Optimal input experiment design and parameter estimation in core-scale pressure oscillation experiments

Abstract

International audienceThis paper considers Pressure Oscillation (PO) experiments for which we find the minimum experiment time that guarantees user-imposed parameter variance upper bounds and honours actuator limits. The parameters permeability and porosity are estimated with a classical least-squares estimation method for which an expression of the covariance matrix of the estimates is calculated. This expression is used to tackle the optimization problem. We study the Dynamic Darcy Cell experiment set-up (Heller et al., 2002) and focus on data generation using square wave actuator signals, which, as we shall prove, deliver shorter experiment times than sinusoidal ones. Parameter identification is achieved using either inlet pressure/outlet pressure measurements (Heller et al., 2002) or actuator position/outlet pressure measurements, where the latter is a novel approach. The solution to the optimization problem reveals that for both measurement methods an optimal excitation frequency, an optimal inlet volume, and an optimal outlet volume exist. We find that under the same parameter variance bounds and actuator constraints, actuator position/outlet pressure measurements result in required experiment times that are a factor fourteen smaller compared to inlet pressure/outlet pressure measurements. This result is analysed in detail and we find that the dominant effect driving this difference originates from an identifiability problem when using inlet–outlet pressure measurements for joint estimation of permeability and porosity. We illustrate our results with numerical simulations, and show excellent agreement with theoretical expectations

LPV system identification for control using the local approach

“…As already mentioned, it is clear from Proposition 5 that the robustness condition (27) in the LPI case can be checked by computing J ub wc (ω) at all frequencies in a fine grid Ω of the interval [0 π] and by verifying that J ub wc (ω) < η rob at all these frequencies. This also means that the LMI optimization problem with multiple LMI constraints (37) will in practice only be used for the LPV case.…”

“…This is summarized in the following proposition. Using the quantity J ub wc computed with this convex optimization problem, we will be able to verify the robustness condition (27) and, consequently, whether the controllerĈ(p(t)) designed based on the identified modelḠ(T (p(t))θ) will achieve sufficient performance with the unknown true system Ḡ (T (p(t))θ 0 ) for all scheduling sequences p ∈ P (in the LPI situation) or p ∈P (in the LPV situation).…”

“…This also means that the LMI optimization problem with multiple LMI constraints (37) will in practice only be used for the LPV case. In this case, verifying (27) consists in checking thatJ ub wc (ω) < η rob /|W i (e jω )| at all frequencies in a fine grid Ω of [0, π].…”

“…We will therefore proceed to optimal experiment design to reduce the size of this uncertainty in an optimal way. We follow the procedure in Section 5 to design a second LPV identification experiment (with minimal input energy J ) which combined with the first one will yield a model with a smaller uncertainty region U for the LPI controller designed based on this new model to be guaranteed to satisfy (27) with the new uncertainty region U . For this purpose, as proposed in Section 5.3, we suppose that the local LTI identification experiments of the second LPV identification experiment are all of duration N p = 1000 and that they can only be performed at the M grid = 17 Figure 3.…”

A method for simultaneously determining axial permeability and transverse permeability of tight reservoir cores by canister degassing test

“…The most widely used unsteady-state technique is pressure pulse decay | 1221 ZHAO et Al. method, [3][4][5][6] oscillating pulse method, [7][8][9] canister degassing test method, 1,10,11 and the Gas Research Institute (GRI) method. 12,13 These methods have several advantages: (a) short experimental duration and (b) the pressure signal or the cumulative flow as opposed to instantaneous flow rate is measured.…”