Method for estimating well surface pressure from tube waves induced in a well

Abstract

A method for estimating well surface pressure includes measuring pressure in a well during pumping a fracture treatment. For each of at least two changes in flow rate of the pumping, a well pipe friction pressure loss, a perforation friction pressure loss and near wellbore pressure loss are determined using events in the measured pressure induced by the at least one change in flow rate. Fracture pressure is determined at times of each of the at least two changes in flow rate from the measured pressure using the determined well pipe friction pressure loss, determined perforation friction pressure loss and near wellbore pressure loss. Fracture pressure is interpolated between the determined fracture pressures to estimate the surface pressure at times intermediate the determined fracture pressures.

Description

PATENT APPLICATION ATTORNEY DOCKET NO. SEI-24-03PCT METHOD FOR ESTIMATING WELL SURFACE PRESSURE FROM TUBE WAVES INDUCED IN A WELL BACKGROUND

[0001] This disclosure relates to the field of treatment of subsurface wells used to extract valuable minerals such as hydrocarbons from subsurface formations. More particularly, the present disclosure relates to specific uses for parameters determinable using pressure measurements made during pumping fluid treatments such as hydraulic fracturing treatments used to enhance fluid production rates and total fluid recovery from such wells.

[0002] US Patent Application Publication No. 2023 / 0228185 filed by Dunham et al. discloses a method for using induced tube waves caused by flow rate changes of fluid in a pipe (a well) to determine frictional fluid pressure loss along the pipe and through perforations in the pipe that connect the well hydraulically to formations outside the well. By being able to calculate such frictional fluid pressure losses, it is possible to determine the fluid pressure at the mouth of one or more fractures in the formations. Such pressure is important in conducting and evaluating hydraulic fracture treatment parameters such as fluid density, fluid viscosity, pumping rate, pumping pressure and proppant concentration during pumping.

[0003] Such friction pressure losses may be used to determine further properties of the well and the surrounding formations.

[0004] A parameter of interest in hydraulic fracturing treatment operations is the fluid pressure in various parts of the well distal from the surface while such treatment is being pumped. Pressure can be measured proximate the surface in the well; other pressures such as those in the mouth of a fracture adjacent to the well and pressure along the well during the fracturing procedure may be useful in identifying faults in the well or in the fracture pumping procedure.

[0005] It is desirable to have a method for determining pressure at various places along the well that does not require special sensors.PATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT SUMMARY

[0006] One aspect of the present disclosure is a method for estimating well surface pressure. A method according to this aspect includes measuring pressure in a well during pumping a fracture treatment. For each of at least two changes in flow rate of the pumping, a well pipe friction pressure loss, a perforation friction pressure loss and near wellbore pressure loss are determined using events in the measured pressure induced by each change in flow rate. Fracture pressure is determined at times of each of the at least two changes in flow rate from the measured pressure using the determined well pipe friction pressure loss, determined perforation friction pressure loss and near wellbore pressure loss. Fracture pressure is interpolated between the determined fracture pressures to estimate the surface pressure at times intermediate the determined fracture pressures.

[0007] A non-transitory computer readable medium according to another aspect of the present disclosure includes logic stored thereon capable of causing a programmable computer to perform actions according to the foregoing method.

[0008] Some implementations further comprise determining a difference between the estimated surface pressure and the measured pressure, and using the difference to identify at least one of a change in perforation efficiency, a fault in well plug integrity, perforation erosion in excess of theoretical erosion, presence of a dominant fracture, proppant effect on pipe friction factor and a fault in friction reducer in the pumped fracture treatment.

[0009] In some implementations, the interpolating is linear.

[0010] Some implementations further comprise determining an internal stress shadow from the determined fracture pressure at a first one of the rate changes and a determined fracture pressure at an end of the pumping a fracture treatment.

[0011] Some implementations further comprise comparing the estimated surface pressure to the measured pressure, and determining existence of a leak in the well when the estimated surface pressure exceeds the measured surface pressure.PATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT

[0012] Some implementations further comprise determining an amount of fluid moving through the leak, and correcting an amount of fluid being pumped into a formation outside the well using the determined amount of fluid flowing through the leak.

[0013] Other aspects and possible advantages will be apparent from the description and claims that follow. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] FIG.1 shows example implementations of equipment that may be used to acquire signals usable with a method according to the present disclosure.

[0015] FIG.2 shows a graph of measured pressure in a well compared to predicted pressure in the well.

[0016] FIG. 3 shows a graph of measured pressure in the well compared to predicted pressure in the well in the case of a plug in the well leaking.

[0017] FIG. 4 shows a graph of measured pressure in the well compared to predicted pressure in the well in the case of unexpectedly high erosion of perforations in a well casing.

[0018] FIG. 5 shows a graph of pressure measurements made in a well proximate a wellhead after a change in fluid flow rate so as to induce water hammer.

[0019] FIG. 6 shows a graph of the result of an example of a computer simulation of wellhead pressure response, obtained by solving the water hammer equations.

