Response surface method for assessing energy production from geopressured geothermal reservoirs
- Esmail Ansari^{1}Email authorView ORCID ID profile and
- Richard Hughes^{1}
Received: 5 August 2016
Accepted: 24 October 2016
Published: 2 November 2016
Abstract
Developing low-enthalpy geothermal resources along the US Gulf Coast is attractive for reducing global warming and providing clean energy. In this work, synthetic yet representative models for typical geopressured geothermal reservoirs located along the US Gulf Coast are considered. A Box–Behnken experimental design is used to select a small set of these models to perform detailed reservoir simulation runs. Full quadratic linear models are fit to the simulation results, and their sufficiency is confirmed by comparing them to kriging response surfaces. To achieve a higher degree of efficiency in using the response surfaces, Hammersley sequence sampling (HSS) method is used instead of traditional Monte Carlo sampling. HSS ensures that the factor space is sampled more uniformly and the response distribution is converged in less time. By evaluating these proxy models in the sampled factor space, the sensitivity and uncertainty of the response to the factors can be assessed. In this work, the sensitivity and uncertainty of engineered convection is assessed. For quantifying engineered convection, five uncertain reservoir attributes were selected. The response was defined as the net extracted enthalpy. In particular, two different designs for harvesting energy from geothermal reservoirs were compared using the response surfaces. In the modeled systems, results show that the regular design is more effective than the reverse design for extracting energy from geopressured geothermal reservoirs.
Keywords
Geothermal reservoir Experimental design Response surface Sampling Forced convection Heat extractionBackground
Reducing greenhouse gases and providing the world’s future energy require searching for clean alternative energy resources that can substitute for fossil fuel. Geopressured geothermal reservoirs along the US Gulf Coast are an alternative energy resource which have been considered as marginal and have not been developed extensively. The information available about these resources comes from well test data performed at the time of their development (John et al. 1998). Assessing the uncertainties associated with the commerciality of these reservoirs by simulating and history matching each case independently is an expensive and time-consuming process and should be reserved for the project design stage. One way for quickly assessing these assets is to study the sensitivity of produced energy to uncertain features using reservoir modeling.
Though computational speed and memory for solving problems is improving over time, detailed modeling for history matching of each reservoir or using Monte Carlo simulations is not feasible because running many models are numerically expensive. To overcome this limitation, there are three alternatives as follows: (1) procedures that efficiently create the history matched models; (2) surrogate reduced order models; and (3) statistical proxy models. The first approach uses efficient gradient-based or gradient-free algorithms for history matching production data and efficiently makes field development plans (Shirangi and Durlofsky 2015; Shirangi 2014). The second approach is to build surrogate reduced order models using piecewise linearization algorithms. These algorithms increase the efficiency of the Newton loop by creating the Jacobian matrices around previously simulated points instead of traditionally solving the flow equations (Ansari 2014; He and Durlofsky 2014; Cardoso and Durlofsky 2010). The third approach, which is used in this work, is to run the detailed model using specific combinations of factors sampled by experimental design and then fit a proxy response surface to the factor space (Ansari 2016). Experimental design and response models are popular and used widely (Fisher and Genetiker 1960; Mishra et al. 2015). Schuetter et al. (2014) compare the use of Box–Behnken sampling and quadratic polynomial regression with Latin Hypercube sampling, multivariate adaptive regression spline (MARS), and additivity variance stabilization (AVAS) techniques for geological CO\(_{2}\) sequestration. They conclude that the model developed using Box–Behnken and quadratic polynomials performs the best. Following Schuetter et al. (2014), this work uses Box–Behnken experimental design. Experimental design and response surfaces have also been used in the context of geothermal reservoir engineering (Hoang et al. 2005; Quinao and Zarrouk 2014). Response surface models are fast and have adequate accuracy to represent the detailed model. Response surface models can be efficiently run thousands of times for uncertainty assessments. Traditionally, simple linear models are used to represent the actual model (Montgomery et al. 2012). For most of the cases, quadratic linear models (polynomials) are adequate. Once the proxy response surface model is constructed using a very small, yet statistically representative, set of detailed model runs, it can be used for sensitivity analysis and uncertainty assessments using sampling methods such as Monte Carlo (MC) or Hammersley sequences sampling (HSS).
This work compares different energy extraction designs for geopressured geothermal reservoirs and identifies the better technique. It further quantifies the uncertainty in the selected design using the developed proxy model and sampling methods.
This paper proceeds as follows: we first introduce design of experiments, response surface modeling, and sampling. Then, we apply these methods to compare two different heat extraction designs: regular design and reverse design. We select the best design for extracting energy by comparing their energy output. Regular design shows better performance than reverse design. For the regular design, the uncertainty of the factors is used to obtain the uncertainty in the cumulative energy produced.
