TPG4160 Reservor Smulaton 2017 page 1 of 9 ONE-DIMENSIONAL, ONE-PHASE RESERVOIR SIMULATION Flud systems The term sngle phase apples to any system wth only one phase present n the reservor In some cases t may also apply where two phases are present n the reservor, f one of the phases s mmoble, and no mass exchange takes place between the fluds Ths s normally the case where mmoble water s present wth ol or wth gas n the reservor By regardng the mmoble water as a fxed part of the pores, t can be accounted for by reducng porosty and modfyng rock compressblty correspondngly Normally, n one phase reservor smulaton we would deal wth one of the followng flud systems: 1 One phase gas 2 One phase water 3 One phase ol Before proceedng to the flow equatons, we wll brefly defne the flud models for these three systems One phase gas The gas must be sngle phase n the reservor, whch means that crossng of the dew pont lne s not permtted n order to avod condensate fallout n the pores Flud behavor s governed by our Black Ol flud model, so that ρ g ρ gs B g constant B g One phase water One phase water, whch strctly speakng means that the reservor pressure s hgher than the saturaton pressure of the water n case gas s dssolved n t, has a densty descrbed by: ρ w ρ ws B w constant B w One phase ol In order for the ol to be sngle phase n the reservor, t must be undersaturated, whch means that the reservor pressure s hgher than the bubble pont pressure In the Black Ol flud model, ol densty s descrbed by: ρ o ρ os + ρ gs R so B o For undersaturated ol, s constant, and the ol densty may be wrtten: ρ o constant B o General form Thus, for all three flud systems, the one phase densty may be expressed as: Norwegan Unversty of Scence and Technology Department of Petroleum Engneerng and Appled Geophyscs 12117
TPG4160 Reservor Smulaton 2017 page 2 of 9 ρ constant B, whch s the model we are gong to use for the flud descrpton n the followng sngle phase flow equatons Partal dfferental form of sngle phase flow equaton We have prevously derved the contnuty equaton for a one phase, one-dmensonal system of constant crosssectonal area to be: ( x ρu ( t φρ The conservaton of momentum for low velocty flow n porous materals s assumed to be descrbed by the sememprcal Darcy's equaton, whch for one dmensonal, horzontal flow s: u k P µ x Usng the flud model defned above: ρ constant B, and substtutng the Darcy's equaton and the flud equaton nto the contnuty equaton, and ncludng a source/snk term, we obtan the partal dfferental equaton that descrbes sngle phase flow n a one dmensonal porous medum: x k µb P x q φ t B The left hand sde of the equaton descrbes flud flow n the reservor, and njecton/producton, whle the rght hand sde represent storage (compressbltes of rock and flud In order to brng the rght hand sde of the equaton on a form wth pressure as a prmary varable, we wll rearrange the term before proceedng to the numercal soluton Chan rule dfferentaton yelds: t φ B 1 B φ (1/ B + φ t t We wll now make use of the compressblty defnton for porosty's dependency of pressure at constant temperature: or c r 1 dφ φ, dφ φc r, and the flud model above: Norwegan Unversty of Scence and Technology Department of Petroleum Engneerng and Appled Geophyscs 12117
TPG4160 Reservor Smulaton 2017 page 3 of 9 ρ constant B whch mples that: B f (P The rght hand sde may then be wrtten: φ t B 1 B, φ (1/ B + φ 1 t t B Thus, the flow equaton becomes: k P x µb x q φ c r B dφ P d(1/ B P + φ t t φc r P d(1/ B P + φ B t t + d(1/ B Recall that the flud compressblty may be defned n terms of the formaton volume factor as: d(1/ B c f B Then, an alternatve form of the flow equaton s: P t k P x µb x q φ B c + c P r f t φc T B However, normally t s more convenent to use the frst form, snce flud compressblty not necessarly s constant, and snce formaton volume factor vs pressure data s standard nput to reservor smulators P t Dfference form of the flow equaton We wll now use the dscretzaton formulas derved prevously to transform our partal dfferental equaton to dfference form For convenence, we wll now drop the tme ndex for unknown pressures, so that f no tme ndex s specfed, t + Δt s mpled Left sde term The sngle phase flow term, s of the form: k P x µb x x f (x P x, whch we prevously derved the followng approxmaton for: Norwegan Unversty of Scence and Technology Department of Petroleum Engneerng and Appled Geophyscs 12117