[0020] FIG.7 shows a graph of a pressure within a well with respect to time after changing fluid flow rate.

[0021] FIG.8 shows a graph of a pressure profile with respect to depth within the well at time t = 2.5 seconds, prior to the tube wave reaching the bottom of the well.

[0022] FIG.9 shows a graph of influence of pipe friction factor on tube wave response.

[0023] FIG.10 shows a graph of influence of perforation friction coefficient on tube wave response.PATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT

[0024] FIGS. 11A and 11B show constrained and unconstrained perforation efficiency calculations indicating a plug leak in a well.

[0025] FIG.12 shows a graph of calculated flow area with respect to time indicating onset of a leak.

[0026] FIGS.13A and 13B show graphs of calculating flow entering a formation through perforations and flow lost to a plug leak in a well.

[0027] FIG.14 shows an example implementation of a computer system that can be used in accordance with the present disclosure. DETAILED DESCRIPTION

[0028] FIG. 1 is a schematic diagram of an example well data acquisition system (“system”) that may be used in some implementations. The system 100 comprises components associated with a well including fluid pump(s) 101, such as hydraulic fracturing fluid pumps or other fluid treatment pumps; sensors such as hydrophones or pressure transducers 102 in fluid pressure communication with the well; a data acquisition and processing apparatus 103 (described in more detail below); a well pipe 104, e.g., a casing or liner disposed in a well drilled through a reservoir formation; a plug or wellbore bottom 106; fracture network 107 in hydraulic communication with the well through perforations 108 made in the well pipe (e.g., casing or liner) 104. A nearby well 109 may be present in the area of interest. One or more water hammer pulses 105 may be generated by the pumps 101, such as by a step change in the rate of pumping, or a pressure pulse may be generated by other means such as a fluid pressure pulse generator. The pulse(s) travel along the well in the form of tube waves. The sensors 102 may be nonintrusive devices such as pressure transducers, accelerometers, and hydrophone(s), any or all of which may be disposed in a location on or near the top of the well (e.g., the wellhead) to measure pressure, pressure time derivative and / or particle motion of fluid in the well continuously before, during, and after pumping of a treatment such as an hydraulic fracture treatment. Characteristics of such data may be analyzed as explained below to obtain parameters suchPATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT as frictional pressure loss along the well and through the perforations also as will be explained in more detail below.

[0029] As will be readily apparent from the disclosure in US Pat. App. Pub. No. 2023 / 0228185 filed by Durham et al. and incorporated herein by reference, each such flow rate change and accompanying pressure change may induce tube waves in the well (104 in FIG. 1). Characteristics of the tube waves may be used, as explained in the ‘185 publication, to determine frictional pressure loss in the well pipe and in the pipe perforations. Thus, during pumping a fracture treatment stage at varying flow rates, it is possible to determine pipe friction and perforation friction, and consequent pressure loss at each of such times over the course of pumping the fracture stage. The measured pressure in the well and the determined frictional pressure losses at such times may be used to calibrate other pressure parameters in the well. Such calibrated pressure parameters may be used to estimate or predict the measured pressure (surface pressure) at times intermediate the times at which the friction parameters and associated pressures are determined. The predicted pressures may be compared to the measured pressures to evaluate certain mechanical properties of the well and whether certain mechanical faults in the well exist.

[0030] Surface pressure in a well during pumping fluid into a formation fracture adjacent to the well is a result of other well pressure parameters, and may be represented by the following equation: ^^^^^^^ ൌ ^^^^^^^^^^ െ ^^ு^^^^ ^ ^^^^^^^^^^ ^ ^^^^^^^^^^ ^ ^^ே^^^^^^ (1)^^ு= hydrostatic pressure of the column of fluid in the well, ^^^^^^= pipe friction pressure loss, ^^^^^^= perforation friction pressure loss, ^^ே^^= near wellbore (NWB) friction pressure loss.

[0031] In a method according to the present disclosure, to calculate the expected surface pressure vs. time (^^^^^^^) it is necessary to calculate the variation of all the foregoingPATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT pressure components with respect to time. The foregoing may be obtained by performing the following actions: 1. perform the method disclosed in the ‘185 publication for each of two or more fluid flow rate changes (“steps”) during pumping a “stage” of the fracture treatment, and calculate the friction pressure loss components at each of these flow rate changes, namely, pipe friction pressure loss, perforation friction pressure loss and NWB friction pressure loss. A more detailed description of the foregoing friction pressure loss determinations follows below with reference to FIGS.5 through 10. 2. The foregoing friction pressure loss components may then be used to calculate the fracture pressure at the time (p, superscript step) of each of the one or more fluid flow rate changes. The foregoing may be calculated using the following equation: ^^^௧^^^^^^ ൌ ^^ ^௧^^ ^ ^ ^௧^^ ^௧^^ ^௧^^^௧^^ௌ^ு െ ^^^^^^ െ ^^^^^^ െ ^^ே^^(2) in which ^^^௧^^= represents the measured (surface)^௧^^ௌ pressure, ^^ு= hydrostatic pressure, which may be obtained as the product of true vertical depth of the well and the fracture fluid slurry density, which may be measured or obtained from the fracture treatment operator, ^^^௧^^^^^^= pipe friction pressure loss obtained as explained above, ^^^௧^^^^^^= perforation friction pressure loss obtained as explained above, and ^^^௧^^ே^^= NWB friction pressure loss obtained as explained above.

