Patent Application: US-201313869269-A

Abstract:
a very simple model has been presented which is able to reproduce slug flow from the instability of a flow with average hold - up and slip . the disclosure demonstrates that slug flow may be modeled as two different , stable solutions to the multiphase flow which coexist at different points in the line , moving with a celerity of u g . by using a white - noise inlet condition which preserves the average hold - up in the pipeline , a series of stable slug and stratified regions can be created without any need to resort to a lagrangian slug tracking scheme . a quite good fit to field data was obtained with minimal effort by adjusting the slip relation . at present , the model merely demonstrates a potential , very attractive , flexible , and easy - to - implement alternative to lagrangian slug tracking .

Description:
turning now to the detailed description of the preferred arrangement or arrangements of the present invention , it should be understood that the inventive features and concepts may be manifested in other arrangements and that the scope of the invention is not limited to the embodiments described or illustrated . the scope of the invention is intended only to be limited by the scope of the claims that follow . flow assurance can be defined as any issue arising in the production system between the reservoir and the central facility which has the potential to impede production . this could include : ‘ flow assurance ’ is in fact a well known term covering all the issues mentioned above . ‘ determining flow assurance ’ as used herein thus means determining the likelihood of occurrence of any one of a number of undesirable issues connected with multiphase pipeline flow and / or estimating or calculating the extent of any such issue . the issues involved are well known in the oil and gas technical field , but include specifically the issues mentioned in the previous paragraph . in other words , ‘ determining flow assurance ’ can mean determining the likelihood ( risk ) of any of the following and / or estimating or calculating the extent of any of the following : pipeline or other corrosion or erosion , hydrate formation or paraffin formation ( e . g . resulting from phase change ), sand deposition , generation or deposition of production fines , hydrate crystals or wax particles , precipitation of scale or asphaltenes , or combinations of any of these issues ( e . g . erosion due to deposited sand .) in addition , there are flow assurance issues that result explicitly from multiphase flow itself these would include : one of the defining characteristics of multiphase flow is the presence of a definitive flow regime , understood as the large - scale variation in the physical distribution of the flowing gas and liquid phases in a flow conduit . multiphase pipe flow is generally considered to fall into one of four basic regimes : stratified flow : a continuous liquid stream flowing at the bottom of the pipe , with a continuous stream of gas flowing over ; stratified - wavy flow is sometimes differentiated from stratified smooth flow . slug flow : stratified flow , punctuated by slugs of highly turbulent liquid . plug flow is a form of slug flow that occurs at lower velocities . annular flow : a thin liquid film adhering to the pipe wall , and a gas stream containing entrained liquid droplets . bubble flow : a continuous liquid flow with entrained gas bubbles . u r = fixed - frame linear liquid velocity in the slug body ( m / s ) u r ′= moving - frame linear liquid velocity in the slug body ( m / s ) u sb = fixed frame superficial liquid velocity in the slug body ( m / s ) u sb ′= moving frame superficial liquid velocity in the slug body ( m / s ) u ss ′= moving - frame superficial liquid velocity in the stratified region ( m / s ) in one embodiment , the disclosure provides a novel approach to slug flow modeling where the fundamental transient nature of hydrodynamic slug flow is accounted for in the model . the momentum equations in the transient multiphase model are greatly simplified by using a single momentum equation with inertial terms removed for the gas - liquid mixture . the momentum equation is used to obtain the pressure profile , which is — in turn — used to find the mixture velocity . it is critical that the multiphase mixture model be regime - free , with a single , correlational approach , along the lines of a drift - flux model , used to obtain the hold - up and pressure drop across the board . using a single momentum equation with multiple energy equations provides an accurate and rapid model with reduced computational requirements . from a flow assurance perspective , it is important to track the temperatures of each phase . temperature differences between phases can deviate significantly from mixture energy models , particularly during transient operations such as shutdown / restart . computing speeds have reached a point now where it is practical and effective to use a pure compositional tracking approach in transient multiphase flow . while look - up tables for fluid properties were — at one time — required to give reasonable simulation times , a more accurate assessment of flow assurance and fluid properties can be obtained by using a more detailed model . in order to formulate a model for slug flow , we must first develop a ‘ point model ’ for an individual pipeline segment , or computational cell , in a pipeline . this pipeline segment must then be joined with other pipeline segments upstream and downstream of it to form a ‘ steady - state ’ model for the pipeline . lastly , these steady - state solutions must be implemented into a transient scheme ; a proper model of hydrodynamic slug flow absolutely requires that slugging not be treated as a pseudo - steady - state , but as an inherently transient phenomenon . first , we must define some terms . the gas and liquid hold - ups are defined by : h g ≡ a g / a ; h l ≡ a l / a ; h g + h l = 1 the superficial gas and liquid velocities are defined by : u sg ≡ q g / a ; u sl ≡ q l / a ; u m ≡ u sg + u sl the velocity difference , or slip velocity , between the gas and the liquid is defined as u s ≡ u g − u l ≡ u sg / h g − u sl / h l ≡ u sg /( 1 − h l )− u sl / h l u s h l 2 +( u m − u s ) h l − u sl ≡ 0 r ( u l )≡ u s h l 2 +( u m − u s ) h l − u sl the zeros of this hold - up function f ( h l *)= 0 are solutions to the holdup equation , or the ‘ steady - state ’ solution to the multiphase flow . for a constant slip velocity u s , the hold - up equation is quadratic and can easily be solved analytically . if the slip velocity is a function of the hold - up , i . e ., it is quite likely that there is no analytic solution to this equation , and it must be solved numerically . regardless of the functional form of u s ( h l ), if the superficial gas and liquid velocities are both positive , then there is always at least one physically - realizable ( i . e ., 0 ≦ h l ≦ 1 ) solution to the equation . this can be seen by examining f ( h l ) at the limits of h l = 0 and h l = 1 ; since f ( h l = 0 )=− u sl & lt ; 0 , and f ( h l = 1 )= u sg & gt ; 0 , there must be some point 0 & lt ; h l *& lt ; 1 which satisfies the equation f ( h l *)= 0 . the existence of at least one solution for 0 & lt ; h l & lt ; 1 can be demonstrated mathematically and hold - up h l * provides a steady - state solution for the hold - up equation . with simpler , regime - free transient multiphase models there is the possibility of folding other flow assurance models directly into the transient simulator without negatively impacting required simulation time . one example is combining a slug capturing model with a kinetic hydrate formation model . the following examples of certain embodiments of the invention are given . each example is provided by way of explanation of the invention , one of many embodiments of the invention , and the following examples should not be read to limit , or define , the scope of the invention . the stability of these ‘ steady - state ’ solutions can be examined through the application of mass conservation for a particular point in the pipeline . the mass conservation equation for the liquid phase is given by : d ( ρ l δz · a · h l )/ dt =( ρ l · a · u sl ) in −( ρ l · a · u sl ) out if the liquid density , pipe cross - sectional area , and section length are constant , this simplifies to a volume conservation equation : δ z · d ( h l )/ dt =( u sl ) in −( u sl ) out consider a single cell , with a constant inlet superficial velocity ( u sl ) in , with ( u sl ) out as a function of the hold - up in that cell : u sl ( h l )≡ u s ( h l ) h l 2 +( u m − u s ( h l )) h l note that the hold - up equation is used not to determine the hold - up from the superficial velocities and the slip velocity ; now the superficial velocity is determined from the hold - up , the slip velocity , and the mixture velocity . the volume conservation equation can be written as : δ z · d ( h l )/ dt = u sl −[ hd s ( h l ) h l 2 +( u m − u s ( h l )) h l ]=− f ( h l ) let h l * be a zero of f ( h l ) and therefore a solution to the hold - up equation . then , if [ d ( f ( h l ))/ dh l ] hl = hl * & gt ; 0 the steady - state solution h l * is stable , and the transient equation will migrate to the steady - state hold - up h l * and remain there . likewise , if [ d ( f ( h l ))/ dh l ] hl = hl * & lt ; 0 the steady - state solution h l * is unstable , and the transient equation will migrate away from the steady - state hold - up . from a graphical point of view , if f ( h l ) crosses the hold - up axis ( h l = h l *) with a positive slope , then the solution is stable ; if f ( h l ) crosses the hold - up axis with a negative slope , then the solution is unstable . of course , it is entirely possible that