Inverse-Problem Description and Terminology


Well-test analysis is the process by which hydraulic parameters of interest such as hydraulic conductivity (K) and specific storage (Ss) are estimated from measured pressure and flow-rate data.  This problem of inferring K, Ss, etc. from a measured response is generally known as an inverse problem.  An inherent quality of inverse problems is that the parameters estimated via this process have some degree of uncertainty associated with their values.  To demonstrate how this uncertainty originates and to better understand the well-test analysis process, consider the following simple example.

Assume that you wish to know the values of two parameters, A and B, which cannot be measured directly.  A related parameter, C, can be measured directly and the total system response – call it D – can also be measured.  After measuring C and D, some analysis indicates that the relationship among the three parameters A, B, and C, and the response D is described by the following equation:  A + B + C = D. In our attempt to estimate A and B, their values will be adjusted and combined with the measured value C such that A + B + C matches the measured response D.

No discipline, including well-test analysis, is complete without some specialized terminology.  In the language of inverse problems, A and B are fitting parameters, C is a non-fitting parameter, and D is a constraint.  The equation A + B + C = D is the conceptual model.  The constraint D serves to limit the possible values of A and B. Obviously, without this information, no unique estimates of A and B can be made.  Constraints used in well-test analysis are typically some combination of pressure and flow-rate measurements.  The non-fitting parameters in well-test analysis (C in this example) include things such as wellbore radius and volume, length of the tested interval, fluid density, etc.

Assuming that values for C and D have been obtained, we see that there is still a problem uniquely determining values for A and B.  To understand the source of this uncertainty, assume that C has a value of 5 and that D has a value of 50.  Let’s further assume that A and B can only vary between values of 0 and 100 (note that this range restriction is an additional constraint).  The mathematical area defined by the possible values of A and B is known as the parameter space (Figure 1).  However, not all of the A and B combinations within this parameter space will satisfy the constraint that A + B + 5 = 50.  The optimal solutions, i.e., those solutions that satisfy both constraints fall within the red region in Figure 1 – a line segment in this example.  This region of optimal solutions is typically referred to as the minimum.

Why the term “minimum?”  When the A and B combinations contained within the minimum are input into the model, the difference between the model output (the estimated value of D) and the measured D value is minimal relative to the difference that would be observed using other A and B combinations within the parameter space – hence, the term “minimum.”  nSIGHTS quantifies the quality of the fit to the constraint as the sum of the squared errors (SSE).  The difference between each constraint data point and its simulated value is squared and these squared differences are then summed.  The SSE value is sometimes referred to as the value of the objective function.  nSIGHTS utilizes a non-linear regression algorithm to search for optimal solutions, i.e., to minimize the stated objective function by minimizing the SSE.

Figure 1. Parameter space and minimum for the conceptual model A + B + 5 = 50

We see in this example that the minimum is not a single-valued combination of A and B – there is a range of possible values.  This range is the result of the correlation that exists between the fitting parameters A and B.  A change in the estimate of A can be compensated by a corresponding change in the estimate of B such that the constraints are still satisfied and the minimal value of the objective function (SSE) is unchanged.  For a given conceptual model, this correlation among fitting parameters is typically the largest source of uncertainty in the estimate of the fitting parameters.

Noise in the data is another source of uncertainty.  Assume that measurements of D can only be resolved to a value somewhere between 45 and 55, i.e., our constraint becomes 45 < A  + B + 5 < 55.  The new minimum is the area between the blue lines shown in Figure 2 – the uncertainty is increased.  Now assume that C cannot be measured exactly – let’s assume that it’s 5 ± 1.  The situation now becomes 45 < A + B + (5 ± 1) < 55.  The possible solutions are now contained between the green lines in Figure 3.  The additional uncertainty results from the correlation between the fitting and non-fitting parameters.

Figure 2: Minimum for the conceptual model 45 < A + B + 5 < 55.

Minimum for the conceptual model 45 < A + B + (5 ± 1) < 55

Figure 3: Minimum for the conceptual model 45 < A + B + (5 ± 1) < 55.

In review, for a given conceptual model, uncertainty can result from correlation among fitting parameters, noise in the data, and correlation among fitting and non-fitting parameters.  Also note that dramatic changes in the estimates of the fitting parameters can result by changing the conceptual model itself.  If we decided that A + (B x C) = D better represented what we observed in the system, then the values of A and B may be very different than those obtained using the previous conceptual model.

Given that uncertainty in the estimates of the fitting parameters is an inherent part of the well-test analysis process, there are several things that an analyst can do to address this uncertainty.  The most straightforward response to the uncertainty is to quantify it.  To see one method of quantifying uncertainty, please go to the Quantifying Uncertainty page.

It is also possible to use one’s knowledge of the sources of the uncertainty to minimize it.  To see an example of this, please go to the Minimizing Uncertainty page (coming soon…).