[0032] To obtain values of ^^^^^^^^^^ at times intermediate the times of the rate changes, the determined values of ^^^^^^^^^^may be interpolated. In the present example implementation, the interpolation may be based on the assumption that the fracture pressure changes linearly between the times of the flow rate changes. Using such assumption, an interpolated value of ^^^^^^^^^^may be calculated for any time intermediate the times of the flow rate changes, where an explicit value of fracture pressure may be determined as explained above. Once the intermediate values of ^^^^^^^^^^ are determined, it is then possible to calculate the other pressure components change with respect to time during the fracture treatment pumping. The main variables in the foregoing pressure components are slurry density, slurry pumping rate, and the pipe friction factor. The flow rate may bePATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT obtained from the fracture treatment operator or a separate flowmeter placed in the pumping line from the pumping unit (101 in FIG.1). The fluid slurry density and the pipe friction factor variations may be calculated based on the fracture treatment data obtained from the fracture treatment operator and previously calibrated pipe friction functions. This action will provide the following: pipe friction coefficient during the stage ^^^^^^, slurry density during the stage ^^^^^^

[0033] It then being known how the fluid slurry density and pipe friction factor change during the fracture treatment, known equations for pipe friction, perforation friction, NWB friction and hydrostatic pressure may be used to calculate the variation of the foregoing pressure components at any and all times during the fracture treatment. These equations have certain unknown parameters. Frictional pressure loss calculations made at the times of fluid flow rate changes, made as explained above with reference to the ‘185 publication may be used to calculate the unknown parameters in these equations. The foregoing will provide the following parameters: ^^ு^^^^, ^^^^^^^^^^, ^^^^^^^^^^, ^^ே^^^^^^ (3)

[0034] Finally, the calculated variation of the pressure components during the fracture treatment may be used to calculate the expected surface pressure during the treatment at times intermediate the times of the flow rate changes. The following equation may be used: ^^^^^^^ ൌ ^^^^^^^^^^ െ ^^ு^^^^ ^ ^^^^^^^^^^ ^ ^^^^^^^^^^ ^ ^^ே^^^^^^ (4)in an individual stage of a fracture treatment. A high internal stress shadow indicates that fractures within the stage are growing close to each other (high efficiency). A low internal stress shadow determines the bottomhole fractures are far from each other which means some of the fractures are not growing (low efficiency). The internal stress shadow σintshadowcan be calculated using ^^^^^^values calculated above for surface pressure prediction as follows: ^^^^௧^^^ௗ^௪^^^^ ൌ ^^^^^^^^^^ െ ^^^^^^^^^ ൌ 0^ (5)PATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT Mainly the end of the fracture treatment pumping (stage) value may be used: ^^^^ௗ^^௧^^^ௗ^௪ ൌ ^^^^^^൫^^ ൌ ^^^^ௗ ^^ ^௧^^^ ൯ െ ^^^^^^^^^ ൌ 0^ (6)to determine wellbore conditions, for example and without limitation: Change in perforation efficiency; Perforation efficiency is the ratio of the number of perforations communicating fluid to the formations outside the well with respect to the total number of perforations shot for the particular fracture treatment stage. Perforation efficiency can be determined at each pressure step using the above cited method disclosed in the ‘185 publication. After calculating how ppipe, pfrac, pNWB, and pH change with time and with the composition of the fluid injected into the well between the pressure steps, pperf(t) is adjusted between those times to match ps(t) using Eq. (1). The orifice equation relates perforation area to perforation pressure drop, allowing computation of the total flow area (A(t)) consistent with the pperf(t) required to match the surface pressure. Eff(t) is then computed as eff(t) = A(t) / Σ(Aperf(t)*Nperf where Aperf(t) is the predicted area of a perforation still receiving fluid at a time (t) determined from measured areas at steps and the erosion as described above, and Nperf is the total number of perforations shot. Plug integrity issues; Loss of plug integrity causes either a slow or a sudden increase in flow area out of the current stage through the plug and out through previous stage perforations and into previous stage fractures. This in turn results in a drop in rate through each perforation, and thus a drop in the pperfrequired to match ps. Such a drop cannot be explained by erosion and thus can be flagged as a possible plug integrity issue. When such an issue is detected while pumping, it allows decisions to be made related to whether to continue the stage or adjust operational parameters. When these issues occur slowly they cannot be identified from pumping data alone. Casing breach; Similar to a loss of plug integrity, a casing breach increases the flow area out of the well, and consequently decreases pperf. In this case however the fluid is exiting the well within or above the current stage. Excessive perforation erosion; Occasionally, perforations erode much more rapidly than expected. This causes a larger increase in total flow area and a consequent larger pperfPATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT pressure decrease over time required to match ps(t). This develops slowly over time and can affect the entire stage. Identifying this can provide guidance in adjusting pumping parameters during the stage, or to change stage designs for subsequent stages. Presence of a dominant fracture; Similar to an excessive erosion issue, a dominant fracture is usually associated with anomalously large erosion of the fracture that receives the anomalously large flow volume and thus leads to a pperf reduction over time that is larger than expected. This can be observed in FIG.4. Also, in this situation the calculated NWB pressure drop is expected to be significantly smaller due to having a dominant fracture and lower internal stress shadowing between active fractures. Calculating the proppant effect on pipe friction; Proppant loading causes an increase in pipe friction which is a function of proppant volume per volume of total fluid. By using information about proppant volumes and characteristics (for example, grain size or distribution) friction pressure can be adjusted for these effects. Friction reducer issues. Friction reducer is added as a liquid or powder to fracture treatment fluids to reduce pipe friction. The effects of friction reducers are a function of the type of chemical used and the volume or mass per unit volume of the fracture treatment slurry. Over time, corrections for these effects on friction have been developed. If when using a known friction reducer these calculations fail to correct the pipe friction pressure for the known effects, this can be flagged as a friction reducer issue.