Methods
Design of experiments (DOE)
In reservoir engineering, factors can be categorized into three types: controllable, observable, and uncertain. The controllable factors may be engineered or selected, such as well location. The observable factors may be accurately measured such as the reservoir thickness at each well location. However, some uncertain factors, such as porosity far from wells, can neither be measured nor engineered. These factors are the important factors, on which the sensitivity analysis should be based.
Response surface methods
Once the results of runs suggested by the designs are obtained, response surfaces are used to determine the correlation between the factors and the response (Montgomery and Myers 1995). Two widely used formulations for the response surfaces are regression and kriging.
Regression
Kriging
Sampling
Once the proxy models are constructed, a sampling method is needed to sample the factors and to translate the uncertainty from the input to the response. For doing this, a Monte Carlo or quasi Monte Carlo method, such as HSS, is generally used (Kroese et al. 2011).
Unlike straight Monte Carlo which samples n-dimensional space randomly, Hammersley sequence fills the space more uniformly (Fig. 2). This characteristic is known as low-discrepancy sequence sampling. In Hammersley sequences, the design point p (which is less than the total dimension n) is conditioned on the previous \(p-1\) points and the total dimension n, thus making the sample points dependent. The points generated in low-discrepancy sampling methods are highly ordered and exhibit much more regularity. The result of sampling using these sequences converges more efficiently than multidimensional Monte Carlo (Kroese et al. 2011). The only drawback of Hammersley sequences is that the number of points should be specified before simulation and if, due to lack of accuracy of the results, the number of points changes, the process needs to be repeated discarding previous results. The Hammersley sequences span the n-dimensional space with a small yet representative sample. The procedure of obtaining a Hammersley sequence is described below.
Regular vs. reverse design
Characteristics of the base hypothetical model [after Plaksina et al. (2011)]
Properties | Base value | Unit |
---|---|---|
Temperature of top cell | 135 | C |
Matrix compressibility | \(2.0\times 10^{-5}\) | \(\mathrm {kPa^{-1}}\) |
Dip angle | 15 | Degree |
Reservoir length | 2000 | m |
Cross-section width | 100 | m |
Reservoir thickness | 30 | m |
Permeability | 300 | md |
Porosity | 0.2 | – |
Rock heat capacity | \(2.6\times 10^6\) | \(\mathrm {J/(m^3C)}\) |
Rock bulk density | 2.3 | \(\mathrm {g/cm^3}\) |
Thermal heat conductivity | 172, 800 | \(\mathrm {J/(m\,day\,C)}\) |
Water thermal expansion | \(8.8\times 10^{-4}\) | \(\mathrm {C^{-1}}\) |
Water compressibility | \(4.5\times 10^{-7}\) | \(\mathrm {kPa^{-1}}\) |
Water molecular weight | 0.01802 | \(\mathrm {kg/gmol}\) |
Water molar density | 55, 500 | \(\mathrm {gmol/m^3}\) |
Water density | 1.02 | \(\mathrm {g/cm^3}\) |
The equilibrium state obtained from natural convection simulations (1000 years of simulation without injection or production) served as the initial condition for the forced convection. For the natural convection period, the temperature of the reservoir boundary is the same as its surrounding cap/base rock. As sediment cools down by cold water injection, it starts to gain conductive heat from the cap/base rock. A model, developed by Vinsome and Westerveld (1980), is used for peripheral boundary heat gain. The model is based on a semi-analytical impermeable heat conduction formulation which adequately describes the boundary condition at the interface. This model ensures adequate accuracy because in practice the thermal conduction coefficients between the reservoir and the cap/base rock are not precisely known.
Figure 4 compares three different boundary conditions for the base case considered in Table 1. The first case assumes that the reservoir is sealed and there is no heat conduction between the reservoir and cap/base rocks. In the second boundary condition, the temperature of cap/base rocks does not change as the reservoir cools down. The third case is the semi-analytical Vinsome and Westerveld's (1980) model. The heat-insulated boundary condition shows much lower produced energy, and the constant temperature boundary condition shows much higher produced energy than that shown by the more realistic boundary condition proposed by Vinsome and Westerveld (1980). Cumulative produced energy from the semi-analytical model is 38% more than the insulated reservoir and 22% less than constant temperature boundary condition after 30 years.
Levels of factors in the Box–Behnken design [Plaksina et al. (2011)]
Levels | Length (m) | Thickness (m) | Dip angle | Porosity | Permeability (md) |
---|---|---|---|---|---|
+1 | 2000 | 30 | 0 | 0.15 | 200 |
0 | 3000 | 40 | 15 | 0.20 | 500 |
\({-}\)1 | 4000 | 50 | 30 | 0.25 | 800 |
Results and discussion
Regular design
The regular design was modeled in detail. A quadratic linear response surface model was fit to the simulation results. Our experience shows that using a second-degree polynomial instead of a first degree results in a better polynomial fit. The factors and interaction terms with p values less than 0.05 were selected as important (Table 3). Then, each factor was assigned a specific distribution. Both Monte Carlo and Hammersley sequence sampling methods were used to sample these factors’ distribution and translate the uncertainty from the factors to the response (net extracted energy).
