Innovations in pressure transient test methods and analysis : de-convolution and reservoir surveillance 26 th November 2013 Society of Petroleum Piers Johnson C.Eng Managing Director of OPC Engineers – London
What we will cover in this talk… • A Brief history of well testing and techniques • Gauges • De-convolution: Theory and Application • Permanent Downhole Gauges (PDH) and using De- convolution • Other Applications of De-convolution (multi-stage fraccing in horizontal wells) 26/11/2013 OPC 2
Well Testing as we used to know it 26/11/2013 OPC 3
A little bit of operational history…. • Onshore Well Testing (1950’s) Drill Stem Tests (DST’s) • Open hole testing with an MFE (Multi Flow Evaluator – mechanically operated tool) and mechanical pressure gauges – Amerada gauge. (1960’s) • Burners allowed offshore testing (1970’s) • Introduction of Electronic gauges and powerful personal computers (1980’s) 26/11/2013 OPC 4
A little bit of operational history …. • Sub Sea Testing equipment (1980’s) • Deep water Sub Sea Testing equipment (1990’s) • Permanent down hole gauges (1990’s) • Wireless communication with gauges (2000’s) 26/11/2013 OPC 5
And some theoretical history…. • Darcy’s law (flow in a porous medium) 19 th Century • Drawdown analysis. • Build up analysis in the 1950’s and 1960’s. Miller Dyes & Hutchison – MDH Plot, One single constant rate drawdown followed by a build up (Horner Plot) 1960’s – Straight line analysis generally by hand. • Many flow periods at different rates with many build ups. (Superposition theory) 1970’s • Derivative Type Curve Analysis (Bourdet et al + Computers & Software ) 1980’s • De-convolution (a few clever people!) 2000’s 26/11/2013 OPC 6
Gauges and Gauge Performance – and cost 26/11/2013 OPC 7 Schlumberger
Gauge and Well Costs over the last 30 years Diagrammatic only 100 100 90 90 80 80 70 70 60 60 Costs Costs 50 50 40 40 Cost of Wells 30 30 Cost of Gauges 20 20 10 10 0 0 1980 1985 1990 1995 2000 2005 2010 2015 Year 26/11/2013 OPC 8
Exploration well test Gauge Data SET PACKER 10000. FINAL BUILD UP CLEAN UP POOH 6000. PSI 2000. INITIAL FLOW MAIN FLOW 0. 0. 20. 40. 60. 80. REVERSE CIRCULATE PRESSURE TEST TUBING WHEN RUNNING IN HOLE (RIH) 600. STB/D -200. 0. 20. 40. 60. 80. Time (hours) BIGASCI Example Data BIGASCI Example Data 26/11/2013 OPC 9
Permanent Down hole Gauge Data 26/11/2013 OPC 10
De-Convolution in Simple Terms Convolution and De-convolution can be thought of as simplified multiplication and division. Given: “P” is a set of measured pressure data “q” is a measured rate -history “ p d ” is the theoretical reservoir type-curve or drawdown function (as shown in next slide) P q p Convolution: d p d P q De-convolution: Note that these equations are just a schematic of the process. The “multiplication” operator is the convolution integral, and the “division” operator is a messy iterative solution as shown in the following slides... 26/11/2013 OPC 11
Some explanation of terms: for a Draw-Down Type-Curve Model (or response function) The pressure change at the well bore, caused by production at a constant rate is characterised by a “type - curve model”, p d (t) 5000. P i Note: the “d” in “p d ” stands for “draw - down”. This “p d ” has q * p d (t) 4900. dimensions of psi/stb/d. 4800. 0. 50. 100. 150. t 80. +q 40. -20. 0. 50. 100. 150. Time (hours) 26/11/2013 OPC 12
Draw-Down Type-Curve Model B kt p ( t ) f , x S d 2 kh c L t Given the following definitions: t = time k = a characteristic permeability h = a characteristic thickness L = a characteristic length S = a constant “skin” x = a list of model parameters α,β = conversion factors For radial flow in an infinite reservoir, the above general equation translates into the classic drawdown equation as follows: B k p ( t ) 162 . 6 log( t ) log 3 . 23 0 . 87 S d 2 kh c r t w 26/11/2013 OPC 13
In order to understand Deconvolution, we need to understand Convolution. The “Convolution” integral describes the pressure change for an arbitrary rate- history, “q(t)”: t dp t d P p t q ( ) d i d dt 0 For “q(t)” made up of ‘n’ constant -rate flow-periods which start at times “ t i ” and are less than “t”: n P p t q q p t t i d i i 1 d i i 1 “Convolution” is just “Superposition” by another name. 26/11/2013 OPC 14
