from that expectation value according to a distribution model such as a normal distribution. Soft constraints may comprise but are not limited to the residual noise being normally distributed about the modeled value, or that a gradient may conform to a fluid density measurement within a defined tolerance or that the gradients may conform to an equation of state model within a defined tolerance. Tolerances may be estimated by sensitivity analysis to the parameters of the constraint. (i.e., pressure may admit an error with a standard deviation of 0.25 psi or that depth measurements may incur a relative error of 2%), requiring a variety of techniques to provide pressure measurements. Constraints may be defined from field knowledge, general knowledge discovered knowledge within the testing zone, or based on previous characterization within the testing zone such as form other logs such as electromagnetic logs, acoustic logs, nuclear magnetic resonance logs, or mud gas logs or nuclear logs each log type either with azimuthal resolution image logs or omnidirectional direction. Constraints may also be obtained from other fluid measurements.

Further, noise may be estimated from depth-pressure measurement set, known prior performance by a monte carlo solution estimated, or combinations therein. Although the methods described herein may be utilized for depth-pressure measurements to form pressure gradients using a linear gradient approximation, the method may be applied to any gradient data for any fluid property such as temperature, pressure, composition, density, compressibility, bubblepoint, gas to oil ratio (GOR), water saturation, and/or the like. Further a higher order model including nonlinear behavior may also be used such as multi-linear, polynomial, exponential, tangential, Equation of State such as but not limited to a Cubic Equation of State, a PC-SAFT Equation of State or empirical model such as a machine learning model defined to historic data including data from within the current test such as data acquired at previous depths, or any combinations therein. Additionally, the relationship between the various gradients may be simultaneously obtained by this method or may be used as internal constraints for other gradient information. For example, an asphaltene gradient may be used as a constraint for a viscosity gradient, or a bubble point gradient. A density gradient may be used as a constraint for a pressure gradient. A such, additional gradients may be determined by sensors in downhole fluid sampling tool 100 (e.g., referring to FIG. 1), assumed from field knowledge, obtained from prior log information, or measured at the surface near real time or post job from samples. Samples may be fluid or rock samples. In addition to setting constraints, modeled noise may be utilized in interpreting depth-pressure measurements.

When compared to standard operations, modeled noise may be incorporated to forming pressure gradients. Traditionally, noise may be derived using a least square fitting method or based on statistical set. In this disclosure a solution may be delineated by identifying a well-constrained model from a well-constrained noise. Additionally, all plausible (as per the constraints) solutions may be enumerated based on such delineations. Solution constraints may be gathered either from petrophysical knowledge (e.g., gradient constraints) or historical data (e.g., noise envelope constraints).

Systems and methods of the present disclosure are improvements over current technology in that noise models are transferred into automatically computing pressure gradients. Also, with its formal characterization of the solution space, this method allows for precise quantification of solution uncertainty and subsequently optimizing further data collection. Specifically, new pressure samples may be selected in locations of greatest interpretation ambiguity so as to reduce such an uncertainty. Thus, systems and methods presented in this disclosure provide uncertainty-aware gradient interpretation and subsequently exploit the characterized interpretation uncertainty to make intelligent future sampling requests in the interest of narrowing down said interpretation uncertainty. As such, depth-pressure measurements may be autonomously interpreted.

The preceding description provides various embodiments of systems and methods of use which may contain different method steps and alternative combinations of components. It should be understood that, although individual embodiments may be discussed herein, the present disclosure covers all combinations of the disclosed embodiments, including, without limitation, the different component combinations, method step combinations, and properties of the system. The systems and methods may include any of the various features disclosed herein, including one or more of the following statements.

Statement 1: A method may comprise obtaining one or more pressure measurements at one or more depths with a downhole fluid sampling tool, forming a depth-pressure measurement set form the one or more pressure measurements, creating a solution novelty threshold from at least the depth-pressure measurement set, constraining a solution space with the solution novelty threshold, and finding a solution-space-inscribed simplex within the solution novelty threshold. The method may further comprise generating a simplicial decomposition for a convex hull of the solution-space-inscribed simplex up to the solution novelty threshold, identifying at least one inscribed simplex within the convex hull of the solution-space-inscribed simplex, determining a novel simplex interior with the at least one inscribed simplex, and forming a plurality of solutions with the novel simplex interior.