TPG4160 Reservor Smulaton 2017 page 4 of 9 x f (x P x 2 f (x +1/2 (P +1 P (Δx +1 + Δx 2 f (x 1/2 Thus, n terms of the actual flow equaton above, we have: k P x µb x k 2 +1/2 (P P 1 (Δx + Δx 1 + O(Δx Δx (P +1 P (Δx +1 + Δx 2 k (P P 1 (Δx 1/2 + Δx 1 + O(Δx Δx We shall now defne transmssblty as beng the coeffcent n front of the pressure dfference appearng n the approxmaton above: Transmssblty n plus drecton Tx +1/2 2 k Δx (Δx +1 + Δx +1/2 Transmssblty n mnus drecton Tx 1/2 2 k Δx (Δx 1 + Δx 1/2 Then, the dfference form of the flow term n the partal dfferental equaton becomes: k P x µb x Tx +1/2 (P +1 P + Tx 1/2 (P 1 P Usng Tx +1/2 as example, the transmssblty conssts of three groups of parameters: 2 Δx (Δx +1 + Δx constant, k +1/2 k f (x, 1 +1/2 1 f (P We therefore need to determne the forms of the two latter groups before proceedng to the numercal soluton Startng wth Darcy's equaton: q ka P µb x For flow between two grd blocks: Norwegan Unversty of Scence and Technology Department of Petroleum Engneerng and Appled Geophyscs 12117
TPG4160 Reservor Smulaton 2017 page 5 of 9 q + 1 2 + 1 1 2 Δx 1 2 Δx +1 We wll assume that the flow s steady state, e qconstant, and that k s dependent on poston The equaton may be rewrtten as: q dx k A µb Permeablty We now ntegrate the equaton above between block centers: +1 q dx A k µb +1 The left sde may be ntegrated n parts over the two blocks n our dscrete system, each havng constant permeablty: +1 q dx q Δx + Δx +1 k 2 k k +1 We may wrte, defnng an average permeablty, : yeldng q Δx + Δx +1 2 k q 2 k +1 Δx + Δx +1 k k Δx + Δx +1 Δx + Δx +1 k k +1 whch s the harmonc average of the two permeabltes In terms of our grd block system, we then have the followng expressons for the harmonc averages: and k k +1/2 Δx + Δx +1 Δx +1 + Δx k +1 k k k 1/2 Δx + Δx 1 Δx 1 + Δx k 1 k Norwegan Unversty of Scence and Technology Department of Petroleum Engneerng and Appled Geophyscs 12117
TPG4160 Reservor Smulaton 2017 page 6 of 9 Flud moblty term We want to ntegrate the rght hand sde: A +1 µb Replacng the flud parameters by moblty λ 1 µb, and lettng be a weak functon of pressure, and assumng the pressure gradent between the block centers to be constant, we fnd that the weghted average of the blocks' moblty terms s representatve of the average Frst, we wll defne the flud moblty term as Then, the average moblty terms are: ( λ +1/2 Δx +1λ +1 + Δx λ ( Δx +1 + Δx and ( λ 1/2 Δx 1λ 1 + Δx λ ( Δx 1 + Δx Rght sde term The dscretzaton of the rght sde term φ c r B + d(1/ B P t s done by usng the backward dfference approxmaton derved prevously: ( P t P t P Δt We wll now defne a storage coeffcent as: Cp φ Δt c r B + d(1/ B and the rght sde approxmaton becomes: φ c r B + d(1/ B P t Cp (P P t Thus, the dfference form of the sngle phase flow equaton s (for convenence, the approxmaton sgn s hereafter replaced by an equal sgn: Tx +1/2 (P +1 P + Tx 1/2 (P 1 P q Cp (P P t, 1, N Norwegan Unversty of Scence and Technology Department of Petroleum Engneerng and Appled Geophyscs 12117
TPG4160 Reservor Smulaton 2017 page 7 of 9 Boundary condtons and producton/njecton terms We have prevously dscussed the two types of boundary condtons we can assgn, the pressure specfcaton (Drchlet condton and the rate condton (Neumann condton For the smple one phase equaton that we consdered ntally, we assumed these to be specfed at ether the end of the system and derved correspondng approxmatons of the flow term for these grd blocks However, n reservor smulaton the boundary condtons normally are no flow boundares at the end faces of the reservor, and producton/njecton wells where ether rate or pressure are specfed, located n any of the grd blocks No flow boundares No flow at the boundares are assgned by gvng the respectve transmssblty a zero value at that pont Ths s the default condton For our one-dmensonal system, ths type of condton would for example be appled to the two end blocks so that: Tx 1/2 0 Tx N+1/2 0 Producton/njecton wells We wll now ntroduce a well term n our dfference equaton, so that t becomes: Tx +1/2 (P +1 P + Tx 1/2 (P 1 P q Cp (P P t, 1, N The well rate term wll be zero for all blocks that do not have a well n t, and nonzero where there s a well Snce our equaton s formulated on a per volume bass, the flow rate must also be on a per volume bass It s defned as postve for producton wells and negatve for njecton wells Constant well producton rate, Q For a constant well rate of volume rate becomes: at surface condtons, whch s the most common well rate specfcaton, the per q Q AΔx If the well s specfed to have a constant well rate of at reservor condtons, the per volume rate becomes: q Q B AΔx Constant well bottom-hole pressure For a well producng or njectng at a constant bottom hole pressure, P bh, the well rate s computed the followng equaton: q Q AΔx WC λ (P P bh AΔx wc λ (P P bh, Norwegan Unversty of Scence and Technology Department of Petroleum Engneerng and Appled Geophyscs 12117