So… what is De-Convolution? • De-convolution is a mathematical solution that characterises the reservoir. • The objective of “de - convolution” is to: – find the draw-down response function, p d , such that... – a best-fit is obtained between a type-curve simulation using p d and a set of pressure measurements, p(t) • And the result from de-convolution is a derivative plot of the response function, p d • Note: p d is a completely arbitrary function! Which makes this exercise difficult. • This is possibly better explained as follows…. 26/11/2013 OPC 15
Superposition 26/11/2013 OPC 16
Deconvolution Wrong Reservoir (dinosaur) model RIGHT Reservoir (dinosaur) model 26/11/2013 OPC 17
So, Convolution is like a Multi-rate Simulation Type- curve simulation computes “p(t)” by convolving a rate-history (q n ) with a draw-down response, p d P i 4800. 4400. p(t) 4000. 0. 50. 100. 150. 200. 250. t q1 t q2 t q3 300. q1 q2 t -100. 0. 50. 100. 150. 200. 250. Time (hours) t > t q1 note: q 0 =0 P i -p(t) = [q 1 -q 0 ]p d (t-t q1 ) Drawdown t > t q2 P i -p(t) = [q 1 -q 0 ]p d (t-t q1 ) + [q 2 -q 1 ]p d (t-t q2 ) Build up 26/11/2013 OPC 18
De- convolution Recovers “ p d ” Given pressure points: p d ( Δ t) Draw-Down Response (PSI/STB/D) p(t 11 ) to p(t q2 ) in 1 st flow 10 0 p(t 21 ) to p(t q3 ) in 2 nd flow P i -p(t 11 ) = [q 1 -q 0 ]p d (t 11 -t q1 ) 10 -1 derivative of p d 10 -2 10 -2 10 -1 10 0 10 1 10 2 P i -p(t q2 ) = [q 1 -q 0 ]p d (t q2 -t q1 ) Delta-T (hr) P i -p(t 21 ) = [q 1 -q 0 ]p d (t 21 -t q1 ) + [q 2 -q 1 ]p d (t 21 -t q2 ) Solve this system of equations to recover p d from the measured P i -p(t q3 ) = [q 1 -q 0 ]p d (t q3 -t q1 ) + [q 2 -q 1 ]p d (t q3 -t q2 ) rates and pressure points 26/11/2013 OPC 19
So what De- convolution does… Given pressure points: Draw-Down Response (PSI/STB/D) p(t 11 ) to p(t q2 ) in 1 st flow 10 0 p(t 21 ) to p(t q3 ) in 2 nd flow P i -p(t 11 ) = [q 1 -q 0 ]p d (t 11 -t q1 ) 10 -1 10 -2 10 -2 10 -1 10 0 10 1 10 2 P i -p(t q2 ) = [q 1 -q 0 ]p d (t q2 -t q1 ) t q2 -t q1 t q3 -t q1 Delta-T (hr) P i -p(t 21 ) = [q 1 -q 0 ]p d (t 21 -t q1 ) + [q 2 -q 1 ]p d (t 21 -t q2 ) …..is allow the de - convolved response to span the entire test P i -p(t q3 ) = [q 1 -q 0 ]p d (t q3 -t q1 ) + [q 2 -q 1 ]p d (t q3 -t q2 ) duration! 26/11/2013 OPC 20
Is De- convolution useful then…? • Does de-convolution derive more information from the test data? – No – a skilled engineer can eventually get the same answers using a regular analysis given some time… • Does de-convolution make the analysis more straight-forward? – Yes – gives a direct “view” of the underlying model controlling the well. Get to the right answer faster. • Does de-convolution help estimate reserves? – Oh Yes! – provides a way of defining an “equivalent” radius-of-investigation for the whole test. 26/11/2013 OPC 21
De-convolution extends duration of analysis data Well production history spans 5000 hours because Permanent Down hole gauges record the pressures and rates are recorded at surface (usually). Down hole Rate measurement devices are good too (another presentation). Response function (or Deconvolved Type curve) spans 5000 hours too! 10 0 So we are no longer relying on build ups alone. 10 -1 10 -2 However…. 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 Delta-T (hr) 26/11/2013 OPC 22
De- convolution doesn’t always work! Given real pressures: Draw-Down Response (PSI/STB/D) p(t 11 ) to p(t q2 ) in 1 st flow 10 0 p(t 21 ) to p(t q3 ) in 2 nd flow P i -p(t 11 ) = [q 1 -q 0 ]p d (t 11 -t q1 ) 10 -1 10 -2 10 -2 10 -1 10 0 10 1 10 2 P i -p(t q2 ) = [q 1 -q 0 ]p d (t q2 -t q1 ) Delta-T (hr) Changes in behaviour P i -p(t 21 ) = [q 1 -q 0 ]p d (t 21 -t q1 ) + [q 2 -q 1 ]p d (t 21 -t q2 ) between flow periods “pollutes” the algorithm which destroys the P i -p(t q3 ) = [q 1 -q 0 ]p d (t q3 -t q1 ) + [q 2 -q 1 ]p d (t q3 -t q2 ) solution for p d 26/11/2013 OPC 23
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