Statement 2: The method of statement 1, wherein identifying the at least one inscribed simplex within the convex hull of the solution-space-inscribed simplex is performed by adding a new simplex point that maximizes a hyperplane in a null space of a new simplex, wherein the new simplex is at a lower dimension than the inscribed simplex.

Statement 3: The method of any preceding statements 1 or 2, further comprising forming a simplex-decomposition-generating algorithm with the simplicial decomposition.

Statement 4: The method of statement 3, further comprising applying the simplex-decomposition-generating algorithm to each of the at least one inscribed simplex.

Statement 5: The method of statement 4, further comprising finding one or more interior points within each simplex from the novel simplex interior.

Statement 6: The method of any preceding statements 1, 2, or 3, wherein the solution novelty threshold is obtained by a synthesis of at least one threshold.

Statement 7: The method of statement 6, wherein the at least one threshold comprises a gradient slope differentiation threshold, a gradient offset differentiation threshold, or a partitioning differentiation threshold.

Statement 8: The method of any preceding statements 1, 2, 3, or 6, wherein finding the solution-space-inscribed simplex is at least based in part on one or more inequalities.

Statement 9: The method of statement 8, wherein the one or more inequalities are measurement constraints.

Statement 10: The method of statement 8, wherein the one or more inequalities are physical fluid gradient constraints.

Statement 11: A system may comprise a downhole fluid sampling tool configured to obtain one or more pressure measurements at one or more depths and an information handling system. The information handling system may be configured to form a depth-pressure measurement set form the one or more pressure measurements, create a solution novelty threshold from at least the depth-pressure measurement set, constrain a solution space with the solution novelty threshold, and find a solution-space-inscribed simplex within the solution novelty threshold. The information handling system may be further configured to generate a simplicial decomposition for a convex hull of the solution-space-inscribed simplex up to the solution novelty threshold, identify at least one inscribed simplex within the convex hull of the solution-space-inscribed simplex, determine a novel simplex interior with the at least one inscribed simplex, and form a plurality of solutions with the novel simplex interior.

Statement 12: The system of statement 11, wherein the information handling system is further configured to add a new simplex point that maximizes a hyperplane in a null space of a new simplex, wherein the new simplex is at a lower dimension than the inscribed simplex.

Statement 13: The system of any preceding claim 11 or 12, wherein the information handling system is further configured to form a simplex-decomposition-generating algorithm with the simplicial decomposition.

Statement 14: The system of statement 13, wherein the information handling system is further configured to apply the simplex-decomposition-generating algorithm to each of the at least one inscribed simplex.

Statement 15: The system of statement 13, wherein the information handling system is further configured to find one or more interior points within each simplex from the novel simplex interior.

Statement 16: The system of any preceding statements 11, 12, or 13, wherein the solution novelty threshold is obtained by a synthesis of at least one threshold.

Statement 17: The system of statement 16, wherein the at least one threshold comprises a gradient slope differentiation threshold, a gradient offset differentiation threshold, or a partitioning differentiation threshold.

Statement 18: The system of any preceding statements 11, 12, 13, or 16, wherein finding the solution-space-inscribed simplex is at least based in part on one or more inequalities.

Statement 19: The system of statement 18, wherein the one or more inequalities are measurement constraints.

Statement 20: The system of statement 18, wherein the one or more inequalities are physical fluid gradient constraints.

It should be understood that the compositions and methods are described in terms of “comprising,” “containing,” or “including” various components or steps, the compositions and methods can also “consist essentially of” or “consist of” the various components and steps. Moreover, the indefinite articles “a” or “an,” as used in the claims, are defined herein to mean one or more than one of the elements that it introduces.