[0037] FIG. 2 is a graph comparing predicted surface pressure to measured surface pressure. The internal stress shadow at the end of pumping ^^^^^^௧ௗ^^^ௗ^௪= 350 pounds per square inch (psi) which shows multiple fractures are growing (normal) in the formations adjacent to the well.

[0038] FIG.3 is a graph of measured pressure compared to predicted surface pressure in which it has been determined that the bottom-of-stage well casing plug has leaks; the measured surface pressure is lower than the predicted surface pressure. When there is a plug issue some of the fluid can leak past the plug; the perforations of the prior treatment stage (deeper in the well, if such is the case) can become active. This will cause a rapid drop of the measured pressure. This rapid drop of the measured pressure will result in aPATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT deviation from the predicted pressure. In this example, at a time of around 2000 seconds, the plug started to leak and the measured pressure became much lower than the predicted pressure. Usually, a plug leak causes a fast drop in the measured pressure, but later in the stage the measured pressure comparison to the predicted pressure will look normal.

[0039] FIG.4 shows a graph of measured pressure compared to predicted pressure wherein ^^^^^^௧ௗ^^^ௗ^௪= 0 psi, which indicates mainly that only one fracture is growing, which itself indicates very low perforation cluster efficiency and possibility of a fracture “hit.”

[0040] In the present example, the measured pressure is always lower than predicted surface pressure which indicates extreme erosion of the perforations in the well casing. Erosion of the perforations causes a decrease of the perforation friction and thus causes a decrease of the measured surface pressure with reference to the predicted pressure. In the present example implementation, normal erosion is included in calculation of the predicted surface pressure. Normal erosion may be included in the calculation of surface pressure using, e.g., empirical data from similar fracture pumping operations or laboratory test data concerning perforation erosion as a result of fracture fluid pumping through such perforations. If the erosion is higher than the normal amount based, e.g., on the erosion calculation, the measured pressure will show a faster decline and will have a lower value than the predicted surface pressure.

[0041] A method for predicting, estimating or calculating an expected pressure in a well and comparing it to the measured well pressure may provide indication of mechanical problems with the well and / or indicia of effectiveness of the fracture treatment being pumped.

[0042] The method set forth above with reference to the ‘185 publication, to determine the pipe friction factor f and the perforation friction coefficient kperf using water hammer is more fully set forth below with reference to FIGS.5 through 10. Water hammer propagates as tube waves in a well. Water hammer is generated by rapid changes in flow rate or pressure at the wellhead or elsewhere in the well (see, Paillet & White, 1982; Wylie et al., 1993). Such a method assumes that there is initially steady flow within the well and out through the perforations from fluid injection into a well at the wellhead at rate Q0. ThePATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT injection rate is then rapidly changed, e.g., decreased by some amount and then held constant at the changed rate. The flow rate change, if rapid enough, generates water hammer oscillations, which appear in wellhead pressure measurements such as the data shown in FIG.5.

[0043] Water hammer can be modeled using computer simulations that solve the equations governing tube wave propagation (see, Wylie et al., 1993): ఘ డொ డ^ ସఛ ^ డ௧ ^ డ௫ൌ െ^ , (7)where x is measured depth along the well, t is time, Q(x,t) is volumetric flow rate, and p(x,t) is the fluid pressure minus hydrostatic pressure. Initial conditions on Q(x,0) and p(x,0) as well as two boundary conditions, one at x=0 and a second at x=L (defined below as the bottom of the well), are also required. The initial conditions correspond to steady flow at the injection rate: ^^^^^, 0^ ൌ ^^^, (9)^^, 0 ൌ ^^଼^ఘ మ ^^^ ^ொబ^ െ^^, (10) where p0 is the isspecified as a (decreased) by an amount ΔQ, such that ^^^0, ^^^ ൌ ^^^ െ Δ^^^^^^^^, (11)where H(t) is the unit step function. It is also possible that the change (decrease) in flow rate occurs over a short, but finite time, in which case H(t) would correspond to a smoothed- out version of the unit step function.