TPG4160 Reservor Smulaton 2017 page 8 of 9 where WC s the the well constant, or the productvty or njectvty ndex of the well, and the same on a per volume bass The well constant may be specfed externally, based on productvty or njectvty tests of the well, or t may be computed from Darcy's equaton If the well s n the mddle of the grd block, one may assume radal flow nto the well, wth block volume as the dranage volume: WC 2πk h ln( r e r w, where r w s the wellbore radus, and the dranage radus may theoretcally be defned as: r e ΔyΔx π However, n reservor smulaton ths formula s normally wrtten as: r e c ΔyΔx Where the value c may vary dependng on well locaton nsde the grd block A commonly used formula s the one derved by Peaceman: r e 020 ΔyΔx For the smple lnear case, wth a well s at the end of the system, at the left or rght faces, the well constant would be computed from the lnear Darcy's equaton: WC k A Δx / 2 Soluton of the dfference equaton Now we have a set of N equatons wth N unknowns, whch must be solved smultaneously In dervng the dfference equaton we have mplctly assumed that all terms of the equaton are evaluated at tme Ths assumpton apples to the coeffcents as well as the pressures on the left sde of the equaton However, one may queston the numercal correctness of ths snce the approxmaton of the tme dervatve on the rght hand sde then becomes a frst order backward dfference If nstead the terms were to be evaluated at, the tme dervatve would become a second order approxmaton, central n tme, and thus a more accurate approxmaton Such a formulaton s known as a Crank-Ncholson formulaton Snce the pressure soluton of such a formulaton often exhbts oscllatory behavor, t s normally not used n reservor smulaton, and we wll therefore not pursue t further here Snce the left and rght hand sde terms of the equaton are at tme, the coeffcents are functons of the unknown pressure In the transmssblty terms, both vscosty and formaton volume factor are pressure dependent, and n the storage terms the dervatve of the nverse formaton volume factor depends on pressure Therefore, an obvous procedure would be to terate on the pressure soluton, lettng the coeffcents lag one teraton behnd and updatng them after each teraton untl convergence s obtaned However, n sngle phase flow the pressure dependency of the coeffcents s small, and such teraton s normally not necessary For now we wll therefore make the approxmaton that the transmssbltes and the storage coeffcents wth suffcent accuracy can be evaluated at the block pressures at the prevous tme step The set of equatons may be rewrtten on the form: where a P 1 + b P + c P +1 d, 1,,N Norwegan Unversty of Scence and Technology Department of Petroleum Engneerng and Appled Geophyscs 12117
TPG4160 Reservor Smulaton 2017 page 9 of 9 a 1 0 a Tx 1/ 2, 2,,N b 1 Tx +1/ 2 Cp b Tx 1/2 Tx +1 / 2 Cp 2,,N 1 b N Tx 1/ 2 Cp c Tx +1 / 2, 1,,N 1 c N 0 d 1 3 4 αp 1 t 2P L d Cp P t + q, 1,,N In order to account for producton and njecton, the followng modfcatons would have to be done for grd blocks havng producton or njecton wells: Rate specfed n a well n block In ths case, no actual modfcaton has to be made, snce s already ncluded n the term However, after computng the pressures, the actual bottom hole pressure may be computed from the well equaton: q wc λ (P P bh Bottom hole pressure specfed n a well n block Here, we make use of the well equaton, wth beng constant: q wc λ (P P bh, and nclude the approprate parts n the and terms: b Tx 1/2 Tx +1/2 Cp wc λ d C p P t + wc λ P bh The well constants are computed as specfed above Well head pressure specfed for a well n block Frequently, we want to specfy a wellhead pressure,, nstead of a bottomhole pressure n a well, reflectng condtons of surface equpment In order to nclude such a condton n our equaton, we need to convert t to a bottom hole pressure condton A well bore model s therefore needed to compute pressure drop n the well bore as functon f rate, frcton, etc Norwegan Unversty of Scence and Technology Department of Petroleum Engneerng and Appled Geophyscs 12117
TPG4160 Reservor Smulaton 2017 page 10 of 9 Fnally, the lnear set of equatons, ncludng boundary condtons and well rates an pressures, may be solved for average block pressures usng for nstance the Gaussan elmnaton method for the tme step n queston We then update the coeffcents and proceed to the next tme step Norwegan Unversty of Scence and Technology Department of Petroleum Engneerng and Appled Geophyscs 12117