the hold - up function f ( h l ) can cross the hold - up axis at more than one point , i . e ., f ( h l ) can have more than 1 physically - realizable solution . in fact , if f ( h l ) is a continuous function of h l , any odd solutions is at least topologically possible ( even numbers of crossings are not possible if u sl , u sg & gt ; 0 ). if the slip velocity u s is constant , then f ( h l ) is quadratic in hold - up . since a quadratic equation can only have , at most — 2 real roots , there can only be a single crossing between 0 & lt ; h l & lt ; 1 , since f ( h l = 0 )& lt ; 0 and f ( h l = 1 )& gt ; 0 . hydrodynamic slug flow is characterized by high - hold - up slugs of liquid with little slip between the gas and liquid phases separated by low - hold - up stratified regions characterized by high slip between the phases . the gas bubble in the separated region travels at a characteristic speed u g which can be related to mixture velocity via a ‘ drift - flux ’ relation : u s =( u g − u m )/ h l =[( c o − 1 ) u m + u o ]/ h l since u m is constant in incompressible slug flow , this has the form where c df is a constant for constant u m . the hold - up function then becomes f ( h l )=( u m + c df ) h l −( u sl + c df ) note that the drift - flux model exhibits a fundamentally wrong behavior in the limit of h l = 0 , as f ( h l = 0 )=− u sl , and should not be used at low u sl s . the drift - flux model is , however , stable for all steady - state hold - ups ( as u m + c df & gt ; 0 everywhere ). this finding also implies that detailed transient modeling of slug flow , including growth , merging , and disappearance of slugs using the drift flux approach under these conditions is simply not possible . thus , the drift - flux model , while giving an excellent average picture of the average hold - up , is not the proper starting point for any kind of transient analysis of hydrodynamic slug flow . let us consider the following form for the slip velocity u s ( h l ): introducing this into the hold - up function f ( h l ), we obtain the following : f ( h l )= a · h l 3 +( b − a )· h l 2 +( u m − b )· h l − u sl fig1 gives the form of this equation for a specific a , b , u sl , and u sg . note that this cubic equation has several very interesting features : at low u sl , there is only one low - hold - up , high - slip solution , which is stable ; as u sl increases above a critical threshold , a two additional hold - up solutions appear — one intermediate hold - up which is unstable , and another high hold - up , low - slip solution which is stable ; as u sl increases still further , the low - and intermediate hold - up solutions disappear , leaving only the single high hold - up , low - slip solution . it is the thesis of this paper that the low hold - up solution which appears at low u sl corresponds to stratified flow , and the high hold - up solution that occurs at high u sl corresponds to bubble flow . at intermediate u sl , there is a possibility that the low - and high - hold - up solutions will both coexist in the pipeline at the same time ; this is hydrodynamic slug flow . it should be pointed out , however , that if we invert the f ( h l ) equation to find u sl ( h l ) as before , we obtain u sl ( h l )= a · h l 3 +( b − a )· h l 2 +( u m − b )· h l let us number the three solution hold - ups which satisfy f ( h l )= 0 as h l1 *, h l2 *, and h l3 *. ( where h l1 * & lt ; h l2 *& lt ; h l3 *) we obtain , at steady - state : this is clearly not the case in slug flow , where the superficial liquid velocity in the slug body is considerably higher than that in the stratified region . this unphysical result must be rectified before we can continue . although the cubic form of f ( h l ) has many appealing properties , there is one last step that must be addressed in order to formulate our transient slug model . while it is true that the superficial velocities in slug flow are not all equal in a reference frame that is fixed with the pipe , in a moving reference frame they can — in fact — be made to be equal . consider fig2 , which shows slug flow in a fixed frame , and also from a reference frame which moves at the velocity of the gas bubble in the stratified region , u g . the linear velocities in the new reference frame are : obviously , the hold - ups are not a function of reference frame ; however , the superficial velocities are . this can be seen by the following : in the moving reference frame , u sbu ′= u sst ′= u m ′, by definition . the gas velocity u g can be calculated from a drift - flux formulation . let us set : thus , u o is determined from a drift - flux type relation ; u o is then used to obtain both u g and u m ′. once u g is known , we can calculate the superficial velocities in the fixed - frame stratified and bubble regions by : lastly , it should be mentioned that the slip velocity , like the hold - up , is frame - invariant . in one embodiment , a simple , time - dependent , two - phase ( gas - liquid ) hydrodynamic slug flow model is described which is capable of producing hydrodynamic slugging from first principles . the slug model correctly predicts transition from stratified to slug flow via an interface instability . the model is capable of producing slug lengths and frequencies , as well as slug void fraction , from first principles . also , flow regime transitions are effectively captured . the model also captures slug initiation on uphill pipe sections and slug decay on downhill sections . because of its simplicity , the model runs extremely fast compared to other multiphase flow simulators the hydrodynamic slug flow model can now be modeled as a function of two competing processes : slug formation occurs naturally when conditions favorable for slug formation exist . this includes the existence of at least two stable solutions to the hold - up function f ( h l ). slug propagation is a wave - like phenomenon , with wave celerity ( slug speed ) equal to the gas bubble velocity u g . in a fixed frame of reference , we have : δ z · d ( h l )/ dt =( u sl ) in −( u sl ) out the above equation set must be upwinded to assure stability of the solution , with the upwinding of the u sl terms ( which move from right ). it was found that a third - order upwind scheme for the right - hand - side terms is required to combat the numerical diffusion which would otherwise destroy the slugs ( courant , et al ., 1952 ). finally , the pipeline is simulated as either a once - through or with periodic boundary conditions , i . e ., whatever leaves out of the right - hand side of the pipeline is reintroduced to the left - hand side . a drift - flux model is employed to calculate u o ( we take c o = 1 ) u o is given by : u o = 0 . 4 ·( gd ) 1 / 2 ·(( ρ l − ρ g )/ ρ l ) 1 / 2 for a 20 - inch oil pipeline at typical operating conditions , u o ˜ 1 m / s . once u o is known , the gas velocity and the the average hold - up can be calculated via the drift - flux model : h l =( u sl + u o )/( u m + u o )= 0 . 4 this drift - flux hold - up is used to initialize the hold - up in the numerical simulation . the slip relation used is : u s ( h l )=− 4 · h l + 4 )& gt ; 0 for all h l ) here the slip velocity is taken as positive for all hold - ups . fig3 shows the hold - up in a 500 m line as a function of distance along the pipeline as a function of time , using the input data given above . in this case , a periodic boundary condition is imposed , such that whatever fluid exits at the right - hand side is reintroduced at the left - hand side . the simulation is initialized at h l = 0 . 4 , and the interface is perturbed with a very small perturbation ( 0 . 41 for a single computational cell was used in this case ). note that even a very small perturbation will eventually grow into a hydrodynamic slug . owing to the third - order upwinding scheme , the slug is very stable , and once it reaches maximum size will continue to move through the domain with no apparent numerical diffusion . the hold - up in the stratified region between slugs is also maintained at h l = 0 . 27 . fig4 gives the behavior of a 1000 m pipeline , again using the same input data as for fig3 . this simulation is now run as a once - through , so that slugs are born near the inlet and grow as they move through the line from left to right . a white - noise random signal of hold - ups , varying between h l =[ 0 , 0 . 8 ] is used at the inlet . as the slugs reach the end of the pipeline , the hydrodynamic slugs have resolved themselves into a more - or - less stable pattern . the longer slugs are around 10 m long (˜ 20 diameters ), or so . fig5 gives the behavior of the 1000 m pipeline , using fig4 as a starting point , and continuing the simulation with periodic boundary conditions . this is meant to simulate an infinitely - long pipeline . the slugs have now completed their growth , with the longest slugs having lengths of ˜ 40 diameters . finally , fig6 shows the impact of the slip equation u s ( h l ). we have restarted the simulation with : u s ( h l )= 4 · h l − 4 (& lt ; 0 for all h l ) now the slugs disappear and drop back into stratified flow . thus , this simple model could potentially be used to model the decay and death of slugs down pipeline declines . all results presented so far were for horizontal pipeline . of course , flow regime , slip velocity , and hold - up are also a function of angle . a very simple model for inclination angle is postulated of the form : so an increase in inclination angle above horizontal results in both a change in f ′( h l ) and an increased hold - up . here : using the same slip relation as figure x - 7 with an additional term to account for angle , only selective hold - ups lead to instability as demonstrate in fig1 . this is a once - through model , in that whatever enters the line at