[0044] The well connects to fracture(s) through one or more perforation clusters. When the wavelengths of tube waves are large compared to the spacing between perforation clusters, the pressure within the well at the entry to each cluster is approximately the same. This allows the collective response of all clusters to be combined into a composite object that isPATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT placed at x=L. It is assumed that the well is hydraulically sealed beyond this, for example by a plug or packer. The location x=L is referred to as the bottom of the well throughout this discussion. Assuming identical clusters, each connecting to one fracture, mass and momentum conservation lead to the following boundary condition at x=L: ^^^^^, ^^^ ൌ ^^^^^^^^^^ ^ ^^^^^^^ െ ^^^௬ௗ , ^^ ൌொ^^,௧^ ே^. (12)

[0045] fractures), pf is the pressure at the mouth of the fracture, phyd is hydrostatic pressure, and pperf(q)=k’perfq2is the pressure loss from perforation friction.

[0046] Any appropriate model for the fracture response can be used to determine pf. When flow rate into a fracture is decreased, following a period of fracture growth, the fracture pressure decays in response to leak-off of fluids, additional fracture extension, and equilibration of pressure within the fracture. The pressure decay can be captured approximately with an evolution equation of the form: ௗ^^ ൌ െ^^ି^^^, (13)time scale Teq. In the method described below, pfcan be treated as a constant if the decay is negligible, or it can be modeled by the equation above or a more sophisticated description of fracture pressure evolution, which may also depend on the flow rate into each fracture, q.

[0047] FIG.6 shows an example of a computer simulation of wellhead pressure response, obtained by solving the water hammer equations, thereby illustrating that the simulation can reproduce the main features observed in the data. Parameter values in this simulation were not chosen or adjusted to match the data in FIG. 5, so an exact agreement is not expected. Parameter values are fluid density ρ = 62.4 lb / ft3, tube wave speed c = 5000 ft / s, well length L = 10,000 ft, friction factor f = 2^10-3, perforation friction coefficient kperf= 0.0064 psi / (bbl / min)2, initial injection rate Q0 = 80 bbl / min, and drop in injection rate ΔQ = 20 bbl / min, and number of clusters Nc= 4. The drop in injection rate is smoothed outPATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT over approximately 0.4 s. FIG.6 and FIG.7, which is a zoomed-in version of part of FIG. 6, show three features of water hammer that will be utilized in this method: 301 and 401, respectively, are the pressure drop at the wellhead that occurs immediately when the injection rate is decreased, 302 and 402, respectively, are an additional depressurization at the wellhead that appears as downward sloping ramping in wellhead pressure, and 303 and 403, respectively, are the first tube wave reflection from the bottom of the well.

[0048] The scientific basis of the method explaining these water hammer features follows. The decrease in injection rate by ΔQ causes an immediate decrease in pressure at the wellhead, Δp1, associated with the generation of a tube wave that propagates down the well. The pressure change is proportional to the change in injection rate (see, Paillet & White, 1982; Wylie et al., 1993): Δ^^^ ൌ ^^்Δ^^, (14)where the proportionality constant is the tube wave hydraulic impedance, ^^ ൌ ఘ^ ் ^ . (15)The above relationship between pressure change and rate change holds only during rapid rate changes, prior to the tube wave propagating too far from the wellhead that it experiences significant attenuation from pipe friction.

[0049] In the absence of pipe friction or other energy loss mechanisms, a tube wave propagates without attenuation down the well, creating a step drop in pressure equal to Δp1and causing the fluid to decelerate from O0 to Q0-ΔQ at all points within the well as the tube wave passes. However, attenuation from pipe friction causes the pressure drop carried by the tube wave to decrease. Because this pressure drop is smaller than Δp1, the flow rate is not fully decreased to Q0-ΔQ. The resulting pressure and flow rate profiles within the well at time t = 2.5 s, prior to the tube wave reaching the bottom of the well, are shown in FIG. 8. The direction of tube wave propagation is marked with the arrows labeled “propagation.”PATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT

[0050] The pressure decrease at the wellhead is now larger than Δp1. The additional depressurization is a consequence of the flow rate behind the tube wave varying along the well. For example, the flow rate is 61 bbl / min at x = 2000 ft whereas flow rate at the wellhead x=0 is Q0-ΔQ = 60 bbl / min. Because the flow rate increases with distance from the wellhead, fluid in this region must expand to satisfy conservation of fluid mass. The expansion of the fluid is accompanied by a decrease in pressure. This depressurization, which can ultimately be attributed to pipe friction following the logic explained above, is the ramp-like feature labeled 302 in FIG. 6 and 402 in FIG. 7. It follows that different values of pipe friction factor f will lead to different slopes of this depressurization ramp. The limiting case of no friction (f=0) will have zero slope, and the slope will increase (down) as f increases. FIG.9 shows simulations for three values of f, illustrating how the slope changes with f.