The p value for all the factors except the permeability is less than 0.001 (Table 3) which means that all the factors have significant effect on the heat production except permeability. For the range of permeability considered for modeling, knowledge of the permeability map is less important for predicting the thermal energy recovery presumably due to the constant well rates assumption. This makes sense because the pressure of a geopressured geothermal reservoir is very high and this pressure constraint can provide the flow rate constraint imposed on the production well for the modeled range of permeability (Table 2).
A low p value and a positive coefficient for the porosity in Table 3 indicate that an increase in porosity would increase the produced energy. The fundamental idea in geothermal reservoirs is to extract the heat stored in the rock and use the fluid as the conduit. The increase in the fluid content of the system increases produced energy because the thermal capacity of the brine is more than the rock.
Summary of the second-order linear regression for the regular design: representing the interaction between two factors
Factors | \(\beta \,\, \mathrm{value}\) | Standard error | t value | Pr (\({>}|\)t|) |
---|---|---|---|---|
(Intercept) | −0.027 | 0.009 | −3.037 | 0.006 |
Thickness | 0.218 | 0.003 | 83.748 | 0.000 |
Length | 0.745 | 0.003 | 289.663 | 0.000 |
Permeability | 0.000 | 0.003 | 0.162 | 0.873 |
Porosity | 0.039 | 0.003 | 15.264 | 0.000 |
ReservoirDip | 0.230 | 0.003 | 88.528 | 0.000 |
I (thickness\(^{2}\)) | −0.007 | 0.005 | −1.277 | 0.214 |
I (length\(^{2}\)) | −0.050 | 0.005 | −9.904 | 0.000 |
I (porosity\(^{2}\)) | −0.001 | 0.005 | −0.245 | 0.809 |
I (permeability\(^{2}\)) | 0.001 | 0.005 | 0.216 | 0.831 |
I (reservoirDip\(^{2}\)) | −0.035 | 0.005 | −6.750 | 0.000 |
Thickness: length | 0.041 | 0.005 | 7.929 | 0.000 |
Thickness: porosity | 0.007 | 0.005 | 1.344 | 0.192 |
Thickness: permeability | 0.001 | 0.005 | 0.220 | 0.828 |
Thickness: reservoirDip | 0.020 | 0.005 | 3.927 | 0.001 |
Length: porosity | 0.007 | 0.005 | 1.316 | 0.201 |
Length: permeability | 0.002 | 0.005 | 0.353 | 0.727 |
Length: reservoirDip | 0.138 | 0.005 | 26.580 | 0.000 |
Permeability: porosity | 0.001 | 0.005 | 0.178 | 0.861 |
Porosity: reservoirDip | 0.004 | 0.005 | 0.734 | 0.471 |
Permeability: reservoirDip | 0.000 | 0.005 | 0.065 | 0.949 |
To test the model, the observation is plotted versus the model prediction (Fig. 8). The observation versus the model prediction falls on the 45\(^\circ\) line which means the model is adequate. This model can be used for the Monte Carlo sampling of factor distributions. For quantifying the uncertainty in produced energy, a distribution was assigned to each factor. It is assumed that length, thickness, reservoir dip angle, and porosity, each follow a normal distribution (Fig. 9). A log-normal distribution is assigned to permeability.
Conclusion
Regular design outperforms reverse design for heat production in the modeled systems. This result is confirmed using polynomial and kriging response surfaces. The heat recovery from the regular design improves as the reservoir length, dip angle, or thickness increase. The results indicate that important criteria in evaluating a set of geothermal reservoirs with adequately high temperature is the size of the reservoir. For the reservoir models within the studied ranges, reservoir dip angle is less important that the reservoir size. The proxy models were efficiently used to construct produced energy distribution from the factor distributions. For having even more efficiency, HSS was used. HSS was about 10 times faster than the Monte Carlo simulation. For the considered problem, the produced energy was between \(7\times 10^{15}\) and \(9\times 10^{15}\) J. Future research should focus on testing the uncertainty in structural and isopach maps of a real reservoir model and compare the results with the results published in this work.
Declarations
Authors' contributions
EA carried out the modeling, coding, and developing the results. EA also drafted the manuscript. The second author, RH supervised the research and guided the interpretation of results. RH also considerably edited and improved the drafts. Both authors read and approved the final manuscript.
Acknowledgements
The R source codes and the datasets for this study are available on request to the authors. The authors gratefully acknowledge financial support for this work from the US Department of Energy under grant DE-EE0005125. We thank Computer Modeling Group for providing reservoir simulation software. We also thank Christopher D. White and the members of the LSU Geothermal team for their comments, suggestions, and ideas supporting our efforts.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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