the inlet exits from the pipeline outlet . there is a 1 - degree inclined section from 100 to 250 m , with a 100 m horizontal section at the inlet and a 250 m horizontal section at the outlet . one can see from the simulation that there are no slugs in the initial stratified region , a creation of slugs on the inclined section , and a dissipation of slugs again on the horizontal section after the incline . in order to test the model at field conditions , we have utilized test 14 from brill , et al . ( 1981 ). the pipeline data are given in table 1 . the pipeline was identified as being in slug flow , with the slugs following a log - normal distribution . the median slug size was measured at 400 feet ( 122 m ), with the longest slug about three times this length at 1200 ft ( 366 m ). gamma densitometer readings indicated slug hold - ups of 0 . 60 , with stratified hold - ups between the slugs of 0 . 20 ; we will adjust our model as much as it possible to match these measured results . the average hold - up is estimated at 0 . 4 , based on the drift - flux model . the inlet hold - up is white noise varying between h l =[ 0 . 2 , 0 . 6 ], such that the average hold - up into the pipeline over time matches the average pipeline hold - up . in reality , the pipeline is most likely chaotic , with inlet slug initiation being influenced by slugs exiting at the outlet over time . the pipeline model has been adjusted to maintain a slug body hold - up of 0 . 6 ; the stratified hold - up between slugs is determined by the slip relation u s ( h l ). here we have used a quadratic slip relationship , chosen because it gave the closest fit to the field data : u s ( h l )= 6 · h l 2 − 12 · h l + 6 the hold - up between slugs produced by this slip relationship is 0 . 27 — somewhat higher than the field measurement of 0 . 2 . fig7 shows the hold - up profile in the pipeline at a specific instant in time . note that there is some consolidation of slugs in the pipeline as one moves from left ( inlet ) to right ( outlet ), and that the profile appears very realistic . fig8 gives the hold - up at the end of the line as a function of time , i . e ., the time trace . slug lengths are determined by measuring the transit time for the slugs and multiplying by the slug velocity , u g = 5 m / s , to obtain the slug lengths . slugs varied in length from 45 m to 750 meters ( 116 to 1928 pipeline diameters ). this is compared to the field measurements , which varied from 30 to 430 m ( 77 to 1105 pipeline diameters ). it should be mentioned that the slug lengths measured in the field were much higher than the 30 diameters maximum obtained in laboratory experiments . fig9 presents a histogram of the number of instances of slugs of a given bin size , plotted against bin size . median slug size predicted by the model is 100 - 150 m — in very good agreement with the field measurement of 120 m . the largest slug predicted by the model was a single instance in the 700 - 750 m bin . this was over twice as large as the largest slug measured in the field , at 430 m . this discrepancy may be due to the much larger number of slugs ( about ten times as many ) measured in the numerical experiments , allowing for the observation of much rarer , much larger slugs . in closing , it should be noted that the discussion of any reference is not an admission that it is prior art to the present invention , especially any reference that may have a publication date after the priority date of this application . at the same time , each and every claim below is hereby incorporated into this detailed description or specification as a additional embodiments of the present invention . although the systems and processes described herein have been described in detail , it should be understood that various changes , substitutions , and alterations can be made without departing from the spirit and scope of the invention as defined by the following claims . those skilled in the art may be able to study the preferred embodiments and identify other ways to practice the invention that are not exactly as described herein . it is the intent of the inventors that variations and equivalents of the invention are within the scope of the claims while the description , abstract and drawings are not to be used to limit the scope of the invention . the invention is specifically intended to be as broad as the claims below and their equivalents . all of the references cited herein are expressly incorporated by reference . the discussion of any reference is not an admission that it is prior art to the present invention , especially any reference that may have a publication data after the priority date of this application . incorporated references are listed again here for convenience : 1 . beggs and brill , “ a study of two phase flow in inclined pipes ,” j . pet . tech ., 607 - 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