[0051] To quantify perforation friction, it is necessary to use information contained within tube waves that are reflected from the bottom of the well. The first reflection is shown at 303 in FIG. 6 and 403 in FIG. 7. The reflection of small-amplitude tube waves from the bottom of the well and associated perforations and fractures is determined by the reflection coefficient (see, Paillet & White, 1982; Wylie et al., 1993; Dunham et al., 2017). ^^ ൌ^ି^^(16) where ZT is theimpedance of the composite object (perforations, fractures, plug) that acts as a reflector at the bottom of the well. The perforations and fractures are connected in series, so their hydraulic impedances are added to get the total hydraulic impedance. At sufficiently high background flow rate, the perforation impedance is greater than the fracture impedance. To simplify the discussion to follow, the fracture impedance is assumed to be negligible in comparison to the perforation impedance. The plug is assumed to be rigid and perfectly sealing, so it does not contribute to the impedance of the composite reflector. As before, assume that there are Ncidentical clusters of perforations, all clusters experience the same pressure within the well and with the total flow rate divided uniformly across them. This means the clusters are connected in parallel, such that the hydraulic impedance of thePATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT composite reflector is Z = Zc / Nc, where Zcis the hydraulic impedance of one cluster (see, Dunham et al., 2017). When the tube waves induce a sufficiently small variation in flow rate q through one cluster of perforations, the hydraulic impedance of that perforation cluster can be obtained by linearizing the perforation friction expression to quantify the ratio of pressure change to flow rate change: ௗ^ ^^^ൌ^^^^ௗ^ൌ 2^^^ᇱ^^^^^. (17)

[0052] ^^ ൌଶ^^ᇲ^^^ொ / ே^మି^^ଶ^^^^^ொି^ ᇲ ^^^ൌ^(18)

[0053] be evaluated at the initial flow rate Q0. For tube waves whose amplitude is too large to justify the use of a linearized theory of wave reflection, computer simulations can be performed to predict the wellhead pressure response. In either case, the reflected wave amplitude is influenced by the perforation friction coefficient kperf. FIG.10 shows simulations for three values of kperf, illustrating how the amplitude of the reflected wave changes with kperf.

[0054] The foregoing described method according to the present disclosure may be used in some implementations to detect and characterize leaks in a plug disposed in the well at the bottom of a zone undergoing fracture treatment. The characterization of the leak may be used to correct the calculated perforation efficiency and / or to guide the well operator as to possible corrections to the fracture treatment procedure.

[0055] To identify the effect of increased flow during pumping a fracture treatment, the predicted surface pressure (PSP), determined as explained above, may be compared to the measured treatment pressure. If the PSP suddenly rises above the measured treatment pressure, this is an indication of a sudden increase in flow area. Such sudden increase in flow area is commonly due to a leak in a well plug disposed in the well below the perforations exposed for the particular treatment stage. An example increase in flow area based on comparing PSP and measured pressure is shown in the graph FIG. 11A. ThisPATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT behavior can also be observed in an unconstrained perforation efficiency graph in FIG. 11B.

[0056] Once the point in time of increased flow area (leak) is identified, the unconstrained perforation efficiency may be used to calculate the effective flow area of the leak using the formula: ^^ ൌ ^ ௨^^^^^௧^^^^^ௗ^^^௪ ^^^^^^^^^^^^^^^^^^^^^^^ ∗ ^^^^^^ ∗ ^^^^^^ (19)efficiency, Aperf = area of a single perforation and Nperf= number of perforations in the stage. The calculated flow area is shown in FIG.12. A sudden increase in Aflow at the leak point LP confirms the issue. This increase can be quantified using a called Cleak,defined as: ^^ ^^^௪ ^^^^ ^^௧^^ ௧^^ ^^^^ ௗ^௧^^௧ ^^ൌ ^^^ି^^^௪ ^^^^ ^^^^^^ ௧^^ ^^^^ ௗ^௧^^௧^^^ ^^^^^௪ ^^^^ ^^௧^^ ௧^^ ^^^௨^ (20)After identifying the leak and calculating the effective flow area, the next action is to model how the leak area and the current stage flow area evolve. Possible assumptions include: 1. Constant Ratio Assumption: The ratio of leak area to the treatment stage flow area remains constant (both areas change proportionally). 2. Constant Leak Area Assumption: The leak area remains fixed, and any change in total flow area is due to the stage. 3. Constant Stage Area Assumption: The stage flow area remains constant, and the leak area changes. 4. Pressure-Driven Assumption: Changes in leak behavior are driven by pressure changes in the volume receiving leaked fluid.