Therefore, the present embodiments are well adapted to attain the ends and advantages mentioned as well as those that are inherent therein. The particular embodiments disclosed above are illustrative only, as the present invention may be modified and practiced in different but equivalent manners apparent to those skilled in the art having the benefit of the teachings herein. Although individual embodiments are discussed, the invention covers all combinations of all those embodiments. Furthermore, no limitations are intended to the details of construction or design herein shown, other than as described in the claims below. Also, the terms in the claims have their plain, ordinary meaning unless otherwise explicitly and clearly defined by the patentee. It is therefore evident that the particular illustrative embodiments disclosed above may be altered or modified and all such variations are considered within the scope and spirit of the present invention. If there is any conflict in the usages of a word or term in this specification and one or more patent(s) or other documents that may be incorporated herein by reference, the definitions that are consistent with this specification should be adopted.

Claims

1. A method comprising:

obtaining one or more pressure measurements at one or more depths with a downhole fluid sampling tool;

forming a depth-pressure measurement set from the one or more pressure measurements;

creating a solution novelty threshold with a synthesis of at least one threshold;

generating a simplicial decomposition for a convex hull of a solution space up to the solution novelty threshold;

identifying at least one simplex within the convex hull of the solution space;

determining a novel simplex interior of the at least one simplex; and

forming a plurality of solutions within the novel simplex interior of the at least one simplex.

2. The method of claim 1, wherein identifying the at least one simplex within the convex hull of the solution space is performed by adding a new simplex point that maximizes a hyperplane in a null space of a new simplex, wherein the new simplex is augmented by one dimension when the new simplex point is added.

3. The method of claim 1, further comprising forming a simplex-decomposition-generating algorithm with the simplicial decomposition.

4. The method of claim 3, further comprising applying the simplex-decomposition-generating algorithm to each of the at least one simplex.

5. The method of claim 4, further comprising finding one or more interior points within each simplex from the novel simplex interior.

6. (canceled)

7. The method of claim 1, wherein the at least one threshold comprises a gradient slope differentiation threshold, a gradient offset differentiation threshold, or a partitioning differentiation threshold.

8. The method of claim 1, further comprising, finding a solution-space-inscribed simplex within the solution novelty threshold.

9. The method of claim 1, wherein the solution space is at least based in part on one or more inequalities.

10. The method of claim 9, wherein the one or more inequalities are physical fluid gradient constraints.

11. A system comprising:

a downhole fluid sampling tool configured to obtain one or more pressure measurements at one or more depths; and

an information handling system configured to: form a depth-pressure measurement set from the one or more pressure measurements; create a solution novelty threshold with a synthesis of at least one threshold; generate a simplicial decomposition for a convex hull of a solution space up to the solution novelty threshold; identify at least one simplex within the convex hull of the solution space; determine a novel simplex interior of the at least one simplex; and form a plurality of solutions within the novel simplex interior of the at least one simplex.

12. The system of claim 11, wherein the information handling system is further configured to add a new simplex point that maximizes a hyperplane in a null space of a new simplex, wherein the new simplex is augmented by one dimension when the new simplex point is added.

13. The system of claim 11, wherein the information handling system is further configured to form a simplex-decomposition-generating algorithm with the simplicial decomposition.

14. The system of claim 13, wherein the information handling system is further configured to apply the simplex-decomposition-generating algorithm to each of the at least one simplex.

15. The system of claim 13, wherein the information handling system is further configured to find one or more interior points within the novel simplex interior of each simplex.

16. (canceled)

17. The system of claim 11, wherein the system includes a gradient slope differentiation threshold, a gradient offset differentiation threshold, or a partitioning differentiation threshold.

18. The system of claim 11, wherein the information handling system is further configured to find a solution-space-inscribed simplex within the solution novelty threshold.

19. The system of claim 11, wherein the solution space is at least based in part on one or more inequalities.

20. The system of claim 19, wherein the one or more inequalities are physical fluid gradient constraints.

21. The system of claim 19, wherein the one or more inequalities are measurement constraints.

22. The method of claim 9, wherein the one or more inequalities are measurement constraints.