[0058] Once a method is selected for modeling flow areas after the leak, it is possible to calculate the volume of fluid that stays in the current stage, and the volume of fluid lost toPATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT the leak. An example of the foregoing calculation is shown in FIGS.13A and 13B, using Assumption 1 to distribute the fluid between the stage and the leak after the leak occurs.

[0059] After estimating the fluid distribution, the perforation efficiency of the current stage is updated to account for the leak. In the example, the initial (uncorrected) end-of-stage perforation efficiency was 100% (as calculated by SAFA). After accounting for the leak, the updated efficiency was corrected to 72%.

[0060] Up to this point, the method has focused on calculating leak effects at the end of the fracture treatment stage. However, if detected during treatment, it is possible to use Assumption 1 or 2 (from Step 3) to predict how much fluid will be lost going forward. This provides the well operator with real-time decision-making power: If the leak loss is small, the treatment may continue as planned; if the leak loss is significant, the well operator may choose to halt the stage and take corrective action., for example, setting another plug in the well.

[0061] It will be appreciated that any or all aspects of the foregoing methods may be implemented in any form of programmable computer which is capable of reading a non- transitory computer readable medium having stored on it logic operable to cause the programmable computer to implement the actions described above for determining pressure. Thus, another aspect of this disclosure is a non-transitory computer readable medium having such instructions stored on it. FIG.14 shows an example computing system 200 in accordance with some implementations. Actions described above with reference to example implementations may be carried out on a computer or computer system, wherein pressure measurements made as described may be entered into the computer or computer system and processed in the computer or computer system as explained above. The computing system 200 may be an individual computer system 201A or an arrangement of distributed computer systems. The individual computer system 201A may include one or more analysis modules 202 that may be configured to perform various tasks and controls according to some implementations, such as the tasks explained with reference to FIGS.2- 13B. To perform these various tasks, the analysis module 202 may operate independently or in coordination with one or more processors 204, which may be connected to one orPATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT more storage media 206. A display device 205 such as a graphic user interface of any known type may be in signal communication with the processor 204 to enable user entry of commands and / or data and to display results of execution of a set of instructions according to the present disclosure.

[0062] The processor(s) 204 may also be connected to a network interface 208 to allow the individual computer system 201A to communicate over a data network 210 with sensors, one or more additional individual computer systems and / or computing systems, such as 201B, 201C, and / or 201D. Note that computer systems 201B, 201C and / or 201D may or may not share the same architecture as computer system 201A, and may be located in different physical locations, for example, computer systems 201A and 201B may be at a well drilling location, while in communication with one or more computer systems such as 201C and / or 201D that may be located in one or more data centers on shore, aboard ships, and / or located in varying countries on different continents.

[0063] A processor may include, without limitation, a microprocessor, microcontroller, processor module or subsystem, programmable integrated circuit, programmable gate array, or another control or computing device.

[0064] The storage media 206 that captures data in a tangible medium may be implemented as one or more computer-readable or machine-readable storage media. Note that while in the example implementation of FIG.14 the storage media 206 are shown as being disposed within the individual computer system 201A, in some implementations, the storage media 206 may be distributed within and / or across multiple internal and / or external enclosures of the individual computing system 201A and / or additional computing systems (e.g., 201B, 201C, 201D), or over a network (“cloud”). Storage media 206 may include, without limitation, one or more different forms of memory including semiconductor memory devices such as dynamic or static random access memories (DRAMs or SRAMs), erasable and programmable read-only memories (EPROMs), electrically erasable and programmable read-only memories (EEPROMs) and flash memories; magnetic disks such as fixed, floppy and removable disks; other magnetic media including tape; optical media such as compact disks (CDs) or digital video disks (DVDs); or other types of storagePATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT devices. Note that computer instructions to cause any individual computer system or a computing system to perform the tasks described above may be provided on one computer- readable or machine-readable storage medium, or may be provided on multiple computer- readable or machine-readable storage media distributed in a multiple component computing system having one or more nodes. Such computer-readable or machine- readable storage medium or media may be considered to be part of an article (or article of manufacture). An article or article of manufacture can refer to any manufactured single component or multiple components. The storage medium or media can be located either in the machine running the machine-readable instructions or located at a remote site from which machine-readable instructions can be downloaded over a network for execution.

[0065] It should be appreciated that computing system 200 is only one example of a computing system, and that any other implementation of a computing system may have more or fewer components than shown, may combine additional components not shown in the example implementation of FIG. 14, and / or the computing system 200 may have a different configuration or arrangement of the components are shown in FIG. 14. The various components shown in FIG. 14 may be implemented in hardware, software, or a combination of both hardware and software, including one or more signal processing and / or application specific integrated circuits.

[0066] Further, the acts of the processing methods described above may be implemented by running one or more functional modules in information processing apparatus such as general purpose processors or application specific chips, such as ASICs, FPGAs, PLDs, or other appropriate devices. These modules, combinations of these modules, and / or their combination with general hardware are all included within the scope of the present disclosure.

[0067] A method for determining pressure according to the present disclosure may eliminate the need for sensors outside the well pipe, there being only needed a pressure sensor that may be conveniently located, e.g., near the surface or at surface connected to fluid flow lines into the well.PATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT

[0068] In light of the principles and example implementations described and illustrated herein, it will be recognized that the example implementations can be modified in arrangement and detail without departing from such principles. The foregoing discussion has focused on specific implementations, but other configurations are also contemplated. In particular, even though expressions such as in “an implementation," or the like are used herein, these phrases are meant to generally reference implementation possibilities, and are not intended to limit the disclosure to particular implementation configurations. As used herein, these terms may reference the same or different implementations that are combinable into other implementations. As a rule, any implementation referenced herein is freely combinable with any one or more of the other implementations referenced herein, and any number of features of different implementations are combinable with one another, unless indicated otherwise. Although only a few examples have been described in detail above, those skilled in the art will readily appreciate that many modifications are possible within the scope of the described examples. Accordingly, all such modifications are intended to be included within the scope of this disclosure as defined in the following claims. References cited in this disclosure Dunham, E. M., Harris, J. M., Zhang, J., Quan, Y., & Mace, K. (2017). Hydraulic fracture conductivity inferred from tube wave reflections. SEG International Exposition and Annual Meeting. SEG-2017-17664595. Paillet, F. L., & White, J. E. (1982). Acoustic modes of propagation in the borehole and their relationship to rock properties. Geophysics, 47(8), 1215-1228. Wylie, E. B., Streeter, V. L., & Suo, L. (1993). Fluid transients in systems. Englewood Cliffs, NJ: Prentice Hall.

Claims

PATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT CLAIMS What is claimed is:

1. A method for estimating well surface pressure, comprising: measuring pressure in a well during pumping a fracture treatment; for each of at least two changes in flow rate of the pumping, determining a well pipe friction pressure loss, a perforation friction pressure loss and near wellbore pressure loss using events in the measured pressure induced by the at least one change in flow rate; determining fracture pressure at times of each of the at least two changes in flow rate from the measured pressure, determined well pipe friction pressure loss, determined perforation friction pressure loss and near wellbore pressure loss; and interpolating fracture pressure between the determined fracture pressures to estimate a surface pressure in the well at times intermediate the determined fracture pressures.

2. The method of claim 1 further comprising determining a difference between the estimated surface pressure and the measured pressure, and using the difference to identify at least one of a change in perforation efficiency, a fault in well plug integrity, perforation erosion in excess of theoretical erosion, presence of a dominant fracture, proppant effect on pipe friction factor and a fault in friction reducer in the pumped fracture treatment.

3. The method of claim 1 wherein the interpolating is linear.

4. The method of claim 1 further comprising determining an internal stress shadow from the determined fracture pressure at a first one of the rate changes and a determined fracture pressure at an end of the pumping a fracture treatment.

5. The method of claim 1, further comprising comparing the estimated surface pressure to the measured pressure, and determining existence of a leak in the well when the estimated surface pressure exceeds the measured surface pressure.PATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT 6. The method of claim 5 further comprising determining an amount of fluid moving through the leak, and correcting an amount of fluid being pumped into a formation outside the well using the determined amount of fluid flowing through the leak.

7. A non-transitory computer readable medium having stored thereon logic operable to cause a programmable computer to perform actions, comprising: accepting as input to the computer measurements of pressure in a well made during pumping a fracture treatment; for each of at least two changes in flow rate of the pumping, determining a well pipe friction pressure loss, a perforation friction pressure loss and near wellbore pressure loss using events in the measured pressure induced by the at least one change in flow rate; determining fracture pressure at times of each of the at least two changes in flow rate from the measured pressure, determined well pipe friction pressure loss, determined perforation friction pressure loss and near wellbore pressure loss; and interpolating fracture pressure between the determined fracture pressures to estimate the surface pressure at times intermediate the determined fracture pressures.

8. The computer readable medium of claim 7 further comprising determining a difference between the estimated surface pressure and the measured pressure, and using the difference to identify at least one of a change in perforation efficiency, a fault in well plug integrity, perforation erosion in excess of theoretical erosion, presence of a dominant fracture, proppant effect on pipe friction factor and a fault in friction reducer in the pumped fracture treatment.

9. The computer readable medium of claim 7 wherein the interpolating is linear.

10. The computer readable medium of claim 7 further comprising logic operable to cause the programmable computer to perform determining an internal stress shadow from the determined fracture pressure at a first one of the rate changes and a determined fracture pressure at an end of the pumping a fracture treatment.PATENT APPLICATION ATTY DOCKET NO. SEI-24-03PCT 11. The computer readable medium of claim 7, further comprising logic operable to cause the programmable computer to perform comparing the estimated surface pressure to the measured pressure, and determining existence of a leak in the well when the estimated surface pressure exceeds the measured surface pressure.

12. The computer readable medium of claim 11 further comprising logic operable to cause the programmable computer to perform determining an amount of fluid moving through the leak, and correcting an amount of fluid being pumped into a formation outside the well using the determined amount of fluid flowing through the leak.

Images